»STATISTIGAL INQUIRIES«Issued by »The Statistical Department«, Denmark

Income- enditure

Relations of Danish Wage

and Salary Earners

DrTDEPAR

By Erling Jørgensen

Published by »The Statistical Departmentein Collaboration with »The Institute of Statisticse

and »The Institute of Economics«of the University of Copenhagen

KØB E N HAVN

1965

PRINTED IN DENMARK BY AARHUUS STIFTSMOGTRYKKERIE A-S

CONTENTSList of tables VIIList of figures VIIIPrefaceChapter I. Background and main results

Ia. Background of the study 3Ib. Main results of the analysis 4Ic. The report 10

Chapter II. Review of the survey materiallia. Introductory remarks 12lIb. Concepts and methods of the survey 12

Collecting the information 12

Income and expenditure concept; unit of analysis 13

Period of the survey 15

Method of selection 15lic. Estimating mean values and their standard errors.

Accuracy of survey results 18

Processing of the material 19

Chapter III. Objectives of the analysis. Engel functionslila. Introductory remarks 22IlIb. Choice of model

Determinants of expenditure 22Engel curves and household survey data 23What, then can the results be used for 28

hIc. Use of estimated Engel curves on the macro level 29

Chapter IV. Methods of analysishVa. Introductory remarks 31IVb. Choice of Engel functions and specification of the variables

Criteria of selecting Engel functions 31Description of the functions selected 32Specification of the variables 33Zero-observations 36Grouping problems 38

IVc. Variance assumptionsGeneral remarks 40Testing the hypothesis V = 2 2 42

IVd. Calculation of estimates of parametersFour linear functions 45The log-normal distribution function 46

VI

IVe. Tests for goodness of fitThe tests used 48Test for number of runs and for length of run; the d-test 49The F-test 50The X2-test 52The coefficient of correlation 52

JVf. Planning the computation programme and carrying out the computation 52

Chapter V. Main resultsVa. Introductory remarks 56Vb. Examination of test-results 57

Coefficient of correlation, between calculated and observed values 57The X2-test 60The F-test 64The test for number of runs and for the longest run; the d-test 71

Summary of test-results 77Vc. Analysis of estimates of the parameters

Regression analysis versus two-way cross-tabulation 77Interpretation of main results 80Interpretation of the estimates of the slope of the regression line 81Are the Engel curves for different social groups identical? 85An important reservation 86Conclusions 88

Chapter VI. Further analysis. The concept of unit-consumers, multiple regressionanalyses, etc.

VIa. The unit-consumer concept 90VIb. Multiple regression analyses 100

Danish summary 111

Literature 117Index 119

Appendix A. Main results 123

Appendix B. Correlation coefficient between residuals of different expenditure items 174Appendix C. The basic material 181

Appendic D. Detailed list of expenditures separately for each of 13 main expenditureitems. 240

LIST OF TABLES1,1 Income elasticities for 13 expenditure items; average values for 12 groups of wage and

salary earners 7

II,! Average income, saving and expenditures in 1955 in kroner per household 20IV,! Values of income elasticity, e, and marginal propensity to consume, m, for five Engel

functions 33IV,2 Average coefficient of variation, separately for 13 expenditure items within each of 12

groups of wage and salary earners 44IV,3 x2-test for constancy of coefficients of variation 46IV,4 Parameter estimates in the three-parameter case of the function

log i = log x + log I + fi log y);. Higher public servants and salaried employeesin the Capital 55

V, i Correlation coefficients (R x 100) between observed and calculated expenditures 58V,2 x2-test 62V,3 F-test for linearity (ratio between variances "around" the regression and variances within

groups) 66V,4 N-test for number of runs 68V,5 1-test for the longest run 73

V,6 d-test for size and direction of deviations from the regression 74V,7 Number of significant test results among 12 groups of wage and salary earners. Signifi-

cance level 95% (x2, F, 1) and 5% (N, d) 76V,8 Gain of regression. Standard errors in the distribution of expenditures and in the distribu-

tion of deviations from the double logarithmic Engel function, log y a + b (log xlog x) 79V,9 F-test for parallelism of Engel curves 83

10 Calculated income elasticities for 13 expenditure items for each of 12 groups of wage andsalary earners 85

V,!! Average income elasticities for 13 expenditure items 89VI,! a Average expenditure per person on dwelling in certain income groups separately for

different types of household 93lb Average expenditure per person on fuel and light in certain income groups separately

for different types of household 93VI,! c Average expenditure per person on food in certain income groups separately for different

types of household 94VI,ld Average expenditure per person on tobacco in certain income groups separately for diffe-

rent types of household 94VI,! e Average expenditure per person on clothing in certain income groups separately for

different types of household 95VI,lf Average expenditure per person on footwear in certain income groups separately for

different types of household 95VI,!g Average expenditure per person on washing and cleaning in certain income groups

separately for different types of household 96VI,!h Average expenditure per person on durables, excl, vehicles, in certain income groups

separately for different types of household 96

VI,! i Average expenditure per person on personal hygiene in certain income groups separatelyfor different types of household 97

VI,lj Average expenditure per person on books, newspapers, etc. in certain income groupsseparately for different types of household 97

VI,lk Average expenditure per person on sports, holidays, hobbies in certain income groupsseparately for different types of household 98

VI,ll Average expenditure per person on transport incl, own car in certain income groupsseparately for different types of household 98

VI,! m Average expenditure per person on union fees, subscriptions etc. in certain income groupsseparately for different types of household 99

VI,ln Average »expenditure« per person on savings in certain income groups separately fordifferent types of household 99

VI,2 Correlation between each two of thirteen expenditure items 104

VI,3 The biggest positive and negative correlation coefficient separately for each group ofwage and salary earners 106

VI,4 The highest and lowest values of the correlation coefficient of deviations, separately foreach social group 108

VI,5 Unexplained variance in the regression analysis 110

LIST OF FIGURES1,1 Income per person and expenditure on clothing. Average values of 154 groups of 3 ob-

servations among lower public servants and salaried employees in the capital 8

1,2 Income per person and expenditure on personal hygiene. Average values of 112 groupsof 3 observations among higher public servants and salaried employees in the capital 8

1,3 Income per person and expenditure on durable goods (excl, motorcars). Average values of51 groups of 3 observations among skilled workers in the rural districts 9

111,1 Expenditure on theatre and cinema 27

IV,! Two groups of expenditures on tobacco 37

IV,2 The variance of the expenditure on clothing plotted against the square of this expenditure 41

IV,3a Type of systematic deviations which will be revealed by the run tests, and may be not byother tests 50

IV,3b Type of systematic deviations which will be revealed by the F-test, but may be not bythe run tests 50

V, la Untransformed expenditure observations plotted against log x 64

lb Logarithmically transformed expenditure observations plotted against log x 64

V,2a Frequency of expenditure for given income 65

V,2b Frequency of log expenditure for given income 65

V,3 Five Engel functions. Income and expenditure on food. Skilled workers in the provincialtowns 81

V,4a Mean points for the twelve Engel curves for expenditures on books, newspapers etc 87

V,4b Mean points for the twelve Engel curves for expenditures on sports, holidays etc 87

1 Income and expenditure on sport, holidays and hobbies for four different types of house-holds 101

VI,2 Income and expenditure on food per person for four different types of households 101

PREFACEThe present analysis of the expenditure behaviour of Danish households of wage andsalary earners is essentially based on the model used by Prais and Houthakker in theiranalysis of a similar English material (The Analysis of Family Budgets, Cambridge,1955).

The scope of the analysis was determined by the Statistical Department, the Instituteof Statistics of the University of Copenhagen and the Institute of Economics of the Uni-versity of Copenhagen in concert. Mr. Kjeld Bjerke, lecturer, chief of division, the Sta-tistical Department, and the heads of the two institutes, Professor Anders Hald, dr. phil.,and Professor P. Nørregaard Rasmussen, dr. polit., having met regularly to discussproblems arising in the course of the work; this committee has also gone through thefinal report.

The day-to-day work was directed by Mr. Erling Jørgensen, cand, polit., The StatisticalDepartment, assisted for a brief period by Mr. Abling Thomsen, cand, polit., the StatisticalDepartment. Mr. Erling Jørgensen has also prepared the manuscript of the presentreport. Computations have been carried out by The Danish Institute of ComputingMachinery, the staff of which has rendered valuable assistance in the programmingprocess. Mrs. Lis Taxøe Jensen, the Statistical Department, has displayed great skilland care in preparing the many tabulations and examples to be used in the analysis.

The Translator of the Statistical Department, Mr. Vagn K. Sandberg, has translatedthe draft manuscript into English, and Mr. Niels Thygesen, cand, polit., has carriedout a terminological revision of this manuscript.

The expenditure in connection with the computations carried out by The DanishInstitute of Computing Machinery has been covered by a grant from the Danish StateResearch Foundation, and a grant has been received from the Rask-ørsted Foundationto cover the cost of the terminological revision of the manuscript. The expenditureincidental to the publishing of the book has been defrayed by the Statistical Department.

The analysis was started in the summer of 1959 and was concluded with a provisionalreport at the end of 1962.

Copenhagen 1964.

Chapter L

BACKGROUND OF THE STUDY ANDTHE MAIN RESULTS

I a. Background of the study.

The survey of income, consumption and saving patterns in 1955 of households of Danishwage and salary earners which was carried through at the beginning of 1956 is the mostcomprehensive and the most detailed of the consumption surveys undertaken by theDanish Statistical Department since it started making this kind of surveys in 1897').The primary object of the surveys was originally to procure information of the "Con-ditions of life in the different classes of society, including nutrition and consumption"2),but after the system of adjusting salaries, wages, benefits and other payments in accord-ance with a price index became generally adopted, the consumption surveys were pri-marily undertaken in order to provide the basic material for constructing a system ofweights to be used in the calculation of price indices. During recent years, however, thegenerally descriptive purpose, which was the primary one in the first consumption sur-veys, seems to be gaining ground again. One of the reasons for this development is thefact that it has been realized that the basic material which is obtained by means of aconsumption survey carefully planned and carried outin this connection the sub-stantial advances in survey techniques of the last decades should be borne in mind-provides information on essential economic relationships, particularly in connectionwith spending in relation to income, which cannot be illustrated so completely in anyother way3).

The Danish consumption survey for 1955 has, in fact, been subject to a more detailedprocessing than any of the previous surveys. Thanks to the scope and quality of the1955 survey it has been possible, through this detailed processing, to arrive at resultswhich are of direct interest to private institutions and persons as well as to public authori-ties.

A general outline of the 1955 survey, its planning and main results, was given in Sta-tistiske Efterretninger in 19574). Food consumption was dealt with separately in an article

')A complete list of publications on Danish consumption surveys will be found p. 118.Act concerning the Central Statistical Bureau 1895.Cf. I.L.O. (II).Statistiske Efterretninger 1957, No. 83.

4

in Statistiske Efterretninger in i 958). The data collected concerning the saving and personalwealth of the households of wage and salary earners were subject to a special analysis,the results of which were given in a volume of the series of Statistiske Undersøgelser in19606). Two volumes in the same series dealt with the data collected on the distributionand composition of the wage and salary incomes7).

The greater part of the information obtained from the households of wage and salaryearners concerned their consumption expenditure in the year 1955, and it was decidedto subject the consumption behaviour of these households to a more detailed analysis.It is the results of this study which are contained in the present publication.

I b. Main results of the analysis.

The analysis aimed at giving a precise description of the relationship between the dis-posable income of the Danish households of wage and salary earners and their expen-diture on some essential items in the year 1955. This relationship between disposableincome and the expenditure on given items is undoubtedly of considerable importancein helping to explain differences in consumption behaviour from one household toanother, although, of course, many other factors must be included if all such differencesare to be explained, such as type of household, residential and social classification, etc.However, the income-expenditure relationship is of great importance in another con-nection, namely for forecasting the development of consumption in response to givenchanges in income, for one household or group of households or for all households as awhole8).

The analytical work thus consisted mainly in deriving the best possible descriptionof the income-expenditure relationships. Such income-expenditure relationships oftengo by the name of Engel curves after the German economist and statistician, ErnstEngel9). More specifically, the work in connection with the analysis has consisted incalculating estimates of the parameters of five types of functions selected in advance andthen comparing these functions by means of a number of tests for goodness of fit inorder to find the Engel function most suitable for each expenditure item.

This is on the whole the same type of analysis as was adopted by J. S. Prais and H. S.Houthakker in their study of British family budgets from 195510). In fact, in many re-spects the present inquiry may be considered an application to Danish data of the ana-lytical tools which Prais and Houthakker present and discuss in their work.

Statistiske Efterretninger 1958, No. 46.Opsparing i lønmodtagerhusstandene 1955, Statistiske Undersøgelser, No. 3, Copenhagen 1960. (Sum-mary in English).Lønmodtagerindkomster, Fordeling og sammensætning, Statistiske Undersøgelser, No. 6, Copenhagen1962 and An Analysis of the Personal Income Distribution for Wage and Salary Earners in 1955.Statistical Inquiries, 1964.Cf. chapter III, p. 28, and Erling Jorgensen (10), pp. 54-61.Cf. Ernst Engel (10)PraisJ. S. and Houthakker H. S. (10)

5

Prais and Houthakker's study contains a discussion of alternative approaches to ana-lyses of budget data as well as a list of literature dealing with family budget studies11).These background problems, therefore, will not be elaborated here.

As regards the main lines of the present study a few points deserve mention in thissummary of methods and results.

To eliminate the most disturbing influences deriving from differences in the size ofthe households interviewed, all expenditure and income amounts were converted intoamounts per person for each of the 3100 households for which data were obtained.

The income concept used, which is the independent variable of the Engel curve, wasthen determined as disposable income (all cash receipts less paid personal taxes) perperson, and Engel functions were derived for 13 main expenditure items. In the appendixwill be found a detailed classification of these expenditures which together amount to85 per cent of total consumption expenditures for all households of wage and salaryearners. The 13 main items were the following:

DwellingFuel and lightingFood (incl, regular eating out, beer, wine, and liquor within the usual householdconsumption)TobaccoClothingFootwearWashing and cleaningDurable goods (excl, motor vehicles)Personal hygieneBooks, newspapers, etc.Sports, holidays, hobbies, etc. (incl, visits to restaurants, theatres, cinemas, andbeer, wine, and liquor outside the usual household consumption)Transport (incl, motor vehicles)Subscriptions, union fees, insurance premiums, etc. (excl, life and pension insurance)

The calculations were carried out separately for 12 groups of wage and salary earners,namely four social groups within each of the three regions of the country.

The three regions were the followingThe capital incl, suburbsThe provincial towns incl, suburbsUrban districts in the rural municipalities

In regions I and 2 the following social grouping was usedHigher public servants and salaried employeesLower public servants and salaried employeesSkilled workersUnskilled workers

11) Prais J. S. and Houthakker H. S. (10) p. 169.

6

In region 3, the urban districts in the rural municipalities, separate calculations werealso carried out for the social group of Agricultural workers. In this region no calculationswere carried out for the group of Higher public servants and salaried employees.

The five types of functions used were the following, x denoting disposable income perperson and y the expenditure per person on the item in question:

(1,1) log i = a + (log y - log ),(1,2) log=a+(v-'_i),(1,3) =a+,9(logvÏ),(1,4) = a + (v-1

(1,5) log = log x + log [1 (a + log y)]

a bar denoting average value and J (t) denoting the cumulative distribution functiont2

of the normal distribution (t)I

lV2

Considerable parts of the report on the analysis are devoted to a discussion of the esti-mation procedure so that it may be said that an evaluation of the methods of analysis wasanother main objective of the analytical work besides the calculation of the results of theanalysis.

The tests applied showed almost consistently that the double-logarithmic function (1,1)gave the best description of the Engel curve for all 13 expenditure items as a whole. Thisresult is in a way surprising because it implies that the income elasticity12) of the house-holds in their demand for each of the 13 commodity groups is constant over the incomerange (for given social group, since the calculations have, as mentioned, been carriedout separately for 12 groups of wage and salary earners). It might have been expectedthat commodity groups which are considered necessities in the higher income groupswould be regarded as luxuries in the lower income groups. This, however, is not con-firmed by the estimates of the income elasticities.

It might then be thought that this constancy of the income elasticity would hold goodonly in the case of the individual groups of wage and salary earners, which do not eachof them cover any wide income interval, but that the matter would be different if ailhouseholds were grouped together, in other words, that the estimates of should bedifferent for the different groups. However, there proves13) to be a remarkable stabilityas regards the mentioned estimate of , when we move from one group of wage andsalary earners to another. In the case of six expenditure items a hypothesis of constantincome elasticity through all 12 household groups can be maintained, and in the caseof the remaining 7 items the deviations, though statistically significant, are not verygreat. The demonstration of this stability in the income elasticity of the households in

Note that the income elasticity is identical with the parameter 1 in the double-logarithmic Engelfunction (I, 1).Cf. chapter V p. 83.

their expenditure on the most important items is one of the most conspicuous results ofthe analysis'4)

This stability renders it justifiable to calculate the average income elasticity for the12 groups of wage and salary earners for each of the 13 expenditure items. These averageelasticities are shown in table 1,1, where the expenditure items have been arrangedby size of the average income elasticity.

Table 1,1.Income elasticities for 13 expenditure items; average values for

12 groups of wage and salary earners.

7

It will be seen from this table that the expenditure items fall into three clearly definedgroups:

A group, which might be called necessities, in which the elasticity is just over 0.5,consisting of three items, food, footwear, fuel and lighting.

A second group, which might be called neutral commodities, with an elasticityclose to unity. This group includes 8 items, among which the two important items ofdwelling and clothing.

Finally, there is the third group, which might be called luxuries, in which theelasticity is significantly higher than unity; this group consists of the two items of transport(incl, motor vehicles) on the one hand and sports, holidays, hobbies, etc. on the other

As already mentioned, the main objective of the analysis has been to give a descrip-tion of the relationship between the income of the households of wage and salary earnersand their expenditures on important items. The analytical method employed, whichconsists chiefly in linear regression analysis, seems to yield satisfactory results in the case

14) This result invites the postulate that the income elasticities found for the population of wage andsalary earners have general validity for all population groups. Concerning the consequences of thispostulate, see Erling Jørgensen (12).

Item Average incomeelasticity

Fuel and lightingFootwearFood

0.510.560.61

Subscriptions, union fees, etc. 0.82Personal hygiene 0.86Washing and cleaning 0.86Dwelling 0.89Books, newspapers, etc. 0.98Tobacco 0.98Durable goods (excl, motor vehicles) 0.99Clothing 1.04

Transport (incl, motor vehicles) 1.39Sports, holidays, hobbies, etc 1.50

Expenditure per person. Danish Kroner

1800

1700

1600

1500

1400

1300

1200

1100

1000

900

800

700

600

500

400

300

200

100

log y = 2.689 . 1.115 (log o - 3.745)

0 1 2345678910 15 20 25

Fig. I, 1. Income and expenditure on clothing. Average values of 154 groups of 3 observations among lower public servantsand salaried employees in the capital.

Expenditure per person. Danish Kroner

log y -- 2178 0807 (log o -.3806)

0 1 2 3 4 5 6 7 8 9101112131415 20 25

Fig. 1,2. Income and expenditure on personal hygiene. Average values of 112 groups of 3 observations among higher publicservants and salaried employees in the capital.

Income Pr. personin 1000 Danish Kren

Income pr. personin 1000 Danish Krone

450

425

400

375

350

325

300

275

250

225

200

175

150

125

100

75

50

25

o

Expenditure per person. Dar'ish Kroner

9

log y = 2.114 -V 1.068 (log x - 3.488)

Income per person. Danish Kroner

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Fig. I, 3. Income and expenditure on durable goods (excl, motorcars). Average values of 51 groups of 3 observations amongskilled workers in the rural districts.

of most expenditure items, cf. fig. I, i and fig. 1,2. For a few items, however, particularly fordurable goods and transport incl, motor vehicles, the residual variation in expenditure fromhousehold to household is very high; the introduction of the disposable income of thehouseholds as explanatory variable has not reduced the variation appreciably. Fig. 1,3demonstrates the high residual variation as regards durable goods in the group of skilledworkers in the rural municipalities.

It may probably be concluded that the analysis of the expenditures of the householdson these items will have to be tackled differently, by including information on type ofhousehold and other environmental factors and particularly on income changes and theconsumption behaviour in earlier periods. Such a dynamic analysis has, however, beenoutside the scope of the present study, but it must be admitted that in the case of durablesand transport the results presented here are rather unsatisfactory.

In a few respects the report goes beyond the objective of the analysis as set out above.In a concluding chapter it is examined to what extent the 13 expenditure items are cor-related, i.e., whether households which spend much or little on one item display acharacteristic expenditure behaviour as regards one or more of the other items. It wasattempted to discover, e.g. whether households with a high consumption of tobaccohave a lower consumption of food than households with a low consumption of tobacco.It was also tried to outline the importance of differences in type of households (size and

800

700

600

500

400

300

200

100

o

10

composition of household) to the consumption behaviour of households for given in-come classes.

As regards the first problemthe interrelationships of the 13 expenditure itemsthecalculations show only a slight correlation. Only in the case of the two items of dwellingand fuel and lighting was there a significant (positive) correlation. This result is a con-sequence of the design of the analysis, since the grouping of the many goods and servicesfor which information was obtained into a moderate number of main expenditure itemsaimed precisely at a grouping with only a slight positive or negative correlation betweenthe individual groups. This attempt to arrive at stable relationships between income anda few groups of expenditures at the same time ruled out a description of the consumptionbehaviour of the households towards individual goods and services; if such a descriptionwere to be attempted, the expenditure on other closely related goods and services wouldundoubtedly have to be taken into account.

As regards the importance of type and size of the household to the consumption be-haviour of the households, the examinations show that the size (i.e. number of persons)of the household was the dominant factor, and that the conversion into amountsper person from amounts per household eliminated the greater part of this "disturbing"influence. In the case of certain expenditure items, among them dwelling and tobacco,other influences made themselves felt; a general influence, as was to be expected, wasthe "economies of scale" effect, i.e. the expenditure per person falls as the number ofpersons per household rises.

I e. The report.

After this introductory survey of the background and plan of the analysis and of someof its main results, chapter II will present a review of the basic material. This review con-sists of a description of the practical work of carrying through the survey of consumptionand saving, i.e. the collection and processing of the basic material, and also a descriptionof the inaccuracy attaching to the figures derived from the basic material. Chapter IIalso contains a brief summary of average expenditure per household on the main expen-diture items. In chapter III the aim of the analysis will be defined, various models for ananalysis of the expenditure behaviour of the households being discussed, a discussionwhich concludes in a statement of the reasons for choosing the Engel curve approach asthe main subject of the analysis. Chapter IV contains a detailed discussion of the methodsof analysis. What types of functions are to be chosen as a basis for deriving Engel curvesfor the different expenditure items? How are the variables to be specified? How is thesuitability of the functions employed in the description of the income-expenditure rela-tionship to be tested?

In chapter V the results of the analysis are presented. The double-logarithmic Engelcurve was, according to the test made, found to be the "best" of the 5 types of functiontested.

Finally, chapter VI suggests examples of some further calculations which should make itpossible to achieve a more exhaustive description of the consumption behaviour of the

11

households than has been possible with the main tool of the present analysis, the Engelcurve. In order to explain the variations observed in the expenditures of the householdson a given item, differences in the size and composition of the households will be discussedas well as the expenditures of the households on one or more other items.

An appendix to the report contains partly the basic material and a detailed descriptionof the expenditure items comprised by each of the 13 main items and partly tablesshowing the results of the computations. These tables fall into two parts, the results ofthe main analysis, cf. chapter V and the results of the further calculations, cf. chapter VI.

A list of the literature used will be found on pages 117-118.

') Cf. references p. 118.

Chapter II.

REVIEW OF THE SURVEY MATERIAL

lia. Introductory remarks.

The pt esent analysis of the consumption patterns of Danish wage and salary earners in1955 is based on the family budget survey of households of Danish wage and salaryearners undertaken in 19561).

This survey comprised a total of 3100 households, selected by stratified samplingamong all households of wage and salary earners; the sampling procedure is decribedbelow.

The questionnaire used in the survey was very detailed as it was desirable to collectinformation on household expenditures for a large number of consumer goods cf. thedetailed list in the appendix p. 240. For the purpose of the present analysis, however, onlymain expenditure items are of interest as the emphasis of the analysis is on the consump-tion pattern as a whole rather than on consumption of individual commodities.

JIb. Concepts and methods of the survey.

1. Collecting the information.

The consumer survey has been carried through by personal interviews. This method hasbeen chosen in preference to the far cheaper one of mailing questionnaires to the house-holds for two reasons. Firstly because some of the questions were so complicatedthat the interpretation of an interviewer was considered necessary in order to ensurethat the households would understand them, and secondly to reduce the non-responserate to a minimum. Both as regards the quality of the information collected and asregards the response rate, gratifying results were achieved. Only 61 questionnaires outof a total of 3161 had to be rejected owing to unsatisfactory completion, and only 473households, or less than 12 per cent of all households approached, refused to cooperate.(Besides, 345 other households could not be contacted because of illness, change ofaddress, etc.).

The survey comprised income and assets of the household as well as its expendituresand savings during 1955, the expenditure being broken down into various items, andsaving being distributed by the various forms of saving.

13

Total saving, defined as net change in assets, was calculated on the basis of the figu-res for changes in debts payable and receivable in the course of the year. The interviewerchecked the figures against the difference between income and total consumption. Whereappreciable discrepancies were found, the household was contacted again, and sub-stantial errors in the figures for saving as well as in the various consumption items wereeliminated. On the whole, it may perhaps be concluded that both the interview methodand the fact that the total budget of the household was included in the survey havehelped in keeping what might be called errors of measurement at a minimum in thedata collected for consumption, saving and personal wealth.

2. Income and expenditure concepts; unit of analysis.

The purpose of the 1955 consumer survey was to illustrate expenditures and savings inhouseholds of wage and salary earners. Hence it follows directly that it is the householdwhich is the relevant unit of analysis both as regards consumption and saving. Thisgives rise to the problem of defining the household concept on which the survey was tobe based.

In drawing up such a definition there are two considerations to keep in mind. Firstly,the household should be defined in such a way that it contains thoseand only those-persons who behave as a unit both in relation to the earning of income (income unit)and to the spending of income (spending unit). Secondly, the household unit adoptedshould be practical for the purpose of selecting, collecting and processing the surveymaterial.

Without going into detailed definitional problems it should be emphasized that thesetwo considerations may in fact be irreconcilable. The consideration that the personsincluded in the household should act as one income and spending unit might lead to theselection of a household concept which will prove to be impractical in the selection ofthe sample or in the collection and processing of the material. Moreover, even if weinsist only on the point that the household should act as one income and spending unitwe are not assured of an unambiguous definition. Thus with regard to board and lodg-ing, domestic servants take part in the consumption of the household, but their incomesare not included in the joint income of the household. On the contrary, they are paidout of this income; domestic servants in some respects form part of the spending unit,but not of the income unit. If it is desired, e.g., to inquire into the relationship betweenthe income of the household and its food consumption, information supplied by thehouseholds in which there are domestic servants will give misleading results.

Further, it may be mentioned that the household concept which would be mostrelevant in an analysis of consumption, will not necessarily be the one that is mostrelevant in an analysis of saving, since it may very well be imagined that persons whoact as one unit as regards consumption will not make their saving decisions in common;examples are: households in which there are boarders and/or older children living athome who pay a certain amount towards the joint consumption of the household, butotherwise dispose independently of the rest of their income.

14

The household concept actually used was the following: those persons (and onlythose) who take part in the joint consumption, i.e., husband, wife, and children withoutan income of their own are included; also included were children living at home whohad incomes of their own and others who stayed permanently with the household,provided that these persons did not spend more than 50 per cent of their incomes outsidethe households.

As regards income, consumption and saving the following concepts were used:

Income:

Cash wages and salaries.Contributions to pension schemes withheld out of the salariesof public servants and salaried employees.Interest and Dividends.Pension, incl.old-age pension.Disablement pension.Contributions from separated or divorcedspouse.Unemployment relief.Contributions to housekeeping made by children andrelatives.Payment by lodgers for board and lodging.Amounts received under in-surance policies.Gifts.Inheritance, scholarships.Sales of motor car, moped, bi-cycle, furniture, clothing, etc.2) .Savings certificates received3).

Consumption expenditure:

Expenditures on purchases of all consumer goods, including all expenditures in con-nection with purchases of durable consumer goods (motor cars, motor cycles, furniture,household appliances, radio and television sets2), etc.), i.e. both initial payments ondurable consumer goods acquired in the course of the year and instalments on hire-purchase debt relating to acquisitions in this or previous years; taxes, subscriptions, etc.-Also cash contributions to relatives and gifts.

Saving:

Amounts spent on increasing, or received by reducing, the below-mentioned items:Cashin--hand.--Bank and savings bank deposits.Bonds and shares.Premium

bonds.Private mortgage deeds.Compulsory saving and savings certificates.-Value of real property.Business assets.Other assets.Life and deferred annuityinsurance (incl, contributions of public servants to pension funds).

Amounts spent on reducing, or received by increasing, the below-mentioned items:Debt to bank and savings bank not secured by mortgage in real property.Mortgage

debt in real property.Other debt apart from hire-purchase debt, etc.Only a few comments are necessary in connection with these definitions. As mentioned

above, saving was calculated also as the difference between income and consumptionin the course of the year. Since this method of calculation must, of course, give the sameresult as the calculation according to the above definition4)if the figures are correct

In the case of purchases of motor vehicles, the value of any motor vehicle traded in has been setoff against the value of the new vehicle.In connection with the imposing of new indirect taxes in 1955 saving bonds were issued to all per-sons with assessed income of kr. 4000 or more. The face value of the bonds was increasing with in-creasing income of the persons concerned.Cf. Statistiske Undersøgelser, No. 3, Opsparing i Lønmodtagerhusstandene. 1955, Copenhagen1960, pp. 11-16.

15

the interviewers were able to get a good check on the data collected by comparingthe amounts of saving resulting from the two definitions.

Besides, it should be emphasized that the definitions used are based on a "cash pointof view". Income comprises all cash payments to the household, incl, gifts and amountsreceived under insurance policies. On the other hand, consumption contains, as a generalrule, all amounts actually paid by the household; this involved, for instance, that inthe case of purchases of durable goods, only the initial cash payment and any instalmentspaid during the survey period were included.

Period of the survey.

In the choice of survey period two conflicting considerations have to be taken into account.Firstly, it is desirable that the households interviewed should be able to remember, atthe time of the interview, the size of their income during the survey period and, in parti-cular, how they have spent this income. For this reason, it would be desirable to haveas brief a survey period as possible. On the other hand, however, it is desirable thataccidental fluctuations should not be allowed to have too much influence on the results,neither as regards the income earned nor as regards the spending of it. If both the earningof the income and the consumption took place at a regular rate, this consideration wouldnot give rise to any problems, but since particularly some consumption expendituresoccur irregularly, it would be reasonable to make the survey period so long that theseirregularities will be smoothed out. Since seasonal factors must be presumed to play adominant part in these fluctuations, it was found reasonable to use the year as the sur-vey period.

Especially as regards income earned experience shows that most households will havea precise idea of it only for a period of one year and only once a year, namely when theyfill in their income tax returns. Therefore the survey was carried out immediately afterthe date for delivering of the income tax returns, viz, the 1st of February.

Method of selection.

The selection of a sample of basic sampling units on the basis of probability theory (i.e.,in such a way that it becomes possible to calculate the standard error of the results)requires, firstly, a specification of the population from which the sample is to be drawn(setting up a frame for the selection), and secondly, the choice of a sampling design basedon random selection (i.e., a selection by which all the elements of the population havea specified probability of being selected).

As regards the setting up of a frame for the consumer survey, the population census onthe 1st October, 1955, provided a complete "list" of all households in Denmark. In view ofthe main object of the survey, which was an analysis of the consumption patterns ofhouseholds of wage and salary earners, it was decided to exclude from the frame allrural municipalities without urban areas because there are very few wage and salaryearners in those municipalities. The few wage and salary earners who were to be foundthere were considered to be represented by the households of wage and salary earners

16

selected in the rural minicipalities with urban areas. The frame was accordingly thosehouseholds in the whole of Denmark, except in the "purely" rural municipalities,which were recorded in the population census schedules as having a wage or salaryearner as head of household.

The choice of sampling design was influenced by a number of factors, the most importantof which will now be briefly discussed.

The guiding principle in the considerations which preceded the choice of samplingdesign was that the standard error of the estimates calculated on the basis of the sampledrawn should be below a certain limit, and that the costs of the survey should be heldat a minimum given this maximum standard error5).

However, the sampling design which gives the lowest standard error for one of theestimates, e.g. for total food consumption expenditure per household, will not always at thesame time give the lowest standard error for all the other estimates. As soon as a surveyis to form the basis of a calculation of several estimates, it is therefore necessary to specifyone of the quantities which it is desired to estimate on the basis of the sample as thedecisive one in the choice of sampling design. One may then hope that this design willalso be favourable as regards the other quantities to be estimated. Alternatively, all thequantities to be estimated must be arranged by order of priority and an overall evalua-tion must be made for the purpose of arriving at a design which minimizes the sum ofthe standard errors for all the quantities estimated, the individual standard errors beingassigned weights corresponding to their order of priority.

One of the objects of the 1955 survey was to provide the basis for calculating a systemof weights for the Danish price index. Therefore the estimation of average expenditureon the main items of goods and services which are covered by the price index wereassigned a high priority. As estimates made on the basis of preceding consumer survey(1948) showed that there was a high correlation between the total expenditure of ahousehold and expenditures on certain main items, the desired end was assumed to beattained by fixing certain limits of the standard error for the total expenditures perhousehold for each of twelve groups of wage and salary earners6).

Ina following section an account will be given of the calculation of these standard errors.With the mentioned point of departure (that the survey should be planned with a

view to minimizing the standard error for the total consumption expenditure), the sampl-ing design was otherwise determined by a number of practical and theoretical considera-tions.

Firstly, already the choice of method of enumeration places certain restrictions onthe sampling procedure. The decision to carry through the survey by means of inter-viewers who are to call on each sample household up to six times, makes it natural toassign to each interviewer as many households as he is able to call on within the periodof the survey. This procedure ensures that interviewers gain a maximum of experiencein taking interviews. It may also be mentioned that the possibilities of supervision forthe central authorities will be considerably reduced if there are too many interviewers.

See E. Lykke-Jensen: (13), pp. 16-18.Viz, four social status groups separately within three district categories; cf. below p. 34.

17

Consequently, it was desirable that the households should be selected in clusterswithin geographical areas, whereby the transport costs of the interviewers would beconsiderably reduced. Each cluster corresponds to the capacity of one interviewer, inthis survey approximately twenty households.

Besides, the very form of the frame will play a part in the considerations concerningthe method of selection. In this case, as already mentioned, the schedules from the 1955population census provided the frame from which the sample was drawn, and as theseschedules are arranged by municipalities (in Copenhagen by "roder" (tax collectiondistricts), in Frederiksberg and Gentofte by parishes), it seemed natural to base thesampling on whole municipalities (parishes or "roder"). As it was possible to groupthese municipalities in accordance with the criteria which were considered relevant tothis inquiry, viz, distribution by industry and degree of urbanization, it was foundreasonable to use stratified sampling. Finally, the desirability of illustrating the con-sumption patterns of the individual social status groups separately within each of thethree district categories, (the capital, provincial towns with suburbs, and rural muni-cipalities 'with urban areas) made it natural to conduct the survey in such a way thatit would be possible to calculate separate estimates for each status group within thesethree district categories.

The result of these considerations was accordingly that the sampling was made intwo stages within each of the three mentioned district categories. At the first stage muni-cipalities ("roder", parishes) were drawn by random selection from strata of uniformmunicipalities already formed, the probability of selection of each municipality ("rode",parish) being proportionate to the number of households in the municipality. Actually,the selection ought to have been made in proportion to the number of households ofwage and salary earners, but this number was unknown. As the households of wage andsalary earners constituted a more or less constant share of the total number of house-holds within each stratum of municipalities this procedure seems permissible. At thesecond stage households of wage and salary earners (basic sampling units) were drawnfrom each municipality in the first stage sample of municipalities, households belong-ing to different status groups7) drawn with different probability.

In the capital 16 first stage units were selected, comprising about 36000 householdsof wage and salary earners, from which were drawn 1262 second stage or final units,i.e. individual households. In the provincial towns the numbers of first and secondstage units were 17 and 920 respectively, the sample of first stage units comprising about85000 households. In the rural districts the numbers were 26, 918 and about 4000 re-spectively. Whereas the final sample of 3100 basic sampling units comprised only about0.45 per cent of all households of wage and salary earners, the number of such house-holds in the first stage sample of municipalities comprised about 18 per cent of the totalnumber8).

Finally, it should be mentioned that the definition adopted of the basic unit of ana-lysisall members of the expenditure unitdid not quite correspond to the units

Higher salaried, lower salaried, skilled and unskilled, cf. p. 5.A similar approach was used in the Danish labor force surveys in 1951 and 1952, cf. The DanishLabor Force Surveys. Statistical Review, New-Series vol. 2, No. 7, pp. 259-267.

18

selected at the second stage (the sampling units), as these units had been determinedby the choice of the frame of the survey, namely the schedules from the 1955 populationcensus. Since, according to the definition used in the population census, the householdcomprises all persons staying permanently in the household, with the exception oflodgers providing their own food, whereas in the consumer survey the household com-prises only the persons who contribute at least fifty per cent of their income towardsthe consumption of the household9), the population census household will in some casescomprise more persons than the basic sampling unit of the consumer survey. This factleads to certain complications in estimating averages for the whole country and alsoin estimating the true standard errors of these averages, but in the following this hasnot been taken into account as we have assumed that the inaccuracy introduced hereby isinsignificant compared with the inaccuracies which arises in the course of the collectionand processing of the questionnaires

lic. Estimating mean values and their standard errors.

1. Accuracy of the results of the survey.

The estimates based on the 1955 consumer survey are subject to a ceitain degree ofinaccuracy. This inaccuracy consists of two components. The first originates in the col-lection and processing of the material, i.e., wrong or inadequate information, errors incoding and punching, etc. The errors of this type are often called systematic errors(bias), cp. the following section. The other component is called sampling error, and itoccurs because only a sample of households and not the entire population is observed.

As the sample of households of wage and salary earners has been selected by strati-fied two-stage sampling, the sampling error of the estimates for each of the twelve groupsof wage and salary earners will consist of two elements; firstly, the error due to thevariation, within strata, among the sampling units at the first stage, municipalities,and secondly the error which is due to the variation among the sampling units at thesecond stage, i.e., among the individual households within municipalities.

While it is impossible to arrive at more precise estimates of the systematic errors, thesampling method adopted makes it possible to form estimates of the two elements ofthe sampling error10).

The calculation have shown that the error element due to variation among individualhouseholds within municipalities is dominant.

Table 11,1 shows estimates of average expenditure per household for 14 expendi-ture groups; the table shows also the average saving and cash income per householdfor each of the twelve groups of wage and salary earners. The standard sampling errorof the estimated total expenditure per household is estimated at kr. 140 or approximatelyi per cent of the total expenditure; 70 per cent of the standard error is due to variationbetween households within first-stage sampling units.

S) Cfr. the exact definition, p. 14 above.10) Cf. Statistiske Undersøgelser No. 3, Opsparing i lønmodtagerhusstandene 1955, Copenhagen 1960,

pp. 3-4.

") Cf. Prais S. J. and Houthakker H. S. (10) p. 42.

19

In the regression analyses which form the greater part of the present inquiry the ob-servations for each of the twelve groups of wage and salary earners into which the 3100households observed have been divided, have been treated as deriving from a simplerandom selection. The estimates of the standard errors of the parameter estimates willtherefore become a little too high since the stratification effect is ignored, and besides,some bias may be expected to occur in the estimation of the parameters because thedeviation of the observations from the regression line is evaluated on an assumption ofsimple random selection, whereas the actual procedure is two-stage stratified sampling,cf. chapter 3, page 23. Howewer, this bias must be considered insignificant in relationto the total variance in the distribution of the deviations from the regression line of theobservations.

2. Processing of the material.

The inaccuracy of the estimates, discussed above, refers only to the sampling error, i.e.,the error which will inevitably occur when estimates for the whole population are tobe made on the basis of a sample of the population. With a given standard deviationin the distribution of the elements of the population the sampling error depends on thesize of the sample and the sampling methods; it has been attempted, within the givencost framework, to make this sampling error as small as possible.

However, the estimates can also be subject to another type of error, which also occursin complete enumerations, namely the so-called systematic errors, i.e., errors causedby wrong or inadequate completion of questionnaires and from the processing of thematerial, that is, errors in the scrutiny, coding and punching of the material received.

In the paragraph above on the enumeration method it was mentioned that the sur-vey was conducted through interviews, partly to induce the sample households to co-operate, and partly to reduce the number of wrong answers. The 160 interviewers hadreceived thorough instruction concerning the survey through letters and at speciallessons at which officers from the Statistical Department went through the problemsin connection with the completion of the questionnaires A provisional scrutiny of theanswers could therefore be made by the interviewers themselves at the time of the inter-view, the interviewers making a rough comparison of incomes and expenditures. Incases of discrepancy the interviewer was to take care that the household interviewedprovided, whereever possible, the necessary supplementary information. There is reasonto believe that thereby more correct figures have been obtained for the size of incomeand for items of expenditure which people might otherwise fail to state correctly.

It is obviously extremely difficult to indicate, even with rough approximation, themagnitude of errors which have arisen owing to people giving wrong answers to theinterviewer. Experience from similar surveys abroad supports a belief that such incor-rect statements are particularly frequent within the field which is often designated con-spicuous consumption, i.e., such items as tobacco, liquor, consumption in restaurants,etc").

20

Table 11,1. Average income, saving and expenditures in 1955 in kroner per household.

S) Excluding life insurance, deferred, annuity insurance, etc. which have been included in saving.) Personal taxes not included.

A detailed comparison with figures for total consumption per household for the wholecountry obtained from production statistics and importexport figures for all expenditureitems showed an over-all agreement, which confirms our impression that deliberatelyincorrect answers occurred only in few cases.

After the material was received at the Statistical Department it was subjected toa thorough scrutiny, in the course of which particularly the information concerningpersonal assets and liabilities as well as the changes in these items were critically exam-ined as it turned out that it was these items in the questionnaire which had caused thegreatest difficulty. The questionnaires which were found to be inadequately completedwere returned with a request for supplementary information. A few questionnaires(61 in all) had to be rejected altogether, i.a., because the quality of the informationprovided was on the whole found to be too poor.

Higherpublic

servantsand

salariedemployees

Lowerpublic

servantsand

salariedemployees

Skilledworkers

Unskilledworkers

Higherpublic

Servantsand

salariedemployees

Lowerpublic

Servantsand

salariedemployees

Number of households in the sample 336 469 206 251 212 341

Average number of persons 2,9 2,3 2,8 2,6 3,0 2,6Imen 1,5 1,0 1,5 1,3 1,5 1,2

of whomIwomen 1,4 1,3 1,3 1,3 1,5 1,4

Expenditure on:1. Dwelling 1789 1183 1160 995 1514 1044

2. Fuel and lighting 867 552 618 570 967 679()Food 4318 3235 4160 3761 3822 29784. Tobacco 739 575 862 792 645 506

5. Clothing 1682 1194 1246 1044 1681 1120

6. Footwear 353 267 295 255 338 2577. Washing and cleaning 363 251 296 253 264 191

8. Durable goods (excl, motor vehicles) 1011 705 803 642 1111 791

9. Personal hygiene 451 354 373 332 383 28910. Books, newspapers, etc 727 493 449 368 660 32011. Sports, holidays, hobbies (incl. cinemas,

theatres, restaurants, etc.) 1975 1112 1210 925 1461 812

12. Transport, (incl, motor vehicles) 1489 726 902 639 1444 519

13. Union fees, subscriptions') 244 225 413 394 248 227

14. Other expenditure2) 1744 1044 1085 859 1502 868

15. Saving 1384 554 475 405 1433 617

"16, Total cash income 22606 13921 16111 13437 20866 12530

Capital and suburbs Provincial towns

21

After this scrutity the information in the questionnaires was transferred to punchcards. Both the punching operation and the subsequent mechanical processing werechecked; in the case of the punching operation the check consisted in a complete veri-fication of all punch cards, and in the case of the tabulation in check runs on the sums;the risk of error at these two stages is very small.

The figures which have been worked out on the basis of the punch card materialmust be presumed to be subject to rather few systematic errors compared with earlierDanish surveys. The errors of this type which might exist originate from the first twostages of the process: the interviewer's collection of information and the scrutiny ofthis information. As mentioned, special efforts have been made in this survey to limitthe possibilities of error at these two stages, cf. the section concerning the enumerationmethod.

and their suburbs Rural districts with urban areas

All Capital Prov. towns

Ruraldistricts

withLowerpublic Agri- households total total urban

Skilled Unskilled servants Skilled Unskilled cultural areasworkers workers and

salariedemployees

workers workers workers

(weightedaverage)

(weightedaverage)

(weightedaverage)

total

(weightedaverage)

154 213 322 155 281 160 3100 1262 920 9183,0 2,9 2,7 3,3 3,3 3,3 2,8 2,6 2,9 3,21,5 1,4 1,3 1,7 1,7 1,8 1,4 1,3 1,4 1,71,5 1,5 1,4 1,6 1,6 1,5 1,4 1,3 1,5 1,5

993 824 792 884 695 503 1027 1217 997 771747 647 708 745 706 603 683 629 719 715

3409 3196 2746 3181 3208 2879 3467 3838 3288 3139659 517 478 511 477 349 616 751 567 472

1012 981 936 883 873 624 1108 1240 1103 904232 225 217 205 195 144 251 285 247 202218 173 175 189 163 150 226 282 199 175749 533 665 555 482 359 690 761 711 546287 249 230 228 200 153 300 368 283 215262 213 263 224 201 148 357 479 302 242

756 610 655 580 453 302 903 1222 793 556492 404 697 829 567 304 712 873 578 650387 425 201 356 395 337 344 334 354 341764 648 667 597 551 366 887 1111 828 619799 468 581 352 562 294 635 626 703 555

12780 10876 10967 11192 10253 7743 13569 15810 12934 10928

Chapter III.

OBJECTIVES OF THE ANALYSIS. ENGEL FUNCTIONS

lila. Introductory remarks.

The basic material available from the 1955 consumer survey is, as mentioned above,very comprehensive. For each of the 3100 households included in the survey approx-imately fifty punch cards (80 columns) were prepared. A complete description of thismaterial, including an analysis of the relationships among the many quantities of whichit is made up, is naturally out of the question. In order to keep the analytical workwithin reasonable limits, it is necessary to concentrate on some essential, well-definedproblems. More precisely: among the many possible models which could be tested bymeans of this material, a few are to be selected which are of substantial interest fromthe points of view of economic theory, social policy, etc. The analysis then consists inconfronting these models with the information collected.

From the point of view of economic theory, interest would focus on a model capableof explaining the consumption expenditures of households as a function of quantitiesfamiliar from economic theory as determinants of consumer behaviour. Hereby it might bepossible to evaluate consumption, once information on those quantities to which it isfunctionally related becomes available. If the quantities in question may be more con-fidently predicted than consumption itself, such functional relationships will be usefulin predicting consumption.

From the point of view of statistical theory the greatest interest will attach to esti-mation procedures; how are the best estimates of the parameters in the chosen modelsto be computed? What tests are applicable for purposes of comparing the estimates?

The computional work involved in the analysis has been carried out on an elec-tronic computer. Accordingly it has been possible to choose more labour-consumingmodels and methods of calculation than if only the traditional calculating facilities hadbeen available.

IlIb. Choice of model.

1. Determinants of expenditure.

According to traditional economic theory the expenditure of a household on a givencommodity is determined primarily by the income of the household and by the priceof the item in question. Prices of other commodities, expenditure of the household onother commodities as well as expenditure of other households on this and other commodi-

23

ties may also appear as important arguments. Other factors are of course, the compositionof the household, its geographical location and social status. Also previous income andincome change of the household as well as its assets might play an important role indetermining the consumption behaviour.

As the present analysis is based upon a consumer survey relating to a given point oftime and a given market, prices may be considered constant, independent of othervariables as e.g. income and expenditure. All other variables mentioned above, however,can be found in the basic material of this inquiry, and if it was possible to set up a simplemodel of the relationships of these variables the parameters of such a model might beestimated.

However, there is no presumption that the relationships between the variables aresimple at all. If the analysis is to be practicable, a relatively simple type of functionmust be chosen and the number of variables must be further reduced. Of the quantitiesmentioned above there is a strong presumption that household income is dominant inthe determination of the expenditure pattern, while the household expenditure on othercommodities plays a less prominent part, cf. chapter VI, p. 110. Therefore, it we furtherdisregard income changes and assets as well as household expenditure on other commo-dities (and the consumption pattern of other households), a relationship remains withingiven social groups of households containing solely the two variables expenditure of thehousehold and its income.

In a discussion of the relationship between these two quantities it is natural to startby emphasizing that expenditure is in the nature of a dependent variable to income,while income may reasonably be considered an independent or a determining variablecf. Prais & Houthakker (10) p. 80. It is quite obvious, however, that very often there isan influence the other way round, planned or incurred expenditures determining tosome extent the income-earning behaviour of the household. On the whole this influencemay be considered weak as compared to the influence of income on expenditure andespecially as regards households of wage and salary earners as their possibilities forincreasing income in the short run are rather limited.

Assuming that all households (household size and composition, social and geographicalgroup held constant) show identical income-expenditure relationships except for randomvariation, a description of the "average" household of wage and salary earners will beof the form

(111,1) y = f (x) +

where y denotes household expenditure on a given item, x the household income andL the effects on y from omitted determining variables plus random effects.

2. Engel curves and household survey data.

Formula (111,1) is the general expression of the Engel curve for a given expenditureitem indicating the relationship between a household's income and its expenditure onthat item. It was decided to place the Engel curve in the centre of the analysis, and thegreater part of this and the following chapter is therefore devoted to a discussion of

24

methods of determining parameters in Engel functions by means of a household budgetsurvey material.

Before discussing the question of the type of Engel functions a few remarks shouldbe made in connection with the general approach of the analysis, which is indicatedby the choice of the Engel curve as the main object of investigation.

Is it at all possible conceptually to estimate Engel curves on the basis of householdbudget surveys? Or stated more precisely: Assuming that all information in a surveyprovides reliable measurements of the incomes and expenditures of a number of indi-vidual households in the population group investigated, is it then possible to estimatetrue Engel curves based on that survey? Obviously, the degree of interest which wouldattach to the analysis from the point of view of economic theory depends very muchon the answer to this question.

It is important to realize from the outset that our basic material does not allow ofany direct testing of an Engel function relating to an individual household. For a givenhousehold only one set of income and expenditures is known, whereas several differentsets of such observations would be necessary to enable us to test any hypothesis concerningthe income-expenditure relationships of this household.

However, it may be possible to make up for this defect by inserting observations ofthe expenditures of other households on the commodity, these other households beingselected in such a way that the relevant values of the income scale will be represented.Thus, instead of studying each household's expenditure reaction to various income levels,the relationship between the expenditures and incomes of many different householdsfor one period is studied and it is postulated that by doing so the Engel curve of the"average" household in 1955 as written in (111,1) above will be obtained.

On the face of it, this postulated Engel curve is merely a description of the incomesand expenditures of various households. Such a description is, of course, valuable initself since it enables us to make a statement of the following form: in the Danish popu-lation of wage and salary earners in 1955, households with an income of x1 kr. spent anaverage of fj (x1) kr. on the i'th commodity, and households with an income of x2 kr.spent fj (x2) kr. apart from random deviations. Obviously the significance of the ana-lysis as seen from the viewpoint of economic theory will be higher if this description ofthe income and the expenditure on certain commodities or groups of commodities of3100 households will be a useful approximation to the Engel curves for the Danishpopulation of wage and salary earners in 1955.

The Engel function, as defined in (111,1), is static, i.e., it gives an expression for theexpenditure behaviour which the "typical" household will display, ceteris paribus, atalternative levels of income after any initial adjustment processes have been completed.This function, accordingly, entirely disregards the time factor and also the processwhereby the households passes from one income level to the other. More concretelythis process might be exemplified as follows: a household whose income rises will notadjust its expenditure behaviour to the new income level until some time has passed;hereby the saving of this household may temporarily be higher than the average forhouseholds whose incomes are permanently on this higher level. Conversely, a house-hold which passes from a higher to a lower income will try to maintain consumption-

25

thereby reducing savingthan the average for households whose incomes are permanentlyon this lower level. Furthermore, the related 'more general' question arises whether thereaction of an individual household to changes in income will depend on income andconsumption changes in neighbouring households').

These and other dynamic elements in the consumption behaviour of the householdshave been left out of account in the Engel functions of the type shown in (111,1)butthey are included in the estimates of the income-expenditure curve which can be madefrom the observations of the 3100 households in the basic material, and probably insuch a way that the estimates are influenced systematically. It is thus highly probablethat among the high-income households in the survey there will be relatively manywho have experienced an appreciable increase in income since the immediately pre-ceding period, while, conversely, there will be relatively many households withdeclining incomes at the lower end of the scale. The consumption expenditures observedfor the high income groups will therefore tend to understate the "true" (static) propen-sity to consume, while among the low income groups the "true" propensity to consumewill be lower than the observed expenditures, i.e., the Engel curve which is estimatedwill rise more slowly than a "true", static Engel curve.

The postulate: that the observed relationships between income and expenditure forthe 3100 households in 1955 are identical with the Engel curves as defined by (111,1)has, however, other weaknesses.

It does not allow for the dependence of the individual household on the consumptionbehaviour of other households. That such interdependence among the consumptionexpenditures of the individual households exists has long be recognized in demandtheory2). The Engel curve is based on a ceteris paribus assumption and answers ques-tions of the type: what amount would a household spend on the i'th commodity if itsincome rises by kr. 1000, kr. 2000, etc., assuming that the other factors in the household situ-ation are unchanged? The most important factors here are: household size, residence,social status and the relative income position of the household in relation to its "neigh-bours". The curves we can estimate from the available observation material, however,refer to households with, frequently, highly deviating environmental factors, and inobservations of expenditures for households at different income levels it is thereforeimpossible to maintain the mentioned ceteris paribus assumption. We may assume thatexpenditures on durable goods are highly susceptible to the influence of environmentalfactors, whereas, e.g., the expenditure on typical necessaries is less dependent on theconsumption behaviour of other households.

The so-called layereffect3), may also lead to a wrong evaluation of the "true" Engelcurve. If, e.g., we imagine that wage and salary earners in the rural districts, who are onan average at a lower income level than wage and salary earners in the towns, have aconsiderably lower expenditure on theatre and cinema than urban wage and salary

Cf. Duesenberry, J (3), Friedman, M. (7), Modigliani, F. (15) for a theoretical discussion of thisaspect of the consumption pattern; Danish empirical studies on the subject are found in Opsparing ilønmodtagerhusstandene 1955, Copenhagen 1960.Cf. Duesenberry, J. (3), Friedman, M. (7), Stone, R. (17).

2) Cf. WoId (19) p. 68.

26

earners on a corresponding income level, the observed income-expenditure relationshipsmay come out as illustrated in fig. 111,1.

The income-expenditure curve, I + 2, which is drawn as representing the house-holds in both groups, exaggerates the income elasticity of the "average household ofwage and salary earners" in the demand for theatre and cinema, because there is amarked shift in expenditure level from rural households to urban households. Thisshift may be due to the fact that these goods are not equally accessible to the two house-hold types.

Now, the consumption survey of wage and salary earners in 1955 was so compre-hensive that it was possible to make separate calculations for twelve different groupsof wage and salary earners defined by residence and social status; to this should beadded that adjustments were made also for observed differences in the size of the house-holds. It may, perhaps, therefore be permissible to conclude that the shifting effectsare smaller in this analysis than in most other similar analyses, in which the numberof observations is most frequently so small as to render impracticable a breakdown intohomogeneous subgroups. It should be emphasized however that this effect may stilldisturb the estimated Engel curves, cf. chap. V pg. 86.

In conclusion it must be underlined, therefore, that one cannot accept, withoutqualifications, Engel curves calculated on the basis of household surveys as representingthe Engel curves as defined by (111,1).

If estimated Engel curves, based on household surveys, are to be used, e.g. for pre-diction of expenditure on certain commodities, income being known or guessed at,great care must be shown. Comparison should always be made to income-expenditurerelationships calculated on the basis of other types of data primarily time-series data4).

On the other hand if time series data alone are used we are precluded from drawingconclusions as regards the situation at a specified time; we have instead to refer thecalculated values to the whole of the period covered by the time series. Hereby the riskof introducing disturbing influences from other factors has grownchanged pricerelations, income level and distribution, etc.so that the calculated values will, forthat reason, become unreliable.

As will be shown in the last section of this chapter some of the biases mentionedabove should not be excluded if the Engel-curve estimates are to be used on the macro-level; what are considered biases in one conception of the Engel curves are in otherinterpretations of the Engel curve rightly considered as true elements of the relation-ships.

In addition to these conceptual difficulties, which may cause serious biases in theestimates of the relationships between y and x, the estimated parameters in the functionsof type (111,1) p. 23, are attended with errors from other sources. One important sourceof error is inaccuracy in the measurement of the independent variable, the house-hold income. These errors of measurement are partly systematic and seem on the wholeto lead to an understatement of income, a phenomenon which is well known from taxincome statistics and which it has hardly been possible to avoid entirely in this survey

4) Cf. Wold (19) p. 50.

Expenditure

2. rural households

Fig. III, 1. Expenditure on theatre and cinema.

either5). The occurence of inaccuracy in the independent variable even if there is nosystematic error of measurement, leads to a systematic error in the evaluation of theslope of the regression lines. If the amount of inaccuracy can be estimated, it will bepossible to adjust for it in the evaluation of the slope6), but this is not possible in ourcase, and the mentioned adjustment therefore cannot be made.

One of the requirements for determining unbiased estimates of the parameters of anEngel function of the form (111,1) by means of the regression analysiswhich will bethe main tool in the followingis that this function is specified in such a way that e isindependent of x. This involves either (1) that x is a quantity given in advance andaccordingly not subject to variation in our experimental set-up, or (2) that any variationin x is independent of e, which is an expression of the unexplained variation in y.

The observed x values do not fulfil the requirement mentioned under (1), alreadybecause x is subject to a considerable error of measurement, cf. above. On the otherhand, it is not quite clear whether the variations in x are of such a nature that not eventhe requirement under (2) is fulfilled. The variation in x due to errors of measurementmay perhaps to a great extent be presumed to be independent of e, but it is possible

1. urban households

27

It turns out, indeed, that on an average for all households observed the sum of expenditures andsavings exceeds the recorded incomes by kr. 145, or slightly over one per cent of the average re-corded income.Cf. Hald, A. (8) p. 615 and Stone, R. (17) p. 296.

Income

28

that there are some causes of income variations which also affect yor which have theirorigin in y. If, for instance, a household's purchases of a motor car or other durablesinfluence the "income-earning" behaviour of this household, there will be a risk of bias7).

In an evaluation of the conclusions which can safely be drawn from the estimatedEngel curves, it is important to take the above-mentioned considerations into account.The fact is that it is not the "true" Engel curves we arrive at, and therefore care mustbe shown if the results are to be utilized in drawing further conclusions. Or, in otherwords, the validity of the analysis depends on the interpretation of the estimates.

3. What, then, can the results be used for?

Firstly, the estimated Engel curves give a more precise description of the income-expen-diture relationships of the households of wage and salary earners in the year 1955 thanwould be possible by the mere presentation of summary averages of expenditures atdifferent income levels. As the computations are made separately for twelve residentialand social status groups, this description will give, in addition, useful illustration ofexisting differences in expenditure behaviour among these twelve groups.

Such a description of the expenditure pattern of the households of wage and salaryearners is obviously of interest in many respects; questions concerning the marketingconditions of certain commodities or groups of commodities in the different parts ofthe country, and questions in connection with differences of the consumption patternsof the different social status groups and income gruops seen in relation to existing orcontemplated excise duties are two important fields. More generally, it may be mentionedthat the official Danish statistics concerning the disposal of national income are con-siderably less developed than statistics concerning the formation of national income, forwhich reason any supplementing description of the kind mentioned here will be veryuseful. However, it is true of all the fields where the results could be applied that theywould be substantially more valuable if they covered the whole population, whereasthis survey, as already mentioned, covers only households of wage and salary earners.

From the point of view of economic theory, however, it is quite as interesting toascertain whether the estimates are of any value except from a purely descriptive pointof view. Are they of value in the analysis of demand at the microlevel? And can theybe used as a basis for forecasts of total consumption at the macro level?

The conclusion of the considerations stated above concerning this problem is thatthe estimates shown in the following chaptersand estimates from other similar familybudget surveysform a very valuable supplement to the existing empirical basis ofdemand theory. The results are primarily applicable at the micro level, i.e., in ananalysis where the point of view is that of the individual household or group of households,whereas it would be more questionable to draw inferences for the analysis of the demandof the total population for the different groups of commodities.

By definition, Engel curves are expressions of the behaviour of individual householdsat alternative income levels and, subject to the above reservations as regards the inter-

7) Cf. Wallander, Jan (18) p. 52.

29

pretation of the estimates of Engel curves calculated here, the results can thereforedirectly only say something about the conditions of an individual household, or groupof households, under alternative assumptions as regards the income of the household orgroup of households. These reservations, as will be remembered, involved especiallyfour problems:

The estimates of the Engel curves were to a certain extent influenced by dynamicadjustment processes, whereas the "true" Engel curves are static.

The estimates were influenced by the fact that for some groups of commoditiesconsumer behaviour was to a considerable degree determined by environmental factors(the interdependence effect), whereas the "true" Engel curves are based on the usualceteris paribus assumption.

The layer effect.The inaccuracy introduced by any errors of measurement.

With these reservations in mind the estimates should, however, be useful as a basisfor conclusions concerning the consumption behaviour of individual households. How-ever, it should be emphasized that the estimates refer to the year 1955 so that statementsconcerning consumption in any subsequent period will be attended with a furtherunknown error. The applicability of the Engel curve estimates on the macro level willbe discussed in the final section of this chapter.

hIc. Use of estimated Engel curves on the macro level.When it is attemptedon the basis of the estimates of Engel curves concerning theconsumption behaviour of individual householdsto draw conclusions as regards the con-sumption of all households, i.e., total national consumption expenditure, the problem ofthe environmental influence on consumption is again brought to the fore. In estimatingEngel curves of the type of formula (111,1) above, one of the main problems is how toavoid too much disturbing influence from the behaviour of other households. The objectis to estimate independent income-expenditure relationships for individual house-holds. But the relationships between total income and total expenditure, i.e. a functionor curve illustrating alternative values of income and expenditure of all households,must necessarily take into account the effects on expenditure of the j'th householdbrought about by a change in the income of other households. Or in other words, on themacro-level a possible interdependence effect must be taken into account. A curveexpressing the relationship between the whole population's total consumption of certaincommodities and its total income will therefore be biased if it is formed by simple aggrega-tion of the Engel curves of individual households.

The occurrence of interdependence among the consumption behaviour of the indi-vidual households naturally makes it difficult to say anything about the developmentof total consumption under alternative assumptions as to the development in total in-come on the basis of knowledge of Engel curves for individual households. For distinctnecessaries, where the interdependence effect is probably moderate, this drawbackmay not be of decisive importance.

30

An expression of the magnitude of the interdependence effect can be estimated undervery simplified assumptions8), and in such cases we might be able to arrive at a betterestimate of the "macro Engel curve". No such estimate of the interdependence effect hasbeen made in this survey.

Assuming that a satisfactory estimate had been calculated of the "true" relation betweenthe population's total income and its total expenditure on various commodities, thisestimate would still be subject to the limitation that it would be valid only for the surveyperiod, viz, the year 1955. Such an estimate would not be directly applicable as a basisfor a calculation of a consumption forecast, because it does not, of course, contain anyelements of shifts in the trend of consumption owing to changes in fashion, taste, etc.These trend factors, which are often of great importance, are, on the other hand, con-tained in the time series covering the development, over long periods, in total consump-tion, total income, etc., which are the usual basis of forecasts. However, it must berealized that such trend factors can be extremely unstable, and as they appear oniyimplicitly in the relationships an adjustment for structural change is impossible.

If time series data concerning total income and total consumption of certain groupsof commodities were supplemented with Engel curves for all groups of the population,based on household survey material and estimated with due regard to the above-men-tioned reservationsand preferably estimated for several consecutive survey periods-then forecasts on consumption expenditures could be improved considerably.

8) Cf. PraisJ. S. and Houthaker, H. S. (10) p. 18.

METHODS OF ANALYSIS

TVa. Introductory remarks.

The income-expenditure relation = f (r), is the main object of the present analysis;j denoting household expenditure on a given expenditure item and r household income.The question is now how this relation is to be estimated on the basis of the availableobservations.

In this situation it might be imagined that the form of the Engel function of a givencommodity was given in advance or had been arrived at on the basis of, e.g., studies ofthe "expenditure process". The task would then "merely" consist in determining theparameters of this function, and the results of the analysis -would then be of the follow-ing type: in the Danish population of wage and salary earners in the year 1955 the para-meters of the Engel function for the expenditure on the i'th commodity assumed thefollowing values

However, no such "true" Engel function is given in advance. This is so because amore general theory on the basis of which a specific function could be derived does notexist. The first step of the present analysis therefore consists in selecting a functionalform. Next comes the comparison of the different functions selected by means of suitabletests for goodness of fit.

In short it may be said that the analysis consists of three stages: 1) the selection ofa number of functions, 2) the estimation of parameters of the selected functions, and 3) acomparison of these functions and the data by means of various tests for goodness of fit.

IVb. Choice of Engel functions and specification of the variables.

1. Criteria for selecting Engel functions.

In selecting Engel functions one may adopt two different points of view. Firstly, onthe basis of the existing theory of consumer behaviour, try to set up, for each expenditureitem, a model which fulfils the theoretical requirements to the greatest possible extent.Or secondly, on the basis of the available observations, select one or more functionsshowing a satisfactory goodness of fit, whether or not these functions can be justifiedby the theory of consumer behaviour.

Of course, it would be most satisfactory to choose the former approach, but it mustbe acknowledged that the theory of consumer behaviour does not at present offer suf-ficient guidance for the selection of "true" Engel functions. But economic theory cantell that the "true" Engel curve for a given commodity has certain characteristics. This

32

information can then be utilized as a supplementary criterion for selecting among al-ternative functions. This criterion being a supplement to the selection by means ofdifferent tests for goodness of fit.

If a given type of function deviates from the characteristics of the true Engel curvefor extreme values of income, whereas the function otherwise "behaves" satisfactorily,this, however, should not exclude the use of the function in question.

Moreover, it is to be observed that computational problems in connection with thedetermination of the parameters of the function should not be too complicated, and thisrequirement naturally limits the types of functions which can be used.

2. Description of the functions selected.

In the present survey the following five functions were selected, in which y is the house-hold income, the expenditure on a given expenditure item, and a, fi and are para-meters:

logi= a+fi(logv_i)log = a + fi (

I)

=a+fi ('ogriogv)

= a + fi ( -log i = log x + log [cP (a + fi log r)J

Functions (IV, 1) to (IV, 4) find little justification in economic theory, whereas func-tion (IV, 5) to a somewhat greater extent can be justified on the basis of studies of the"consumption process".

Functions (IV, I) to (IV, 4) are two-parameter functions, which are linear in the twovariables or in simple transformations of these variables. This means that we can usethe computationally very convenient techniques of linear regression analysis. Thesefunctions represent to some extent alternative hypotheses as regards the income elasti-

city of the expenditure, e = or the marginal propensity to consume m =dv i dr

and can thus be used for testing those hypotheses concerning the characteristics of the"true" Engel curves which are related to e and m as suggested by economic theory.

Table IV, I show the values of e and m for the five functions.According to function (IV, 1) the income elasticity is a constant, being identical

with the parameter fi. According to function (IV, 2), e is inversely proportional to in-come, whereas according to function (IV, 3), e is inversely proportional to expenditureitself. If one considers the marginal propensity to consume, m, it will be found that ac-cording to function (IV, 3) m is inversely proportional to income and according to func-tion (IV, 4) inversely proportional to the square of income. Among other characteristics

log17= a+ p (log y logy)

Il ilog 17 = a + P I - - -\v V

?7=a+p(logv_logv)

= a + fi ( - D

log 17 = log x + log [ (a + log y)]

33

Table IV, 1.Values of income elasticity, e, and marginal propensity to consume, m, for five Engel functions.

of functions (IV, 1) to (IV, 4) which are interesting from the point of view of economictheory may be mentioned that functions (IV, 1) and (IV, 2) reflect one feature of trueEngel curves: that expenditure can never be negative, while functions (IV, 2), (IV, 4)and (IV, 5) reflect a theoretically desirable property of Engel curves of certain commod-ities, viz, that expenditure asymptotically tends towards a saturation expenditure. Con-cerning function (IV, 5), it should be mentioned firstly that it contains many of thequalities which can be said to be characteristic of the "true" Engel curves. Expenditurecan never be negative; the income elasticity is falling with rising income, and the mar-ginal propensity to consume is first rising and then falling. Secondly, the use of function(IV, 5) as model for the consumption behaviour can be justified by analogy to certainbiological experiments2).

In the actual estimation procedure it was decided to fix the parameter p at a givenvalue inter aha because the 3-parameter estimation met with serious difficulties, cfr.chap. IV, p. 55.

3. Spec jfication of the variables.

After the functions have been chosen, the observations must be put into a form suitablefor computation, and here a number of problems arise. The following deserve specialattention: 1) the precise specification of the two variables, y and 17, the household in-come and the expenditure on a given commodity or group of commodities, and 2) pro-blems concerning the grouping of commodities and households. In the foregoing theinterpretation of the Engel functions and the selection of certain functions have been

2) Aitchison and Brown (1), page 128, and the same authors in The Review of Economic Studies,No. 57, 1955.

Function e m

P

V V2

17 V

P P

V17 V2

(a + log y) 17[cp(a-]--logv)](a + log y) y [(a+ logy)]

34

discussed on the assumption that all variables other than income and expenditure,were "under control" (p. 23). Accordingly it has been assumed that the parameters ofthe functions selected were valid only for households in a certain area with that parti-cular social status, of given size, etc.

Now, it is obvious that in real life one cannot estimate the parameters under suchrestricted assumptions. In this field it is impossible to make laboratory experiments inwhich all variables other than those examined are kept under control. It is thereforeonly with rough approximation that one can isolate and measure the influences due tothe factors which are of interest in any given inquiry.

In the present survey of the income and expenditure in 1955 of households of wageand salary earners the factors which may be expected to influence the expenditure be-haviour of households apart from the dynamic factors discussed above (p. 25) will espe-cially be residential differences (whether the household lives in a rural district or in aprovincial town or in the capital), differences as to social status (whether the householdbelongs to, for instance, the group of higher salaried employees or the group of unskilledworkers), and differences in size and types of households.

The available basic material is so comprehensive that it is possible to make separatecalculations for several subgroups of wage and salary earners, and by using domicileand social status as criteria in this subgrouping the greater part of the variation in ex-penditure attributable to differences in these two respects will be eliminated. The fol-lowing subdivision was used in the survey:

Higher public servants and salaried employees in the Capital.Lower public servants and salaried employees in the Capital.Skilled workers in the Capital.Unskilled workers in the Capital.Higher public servants and salaried employees in the Provincial towns.Lower public servants and salaried employees in the Provincial towns.Skilled workers in the Provincial towns.Unskilled workers in the Provincial towns.Lower public servants and salaried employees in the Rural districts.Skilled workers in the Rural districts.Unskilled workers in the Rural districts.Farm workers in the Rural districts.

By a further subdivision into subgroups by e.g. the size and composition of households,the number of observations would be so small in many subgroups that in spite of thesubgroups being more homogeneous it would not be possible to calculate the parameterswith reasonable accuracy3).

The size of household may, however, be taken into account, if for all households yand i represent income per person and expenditure per person, respectively4). Differences in

S) This in fact is the same thing as to say that no criteria for further subdividing are of "significant"importance for the stability of the relations in question.

4) Cf. S. J. Prais and H. S. Houthakker (10) pp. 88-93.

5) S. J. Prais and H. S. Houthakker, (10) p. 80.

35

type of household will probably still make themselves felt, but this influence can now,with good approximation, be considered as being of a random nature.

In a following part of this chapter the stochastic element of the model, will be takenup for discussion in greater detail (cf. p. 40), and the discussion at this point can thereforebe finished with a few further remarks concerning the definition of and y, the dependentand the independent variable of the Engel functions.

The dependent variable, i, the expenditure per person on a given commodity group,is defined as the value of the goods (and services) in this commodity group which thehousehold has bought during the survey period; there is one important exception to thisrule, viz, with regard to goods bought on the hire purchase system. In the case of thesegoods (particularly certain durable goods, furniture, radios, refrigerators and not leastown means of transport, motor-cars, motor-cycles, etc.) i is defined as the amount spent bythe household during the survey period in connection with the purchase of these goods,i.e., down-payment plus any instalments.

The independent variable, y, is defined as household disposable income per person,i.e., income earned less personal taxes paid.

The use of disposable income as the independent variable rather than total income is ofvarying importance for the different household types. An examination of all groups ofwage and salary earners as regards the taxes paid as a percentage of total income incertain income groups, separately for 5 household types, showed that this percentagefalls for a given total income with growing size of household (allowances for dependentsand children).

For a given household type the tax percentage naturally increases with rising incomeowing to the taxation system.

These facts must be borne in mind when reading the discussion of the income-ex-penditure relationship in the following chapter.

Disposable income and total income are thus strongly correlated, but differences inhousehold type, and income level and changes in income give rise to systematic devia-tions between the two income concepts.

In the present survey well-founded estimates of the incomes of the individual 'house-holds in the survey period have been obtained by checking information on income withinformation on expenditure + savings, see chapter II; it was therefore decided to usethis estimate of household income (less personal taxes paid) as the independent variable,cf. what has been said above concerning the importance of errors in the measurementof the independent variable (p. 27). In the corresponding British inquiry, householdincome could not be used as the independent variable because information about thesize of this income had been collected only for part of the material; instead the sum of allrecorded expenditures was used as the independent variable5).

In their arguments to justify this procedure, however, Prais and Houthakker tend toconclude that the sum of expenditures actually is a "better" independent variable thanthe income concept defined above because it enables them to arrive at more stablerelationships:

36

"The true determinants of the expenditure pattern of a household in a dynamicsituation are a complicated function of past, present and expected incomes, and thoughthis function can analytically be formulated in a precise way it is of little help here. Thesuccess of an empirical analysis must depend on the choice of some simple, readily ob-tainable, measure which substantially represents the facts. ... The use of total expenditureas the determining variable in the Engel curve can be justified on the assumption thatwhile total expenditure may depend in a complicated way on income expectations andthe like, the distribution of expenditures among the various commodities depends onlyon the level of total expenditure"6).

A priori the greatest interest seems, however, to attach to an elucidation of the rela-tionship between income and expenditure rather than the relationship between the sumof all expenditures and components of this sum (cf. the discussion of the uses of the Engelcurve functions on page 22).

If in many cases the use of total expenditure as the independent variable leads tobetter goodness of fit than the use of income, an important reason, of course, is the dif-ference in saving behaviour of the households. It is obvious that differences in savingbehaviour "disturb" the functional relationship between income and expenditure,but this does not mean that income is a bad independent variable. It means thatone should explain both saving and consumption in the same model.

Any satisfactory model illustrating the relationship between income, expenditureson the different commodity groups, and saving would naturally also have to takeincome changes into account. This has not been done in the present survey, and there-fore the relationship between y and will to a certain extent be less stable than if theavailable information on the influence of income changes had been utilized7). In Appen-dix D, which presents some of the basic material, information is given on saving as wellas on income changes of households from 1953 to 1954 and from 1954 to 1955.

4. Zero-observations.

Three of the functions selected (IV,l), (IV,2) and (IV,5) namely the functions in whichthe dependent variable is the logarithmic transformation of expenditure, s, are only de-fined for s > O. Therefore a problem arises if zero observations of s are found in thebasic material (assuming that negative values cannot occur). The occurrence of zeroobservations not only creates a purely computational problem, but also raises thefundamental question of whether the functions selected can be used at all in the de-scription of the observed relationship between income and expenditures on variouscommodity groups.

Assuming that zero observations occur with the same frequency in all income inter-vals, one may get a picture as shown in figure IV, 1.

) S. J. Prais and H. S. Houthakker, (10) p. 81.?) Cf. Opsparing i lenmodiagerhussiandene, Det Statistiske Departement, København, 1960, p. 31, where

it is shown that households with increasing incomes save significantly more than households withfalling incomes.

Expenditure

on tobucco

8) S. J. Prais and H. S. Houthakker (10) p. 50.

Income

Fig. IV, 1. Two groups of Expenditures on tobacco.

Here it is evident that "true" zero observations exist and that it is therefore necessaryto split the observation material into two groups before it is possible to give a satisfactorydescription of the relationship between u and i.

In one group, which contains all non-smokers (or rather all who do not buy tobacco)the function = O applies and for the remainder group one can then try to use thefunctions selected.

However, it turned out that a hypothesis concerning the occurrence of "true" zeroobservations can only be confirmed in exceptional cases. For most commodity groupsthe occurrence of zero observations is limited to the lowest income intervals, and it isperhaps then permissible to assume that the zero observations may be due to randomdeviations from the true values. If this is the case, it will not be justified to split up thematerial; instead it is necessary to work out a computational technique which permitsthe occurrence of zero observations. Here several possibilities seem open.

Firstly, one can assign an arbitrarily low value to the zero observation households,e.g., as suggested by Prais and Houthakker8) it = 0.25 m, m being the unit of measure-

37

38

ment. Assuming that all observations of rj <0.5 have been recorded as 0, and assuminga rectangular distribution of these observations their mean value will then be 0.25 m.This method leads to biases in the estimates of the parameters; especially a log y willbe too big (compared with the corresponding uncorrected estimates in the other 3 Engelfunctions).

Even if, by means of suitable reductions of all > 0.5, one might be able to avoid asystematic bias of the expenditure average, this method would nevertheless introduce aconsiderable element of arbitrariness into the calculation of the parameters of the func-tions and would therefore not be very satisfactory.

Another way out would be to try to estimate parameters direct from the functions,of which (IV, 1) and (IV, 2) have been formed by logarithmic transformation, i.e., inthe functions

The parameters a* and fi can be estimated by an iterative process, where each stageof the iteration is a linear regression.

An examination of several examples showed that in the successive stages of iterationthe estimates of a* and fi did not converge; the result of changes in one parameter seemedexactly to offset the result of the changes in the other parameter so that the results ofthe computations showed a continued oscillation. This method was therefore abandonedand instead it was decided not to use individual observations but to follow the methodadopted in the British inquiry: to carry out the calculations on the basis of a groupedmaterial. This, however, raises the problem of how to group the observations9).

5. Grouping problems.

In the grouping of the households the zero expenditure observations will in most casesbe grouped together with positive i values, and the group averages will therefore, exceptin very few cases, be higher than zero.

In the present inquiry the observations have been grouped in the following way:Within a given social group (see the list of social groups above p. 34) the households

are arranged according to size of income per person. The households are then grouped inthrees so that the one or two excess households (if the number of households is notdivisible by three) are rejected "from the middle" of the income range as it must beconsidered valuable to fully utilize the relatively few observations at the outer limits ofthe field of observation. The values of y and which are accordingly included in thecalculations are always the arithmetical average of the three household values observed

) Cf. S. J. Prais and H. S. Houthakker (10) pp. 50-51, and concerning the computational conse-quences of grouping, pp. 59-62.

39

for each group. Hereby it is achieved that transformations into logarithms or reciprocalvalues can be confined to the group averages.

In the very few cases where a group value of becomes equal to zero, it is rejected.Also rejected are a few individual households where the expenditure on certain

necessities (food and dwelling) was extraordinarily low, namely households with an ob-served expenditure on food of zero or households whose expenditure on food was belowkr. 300 at the same time as their expenditure on dwelling was kr. 0. In the case of thesehouseholds (most often households of single persons who receive board and lodging aspart of their remuneration), there were so severe errors of measurement that their ex-clusion from the observation material was deemed unavoidable.

However, the material is also grouped in another way: the several hundred individualcommodities and services are grouped into main commodities or commodity groups,so that only expenditure on these commodity groups are considered. Unlike the above-mentioned grouping of the individual households, this grouping of commodities is in-dispensable, if an overall description of the expenditure pattern is aimed at. The problemin this connection is not, therefore, whether a grouping is to be undertaken, but how thematerial is to be grouped and how far this grouping is to be carried.

Here, there are several, more or less conflicting, points of view to be considered. Adetailed classification of commodities will be desirable if the principal interest attachesto the marketing possibilities of the individual commodity. If the main interestas in thisanalysisattaches to an overall picture of the relationship between income and con-sumption expenditures, rather few groups should be considered. Another point has tobe made; in order to arrive at a stable functional relationship between r and it wouldbe desirable to group the material into groups which are felt by the households to be"natural", i.e., that in spending their income the households think in terms of and ac-tually distinguish among these categories of consumption expenditures. The breakdowninto "natural" budget items which should contribute to stability in the consumptionfunctions is, at the same time well in line with the aim of obtaining an overall descrip-tion of the consumption behaviour. On the other hand, it must be borne in mind thatthis procedure may group together commodities with different income elasticities,although from other points of view a grouping which leads to a higher degree ofhomogeneity within the individual expenditure groups might be desirable.

In the present analysis the following grouping has been used:

I. Dwelling.Fuel and lighting.Food.Tobacco.ClothingFootwear.Washing and cleaning.Durable goods (excl, motor vehicles).Personal hygiene.Books, newspapers, etc.

40

Sports, holidays, hobbies, etc.Transport (incl, motor vehicle).Subscriptions, union fees, etc.

For all households together these groups comprise close to 90 per cent. of total con-sumption expenditures. The items which have been excluded are, inter aha, expenditureson education, domestic servants, gifts and charities. A detailed description of the 13expenditure items will be found in the appendix.

IVc. Variance assumptions.

1. General remarks.

Now the actual calculation of the estimates of the parameters of the Engel curves is insight. The alternative Engel functions have been set up; the dependent and the inde-pendent variables of these functions have been defined, and finally the problems relatingto the grouping of commodities and households have been dealt with (whereby one alsoarrived at a workable procedure as regards the treatment of zero observations).

Before the calculation of parameter estimates of the Engel functions can be made itmust be specified how the stochastic element enters. As mentioned above, the Engelfunctions which were chosen are of the form = f (y) where f (y) is characterized by meansof the parameters a and j9 (and further of x for the log-normal distribution function); cf. p.32 above. However, inserting in the model the actual income and expenditure observationsx and y for y and , f(x) does not exhaustively describe a given household's expenditureon a given commodity group; each expenditure observation contains a stochastic elementand it is necessary to specify the properties of this stochastic element, r.

The simplest approach is to assume r to be independent of x and normally distributedwith mean value O and variance V { r = a2 = V yx . If these assumptions are ac-cepted, efficient estimates of the parameters of the five models will appear from a simpleleast-squares regression analysis10).

1f on the other hand, these simple assumptions are not fulfilledand this they arenot always in this analysisthen estimation of parameters carried out on these errone-ous assumptions as to the distribution of the stochastic element will involve a loss in theefficiency of the parameter estimates; these estimates will accordingly have an unneces-sarily high standard error. Moreover, an estimation on erroneous assumptions as regardsthe variance of the distribution of e will make it difficult to apply the proposed tests forgoodness of fit. If one accepts theinefficientparameter estimates obtained in thisway, one may all the same be able to use the different tests for goodness of fit if certaincorrections in the variance estimates are made"). Prais and Houthakker, in theiranalysis of the material of the British family budget surveys, have disregarded thesecomplications and have everywhere estimated on the variance assumption mentionedabovealso in cases where this assumption obviously does not hold good.

10) The homoscedastic case of J. Aitchison and J. A. C. Brown (1), p. 46 and S. J. Prais and H. S.Houthakker (10), p. 78.

") Cf. S. J. Prais and H. S. Houthakker (10) p. 57 and p. 96.

41

However, it seems to be a more satisfactory alternative to try to specify the model insuch a way that the least-squares regression estimator becomes the efficient estimator;thereby a correction to the testing procedure is, at the same time, avoided.

The efficient, least-squares regression estimate of the parameters will be achieved byweighting all y for given x values by the reciprocal value of their variance V y xObservations to which a high degree of variability attaches will then be included in thecalculations with less weight than observations where V y x is small. If, therefore,we know the true value of V y x for all x, such a weighted calculation of the esti-mates will give the desired result. Now, this true value is unknown, and the problemthen becomes to form suitable estimates of V y x for all x.

When plotting estimates of V x against y2 it was found in a number of casesthat there seemed to be reason to assume that V y x i.e., that the varianceof y for given x increases proportionally to the square of the dependent variable, cf. fig.IV,2.

If this assumption could be maintained, it would mean that the residual variance inthe functions in which the logarithm of y is used as a dependent variable would be-come constant. This can be shown in the following way. The variance assumptionVy j x = i22 means that e is included multiplicatively in the Engel function, i.e.,that a given sample of expenditure observations can be described by a function of theform y = f (x) (1 + e). If, now, both sides of the equation are transformed logarithmi-

Fig. IV,2.The variance of the expenditure on clothing (per person) within groups often households,arranged by the size of income per person, plotted against the square of this expenditure.

Higher salaried employees and civil servants in provincial towns.

42

cally, the result will be a function of the form log y = log f (x) + log (1 + s). Thestochastic element of this quantity, log (1 + s), will then be independent of y and xand the simple least squares regression estimator is efficient and unbiased. As will beshown later rather simple efficient and unbiased estimators can be devised for the re-maining Engel functions in this case, too.

It is evident that this convenient property of the hypothesis concerning the distributionof the stochastic element will further increase the interest in having the hypothesistested, and it was therefore decided to examine this problem in greater detail.

2. Testing the hypotesis V = a2 i2.

Assuming V yx = a2 2, it follows that the coefficient of variation, y, in the distrib-ution of y for given x is constant since

V yx

In other words, the hypothesis can be tested by means of a test for the constancy ofthe coefficient of variation. If a test for such constancy does not show significant resultsfor too many expenditure items, it would seem justified to maintain the hypothesis.

As mentioned above, the observations have everywhere been grouped in threes. Foreach group an estimate, c, of y has been calculated, the c-value for group number mbeing calculated by the formula

S

cm =Ym

A test for the constancy of ' can be developed if one can construct a theoretical c-dis-tribution derived from groups of three observations which are known beforehand tofollow the variance hypothesis being tested and compare the observed c-values withthis theoretical distribution.

In an article Hendricks'2) has derived an approximation formula for the distributionof c, assuming yto be normally distributed and the number of observations per group = n.

Assuming n = 3, Hendricks formula can be developed into the following expression'3

(IV, 8) p cF dc

c2

(i + 2 (1+ c2)) exp { 2 (3+2 c2)} dc

and by integrating one obtains

(IV, 9) P cH 13 +*c2

exp { 2

2.3c2c2)}

P c and p c dc denote the cumulative distribution function and the distributionfunction respectively for c, denote the mean value of y and a2 the mean value of 52

') Hendricks (9), cf. also Mc. Kay (14).13) Karl Vind, cand, polit., the Statistical Institute of the University of Copenhagen, has derived

(IV, 8) and (IV, 9) and has taken part in the preparation of section IV, b, 2.

(IV, 8) and (IV, 9) are approximative, a good approximation depending on

(_-i/if) and ( \ being small, where (u) denotes the cumulative

a aj/32c2Jnormal distribution function. Not least the second of these assumptions is critical, sinceit implies that y, i.e. the true c-value in a given expenditure group, must not be higher

than about 0.5. For y = -- = 0.5, 1 (u) will fluctuate around6

V3 + 2.0,520.0006, where u . For y = 0.6, (u) will fluctuate around 0.005. For y

y V3 + 2 e2higher than 0.6, (u) will grow steeply.

If now y is assumed equal to the observed average of the c-values from all groups ofthree, the test hypothesis that the observed c- values are distributed around the truevalue as indicated by the distribution function (IV, 8) above can be tested. By groupingthe observed c-values in suitable intervals and calculating the expected number in thesame intervals according to the theoretical distribution function shown above (formula(IV, 8)), the hypothesis can be tested by means of a z2 - test.

Before starting these calculations it is necessary to ascertain whether the assumptionsunder which the distribution function (IV, 8) was derived can be considered fulfilledin the present case. As mentioned, the assumptions of formula (IV, 8) and (IV,9) impliedthat y, the "true" value of c should not be higher than approx. 0.5. A glance at theobserved average c-values, cf. table IV,2, page 45, will show that this assumption in severalcases cannot be considered fulfilled. What then? Is the approximation in formula (IV, 8)above nevertheless satisfactory or must the attempt to test the hypothesis be abandonedin those cases where > 0.5? This problem has been investigated experimentally. Bymeans of random sampling numbers were formed distributions of c-values with given yand then it was tested whether the theoretical c-distribution (IV, 8) differs systemati-cally from the experimental c-distributions.

In the present case c-distributions were constructed from 100 groups of normally dis-tributed random sampling numbers each group consisting of three numbers. For eachgroup an estimate of was calculated by means of two of the three numbers; the third oneis then taken as an independent estimate of . By choosing a suitable mean of the randomsampling numbers a series of c-quantities with given y was produced. In the presentcase c-distributions were formed with y = 0.25, 0.33, 0.50, 0.67 and 1.0. These c-dis-tributions with known y-values were then compared with the distributions calculatedon the basis of the theoretical distribution (IV, 9) to ascertain the degree of approxima-tion.

It turned out that the distribution formulas (IV, 8) and (IV, 9) produced c-distri-butions which did not differ significantly from the empirically derived "true" c-distri-butions even for y = 0.67. However, it should be mentioned that this result is basedon only one series of 100 groups, so that it cannot without hesitation be consideredgenerally valid.

When y tends towards 1.0 (IV, 8) and (IV, 9) is clearly useless.

43

44

Table IV,2. Average coefficient of variation, separately for 13

The validity of the theoretical c-distribution formula for practically all items havingthus been substantiated, the Z2-test for the postulated variance assumption can now becarried out in the way mentioned above. These test calculations were carried out forall expenditure items separately for each of the 12 social groups into which the materailhas been divided.

Table (IV,3) shows the result of these calculations. The table indicates the calculatedx2-values; in brackets after each x2-value has been given the number of degrees of free-dom. All values which are outside the interval

X2 <X2<X2.525 .975

have been italicized. It wi]l be seen that most of x2-values fall within this interval, butthe table shows that the items of durable goods and transport display many signi-ficant x2-values, which seems to indicate that the c-values for these items cannot beconsidered distributed at random with a constant true value.

Also among the other expenditure items are there some significant x2-values (tobacco,sports, holidays and hobbies), especially in the group of lower salaried employees.

In this group special factors make themselves felt as regards the distribution by house-hold type which causes a higher variation in the amounts of expenditure on the variouscommodity groups. The many households consisting of single, often relatively young,employees, have in many cases special arrangements as regards their consumption offood and dwelling, and this again leads to an anomalous behaviour as regards theirexpenditure on other items.

Income Dwelling Fuel & ight Food Tobacco Clothing

The capital:Higher salaried employees 0.0100558 0.4118768 0.3580842 0.1916989 0.6973217 0.4142853Lower salaried employees 0.0099754 0.4138525 0.5104255 0.2391395 0.8289576 0.4674581Skilled workers 0.0105403 0.4640940 0.3941377 0.1949627 0.5843378 0.4060097Unskilled workers 0.0118760 0.4802885 0.4158627 0.2 140957 0.6281060 0.5241140

Provincial towns:Higher salaried employees 0.0229228 0.4123087 0.4247304 0.2288049 0.6666245 0.4061404Lower salaried employees 0.0105639 0.4859840 0.5399429 0.2534217 0.7225977 0.4221963Skilled workers 0.0148521 0.4074703 0.4089206 0.1767807 0.6510627 0.4185785Unskilled workers 0.0115769 0.4168361 0.3582310 0.2310122 0.6795800 0.4349383

Rural districts:Lower salaried employees 0.0085254 0.5753355 0.5261609 0.3213999 0.7005913 0.5041757Skilled workees 0.0149660 0.4184816 0.3557065 0.1971511 0.6123814 0.4990967Unskilled workers 0.0086862 0.45 13609 0.3595431 0.2094953 0.608437 1 0.4006761Agricultural workers 0.0138114 0.5993484 0.3452973 0.2153365 0.6443781 0.4770866

expenditure items within each of 12 groups of wage and salary earners.

As mentioned above, page 39, it was deemed necessary, before starting the maincomputation programme, to reject the households in which the expenditures on foodand dwelling were near zero. 29 households out of a total of 39 households rejected be-longed to the group of "lower employees and public servants", and of these 29 house-holds, 25 belonged to the above-mentioned household type of one person. The f-valuesin the table were calculated before the 39 households were rejected.

lyd. Calculation of Estimates of Parameters.

1. Four linear functions.

Maximum-likelihood estimates of the parameters of the four linear relations can nowbe calculated in a simple manner, assuming that the above-mentioned variance as-sumptions are valid. As regards the type of function log y = f (x) + e, in which V ylx

.

is assumed constant, maximum-likelihood estimates of the parameters can be ob-tained by means of a simple, unweighted least squares estimation14). For the othertype of function y = f (x) (1 + e) the maximum-likelihood estimate can be obtainedby means of an iterative calculation. As mentioned above page 41, it is a prerequisitefor obtaining the efficient estimate of the parameters that the observations (x, y) shouldbe weighted by the reciprocal value of the variance V yx .

14) Cf., e.g., A. Hald, (8) § 18,3, p. 528.

45

Footwear Washing &cleaning

Durablesexcl.

vehicles

Personalhygiene

Books,newspapers

etc.

Sports,holidays,hobbies

Transportincl, own car

Union fees,subscriptions

etc.

0.4112018 0.4458214 0.7144320 0.3626326 0.4883780 0.4814214 0.7731124 0.39640750.4372635 0.4506974 0.8814987 0.3819749 0.5080054 0.5579195 0.6946542 0.43510820.3462304 0.4120135 0.7127083 0.3624174 0.4516408 0.4951017 0.7926875 0.32059710.4440553 0.4946532 0.8143076 0.4150490 0.5411231 0.6219904 0.8325174 0.4192661

0.3602831 0.4769174 0.8262957 0.3476057 0.5117822 0.5397854 0.9513031 0.23637460.4233208 0.5308037 0.8837792 0.3897127 0.5188425 0.5658513 0.8252526 0.34844790.3699763 0.4177406 0.7241689 0.3894987 0.4751481 0.5430097 0.8953278 0.30668530.3959437 0.4723487 0.8089104 0.3510550 0.4778457 0.5694717 0.8622860 0.3109550

0.4252238 0.5500041 0.8916780 0.4078307 0.5945628 0.6114558 0.9503679 0.35851700.4358140 0.4724081 0.7867449 0.3860195 0.4983964 0.6142276 0.8669539 0.35109810.4227025 0.4618598 0.7643399 0.3981474 0.5260356 0.631437 0.8663685 0.29879650.4169857 0.4591317 0.7869866 0.3807334 0.5176211 0.6789559 0.7583866 0.4023374

46

Significant values are italized. Figures in brackets are degrees of freedom.

As it is now assumed that V ' x = u2.rj2 i2. [f (x)]2 this will mean that the

weight is2 2

or since u2 is constant, simply w - 2u [f (x)] [f (x)]

Now, however, this weight depends on the parameters which are to be estimated andtherefore it is necessary to proceed step by step by means of an iterative process. Theinitial values for the parameters a and are calculated by a simple, unweighted re-gression (which yields unbiased, but not efficient estimates) and on this basis the valuesof f (x) and thus of w for the first stage of the iteration are calculated; thereafter thesevalues of w are used at the next stage of the iteration, which consists in a weighted re-gression analysis15). This stage gives new estimates of a and , which are used to computenew values for f (x) and thus for w, which again are used in the next stage, etc. Theiteration process is carried on until the changes in the estimates a and b of a and be-come sufficiently small (in terms of Sa and sb), a situation which will often occur veryquickly since the calculated initial values are not very far off the mark.

2. The log-normal distribution function.

The computational procedure adopted in deriving the maximum-likelihood estimatesfor the three parameters a, and x of the function s = (a + log r) has been dealt

15) Cf. A. Hald, (8) § 18,6, P. 552.

Table IV,3. X2-test for constancy

Groups of wage and salary earners

Expenditure

Dwelling Fuel & light Food Tobacco Clothing

The capital:Higher salaried employees 5,5 (6) 2.4 (5) 3.7 (2) 20.2(10) 6.9 (6)Lower salaried employees 10.2 (7) 17.6(9) 7.6 (3) 74.3 (12) 14.8 (9)Skilled workers 6.2 (6) 9.4 (4) 3.4 (2) 11.5 (7) 9.1 (4)Unskilled workers 5.0 (6) 13.7 (5) 2.8 (2) 8.3 (8) 3.8 (7)

Provincial towns:Higher salaried employees 5.0 (4) 2.9 (4) 1.4 (2) 20.6(8) 7.2 (4)Lower salaried employees 13.9 (7) 16.2(8) 18.4 (3) 41.9(10) 3.9 (6)Skilled workers 2.8 (3) 0.4 (3) 1.5 (1) 8.3 (5) 4.1 (4)Unskilled workers 5.3 (4) 5.4 (5) 6.0 (2) 24.3(8) 4.3 (5)

Rural districts:Lower salaried employees 20.8(9) 28.3(7) 45.3(5) 30.7(10) 27.9 (7)

Skilled workers 10.9 (4) 3.4 (3) 2.2 (2) 13.3(6) 12.1 (5)

Unskilled workers 1.8 (6) 5.1 (5) 1.2 (2) 25.2(9) 4.4 (6)Agricultural workers 9.9 (6) 7.1(3) 6.2 (2) 13.6(6) 5.0 (5)

of coefficients of variation.

groups

1) Cf. J. Aitchison and J. A. C. Brown, (1) p. 82.

47

with by Aitchison and Brown16). Inserting for ii and r the observations y and x andassuming also in this case that V y[x = ci2 2 the function can be logarithmicallytransformed into the computionally more convenient form(IV, 10) log y = log + log 'P (a + log x) + g

Also here the calculations must be carried Out by means of iteration, but in this case, un-like that of functions (3) and (4), it is difficult to achieve convergence as it is not possible tofind good initial values for the three parameters in any simple way. Aitchison and Brown

suggest that one should guess at an initial value for ic, k0 say. By plotting against log x

on probability paper one should obtain a straight line ( (a + log x) representing thecumulated log-normal distribution). If the value for k0 has been fixed wrongly the curvedepicted will not be a straight line, and new k0 values should then be guessed untilaccording to a graphical inspection the curve in the diagram seems to be a straight line.Initial values of a and are then read from the diagram, after which the iteration canbe commenced. As the computations include estimation of parameters separately for12 social groups and 13 expenditure items, the work in connection with this graphic"targeting" would become of quite considerable dimensions; moreover, exampleswhich have been worked out seem to show that the shape of the curve on probabilitypaper was almost unaffected by large variations of ko The variances of these estimates

Footwear Washing& cleaning

Durablesexcl.

vehicles

Personalhygiene

Books,newspapers

etc.

Sports,holidays,hobbies

Transportincl, own car

Union fees,subscription

etc.

8.7(6) 23.6(7) 34.5(10) 8.7(6) 9.9(7) 10.9(7) 55.8(10) 14,4(6)18.1 (8) 15.2 (8) 100.0 (11) 14.2 (6) 23.0 (9) 44.9(11) 23.5(11) 20.9 (8)6.6 (5) 8.9 (4) 20.1(8) 8.4 (5) 2.3 (5) 12.5 (6) 7.8 (7) 0,7 (4)8.8 (5) 8.3 (6) 40.5(10) 4.2 (5) 8.7 (7) 11.3 (8) 40.7 (10) 15.7(5)

7.0 (5) 10.9 (6) 34.7(8) 5.8 (5) 4.0 (6) 16.5(6) 17.2(7) 5.2 (4)11.8(6) 7.1 (8) 52.7(11) 7.6(6) 10.5(7) 32.8(9) 53.7(11) 13.9(5)0.9 (3) 2.2 (4) 14.7 (5) 6.3 (3) 5.8 (5) 5.3 (5) 24.8 (5) 5.1(3)4.1 (4) 7.5 (5) 35.5(8) 5.0 (5) 12.4 (6) 24.7(6) 52.9(8) 5.7 (4)

16.0(6) 16.5(8) 57.5(11) 9.2 (6) 27.4(9) 37.5(9) 84.3(11) 7.9 (5)2.3 (4) 14.1 (5) 15.5(5) 4.2 (3) 5.9 (5) 30.3(6) 26.0 (5) 2.3 (3)6.1 (4) 1.9 (6) 44.0 (10) 3.8 (6) 12.5 (7) 48.2 (9) 59.3(10) 9.5 (3)4.1(4) 4.9 (5) 15.9(5) 4.7 (3) 4.5 (5) 21.9(6) 15.6(6) 3.6 (4)

48

of initial parameter values are thus very big and it was therefore deemed desirable towork out a method of estimation by which the initial values could be computed me-chanically.

As initial value for k war chosen ymax = the highest average value observed for y inthe groups of three households into which the basic material had been grouped.

Then initial values for a and were calculated by linear regression since (IV, 5)

implies that u = a + log x, where di (u) =

As a result of the method of calculation adopted the whole computation programmefor function (IV, 5) became "automatic". Naturally, there was no guarantee that theiteration would always converge; there was no possibility of ensuring in advance thatthe initial parameter estimates a0, b0 and k0 would fall within the region of conver-gence'7). It turned out in fact that in some cases (19 out of 156) the iteration processdiverged. As will be shown later in this chapter it also turned out that a fixed value ofthe parameter j3 had to be chosen to ensure workability of the estimation procedure.

All estimates of the parameters have been shown in appendix A; extracts of the resultsare shown and commented upon in chapter V.

IVe. Tests for Goodness of Fit.

Cf. J. Aitchison and J. A. C. Brown (1) p. 75.Cf. Durbin and Watson: (4).

1. The tests used.

One of the purposes of the present analysis was, as already mentioned, to find that oneof the chosen relations which according to the available observations would showforeach expenditure itemthe best goodness of fit. On the basis of such an investigationit would be possible to conclude that among the five given types of Engel curves, onetype gives the best fit if we consider the i'th expenditure item; in the case of the j'thexpenditure item it may be another function type which gives the best fit, and so on.To be able to draw such a conclusion one must test the goodness of fit of the five func-tions. These tests consist in various comparisons of the calculated function values, f (x),with the observed values of household expenditures y. The function which passes mostof these tests can then be said to give the best description of the relationship betweeny and i from the point of view of the available observations x and y.

In the present analysis the following tests have been used in evaluating the chosenfunctions:

Test for number of runs above and below the curve and test for length of run.Durbin and Watson's d-test'8).F-test for the ratio between the variance in the distribution of deviations from thecurve and the variance within groups,x2-test for normality of residuals.

1) Cf. A. Hald (8), p. 346 and Prais and Houthakker (10), p. 53.2O) Cf. A. Hald (8), table 13.5, p. 348.21) Durbin and Watson (4).

49

Moreover, the coefficient of correlation, R, between observed and calculated expendi-tures was computed to give a rough indication of closeness of fit; it should also be men-tioned that the estimate of the standard deviation, sj, of the parameter estimate b makesit possible by means of a t-test to test in a simple manner the hypothesis = o againstthe alternative hypothesis fi 0.

In the following a brief description of the various tests will be given.

2. Test for number of runs and for length of run and the d-test.

If a given function expresses the true relationship between r and i the observed devia-tions from this relationship shown by the observations x and y are of a purely randomnature. In that case the number of runs of residuals with the same sign, runs below andabove the curve, will follow a distribution 19) which is approximately normal whenboth the number of positive residuals, P, and that of negative residuals, Q, exceed 10,and in which mean value and standard deviation depend solely on the number of ob-servations. Given the number of observations, therefore, significance limits for runsabove and below the curve can be estimated. Similarly, limits of significance can bederived for the longest run 20). If the upper limit of significance is exceeded by the testfor the number of runs (which is analogous with the lower limit of significance beingexceeded by the test for the longest run), this means that the residuals change signs"too often"; this may be caused by a negative correlation between two successive ob-servations. Since such a hypothesis is not relevant for the present survey, moderatetransgressions of the upper limit of significance are not considered important. On theother hand, transgressions of the lower limit of significance (or for the second test, theoccurrence of too long a run) must be considered more important because this maymean that the calculated curve deviates systematically from the observations over greateror smaller parts of the range of variation.

These tests give the same result whether the residuals in question are large or small; thetests respond only to their signs. Durbin and Watson's21) d-test has been designed so as tocover both the sign and the size of the residuals. The test is based on the quantity d, which

is defined as d = (tk - tk4)2/E tk2, where tk = yk - f (xk). A high d-value means fre-

quent changes of signs, and the transgression of the upper limit of significance is thus evi-dence in favour of a hypothesis of negative correlation between successive observations, ahypothesis which, as already mentioned, is not considered relevant in this case. A lowd-value, on the other hand, indicates too few changes of signs, and the transgression ofthe lower limit of significance will therefore tend to substantiate that the model in ques-tion does not express the true functional relationship but deviates systematically from it.The limit of significance is given as a zone; d-values above and below that zone giveclear evidence, whereas d-values within the zone do not allow of any universal con-clusion.

50

The tests for runs and the d-test will, of course, point in the same direction since theyaim at the same alternative hypothesis.

3. The F-test.

The F-test which compares the variance in the distribution of the residuals, y - f (x),y being the average in the groups of three observations, with the average variancewithin the groups, is suitable as a test of different alternative hypotheses although in thiscase, too, only one limit of significance (the upper one) is relevant.

The test hypothesis is here again that the chosen model = f (r) expresses the truerelationship between y and x and that the residuals, y - f (x), are everywhere distributedwith mean value O and variance ci2 = V yx . If the test hypothesis is correct, theestimate, s22, of the variance of the residuals will have the same true value as the estimate,

of the variance within groups, and the ratio between the two variances F = will

follow a F-distribution.

y

X

Figure IV, 3a. Type of systematic deviations which will berevealed by the run tests, and may be not by other tests.

y

X

Figure IV, 3b. Type of systematic deviations which will berevealed by the F-test, but may be not by the run tests.

51

Significant F-values can now be cited in support of several alternative hypotheses.Firstly, the occurrence of the type of systematic deviations which is tested directlythrough run tests and the d-test will also manifest itself in significant F-values. However,the two types of tests do not measure the deviations from the hypothesis in the same way;confer the example below22). The run tests may reveal a systematic tendency in thedeviations of the observations from the curveeven if the individual deviations arevery small as illustrated in figure IV,3a, where the true "relationship" has been plottedtogether with the chosen function. It is not certain that the F-test will be able to revealsuch small systematic deviations. Conversely, it is possible that the run tests will notreveal the type of deviation between the observations and the chosen function whichhas been illustrated in figure IV,3b, whereas it is likely that this pattern of deviationswill lead to significant results of the F-test.

However, if it is assumed that this type of alternative hypothesis is not relevant, signif-icant F-values can, secondly, substantiate hypotheses which say something of thedistribution of the stochastic element, e. If the hypothesis of constant residual variancecannot be upheld then the ratio between s22, and s12 will not follow a F-distribution,and significant F-values can then be taken as an expression of the fact that the assumptionsof applying this test have not been fulfilled.

It is obvious that the F-test cannot be used without qualification in the case of thefunction types with untransformed expenditure, y, as dependent variable since it ap-peared from the analysis made that V yjx was not constant; in that case this variancewas expressed as a simple function of the level of expenditure namely V yx = a2

-' a2 [f (x)]2.In estimating the parameters this variance assumption was taken into account, and

it is therefore necessary to do the same thing here so that in the calculation of s22 andthe observations are weighted by their reciprocal variance, i.e., the same weight as

was used in the parameter estimation w 1 23).[f(x)]2

Thus significantly high F-values may be taken to indicate that the chosen varianceassumption V yjx = a2 n2 has not been correct, but even then, there remains thealternative hypothesis that the chosen function deviates systematically from the "true"one, the hypothesis shown in figure IV, 3b.

22) Cf. S. J. Prais and H. S. Houthakker (10) p. 52.52) As regards the actual calculation of and si2 it should be mentioned that whereas s can nat-

urally always be taken direct from the regression analysis, the matter is a little more difficult asregards the estimation of ai2. The estimate si2 cannot be calculated direct in those cases wherelog y is the dependent variable as a number of individual observations are zero. However, it holdsgood, with good approximation, that sj = 5i2(log ) M2. c2 where c2 is the average square ofcoefficient of variation in the distribution of y-values and M = 0.4343. The approximation issatisfactory if < 0.3, which is not always the case in our material.

In cases where the untransformed y-values appear as dependent variable it would be possibleto form estimates of the inner variance si2 direct on the basis of individual observations. But thisdoes not become necessary since we have an estimate of the coefficient of variation which is alsoan estimate of ai2 cp. (IV, 6).

52

4. The 2 -test.

To test the hypothesis that the residuals are normally distributed the f-test has beenused. The observed deviations are grouped, and by comparing this grouped distributionof deviations with a normal distribution with the same mean value (0) and variance isit possible to calculate a x2-quantity24). Also in this case it will, of course, be necessaryto insert the variance assumption used in the parameter estimation. Significantly highf-values support the alternative hypothesis that the deviations of the observations fromthe function are not normally distributed. Assuming that the tests referred to above havegiven insignificant results, it has, prior to the f-test, been possible to substantiate the fol-lowing hypothesis: The relationship suggested by the observations does not deviate system-atically from the proposed functional relationship (test for number of runs and size oflongest run, d-test and F-test), and the deviations of the observations from this functionhave everywhere a variance which is of the same magnitude as the variance withingroups (F-test). A significant f-value will then indicate that the residuals are not nor-mally distributed.

This alternative hypothesis can be further specified in the present case. For if it istrueas the test calculations seem to showthat the residuals are normally distributedin the case of the functions where the dependent variable is a logarithmic transformationof the expenditure, then the deviations from the function in the cases where the depen-dent variable is the untransformed value of expenditure will be log-normally distributed.And this is true even if one uses the weighted calculation method in the parameter estim-ation. Use of weights in the calculation influences the magnitude of the residuals, butnot the form of their distribution. One can thereforewith good support in the otherresults of the test calculationsadvance the assertion that significant f-values supportthe alternative hypothesis that the residuals are log-normally distributed25).

A comparison of the calculated values of the coefficient of correlation for the differentfunctions gives an impression of which of the five functions has the closest fit for eachexpenditure item in each group of wage and salary earners. However, such a directcomparison will only be possible if the residuals are at the same level, i.e., if the varianceassumption used in the parameter estimation, V yx = a2 [f (x)]2, is also used here asa basis for assigning weights. In the following chapter the results of this calculation aswell as of all the test calculations referred to above will be shown.

IV. f. Planning the Computation Programme and CarryingOut The Computations.

After specifying the five functional relationships and the procedure to be used in theirestimation the practical part of the analysis can be started. This part comprises work-ing out a detailed computation programme and the corresponding code for feeding it

Chapter V, p. 60.Chapter V, p. 64.

5. The coefficient of correlation.

i. e. DAnsk Sekvens Kalkulator.Cf. J. Aitchison and J. A. C. Brown (1) p. 82, and above p. 47.

53

into DASK26) and (at length) computation and printing of parameter estimates and theirstandard errors and of test resultsaccording to the tests referred to above.

It will be understood from what has been mentioned above that the computationprogramme is rather comprehensive. For each of the twelve groups of wage and salaryearners into which the basic material has been divided, parameter estimates and theirstandard errors are to be calculated separately for five Engel functions for a total of 13expenditure items. This gives a total of 12 x 5 x 13 = 780 sets of calculations; thesame number of tests have to be made. Coding the programme, therefore, was boundto require a great amount of work, and the fact that part of the computation programme(especially the part concerning the log-normal distribution and the regression analysiswith the above-mentioned special variance assumptions and some of the test calculations)had not previously been performed on the DASK rendered the coding even more difficult.

In planning the computation programme it was, of course, necessary to consider howthe individual operations were to be performed on the DASK, so that the computationprogramme could as far as possible be adapted to the capacity of that electronic com-puter. During the greater part of the analysis a very useful contact was established withthe division concerned at the Danish Institute of Computing Machinery.

In the present investigation it had been attempted to guard against "unforeseen"difficulties by working out in advance, at the consumer survey section of The StatisticalDepartement, examples of all the computing operations which were to be performedaccording to the computation programme.

In the course of this preliminary work several sources of error were traced, and severalcorrections had to be made in the computation programme and in the input tapebutnevertheless the performance of the computation programme on the DASK presentedseveral unpleasant surprises, two of which deserve to be mentioned because they seemto be of a certain theoretical interest.

The code for the computation programme was made so flexible that the compu-tation process could be stopped at all "vital" points and any necessary corrections bemade without re-running the whole programme. As a control measure which could beexpected to be very effective an inspection was introduced after all computations hadbeen run for the first of the twelve groups of wage and salary earners. Hereby it wasexpected that the weak spots which might have excaped the attention during the cal-culations of the examples mentioned above would be revealed.

This inspection of the results of the computations for the first group of wage and salaryearners showed that as regards function type (IV, 5) log ij = log x + log '2 (a + j log y)the computation programme did not work in five cases out of the thirteen expenditureitems comprised by the programme because the iterative process for the calculation of est-imates of x, a and did not converge27). This made essential changes in the programmenecessary, see below. For the other four functions the programme performance seemedfully satisfactory. Thanks to the flexibility of the code worked out for the computationprogramme the computations could be continued for these four functions, where the

54

results for the first group of wage and salary earners had been found satisfactory, whilethe corrections to the programme for the fifth function type were considered. But in thecourse of the continued run of the accepted programme for the four linear functionsit proved impossible in several cases as regards, functions (IV, 3) and (IV, 4), to obtainconvergence in the estimation of the parameters.

The reason for the lack of convergence in the case of function (IV, 5), must be foundin the fact that the variance V y Ix F in the distribution of the observed expenditurevalues is so big that the functional relationship between x and y can be satisfactorilydescribed by means of two parameters. Or in other words: With the given variancein the distribution of the observations one of the three parameters can be selectedarbitrarily within a wide interval, and the other two parameters can be estimatedconditioned by this arbitrarily selected parameter value without this causing any appre-ciable rise in the unexplained variance of y. The iterative procedure for determiningthe three parameter estimates becomes highly unstable, and in several cases convergencewill consequently not be obtained. And equally regrettable: in the cases where the itera-tion process did converge the standard deviations of the parameter estimates were sogreat that the estimates had to be considered almost useless, cf. table IV,4, whichshows the result of the computations for eight expenditure items in the group of higherpublic servants and salaried employees in the capital.

The table emphatically demonstrates that the results obtained are not very useful;the standard deviation of the estimates, apart from the food item, are of the same orderas the estimates themselves.

The "solution" chosen to this problem consisted in arbitrarily fixing in advance aconvenient value of the parameter , namely unity; it should be added that unity doesnot significantly deviate from any of the calculated j3-values in the three-parametercalculation, cf. table IV,4. This solution was also chosen by Aitchison and Brownin their analysis of British household budgets28). However, it does not appear from theirreport whether they had previously experienced just as disappointing results in thethree parameter estimation as was the case in the present analysis.

The reason for the cases of lacking convergence which occurred in the iterative cal-culation of parameter estimates in functions

(IV,3) = a--(logvlogv)and

(IV, 4)

was to be found in an unfortunate property of the estimation procedure adopted. Asmentioned above, p. 44, it was found that it applied to the functions in which the un-transformed value y was the dependent variable that V yx

F= a2 2 for which reason

it was decided to carry through the regression analysis with2

as weights; the test[f(x)]

calculations were carried through in accordance with the same principle in ordei to

) Cf. J. Aitchison and J. A. C. Brown (1) p. 130.

Table IV,4. Parameter estimates in the three-parameter case of the function

55

obtain everywhere as efficient estimators as practically possible. If, however f (x) assumesvery low values, in the extreme case zero, the weight factor for the value or values inquestion will completely dominate the calculation, and the iteration process, which wasmentioned on p. 46, will become quite unstable. If the initial parameter estimates orone of the subsequent estimates in the iteration process results in such low values off (x), there will be a risk that the iteration will stop. The reason why the group of wageand salary earners which was computed first, viz, higher public servants and salariedemployees, passed through without any stop as the only one of all the groups, is thatthe x-values in this group are rather large so that no f (x) values came close to zero.

The correction which was made in the computation programme to enable calculationsof parameter estimates to be made in the 41 cases (out of a total of 143) in the other 11groups of wage and salary earners in which the iteration process did not converge con-sisted in rejecting the special variance assumption and estimating on the assumptionof constant variance, i.e. applying the estimation procedure from the function types inwhich the logarithmically transformed expenditure, log y, is the dependent variable.The parameter values and test values thus estimated are not efficient, but this estimationprocedure were preferred rather than rejecting observations or omitting estimation29).

29) As mentioned p. 40, Prais and Houthakker, in their analysis, have everywhere used this estimationprocedure.

log = log x + log [ (a + log r)]. Higher public servants and salaried employeesin the Capital.

Expenditure item k 'k a 'a b Sb

Dwelling convergence not obtainedFuel and light 7600 63000 4.796 1.1 0.793 1.4Food 2900 950 6.192 2.0 1.62 0.64Tobacco 6700 820 7.517 5.3 1.87 1.7Clothing 3200 5600 6.431 2.6 1.44 1.0Foot wear 230 220 5.084 4.8 1.34 1.6Washing and cleaning 380 320 7.414 3.3 1.84 1.1Durables convergence not obtainedPersonal hygiene 1500 4800 5.460 2.2 1.10 1.1

Books, newspapers etc. convergence not obtainedSports, holidays, hobbies 13000 30000 7.834 2.3 1.63 0.93Transport (incl, own car) convergence not obtainedUnion fees, subscriptions etc. 660 2100 4.861 3.0 1.03 1.4

Chapter V.

MAIN RESULTS.

Va. Introductory remarks.

In this chapter some of the main results of the Engel curve analysis will be discussed.In the next chapter some results of certain further calculations will be put forward,but as already mentioned the main object of this inquiry has been the Engel curveanalysis.

In the appendix is shown, separately for each of the twelve groups of wage and salaryearners, parameter estimates and test results for five Engel functions (1,1) to (1,5) of thethirteen expenditure items.

However, it may perhaps be natural to try to boil down the somewhat overwhelmingabundance of figures in these tables.1) How can the result of the analysis be summed up?

In doing so the point of departure will be taken in the description of the object ofthe analysis which was given in chapter IV, p. 31, where the object was characterizedby the following two steps:

I) calculation of parameter estimates in the selected models and2) testing these models by various tests for goodness of fit.It will first be examined whether the tests for goodness of fit point in the same direction

or, in other words, whether it is possible to classify the selected Engel functions on thisbasis. Then follows an interpretation of the calculated parameter values.

The following tests were used (cf. p. 48 above):The coefficient of correlation between calculated and observed values (no proper

test, but the size of the coefficient of correlation can be taken as a measure of the"closeness of fit" of the relations proposed.2)

Test for number of runs.Test for longest run (where a run is defined as a series of positive or negative de-

viations between calculated and observed values).The d-test, in which both the signs and the numerical values of the deviations enter.The F-test, the ratio of the variance within groups and the variance of the residuals.The x2-test for the normality of the distribution of the residuals.

The proper tests fall into three categories. 10. The x2-test is concerned with the formof the distribution of the deviations between the calculated and the observed values,the test hypothesis being that these deviations are normally distributed. 2°. The variance-

More specifically 5 (models) x 12 (groups of wage and salary earners) X 13 (expenditure items) =780 less 19 cases in which the estimation procedure failed (model (1,5)) = 761 sets of parameterestimates and corresponding test results.Cf. Prais & Houthakker, (10), p. 95.

N

V (logy1 logy)2. (logY1 - logY)2i I

N(y - ) (Y - Y) wj

(V,2) R2IN N(yj - w1 (Y1 - Y)2w1

where N is the number of groups of three observations in the given social group,Y1= f(xj) for the models with untransformed dependent variable and log Yj = f(xj)

for the models with logarithmically transformed dependent variable; wj= [f( )]2

Formula (V, 1) was used in the case of the Engel functions in which log y is the de-pendent variable and formula (V,2) in the case of the other functions, in which theuntransformed y values appear as dependent variables.3)

57

ratio test, the F-test, compares the deviations of the observed group averages from thecalculated function values with the variation within the groups. The test hypothesisis that the corresponding variances are equal, and a significant F-value is interpretedas an indication that the calculated Engel function deviates systematically from the"true" Engel curve. 3°. The tests for number of runs and length of runs and the d-test,measure systematic tendencies in the signs of the deviations. The test hypothesis statesthat there is no such systematic tendency, and that the signs of successive deviationschange at random. In the following the result of these tests will be examined separatelyfor each test.

However, it may be disclosed already here that the double-logarithmic functionclearly stands out as the one of the five selected functions which gives the best fit foralmost all expenditure items. A considerable part of the following interpretation of theresults will therefore be based on the estimated parameters of this function.

Vb. Examination of test results.

1. Coefficient of correlation, R, between calculated and observed values.

Table V, i shows the calculated coefficients of correlation between the observed and cal-culated expenditures.

The R-values have been calculated by one of the following formulae:N

(log y - log y) (log Y1 - log Y)(V,l) R1=1

8) For estimation purposes the function t = xD (c + 3 log y) was logarithmically transformed intothe form log i log x + log I (a + 3 log y) and all test calculations, accordingly, were basedon the deviations of logarithmically transformed expenditures. The R-values in table V, 1, howeverare based on formula (V,2).

58

Table V,l. Correlation coefficients (R X 100)

1 2

Groups of wage and salary earners*)

3 4 5 6 7 8 9 10 il 12

(l)logy=a+b(logxlogx)Dwelling 79 81 79 80 83 80 81 68 77 81 80 70

Fuel & light 76 49 57 34 69 51 36 60 57 45 76 78

Food 82 84 94 91 89 88 86 89 85 92 94 88Tobacco 53 51 72 72 57 60 73 80 65 70 82 71

Clothing 80 84 84 80 84 86 83 76 86 82 84 83Footwear 54 70 70 69 56 69 79 67 81 69 64 66

Washing & cleaning 76 77 72 75 75 78 75 71 77 78 82 67

Durables excl, vehicles 36 46 67 56 54 55 64 72 58 52 52 54Personal hygiene 82 84 77 78 83 80 85 88 86 76 87 81

Books, newspapers etc. 81 74 76 75 76 81 76 84 70 74 80 75

Sports, holidays, hobbies 90 84 91 84 90 89 91 85 85 83 86 77Transport incl, own car 41 56 72 70 62 64 66 70 65 73 76 51

Union fees, subscriptions etc 69 66 90 85 85 81 88 90 81 88 93 80

(2)logy=a+b (!_)

Dwelling 75 78 76 78 82 79 80 64 74 82 77 65

Fuel & light 74 46 62 41 68 49 45 54 61 57 79 74

Food 83 84 92 87 87 86 84 86 83 88 91 89Tobacco 53 54 69 66 56 59 71 78 68 67 78 67

Clothing 79 82 86 78 82 85 83 76 81 81 79 83

Footwear 54 67 72 68 55 68 79 69 77 68 61 66Washing & cleaning 77 77 66 67 69 75 71 68 69 74 77 54Durables excl, vehicles 40 47 65 58 51 56 65 73 60 52 57 58

Personal hygiene 80 87 78 80 01 80 86 86 82 73 83 80

Books, newspapers etc 76 74 73 71 73 77 76 83 68 73 79 80Sports, holidays, hobbies 89 82 88 83 88 85 92 83 81 77 83 70

Transport incl, own car 46 56 69 67 59 64 64 63 60 71 67 41

Union fees, subscriptions etc 68 68 87 85 84 79 87 83 77 82 90 79

(3) y = a + b (log x - i- ;)

Dwelling 73 76 79 79 87 84 79 63 74 81 77 63

Fuel & light 77 57 66 57 74 62 56 52 77 63 82 72

Food 84 86 93 90 88 89 85 88 86 90 93 90

Tobacco 62 67 72 73 69 70 73 84 81 67 68 71

Clothing 79 84 87 85 85 84 82 87 83 83 79 80

Footwear 58 69 70 74 58 72 80 72 79 70 68 67

Washing & cleaning 77 76 70 74 74 82 73 73 72 73 78 53

Durables excl, vehicles 46 58 63 72 51 47 74 82 76 21 66 66

Personal hygiene 79 84 78 86 84 76 86 90 84 78 86 82

Books, newspapers etc. 66 75 73 78 80 42 75 91 68 69 82 90Sports, holidays, hobbies 85 45 84 85 75 74 91 78 77 41 81 71

Transport incl, own car 46 47 71 70 65 64 62 55 45 73 16 37

Union fees, subscriptions etc. 67 72 88 88 86 81 89 82 79 82 90 83

between observed and calculated expenditures.

59

Groups of wage and salary earners*)

1 2 3 4 5 6 7 8 9 10 11 12

Il l\-Dwelling 70 67 76 78 85 81 73 50 66 79 67 64Fuel & light 73 53 69 53 69 56 60 42 73 66 80 71Food 81 83 89 84 79 83 80 79 79 84 87 87Tobacco 62 72 67 60 63 67 69 78 74 64 73 66Clothing 74 68 84 83 80 80 78 86 76 77 69 72Footwear 57 66 73 75 60 72 78 79 71 71 66 64Washing & cleaning 75 72 65 67 70 76 69 71 66 65 73 46Durables excl, vehicles 47 62 60 71 48 52 73 77 76 42 68 66Personal hygiene 75 78 79 88 79 73 85 84 79 73 80 76Books, newspapers etc 56 70 68 70 76 41 73 85 67 66 75 88Sports, holidays, hobbies 79 69 77 81 69 65 84 71 74 50 52 64Transport incl, own car 54 42 67 66 60 58 24 52 46 65 45 29Union fees, subscriptions etc 63 67 81 86 83 74 84 78 86 72 88 78

(5) log y log k + log D (a + log x)

Dwelling 72 78 78 79 83 81 80 65 76 27 75 59Fuel & light 76 56 64 55 73 62 53 53 74 59 80 71Food 84 85 94 - 88 89 85 89 86 - 94 90Tobacco 62 65 72 - 68 68 70 79 76 66 47 70Clothing 79 - 84 82 - 84 82 76 61 - 82 82Footwear 58 69 69 73 58 71 79 68 81 68 68 67Washing & cleaning 77 76 71 76 72 79 71 69 73 76 79 52Durables excl, vehicles 43 50 66 68 48 46 - 74 58 59 59 60Personal hygiene 79 83 76 81 82 76 84 88 86 78 86 83Books, newspapers etc. 70 75 73 79 72 79 74 - 71 21 82 84Sports, holidays, hobbies 64 55 67 - - 65 89 - - 56 58 -Transport incl, own car 34 43 42 66 55 - - - 30 - - 34Union fees, subscriptions etc 67 72 49 61 84 82 89 62 80 86 - 82

(4) y = a + b

*) 1, Higher public servants and salaried employees. The capital. 2. Lower public servants and salaried employees.The capital. 3. Skilled workers. The capital. 4. Unskilled workers, The capital. 5. Higher public servants andsalaried employees. Provincial towns. 6. Lower public servants and salaried employees. Provincial towns. 7. Skilledworkers. Provincial towns. 8. Unskilled workers. Provincial towns. 9. Lower public servants and salaried employees.Rural districts. 10. Skilled workers. Rural districts. 11. Unskilled workers. Rural districts. 12. Agriculturalworkers. Rural districts.

60

When considering the R-values given in table V, i it must be borne in mind that theR-values from the two formulae cannot be compared directly. Whether one of the for-mulae generally leads to systematically higher or lower R-values than the other one is,however, very difficult to determine. According to an unpublished paper by Theil,which is referred to in Prais and Houthakker4), the logarithmic form seems to resultin higher R-values than does the use of untransformed y observations, but Thejl'scalculations do not aim at weighted R-values calculated by means of formula (V,2),and his conclusions do not, therefore, apply to our case. Since the "transformation"effect seems to be moderate even when unweighted calculations are used, any systematicdifferences in the R-values caused by this transformation effect will be disregarded inthe following.

Going through the table item by item it will be found that none of the five Engelfunctions stands out as the best in all cases, but on the other hand it is noteworthy thatthe double-logarithmic function more frequently than any of the other functions hasthe highest R-value. Counting for each of the 13 categories of expenditures the numberof social groups (Out of a total of 12) in which this function has the highest R-value,it ranks first in 6 categories and tied first in another 3.

It must be emphasized, however, that, considered in isolation, the R-values shownare not a suitable criterion for deciding which of the five functions offers the best de-scription of the observations.

For one thing, the above-mentioned reservations regarding comparisons between theR-values of the different types of functions must be taken into account, and for anotherit must again be emphasized that the R-test is no proper test, since it is not possible toset up test hypotheses as regards closeness of fit, which may be accepted or rejected at agiven significance level for R.

2. The 2 -test.

By grouping the differences, t, between the observed and calculated expenditures into kgroups of size s, where st2 is the variance in the distribution of these differences orresiduals, the following grouped distribution will appear:

4) Cf. J. Prais and H. S. Houthakker (10), p. 96.

Interval Number of residuals

t < - 2 St n'- 2 St < t < - lIst n2

t>2st

61

This distribution is then compared with a similarly grouped, normal distribution withthe same mean and variance and with the same number of elements as the empirical

distribution, L = n3.

For each group in the distribution of the residuals is calculated the difference(nj - L 1) between number of elements in the empirical and in the theoretical, normaldistribution, e3 denoting the expected frequency in the j'the group. The quantity

(n - Lej)2/Lej

will then be approximately 2-distributed with k-m-1 degrees of freedom if the testhypotesis concerning the normality of the residuals is correct; m is the number of para-meters in the given Engel function, and k is the number of groups after it has been ensured,through a suitable merging of too small groups, that everywhere

L ej> 5

The 2-values have been shown in table V,2.It holds good for all expenditure items, with the exception of the expenditure on tobacco,

that the functions in which log y is the dependent variable, show the fewest significantx2-values.

In the case of several expenditure items there is only one social group out of twelvewhich has significant 2-values for these two functions.

It thus seems evident that this test points to one of the logarithmically transformedEngel functions as being the bestwhich of them must be left open until further evidencecan be put forward as they do almost equally well.

In the case of the types of function in which the untransformed value, y, appears asthe dependent variable it does not seem possible, however, to accept the hypothesisthat t is normally distributed. It has been taken into account, in connection with theparameter estimation and subsequent tests for these two functions, that the varianceof y (for given x) increases with the value of y. According to the investigations madethis relationship could be described with good approximation by the formula

V y x = 2 [f(x)]2 s2 [f(x)]2

wherefore the tests have been based on the quantity:

f(x) - yf(x)

Hereby was obtained that the variance in the distribution of t could be consideredconstant and independent of x. On the other hand, as the x2-tests show, the distributionsobtained are evidently not normal.

It will be found, however, that the results achieved correspond closely to what wasto be expected from the discussion on this subject in chapter IV, p. 52. It was concluded

62

Table V,2.

1 2 3 4

Groups of wage and salary earners*)5 6 7 8 9 10 11 12

(l)logy=a+b(logx-logx)Dwelling 9.9 5.2 4.5 4.1 5.4 9.4 7.0 4.6 5.0 0.6 3.8 6.6Fuel & light 6.3 25.5 10.8 8.0 3.8 15.0 6.5 3.1 23.4 2.3 17.7 8.6Food 8.9 10.2 1.4 8.5 7.1 5.8 3.9 2.8 2.8 9.4 9.0 5.8Tobacco 21.5 30.4 6.1 14.6 9.0 32.0 5.8 4.9 11.1 5.9 11.7 1.6Clothing 8.4 8.5 8.7 13.3 6.4 6.1 2.1 6.3 5.5 3.0 3.6 3.2Footwear 10.0 9.4 9.8 7.3 5.0 10.9 2.8 1.6 6.7 8.6 8.9 2.4Washing & cleaning 12.4 13.8 5.0 9.0 3.8 5.6 0.7 7.8 3.7 1.1 9.0 8.8Durables excl, vehicles 3.9 8.2 12.8 9.7 14.6 12.6 2.2 3.2 6.8 3.7 7.3 1.3Personal hygiene 5.1 9.1 6.6 5.0 6.8 7.0 3.7 7.0 14.9 11.5 3.8 5.6Books, newspapers etc 10.6 9.9 0.9 3.4 7.4 12.6 4.3 2.8 8.3 1.5 5.2 6.1Sports, holidays, hobbies. . 6.0 3.9 4.5 3.3 10.2 8.5 5.0 14.0 4.6 7.1 5.2 1.8Transport incl, own car. . 25.7 40.3 3.8 5.1 6.7 7.1 0.5 4.6 5.7 4.9 4.0 9.7Union fees, subscriptions etc. 3.3 10.2 5.0 8.0 4.1 6.4 13.0 4.9 8.1 6.6 10.6 3.595% significance level 14.1 16.9 11.1 14.1 11.1 14.1 7.8 11.1 14.1 7.8 14.1 7.8Degrees of freedom 7 9 5 7 5 7 3 5 7 3 7 3

(I i(2) log y = a + b x

Dwelling 5.9 4.8 3.3 9.9 4.6 3.9 5.3 3.2 7.7 4.1 7.4 2.6Fuel & light 7.4 16.7 4.5 13.7 6.0 9.6 9.6 3.2 20.7 2.9 8.2 4.7Food 8.8 10.9 0.9 7.3 14.0 4.3 0.9 1.8 2.4 2.2 2.9 1.8Tobacco 12.9 38.8 1.1 14.6 10.6 18.9 6.4 9.7 13.4 3.7 10.6 0.6Clothing 9.2 3.2 6.6 19.7 5.4 7.7 3.2 9.4 4.5 6.2 1.5 6.3Footwear 6.5 5.8 3.6 7.0 3.0 15.5 5.4 1.9 9.0 3.7 11.7 2.9Washing & cleaning 8.6 7.8 5.0 9.8 13.5 5.8 1.2 1.4 3.5 3.2 10.5 1.7Durables excl, vehicles 11.5 9.1 5.5 15.0 10.6 17.6 0.1 3.0 2.9 2.6 7.8 1.1Personal hygiene 6.6 7.0 8.4 8.8 4.5 10.4 2.7 2.9 6.2 10.6 14.1 5.3Books, newspapers etc 11.6 6.7 3.0 9.0 6.7 15.2 4.1 3.3 8.1 1.3 8.8 3.3Sports, holidays, hobbies. . . 4.6 5.0 4.0 2.2 3.0 2.5 0.4 4.2 6.7 2.3 4.9 6.6Transport incl, own car 16.9 34.0 5.1 7.6 11.8 8.7 1.2 3.4 5.5 2.0 4.5 8.1Union fees, subscriptions etc. 6.9 8.2 4.2 10.2 11.9 7.0 7.8 3.1 10.8 2.2 9.4 3.495% significance level 14.1 16.9 11.1 14.1 11.1 14.1 7.8 11.1 14.1 7.8 14.1 7.8Degrees of freedom 7 9 5 7 5 7 3 5 7 3 7 3

(3) y = a + b (log x - log x)Dwelling 17.2 18.9 2.7 12.9 12.6 13.5 2.5 12.9 9.5 5.4 11.3 4.9Fuel & light 14.8 22.0 8.0 8.8 3.6 3.8 6.9 12.0 8.1 3.3 17.7 20.2Food 13.1 10.7 7.1 3.5 12.5 4.0 1.4 1.5 8.9 3.4 5.9 0.8Tobacco 3.6 11.6 6.9 10.7 6.5 6.0 6.9 3.8 12.5 5.4 - 3.2Clothing 4.3 13.2 2.4 9.0 7.1 11.9 2.6 9.0 19.4 0.5 15.2 2.1Footwear 8.6 13.8 18.9 2.2 3.1 21.5 2.7 4.7 13.9 4.0 5.1 8.5Washing & cleaning 10.4 25.1 13.5 14.7 8.0 14.3 1.3 11.1 14.0 7.0 31.4 3.2Durables excl. vehicles 39.6 60.0 3.9 5.4 31.9 91.1 9.3 10.4 38.8 - 26.2 2.7Personal hygiene 17.7 36.9 1.8 13.9 13.8 36.9 4.8 7.9 18.6 4.5 27.4 3.6

Z2-test.

63

*) 1. Higher public servants and salaried employees. The Capital. 2. Lower public servants and salaried employees.The Capital. 3. Skilled workers. The Capital. 4. Unskilled workers. The Capital. 5. Higher public servants andsalaried employees. Provincial towns. 6. Lower public servants and salaried employees. Provincial towns. 7. Skilledworkers. Provincial towns. 8. Unskilled workers. Provincial towns. 9. Lower public servants and salaried employees.Rural districts. 10. Skilled workers. Rural districts. 11. Unskilled workers. Rural districts. 12. Agriculturalworkers. Rural districts.

1 2 3

Groups of wage and salary earners*)4 5 6 7 8 9 10 11 12

Books, newspapers etc 50.5 19.7 8.1 18.2 16.8 - 11.0 3.9 43.2 4.8 13.9 4.0Sports, holidays, hobbies. . . 8.6 - 14.2 9.7 - - 2.8 - - - 35.7 29.5Transport incl. own car 80.6 229.8 44.3 60.7 64.5 100.5 7.8 52.5 - 4.1 - 34.4Union fees, subscriptions etc. 8.6 14.2 2.7 11.7 8.3 8.5 5.4 1.8 11.5 1.5 12.7 3.295% significance level 14.1 16.9 11.1 14.1 11.1 14.1 7.8 11.1 14.1 7.8 14.1 7.8

Degrees of freedom 7 9 5 7 5 7 3 5 7 3 7 3

(4) y = a + b (!_ I)

Dwelling 27.8 33.4 6.8 - 4.8 14.7 8.2 - 14.1 8.4 - 3.3Fuel & light 25.0 34.8 9.6 10.9 1.2 12.4 10.3 11.8 14.7 3.3 20.7 3.2Food 8.0 10.1 1.7 9.7 33.4 9.0 3.1 4.6 14.8 5.9 33.7 8.3Tobacco 15.0 - 12.8 - 11.9 12.2 5.5 18.3 - 3.9 - 2.9Clothing 7.6 - 1.0 9.4 9.9 19.5 1.1 8.0 47.1 8.2 - -Footwear 5.8 14.9 13.3 8.3 10.7 27.3 1.3 5.8 22.1 3.8 3.6 7.0Washing & cleaning 7.6 29.6 16.9 16.7 17.9 37.7 5.5 31.9 36.8 19.3 42.1 10.0

Durables excl, vehicles 42.1 - - 9.8 32.7 98.4 11.8 19.6 47.0 52.0 - 5.1

Personal hygiene 35.7 54.5 6.9 8.6 12.4 42.7 8.1 12.2 - 6.4 30.6 9.3Books, newspapers etc 68.4 - 13.7 32.6 25.1 - 9.2 6.6 - 10.0 28.1 2.4Sports, holidays, hobbies. . . 35.1 - 20.7 16.8 - - 13.6 25.2 - - - 45.4Transport incl, own car. 86.4 - 56.7 77.4 47.2 - - 88.7 - 5.5 - 44.3Union fees, subscriptions etc. 23.1 31.5 3.3 - 23.4 18.4 10.3 - -. 11.0 - 13.195% significance level 14.1 16.9 11.1 14.1 11.1 14.1 7.8 11.1 14.1 7.8 14.1 7.8

Degrees of freedom 7 9 5 7 5 7 3 5 7 3 7 3

(5) log y = log k + log C1 (a + log x)

Dwelling 6.0 6.6 7.5 7.9 2.5 6.5 3.5 4.5 2.8 25.2 4.2 2.9Fuel & light 6.3 20.8 8.4 8.8 4.1 15.0 2.3 4.1 20.9 1.4 14.5 10.6Food 13.7 9.4 2.4 - 9.0 10.7 0.6 4.8 8.9 - 14.8 3.6Tobacco 19.0 32.7 4.9 - 8.8 25.5 5.6 8.5 11.0 2.8 14.0 0.1

Clothing 12.3 - 4.4 12.5 8.4 1.5 5.2 34.5 - 4.2 3.1

Footwear 13.2 10.1 9.4 9.1 2.0 14.2 5.5 1.8 4.9 4.2 9.1 4.0Washing & cleaning 9.1 12.4 6.1 11.6 5.8 9.6 0.5 7.4 8.8 2.5 8.1 4.9Durables excl, vehicles 5.9 8.9 4.8 10.4 13.0 13.5 - 2.2 7.9 3.1 7.9 1.9

Personal hygiene 8.7 8.2 7.2 2.7 8.4 7.1 3.4 1.9 13.8 5.9 5.0 13.7

Books, newspapers etc 9.4 8.4 2.3 3.6 9.8 11.8 7.8 - 8.9 23.2 11.2 6.6Sports, holidays, hobbies. . . 19.6 16.4 6.3 - - 8.0 3.7 - - 8.8 11.3

Transport incl, own car 29.3 39.3 3.6 6.0 8.8 - - - 3.7 - - 10.3Union fees, subscriptions etc. 4.4 12.0 29.4 23.4 2.1 10.5 7.9 34.5 11.2 2.3 - 3.095% significance level 14.1 16.9 11.1 14.1 11.1 14.1 7.8 11.1 14.1 7.8 14.1 7.8Degrees of freedom 7 9 5 7 5 7 3 5 7 3 7 3

64

there that the logarithmically transformed dependent variables, log y, were normallydistributed, and the untransformed values, y, consequently followed the log-normaldistribution; a graphic presentation of these two cases has been given in fig. V,la andV, lb, where the "true" Engel curves have also been drawn.

Expenditure on a given Item

y

log x

Expenditure on a given item

log y

log x

Fig. V, lb

log x

The distributions of residuals from the Engel curve for a given value of x can be illu-strated as shown in fig. V,2a and V,2b.

The distribution of residuals according to fig. V, lb is normal and independent of thechosen value of x. The distribution according to V, I a is log-normal and the variancein the distribution, V y x , increases proportionally to [f(x)]2. The correction madein the tests consisted, as mentioned, in a division of all {f(x) _y] by f(x). Thereby itis achieved that the variance in the corrected distribution of residuals becomes inde-pendent of x, but the distribution form of the deviations is not changed.

We are thus in the situation of having set up an explicit alternative hypothesis whichmay be assumed to hold good for the functions for which the test hypothesis was rejected.

3. The F-test.

The variance-ratio test has been calculated as the ratio of the variance of the deviationsof the group averages from the calculated values to the variance within groups,

F 2' say.

The number of degrees of freedom for the two independent variance estimates areL

L-2 for s22, where Lis the number of group averages, and E (nj - I) = 2 L for s12,J 1

yFig. V, 2a Frequency of expenditure for given income

og yFig. V, 2b Frequency of log expenditure for given income

y

log y

65

66

Table V,3. F-test for linearity (ratio between

1 2 3

Groups of wage and salary earners*)4 5 6 7 8 9 10 11 12

(1) log y = a + b (log x - log x)

Dwelling 1.45 1.24 1.05 1.14 0.73 1.09 0.83 1.38 0.92 0.95 1.31 1.31

Fuel & light 1.30 1.31 0.78 1.92 0.96 1.00 1.21 1.12 1.25 1.30 1.04 1.36

Food 1.59 1.19 0.67 1.02 0.78 1.20 1.67 0.83 1.12 1.17 0.85 1.21

Tobacco 1.57 1.43 1.27 1.32 1.45 1.56 1.07 0.86 1.60 0.81 1.22 1.56

Clothing 1.20 1.14 1.23 1.06 1.07 1.10 1.25 1.84 0.95 0.99 1.47 1.54

Footwear 1.19 0.99 0.96 0.92 1.34 1.10 0.79 1.46 0.79 0.89 1.25 0.87Washing & cleaning 1.10 1.02 1.27 1.11 1.29 1.22 1.52 1.21 1.08 0.99 0.91 2.07Durables excl, vehicles 2.16 1.89 1.76 1.80 2.10 1.97 2.88 1.33 1.85 2.41 1.75 1.39

Personal hygiene 1.05 1.37 1.25 1.30 0.89 1.47 0.68 1.01 0.94 1.48 0.94 0.94Books, newspapers etc. 0.97 1.46 0.91 1.39 1.48 1.14 1.04 0.90 1.37 1.20 0.96 1.54Sports, holidays, hobbies 0.91 1.20 0.86 1.14 0.95 1.28 0.87 1.20 1.55 1.51 1.23 1.54

Transport incl, own car 2.84 2.50 2.04 1.36 2.97 2.24 1.92 1.85 2.05 1.91 1.54 2.20Union fees, subscriptions etc.. . 1.17 1.29 0.77 1.28 0.95 1.50 0.66 0.91 1.38 1.05 0.79 1.27

95% significance level 1.31 1.26 1.36 1.41 1.40 1.31 1.48 1.40 1.32 1.48 1.34 1.47

f1 110 152 81 66 68 109 49 68 100 49 91 51

f0 224 308 166 136 140 222 102 140 204 102 186 106

(2) logy = a + b ( - I)

Dwelling 1.69 1.43 1.16 1.21 0.76 1.15 0.89 1.55 1.02 0.91 1.51 1.47

Fuel & light 1.37 1.36 0.70 1.81 0.99 1.04 1.10 1.25 1.18 1.11 0.93 1.58

Food 1.53 1.18 0.87 1.48 0.93 1.40 1.86 1.03 1.22 1.66 1.20 1.17Tobacco 1.56 1.38 1.38 1.54 1.48 1.60 1.15 0.94 1.51 0.87 1.43 1.74

Clothing 1.26 1.26 1.14 1.13 1.16 1.23 1.29 1.82 1.32 1.05 1.85 1.55

Footwear 1.19 1.07 0.90 0.96 1.36 1.13 0.82 1.42 0.95 0.92 1.33 0.88Washing & cleaning 1.09 1.04 1.47 1.38 1.53 1.38 1.71 1.29 1.38 1.13 1.13 2.65Durables excl, vehicles 2.08 1.88 1.86 1.73 2.19 1.92 2.78 1.30 1.80 2.38 1.61 1.31

Personal hygiene 1.14 1.37 1.18 1.22 1.00 1.49 0.63 1.10 1.18 1.64 1.20 0.95Books, newspapers etc. 1.21 1.45 1.01 1.59 1.65 1.39 1.03 0.95 1.47 1.21 1.01 1.25

Sports, holidays, hobbies 0.98 1.35 1.16 1.21 1.09 1.63 0.85 1.34 1.92 1.99 1.53 1.91

Transport incl, own car 2.70 2.50 2.20 1.47 3.15 2.25 1.99 2.18 2.30 2.03 2.02 2.47Union fees, subscriptions etc.. . 1.20 1.24 0.98 1.32 1.06 1.60 0.75 1.47 1.69 1.47 1.13 1.33

95% significance level 1.31 1.26 1.36 1.41 1.40 1.31 1.48 1.40 1.32 1.48 1.34 1.47

f1 110 152 81 66 68 169 49 68 100 49 91 51

fe 224 308 166 136 140 222 102 140 204 102 186 106

(3) y = a + b (log x - lï)Dwelling 1.63 1.61 1.03 1.30 0.64 1.07 0.87 1.65 1.04 1.02 1.58 1.63

Fuel & light 1.18 1.11 0.68 1.18 0.84 0.75 0.87 1.40 0.74 1.03 0.91 1.72

Food 1.38 1.10 0.75 1.23 0.92 1.21 1.81 0.94 1.15 1.35 0.94 1.09Tobacco 0.96 0.83 1.29 1.26 1.00 1.10 1.12 0.89 1.21 0.98 - 1.72Clothing 1.11 1.25 1.09 0.95 1.07 1.46 1.24 1.74 1.39 0.84 2.02 1.78

Footwear 1.07 1.00 1.03 0.81 1.32 1.09 0.76 1.38 0.94 0.78 1.02 0.91Washing & cleaning 1.00 1.10 1.43 1.19 1.61 1.33 1.73 1.72 1.47 1.37 1.21 3.24

Durables excl, vehicles 2.16 1.96 1.71 1.24 2.75 2.60 2.57 1.33 1.96 - 1.77 1.24

Personal hygiene 1.21 1.66 1.23 1.21 0.99 2.09 0.66 1.22 1.36 1.34 1.17 1.00

Books, newspapers etc. 1.77 1.42 1.12 1.64 1.74 - 1.29 0.88 2.38 1.39 1.14 1.15

variances "around" the regression and variances within groups).

67

*) 1. Higher public servants and salaried employees. The Capital. 2. Lower public servants and salaried employees.The Capital. 3. Skilled workers. The Capital. 4. Unskilled workers. The Capital. 5. Higher public servants andsalaried employees. Provincial towns. 6. Lower public servants and salaried employees. Provincial towns. 7. Skilledworkers. Provincial towns. 8. Unskilled workers. Provincial towns. 9. Lower public servants and salaried employees.Rural districts. 10. Skilled workers. Rural districts. II. Unskilled workers. Rural districts. 12. Agriculturalworkers. Rural districts.

1 2 3 4Groups of wage and salary earners*)

5 6 7 8 9 10 11 12

Sports, holidays, hobbies 1.10 - 1.62 1.39 - - 1.19 - - - 2.53 3.10Transport incl, own car 4.14 5.34 2.50 2.24 3.03 3.72 2.09 4.39 - 1.96 - 3.49Union fees, subscriptions etc 1.26 1.16 0.84 1.33 1.05 1.60 0.61 1.40 1.76 1.38 1.22 1.28

95% significance level 1.31 1.26 1.36 1.41 1.40 1.31 1.48 1.40 1.32 1.48 1.34 1.47f1 110 152 81 66 68 169 49 68 100 49 91 51f2 224 308 166 136 140 222 102 140 204 102 186 106

(4) y = a + b ( - 1)

Dwelling 2.07 2.39 1.37 - 0.95 1.18 1.20 - 1.47 1.26 - 1.83Fuel & light 1.55 1.28 0.67 1.32 1.12 1.83 0.85 1.70 1.00 1.02 1.20 2.02Food 1.81 1.43 1.42 2.33 1.71 1.18 2.53 1.73 1.79 2.60 2.22 1.60Tobacco 1.08 - 1.73 - 1.36 1.57 1.53 1.41 - 1.22 - 2.25Clothing 1.48 - 1.44 1.34 1.67 1.16 1.73 2.38 2.28 1.29 - -Footwear 1.15 1.15 1.04 0.91 1.41 1.76 0.90 1.50 1.42 0.85 1.21 1.00Washing & cleaning 1.21 1.38 1.96 1.69 2.42 0.88 2.20 2.48 2.15 2.20 1.82 4.53Durables excl, vehicles 2.41 - -. 1.45 3.01 1.69 3.21 1.86 2.14 6.03 -. 1.31Personal hygiene 1.60 2.36 1.30 1.39 1.48 1.41 0.79 2.21 - 1.99 2.09 1.43Books, newspapers etc. 3.13 - 1.44 2.51 2.40 2.02 1.59 1.44 -. 1.63 1.70 1.28Sports, holidays, hobbies 1.99 - 2.86 2.01 - 1.78 2.50 2.84 - - - 4.99Transport incl, own car 3.73 - 3.40 2.96 3.34 1.80 - 6.38 - 3.10 - 4.34Union fees, subscriptions etc.. . 1.51 1.45 1.44 - 1.67 1.60 1.01 - - 2.61 - 1.9095% significance level 1.31 1.26 1.36 1.41 1.40 1.31 1.48 1.40 1.32 1.48 1.34 1.47

f1 110 152 81 66 68 169 49 68 100 49 91 51f5 224 308 166 136 140 222 102 140 204 102 186 106

(5) log y = log k + log t (a + log x)Dwelling 1.45 1.24 1.04 1.11 0.75 1.07 0.80 1.38 0.91 8.07 1.31 1.29Fuel & light 1.29 1.32 0.74 1.85 0.95 1.00 1.14 1.12 1.21 1.21 0.97 1.35Food 1.47 1.13 0.65 0.76 1.16 1.61 0.80 1.07 - 0.84 1.11Tobacco 1.54 1.40 1.27 1.43 1.54 1.05 0.85 1.55 0.79 3.83 1.54Clothing 1.16 - 1.16 1.04 - 1.08 1.20 1.79 7.35 - 1.49 1.49Footwear 1.17 0.98 0.91 0.90 1.33 1.09 0.77 1.42 0.79 0.87 1.24 0.85Washing & cleaning 1.06 1.00 1.29 1.15 1.31 1.21 1.52 1.20 1.10 0.99 0.91 2.14Durables excl, vehicles 2.11 1.86 1.73 1.75 2.07 1.94 - 1.30 1.82 2.33 1.70 1.33Personal hygiene 1.03 1.32 1.19 1.25 0.89 1.41 0.63 0.98 0.93 1.44 0.94 0.90Books, newspapers etc. 0.99 1.42 0.91 1.38 1.47 1.14 1.02 - 1.36 8.37 0.94 1.39Sports, holidays, hobbies 3.48 3.42 2.86 - - 2.89 0.82 - - 2.58 2.52 -Transport incl, own car 2.79 2.47 2.82 1.34 2.91 - - - 2.64 - - 2.21Union fees, subscriptions etc 1.15 1.26 1.78 8.14 0.93 1.45 0.64 20.7 1.38 1.06 1.24

95% significance level .. 1.31 1.26 1.36 1.41 1.40 1.31 1.48 1.40 1.32 1.48 1.34 1.47f1... 110 152 81 66 68 169 49 68 100 49 91 51f2... 224 308 166 136 140 222 102 140 204 102 186 106

68

Table V,4. N-test

Groups of wage and salary earners*)1 2 3 4 5 6 7 8 9 10 11 12

(1) log y = a + b (log x - log x)

Dwelling 49 46,3 76 65,5 36 26,6 38 33.2 35 27.3 48 46.0 24 18.9 34 27.5 44 41.9 25 19.4 44 37.9 21 20.3Fuel & light 63 46.3 67 65.4 33 26.3 38 32.9 39 27.3 63 45.2 25 19.3 32 27.3 42 40.4 19 19.4 39 37.9 27 20.1Food 63 46.5 73 65.5 35 26.8 40 32.9 36 27.6 48 45.8 28 19.4 32 27.6 42 41.9 29 18.9 46 37.9 27 20.1Tobacco 47 45.8 70 61.7 37 26.7 50 33.4 43 27.3 54 44.7 18 19.3 43 27.2 47 41.7 21 18.7 49 37.8 26 20.1Clothing 48 46.1 81 65.6 40 26.5 46 32.6 36 27.7 60 45.9 30 19.3 35 27.5 55 41.9 22 19.3 46 37.7 28 20.3Footwear 58 46.2 66 65.6 33 26.7 43 32.9 34 27.7 57 46.0 29 19.4 45 27.7 51 41.8 28 19.1 47 37.6 22 20.3Washing & cleaning 45 46.3 81 65.2 37 26.8 33 33.4 36 27.5 45 46.0 21 19.4 35 27.6 48 41.9 30 19.4 46 37.6 23 20.3Durables excl, vehicles 59 46.5 81 65.6 36 26.3 49 33.2 39 26.9 47 46.0 23 19.3 33 27.6 50 41.8 23 19.4 45 37.9 30 20.3Personal hygiene 57 46.4 95 65.4 35 26.8 41 33.4 37 27.6 51 45.7 2119.1 33 27.5 6441.6 21 18.6 51 37.9 32 19.8Books, newspapers etc 61 45.9 70 65.4 34 26.7 40 33.4 36 27.3 59 45.9 29 19.4 35 27.6 45 41.8 28 19.4 47 37.9 31 20.3Sports, holidays, hobbies 55 46.4 76 65.6 34 26.7 49 33.2 41 27.6 62 45.6 28 19.3 36 27.3 58 41.9 29 19.3 51 37.9 22 20.2Transport incl, own car. . 49 46.1 80 62.9 3626.8 51 33.2 3927.5 45 45.7 23 19.4 39 27.6 50 41.9 30 19.4 49 37.9 25 20,3Union fees, subscriptions etc 58 46.5 72 64.9 36 26.5 31 32.9 35 27.7 60 46.0 32 18.2 34 27.5 51 41.8 20 19.3 36 37.9 22 19.8

(2) logy = a + b ( - I)

Dwelling 49 46.3 69 65.6 35 26.8 39 33.2 37 27.5 48 46.0 28 19.4 30 27.6 45 41.7 25 19.3 35 37.9 21 20.1Fuel & light 57 46.3 67 65.5 33 26.5 42 32.4 37 27.3 61 45.4 26 19.1 28 27.3 43 40.7 20 19.4 40 37.9 19 20.1Food 52 46.5 70 65.1 31 26.8 33 33.3 28 27.2 53 46.0 28 19.4 29 27.7 43 41.9 25 19.4 35 37.9 29 20.2Tobacco 51 45.5 72 61.3 37 26.7 43 33.4 42 27.2 55 43.7 19 19.3 41 26.9 55 41.4 25 18.9 51 37.8 25 20,3Clothing 48 45.8 77 65.6 33 26.8 43 32.9 33 27.3 55 45.8 28 19.4 36 27.3 39 41.9 25 18.9 45 37.9 29 20.3Footwear 62 46.1 70 65.6 34 26.8 42 32.9 33 27.7 58 46.0 28 19.4 45 27.7 49 41.9 28 19.1 47 37.4 23 20.1Washing & cleaning 44 46.2 80 64.9 33 26.7 33 33.4 35 27.3 43 46.0 25 19.4 33 27.5 43 41.7 30 19.3 39 37.6 23 20.3Durables excl, vehicles 62 46.3 82 65.6 35 26.5 48 32.9 39 27.2 57 46.0 26 19.4 34 27.7 50 41.9 23 19.3 44 37.9 35 20.3Personal hygiene 55 46.4 87 65.6 37 26.7 44 33.4 29 27.7 54 45.8 19 19.4 34 27.6 43 41.6 22 18.9 40 37.7 24 20.1Books, newspapers etc. 51 45.9 72 65.6 33 26.7 35 33.4 35 27.5 51 45.8 25 19.3 31 27.7 37 41.6 26 19.4 40 37.8 28 20.3Sports, holidays, hobbies 49 46.5 65 65.5 29 26.8 45 33.4 31 27.6 41 46.0 26 19.4 37 27.6 49 41.9 25 19.1 45 37.9 24 20.3Transport incl, own car. . 5446.2 82 63.5 3426.6 43 33.1 4027.6 5045.8 23 19.4 33 27.7 4941.9 29 19.1 39 37.6 23 20.2Union fees, subscriptions etc 61 46.3 78 65.1 33 26.7 30 32.9 28 26.9 51 45.8 29 18.6 25 27.7 51 41.9 21 19.4 37 37.8 25 20.1

(3)y=a+ b(logx-logx)Dwelling 45 455 70 64.3 37 26.7 37 32.9 39 27.7 52 44.7 26 19.4 36 26.6 43 41.2 23 19.4 35 37.0 23 19.1Fuel & light 57 46.1 65 64.3 31 26.6 42 33.3 35 27.7 63 46.0 24 19.4 30 26.6 47 41.9 20 19.4 4436.4 2117.8Food 56 46.4 76 65.6 33 26.8 37 33,4 32 27.2 53 46.0 28 19.3 29 27.5 42 41.6 26 19.4 43 37.7 31 20.3Tobacco 55 46.3 74 65.5 37 26.5 49 32.9 38 27.6 61 46.0 23 18.6 41 27.5 52 41.0 23 18.3 39 37.0 25 19.6Clothing 58 46.2 73 64.8 33 26.8 41 32.9 33 27.7 51 44.7 28 19.1 34 26.6 29 41.0 23 19.4 49 37.6 27 20.6Footwear 60 46.5 70 64.8 35 26.0 47 33.4 32 27.3 60 45.6 31 19.1 45 27.6 49 41.8 28 19.3 49 37.9 22 19.1Washing & cleaning 4846.3 78 65.5 31 25.7 33 31.8 35 26.9 49 44.7 23 19.3 33 26.9 41 41.0 26 18.2 35 34.8 23 20.6Durables excl. vehicles 64 44.4 80 61.3 35 26.3 48 33.2 29 255 35 350 25 18.2 31 27.3 44 38.4 17 16.9 42 36.0 31 19.3Personal hygiene 51 458 84 63.2 3426.7 39 32.9 35 27.3 52 43.4 2119.1 32 27.5 37 39.3 22 19.3 4437.4 26 20.1Books, newspapers etc. 53 43.6 74 64.5 33 257 37 33.1 35 26.6 45 44.7 26 18.6 35 27.3 35 39.3 24 18.6 42 37.4 28 19.6Sports, holidays, hobbies 51 45.8 63 64.5 25 26.5 43 32.6 37 26.9 37 42.0 2119.3 29 26.3 39 40.1 13 16.9 31 36.4 18 16.6Transport incl, own car. . 43 40.1 6652.3 31 254 33 27.7 34259 40 37.2 17 18.2 27 24.6 37 350 23 19.4 17 23.1 21 16.2Union fees, subscriptions etc. 57 46.1 72 656 31 26.8 28 33.3 28 26.6 53 450 31 18.9 25 27.6 51 41.2 19 19.3 3537.2 25 20.

br number of runs.*)

Groups of wage and salary earners*)1 2 3 4 5 6 7 8 9 10 11 12

(4) y = a + b -)welling 43 45.3 55 63.2 27 26.5 37 32.4 27 27.5 39 44.7 24 18.2?uel & light 49 44.0 61 63.5 36 26.6 36 33.3 33 27.5 63 45.0 24 19.4ood 50 45.8 60 65.4 19 26.8 19 33.3 20 23.4 35 45.2 21 18.6

['obacco 61 45.9 72 65.4 31 25.4 37 31.8 30 25.9 49 44.7 23 19.11othing 49 46.2 61 63.5 29 26.8 33 33.4 25 27.2 43 43.7 21 19.3ootwear 58 46.5 65 64.9 34 26.3 39 33.4 35 27.6 55 45.0 29 19.3

ATashing & cleaning 46 46.2 61 64.8 31 26.0 29 31.4 29 25.5 35 42.5 19 18.2)urables excl, vehicles 62 43.6 71 61.7 31 25.0 42 32.9 27 25.1 36 37.2 23 17.8ersona1 hygiene 37 43.2 57 59.9 35 26.6 35 32.6 27 26.9 39 42.0 17 19.3

3ooks, newspapers etc 49 43.6 74 62.4 25 24.6 37 32.4 21 259 31 37.2 23 19.1;ports, holidays, hobbies 39 44.7 55 61.7 15 26.3 31 33.2 33 26.3 33 38.0 19 18.2L'ransport incl, own car. 45 41.2 5851.5 2925.0 29 27.7 28 24.6 3635.0 17 15.7Jnion fees, subscriptions etc 51 46.1 74 65.1 23 26.6 26 33.2 29 25.9 47 45.2 27 19.4

28 25121 26.927 27.531 26.331 26.637 26.632 26.321 26.921 27.227 26.927 25.123 25.121 26.6

69

31 37.040 35.727 35.439 35.437 33.845 37.931 34.342 35431 37.035 37.027 30.719 24.129 357

23 19.517 19.823 19.521 19.523 18.821 19.521 19.133 19.516 19.127 19.817 17.317 16.023 19.1

) The italicized figures denotes the lower 5 per cent significance level. I. Higher public servants and salaried employees. The Capital. 2. Lower publicervants and salaried employees. The Capital. 3. Skilled workers. The Capital. 4. Unskilled workers, The Capital. 5. Higher public servantsnd salaried employees. Provincial towns. 6. Lower public servants and salaried employees. Provincial towns. 7. Skilled workers. Provincial towns,

Unskilled workers. Provincial towns. 9. Lower public servants and salaried employees. Rural districts, 10 Skilled workers. Rural districts.1. Unskilled workers. Rural districts. 12, Agricultural workers. Rural districts.

(5) log y = log k + log (a + log x))welling 53 46.1 78 656 37 26.5 38 33.2 34 27.3 48 46.0 26 19.1 32 27.5 44 41.9 1118.6 44 37.9 21 20.3ue1 & light 61 46.1 67 65.1 33 26.5 38 33.1 37 27.5 65 450 25 19.3 32 27.3 42 40.4 19 19.4 41 37.7 27 20.1

?ood 63 46.4 75 65.4 34 26.8 - - 34 27.7 53 46.0 28 19.4 34 27.5 44 41.9 - - 46 37.8 2920.3Fobacco 49 45.9 70 62.1 37 26.7 - - 43 27.2 54 44.1 18 19.3 44 26.9 51 41.4 25 19.0 33 37.9 28 20.3lothing 50 46.3 - - 40 26.5 48 32.9 - - 60 459 30 19.3 38 27.5 31 40.1 - - 46 37.8 28 20.2

?Ootwear 60 46.2 66 65.6 33 26.7 41 32.9 34 27.7 58 46.0 29 19.4 45 27.7 49 41.9 28 19.1 47 37.4 22 20.3?Vashing & cleaning 46 46.2 82 651 37 26.8 31 33.2 36 27.6 46 46.0 21 19.4 35 27.6 47 41.9 30 19.4 48 37.7 23 20.3)urables excl, vehicles 61 46.5 81 65.6 36 26.3 49 33.2 39 27.2 47 46.0 - - 33 27.7 52 41.6 23 19.4 45 37.9 30 20.2ersonal hygiene 55 46.4 94 65.5 36 26.7 41 33.4 39 27.7 49 46.0 23 19.4 33 27.6 62 41.6 21 18.6 47 37.9 32 19.8ooks, newspapers etc 59 45.9 70 655 34 26.7 38 33.4 36 27.3 55 46.0 29 19.4 - 42 41.8 15 19.4 49 37.9 31 20.3ports, holidays, hobbies 33 46.1 51 656 29 26.8 - - - - 45 46.0 28 19.4 - - - 27 19.4 33 37.8L'ransport incl, own car. 51 46.2 82 62.8 37 26.8 51 33.2 39 27.5 ------42 41.9 - - - - 25 20.3Jnion fees, subscriptions etc. 63 46.5 76 64.3 15 26.6 25 32.9 35 27.6 62 46.0 34 18.6 19 27.5 53 41.8 22 19.4 - - 24 20.1

35 40.7 17 18.233 41.0 24 19.133 41.0 17 19.453 40.7 25 18.023 39.7 19 18.943 41.0 25 19.437 39.3 17 16.941 37.9 15 16.333 38.9 2119.435 36.3 24 19.133 357 15 16.327 31.4 2118.939 40.1 15 18.2

i (y) =

The F-values thus calculated have been shown in table V,3.Can any conclusions be drawn from the F-test as regards the determination of the

"best" algebraic formulation(s) of the Engel curve? Does the F-test point in the samedirection as the X2-test and as the more dubious evidence from the correlation coefficients,i.e. to one of those functions in which the dependent variable is the logarithmicallytransformed expenditure, log y?

As mentioned above there are several reservations to be made. For expenditureitems where > 0.5 the F-test fades away as regards functions (1,1), (1,2), and (1,5). Asregards functions (1,3) and (1,4), parameter estimation has failed in some cases, cf.chapter IV, p. 54, and, accordingly, no testing is possible. These cases coincide with theinvalidation due to high a-values of the F-test performed on functions (1,1) and (1,2).Despite these substantial reservations concerning the results shown in the table, itnevertheless seems justified to draw the general conclusion that the F-test confirms theconclusion of the 2-test to the effect that the logarithmic functions pass the testsmore easily than the functions in which the expenditure enters untransformed.

For all expenditure items (except footwear) there are significant F-values for all fivetypes of functions in one ore more of the twelve social groups, but in the case of functions(1,1) and (1,2) frequently only one of the social groups gives significance. For the double-logarithmic function (1,1) the items of dwelling and footwear and fuel and lighting thus show

70

where ni = 3 is the number of observations in the j'th group. s22 has been calculated

direct on the basis of [log y - f(x)] and (Y ), respectively for the two types of

functions.As regards s2 it has not been possible to use the individual observations in those

cases where log Y was the dependent variable since several observations of Y were zero.This zero-observation problem, which was apparently solved by the grouping of theobservations, cf. the discussion in chapter IV, p. 36, thus crops up again here in a new

form. However, for moderate values of the average coefficient of variation = 8y12

si2, may be found with good approximation as (log y) M2 . , where M = 0.4343.This approximation is not very good in those cases where > 0.5, which must be

borne in mind when the F-tests are examined below. For the three items durable goods,transport and sports, holidays and hobbies several of the calculated c-values are very high,and the F-test in the case of these items must be considered dubious as regards the twofunctions in which log y is the dependent variable.

As regards the functions in which y is the dependent variable, s2 has everywhere beenconsidered equal to , since

71

only one significant F-value, food, house-cleaning and washing, two significant values, andclothing, personal hygiene, sports, holidays and hobbies, and subscriptions, etc. three significantvalues. The semi-logarithmic function (1,2) displays somewhat poorer results. For theremaining functions the number of significant results are much higher.

4. The test for number of runs and for the longest run; the d-test.

The last category of tests for goodness of fit tests the hypothesis that the observationsare distributed at random around the Engel curve determined by the calculated para-meter estimates.

The two runs test consist of a simple count of the number of changes of signs (testfor number of runs, here called N-test) and of the number of elements in the longestrun (test for longest run, here called l-test). A run is accordingly defined as a series ofsuccessive deviations with the same sign, cf. e.g. A. Hald, (8), p. 342, and Prais andHouthakker, (10), pp. 53-55. If the number of positive and negative deviations, P andQ, respectively are greater than about ten, D is approximately normally distributed

with mean M D F i + 2 PQ

2PQ(2PQN)and variance V DF N2 (N - 1)where D is number of runs and N the number of observations.

If the test hypothesis is rejected in favour of an alternative hypothesis because thecalculated Engel curve "misses the mark" in a greater or smaller area of the field ofobservation, then the number of runs will be too small.

The limit of significance can be chosen as

M{D.-2VV{DFIn table V,4 the number of runs, D, and the limit of significance thus found has been

shown.The test for the longest run, the l-test, which is not, of course, unrelated to the N-test,

is derived from the knowledge of the distribution function for the number of runs of agiven length.

In table V,6 all the longest runs have been shown, and for each set of calculationshas been given the 5 per cent significance limit.

A glance at table V,4 and table V,5 will show how the results of the two tests of runscorrespond so that they should rightly be considered as one test. The conclusion ofthis combined test points in the same direction as the previous tests: among the fiverelations tested the two in which the dependent variable is a logarithmic transformationof the expenditure performs better than the other three functions, and as indicated alsoby the F-test, the double-logarithmic function seems to do best.

The final test, the d-test, is based on the following quantity, cf. Prais and Houthakker(10), page 53

d = E (tk tk )2/E tk2

72

Table V,5. l-test for

1 2

Groups of wage and salary earners*)3 4 5 6 7 8 9 10 11 12

(1)logy=a+ b(logxlogx)Dwelling 9845676610756FueI&light 77475957910106Food 8 7 4 5 8 6 5 13 11 6 6 7

Tobacco 7 9 5 6 4 6 6 5 8 6 6 8Clothing 7956464712565Footwear 10 9 6 7 5 6 6 6 5 5 8 5

Washing & cleaning 9 6 5 7 5 10 7 7 7 4 9 8Durables excl, vehicles 10 5 5 7 7 7 5 10 6 9 6 5

Personal hygiene 11 5 6 6 5 9 6 5 7 7 8 6

Books, newspapers etc. 7 8 6 6 6 6 5 5 6 4 6 4Sports, holidays, hobbies 6 5 6 4 7 6 8 7 5 5 6 7

Transport incl, own car 10 7 4 4 6 8 6 7 8 5 9 6

Union fees, subscriptions etc. 6 9 7 9 7 6 5 8 6 6 6 6

95% significance level 11 12 10 10 10 11 9 10 11 9 10 9

(2) logy = a + b ( -Dwelling 9 11 6 6 6 7 4 6 8 5 11 6

Fuel&light 7747595109101414Food 8 8 6 11 8 9 7 9 9 6 9 5

Tobacco 7 9 5 6 4 6 6 5 6 5 8 6

Clothing 8965665715474Footwear 10966566610586Washing & cleaning 12 6 4 7 7 10 7 7 7 4 9 6

Durables excl, vehicles 8 6 5 7 7 7 4 10 5 9 7 4

Personal hygiene 11 6 5 6 5 6 6 5 8 5 10 7

Books, newspapers etc 11 8 11 7 6 8 4 6 7 4 6 6

Sports, holidays, hobbies 6 15 13 6 8 8 6 5 12 7 6 7

Transport incl, own car 9 6 5 7 6 8 6 7 9 5 10 8

Union fees, subscriptions etc. 6 6 6 10 8 12 6 7 7 6 6 4

95% significance level 11 12 10 10 10 11 9 10 11 9 10 9

(3) y = a + b (log x - log x)

Dwelling 11 18 6 6 6 7 7 6 8 7 11 7

Fuel & light 10 10 4 7 5 8 5 10 9 10 14 19

Food 886889799665Tobacco 6 9 6 6 5 7 6 6 6 8 8 6

Clothing 7966675915487Footwear 10 11 8 5 5 6 6 5 10 5 5 6

Washing & cleaning 12 6 9 7 7 10 7 6 7 4 9 8Durables excl, vehicles 8 6 5 6 8 13 6 10 11 9 7 4Personal hygiene 12 10 5 6 5 10 6 5 15 5 8 7

Books, newspapers etc. 9 8 11 7 6 9 4 6 12 8 7 8

Sports, holidays, hobbies 7 14 14 8 9 13 8 12 16 16 14 7

Transport incl, own car 14 11 7 12 12 9 10 11 14 9 20 9

Union fees, subscriptions etc. 10 9 6 11 8 12 5 8 7 6 6 4

95% significance level 11 12 10 10 10 11 9 10 11 9 10 9

the longest run.

73

*) 1. Higher public servants and salaried employees. The Capital. 2. Lower public servants and salaried employees.The Capital. 3. Skilled workers. The Capital. 4. Unskilled workers. The Capital. 5. Higher public servants andsalaried employees. Provincial towns. 6. Lower public servants and salaried employees. Provincial towns. 7. Skilledworkers. Provincial towns. 8. Unskilled workers. Provincial towns. 9. Lower public servants and salaried employees.Rural districts. 10. Skilled workers. Rural districts. Il. Unskilled workers. Rural districts. 12. Agriculturalworkers. Rural districts.

Groups of wage and salary earners*)1 2 3 4 5 6 7 8 9 10 11 12

(4) y = a + b ( - I)

Dwelling 11 21 11 6 12 11 7 7 14 12 11 9Fuel & light 10 11 4 10 8 11 6 16 19 9 15 19Food 8 20 13 17 26 21 16 10 23 13 17 10Tobacco 6 9 9 10 10 11 6 8 8 9 10 10Clothing 12 17 7 10 13 18 7 11 24 9 9 7Footwear 101687556614696Washing & cleaning 12 15 9 15 13 17 7 10 10 7 13 8Durables excl, vehicles 8 9 8 9 13 13 7 11 11 11 7 4Personal hygiene 15 31 6 9 10 18 7 17 15 6 12 19Books, newspapers etc. 10 8 11 7 11 17 5 10 13 8 14 8Sports, holidays, hobbies 11 14 20 11 9 15 8 10 15 16 14 7Transport incl, own car 14 13 7 14 13 9 10 11 14 12 22 12

Union fees, subscriptions etc. 10 8 15 11 8 16 6 14 14 15 15 8

95% significance level 11 12 10 10 10 11 9 10 11 9 10 9

(5) log y = log k + log (a + log x)

Dwelling 98456746102456Fuel&light 7 7 4 7 5 9 5 7 9 10 11 6Food 876-875910-65Tobacco 795-466585184Clothing 7-55-64717-65Footwear 10 9 6 7 5 6 6 6 9 5 8 5Washing & cleaning 9 6 5 10 5 10 7 7 7 4 9 8Durables excl, vehicles 6 5 5 7 7 7 - 10 5 9 6 5Personal hygiene 11 5 6 6 5 5 6 5 7 7 8 6Books, newspapers etc 7 8 6 7 6 6 5 - 6 14 6 4Sports, holidays, hobbies 21 23 13 - - 12 8 - - 7 19 -Transport incl, own car 9 7 5 4 6 - - - 11 - - 6Union fees, subscriptions etc. 6 9 17 18 7 5 5 20 6 6 - 4

95% significance level 11 12 10 10 10 11 9 10 11 9 10 9

74

Table V,6. d-test for size and

1 2 3 4Groups of wage and salary earners*)

5 6 7 8 9 10 11 12

(1) log y a + b (log x - log x)Dwelling 2.24 1.76 2.03 1.83 2.15 1.77 2.34 1.77 1.82 1.65 1.91 1.57Fuel & light 2.03 1.97 1.86 1.43 2.26 2.20 1.42 1.64 1.59 1.43 1.54 1.95Food 1.85 2.06 2.11 1.76 1.97 1.90 2.46 1.80 1.57 2.03 1.74 1.58Tobacco 1.86 1.91 1.70 2.06 2.35 2.14 1.57 2.44 2.04 2.12 2.28 2.26Clothing 1.69 2.12 2.04 2.08 1.94 2.08 2.44 2.32 2.21 1.95 2.43 2.05Footwear 2.07 1.91 1.94 2.30 1.86 2.05 2.10 2.11 2.19 1.94 2.17 1.83Washing & cleaning 1.65 1.84 1.69 1.89 1.56 1.68 1.74 1.90 1.79 2.30 1.93 1.43Durables excl, vehicles 2.21 2.07 2.07 1.95 2.22 1.84 2.20 1.99 1.92 1.50 1.76 2.33Personal hygiene 1.84 2.27 2.06 1.83 1.97 1.81 1.69 1.62 2.36 1.47 1.96 1.78Books, newspapers etc. 2.04 1.91 2.05 1.88 2.00 1.80 2.14 2.06 1.85 1.55 1.79 1.52Sports, holidays, hobbies 2.14 1.87 2.26 1.97 2.15 2.15 1.90 2.28 2.34 2.37 2.02 1.34Transport incl, own car 1.98 1.96 2.34 2.54 2.41 1.87 1.86 1.79 1.97 2.23 1.85 1.83Union fees, subscriptions etc.. . 2.10 1.85 2.04 1.59 2.13 2.18 2.64 1.57 2.21 1.87 1.86 1.85

(1 1(2) log y = a + b ii

Dwelling 1.89 1.52 1.84 1.75 2.04 1.69 2.14 1.58 1.66 1.72 1.66 1.40Fuel & light 1.94 1.90 2.07 1.52 2.22 2.13 1.54 1.49 1.69 1.67 1.71 1.67Food 1.81 2.03 1.62 1.23 1.72 1.65 2.18 1.47 1.45 1.32 1.23 1.65Tobacco 1.87 1.98 1.58 1.77 2.29 2.09 1.50 2.24 2.18 1.95 1.95 2.02Clothing 1.59 1.90 2.23 1.94 1.80 1.88 2.36 2.27 1.58 1.84 1.91 2.04Footwear 2.06 1.76 2.06 2.19 1.83 1.99 2.05 2.10 1.85 1.86 2.03 1.83Washing & cleaning 1.60 1.77 1.48 1.53 1.32 1.48 1.55 1.72 1.40 1.96 1.55 1.15Durables excl, vehicles 2.29 2.08 1.97 2.02 2.13 1.87 2.27 1.97 1.97 1.56 1.90 2.50Personal hygiene 1.70 2.23 2.18 1.91 1.76 1.79 1.80 1.43 1.89 1.31 1.49 1.72Books, newspapers etc. 1.72 1.91 1.86 1.64 1.78 1.50 2.13 1.89 1.75 1.52 1.69 1.90Sports, holidays, hobbies 1.96 1.64 1.69 1.84 1.86 1.70 1.90 2.08 1.91 1.84 1.60 1.07Transport incl, own car 2.09 1.95 2.18 2.32 2.28 1.85 1.82 1.57 1.76 2.11 1.42 1.66Union fees, subscriptions etc.. . 2.05 1.90 1.63 1.52 1.88 2.06 2.35 1.06 1.82 1.31 1.34 1.75

(3)y==a+b(logx-logxDwelling 1.95 1.51 1.88 1.63 2.12 1.66 2.16 1.71 1.60 1.70 1.67 1.60Fuel & light 2.03 1.88 1.95 1.91 2.22 2.12 1.53 1.53 1.60 1.66 1.69 1.82Food 1.94 2.11 1.88 1.44 1.90 1.83 2.29 1.61 1.57 1.60 1.49 1.68Tobacco 1.75 1.81 1.62 1.80 2.32 1.98 1.39 2.03 2.08 2.08 2.23 2.02Clothing 1.59 1.79 2.16 1.92 1.78 1.77 2.18 2.21 1.46 1.71 1.65 1.90Footwear 2.12 1.86 2.10 2.20 1.83 2.00 2.12 2.04 1.96 1.90 2.08 1.88Washing & cleaning 1.63 1.76 1.33 1.58 1.24 1.31 1.64 1.78 1.42 1.94 1.58 1.17Durables excl, vehicles 2.26 2.15 1.56 2.26 2.31 1.77 2.14 1.74 1.94 2.54 1.97 2.23Personal hygiene 1.70 2.16 2.19 1.87 1.92 1.78 1.83 1.31 1.79 1.24 1.51 1.73Books, newspapers etc. 1.57 1.87 1.82 1.36 1.89 2.07 2.11 1.82 1.84 1.53 1.64 1.99Sports, holidays, hobbies 1.50 1.98 1.00 1.46 2.29 2.02 1.12 1.95 1.91 2.35 0.97 0.76Transport incl, own car 2.20 1.95 1.87 2.12 2.12 1.86 1.34 1.25 2.18 1.45 1.62 1.67Union fees, subscriptions etc.. . 2.05 1.84 1.56 1.37 2.00 2.09 2.48 1.05 1.95 1.22 1.29 1.81

direction of deviations from the regression.

75

Groups of wage and salary earners*)1 2 3 4 5 6 7 8 9 10 11 12

(4) y = a + b ( -Dwelling 1.43 1.18 1.53 1.77 1.50 1.18 1.66 1.57 1.22 1.31 1.73 1.39Fuel & light 1.61 1.75 2.02 1.86 1.90 1.83 1.64 1.37 1.41 1.82 1.35 1.47Food 1.50 1.68 1.02 0.82 1.27 1.18 1.71 1.00 1.08 0.81 0.70 1.20Tobacco 1.64 2.20 1.32 1.91 1.89 1.57 1.18 1.35 2.14 1.81 1.37 1.65Clothing 1.22 1.91 1.67 1.47 1.13 1.16 1.63 1.86 0.91 1.19 2.31 1.94Footwear 2.03 1.61 2.06 1.92 1.74 1.76 1.88 1.94 1.41 1.70 1.78 1.73Washing & cleaning 1.34 1.39 1.01 1.18 0.89 0.88 1.31 1.49 0.99 1.51 1.07 1.00Durables excl, vehicles 2.22 2.07 2.10 2.02 2.16 1.69 2.02 1.50 1.82 1.17 1.78 2.12Personal hygiene 1.35 1.67 1.99 1.62 1.35 1.41 1.58 0.79 1.45 0.85 0.90 1.37Books, newspapers etc. 1.16 2.35 1.55 0.98 1.34 2.02 1.83 1.13 2.01 1.35 1.28 1.85Sports, holidays, hobbies 0.81 1.99 0.57 1.05 1.73 1.78 0.53 1.36 1.23 2.19 1.38 0.49Transport incl. own car 2.22 1.66 1.49 1.84 1.98 1.80 2.24 1.10 1.78 0.91 2.27 1.47Union fees, subscriptions etc.. 1.86 1.56 0.87 1.86 1.38 1.60 1.67 1.94 1.54 0.72 1.66 1.44

(5) log y = log k + log I1 (a ± log x)Dwelling 2.21 1.78 2.01 1.87 2.22 1.83 2.45 1.80 1.90 0.29 1.92 1.56Fuel & light 2.05 1.96 1.95 1.48 2.28 2.21 1.47 1.82 1.63 1.51 1.64 1.94Food 1.93 2.13 2.15 - 2.09 1.97 2.48 1.88 1.65 - 1.74 1.69Tobacco 1.87 1.94 1.71 - 2.35 2.15 1.57 2.45 2.09 2.10 0.75 2.25Clothing 1.74 - 2.13 2.11 - 2.13 2.49 2.57 0.30 - 2.38 2.15Footwear 2.09 1.90 2.10 2.33 1.86 2.10 2.17 2.29 2.20 1.96 2.18 1.89Washing & cleaning 1.69 1.86 1.64 1.84 1.53 1.67 1.72 1.96 1.72 2.29 1.88 1.37Durables excl, vehicles 2.24 2.08 2.11 1.98 2.22 1.96 - 2.05 2.05 1.52 1.80 2.37Personal hygiene 1.86 2.32 2.12 2.06 1.98 1.87 1.82 1.69 2.36 1.44 1.95 1.83Books, newspapers etc. 2.11 1.95 2.06 1.85 2.00 1.80 2.19 - 1.86 0.27 1.82 1.69Sports, holidays, hobbies 0.59 0.71 0.67 - - 0.96 2.29 - - 1.46 0.98 -Transport incl, own car 2.02 1.97 1.68 2.55 2.41 - - - 1.53 - - 1.88Union fees, subscriptions etc.. 2.12 1.90 0.10 0.27 2.17 2.24 2.65 0.09 2.22 1.83 - 1.86

*) 1. Higher public servants and salaried employees. The Capital. 2. Lower public servants and salaried employeesThe Capital. 3. Skilled workers. The Capital. 4. Unskilled workers. The Capital. 5. Higher public servants andsalaried employees. Provincial towns. 6. Lower public servants and salaried employees. Provincial towns. 7. Skilledworkers. Provincial towns. 8. Unskilled workers. Provincial towns. 9. Lower public servants and salaried employees.Rural districts. 10. Skilled workers. Rural districts. 11. Unskilled workers. Rural districts. 12. Agriculturalworkers. Rural districts.

Table V,7. Number of significant test results among 12 groups of wage and salary earners. Significance level 95% (y,', F2, 1) and5% (N, d.)

only li test results. only 10 test results. only 9 test results. only 8 test results, only 7 test results. only 6 test results.

logy=a+b(logxlogx)

x' F N x2

logy=a+bQ)

F N 1 d

y=a+b(logxlogx)

' F N

y=a+b()

x2 F N 1

logy=logk+log(a+logx)

d x' F N 1 d1 d I d

Dwelling 0 1 0 0 0 0 5 1 1 2 5 5 2 3 2 6 6' 8 9 8 1 2 1 1 2

Fuel&light 5 1 3 2 5 2 3 4 4 2 5 2 4 4 1 6 4 4 5 4 5 2 2 2 5

Food 1 2 1 2 1 1 5 2 1 5 1 3 2 0 3 4 12 9 11 10 12 22 02 02 12

Tobacco 4 6 1 0 0 3 9 1 0 1 0' 1' 0 0 134 54 2 4 6 31 71 2' 1' 2'

Clothing 1 3 0 0 0 1 4 2 1 2 2 5 2 1 2 33 6' 7 8 6 l 43 l 1' 1'Footwear 1 0 2 0 0 1 0 0 0 0 3 0 0 0 0 2 4 1 2 2 1 1 0 0 0

Washing&cleaning 1 2 2 0 1 1 6 3 2 8 5 7 3 1 8 10 11 8 7 11 0 2 0 0 3

Durables excl, vehicles 1 10 0 2 0 2 10 0 2 0 7' 9' 2 4 173 8 6 5 2 1' 91 0' 0' 1'

Personalhygiene 2 3 0 1 0 2 3 1 2 3 6 3 1 2 3 8' 7' 9 8 8 1 2 0 0 1

Books, newspapers etc 0 4 0 0 0 1 5 2 2 1 6' 51 3 2 2 6 8' 6 6 7 2' 4' 1' 1 1

Sports, holidays, hobbies. . . 1 3 0 0 1 0 6 3 3 3 3 3 5 7 6 66 66 10 10 8 2' 6' 55 55 6

Transport incl, own car 3 11 1 0 0 4 12 0 1 1 82 10' 8 9 3 6 75 10 10 4 3 6 0 0 1

Union fees, subscriptions etc 1 3 1 0 0 1 3 3 2 4 1 3 5 2 5 74 74 6 7 7 31 5 3 3 3

77

where tk is the deviation of the observed from the calculated expenditure on a givenexpenditure item for the k'th group of observations, where the observations are arrangedby increasing values of income.

As in the z2-test it is always the weighted residuals, which are used in the tests forthe two relations in which the untransformed value of the expenditure, y, is the de-pendent variable.

In their article5) Durbin and Watson have examined the distribution function for dand have calculated significance zones for this quantity under alternative assump-tions as regards number of observations (L) and number of parameters in the functionon the basis of which the residuals have been calculated (in this case 2). Durbin andWatson's tables do not allow of any precise delimitation of the level of significance sinced-values within the calculated significance zones do not permit any conclusion as regardsrejection or acceptance of the test hypothesis.

It will be seen from table V,6 and table V,7 that the d-test permits a more preciseconclusion than the run tests. While both function (1,1) and (1,2) pass the N- and l-testsfairly well, the double-logarithmic function getting on the whole the best marks, thed-test reveals the difference between these two functions more clearly.

In a number of cases the semi-logai ithmic function gives d-values which are clearlybeyond the significance zone presumably because the size of the residuals in the firstand in the last run is in certain cases larger than permissibleeven where the numberof elements in these runs (as determined by the shifts of sign) may not be significant.

5. Summary of test results.

In table V,7 a summary of the tests has been given. For each item of expenditure thenumber of significant test results among the twelve social groups has been counted separa-tely for each of the five Engel functions. Bearing in mind the reservations mentionedabove as regards the validity of the different tests, this summary clearly emphasizesthe conclusion which has gradually emerged in the course of the discussion of the indi-vidual tests. The double-logarithmic Engel function gives the best goodness of fit amongthe five functions tested. This does not, of course, mean that the "true" Engel curvefunction have thus been found, but the selection of the double-logarithmic functionas the "best" may, nevertheless, justify a further analysis of the result of the parameterestimation for this function.

Vc. Analysis of estimates of the parameters.

1. Regression analysis versus two-way cross-tabulation.

Table V,8 shows estimates of the standard deviation in the distribution of log y forall 13 expenditure items, separately for each of the 12 social groups. To make it possibleto assess the effect of the introduction of the disposable income x as explanatory variable,estimates of the standard deviation in the distribution of the residuals from the calculated

5) Cf. Durbin and Watson (4).

78

double-logarithmic Engel function have been shown too. It will be seen that the un-explained part of the variation in expenditures, is reduced by 10 to 60 per cent as onegoes from the simple mean value description to the Engel function. The table showsthat the gain is fairly constant from one social group to another for the same expenditureitemwhereas there are great variations among the expenditure items. The size of thegain depends partly on the slope of the Engel curve and partly on the size of the variancewithin groups. If this variance is small, the gain from the regression analysis will, ceterisparibus, be relatively great, and if the regression line is steep, the gain will, ceteris pari bus,also be great.

The food group has a small variance, and even if the slope of the regression line ismoderate a considerable gain is nevertheless achieved through the regression. The itemsports, holidays, hobbies etc., has a relatively high variance, but since the regression linesare rather steep, the gain is also in this case considerable. Expenditures on fuel, light andfootwear rise slowly with income and the variance is substantial; the regression gain isinsignificant. This is also the case of the item of durable goods, where the extremelyhigh variance almost completely counteracts the effect of the fairly steep regressionline.

Another measure of this regression gain is obtained by calculating the ratio of theestimates of the slopes to their estimated standard errors. This is at the same time atest for the hypothesis fi = O, since this hypothesis can be tested by a t-test. The

quantityb O

follows the t-distribution with N-2 degrees of freedom, where N

is the number of group averages included in the calculation of b. The t-values thuscalculated show that the hypothesis = O, i.e. that the slope of the regression line iszero, cannot be accepted in one single case.

It thus seems justified to conclude that the description of the consumption behaviourof the household has gained considerably in precision by the inclusion of the disposablehousehold income as explanatory variable.

By arranging the observations of expenditures in cross-tables where each household isplaced in a cell according to its disposable income, it is often attempted to includedisposable income as an explanatory variable. The frequent use of this type of tables inpublications dealing with household-surveys is often explained by the wish to presentthe material in a clear manner without adopting a definite hypothesis concerning the

form of the relationship between the two variable expenditure and income.The method used here for the description of the expenditure-income relationships,

however, has obvious advantages over this frequently used grouping method. As shownby Amundsen6), information is lost to a considerable extent when the basic materialis split up into the many cells of such a table since only the observations of the individualcell is used in the calculation of the average expenditure figure of this cell.

The parameter estimates have thus made it possible to give a more precise descriptionof the observations than the usual description by averagesbe they overall averagesor "cell averages" in a cross table.

6) Amundsen (2).

Table V,8. Gain of regression. Standard errors in the distribution of expenditures and in the distribution of deviations from thedouble logarithmic Engel function, log y = a + b (log x -

Fuel Washing Durables Peonal Books, Sports, Trans- UnionGroup of wage and salary earners Dwelling &

lightFood Tobacco Clothing Footwear &

cleaningexcl.

vehicleshygiene newspa-

pers etc.holidays,hobbies

port incl.own car

fees,sub-scription

The capitalHigher public servants Slog y 0.23 0.18 0.12 0.29 0.21 0.15 0.20 0.31 0.19 0.24 0.30 0.40 0.17and salaried employees Slog y Ix 0.14 0.12 0.069 0.25 0.13 0.13 0.13 0.29 0.11 0.14 0.13 0.36 0.13

Lower public servants Slog y 0.22 0.20 0.12 0.32 0.25 0.17 0.22 0.36 0.23 0.25 0.31 0.37 0.19

and salaried employees 5log y x. 0.13 0.17 0.067 0.27 0.14 0.12 0.13 0.32 0.12 0.17 0.17 0.31 0.14

Skilled workers Slog y 0.22 0.14 0.14 0.25 0.24 0.14 0.19 0.35 0.17 0.20 0.32 0.43 0.20

Slog y x 0.13 0.12 0.050 0.18 0.13 0.10 0.14 0.26 0.11 0.13 0.13 0.30 0.086

Unskilled workers Slog y . 0.23 0.18 0.15 0.29 0.26 0.17 0.21 0.35 0.21 0.28 0.34 0.38 0.25

Slog y Ix. . 0.14 0.17 0.062 0.20 0.16 0.13 0.14 0.29 0.13 0.18 0.19 0.27 0.13

Provincial towns.Higher public servants 5iog y 0.19 0.16 0.12 0.27 0.22 0.14 0.22 0.37 0.18 0.27 0.33 0.54 0.17

and salaried employees Slog y x 0.11 0.12 0.056 0.22 0.12 0.11 0.15 0.32 0.097 0.17 0.14 0.43 0.093

Lower public servants Slog y 0.23 0.18 0.15 0.32 0.25 0.17 0.27 0.41 0.23 0.29 0.38 0.46 0.21

and salaried employees Slog y I x. . 0.14 0.16 0.072 0.25 0.13 0.13 0.17 0.34 0.14 0.17 0.18 0.35 0.12

Skilled workers 5log y 0.17 0.14 0.13 0.27 0.25 0.16 0.22 0.44 0.17 0.21 0.34 0.34 0.17

Slog I . . 0.10 0.14 0.067 0.19 0.14 0.096 0.15 0.34 0.090 0.14 0.14 0.34 0.079

Unskilled workers Slog y 0.20 0.14 0.13 0.28 0.25 0.18 0.21 0.36 0.21 0.24 0.34 0.44 0.19

5log y x . 0.14 0.11 0.060 0.17 0.17 0.13 0.15 0.25 0.10 0.13 0.18 0.32 0.086

Rural districtsLower public servants 5logy 0.22 0.19 0.14 0.33 0.26 0.18 0.25 0.39 0.23 0.27 0.38 0.48 0.20

and salaried employees Slog y Ix . 0.14 0.16 0.074 0.25 0.13 0.11 0.16 0.32 0.12 0.19 0.20 0.36 0.12

Skilled workers 5log y 0.21 0.13 0.13 0.23 0.23 0.17 0.21 0.39 0.20 0.22 0.36 0.50 0.20

Slog y x . 0.12 0.12 0.052 0.17 0.13 0.12 0.13 0.33 0.13 0.15 0.20 0.35 0.096

Unskilled workers Slog y 0.25 0.18 0.15 0.32 0.25 0.16 0.23 0.32 0.21 0.25 0.38 0.45 0.21

Slog y z 0.15 0.12 0.052 0.19 0.14 0.12 0.13 0.27 0.10 0.14 0.19 0.29 0.076

Agricultural workers sj y 0.27 0.19 0.14 0.31 0.29 0.14 0.25 0.31 0.18 0.29 0.37 0.36 0.21

Slog y x. . 0.19 0.12 0.068 0.22 0.17 0.11 0.19 0.26 0.11 0.19 0.24 0.31 0.13

80

2. Interpretation of main results.

The next stage of the analysis, tackles the central problem: How can the parameterestimates be interpreted on the basis of economic theory?

Also in this connection it is the double-logarithmic Engel function which will beselected for treatment; primarily because this function emerged as the best among thefive types tested for goodness of fit, but also because the parameter estimate b of theslope is at the same time an estimate of the income elasticity in the demand for a givenexpenditure item.

The estimate a of the parameter indicates the average value of the logarithmicallytransformed expenditure observations. This estimate, together with log x, the averagevalue of the transformed income observations, determines the coordinates of the centreof gravity, the mean, of the observations and can accordingly be interpreted as anestimate of the level of expenditures within the social group in question.

With the modifications which result from the use of transformations other than thelogarithmic one, this interpretation can be extended to cover all four linear regressionmodels: the estimate a in conjunction with the average transformed or untransformedincome observations indicates the level of the expenditure in the social group in question.In the case of the cumulative log-normal distribution function, the situation is different.

The estimates k and a in this function, log y == log k + log cl (a + log x), can herebe interpreted as regulators of the unit of measurement in terms of which the expenditureand the disposable income are to be measured. It is postulated that one and the sameEngel curve can describe the relationship between the disposable income and anyexpenditure item, although for any given expenditure item an adjustment must be madeof the two units of measurement on the x- and y-axis determined by the values of the para-meters and 7). The estimate k may be interpreted as an estimate of the saturationexpenditure, , on the given item for the given social group, i.e. the total expenditurewhich households of that group would spend on that item, if the income tended towardsinfinity.

Also the functionslog = a + (v' - v1)

and= a + 1 (v' - ir1)

have a saturation expenditure, namely

antilog (a -and

a - fi v1respectively.

The estimates of these two expressions and of are widely different; this is only areflection of the fact that the different Engel functions deviate considerably from oneanother as soon as we move outside the range of the observation cf. fig. V,3; however,7) Cf. Aitchison and Brown (1), p. 131.

Expenduure on food in 1000 kr.

(1,1)

(")_._--3)-

81

income in 1000 kr.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Fig. V, 3. Five Engelfunctions. Income and expenditure on food. Skilled workers in the provincial towns.

what determines the suitability of an Engel function in the description of the data isprimarily the goodness of fit of this function and not, for instance, whether such aconceptually vague quantity as the saturation expenditure deviates more or less fromsome preconceived ideas about it.

The tables of results in appendix A, pp. 126-173, show that the expenditure levelvaries systematically from one social group to another for all expenditure items, aswas in fact to be expected since the social grouping is at the same time to a high degreea grouping by income. Since the expenditure is rising with rising income for all 13expenditure items, the parameter estimates a, which illustrate the level of expendi-tures, therefore show the highest values for the social groups of higher public servantsand salaried employees and the lowest for the groups of unskilled workers and farmworkers.

3. Interpretation of the estimates of the slope of the regression line.

However, it is of greater interest to examine the parameter estimate b or the two para-meter estimates in conjunction, i.e. the whole Engel curve.

It will then be natural to examine first whether the division into twelve groups ofwage and salary earners has been appropriate, or in other words, whether the twelveEngel curves for a given expenditure item which are estimated separately for each ofthe twelve groups of wage and salary earners, are significantly different.

The reason for dividing the observations into these twelve groups was an assump-tion that differences in geographical and social grouping would be reflected in cha-

82

racteristic differences in consumption behaviour. To the extent that this geographicaland social grouping is at the same time a grouping by income levels, this will naturallyinfluence the level of the consumption expenditure as shown by the estimate a. Butthe question is whether this grouping is anything more than a grouping by income.May we consider the twelve Engel curves as being generated by the same consumptionprocess, i.e. be considered estimates of the same Engel curve?

This problem can be examined by comparing the twelve regression lines for eachexpenditure item, a comparison which, like the calculation of the parameter estimates,is a standard feature of the regression analysis8).

Such a comparison has been made in the case of the double-logarithmic functionfor all 13 expenditure items.

First, it is examined whether the estimates of the slope b1, b2, ... b12 calculated fora given expenditure item can be considered as estimates of the same "true" slope ,

and if this hypothesis can be upheld, it is examined whether the parallel Engel curvescan be considered as estimates of one single Engel curve. The former test is a test forthe parallelism of the Engel curves, and the latter a test for their identity.

The test for the parallelism of the Engel curves is performed as an F-test, in which avariance calculated on the basis of the variation among the 12 estimated b-values is com-pared with an expression of the inner variance:

s22. 12

Ff2, fi - -i, [f1 = (n - 1) and f2 = 12 - 1]1

12 -s22 = (b - b)2 (log x - log xk)2

11 i

Cf. for instance, A. Hald (8), pp. 579-584.Cf. A. Hald (8), p. 580.

12E(nk-2) s2jogyjx

S2= 112

(nk - 2)i

In these relations log xk and log yk are the coordinates of the mean of the observationsof the k'th group of wage and salary earners; k is the number of observations in thisgroup.

The average slope 1 for a given expenditure item has been calculated as a weightedaverage of the twelve individual slopes 9)

12E (logx - log xk)2

12E E (logx - log xk)21

where

and

83

If the hypothesis is correct (concerning the parallelism of the Engel curves) it will betrue that 15 is normally distributed around fi with the variance

2alogy X= 12

(log xk - log Xk)2i

and the 12 x 13 calculated slopes b have been shown in table V, 10.Table V,9 shows that in 6 out of 13 cases the hypothesis concerning the parallelism

of the Engel curves cannot be rejected, whereas it must be rejected in the 7 remainingcases.

Bearing in mind that the estimate b of the slope in the double-logarithmic Engelcurve is also an estimate of the income elasticity of the expenditure on a given item,it seems to be a reasonable a priori hypothesis that fi varies from the higher to the lowersocial groups because these groups are at different income levels. If y is observed overa sufficiently wide income interval, the income elasticity, i.e. the relative increase inthe expenditure in proportion to a given relative increase in income, must be expectedto be decreasing with increasing income.

Actually it is rather strange that the double logarithmic function, according to whichthe income elasticity is assumed to be constant, should, as far as can be seen from thetests made, turn out to be the "best" of the five types of functions within each of thetwelve social groups, since each social group after all spans an income interval of severalthousand kroner. But if we go further and cover the whole scale from agriculturallabourers and unskilled workers to higher public servants and salaried employees, ahypothesis of constant income elasticity seems to be contrary to all sensible a prioriassumptions.

It is this property of which is utilized in forming the estimate s22.The F-tests appear from table (V,9).

Table V,9. F-test for parallelism of Engel curves.

Dwelling 1.32Fuel and light 4.22Food 3.46Tobacco 2.38Clothing 1.72Footwear 2.40Washing and cleaning 1.06Durables 3.72Personal hygiene 2.89Books, newspapers etc. 1.45Sports, holidays, etc 1.30Transport 6.09Union fees etc 1.89

F.95 1.89

84

Nevertheless, table V,9 shows, as mentioned, that for the six expenditure items:dwelling, clothing, washing and cleaning, books and newspapers, etc., sport and holidays, andfinally uni on fees, etc., such a hypothesis of common slope, i.e. constant income elasticity,cannot be rejected.

It will not be attempted here to explainor rather to explain awaythis phenomenon.As will be remembered, the purpose of the present inquiry has been laid down as anattempt to describe the observations of incomes and expenditures, since an attempt toexplain the consumption behaviour of the households must for the time being be con-sidered unduly ambitious10).

It should be mentioned, however, that the very rough grouping of the almost endlessnumber of goods and services into only 13 items is undoubtedly one of the decisivecauses of the stability found in the income elasticity between social groups. If, instead,sharply defined individual commodities and services had been considered, lounge suitsof a particular quality, flats of a given size and quality, etc. the income elasticity ofdemand for these goods and services would undoubtedly have been falling with risingdisposable income.

However, the six expenditure items with constant income elasticity are not suchheterogenous items in which have been included many different types of goods andservices. The items dwelling, washing and cleaning, clothing, and partly the item sportand holidays, etc. correspond to rather well defined parts of the budget of any household,and it is actually very interesting that the income elasticity for these items is so stableas shown by table V,9. The need for shelter, clothing, for entertainment, etc. naturallymakes itself felt at all income levels, but the interesting thing is that at any place in the in-come scale the same relative increase in the expenditure is produced by a given relative risein income. Not least in the expenditure on the item sport, holidays, etc. does the incomeelasticity seem to be remarkably constant at a very high level around 1.5. Of coursethe goods and services demandedrestaurants, theatres, cinemas, holiday trips, hobbiesand sportvary widely over the different income classes, social groups and age groups,but for all groups there seems to be a very long, still unfulfilled list of demand in thisfield.

For the seven items for which the hypothesis of the parallelism of the regression lineshad to be rejected table V, 10 confirms that as a general rule this is precisely due to thefact that the income elasticity is falling as the income level increases, cf. the items offood, tobacco and footwear. However, this is not the whole explanationindeed, intwo cases the explanation seems to be the opposite, namely that the income elasticityrises with the income, cf. the items offuel and lighting, and transport (incl, expenditure onmotor vehicles), where b in several cases rises as one moves from a lower social groupto a higher one within the same geographical area. In the case of these seven items thereis also another interesting phenomenon which emerges clearly, viz, the significantinfluence of the social grouping on the income elasticity. If, e.g., one takes the largeitem of food, table V, 10 shows that the wage-earning groups in the capital have a con-siderably higher income elasticity in their demand for food than the groups of salaried em-

'°) Cf. E. Jorgensen (12).

85

ployees, 0.69 and 0.70 respectively compared with 0.52 and 0.53. There is also a markeddifference in the level of b in the case offootwear as we move from wage-earners to salariedemployees.

The item of personal hygiene also exhibits significant differences in the b-values forthe two social groups, but here the groups of salaried employees are at the highestlevel. The breakdown into social groups, and particularly the distinction between salariedemployees and manual workers, thus seems to correspond to a real difference in be-haviour in the case of several important items of consumption. The geographical break-down, on the other hand, seems to be justifiable on the basis of existing differences inexpenditure behaviour only as far as the items of dwelling and fuel and lighting are con-cerned.

4. Are the Engel curves for djfferent social groups identical?

For six expenditure items the hypothesis of parallel regression lines for the twelve socialgroups could not be rejected. In these cases it was subsequently examined whether theseparallel curves could be considered as identical (test for identity). This identity test isperformed in two stages; at the first stage it is examined, by means of an ordinary F-testfor linearity, whether the twelve mean points (log xk, ak) can be considered as beingon a straight line-which, of course, they must if the test hypothesis is correct. If this

Table V, 10. Calculated income elasticities for 13 expenditure items for each of 12 groupsof wage and salary earners.

I. Higher public servants and salaried employees. 2. Lower public servants and salaried employees. 3. Skilled workers4. Unskilled workers. 5. Agricultural workers.

The capital Provincial towns Rural districts

2 3 4 2 3 4 2 3 4 5

Dwelling 0.96 0.93 0.87 0.94 0.80 0.86 0.75 0.70 0.84 0.93 0.98 0.96Fuel & light 0.73 0.50 0.40 0.31 0.57 0.44 0.28 0.43 0.55 0.32 0.64 0.76Food 0.52 0.53 0.69 0.70 0.55 0.64 0.60 0.62 0.58 0.65 0.67 0.65Tobacco 0.80 0.84 0.93 1.07 0.76 0.90 1.07 1.17 1.06 0.93 1.28 1.13Clothing 0.89 1.12 1.02 1.05 0.91 1.02 1.10 1.01 1.12 1.02 0.99 1.24Footwear 0.43 0.62 0.50 0.61 0.38 0.57 0.67 0.63 0.71 0.62 0.50 0.49Washing & cleaning 0.80 0.85 0.70 0.81 0.82 0.98 0.90 0.77 0.94 0.88 0.89 0.87Durables excl, vehicles 0.57 0.89 1.21 0.98 1.01 1.04 1.50 1.37 1.10 1.07 0.79 0.85Personal hygiene 0.81 1.00 0.68 0.83 0.73 0.87 0.77 0.97 0.95 0.81 0.87 0.73Books, newspapers etc. 1.02 0.95 0.77 1.05 1.02 1.11 0.86 1.04 0.93 0.86 0.91 1.10Sports, holidays, hobbies 1.39 1.37 1.51 1.46 1.46 1.56 1.68 1.54 1.59 1.61 1.57 1.43Transport incl, own car 0.86 1.09 1.56 1.34 1.68 1.39 1.58 1.62 1.53 1.96 1.66 0.92Union fees, subscriptions etc 0.63 0.66 0.90 1.10 0.76 0.79 0.80 0.91 0.82 0.94 0.94 0.88

3.81 3.75 3.71 3.67 3.75 3.66 3.58 3.53 3.49 3,49 3.47 3.36k5c

86

proves to be the case, it is finally examined, by means of a t-test, whether the slopeF of the line formed by the twelve mean points is identical with the weighted averageof the twelve individual slopes.

It turns out that none of the six expenditure items pass this test for identity. Theitem of sport, holidays, etc. passes the first stage of the test (concerning the linearity of thetwelve mean points), but shows significance for the second stage of the test; the otherfive items show significance already for the linearity test, see fig. V,4a and fig. V,4b,where the twelve mean points have been plotted for the items of books, newspapers,etc. and sport, holidays, etc. together with the twelve individual Engel functions.

Now, what does this result mean? The immediate interpretation is, of course, thatwe are here confronted with the "layer effect" described by Wolcl11) and commentedon page 25 above. The twelve social groups have the same income elasticity for the sixexpenditure items, but there is a difference in the level of the actual expenditure amongthe social groups. This may be due to differences in environment, in upbringing and inhabits of life, or it may be due to differences in the accessibility of the goods in question,for instance, owing to differences in distance from places where the goods are available.Thus restaurants, cinemas and other forms of entertainment are more easily availablein the towns, cf. the instance of this mentioned on page 26.

If this theory of the layer effect holds good, it may be concluded that the sub-division of the material into geographical andparticularlysocial groups correspondsto a real difference in consumption behaviour not only for the seven items with differentincome elasticities, but also for the six items where the twelve social groups could beconsidered as having the same income elasticity; in the case of the latter items the exist-ence of the layer effect should then be the explanation of the difference in the expenditurelevel from social group to social group'2).

If this interpretation of the results is accepted, the conclusion must be that by sub-division of the data into relevant groups it is possible to derive estimates of the income-expenditure relationships which are less biased than those relationships which could bederived for the total sample

If this subdivision had not been undertaken, the income elasticities for the two itemsof books, newspapers, etc. and sport, holidays, etc., to mention two examples, would havebeen considerably higher than the average of the elasticities of the twelve subgroups(viz. 1.28 against 0.98 and 2.00 against 1.50), cf. fig. V,4a and fig. V,4b, on page 87.

5. An important reservation.

As regards this interpretation of the results it is, however, necessary to make an importantreservation. All the results analysed so far are derived from linear regression analyses.In chapter III and IV the assumptions of the inquiry was discussed and it was foundthat this form of analysis was a suitable analytical tool if due regard is paid to problems

Cf. Wold (19), P. 68.Such layer effects may, of course, also be imagined to exist for the seven items for which theEngel curves of the individual social groups are not parallel.

1,4

log y

The capital xThe provincial towns oThe rural districts A

87

3,3 3,4 3,5 3,6 3,7 3,8 og x

Fig. V, 4a. Mean points for the twelve Engel curves for expenditures on books, newspapers etc. For each of the threeparts of the country the capital (X), the provincial towns (0) and the rural districts (e,,) the four Engel curves may be

considered identical. The average Engel curve for whole country deviates significantly.

og y

2,8

2,7

2,6

2,5

2,4

2.3

2,2

2,1

2,0

1,9

The capital x

The provincial towns oThe rural districts 4

3,3 3,4 3,5 3,6 3,7 3,8 log xFig. V, 4b. Mean points for the twelve Engel curves for expenditures on sports, holidays etc. In three cases only oneEngel curve may be considered identical to another one. The average Engel curve for the whole country deviates

significantly.

88

concerning zero observations and variance assumptions. However, it will probably beuseful to point Out that when we choose a given method, we also choose, to some extent,the results. If, for instance, the independent variable of the regression analysis, thedisposable income per person (or transformations hereof), cannot be considered fullyindependent in relation to the dependent variables, there will be bias in the estimates.

Now, considering again fig. V,4b, which shows the Engel curves of the twelve socialgroups with corresponding mean points for the expenditure item of sport, holidays,etc., it may very well be imagined that the line through the twelve mean pointsandnot the twelve single Engel curves deviating systematically from this mean point re-gression linerepresented the "true" relationship between log x and log y. It is worthnoting in this connection that the weighted average of the b for the twelve social groupswas appreciable lower than the estimate of the slope of the regression line through thetwelve mean points in the case of five out of the six expenditure items that had passedthe test for parallelism'3); i.e. displayed the same type of bias as may exist for sports,holidays etc.

On the whole, we are led to conclude that the immediate interpretation of the resultsof the calculations, namely that the subdivision into special groups seems to correspondto real differences in expenditure behaviour is upheld. This is so especially in the caseof the expenditure items where the Engel curves of the social groups showed significantdifferences both in slope and level; but it must be borne in mind that purely technicalfactorshere the choice of the regression technique as analytical toolcan exercisesome influence which would involve modifications of the conclusions drawn.

6. Conclzesions.

If the results for the double-logarithmic Engel function are to be summarized in a singletable, the average income elasticity can be shown for each expenditure item. It is truethat it was found above that these social groups differed systematically as regards theirexpenditures, but this applied primarily to the level of the calculated Engel curves andnot to their slope. In the case of the six items for which the calculated Engel curves couldbe considered parallel the calculation of such averages is natural, and although theelasticities differ significantly for the remaining seven items it is nevertheless charac-teristic that this difference between expenditure items within each social group is morepronounced than the difference between the individual social groups for the sameexpenditure item, so that also here it does make sense to calculate an average b-value.In table V, 11 have been shown the 13 average values for each of the 12 income elasticitiescalculated as a weighted average of the twelve b-values for each item, the weights beingthe sum of the squared deviations from the mean income, cf. above p. 82.

18) The item of union fees, etc. is influenced by the great expenditures of the wage-earning groups onunion fees, so that the line through the mean points of the twelve social groups for this item willhave less slope than the average of the slopes of the twelve individual lines.

Table V, 11. Average income elasticities for 13expenditure items.

I Fuel & light 0.512 Footwear 0.563 Food 0.614 Union fees, subscriptions etc. 0.825 Personal hygiene 0.866 Washing & cleaning 0.867 Dwelling 0.898 Books, newspapers etc. 0.989 Tobacco 0.98

10 Durables excl, vehicles 0.9911 Clothing 1.0412 Transport incl, own car 1.3913 Sports, holoidays, hobbies 1.50

89

In the table the b-values have been arranged by order of magnitude, and it will beseen how the slopes of the Engel curves, i.e. the income elasticities, of the 13 expenditureitems fall into three clearly distinguishable groups: 3 items which could be called neces-sities ranging from 0.51 to 0.61; 8 items which may be labelled neutral goods rangefrom 0,82 to 1.04; and 2 items belonging in the category of luxury goods with 1.39and 1,5014). Reverting to table V,l0, where all b-values have been shown, it will beseen that this ordering by size of average income elasticities comes very close to theordering by size within each of the 12 social groups, so that stability in the incomeelasticity of the main groups of expenditure items seems to be a general feature.

In the discussion of the usefulness of the Engel curve analysis in the description ofthe relationship between x and y, it was found that some expenditure items were not"explained" much more adequately by the inclusion of the disposable income as anexplanatory variablein particular the items of durable goods and transport (incl.transport by own motor vehicle). Now it is found that the estimated parameter valuesfor these items show wide fluctuations of a random nature from social group to socialgroup. The estimate b range from 0.86 to 1.96 for the item of transport and from 0.57to 1.50 for durable goods.

However, the remaining items seem to show such a high degree of stability that thecalculated average income elasticities should be of value also in a wider context, e.g.in the description of the expenditures of the whole population or groups of the populationon these items15).

1) Cf. E. Jørgensen (12).15) Cf. E. Jørgensen (12), where it has been attempted to utilize the results in such a wider context.

AverageExpenditure item income

elasticity

Chapter VI.

FURTHER ANALYSES.THE CONCEPT OF UNIT CONSUMERS, MULTIPLE

REGRESSION ANALYSES, ETC.

VIa. The unit-consumer concept.

The two variables of the Engel function the dependent variable y, i.e. the expenditureon a given item, and the explanatory variable x, i.e. the disposable income, have beendefined (p. 23, chapter III) as expenditure and income per person in the individualhouseholds. The argument in favour of adopting this definition of x and y was, of course,that differences in the size of households (number of persons per household) are re-flected very clearly in the consumption behaviour of the households. When x and yare measured as income and expenditure per person, most of the effects of this sourceof variation in the income-expenditure relationships will be eliminated, particularly inthe case of such main items as food, clothing and footwear depending primarily on thenumber of persons in the household. As far as items such as dwelling, durable goods,sports, holidays and hobbies are concerned, it may be doubtful how much the unex-plained part of the variation in y according to the Engel curve adopted will be reduced.

It is natural to raise the more general question: Is it possible to set up a model for theEngel curve in which x and y are specified in such a way that the effect from differencesnot alone in size of household but also in the type of persons will be eliminated? Or lessambitiously: Can x and y be specified in a way which provides a better approximationto this ideal than the specification used in this analysis (income and expenditure perperson)? This is the approach adopted by Prais and Houthakker in their attempt tocalculate unit-consumer scales separately for each expenditure item, where the scaleindicates, for each type of person, the weight at which the person in question is to beincluded in the specification of x and y for the individual households.

In a household consisting of a man, aged 47, a wife aged 43, a girl of il and a boyof 8, the specification of x and y according to the method adopted in this inquiry wouldsimply consist in dividing the total disposable income of the household and its totalexpenditure on the given commodity group by the number of persons in the household.The unit-consumer scale, as set up by Prais and Houthakker, indicates for each commod-ity or commodity group how these four persons are to be measured to arrive at the divisorwhich gives the desired specification of x and y. The idea is thus that a standard measure is

(VI, 2) y =k,.ni+k2.n2+...+kt.ntf(m)

(VI, 2) is a multiple regression equation, and by ordinary regression analysis k1, k2 . . ,ktmay be determined if y/f (m) is known. It will be seen from the above that the methodis based on ari assumption that the Engel function originally selected is the correct one,also after the observed values have been converted into consumption and income units,and also on an assumption that the effects from the "scale values" of the t types of per-sons enter linearly, since otherwise it will be extremely difficult to solve (VI, 1).

It is easy to point to defects in this approach: the requirement of a priori knowledgeof the "best" type of function and, be it noted, the best one after the consumption unitadjustment, the requirement that the contributions of the individual types of personsto the total income of a given household and total expenditure on a given commoditygroup enter linearly, etc. And it is difficult to see how these defects could be overcome2).But jf such scales could be determined, it would not only be possible to reduce the re-sidual variance, but also to have scales by which it would be possible to answer suchquestions as: How much extra expenditure for a household on a given item wouldbe caused by a ten-year-old boy? etc.

Cf. Houthakker, H. S. and Prais, J. S. (10), p. 133.Cf. Forsyth. (6).

91

introduced, so that for the household mentioned the value on the income scale may be e.g.1.9 units and on the scale for expenditure on food 3.2 units, etc., the unit chosen being theaverage income and average food expenditure, respectively, of one adult male. Prais andHouthakker suggest a method for calculating estimates of such scales'), which may bebriefly described as follows:

By means of the tests the best Engel function is selected. This function is now assumedto give the best description of the relationship between income and expenditure (on agiven commodity in a given population group).

It it is now imagined that all persons in a household are converted into income units(unit e.g. = average income for one married man between 30 and 40 years of age)and consumption units (average consumption for one married man between 30 and 40years of age), the Engel curve chosen wonid then apply to the relationship between in-come per income unit and consumption per consumption unit. If one distinguishes amongt types of persons, the following relationship will apply

(VI, 1) y ' x

k1. ni k01. ni

where Uj is the number of persons in the household of type i, k1 is this type of person'svalue on the consumption unit scale for the given expenditure item, and k01 this type

person's value on the income unit scale. Denotingx bym

E k01. flj

(VI, 1) can be formulated in the following way:

92

Such a tool would be highly relevant in the design of social welfare policy and taxation.But even if it may, on the face of it seem extremely interesting to obtain replies to such

very general questions as the one mentioned above, it seems that questions of this type aretoo general; they cannot be answered satisfactorily since the consumer scale values ofgiven types of persons depend very much on their income level and on the householdtype to which they belong. And if one tries to include these two factors in the consumerscale calculations, these calculations would, for one thing, become enormously compli-cated and for another the result of the calculations would be very difficult to interpret.

These considerations naturally lead up to an attempt to illustrate the influence of thehousehold type in another way. It seems evident that household type and income levelwill have to be taken into account if the objectiveachieving realistic descriptionsisto be fulfilled.

It seems as if this objective could be more satisfactorily fulfilled by calculating income-expenditure relationships separately for the different household types.

In that case one must abandon the idea of a general model describing the observa-tions. The method suggested is more primitive and moderate in its approach, but throughcareful comparison between the Engel curves of different household types, it will prob-ably be possible to achieve a more realistic description. This "method" of treating theinfluence of the household type corresponds completely to the method used in treatingthe residence and social status effect, i.e., separate calculations for each subgroup andcomparison of the results. However, the method presupposes that there is a sufficientnumber of observations for an adequate description of each of the subgroups to be given.In the present study the material has been divided into 12 groups taking into accountdifferences in residence and social status. A further breakdown of each of these 12 groupsinto a large number of subgroups according to household type would not leave a suffi-cient number of observations in each subgroup. It would therefore be necessary toabandon the original grouping and this in turn would necessitate a correction for theobserved differences among the 12 residential and social groups. Such comprehensivecorrection and regrouping have been outside the scope of the present study, and there-fore these calculations have not been made.

In order to get some idea of the influence of the household type, tables VI,la to VI,lnhave been set up. The tables show for the thirteen expenditure items included in the sur-vey and for savings the average expenditure per person for certain income bracketsfor all social groups as a whole, separately for different household types. In the calcu-lation of averages for the whole country of the expenditures of the 12 groups of wageand salary earners the shares of the individual groups in the total population of wageand salary earners have been used as weights. In brackets after each expenditure averagehas been shown the number of observations on the basis of which the average has beencalculated.

In interpreting the tables the weaknesses of such a tabular description must naturallybe borne in mind, confer the remarks on this subject in chapter V, p. 78.

Thus the number of observations in many of the cells of the table is so small that theaverages are subject to so great inaccuracy that their usefulness is rather limited.

Despite the weaknesses of the table, it is still possible to draw some important con-

Table VI, lb. Average expenditure per person on fuel and light in certain income groupsseparately for different types of household.

93

Table VI, la. Average expenditure per person on dwelling in certain income groupsseparately for different types of household.

Type of household

Disposable income per penon

o-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

I Singleman 5(1) 123(5) 200(52) 197(51) 215(68) 270(49) 612(10)2 Single woman 423 (3) 162 (25) 273 (42) 295 (99) 359 (69) 469 (44) 548 (8)3 Couples without

children 235 (4) 289 (84) 347 (234) 340 (172) 384 (99) 456 (45) 530 (2)4 Couples with 1 child 207 (9) 223 (314) 246 (251) 279 (62) 310 (7) 471 (4)5 Coupleswith2children 170 (43) 187 (438) 222 (144) 279 (16) 237 (3) 220 (2)6 Couples with 3 children 145 (58) 171 (146) 190 (34) 185 (3) 214 (1)7 Couples with 4 or more

children 121 (72) 148(40)8 Single man with I or

more children 93 (2) 207 (2) 188 (5) 432 (3) 278 (2) 313 (1)9 Single woman with I

or more children. . 160 (7) 233 (36) 259 (33) 343 (71) 243 (3) 630 (1)10 Other types 132 (18) 188 (88) 281 (49) 325 (12) 508 (5) 634 (1)

Type of household

Disposable income per person

o-1,999

2,000-3,999

4,000-5,999

6,000-7.999

8,000-9,999

10,000-14,999

15,000and above

i Single man 360 (1) 290 509 (6) 560 (51) 820 (69) 950 (49) 1715 (10)2 Single woman 361 (4) 336 (25) 645 (42) 734 (99) 961 (69) 1129 (44) 1410 (8)3 Couples without

children 203 (4) 319 (84) 431 (233) 564 (172) 754 (99) 1042 (45) 1075 (12)4 Couples with 1 child 143 (9) 297 (314) 431 (251) 518 (61) 685 (7) 1176(4)5 Couples with 2 children 172 (43) 261 (438) 386 (144) 534 (16) 737 (3) 305 (2)6 Coupleswith3children 124(58) 234 (146) 356 (34) 428 (3) 278(11)7 Couples with 4 or more

children 116(72) 199(40)8 Single man with 1 or

more children 216 (2) 109 (2) 356 (5) 690 (3) 525 (2) 160 (1)9 Single woman with 1

or more children. . 195 (7) 343 (36) 420 (33) 440 (7) 515 (3) 1364 (1)10 Other types 150 (18) 233 (88) 415 (49) 434 (12) 1076 (5) 918 (1)

94

Table VI, Ic. Average expenditure per person on food in certain income groups separatelyfor different types of household.

Table VI,ld. Average expenditure per person on tobacco in certain income groupsseparately for different types of household.

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

1 Single man 91(1) 362 (7) 414 (52) 472 (50) 646 (68) 802 (49) 672 (10)2 Single woman 228 (8) 98 (25) 220 (42) 335 (99) 315 (68) 300 (44) 319 (8)3 Couples without

children 116 (4) 188 (84) 272 (234) 405 (172) 411 (99) 452 (44) 357 (1)4 Couples with 1 child 148 (9) 173 (314) 258 (251) 326 (62) 286 (7) 298 (4)5 Coupleswith2 children 78(43) 150 (438) 212 (144) 263 (16) 311 (3) 264 (2)6 Couples with 3 children 87 (58) 137 (146) 206 (34) 405 (3) 364 (1)7 Couples with 4 or more

children 64(71) 144(40)8 Single man with 1 or

more children 28(2) 431 (2) 254 (4) 24(2) 453 (2) 430 (1)9 Single woman with 1

or more children. 18(7) 121 (36) 195 (32) 299 (7) 263 (3) 314 (1)10 Other types 64 (18) 131 (88) 253 (49) 440 (12) 471 (5) 900 (1)

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

i Single man 720(6) 1392(52) 1944(51) 2136(69) 2515(50) 2310(10)2 Single woman 647 (5) 674 (25) 1201 (42) 1687 (99) 1750 (69) 1882 (44) 2027 (8)3 Couples without

children 1056 (4) 1208 (84) 1537 (234)1711 (172)1894 (99) 2053 (45) 1743 (2)4 Couples with I child 799 (9) 1050 (324) 1272 (251)1502 (67) 1588 (7) 1455 (4)5 Couples with 2 children 722 (43) 940(438)1122 (144)1322 (16) 1214(3) 1560(2)6 Couples with 3 children 672(58) 896(146)1156(34) 1518(3) 1174(1)7 Couples with 4 or more

children 599 (72) 849 (40)8 Single man with 1 or

more children 839(2) 1048(2) 1622(5) 2014(3) 2039(2) 2100(1)9 Single woman with 1

or more children. 631 (7) 997 (36) 1368 (33) 1773 (7) 1366 (3) 1724(1)10 Other types 723 (18) 919 (88) 1292 (49) 1585 (12) 1353 (5) 2840 (1)

95

Table VI,le. Average expenditure per person on clothing in certain income groupsseparately for different types of household.

Table VI,lf. Average expenditure per person on footwear in certain income groupsseparately for different types of household.

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

1 Single man 131 (1) 453 (7) 433(52) 530 (51) 657 (69) 1005 (50) 1267 (10)2 Single woman 211 (11) 457 (25) 530 (42) 760 (99) 959 (69) 1152 (44) 1240 (8)3 Couples without

children 274(4) 215 (84) 403 (234) 572 (172) 753 (99) 937 (45) 2489 (2)4 Couples with 1 child 80 (9) 258 (314) 437 (251) 569 (62) 863 (7) 926 (4)

5 Couples with 2 children 136 (43) 259 (438) 464 (144) 827 (16) 592 (3) 336 (2)

6 Couples with 3 children 171 (58) 244 (146) 446 (34) 581 (3) 1064 (1)

7 Couples with 4 or morechildren 124 (72) 297 (40)

8 Single man with 1 ormore children 94(2) 355 (2) 411 (5) 574 (3) 1167 (2) 680 (1)

9 Single woman with 1 ormore children 103 (7) 322 (36) 473 (33) 539 (7) 1052 (3) 636 (1)

10 Other types 123 (18) 251 (88) 359 (49) 578 (12) 551 (5) 971 (1)

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

1 Singleman 35(1) 81(7) 87(52) 118(51) 125(69) 151(50) 144(10)2 Single woman 85 (7) 117 (25) 125 (42) 159 (99) 184 (69) 193 (44) 202 (8)3 Couples without

children 38(4) 48(84) 87(234) 111 (172) 136 (99) 156(45) 259 (2)4 Couples with 1 child. 36 (9) 68 (314) 97 (251) 127 (62) 109 (7) 104(4)5 Coupleswith2 children 43(43) 74(438) 107 (144) 165 (16) 144 (3) 111 (2)6 Couples with 3 children 52 (58) 73 (146) 118 (34) 104 (3) 110 (1)7 Couples with 4 or more

children 41(72) 76 (40)8 Single man with 1 or

more children 52 (2) 94(2) 81(5) 158 (3) 227 (2) 136 (1)9 Single woman with 1

ormorechildren.. 58(7) 104(36) 133(33) 117(7) 167(3) 175(1)10 Other types 41(18) 60(88) 78(49) 115(12) 118(5) 178(1)

96

Table VI, Ig. Average expenditure per person on washing and cleaning in certainincome groups separately for different types of household.

Table VI,lh. Average expenditure per person on durables excl, vehicles in certainincome groups separately for different types of household.

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

1 Single man 53(7) 75(48) 219 (51) 268 (67) 548 (50) 1179 (10)2 Single woman 22 (9) 99 (25) 335 (42) 406 (99) 682 (69) 653 (44) 344 (8)3 Couples without

children 82 (4) 140 (84) 343 (234) 599 (172) 511 (99) 592 (45) 504 (2)4 Couples with 1 child 54(9) 193 (314) 306 (251) 320 (62) 398 (17) 905 (4)5 Couples with 2 children 74(43) 167 (438) 296 (144) 501 (16) 130 (3) 622 (2)6 Couples with 3 children 91(58) 131 (146) 262 (34) 196 (3) 438 (1)7 Couples with 4 or more

children 66 (72) 150 (40)8 Single man with 1 or

more children 32 (2) 80 (2) 138 (5) 244 (3) 502 (2) 222 (i)9 Single woman with 1 or

more children 123 (7) 99 (36) 300 (33) 291 (7) 305 (3) 428 (1)10 Other types 39 (18) 147 (88) 281 (49) 244 (12) 269 (5) 329 (1)

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

1 Single man (1) 107 (7) 150 (52) 180 (51) 213 (69) 297 (48) 247 (10)2 Single woman 24 (6) 38 (25) 92 (42) 147 (99) 165 (69) 199 (44) 207 (8)3 Couples without

children 44(4) 60(84) 88(234) 115 (172) 154 (99) 199 (45) 176 (2)4 Couples with i child 40(9) 63(314) 96(251) 114 (62) 84(7) 103 (4)5 Couples with 2 children 34 (43) 59 (438) 81(144) 120 (16) 99 (3) 81(2)6 Couples with 3 children 36 (58) 53 (146) 79 (34) 41(3) 47 (1)7 Couples with 4 or more

children 32 (72) 49 (40)8 Single man with 1 or

more children 31(2) 86 (2) 133 (5) 123 (3) 135 (2) 65 (1)9 Single woman with i or

more children 59 (7) 74 (35) 101 (33) 124 (7) 95 (3) 256 (1)10 Other types 29 (18) 53 (88) 80 (49) 150 (12) 176 (5) 89 (1)

97

Table VI,li. Average expenditure per person on personal hygiene in certain incomegroups separately for different types of household.

Table VI,lj. Average expenditure per person on books, newspapers, etc. in certainincome groups separately for different types of household.

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

I Single man 113 (1) 105 (7) 129 (52) 159 (51) 162 (69) 206 (48) 219 (16)2 Single woman 99 (6) 146 (25) 185 (42) 302 (99) 296 (69) 320 (44) 413 (8)3 Couples without

children 27(4) 64(84) 117 (234) 170 (172) 214 (99) 243 (45) 142 (2)4 Couples with i child 37(9) 83(314) 124 (251) 155 (62) 121 (7) 169 (4)5 Couples with 2 children 42 (43) 79(438) 122 (144) 164 (16) 244 (3) 120 (2)6 Couples with 3 children 43 (58) 70(146) 114 (34) 75 (3) 173 (1)7 Couples with 4 or more

children 32 (72) 66 (40)8 Single man with 1 or

more children 27 (2) 71(2) 107 (5) 131 (3) 196 (2) 196 (1)9 Single woman with i or

more children 46 (7) 99 (35) 148 (33) 186 (7) 306 (3) 363 (1)10 Othertypes 33(18) 66(88) 120(49) 153(12) 133(5) 207(1)

Type of household

Disposable income per person

Q-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

1 Single man (1) 63 (7) 85 (52) 112 (51) 148 (69) 233 (48) 280 (10)2 Single woman 53 (6) 70 (25) 87 (42) 136 (99) 164 (69) 220 (44) 334 (8)3 Couples without

children 40(4) 55(84) 84(234) 111 (172) 142 (99) 160 (45) 71(2)4 Couples with 1 child 24(9) 46(314) 76(251) 90(62) 120 (7) 97(4)5 Couples with 2 children 25 (43) 45 (438) 71(144) 86 (16) 63 (3) 56 (2)6 Couples with 3 children 24(58) 44 (146) 58 (34) 55 (3) 32 (1)7 Couples with 4 or more

children 18 (72) 44 (40)8 Single man with 1 or

more children 20 (2) 65 (2) 52 (5) 79 (3) 60 (2) 273 (1)9 Single woman with i or

more children 18 (7) 41(35) 81(32) 130 (7) 248 (3) 297 (1)10 Other types 24(18) 40(88) 74(49) 83(12) 119(15) 240(1)

98

Table VI,lk. Average expenditure per person on sports, holidays, hobbies in certainincome groups separately for different types of household.

Table VI,ll. Average expenditure per person on transport incl, own car in certainincome groups separately for different types of household.

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000.-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

I Single man 104(1) 199 (7) 324 (52) 889 (51) 762 (69) 1116 (50) 897 (10)2 Single woman 61(10) 126 (25) 219 (42) 379 (99) 432 (69) 518 (44) 520 (8)3 Couples without

children 42 (4) 105 (84) 227 (234) 429 (172) 750 (99) 1446 (45) 674 (2)4 Couples with 1 child 44 (9) 157 (314) 359 (251) 490 (62) 542 (7) 1609 (4)5 Coupleswith2 children 48 (43) 137 (438) 359 (144) 659 (16) 1359 (3) 3406 (2)6 Couples with 3 children 40 (58) 128 (146) 407 (34) 392 (3) 212 (1)7 Couples with 4 or more

children 54 (72) 206 (40)8 Single man with i or

more children 171 (2) 19 (2) 319 (4) 940 (3) 71(2) 992 (1)9 Single woman with 1 or

more children 50 (7) 86 (36) 144 (33) 297 (7) 336 (3) 264 (1)10 Other types 64 (18) 164 (88) 247 (49) 866 (12) 161 (5) 16 (1)

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

i Single man 851 (1) 305 (7) 574 (52) 730 (51) 1122 (69) 1430 (50) 2774 (10)2 Single women 74(11) 172 (25) 311 (42) 602 (99) 880 (69) 1463 (44) 2061 (8)3 Couples without

children 36 (4) 123 (84) 330 (234) 596 (172) 839 (99) 1288 (45) 1969 (2)4 Couples with i child 52 (9) 189 (314) 360 (251) 741 (62) 1189 (7) 940 (4)5 Couples with 2 children 54(43) 177 (438) 468 (144) 826 (16) 846 (3) 1068 (2)6 Couples with 3 children 62 (58) 165 (146) 397 (34) 760 (3) 1430 (1)7 Couples with 4 or more

children 47 (72) 204 (40)8 Single man with 1 or

more children 33 (2) 285 (2) 367 (4) 399 (3) 547 (2) 1303 (1)9 Single woman with I or

more children 64 (7) 153 (36) 421 (33) 646 (7) 955 (3) 987 (1)10 Other types 66 (18) 169 (88) 352 (49) 705 (12) 571 (5) 1266 (1)

Table VI, ln. Average »expenditure« per person on savings in certain income groupsseparately for different types of household.

99

Table VI, 1m. Average expenditure per person on union fees, subscriptions etc. in certainincome groups separately for different types of household.

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

1 Single man (1) 106 (7) 234 (52) 365 (51) 345 (69) 384 (48) 248 (10)2 Single woman 87 (5) 105 (25) 197 (42) 220 (99) 197 (69) 227 (44) 194 (8)3 Couples without

children 209 (4) 189 (84) 247 (234) 262 (172) 230 (99) 208 (45) 85 (2)4 Couples with 1 child 101 (9) 150 (314) 155 (251) 179 (62) 193 (7) 119 (4)5 Coupleswith2 children 104 (43) 125 (438) 131 (144) 94(16) 132 (3) 187 (2)6 Coupleswith3children 102 (58) 99(146) 117 (34) 164(3) 121 (1)7 Couples with 4 or more

children 67 (72) 87 (40)8 Single man with 1 or

more children 92(2) 114 (2) 134 (2) 197 (3) 292 (2) 467 (1)9 Single woman with 1 or

more children 44(7) 83(35) 111 (32) 238 (7) 168 (3) 124(1)10 Other types 65(18) 107 (88) 146 (49) 179 (12) 112 (5) 78(6)

Type of household

Disposable income per person

0-1,999

2,000-3,999

4,000-5,999

6,000-7,999

8,000-9,999

10,000-14,999

15,000and above

1 Single man 2 (1) 10 (6) 43 (52) 54 (50) 43 (68) 29 (50) 158 (10)2 Single woman 30 (9) 40 (25) 43 (39) 30 (97) 49 (69) 95 (44) 407 (8)3 Couples without

children 52 (4) 25 (84) 34 (234) 29 (172) 72 (99) 77 (45) 181 (2)4 Couples with I child 6 (7) 7 (314) 13 (251) 20 (62) 62 (7) 31(4)5 Couples with2 children 4(38) 4(438) 14 (144) 85(16) 4(3) 47(2)6 Couples with 3 children 2 (14) 12 (34) 40 (3) 1(1)7 Couples with 4 or more

children 1(49) 3 (29)8 Single man with 1 or

11 (2) 1(1) 6(3) 20(2) 10(2) 18(1)more children9 Single woman with 1 or

more children 6 (5) 18 (26) 11(32) 1(6) 25 (3) 170 (1)10 Othertypes 16(10) 9(75) 21(49) 54(12) 197(5) 252(1)

100

clusions from the averages shown. In general it may be said that even if the conversionof all expenditure and income figures from amounts per household into amounts perperson has undoubtedly eliminated also a substantial part of influence of the house-hold type, there still remain systematic effects on the income-expenditure relationshiparising from differences in household type among the observed households. This con-clusion leads to the observation that the existing differences between the residentialand social groups may to some extent be caused by systematic differences in householdtypes between these groups.

Figures VI, 1 and VI,2 show income and expenditure on the items of food and of sports,holidays and hobbies, the most typical necessity item and luxury item, respectively ac-cording to the figures in the tables. For the necessity item the diagram shows that theexpenditure per person for given income per person falls appreciably with the numberof persons (children) in the household. With a given income (per person) the largehouseholds spend a smaller share of their income on food than households with fewpersons. The expenditure items of dwelling and tobacco show the same picture. How-ever, in the case of sports, holidays and hobbies the figure shows an equally appreciableshift in the opposite direction: for a given income per person the expenditure (per per-son) rises with the size of household. The items of clothing and transport present thesame picture, if not quite so markedly.

The conclusion of these results compared with the results of the residential and socialgrouping effect referred to in chapter V above is that future analyses of consumptionsurvey data should presumably place more emphasis on the household type effect and lesson differences in residential and social grouping.

If in the present survey the observations had been divided into only four groups byresidence and social grouping (one group of salary earners and one of wage earners inthe two areas Copenhagen and the rest of Denmark) it should have been possible toundertake Engel curve analyses separately for three household types within each ofthese four groups of wage and salary earners and have the same number of observationsavailable in each Engel curve analysis as in the present survey. Judging from the resultsshown above it seems that such a breakdown of the observations would have givengreater homogeneity in the individual subgroups, and thus presumably more stableEngel curves (i.e. less residual variation) and consequently a more precise descriptionof the observed income-expenditure relationships.

VI.b. Multiple Regression Analyses.

As the main tool in the description of the expenditure behaviour of households of wageand salary earners in this analysis the Engel function has been chosen, in which thedisposable income of the household has been included as the only explanatory variable.A decisive reason for this choice was, of course, that for several purposes it is of interestto know possible expenditure reactions to changes in disposable income.

The object of the analysis has thus been shifted slightly away from the general oneof providing a description of consumption behaviour towards an attempt to show theeffect of income on expenditure. Accordingly it has been attempted to isolate this effect

1800

1700

i 1600

1500

140(5

1300

1200

1100

1000

900

800

700

600

500

2000 4000 6000 8000 10,000 Ineorre per personDan(sh Kroner

Fig. VI, 1. Income and expenditure on sport, holidays and hobbiesper person for four different types of households.

10.000 Innorre per perron.Danish Kroner

Fig. VI, 2. Income and expenditure on food per personfor four different types of households.

0 2000 4000 6000 8000

101

1000 -

O 900-o

800

700 -

- 600-

500-

400 -

300 -

200

100 -

102

by careful specification of the two variables and by a subdivision into residential andsocial grouping.

However, if in an attempt to arrive at a good description of the expenditure behaviourof the households one does not wish to be constrained by the considerations that ledto placing the main emphasis on the influence of income and thereby to the preoccu-pation with the Engel curve, it seems natural to try to make the description more com-plete by introducing more explanatory variables in the expenditure functions in addi-tion to disposable income.

Among such variables which may be imagined to influence the expenditures of thehouseholds on the different expenditure items there are a great many about which thebasic material gives no information. There are the prices of the commodities, but alsopast or expected changes in these prices as well as variables reflecting plans or expec-tions in the households.

However, the observations do contain additional information which might be in-cluded in the description of household expenditures on the different commodity groups;presumably the residual variation of y could be reduced thereby. The information re-ferred to concerns expenditures on one or more other commodity groups, the personalwealth of the households, their saving during the survey period and finally the income changesexperienced by the households during the preceding two-year period. Information onthe year of establishment of the households should be valuable as a supplementary variablein the case of certain commodity groups (dwelling, durable goods).

Appendix C gives data on the expenditures of the individual households on all the13 items, so that in the analysis of one expenditure item it would be very simple to includeinformation on the expenditure of the individual households on the other items; theappendix moreover contains information which makes it possible to include all the otherexplanatory variables suggested above.

The present chapter only takes up the expenditure on one of the other commoditygroups for further discussion. This is not because it is thought that this supplementaryinformation necessarily gives the greatest contribution to reducing the residual variationof y, but becauseas a byproduct of the main analysisit is possible to ascertain foreach expenditure item y1 the expenditure item yt which will give the greatest effect ifadded as supplementary explanatory variable.

The more comprehensive analysis introducing one or more of the data shown in theappendix (income changes, personal wealth, etc.) with a view to providing a moreadequate description or explanation of the expenditure behaviour of the households,therefore still remains. In this connection it should be mentioned that in a separate studyof the saving pattern of households it was found that particularly income changes seemedto have great effect3). Households with rising incomes recorded considerably highersavings than households who had experienced a fall in income. Moreover, differencesin personal wealth were reflected in significant differences in saving (for given incomelevel), households whose personal wealth was around zero saving less than householdswith considerable psoitive or negative wealth4). It is, perhaps, not unreasonable toexpect that these variables would also have an effect in the explanation of the expendi-ture on some of the commodity groups (particularly durable goods).

103

In choosing an expenditure item, y, as supplementary explanatory variable for theexpenditure yj, the criterion for selecting Yh must be some expression of the correlationbetween yj and y. However, this expression must naturally be adjusted for the in-fluence of income, on both variables. The problem is this: If a household with a givenincome, residence, social grouping, household type, etc., has a higher expenditure onitem i than expected according to the "best" Engel function, can this be ascribed to anyappreciable extent to this household's expenditure on item h?

It is here necessary to point out that every expenditure item yj can be described or"explained" exhaustively by means of the income and all other m - I expenditureitems including savings on the basis of the budget relation

(VI, 3) = i1 y

It is obvious that a given household's expenditure on commodity group No. i is uni-quely determined if the disposable income and all other uses of that income are in-cluded in the explanation. There will then be so many constraints on the variablesthat there simply are no more degrees of freedom left. According to the budget relation(VI, 3) the following identity exists

(VI, 4) yj=x h'1

Thepositive or negativecorrelation between the residuals in the relations de-scribing household expenditures on items i and h will determine whether it will beuseful to include yh in the description of yj. It is evident that such a correlation willbe most marked in the case of commodities which are close substitutes or complementsin the consumption of the households. Abnormally high consumption of butter willthus, probably, occur at the same time as abnormally low consumption of margarine,high values for expenditure on petrol will often occur together with high values forexpenditure on purchases of motor cars, etc.

In the breakdown of expenditure items which has been undertaken in the presentanalysis it has been attempted to place commodities which, in the opinion of thehousehold are closely related to one another in the same group, cf. chapter IV, p. 38;the 13 expenditure items represent as far as possible 13 "unrelated" commodity cate-gories, and the mentionedpositive or negativecorrelation cannot therefore be ex-pected to have any high value.

If now the best of the Engel functions studied is taken and for each of the expenditureitems in question are calculated the residuals for the individual households from theexpected expenditure, a table of the correlation between these deviations for each item canbe set up. Such a correlation table for the group of skilled workers in the capital has beengiven in table VI,2. Inthe appendix will be found such tables for all twelve groups of wage

Opsparing i lønmodtagerhusstandene 1955, Statistiske Undersøgelser, No. 3, Copenhagen 1960, p. 31.Same as above, pp. 31-32.

Table VI,2. Correlation between each two of thirteen expenditure items.

Skilled workers. The Capital.

1. Dwelling. 2. Fuel and light. 3. Food. 4. Tobacco. 5. Clothing. 6. Footwear. 7. Washing and cleaning. 8. Durable goods. 9. Personal hygiene. 10. Books, news-papers etc. 11. Sports, holidays, etc. 12. Transportation. 13. Union fees and subscriptions.

1 2 3 4 5 6 7 8 9 10 11 12 13

- 0.369 -0.117 0.003 -0.324 -0.283 -0.046 0.054 -0.047 0.155 0.025 0.017 0.1622 - - 0.110 -0.057 -0.185 -0.146 -0.126 0.015 -0.038 0.205 -0.066 -0.124 0.2253 - - - -0.017 -0.193 -0.037 0.267 -0.165 0.065 0.079 0.048 -0.340 0.1974 - - - - -0.127 -0.089 0.244 0.085 0.102 0.034 0.171 -0.185 -0.0315 - - - - - 0.480 -0.176 0.112 0.265 0.061 0.110 0.010 -0.2076 - - - - - - -0.132 0.056 0.120 -0.077 0.074 -0.077 -0.1227 - - - - - - - -0.055 0.018 0.099 0.204 -0.035 0.0868 - - - - - - - - 0.076 0.152 0.109 -0.182 -0.0399 - - - - - - - - - 0.173 0.116 0.018 -0.055

10 - - - - - - - - - - 0.165 -0.110 0.064il - - - - - - - - - - - -0.219 -0.22712 - - - - - - - - - - - --0.09713 - - - - - - - - - - - - -

Unskilled workers. The Capital.

1 2 3 4 5 6 7 8 9 10 11 12 13

1 - 0.336 -0.097 0.024 0.161 0.122 0.151 0.067 0.169 0.074 0.034 -0.167 -0.0162 - - -0.166 -0.034 -0.171 -0.123 -0.073 0.133 0.079 -0.031 -0.049 0.020 0.1273 - - - 0.133 0.025 -0.086 0.275 -0.143 0.014 0.114 0.049 -0.032 0.1864 - - - - -0.100 -0.030 0.186 -0.104 0.025 0.069 0.024 0.041 0.1295 - - - - - 0.532 0.172 0.196 0.338 0.126 0.182 -0.187 0.0346 - - - - - - 0.265 0.099 0.300 0.053 0.092 -0.032 -0.0667 - - - - - - - -0.034 0.121 0.180 -0.049 -0.026 -0.0298 - - - - - - - - 0.111 -0.023 -0.053 -0.161 0.0409 - - - - - - - - - 0.110 0.083 -0.012 0.114

10 - - - - - - - - - - 0.102 -0.171 0.19011 - - - - - - - - - - - -0.164 0.18612 - - - - - - - - - - - ---0.01613 - - - - - - - - - - - - -

andN

SSDh= E(Yh,Jyh,J)2

r= S P D,, h

s s D1. S S Db

N N

where E (Y1, j - y', j) E (Yh, j - yb, j)N

j jSPDI,b=E(Y,,jyj,j)(Yh,jyb,J)N

and2

IN t

N (E(Yj,j_yj,j))/

N'N 2

(E (Yh, j - Yb,'jN

105

and salary earners, cf. appendix B, p. 174. Table VI,2 shows for each of the 13 expendi-ture items the correlation between the deviations of the individual households from theexpected expenditure on this item and the deviation from the expected expenditure oneach of the other 12 expenditure itemswhere the expected expenditure has been cal-culated according to the double logarithmic Engel function.

An estimate of the correlation between the deviations of the observed values of twoexpenditure items from the calculated values according to the double logarithmic func-tion should be based on the logarithmically transformed expenditures. However, since,i.a. for technical reasons, correlations had to be computed on the basis of the individualhouseholds and not on the basis of the deviations of expenditures of groups of households,the zero observation problem again arose, cf. p. 36 above, and it was therefore decidedto base the calculations on the deviations of the untransformed, observed expendituresfrom the antilogarithm of the calculated value f (x) = a + b (log x - log x). Forexpenditure items No. i and h the estimate of the correlation coefficient, r, is accordinglyobtained in the following way

where Y1, j = antilog f1 (xj) and Yb, j = antilog fh (xj)

Since it is expenditures rather than logarithms of expenditures which are used, theexpenditure deviations will, in the case of the high income brackets, be included withtoo much weight a consequence of which is a certain bias. The primary aim in computingthe correlations has been to find the sets of expenditure items which were "closest" or"farthest", and not to measure how close or how far. In other words the computationsoffer a qualitative rather than a quantitative impression; it has not, accordingly, beenattempted to introduce any correction for the bias in the correlation coefficients andthe tables must therefore be read with this reservation in mind.

It must also be borne in mind that the budget relation (VI, 3) brings about a slightnegative correlation between the expenditure residuals in the equations explaining the in-

106

Table VI,3. The biggest positive and negative correlationcoefficiertt

dividual items. A household with au expenditure considerably above the expected ex-penditure ou au item which weighs heavily in the budget must necessarily display ne-gative deviations from the expected expenditure on one or more of the other items.However, this cannot be seen from the correlation tables, partly because the incomeconcept used, disposable income, is not precisely equal to the sum of the 13 expenditureitems studied, as some small items (domestic help, gifts and charity, etc.) are left out,which is also the case for the part of the disposable income used for saving, and partlybecause of the above-mentioned bias in the correlation estimates.

Only very few of these correlation estimates exceed a level as moderate as 0.5, whichis some sort of a confirmation of the impression that the commodity classification hasbeen reasonable, cf. above. Even if very high correlation values do not occur, there isstill, in the case of some items, a considerable correlation between the deviations of thehouseholds from the expected expenditure behaviour. Consequently the description ofthe expenditure behaviour of the households with regard to these commodity groupswill evidently become more satisfactory (resulting in a lower residual variance of the

Expenditure item

Group of wage

2 3 4

1 Dwelling positive 2 0.309 2 0.395 2 0.369 2 0.336negative 13 -0.100 4 -0.055 5 -0.324 12 -0.167

2 Fuel and light positive 1 0.309 1 0.395 1 0.369 1 0.336negative 11-0.385 11 -0.144 5-0.185 5-0.171

3 Food positive 7 0.365 4 0.262 7 0.267 7 0.275negative 8 -0.045 12 -0.261 12 -0.340 2 -0.166

4 Tobacco positive 3 0.288 3 0.262 7 0.244 7 0.186negative 13 -0.156 6 -0.064 12 -0.185 8 -0.104

5 Clothing positive 6 0.578 6 0.469 6 0.480 5 0.532negative 2 -0.240 2 -0.125 1 -0.324 12 -0.187

6 Footwear positive 5 0.578 5 0.469 5 0.480 5 0.532negative 12 -0.112 2 -0.075 1 -0.283 2 -0.132

7 Washing and cleaning . positive 3 0.365 9 0.236 3 0.267 3 0.275negative 12- 0.027 8 -0.043 5 -0.176 2 -0.073

8 Durable goods positive 1 0.176 6 0.148 10 0.152 5 0.196negative 10 -0.139 12 -0.054 12 -0.182 12 -0.161

9 Personal hygiene positive 5 0.485 5 0.365 5 0.265 5 0.338negative 2 -0.145 2 -0.139 13 -0.055 12 -0.012

10 Books, newspapers etc. positive 9 0.381 13 0.347 2 0.205 13 0.190negative 2-0.181 12 -0.116 12 -0.110 12 -0.171

11 Sports, holidays etc positive 5 0.359 4 0.102 7 0.204 13 0.186negative 2 -0.385 10-0.116 13 -0.227 12 -0.167

12 Transportation positive 13 0.065 4 0.102 9 0.018 4 0.041negative 6-0.114 10-0.116 3-0.340 5-0.187

13 Union fees and subscriptions positive 11 0.141 10 0.347 2 0.225 10 0.190negative 2 -0.156 6-0.006 11 -0.227 6-0.066

separately for each group of wage and salary earners.

and salary earners*)

*) 1. Higher public servants and s3. Skilled workers, the Capital; 4towns; 6. Lower public servantsworkers, provincical towns; 9. Lil. Unskilled workers, rural districts

107

alaried employees, the Capital; 2. Lower public servants and salaried employees, the Capital;Unskilled workers, the Capital; 5. Higher public servants and salaried employees, the provincical

and salaried employees, provincical towns; 7. Skilled workers, provincical towns; 8. Unskilledower public servants and salaried employees, rural districts; 10. Skilled workers, rural districts

12. Farm workers, rural districts.

dependent variable y) if the expenditure on the commodity group for which the tableshows the highest positive or negative correlation is introduced as an explanatory vari-able in addition to disposable income.

Table VI, 3 shows for each expenditure item the two other items displaying the great-est positive or negative correlation coefficient. Only in the case of three items would thesame supplementary determining variable be chosen in all twelve groups of wage andsalary earners if this table were used as the criterion. The items of dwelling and fuel &lighting are so closely related that in the description of one of them the other wouldeverywhere be included as the supplementary variable yi. This is also true, except forone group of wage and salary earners, for the items of clothing and footwear. In the caseof both sets of expenditures there is positive correlation. For the remaining items thepicture changes from one group of wage and salary earners to the other, and especially

5 6 7 8 9 10 11 12

2 0.294 2 0.250 2 0.278 2 0.292 2 0.183 2 0.341 2 0.191 2 0.49112 -0.122 11 -0.160 4-0.302 11 -0.335 8 -0.199 6 -0.205 3-0.174 11 -0.233

1 0.294 1 0.250 1 0.278 1 0.292 1 0.183 1 0.341 10 0.296 1 0.4915-0.192 9 -0.252 11 -0.274 12 -0.288 11 -0.263 11 -0.224 11-0.447 11-0.2282 0.338 13 0.159 13 0.288 13 0.260 11 0.174 13 0.214 7 0.165 10 0.2205 -0.235 8-0.132 8 -0.236 5 -0.388 13 -0.172 8 -0.327 1 -0.174 1 -0.1909 0.368 7 0.204 7 0.293 10 0.211 11 0.315 2 0.254 9 0.228 7 0.2208 -0.095 8 -0.094 1 -0.302 8 -0.135 2 -0.099 8 -0.234 2 -0.253 8-0.1249 0.507 6 0.510 6 0.596 6 0.590 6 0.428 6 0.496 6 0.447 6 0.4583 -0.235 2 -0.252 12 -0.396 3 -0.388 2 -0.222 8 -0.292 13 -0.230 12 -0.1955 0.469 5 0.510 5 0.596 5 0.590 5 0.428 5 0.496 5 0.447 5 0.4583 -0.200 2 -0.223 12 -0.336 3 -0.327 2 -0.227 1 -0.205 3 -0.101 2 -0.146

13 0.371 4 0.204 11 0.467 9 0.482 9 0.291 5 0.429 13 0.210 11 0.33312 -0.093 8 -0.044 2 -0.202 12 -0.196 11 -0.152 12 -0.305 1 -0.125 8 -0.2145 0.130 9 0.052 5 0.191 6 0.088 5 0.251 11 0.054 9 0.240 9 0.290

11 -0.130 3 -0.132 13 -0.242 12 -0.191 1 -0.199 3 -0.327 13 -0.175 11 -0.2455 0.507 6 0.397 10 0.309 7 0.482 6 0.360 7 0.384 6 0.402 5 0.2992-0.182 2 -0.252 12 -0.045 3 -0.326 2 -0.248 8 -0.229 2 -0.146 3 -0.153

11 0.282 11 0.182 9 0.309 4 0.211 7 0.171 9 0.276 2 0.296 7 0.26212 -0.162 1 -0.130 12 -0.223 8 -0.174 1 -0.112 8 -0.188 11-0.091 8-0.1159 0.323 4 0.191 7 0.467 12 0.411 4 0.315 6 0.216 6 0.254 7 0.3333 -0.206 2 -0.165 2 -0.274 1 -0.335 2 -0.263 2 -0.224 2 -0.447 8 -0.242

13 0.049 7 0.014 1 0.062 11 0.411 10 0.130 11 0.136 1 0.122 7 0.32211 -0.182 2 -0.167 5 -0.396 2 -0.288 2 -0.186 7 -0.305 3 -0.173 5-0.1957 0.371 3 0.159 3 0.288 3 0.260 7 0.110 3 0.314 7 0.210 10 0.2428 -0.072 12 -0.139 8-0.242 9 -0.200 11 -0.180 8-0.270 5 -0.230 5 -0.162

108

Table VI,4. The highest and lowest values of the correlationcoefficient of deviations,separately for each social group

S) I. Dwelling. 2. Fuel and light. 3. Food. 4. Tobacco. 5. Clothing 6. Footwear. 7. Washing and cleaning.8. Durables excl, own car, 9. Personal hygiene. 10. Books, newspapers etc.. I I - Sports, holidays, hobbies, etc. 12. Trans-port, incl, own car. 13. Union fees, subscriptions etc.

in the case of the negatively correlated items such typically interrelated expendituresets are lacking.

Table VI,4 gives the five highest coefficients of correlation of each category (positiveand negative) for each of the twelve groups of wage and salary earners. Also here especi-

Social groups

Highest Lowest

Expendituregroup*) Coefficients Expenditure

group*) Coefficient

1 5-6 0.578 2-11 -0.385Higher public servants and salaried 5-9 0.485 2-5 -0.240employees, the Capital 6-9 0.432 2-10 -0.181

9-10 0.381 2-13 -0.15610-11 0.378 4-13 -0.156

2 5-6 0.469 2-11 -0.144Lower public servants and salaried 1-2 0.395 2-9 -0.139employees, the Capital 6-9 0.370 2-5 -0.125

5-9 0.365 10-12 -0.11610-13 0.347 2-6 -0.075

3 5-6 0.480 3-12 -0.340Skilled workers, the Capital 1-2 0.369 1-5 -0.324

3-7 0.267 l-6 -0.2835-9 0.265 11-13 -0.2274-7 0.244 11-12 -0.219

4 5-6 0.532 5-12 -0.187Unskilled workers, the Capital 5-9 0.338 5-2 -0.171

1-2 0.336 10-12 -0.1716-9 0.300 1-12 -0.1673-7 0.275 2-3 -0.167

5 5-9 0.507 3-5 -0.235Higher public servants and salaried 5-6 0.469 3-11 -0.206

employees, provincial towns 6-9 0.423 3-6 -0.2007-13 0.371 2-5 -0.1924-9 0.368 2-9 -0.182

6 5-6 0.510 2-5 -0.252Lower public servants and salaried 6-9 0.397 2-9 -0.252employees, provincial towns 5-9 0.390 2-6 -0.223

1-2 0.250 2-12 -0.1674-7 0.204 2-11 -0.165

Social groups

7Skilled workers, provincial towns

8Unskilled workers, provincial towns

10Skilled workers, rural districts

11

Unskilled workers, rural districts

12

Farm workers, rural districts

Table VI,4. continued.

Expendituregroup*)

Highest

Coefficients Expendituregroup')

Lowest

Coefficient

5-6 0.596 5-12 -0.3967-11 0.467 6-12 -0.3369-10 0.309 2-11 -0.2744-7 0.293 8-13 -0.2423-13 0.288 3-8 -0.236

5-6 0.590 3-5 -0.3887-9 0.482 1-11 -0.3355-7 0.445 3-6 -0.327

11-12 0.411 3-9 -0.3266-9 0.394 2-12 -0.288

9 5-6 0.428 2-11 -0.263Lower public servants and salaried 6-9 0.360 2-9 -0.248employees, rural districts 5-9 0.359 2-6 -0.227

4-11 0.315 2-5 -0.2227-9 0.291 1-8 -0.199

5-6 0.496 3-8 -0.2 375-7 0.429 7-12 -0.3057-9 0.384 3-12 -0.3011-2 0.341 5-8 -0.2923-13 0.314 7-8 -0.274

5-6 0.447 2-11 -0.4476-9 0.402 2-4 -0.2535-9 0.398 5-13 -0.2302-10 0.296 8-13 -0.1756-11 0.254 1-3 -0.174

1-2 0.491 8-11 -0.2455-6 0.458 1-11 -0.2337-11 0.333 2-11 -0.2287-12 0.322 7-8 -0.2145-9 0.299 5-12 -0.195

109

*) 1. Dwelling. 2. Fuel and light. 3. Food. 4. Tobacco. 5. Clothing. 6. Footwear. 7. Washing and cleaning.8. Durables excl, own car. 9. Personal hygiene. 10. Books, newspapers etc. il. Sports, holidays, hobbies, etc. 12. Trans-port, incl, own car. 13. Union fees, subscriptions etc.

ally the negatively correlated items are characterized by great differences in the patternfrom one group of wage and salary earners to the other.

How the extra determining variable is to be fitted into the Engel function to yield themaximum reduction in the residual variance of the dependent variable, will not bediscussed here.

110

One technically simple method would be to let y or a transformation thereof enterlinearly so that the result will be e.g. a function of the following form

log j = a + fi (log y - log r) + (log h - log 17h)

whereby efficient estimates of parameters and of the residual variance on y, can becalculated according to the theory of multiple linear regression.

The results of such calculations as regards the case of footwear in the group of skilledworkers in the provincial towns are shown in table VI, 5, expenditure on clothingentering as supplementary explanatory variable. The residual variance of log yj isreduced by about 35 per cent namely from 0.00928 to 0.00587.

Table VI,5. Unexplained variance in the regression analysis. Skilled workers in provin-cial towns, log y1 = a + b1 (log x - log x) + b2 (log y2 - log y2). Expenditure onfootwear, y1 as a function of income, x and expenditure on clothing, y2.

logy1 = 1.8718 + 0.1909 (Iogx-3.5746) + 0.4326 (logy2-2.4665)S2log i log z, log 112 = 0.005873S log iii log z, log Y2 = 0.0766

logy1 = a' + b' (logy2logy2)Slog 111 logyz = 0.0784

logy1 = a" + b" (log xlog x)Slog yi log z = 0.0965

DANSK RESUMÉ

Undersøgelsen af danske lønmodtagerhusstandes indkomst-, forbrugs- og opsparingsforholdfor året 1955, som gennemførtes i begyndelsen af året 1956, er den største og mestdetaillerede af de forbrugsundersøgelser, Det statistiske Departement har foretaget, sidenman i 1897 påbegyndte denne art af undersøgelser. Forbrugsundersøgelsernes primæreformål var oprindelig at fremskaffe oplysning om »Livsvilkår i de forskellige samfundslag,derunder ernærings- og forbrugsforhold,1) men efter at pristalsreguleringen af lønningerog ydelser og tilskud af forskellig art vandt stærkt frem, har de foretagne forbrugsunder-søgelser her som i mange andre vesteuropæiske lande i første række skullet tjene som red-skab til opstilling af vægte ved prisindeksberegningerne. I de seneste år synes imidlertiddet alment beskrivende, som var det primære sigte med de første forbrugsundersøgelser,atter at komme i første række. Dette skyldes bl.a., at man har erkendt, at det grund-materiale, som tilvejebringes ved en omhyggelig planlagt og udført forbrugsundersøgelse- i denne forbindelse må de senere års betydelige fremskridt indenfor undersøgelsesteknik-ken haves i erindring - rummer oplysninger om væsentlige økonomiske sammenhænge isærvedrørende anvendelsen af den indtjente indkomst, der ikke, eller kun mangelfuldt,kan belyses ad anden vej.2)

Forbrugsundersøgelsen for året 1955 har da også været genstand for en mere omfat-tende bearbejdelse end nogen af de foregående undersøgelser.

En almindelig oversigt over 1955-undersøgelsen, dens tilrettelæggelse og dens hoved-resultater er givet i Statistiske Efterretninger i l957.) Fødevareforbruget blev særskiltbehandlet i en artikel i Statistiske Efterretninger i l958.) De indhentede oplysningervedrørende lønmodtagerhusstandenes opsparings- og formueforhold blev gjort til gen-stand for en særskilt analyse, hvis resultater meddeltes i et hæfte i serien Statistiske Under-søgelser i l96O.) I samme serie behandledes de indhentede oplysninger om lønindkom-sternes fordeling og sammensætning6).

Hovedparten af de indhentede oplysninger fra de adspurgte lønmodtagerhusstandevedrørte disses husstandes forbrugsudgifter i året 1955, og man besluttede derfor atunderkaste lønmodtagernes forbrugsadfærd en mere indgående analyse. Det er resul-taterne fra denne analyse, der indeholdes i nærværende publikation.

1) Lov om Statens statistiske Bureau 1895.2)Jfr. I. L. 0. (11).

Statistiske Efterretninger 1957, nr. 83.Statistiske Efterretninger 1958, nr. 46.Opsparing i lønmodtagerhusstandene 1955, Statistiske Undersøgelser nr. 3, Kbh. 1960.Lønmodtagerindkomster, Fordeling og sammensætning, Statistiske Undersøgelser nr. 6, Kbh. 1962.

112

2. Analysens hovedresultater.

Analysen tilsigtede at give en præcis beskrivelse af sammenhængen mellem de danskelønmodtagerhusstandes disponible indkomst og udgiften til nogle væsentlige udgifts-poster i året 1955. Denne sammenhæng mellem disponibel indkomst og udgiften tilgivne udgiftsposter er utvivlsomt af væsentlig betydning, hvis man vil prove at forklareforskelle i forbrugsadfærd fra den ene husstand til den anden, selvom naturligvis mangeandre forhold spiller ind såsom husstandstype, bopæls- og socialgruppering m.v. Ind-komst-udgiftsrelationen er imidlertid tillige af væsentlig betydning, hvis man vil for-søge at foretage skøn over forbrugets sandsynlige udvikling ved givne, alternative ind-komstforskydninger, hvad enten dette nu drejer sig om den enkelte husstand eller hus-standsgruppe, eller det drejer sig om alle husstande under eet.7)

Hovedvægten i analysen blev derfor lagt på udledning af de indkomst-udgiftsrelationer,som ifølge det foreliggende grundmateriale gav den bedste beskrivelse af sammenhængen.De nævnte indkomst-udgiftsrelationer går ofte under betegnelsen Engelfunktioner efterden tyske økonom og statistiker, Ernst Engel, og det konkrete analysearbejde har be-stået i at beregne skøn over parametrene i fem på forhånd udvalgte funktionstyper ogderefter ved et antal test at sammenligne disse funktionstyper for at finde frem til denfor hver udgiftspost bedst egnede Engelkurve.

For at eliminere de væsentligste forstyrrende påvirkninger hidrorende fra forskelle ihusstandsstørrelse i de undersøgte husstande omregnedes alle udgifts- og indkomstbeløbfor hver af de 3100 husstande til beløb pr. person.

Engel funktionens uafhængigt variable fastlagdes som disponibel indkomst (samtligekontantindtægter minus betalte personlige skatter) pr. person, og der blev udledt Engel-funktioner for følgende 13 udgiftsposter, der tilsammen udgør 85 pct. af totalforbrugetfor samtlige lønmodtagerhusstande.

Bolig.Brændsel og belysning.Fødevarer (incl, regelmæssig fortæring ude og ø1, vin og spiritus indenfor detsædvanlige husholdningsforbrug).Tobak.Beklædning.Fodtøj.Vask og rengøring.Varige goder (excl. motorkøretøjer).Personlig pleje.Bøger, aviser m.v.Sport, ferie, fritid m.v. (incl. restaurationsbesøg, teater, biograf og øl, vin ogspiritus uden for det sædvanlige husholdningsforbrug).Transport (incl. motorkøretøjer).Kontingent og forsikringer m.v. (excl, livs- og pensionsforsikringer).

Beregningerne udførtes særskilt for 12 lønmodtagergrupper, nemlig 4 socialgrupperindenfor hver af 3 landsdele, jfr. resultattabellen i bilag A. De 5 funktionstyper, hvis7) Jfr. Erling Jorgensen (12), side 54-61.

113

parametre man dannede skøn over, var følgende, hvor y betegner den disponible indkomstog s udgiften til en given udgiftspost:

(1) logi = a+ fi(logv logy)

(2)log=a+fi( -i = a + fi (log y - log )

i=a+fi(! -log i = log + log [ (a + log y)]

Væsentlige dele af analyserapporten behandler estimationsproblemer, således atman kan sige, at udredningen af analysemetoderne var et andet hovedformål ved analyse-arbejdet ved siden af selve beregningen af analyseresultaterne.

De foretagne testberegninger viste næsten samstemmende, at funktionstype (1), dendobbeltlogaritmiske funktion, for samtlige 13 udgiftsposter gav den bedste fremstillingaf Engelrelationen. Dette resultat er i en vis forstand overraskende, fordi det indebærer,at indkomstelasticiteten i de pågældende husstandes efterspørgsel efter de 13 udgiftsposterer konstant (for given socialgruppe, idet beregningerne som nævnt er udført særki1t for12 lønmodtagergrupper) og altså uafhængig af indkomstniveauet. Dette følger af, atindkomstelasticiteten er identisk med parameteren i den dobbeltlogaritmiske Engel-funktion. Man måtte vel på forhånd vente, at varegrupper, der i de højeste indkomst-grupper betragtes som nødvendighedsvarer (lav indkornstelasticitet), i de lavere ind-komstgrupper ville gå over til at blive betragtet som luksusvarer (høj indkomstelasticitet).Imidlertid viser det sig8), at der er en bemærkelsesværdig stabilitet til stede, også nårvi går fra lønmodtagergruppe til lønmodtagergruppe for så vidt angår det nævnteparameterskøn over fi, skønnet over indkomstelasticiteten. For 6 udgiftsposters vedkom-mende kan en hypotese om konstant indkomstelasticitet alle 12 lønmodtagergrupperigennem opretholdes, og for de resterende 7 posters vedkommende er afvigelserne omendstatistisk signifikante dog ikke særligt store. Konstateringen af denne stabilitet i lønmod-tagerhusstandenes indkomstelasticitet i udgiften til de væsentligste udgiftsposter er et alde mest iøjnefaldende resultater af analysearbejdet9).

Denne stabilitet gør det forsvarligt at beregne gennemsnit af de 12 lønmodtagergrup-pers indkomstelasticiteter for hver de 13 udgiftsposter. Disse gennemsnitselasticiteter ervist i nedenstående oversigt, hvor udgiftsposterne er ordnet efter gennemsnitsindkomst-elasticitetens størrelse.

Man ser umiddelbart af denne tabel, at de beregnede gennemsnitselasticiteter falderi tre klart afgrænsede størrelsesgrupper:

1. En gruppe, man kunne kalde nødvendighedsvarer, hvor elasticiteten ligger pågodt 0.5, bestående af de tre poster, fødevarer, fodtøj og brændsel og belysning.

Jfr. kap. V, side 83.Dette resultat frister til at postulere, at de fundne indkomstelasticiteter for lønmodtagerbefolkningenhar generel gyldighed for alle befolkningsgrupper. Om konsekvenserne heraf se Erling Jørgensen,(12).

114

10) Jfr. kapitel V, side 85.

J

') Ved beregningen af de 3 gennemsnitselasticiteter er de 13 udgiftsposters andel i totalforbrugetanvendt som vægte.

En anden gruppe, man kunne kalde neutrale varer, hvor elasticiteten ligger på etniveau omkring 1.0, og udgiften derfor stiger med samme procent som indkomsten. Idenne gruppe ligger bl.a. de to vigtigste udgiftsposter bolig og beklædning.

Endelig er der den tredje gruppe, som man kunne kalde luksusvarer, hvor elastici-teten er ca. 1.5, bestående af de to poster transport (incl. motorkøretøjer) og sport,ferie og fritid.

Analysen gav videre til resultat, at de beregnede 12 Engelkurver for hver udgiftspost -nemlig en for hver af de 12 lønmodtagergrupper, hvori materialet var opdelt - ikke kunnebetragtes som sammenfaldende, men at denne opdeling efter bopæl og socialgrupperingsyntes at modsvare faktisk eksisterende forskelle i forbrugsadfærd de tolv grupper imel-lem'0).

Hovedformålet med analysen har som nævnt været at formulere en præcis beskrivelseaf sammenhængen mellem lønmodtagerhusstandenes indkomst og udgifter til væsentligeudgiftsposter. Den anvendte analysemetode, som overvejende består i lineær regressions-analyse, synes at give tilfredsstillende resultater for de fleste udgiftsposter, med »pæne<udledte Engelkurvefunktioner til følge. For enkelte poster, især de to poster varige goderog transport (incl. motorkoretojer), er den uforklarede del af udgiftsvariationen fra husstandtil husstand imidlertid uforholdsmæssig høj og er kun blevet reduceret ganske lidt vedinddragelsen af husstandenes disponible indkomst i undersøgelsesperioden som uafhæn-gig, forklarende variabel.

Man kan formentlig heraf konkludere, at analysen af husstandenes udgifter til disseposter må gå ad andre veje end den her anvendte, med inddragelse of oplysninger om

Udgiftspost Indkomst-elasticitet

NødvendighedsvarerBrændsel og belysning 0.511Fodtøj 0.562Fødevarer 0.608

»Neutrale« varerKontingenter og forsikring 0.821Personlig pleje 0.856Vask og rengøring 0.859Bolig 0.885Bøger, aviser m.v. 0.977Tobak 0.980Varige goder excl. motorkøretøjer 0.989Beklædning 1.035

LuksusvarerTransport incl. motorkøretøjer 1.386Sport, ferie og fritid 1.500

Gnstl.Indkomst-elasticitet'

}

0.59

0.94

115

husstandstype og øvrige milieubetingede faktorer samt ikke mindst af oplysninger ved-rørende indkomstændringer og tidligere perioders forbrugsadfærd. En sådan dynamiskanalyse har imidlertid ligget uden for rammerne af dette arbejde, men det må erkendes,at de her repræsenterede resultater for disse posters vedkommende er utilfredsstillende.

På et par punkter er man gået ud over det ovenfor afgrænsede analyseformål, idet mani et afsluttende kapitel har undersøgt dels, i hvilket omfang de 13 udgiftsposter er korre-lercde, d.v.s. om husstande, der giver meget eller lidt ud til een bestemt udgiftspost udviser,en karaktcrisk udgiftsadfærd med hensyn til een eller flere af dc øvrige poster (har hus-stande med et højt tobaksforbrug et mindre fødevareforbrug end husstande med lavtforbrug af tobak? etc.); dels har man forsøgt at skitsere, hvilken betydning forskelle ihusstandstype (husstandens størrelse og sammensætning) har for husstandenes forbrugs-adfærd i de forskellige indkomstklasser.

Hvad det første problem angår - korrelationen mellem de 13 udgiftsposter - viste deforetagne beregninger, at der kun i ringe grad kunne påvises en sådan korrelation.Kun for så vidt angår de to poster bolig og brændselbelysning fandtes en stærk (positiv)korrelation. Dette resultat falder godt i tråd med hele oplægget til analysen, idet grup-peringen af de mangfoldige varer og tjenester, hvorom oplysninger indhentedes, i etbeskedent antal hovedudgiftsposter netop sigtede mod en gruppering, hvor der kun varen ringe positiv eller negativ korrelation mellem de enkelte grupper. Herved ville mansøge at nå frem til stabile inclkomstudgiftsrelationer, men måtte naturligvis samtidig giveafkald på at beskrive husstandenes forbrugsadfærd overfor enkeltvarer og tjenester.

Med hensyn til husstandstypens betydning for husstandenes forbrugsadfærd viste deforetagne undersøgelser, at selve husstandsstørrelsen var den dominerende faktor, og atman ved den foretagne omregning til beløb pr. person fik elimineret størsteparten afdenne »forstyrrende« påvirkning. For visse udgiftsposter, bl.a. bolig ogfritidsudgjfler, varder imidlertid stadig mærkbare påvirkninger at spore udover persontalseffekten, oggenerelt gjaldt det, som man vel også på forhånd ville vente, at der består economies ofscale, d.v.s., at udgiften pr. person til en given udgiftspost er faldende med persontalletpr. husstand.

Rapportens kapitel I indeholder en oversigt over arbejdets baggrund og tilrettelæggelsesamt over nogle af analysens hovedresultater. Kapitel II er en gennemgang af det an-vendte grundmateriale. Denne gennemgang indeholder dels en beskrivelse af forbrugs-og opsparingsundersøgelsens praktiske udførelse, d.v.s. grundmaterialets indsamling ogbearbejdelse, og dels en udledning af den statistiske usikkerhed, som de fra undersøgelses-materialet udllede tal er behæftet med. I kapitel III afgrænses analyseopgaven, idet for-skellige modeller til beskrivelse af de adspurge husstandes udgiftsadfærd diskuteres, endiskussion der munder ud i en motivering for valget af Engelkurveproblematikken somanalysens hovedemne. Kapitel IV indeholder en detailleret gennemgang af analyse-metoderne. Hvilke funktionstyper skal lægges til grund ved udledningen afEngelkurvcr forde forskellige udgiftsposter? Hvorledes skal de i Engelfunktionen indgående variablenærmere afgrænses? Og ikke mindst, hvorledes skal de anvendte funktionstypers egnethedved beskrivelsen af indkomstudgiftsrelationerne afprøves?

Herefter følger i kapitel V rapportens hovedafsnit, nemlig gennemgangen af analyse-resultaterne, hvorunder især den dobbeltlogaritmiske Engelkurvefunktion, som efter de

116

udførte test for goodness of fit fandtes at være den »bedste« af de 5 afprøvede funktions-typer, kommenteres.

Endelig er der i det afsluttende kapitel VI anført nogle eksempler på nærliggendevideregående beregninger, der skønnes at kunne bidrage til en yderligere præcision i be-skrivelsen af husstandenes forbrugsadfærd, end nærværende analyses hovedredskab,Engelkurven, muliggør. Til »forklaring« af de observerede forskelle i husstandenes ud-gifter til en given udgiftspost fremdrages dels forskelle i husstandenes størrelse og sammen-sætning og dels husstandenes udgifter til een eller flere andre udgiftsposter.

I bilag til rapporten er anført dels en samlet oversigt over analyseresultaterne, derfalder i to afsnit, hovedanalysens resultater, jfr. kap. V og resultater af de videregåendeberegninger jfr. kap. VI, dels det ved analysen benyttede grundmateriale suppleret medvisse yderligere oplysninger, der vil kunne inddrages i eventuelle supplerende analyser.

En liste over den benyttede litteratur er anført side 117-118.

LIST OF LITERATURE

Aitchison, J. and Brown,J. A. C.: The Lognormal Distribution, Cambridge University Press, 1957.

la. Same: "A synthesis of Engel Curve Theory", Review of Economic Studies, vol. XXII, no. 57,1955.

Amundsen, A.: Metoder i Analysen av Forbruksdata, Statistisk Sentralbyrå, Oslo, 1960.

Duesenberry, J.: Income, Saving and the Theory of Consumer Behaviour, Cambridge, 1949.

Durbin and Watson: "Testing for Serial Correlation in Least-Squares Regression", Biometrika37 and 38.

Engel, Ernst: "Die Produktions- und Consumtionsverhaltnisse des Königreichs Sachsen", Zeit-schrift des statistischen Bureau des Koniglich Sachsischen Ministeriums des Innern, 1857.

Forsyth: "The Relationship between Family Size and Family Expenditure", Journal of theRoyal Statistical Society, series A, vol. 123, part 4, 1960.

Friedman, M.: A Theory of the Consumption Function, Princeton, 1957.

Hald, A.: Statistical Theory with Engineering Applications, New York, 1952.

Hendricks, W.: "An Approximation to "Students" Distribution". The Annals of MathematicalStatistics, 1936.

Houthakker, H. S. and Prais, J. S.: The Analysis of Family Budgets With an Application toTwo British Surveys Conducted in 1937-39 and Their Detailed Results. Cambridge UniversityPress, 1955.

I.L.O.: Family Living Studies, A Symposium, Geneva, 1961.

Jørgensen, Erling: "Husholdningsbudgetundersogelser af forbrugsforhold", NationaløkonomiskTidsskrift 1962, 1-2.

Lykke Jensen, E.: Repræsentative undersogelsers teori og metode I. Simpel tilfældig udvælgelse,Copenhagen 1957.

Mc Kay, A. T.: "The Distribution of the Difference between the Extreme Observations and theSample Mean in Samples of n from a Normal Universe". Biometrika, 1935.

Modigliani, F.: Fluctuations in the saving-income Ratio: A Problem in Economic Forecasting.Studies in Income and Wealth, vol. 11, New York, 1949.

Statistical Review, New-Series vol. 2, No. 7.

Statistisk Department (cf. below).

Stone, R.: Measurement of Consumers Expenditure and Behaviour in The United Kingdom1920-38, vol. I, Cambridge, 1954.

Wallander, Jan: Studier i bilismens ekonomi, Uppsala 1958.

Wold, H.: Efterfrågan på jordbruksprodukter och dess kanslighet for pris- och indkomst-forandringer, S.O.U. 1940 : 16, Stockholm, 1940.

«9-

118

Det Statistiske Department:

Statistiske Meddelelser, 4.6.6 Danske Arbejderfamiliers Forbrug, 1. Afdeling, Byarbejdere, Copen-hagen, 1900.

- - 4.11.2 Danske Arbejderfamiliers Forbrug, 2. Afdeling: Landarbejdere,Copenhagen, 1901.

- - 4.40.1 Danske Husholdningsregnskaber, I. Afdeling, Byarbejdere, Copen-hagen, 1912.

- - 4.40.2 Danske Husholdningsregnskaber, 2. Afdeling, Arbejdere og Haand-værkere på Landet, Copenhagen, 1913.

- - 4.40.3 Danske Husholdningsregnskaber, 3. Afdeling, Husmænd og Gaard-mænd, Copenhagen, 1914.

- - 4.54.4 Husholdningsregnskaber for Tjenestemandsfamilier M. Fi., Copen-hagen, 1918.

- - 4.69.5 Husholdningsregnskaber for 1922, Copenhagen, 1925.

- - 4.100.1 Husholdningsregnskaber 1931, Copenhagen, 1936.

- - 4.122.1 Husholdningsregnskaber I Aaret 1/4 1939-29/3 1940 Og i 4 UgersPerioden 11/4-8/5 1942, Copenhagen, 1944.

Statistiske Efterretninger 1915 (no. 24), 1916 (no. 20), 1942 (no. 17), 1949 (no. 58), 1957 (no. 83),and 1958 (no. 46).

Statistiske Undersøgelser, no. 3, Opsparing i Lønmodtagerhusstandene 1955, Copenhagen, 1962.

Statistiske Undersøgelser, no. 6, Lønmodtagerindkomster, Fordeling og Sammensætning, Copenhagen,1962.

Statistical Inquiries, An Analysis of the Personal Income Distribution for Wage and Salary Earners in1955, Copenhagen, 1964.

INDEX

Average expenditure 20, 21 Groups of wage and salary earners 4

Budget relation 103

Calculation of estimates of parameters 45Clothing, expenditure on 8

Coefficient of correlation 52, 57Coefficient of variation 42

constancy of 42average value of 44

Computation programme 52Consumption surveys 3

Correlation between deviations 105

Danish Statistical Department 3

Dansk Sekvens Kalkulator 53Disposable income 35Double-logarithmic function 5Durable goods, expenditure on 9

d-test 49, 71

Elasticity of income 5, 33Engel curves 23Engel functions 23

use of 28variables of 33

Error in the independent variable 27Estimator, efficient 41

Expenditure, average per item 20, 21for different types of household 93 if.

Expenditure concept 14

Expenditure items 4, 7

Frame of the selection 15

F-test 50, 64

Grouping of the observations 38of the expenditures 39

Hendricks, W. 42Houthakker, H. S., and Prais, i. S. 4

Income concept 14

Income elasticity 5

average values of 89Interdependence among households in their

expenditure 25

Layer-effect 25Log-normal distribution 46

transformation of 47

3-parameter case 55

Macro level, Engel curve used on 29Multiple regression analyses 100

Personal hygiene, expenditure on 8

Prais, I. S., and Houthakker, H. S. 4Processing of the material 19

Propensity to consume 33

Regression analysis, versus cross-tabulation 7

modification of 88Regression lines, slopes of 81Response rate 12

Sampling errors 18

Sampling method 15

Saturation expenditure 80Saving concept 14

Social groups 4, 34Statistical Department 3

Survey of income, consumption and saving pat-terns 1955 3

Survey period 15

Systematic errors 18

120

Test, for goodness of fit 48

for identity of regression lines 85

for length of run 49, 71for number of runs 49, 71for parallelism of regression lines 82

Time series versus cross-sectional analysisT-test for

1= 0 78

Types of functions 5

Unit-consumers concept 90

Unit of analysis 13

Variance assumptions 40

x2-test, for constancy of variation 47

26 for normality of deviations 52, 60

Zero-observations 36

APPENDICES

Appendix A. Results of regression analysis.

The results comprise the estimates a, k and b for the five Engel functions

i logy = a + b (logx - gx)

2logy=a+b(3 y=a+b(logxlogx)

Ii i4 y=a±b - -

'X X5 log y = log k + log (a + log x),

x denoting disposable income per person and y denoting expenditure per person on a

given item. Also included are the averages of the dependent variable, log x or and

estimates of the standard errors of a and b, 5a and sj as well as the estimates s and s2denoting the square roots of the variance in the distribution of y within groups and ofthe variance in the distribution of residuals respectively. The tables of result finallycontain the results of the following tests: the correlation coefficient, R, between observedand calculated expenditures, N-test for number of runs and 1-test for number of elementsin the longest run, d-test for size and sign of the residuals, F-test, and x2-test for normalityof the residuals.

The limits of significance (5 or 95 per cent) are given in the following table, separatelyfor each of the twelve groups of wage and salary earners1).

123

1) This table do not include the limits of significance for the N-test the test for number of runs, asthese limits differ from expenditure item to expenditure item within each group of wage and salaryearners, cfr. table V,5.

Limits of significance.

125

Group of wage andsalary earners

Numberof

groupsof 3

house-holds

z -test d-test F-test

l-testdegrees

offreedom

X as

degreesof

freedomd,5

degreesof

freedomF,55

1. Higher sal.public servantsand empl., thecapital 112 7 14.1 110 1.64 111.224 1.31 11

2. Lower publicservants andsal. empi., thecapital 154 9 16.9 512 1.67 153.308 1.26 12

3. Skilled work-ers, the capital 83 7 14.1 81 1.59 82.166 1.36 10

4. Unskilledworkers, thecapital 68 5 11.1 66 1.54 67.136 1.41 10

5. Higher publicservants andsal. empi 70 5 11.1 68 1.55 69.140 1.40 10

6. Lower publicservants andsal. empl 111 7 14.1 109 1.64 110.222 1.31 11

7. Skilled work-ers provincialtowns 51 3 7.8 49 1.46 50.102 1.48 9

8. Unskilledworkers, pro-vincial towns 70 5 11,1 68 1.55 69.140 1.40 10

9. Lower publicservants andsal. empl., ru-raldistr 102 7 14.1 100 1.63 101.204 1.32 11

10. Skilled work-ers, rural di-stricts 51 3 7.8 49 1.46 50.102 1.48 9

11. Unskilledworkers, ruraldistricts 93 7 14.1 91 1.61 92.186 1.34 10

12. Agric. workers,rural distr. . 53 3 7.8 51 1.46 52.106 1.47 9

126

Appendix A. Main results. Higher public servants and salaried

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

1:logy = a + b (1ogx-ij)a 2.768 2.452 3.151 2.354 2.723 2.057b 0.960 0.734 0.524 0.799 0.885 0.429

3.806 3.806 3.806 3.806 3.806 3.806

s1 0.117 0.105 0.0548 0.197 0.116 0.116

s2 0.141 0.120 0.0690 0.247 0.127 0.1270.013 0.011 0.0065 0.023 0.012 0.012

sb 0.070 0.060 0.034 0.12 0.063 0.063

R 0.79 0.76 0.82 0.53 0.80 0.54

N 49 63 63 47 48 58

9 7 8 7 7 10

d 2.24 2.03 1.85 1.86 1.69 2.07F 1.45 1.30 1.59 1.57 1.20 1.192 (f) 9.9 (7) 6.3 (7) 8.9 (7) 21.5 (7) 8.4 (7) 10.0 (7)

2: log y = a + b ( -a 2.768 2.452 3.151 2.354 2.723 2.057b104 -0.261 -0.206 -0.151 -0.231 -0.250 -0.123

1 i0- 1.705 1.705 1.705 1.705 1.705 1.705X

si 0.117 0.105 0.0548 0.197 0.116 0.1160.152 0.123 0.0678 0.246 0.130 0.127

Sa 0.014 0.012 0.0064 0.023 0.012 0.012

Sb 10 0.022 0.108 0.0097 0.035 0.019 0.018

R 0.75 0.74 0.83 0.53 0.79 0.54

N 49 57 52 51 48 62

L 9 7 8 7 8 10

d 1.89 1.94 1.81 1.87 1.59 2.06F. 1.69 1.37 1.53 1.56 1.26 1.19

x2 (f) 5.9 (7) 7.4 (7) 8.8 (7) 12.9 (7) 9.2 (7) 6.5 (7)

3:y = a + b(1ogx-J)a 500.7 251.7 1322 209.2 450.3 111.9

b 1300 525 1880 512 1110 133

log x 3.679 3.697 3.724 3.683 3.682 3.735

s1 0.269 0.242 1.26 0.454 0.267 0.2670.344 0.263 0.148 0.445 0.281 0.277

sa 17 6.6 19 9.4 13 3.0

Sb 110 42 120 61 83 18

employees. The capital. (1.1.) 112 groups of 3 households.

127

459

5910

5711

61

9

556

4910

586

1.65 2.21 1.84 2.04 2.14 1.98 2.101.10 2.16 1.05 0.97 0.91 2.84 1.17

12.4 (7) 3.9 (7) 5.1 (7) 10.6 (7) 6.0 (7) 25.7 (7) 3.3 (7)

2.080 2.448 2.178 1.988 2.800 2.547 2.062-0.231 -0.186 -0.226 -0.272 -0.393 -0.274 -0.179

1.705 1.705 1.705 1.705 1.705 1.705 1.705

0.124 0.195 0.105 0.140 0.138 0.215 0.1170.130 0.281 0.112 0.154 0.137 0.353 0.1280.012 0.027 0.011 0.015 0.013 0.033 0.0120.018 0.040 0.016 0.022 0.020 0.050 0.018

0.77 0.40 0.80 0.76 0.89 0.46 0.684412

1.601.098.6

104.0245

(7)

628

2.292.08

11.5

282.8666

(7)

5511

1.701.146.6

130.2298

(7)

5111

1.721.21

11.6 (7)

82.06220

4961.960.984.6

402.11910

(7)

5492.092.70

16.9

326.61390

(7)

616

2.051.206.9

107.3186

(7)

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

Union fees,subscriptions etc.

2.080 2.448 2.178 1.988 2.800 2.547 2.0620.803 0.574 0.807 1.015 1.387 0.863 0.6343.806 3.806 3.806 3.806 3.806 3.806 3.8060.124 0.195 0.105 0.140 0.138 0.215 0.1170.130 0.287 0.108 0.138 0.132 0.362 0.1260.012 0.027 0.010 0.013 0.012 0.034 0.0120.065 0.14 0.054 0.069 0.066 0.18 0.063

0.76 0.36 0.82 0.81 0.90 0.41 0.69

3.686 3.686 3.689 3.671 3.620 3.635 3.7110.286 0.449 0.241 0.323 0.319 0.495 0.2700.285 0.660 0.266 0.429 0.334 1.01 0.3023.0 19 3.5 3.6 15 36 3.2

19 120 22 24 110 260 20

128

Appendix A. Main results (continued). Higher public servants and

RNidF2 (f)

4: y = a + b (1X

ab10

0.7345II1.951.63

17.2

- X

508.7-290

(7)

0.775710

2.031.18

14.8

253.9-119

(7)

0.845681.941.38

13.1

1328-448

(7)

0.6255

61.750.963.6

209.7-116

(7)

0.7958

7

1.591.1143

456.0-244

(7)

0.586010

2.121.078.6

112.1326

(7)

Sk

RN1

dFz2 (f)

12000

0.7253

92.211.456.05 (9)

1300

0.7661

7

2.051.296.35 (9)

1000

0.846381.931.47

13.70 (9)

2800

0.6249

7

1.871.54

18.97 (9)

6600

0.7950

7

1.741.16

12.26 (9)

110

0.586010

2.091.17

13.15 (9)

I io+ 2.234 2.152 2.021 2.224 2.223 1.977X

si 0.269 0.242 0.126 0.454 0.267 0.2670.388 0.301 0.170 0.472 0.324 0.287

Sa.. 20 7.6 22 10 15 3.1

Sb1O4 28 11 31 14 21 4.5

a -5.687 -5.075 -4.466 -5.258 -5.490 -4.160k 19870 2813 5646 3132 11650 319.8s1 0.117 0.105 0.0548 0.197 0.116 0.116s2 0.141 0.119 0.0664 0.245 0.125 0.125

0.29 0.28 0.16 0.51 0.29 0.36

R 0.70 0.73 0.81 0.62 0.74 0.57N 43 49 50 61 49 581 11 10 8 6 12 10d 1.43 1.61 1.50 1.64 1.22 2.03F. 2.07 1.55 1.81 1.08 1.48 1.15x2 (f) 27.8 (8) 25.0 (7) 8.0 (7) 15.0 (7) 7.6 (7) 5.8 (7)

5: logy = Iogk + 1og (a + logx)

ExpenditureParameter estimates

and test resultsDwelling Fuel & light Food Tobacco Clothing Footwear

salaried employees. The capital. (1.1.) 112 groups of 3 households.

groups

7.6 (7) 42.1 (7) 35.7 (7) 68.4 (7) 35.1 (7) 86.4 (7) 23.1 (7)

129

0.77 0.43 0.79 0.70 0.64 0.34 0.6746 61 55 59 33 51 639 6 11 7 21 9 6

0.774812

1.631.00

10.4

104,4-55.1

(7)

0.4664

82.2621.6

39.6

277.9-154

(7)

0.795112

1.701.21

17.7

131.1-67.0

(7)

0.6653

91.571.77

50.5 (7)

85.26-45.0

0.8551

7

1.501.108.6

398.8-394

(7)

0.464314

2.204.14

80.6

272.0-295

(7)

0.675710

2.051.268.6

109.0-42.8

(7)

1.691.069.13 (9)

2.242.115.93 (9)

1.861.038.73 (9)

2.110.999.41 (9)

0.593.48

19.55 (9)

2.022.79

29.28 (9)

2.121.154.47 (9)

2.207 2.228 2.194 2.281 2.586 2.616 2.067

0.286 0.449 0.241 0.323 0.319 0.495 0.2700.315 0.697 0.305 0.571 0.449 0.956 0.3323.3 20 4.0 4.8 21 31 3.54.6 28 5.6 6.7 29 44 5.0

-5.272 -4.636 -5.278 -5.829 -9.705 -5.440 -4.7891713 1400 2171 4587 2381.108 7003 719.0

0.124 0.195 0.105 0.140 0.138 0.215 0.1170.128 0.283 0.107 1.139 0.257 0.359 0.1250.32 0.55 0.27 0.34 0.051 0.54 0.32

960 990 1000 3500 50.108 7200 320

0.75 0.47 0.75 0.56 0.79 0.54 0.6346 62 37 49 39 45 51

12 8 15 10 11 14 10

1.34 2.22 1.35 1.16 0.81 2.22 1.861.21 2.41 1.60 3.13 1.99 3.73 1.51

Washing & Durables excl. Personal Books, newspapers Sports, holidays, Transport incl. Union fees,cleaning vehicles hygiene etc. bobbies own car subscriptions etc.

130

log y = a + b (

ab10

X

SbiO4RN 69 67 70 72 77 70L 11 7 8 9 9 9dFz2 (f)

Appendix A. Main results (continued). Lower public servants and salaried

ExpenditureParameter estimates

and test resultsDwelling Fuel & light Food Tobacco Clothing Footwear

1: log y = a + b (log x - log x)ablogxSj

RN1

dFx2 (f)

y = a + b (log xablog xs1

s2

sb

768

677

737

709

819

669

1.76 1.97 2.06 1.91 2.12 1.91

1.24 1.31 1.19 1.43 1.14 0.995.2 (9) 25.5 (9) 10.2 (9) 30.4 (9) 8.5 (9) 9.4 (9)

-2.735 2.368 3.152 2.344 2.689 2.063

-0.198 -0.104 -0.118 -0.197 -0.242 -0.132

1.975 1.975 1.975 1.975 1.975 1.975

0.116 0.149 0.0613 0.228 0.129 0.1230.139 0.174 0.0666 0.267 0.145 0.1270.011 0.014 0.0054 0.022 0.012 0.0100.013 0.016 0.0060 0.025 0.014 0.012

0.78 0.46 0.84 0.54 0.82 0.67

2.735 2.368 3.152 2.344 2.689 2.0630.931 0.498 0.533 0.838 1.115 0.6233.745 3.745 3.745 3.745 3.745 3.7450.116 0.149 0.0613 0.228 0.129 0.1230.129 0.171 0.0670 0.273 0.138 0.1220.010 0.014 0.0054 0.022 0.011 0.00980.055 0.073 0.028 0.12 0.059 0.052

0.81 0.49 0.84 0.51 0.84 0.70

1.521.434.8 (9)

1.901.36

16.7

228.0311

(9)

2.031.18

10.9

13121765

(9)

1.981.38

38.8

193.0487

(9)

1.901.263.2

348.61080

(9)

1.761.075.8

107.7161

(9)

- i)453.3

10403.596 3.651 3.652 3.580 3.549 3.6420.268 0.344 0.141 0.525 0.298 0.2820.340 0.362 0.148 0.478 0.333 0.281

13 6.9 16 8.2 11 2.572 37 87 44 58 14

employees. The capital. 1.2 154 groups of 3 households.

131

816

815

955

708

765

807

729

1.84 2.07 2.27 1.91 1.87 1.96 1.851.02 1.89 1.37 1.46 1.20 2.50 1.29

13.8 (9) 8.2 (9) 9.1 (9) 9.9 (9) 3.9 (9) 40.3 (9) 10.2 (9)

2.047 2.396 2.203 1.961 2.653 2.393 2.115-0.188 -0.198 -0.221 -0.211 -0.295 -0.241 -0.150

1.975 1.975 1.975 1.975 1.975 1.975 1.975

0.133 0.236 0.106 0.138 0.155 0.193 0.1240.136 0.323 0.124 0.166 0.180 0.305 0.1380.011 0.026 0.010 0.013 0.015 0.025 0.0110.013 0.030 0.012 0.016 0.017 0.029 0.013

0.77 0.47 0.84 0.74 0.82 0.56 0.688061.771.047.8 (9)

94.10204

8262.081.889.1

204.4686

(9)

8762.231.377.0

119.7333

(9)

7281.911.456.7 (9)

74.49188

6515

1.641.355.0

571.81700

(9)

8261.952.50

34.0

204.9722

(9)

7861.901.248.2

117.0212

(9)

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

Union fees,subscriptions etc.

2.047 2.396 2.203 1.961 2.653 2.393 2.1150.853 0.885 0.999 0.946 1.366 1.087 0.6563.745 3.745 3.745 3.745 3.745 3.745 3.7450.133 0.236 0.106 0.138 0.155 0.193 0.1240.134 0.324 0.124 0.167 0.170 0.305 0.1410.011 0.026 0.010 0.013 0.014 0.025 0.0110.057 0.14 0.053 0.071 0.072 0.13 0.060

0.77 0.46 0.84 0.74 0.84 0.56 0.66

3.601 3.536 3.565 3.582 3.745 3.523 3.6230.306 0.543 0.244 0.3 18 - 0.445 0.2860.321 0.759 0.314 0.379 - 1.028 0.3082.6 15 3.4 2.5 - 20 72

14 79 18 13 - 110 9

132

Appendix A. Main results (continued). Lower public servants and salaried

Parameter estimatesand test results

Expenditure

Dwelling Fuel & lightI

Food Tobacco Clothing Footwear

R 0.76 0.57 0.86 0.67 0.84 0.69N 70 65 76 74 73 70

1 18 10 8 9 9 11

d 1.51 1.88 2.11 1.81 1.79 1.86

F 1.61 1.11 1.10 0.83 1.25 1.00

z2 (f) 18.9 (9) 22.0 (9) 10.7 (9) 11.6 (9) 13.2 (9) 13.8 (9)

4:y= a+ b(I_I\x X

a 487.1 230.8 131.9 275.4 576.3 109.9bio_a -167 -54.0 -315 -ill -291 -27.8

1 i0 2.677 2.449 2.452 1.975 1.975 2.471X

0.268 0.344 0.141 - - 0.2820.414 0.389 0.169 - - 0.302

17 7.4 19 - - 2.8

sb'l04 15 7.0 17 - - 2.6

R 0.67 0.53 0.83 0.72 0.68 0.66N 55 61 60 72 61 65

i 21 11 20 9 17 16

d 1.18 1.75 1.68 2.20 1.91 1.61

F 2.39 1.28 1.43 - - 1.15

z2 (f) 33.4 (9) 34.8 (9) 10.1 (9) - - 14.9 (9)

5: log y = log k + log (a + log x)

a -5.496 -4.251 -4.375 -5.257 - -4.639k 13810 772.5 5447 3440 - 631.9

0.116 0.149 0.0613 0.228 - 0.1230.129 0.171 0.0562 0.270 - 0.122

sa 0.25 0.38 0.15 0.49 - 0.296800 310 940 3100 - 240

R 0.78 0.56 0.85 0.65 - 0.69

N 78 67 75 70 - 66

1 8 7 7 9 - 9

d 1.78 1.96 2.13 1.94 - 1.90

F 1.24 1.32 1.13 1.40 - 0.98

z2 (f) 6.66 (11) 20.84 (11) 9.45 (li) 32.74 (li) - 10.14 (11)

employees. The capital. 1.2 154 groups of 3 households.

groups

61 71 57 74 55 58 7415 9 31 8 14 13 8

133

0.76786

0.5880

6

0.838410

0.75748

0.456314

0.476611

0.72729

1.76 2.15 2.16 1.87 1.98 1.95 1.841.10 1.96 1.66 1.42 - 5.34 1.16

25.1 (9) 60.0 (9) 36.9 (9) 19.7 (9) - 229.8 (9) 14.2 (9)

95.79 351.2 122.6 106.6 571.8 374.0 118.7-32.2 -161 -50.4 -46.7 -339 -185 -34.3

2.795 1.975 3.048 1.975 1.975 1.975 2.639

0.306 - 0.244 - - - 0.2860.360 - 0.375 - - - 0.3453.0 - 4.2 - - - 3.52.5 - 3.3 - - - 3.0

0.72 0.62 0.78 0.70 0.69 0.42 0.67

1.39 2.07 1.67 2.35 1.99 1.66 1.561.38 - 2.36 - - - 1.45

29.6 (9) - 54.5 (9) - - - 31.5 (9)

-5.290 -5.376 -5.678 -5.540 -9.633 -5.906 -4.7481849 4920 6099 2558 1564.108 16390 838.4

0.133 0.236 0.106 0.138 0.155 0.193 0.1240.133 0.322 0.122 0.164 0.287 0.303 0.1390.29 0.51 0.22 0.29 0.046 0.40 0.29

970 4700 2900 1500 29.108 15000 340

0.76 0.50 0.83 0.75 0.55 0.43 0.7282 81 94 70 51 82 76

6 5 5 8 23 7 91.86 2.08 2.32 1.95 0.71 1.97 1.901.00 1.86 1.32 1.42 3.42 2.47 1.26

12.40(11) 8.92(11) 8.28(11) 8.48(11) 16.43(11) 39.25(11) 11.95(11)

Washing & Durables excl. Personal Books, newspapers Sports, holidays, Transport incl. Union fees,cleaning vehicles hygiene etc. hobbies own car subscriptions etc.

134

Appendix A. Main results (continued). Skilled workers.

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

log y = a + b (log x - log x)a 2.616 2.333 3.166 2.450 2.586 1.999b 0.868 0.402 0.685 0.931 1.020 0.502log x 3.705 3.705 3.705 3.705 3.705 3.705s 0.132 0.131 0.0603 0.156 0.116 0.103

0.135 0.116 0.0495 0.176 0.129 0.1010.016 0.014 0.0060 0.021 0.016 0.012

sj 0.084 0.072 0.031 0.11 0.080 0.063

R 0.79 0.57 0.94 0.72 0.84 0.70N 36 33 35 37 40 331 4 4 4 5 5 6d 2.03 1.86 2.11 1.70 2.04 1.94F 1.05 0.78 0.67 1.27 1.23 0.96z2 (f) 4.5 (5) 10.8 (5) 1.4 (5) 6.1 (5) 8.7 (5) 9.8 (5)

2: log y = a + b ( IX

a 2.616 2.333 3.166 2.450 2.586 1.999b104 -0.172 -0.091 -0.138 -0.183 -0.213 -0.106i 10+ 2.171 2.171 2.171 2.171 2.171 2.171X

si 0.132 0.131 0.0603 0.156 0.116 0.103s2 0.142 0.110 0.0563 0.184 0.124 0.0973

0.017 0.013 0.0068 0.022 0.015 0.01251,10 0.018 0.014 0.0072 0.023 0.016 0.012

R 0.76 0.62 0.92 0.69 0.86 0.72N 35 33 31 37 33 341 6 4 6 5 6 6d 1.84 2.07 1.62 1.58 2.23 2.06F 1.16 0.70 0.87 1.38 1.14 0.90z2 (f) 3.3 (5) 4.5 (5) 0.9 (5) 1.1 (5) 6.6 (5) 3.6 (5)

3: y = a + b (log x-i3)a 341.5 207.3 1276 230.9 279.2 94.47b 794 243 2220 593 850 119

log x 3.555 3.624 3.587 3.540 3.521 3.617si 0.304 0.302 0.139 0.360 0.267 0.237sa 0.309 0.249 0.120 0.408 0.278 0.240

14 6.4 20 13 11 2.876 34 110 70 60 15

The capital. (1.3.) 68 groups og 3 households.

135

375

365

356

346

346

364

367

1.69 2.07 2.06 2.05 2.26 2.34 2.041.27 1.76 1.25 0.91 0.86 2.04 0.775.0 (5) 12.8 (5) 6.6 (5) 0.9 (5) 4.5 (5) 3.8 (5) 5.0 (5)

2.023 2.376 2.105 1.900 2.597 2.370 2.337-0.133 -0.241 -0.142 -0.152 -0.298 -0.308 -0.178

2.171 2.171 2.171 2.171 2.171 2.171 2.171

0.120 0.198 0.101 0.135 0.143 0.209 0.09720.145 0.270 0.109 0.136 0.154 0.311 0.09620.018 0.033 0.013 0.016 0.019 0.038 0.0120.019 0.035 0.014 0.017 0.020 0.040 0.012

0.66 0.65 0.78 0.73 0.88 0.69 0.873341.481.475.0

95.11172

(5)

3551.971.865.5

199.1622

(5)

3752.181.188.4

112.2204

(5)

3311

1.861.013.0 (5)

69.39138

2913

1.691.164.0

217.21080

(5)

345

2.182.205.1

104.6831

(5)

3361.630.984.2

177.7396

(5)

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport inCl.own car

Union fees,subscriptions etc.

2.023 2.376 2.105 1.900 2.597 2.370 2.3370.702 1.214 0.679 0.772 1.507 1.561 0.8953.705 3.705 3.705 3.705 3.705 3.705 3.7050.120 0.198 0.101 0.135 0.143 0.209 0.09720.135 0.263 0.112 0.129 0.132 0.299 0.08550.016 0.032 0.014 0.016 0.016 0.036 0.0100.084 0.16 0.070 0.080 0.082 0.19 0.053

0.72 0.67 0.77 0.76 0.91 0.72 0.90

3.583 3.528 3.582 3.573 3.459 3.407 3.5590.276 0.456 0.232 0.311 0.329 0.482 0.1510.330 0.596 0.257 0.330 0.418 0.762 0.2054.0 16 3.7 3.0 14 14 4.8

22 92 20 16 87 100 26

136

Appendix A. Main results (continued). Skilled workers.

logy = log k + log 1 (a + log x)

RN1

dFz2 (f)

5700

0.783742.011.047.51 (7)

250

0.643341.950.748.49 (7)

3200

0.943462.150.652.46 (7)

6900

0.7237

5

1.71

1.274.95 (7)

13000

0.8440

52.131.164.48 (7)

140

0.6933

62.100.919.49 (7)

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

R 0.79 0.66 0.93 0.72 0.87 0.70N 37 31 33 37 33 35

1 6 4 6 6 6 8

d 1.88 1.95 1.88 1.62 2.16 2.10F 1.03 0.68 0.75 1.29 1.09 1.03

x2 (f) 2.7 (5) 8.0 (5) 7.1 (5) 6.9 (5) 2.4 (5) 18.9 (5)

4: y = a + b (i iX

a 436.1 204.9 1285 237.4 286.7 93.00biO-4 -126 -44.3 -362 -89.5 -129 -21.8

1 i0X

s1

s2

3.062

0.3040.355

2.660

0.3020.247

2.850

0.1390.165

3.162

0.3600.473

3.281

0.2670.319

2.712

0.2370.242

16 6.3 27 15 13 2.813 5.7 23 12 10 2.5

R 0.76 0.69 0.89 0.67 0.84 0.73

N 27 36 19 31 29 34

1 11 4 13 9 7 8

d 1.53 2.02 1.02 1.32 1.67 2.06F 1.37 0.67 1.42 1.73 1.44 1.04

x2 (f) 6.8 (5) 9.6 (5) 1.7 (5) 12.8 (5) 1.0 (5) 13.3 (5)

a -5.290 -3.934 -4.787 -5.457 -5.697 -4.246k 7424 532.8 10640 7180 16910 343.7

si 0.132 0.131 0.0603 0.156 0.116 0.103

2 0.135 0.113 0.0486 0.176 0.125 0.0311

Sa 0.42 0.52 0.20 0.49 0.35 0.38

The capital. (1.3.) 68 groups og 3 households.

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own Car

Union fees,subscriptions etc.

137

0.71 0.66 0.76 0.73 0.67 0.42 0.4937 36 36 34 29 37 15

5 5 6 6 13 5 17

1.641.296.19 (7)

2.111.734.88 (7)

2.121.197.29 (7)

2.060.912.31 (7)

0.672.866.36 (7)

1.682.823.64 (7)

0.1017.7829.44 (7)

0.7031

91.331.43

13.5 (5)

95.79-28.2

0.633551.561.713.9 (5)

321.6-165

0.7834

52.191.231.8

111.7-33.8

(5)

0.733311

1.821.128.1 (5)

71.21-22.1

0.8425141.001.62

14.2

205.7-161

(5)

0.7131

7

1.872.50

44.3

100.8124

(5)

0.8831

61.560.842.7

182.8-60.7

(5)

2.890 2.171 2.900 2.900 3.892 4.197 3.014

0.276 - 0.232 0.311 0.329 0.482 0.1510.387 - 0.264 0.374 0.556 0.888 0.2684.8 - 3.8 3.4 19 18 6.44.0 - 3.2 2.9 16 17 5.3

0.65 0.60 0.79 0.68 0.77 0.67 0.81

-4.825 -6.192 -4.772 -5.028 -9.572 -9.546 -9.733815.5 37430 903.6 867.6 1263.108 670.210 117810

0.120 0.198 0.101 0.135 0.143 0.209 0.09720.136 0.260 0.110 0.129 0.242 0.351 0.4100.40 0.59 0.34 0.44 0.068 0.11 0.031

450 57000 440 630 36.108 32.108 12.108

31 31 35 25 15 29 239 8 6 11 20 7 151.01 2.10 1.99 1.55 0.57 1.49 0.871.96 - 1.30 1.44 2.86 3.40 1.44

16.9 (5) - 6.9 (5) 13.7 (5) 20.7 (5) 56.7 (5) 3.3 (5)

138

Appendix A. Main results (continued). Unskilled workers.

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

log y = a + b (log x - i)a 2.602 2.328 3.157 2.449 2.551 1.968b 0.940 0.312 0.702 1.074 1.047 0.613

3.672 3.672 3.672 3.672 3.672 3.672

si 0.133 0.124 0.0615 0.178 0.152 0.1310.142 0.172 0.0620 0.204 0.156 0.125

s 0.016 0.019 0.0068 0.022 0.017 0.0140.080 0.096 0.035 0.11 0.088 0.070

R 0.80 0.34 0.91 0.72 0.80 0.69N 38 38 40 50 46 431 5 7 5 6 6 7

d 1.83 1.43 1.76 2.06 2.08 2.30F 1.14 1.92 1.02 1.32 1.06 0.92x2 (f) 4.1 (7) 8.0 (7) 8.5 (8) 14.6 (7) 13.3 (7) 7.3 (7)

2: logy = a + b ( i)

a 2.602 2.328 3.157 2.449 2.551 1.968b l0- -0.158 -0.0644 -0.115 -0.170 -0.176 -0.103

2.359 2.359 2.359 2.359 2.359 2.359X

si 0.133 0.124 0.0615 0.178 0.152 0.1310.146 0.167 0.0749 0.221 0.161 0.1280.016 0.018 0.0082 0.024 0.018 0.014

Sbl04 0.014 0.016 0.0072 0.021 0.016 0.012

R 0.78 0.41 0.87 0.66 0.78 0.68N 39 42 33 43 43 421 6 7 11 6 5 6

d 1.75 1.52 1.23 1.77 1.94 2.19F 1.21 1.81 1.48 1.54 1.13 0.96X2 (f) 9.9 (7) 13.7 (7) 7.3 (7) 14.6 (7) 19.7 (7) 7.0 (7)

3: y = a + b (log xa 308.1 211.2 1237 223.7 239.1 85.06b 707 222 2010 551 696000 122

3.473 3.586 3.535 3.467 3.431 3.552

si 0.306 0.286 0.142 0.409 0.350 0.3010.349 0.310 0.157 0.459 0.341 0.271

sa 13 7.4 23 13 11 2.6

Sb 60 36 110 58 48 12

The capital. (1.4.) 83 groups of 3 households.

groups

139

337

497

416

406

494

514

31

9

1.89 1.95 1.83 1.88 1.97 2.54 1.591.11 1.80 1.30 1.39 1.14 1.36 1.289.0 (7) 9.7 (7) 5.0 (7) 3.4 (7) 3.3 (7) 5.1 (7) 8.0 (7)

1.993 2.301 2.108 1.907 2.492 2.288 2.295-0.125 -0.176 -0.145 -0.170 -0.248 -0.220 -0.188

2.359 2.359 2.359 2.359 2.359 2.359 2.359

0.134 0.217 0.115 0.155 0.177 0.232 0.1190.158 0.285 0.127 0.195 0.195 0.281 0.1370.017 0.031 0.014 0.021 0.021 0.031 0.0150.015 0.027 0.012 0.019 0.019 0.027 0.013

0.67 0.58 0.80 0.71 0.83 0.67 0.8533

7

1.531.389.8 (7)

85.51155

487

2.021.73

15.0

142.5449

(7)

4461.911.228.8 (7)

94.64231

357

1.641.599.0 (7)

55.44162

4561.841.212.2 (7)

120.8725

437

2.321.477.6 (7)

89.25509

3010

1.521.32

10.2 (7)

119.5392

1.993 2.301 2.108 1.907 2.492 2.288 2.2950.810 0.984 0.828 1.052 1.463 1.342 1.1023.672 3.672 3.672 3.672 3.672 3.672 3.6720.134 0.217 0.115 0.155 0.177 0.232 0.1190.142 0.291 0.132 0.182 0.189 0.270 0.1350.016 0.032 0.014 0.020 0.021 0.030 0.0150.079 0.16 0.074 0.10 0.11 0.15 0.075

0.75 0.56 0.78 0.75 0.84 0.70 0.85

3.520 3.413 3.465 3.426 3.310 3.316 3.4050.309 0.499 0.266 0.356 0.407 0.534 0.2 740.338 0.557 0.293 0.445 0.479 0.800 0.3 163.4 11 3.5 3.3 9.8 12 5.1

16 48 16 15 50 60 23

Washing & Durables excl. Personal Books, newspapers Sports, holidays, Transport incl. Union fees,cleaning vehicles hygiene etc. hobbies own car subscriptions etc.

140

Appendix A. Main results (continued). Unskilled workers.

log y = log k + log 1 (a + log x)

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

R 0.79 0.57 0.90 0.73 0.85 0.74N 37 42 37 49 41 471 6 7 8 6 6 5

d 1.63 1.91 1.44 1.80 1.92 2.20F 1.30 1.18 1.23 1.26 0.95 0.81

x2 (f) 12.9 (7) 8.8 (7) 3.5 (7) 10.7 (7) 9.0 (7) 2.2 (7)

4: y = a + b ( -a 457,2 213.8 1276 347.2 234.9 84.23b104 -140 -29.8 -253 -122 -82.7 -16.5

I 10+ 2.359 2.875 3.266 2.359 4.395 3.276X

St - 0.286 0.142 - 0.350 0.301- 0.329 0.2 16 - 0.405 0.287- 7.9 32 - 13 2.8

b104 - 5.3 19 - 6.3 1.6

R 0.78 0.53 0.84 0.60 0.83 0.75N 37 36 19 37 33 391 6 10 17 10 10 7

d 1.77 1.86 0.82 1.91 1.47 1.92F - 1.32 2.33 - 1.34 0.91

x2 (f) - 10.9 (7) 9.7 (7) - 9.4 (7) 8.3 (7)

RN1

dFz2 (f)

7300

0.7938

51.871.11

7.91 (9)

130

0.5538

7

1.481.858.89

------

(9) -

- 15000

- 0.82- 48- 5- 2.11- 1.04- 12.53 (9)

240

0.7341

7

2.330.909.17 (9)

a -5.414 -3.575 - - -5.695 -4.504k 9976 399.3 - - 16810 465.2

0.133 0.124 - - 0.152 0.131s2 0.140 0.169 - - 0.155 0.124Sa 0.37 0.49 - - 0.42 0.41

The capital. (1.4.) 83 groups of 3 households.

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

141

Union fees,subscriptions etc.

0.76 0.68 0.81 0.79 - 0.66 0.6131 49 41 38 - 51 2510 7 6 7 - 4 18

0.7433

7

1.581.19

14.7

87.41-18.6

(7)

0.724862.261.245.4

150.8-52.6

(7)

0.863961.871.21

13.9 (7)

81.23-28.7

0.7837

7

1.361.64

18.2 (7)

55.60-19.0

0.8543

81.461.399.7

121.9-84.3

(7)

0.703312

2.122.24

60.7

85.23-59.8

(7)

0.882811

1.371.33

11.7

230.8-81.0

(7)

1.841.15

11.60 (9)

1.981.75

10.42 (9)

2.061.252.71 (9)

1.851.383.67 (9)

- 2.55- 1.34- 6.06 (9)

0.278.14

23.40 (9)

3.511 4.341 4.420 4.440 5.467 5.483 2.359

0.309 0.499 0.266 0.356 0.407 0.534 -0.402 0.602 0.313 0.564 0.577 0.918 -4.1 12 3.4 4.2 12 14 -2.3 6.0 1.7 2.1 6.8 7.7 -0.67 0.71 0.88 0.70 0.81 0.66 0.86

29 42 35 37 31 29 2615 9 9 7 11 14 11

-5.057 -5.537 -5.118 -5.703 - -6.443 -9.6181204 6530 1758 3877 - 70590 836.8.108

0.134 0.217 0.115 0.155 - 0.232 0.1190.144 0.287 0.128 0.182 - 0.269 0.3390.39 0.60 0.33 0.42 - 0.61 0.039

800 8200 1000 3600 - 120000 130.108

1.18 2.02 1.62 0.98 1.05 1.84 1.861.69 1.45 1.39 2.51 2.01 2.96 -

16.7 (7) 9.8 (7) 8.6 (7) 32.6 (7) 16.8 (7) 77.4 (7) -

142

Appendix A. Main results (continued). Higher public servants and

ExpenditureParameter estimates

and test resultsDwelling Fuel & light Food Tobacco Clothing Footwear

1: log y = a + b (log x - log x)ab

s1

Sa

Sb

RN 351 6

d 2.15F 0.73

x2 (f) 5.4 (5)

log y = a + b - f)

a 2.697b10-4 -0.1841. 10-l-4 1.957Xs1 0.124

0.1110.013

Sj 10 0.016

R 0.82N 37 37 28 42 33 331 6 5 8 4 6 5

d 2.04 2.22 1.72 2.29 1.80 1.83F. 0.76 0.99 0.93 1.48 1.16 1.36

x2 (f) 4.6 (5) 6.0 (5) 14.0 (5) 10.6 (5) 5.4 (5) 3.0 (5)

y = a + b (log x -Ï)a 400.0 283.3 1116 169.8 398.0 102.8b 898 431 1516 374 964 101

log x 3.592 3.643 3.654 3.592 3.580 3.680Sl 0.294 0.281 0.146 0.426 0.263 0.227

2 0.236 0.258 0.140 0.425 0.273 0.260Sa 12 9.1 19 9.4 14 32

Sb 63 48 100 48 73 5

2.697 2.488 3.085 2.289 2.706 2.0200.800 0.568 0.548 0.763 0.905 0.3793.751 3.751 3.751 3.751 3.751 3.7510.124 0.122 0.0633 0.185 0.114 0.09840.109 0.120 0.0559 0.223 0.118 0.1140.013 0.014 0.0067 0.027 0.014 0.0140.065 0.072 0.034 0.13 0.071 0.069

0.83 0.69 0.89 0.57 0.84 0.5639

5

368

434

364

345

2.26 1.97 2.35 1.94 1.860.96 0.78 1.45 1.07 1.343.8 (5) 7.1 (5) 9.0 (5) 6.4 (5) 5.0 (5)

2.488 3.085 2.289 2.706 2.020-0.130 -0.124 -0.173 -0.206 -0.0863

1.957 1.957 1.957 1.957 1.957

0.122 0.0633 0.185 0.114 0.09840.122 0.0611 0.225 0.123 0.1150.015 0.0073 0.027 0.015 0.0140.017 0.0085 0.032 0.017 0.016

0.68 0.87 0.56 0.82 0.55

salaried employees. Provincial towns. (2.1) 70 groups of 3 households.

143

365

397

375

366

41

7

396

357

1.56 2.22 1.97 2.00 2.15 2.41 2.131.29 2.10 0.89 1.48 0.95 2.97 0.953.8 (5) 14.6 (5) 6.8 (5) 7.4 (5) 10.2 (5) 6.7 (5) 4.1 (5)

1.916 2.462 2.099 1.906 2.638 2.434 2.078-0.175 -0.221 -0.165 -0.228 -0.333 -0.370 -0.172

1.957 1.957 1.957 1.957 1.957 1.957 1.957

0.128 0.219 0.103 0.144 0.149 0.251 0.09530.158 0.324 0.103 0.184 0.155 0.443 0.09820.019 0.039 0.012 0.022 0.019 0.053 0.0120.022 0.045 0.014 0.026 0.022 0.062 0.014

0.69 0.51 0.81 0.73 0.88 0.59 0.8435

7

1.321.53

13.5 (5)

69.58151

397

2.132.19

10.6

243.6682

(5)

2951.761.004.5

107.0207

(5)

3561.781.656.7 (5)

56.25177

31

81.861.093.0( 5)

566.71830

4062.283.15

11.8

94.41033

(5)

2881.881.06

11.9

99.3205

(5)

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.Own car

Union fees,subscriptions etc.

1.916 2.462 2.099 1.906 2.638 2.434 2.0780.816 1.008 0.730 1.022 1.459 1.676 0.7573.751 3.751 3.751 3.751 3.751 3.751 3.7510.128 0.219 0.103 0.144 0.149 0.251 0.09530.145 0.317 0.097 0.175 0.145 0.431 0.09280.017 0.038 0.012 0.021 0.017 0.051 0.0110.087 0.19 0.059 0.11 0.087 0.26 0.056

0.75 0.54 0.83 0.76 0.90 0.62 0.85

3.598 3.530 3.615 3.532 3.751 3.360 3.6050.294 0.504 0.237 0.331 - 0.575 0.2200.373 0.835 0.236 0.436 - 1.00 0.2253.4 28 3.2 3.5 - 21 2.9

17 140 17 17 - 150 15

144

Appendix A. Main results (continued). Higher public servants and

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

R 0.87 0.74 0.88 0.69 0.85 0.58N 39 35 32 38 33 321 6 5 8 5 6 5

d 2.12 2.22 1.90 2.32 1.78 1.83F 0.64 0.84 0.92 1.00 1.07 1.32

x2 (f) 12.6 (5) 3.6 (5) 12.5 (5) 6.5 (5) 7.l( 5) 3.1 (5)

4:y= a+ b()a 393.3 290.2 1144 169.7 396.2 102.4biO-4 -140 -71.9 -256 -56.0 -143 -19.5llo+. .. 2.972 2.483 2.400 2.987 3.074 2.336X

s1 0.294 0.281 0.146 0.426 0.263 0.2272 0.286 0.298 0.190 0.496 0.340 0.269

sa 15 11 26 11 18 3.4Sb l0- 11 9.3 24 7.9 13 3.2

R 0.85 0.69 0.79 0.63 0.80 0.60N 27 33 20 30 25 351 12 8 26 10 13 5

d 1.50 1.90 1.27 1.89 1.13 1.74F 0.95 1.12 1.71 1.36 1.67 1.41

x2 (f) 4.8 (5) 1.2 (5) 33.4 (5) 11.9 (5) 9.9 (5) 10.7 (5)

5: logy = log k + log (a + log x)

a -5.163 -4.500 -4.439 -5.063 - -3.884k 6390 1375 5023 2081 - 237.2

0.124 0.122 0.0633 0.185 - 0.094s2 0.107 0.119 0.0551 0.221 - 0.114S5 0.40 0.42 0.22 0.58 - 0.39Sk 4400 710 1300 2000 - 77

R 0.83 0.73 0.88 0.68 - 0.58N 34 37 34 43 - 341 6 5 8 4 - 5

d 2.22 2.28 2.09 2.35 - 1.86F 0.75 0.95 0.76 1.43 - 1.33x2 (f) 2.53 (7) 4.12 (7) 9.05 (7) 8.82 (7) - 2.01 (7)

salaried employees. Provincial towns. (2.1.) 70 groups of 3 households.

groups

145

-5.198 -5.718 -4.967 -5.752 - -7.399 -5.043

780 17000 590 3300 - 440000 610

0.72 0.48 0.82 0.72 - 0.55 0.84

1.531.31

5.84 (7)

2.222.07

12.99 (7)

1.980.898.45 (7)

2.001.479.89 (7)

- 2.41- 2.91- 8.86 (7)

2.170.932.13 (7)

0.7435

7

1.241.61

8.0 (5)

70.1-24.1

0.5129

82.312.75

31.9

263.9-101

(5)

0.8435

5

1.920.99

13.8

106.2-32.5

(5)

0.8035

61.891.74

16.8 (5)

52.19-26.4

0.7537

92.29

--

566.7-383

0.653412

2.123.03

64.5

158.0-219

(5)

0.8628

82.001.058.3

98.4-32.5

(5)

2.884 3.233 2.808 3.541 1.957 3.493 2.862

0.294 0.504 0.237 0.331 - 0.575 0.2200.458 0.873 0.289 0.512 - 1.05 0.2844.3 31 4.0 4.0 - 3.2 3.63.1 21 3.0 2.7 - 35 2.7

0.70 0.48 0.79 0.76 0.69 0.60 0.8329 27 27 21 33 28 2913 13 10 11 9 13 8

36 39 39 36 - 39 355 7 5 6 - 6 7

1133 11970 1137 3606 - 2087000 12370.128 0.219 0.103 0.144 - 0.251 0.09530.147 0.315 0.0972 0.175 - 0.428 0.09190.40 0.65 0.33 0.42 - 0.70 0.30

0.89 2.16 1.35 1.34 1.73 1.98 1.382.42 3.01 1.48 2.40 - 3.34 1.67

17.9 (5) 32.7 (5) 12.4 (5) 25.1 (5) - 47.2 (5) 23.4 (5)

Washing & Durables excl. Personal Books, newspapers Sports, holidays, Transport incl. Union fees,cleaning vehicles hygiene etc. hobbies Own car subscriptions etc.

146

Appendix A. Main results (continued). Lower public servants and

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

log y = a + b (log x - j)a 2.619 2.405 3.071 2.259 2.628 2.002b 0.856 0.440 0.640 0.896 1.017 0.567

log x 3.656 3.656 3.656 3.656 3.656 3.6560.131 0.157 0.0660 0.203 0.121 0.1200.137 0.157 0.0723 0.253 0.127 0.1260.013 0.015 0.0069 0.024 0.012 0.0120.061 0.070 0.032 0.11 0.057 0.056

R 0.80 0.51 0.88 0.60 0.86 0.69N 48 63 48 54- 60 571 7 9 6 6 6 6d 1.77 2.20 1.90 2.14 2.08 2.05F 1.09 1.00 1.20 1.56 1.10 1.10

x2 (f) 9.4 (7) 15.0 (7) 5.8 (7) 32.0 (7) 6.1 (7) 10.9 (7)

2: logy = a + b

a(

2.619 2.405 3.071 2.259 2.628 2.002bl04 -0.146 -0.0720 -0.108 -0.151 -0.173 -0.9671. l0 2.484 2.484 2.484 2.484 2.484 2.484X

s 0.131 0.157 0.0660 0.203 0.121 0.120

sa 0.140 0.160 0.0781 0.257 0.134 0.1280.013 0.015 0.0074 0.024 0.013 0.012

b104 0.011 0.012 0.0060 0.020 0.010 0.0099

R 0.79 0.49 0.86 0.59 0.85 0.68N 48 61 53 55 55 581 7 9 9 6 6 6d 1.69 2.13 1.65 2.09 1.88 1.99

F 1.15 1.04 1.40 1.60 1.23 1.13

x2 (f) 3.9 (7) 9.6 (7) 4.3 (7) 18.9 (7) 7.7 (7) 15.5 (7)

3:y = a + b(logx-logx)a 311.3 245.3 1021 149.0 294.7 91.32b 755 292 1610 365 813 134

3.448 3.550 3.517 3.447 3.424 3.5270.302 0.362 0.152 0.466 0.280 0.277

s2 0.312 0.313 0.167 0.490 0.338 0.290sa 10 7.5 17 7.9 11 2.6

48 35 80 36 50 12

salaried employees. Provincial towns. (2.2.) 111 groups of 3 households.

147

4510

477

51

959

662

645

860

61.68 1.84 1.81 1.80 2.15 1.87 2.181.22 1.97 1.47 1.14 1.28 2.24 1.505.6 (7) 12.6 (7) 7.0 (7) 12.6 (7) 8.5 (7) 7.1 (7) 6.4 (7)

1.870 2.367 2.071 1.801 2.478 2.148 2.088-0.163 -0.185 -0.149 -0.181 -0.261 -0.240 -0.134

2.484 2.484 2.484 2.484 2.484 2.484 2.484

0.152 0.243 0.115 0.158 0.155 0.236 0.1010.179 0.337 0.140 0.186 0.198 0.354 0.1270.017 0.032 0.013 0.018 0.019 0.034 0.0120.014 0.026 0.011 0.014 0.015 0.027 0.0099

0.75 0.56 0.80 0.77 0.85 0.64 0.794310

1.481.385.8 (7)

50.49149

577

1.871.92

17.6

221.3553

(7)

5461.791.49

10.4

91.68208

(7)

51

81.501.39

15.2

79.57213

(7)

41

81.701.632.5

432.51490

(7)

5081.852.258.7 (7)

42.39443

51

12

2.061.607.0 (7)

99.36201

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays, Transport incl.hobbies own car

Union fees,subscriptions etc.

1.870 2.367 2.071 1.801 2.478 2.148 2.0880.983 1.042 0.866 1.107 1.564 1.389 0.7883.656 3.656 3.656 3.656 3.656 3.656 3.6560.152 0.243 0.115 0.158 0.155 0.236 0.1010.168 0.340 0.139 0.169 0.175 0.353 0.1230.016 0.032 0.013 0.016 0.017 0.034 0.0120.075 0.15 0.062 0.076 0.078 0.16 0.055

0.78 0.55 0.80 0.81 0.89 0.64 0.81

3.411 3.445 3.458 3.656 3.656 3.234 3.4770.350 0.559 0.264 - - 0.543 0.2320.404 0.901 0.381 - - 1.05 0.2932.3 21 3.7 - - 7.9 3.0

11 98 17 - - 52 14

148

log y = log k + log cD (a + log x)

Appendix A. Main results (continued). Lower public servants and

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

R 0.84 0.62 0.89 0.70 0.84 0.72

N 52 63 53 61 51 60

1 7 8 9 7 7 6

d 1.66 2.12 1.83 1.98 1.77 2.00

F 1.07 0.75 1.21 1.10 1.46 1.09

x2 (f) 13.5 (7) 3.8 (7) 4.0 (7) 6.0 (7) 11.9 (7) 21.5 (7)

(1 i4: y = a + bX

a 298.9 248.6 1039 155.9 287.1 91.01

b104 -89.4 -37.2 -197 -43.1 -93.8 -17.5

10 4.3 16 3.198 3.474 4.055 4.523 3.450X

s1 0.302 0.362 0.152 0.466 0.280 0.277

s2 0.372 0.355 0.219 0.555 0.417 0.317

sa 12 8.6 23 9.4 14 2.9

sb 10-t 6.1 5.2 13 4.7 6.7 1.6

R 0.81 0.56 0.83 0.67 0.80 0.72

N 39 63 35 49 43 55

1 11 11 21 11 18 5

d 1.18 1.83 1.18 1.57 1.16 1.76

F 1.52 0.96 2.08 1.41 2.22 1.31

x2 (f) 14.7 (7) 12.4 (7) 9.0 (7) 12.2 (7) 19.5 (7) 27.3 (7)

RN1

dFZ2 (f)

3800

0.8148

7

1.831.076.57 (9)

280

0.6265

92.211.00

15.03 (9)

1500

0.8953

7

1.971.16

10.73 (9)

3300

0.685462.151.54

25.49 (9)

10000

0.8460

6

2.131.088.41 (9)

160

0.7158

6

2.101.09

14.15 (9)

a -5.200 -3.971 -4.595 -5.307 -5.625 -4.379k 6918 685.4 6896 3753 17720 435.3

0.131 0.157 0.0660 0.203 0.121 0.1200.136 0.157 0.0711 0.252 0.126 0.1250.30 0.44 0.16 0.46 0.27 0.31

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

salaried employees. Provincial towns. (2.2.) 111 groups of 3 households.

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.Own car

Union fees,subscriptions etc.

0.88 1.69 1.41 2.02 1.78 1.80 1.601.98 2.83 2.69 - - - 2.35

37.7 (7) 98.4 (7) 42.7 (7) - - - 18.4 (7)

149

1800 14000 1000 3700 2000.108 - 560

1.671.219.69 (9)

1.961.94

13.51 (9)

1.871.417.16 (9)

1.801.14

11.85 (9)

0.962.898.00 (9)

- 2.24- 1.45- 10.51 (9)

0.824910

1.311.33

14.3 (7)

48.56-17.1

0.473513

1.772.60

91.1

199.0-66.8

(7)

0.765210

1.782.09

B6.9

91.60-24.5

(7)

0.4245

9

2.07--

79.57-31.7

0.743713

2.02--

432.5-218

0.6440

9

1.863.72

100.5

246.0-107

(7)

0.815312

2.091.608.5

105.5-23.7

(7)

4.683 4.541 4.110 2.484 2.484 2.484 3.689

0.350 0.559 0.264 - - - 0.2320.492 0.941 0.432 - - - 0.3552.8 22 4.3 - - - 3.81.4 10 2.1 - - - 2.1

0.76 0.52 0.73 0.41 0.65 0.58 0.7435 36 39 31 33 36 4717 13 18 17 15 9 16

0.79 0.46 0.76 0.79 0.65 - 0.8246 47 49 55 45 - 6210 7 5 6 12 - 5

-5.535 -5.694 -5.230 -5.855 -9.450 - -5.0162508 11440 2077 4643 677.4.108 - 1437

0.152 0.243 0.115 0.158 0.155 - 0.1010.167 0.339 0.136 0.159 0.263 - 0.1220.34 0.53 0.26 0.34 0.69 - 0.23

150

Appendix A. Main results (continued). Skilled workers.

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

log y = a + b (log x Ïj)a 2.504 2.368 3.045 2.313 2.467 1.872

b 0.750 0.276 0.601 1.068 1.103 0.670

log x 3.575 3.575 3.575 3.575 3.575 3.575

si 0.112 0.123 0.0521 0.181 0.123 0.108

s2 0.102 0.135 0.0673 0.187 0.137 0.09650.014 0.019 0.0094 0.026 0.019 0.0140.077 0.10 0.051 0.14 0.10 0.073

R 0.81 0.36 0.86 0.73 0.83 0.79

N 24 25 28 18 30 29

1 6 5 5 6 4 6

d 2.34 1.42 2.46 1.57 2.44 2.10

F 0.83 1.21 1.67 1.07 1.25 0.79

x2 (f) 7.0 (3) 6.5 (3) 3.9 (3) 5.8 (3) 2.1 (3) 2.8 (3)

2: log y = a + b ( -a 2.504 2.368 3.045 2.313 2.467 1.872

b104 -0.117 -0.0550 -0.0933 - 0.164 -0.173 -0.105

2.901 2.901 2.901 2.901 2.901 2.901X

s1 0.112 0.123 0.0521 0.181 0.123 0.1080.106 0.129 0.0711 0.194 0.139 0.09780.015 0.018 0.0099 0.027 0.020 0.014

5b l0 0.013 0.016 0.0085 0.023 0.017 0.012

R 0.80 0.45 0.84 0.71 0.83 0.79

N 28 26 28 19 28 28

4 5 7 6 5 6

d 2.14 1.54 2.18 1.50 2.36 2.05

F 0.89 1.10 1.86 1.15 1.29 0.82

x2 (f) 5.3 (3) 9.6 (3) 0.9 (3) 6.4 (3) 3.2 (3) 5.4 (3)

3: y = a + b (log x - log x)a 282.9 232.1 1016 161.0 222.8 67.45b 545 236 1560 515 693 116

log x 3.465 3.512 3.483 3.411 3.414 3.474s1 0.259 0.252 0.120 0.417 0.282 0.249

0.242 0.264 0.162 0.442 0.314 0.21810 8.7 24 11 11 2.1

60 50 140 70 68 13

Provincial towns. (2.3.) 51 groups of 3 households.

151

21

7

235

216

295

288

236

325

1.74 2.20 1.69 2.14 1.90 1.86 2.641.52 2.88 0.68 1.04 0.87 1.92 0.660.7 (3) 2.2 (3) 3.7 (3) 4.3 (3) 5.0 (3) 0.5 (3) 13.0 (3)

1.846 2.230 1.963 1.703 2.345 2.055 2.285-0.135 -0.244 -0.123 -0.136 -0.266 -0.243 -0.124

2.901 2.901 2.901 2.901 2.901 2.901 2.901

0.121 0.203 0.110 0.135 0.151 0.244 0.09770.159 0.338 0.087 0.138 0.140 0.345 0.08490.022 0.047 0.012 0.019 0.020 0.048 0.0120.019 0.041 0.010 0.017 0.017 0.041 0.010

0.71 0.65 0.86 0.76 0.92 0.64 0.8725

726

419

6

254

266

236

296

1.55 2.27 1.80 2.13 1.90 1.82 2.351.71 2.78 0.63 1.03 0.85 1.99 0.751.2 (3) 0.1 (3) 2.7 (3) 4.1 (3) 0.4 (3) 1.2 (3) 7.8 (3)

59.67 61.87 78.30 41.84 67.65 72.2 163.8147 695 169 107 731 414 353

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

Union fees,subscriptions etc.

1.846 2.230 1.963 1.703 2.345 2.055 2.2850.898 1.501 0.767 0.858 1.679 1.576 0.7993.575 3.575 3.575 3.575 3.575 3.575 3.5750.121 0.203 0.110 0.135 0.151 0.244 0.09770.149 0.344 0.0905 0.138 0.141 0.338 0.07940.021 0.048 0.013 0.019 0.020 0.047 0.0110.11 0.26 0.069 0.10 0.11 0.26 0.60

0.75 0.64 0.85 0.76 0.91 0.66 0.88

3.441 3.287 3.454 3.436 3.290 3.332 3.4540.279 0.466 0.252 0.312 0.348 0.563 0.2250.367 0.748 0.205 0.354 0.380 0.813 0.1763.3 11 2.4 2.2 5.8 10 4.3

20 92 14 14 50 74 26

152

Appendix A. Main results (continued). Skilled workers.

logy = log k + log I (a + log x)

RN1

dFx2 (f)

2300

0.8026

42.450.803.56 (5)

180

0.5325

5

1.471.142.39 (5)

1600

0.8528

52.481.610.61 (5)

19000

0.701861.571.055.64 (5)

24000

0.823042.491.201.55 (5)

330

0.792962.170.775.57 (5)

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

R 0.79 0.56 0.85 0.73 0.82 0.80N 26 24 28 23 28 31

1 7 5 7 6 5 6d 2.16 1.53 2.29 1.39 2.18 2.12F 0.87 0.87 1.81 1.12 1.24 0.76x2 (f) 2.5 (3) 6.9 (3) 1.4 (3) 6.9 (3) 2.6 (3) 2.7 (3)

4: y = a + b ( -a 290.4 229.8 1026 159.2 232.7 67.47bb-4 -67.8 -34.7 -202 -61.1 -83.1 -15.2I l0+ 3.650 3.373 3.535 4.265 4.085 3.645X

s 0.259 0.252 0.120 0.417 0.282 0.249

2 0.283 0.261 0.191 0.515 0.371 0.23612 8.6 28 13 14 2.3

sjybO4 8.9 6.6 26 9.1 9.5 1.7

R 0.73 0.60 0.80 0.69 0.78 0.78N 24 24 21 23 21 291 7 6 16 6 7 6d 1.66 1.64 1.71 1.18 1.63 1.88F 1.20 0.85 2.53 1.53 1.73 0.90x2 (f) 8.2 (3) 10.3 (3) 3.1 (3) 5.5 (3) 1.1 (3) 1.3 (3)

a -4.855 -3.399 -4.428 -5.701 -5.797 -4.629k 3229 414.1 5711 12470 22650 517.4s1 0.112 0.123 0.0521 0.181 0.123 0.108

2 0.100 0.131 0.0661 0.185 0.135 0.09480.45 0.69 0.22 0.67 0.45 0.44

Provincial towns. (2.3.) 51 groups of 3 households.

153

1400 - 730 870 3400000 - 1600

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

Union fees,subscriptions etc.

0.73 0.74 0.86 0.75 0.91 0.62 0.8923 25 21 26 21 17 31

7 6 6 4 8 10 5

1.64 2.14 1.83 2.11 1.12 1.34 2.481.73 2.57 0.66 1.29 1.19 2.09 0.611.3 (3) 9.3 (3) 4.8 (3) 11.0 (3) 2.8 (3) 7.8 (3) 5.4 (3)

61.01 41.89 77.42 41.42 57.15 193.4 164.0-18.3 -79.4 -21.6 -13.2 -83.0 -113 -44.3

3.888 5.608 3.847 4.014 5.491 2.901 3.827

0.279 0.466 0.252 0.312 0.348 - 0.2250.414 0.836 0.224 0.394 0.550 - 0.2263.9 9.8 2.6 2.5 7.9 - 5.52.8 11 1.9 1.8 7.7 - 4.0

0.69 0.73 0.85 0.73 0.84 0.24 0.84

1.721.520.56 (5)

- 1.82- 0.63- 3.48 (5)

2.191.027.86 (5)

2.290.823.77 (5)

- 2.65- 0.64- 7.93 (5)

19 23 17 23 19 17 277 7 7 5 8 10 61.31 2.02 1.58 1.83 0.53 2.24 1.672.20 3.21 0.79 1.59 2.50 - 1.015.5 (3) 11.8 (3) 8.1 (3) 9.2 (3) 13.6 (3) - 10.3 (3)

0.71 - 0.84 0.74 0.89 - 0.8921 - 23 29 28 - 34

7 - 6 5 8 - 5

-5.255 - -4.906 -5.154 -7.231 - -4.9911531 - 1016 895.9 1758000 - 2492

0.121 - 0.110 0.135 0.151 - 0.09770.149 - 0.0873 0.136 0.137 - 0.07820.46 - 0.43 0.52 0.54 - 0.38

154

Parameter estimatesand test results

Appendix A. Main results (continued). Unskilled workers.

Expenditure

x)

2 347 3.034 2.215 2.453 1.8570.430 0.617 1.168 1.014 0.6263.526 3.526 3.526 3.526 3.526

2 (f) 4.6 (5)

2: logy = a + b (-)a 2.457b l0 -0.0899

1. l0+ 3.261X

s1 0.122s2 0.152

0.0180.013

R 0.64N 30 281 6 10

dFy2 (f)

3: y = a + b (log x - log x)a 266.0 219.9b 418 204log x 3.411

si 0.2820.363

12

sb 64

327

3213

435

357

456

1.64 1.80 2.44 2.32 2.111.12 0.83 0.86 1.84 1.463.1 (5) 2.8 (5) 4.9 (5) 6.3 (5) 1.6 (5)

2.347 3.034 2.215 2.453 1.857-0.0527 -0.0821 -0.156 -0.140 -0.0876

3.261 3.261 3.261 3.261 3.261

0.103 0.0656 0.183 0.123 0.1090.115 0.0667 0.177 0.166 0.1300.014 0.0080 0.021 0.020 0.0160.010 0.0058 0.015 0.014 0.011

0.54 0.86 0.78 0.76 0.69

0.183 0.123 0.1090.109 0.0597 0.170 0.167 0.1320.013 0.0071 0.020 0.020 0.0160.069 0.038 0.11 0.11 0.083

0.60 0.89 0.80 0.76 0.67

1.58 1.491.55 1.253.2 (5) 3.2 (5)

299

1.471.031.8

984.01450

(5)

415

2.240.949.7 (5)

99.26365

367

2.271.829.4

164.2633

(5)

456

2.101.421.9 (5)

65.59109

Dwelling

log y = a + b (log x log

a 2.457b 0.704log x 3.526si 0.122

0.1440.017

sb 0.091

R 0.68N 341 6d 1.77F 1.38

3.460 3.428 3.272 3.266 3.4140.238 0.151 0.422 0.284 0.2510.281 0.147 0.398 0.374 0.2957.5 18 5.9 9.4 2.4

41 96 29 45 13

Fuel & light Food Tobacco Clothing Footwear

Provincial towns. (2.4.) 70 groups of 3 households.

155

357

3310

335

355

367

397

348

1.90 1.99 1.62 2.06 2.28 1.79 1.571.21 1.33 1.01 0.90 1.20 1.85 0.917.8 (5) 3.2 (5) 7.0 (5) 2.8 (5) 14.0 (5) 4.6 (5) 4.9 (5)

1.753 2.169 1.916 1.688 2.240 1.976 2.275-0.103 -0.190 -0.131 -0.141 -0.207 -0.201 -0.116

3.261 3.261 3.261 3.261 3.261 3.261 3.261

0.135 0.220 0.102 0.137 0.167 0.233 0.09080.153 0.252 0.107 0.133 0.193 0.344 0.1100.018 0.030 0.013 0.016 0.023 0.041 0.0130.013 0.022 0.0093 0.012 0.017 0.030 0.0096

0.68 0.73 0.86 0.83 0.83 0.63 0.8333

7

1.721.291.4 (5)

47.26105

34101.971.303.0 (5)

51.56391

3451.431.102.9

59.02163

(5)

31

61.890.953.3 (5)

32.13102

3752.081.344.2

239.3897

(5)

3371.572.183.4

56.01384

(5)

2571.061.473.1

162.9321

(5)

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

Union fees,subscriptions etc.

1.753 2.169 1.916 1.688 2.240 1.976 2.2750.771 1.368 0.969 1.041 1.538 1,617 0.9133.526 3.526 3.526 3.526 3.526 3.526 3.5260.135 0.220 0.102 0.137 0.167 0.233 0.09080.148 0.254 0.102 0.129 0.183 0.317 0.08650.018 0.030 0.012 0.015 0.022 0.038 0.0100.094 0.16 0.065 0.081 0.12 0.20 0.055

0.71 0.72 0.88 0.84 0.85 0.70 0.90

3.371 3.169 3.331 3.304 3.526 3.289 3.3990.311 0.509 0.235 0.314 - 0.537 0.2090.407 0.586 0.259 0.295 - 1.12 0.248

2,5 6.0 2.1 1.4 - 11 2513 34 10 6,6 - 71 8

156

Appendix A. Main results (continued). Unskilled workers.

log y = log k + log 1' (a + log x)

RN1

dFx2 (f)

1400

0.6532

61.801.384.55 (7)

230

0.5332

7

1.821.124.19 (7)

1800

0.8934

91.880.804.87 (7)

27000

0.7944

52.450.858.51 (7)

10000

0.7638

7

2.571.795.25 (7)

210

0.684562.291.421.81 (7)

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

R 0.63 0.52 0.88 0.84 0.87 0.72N 36 30 29 41 34 451 6 10 9 6 9 5

d 1.71 1.53 1.61 2.03 2.21 2.04F 1.65 1.40 0.94 0.89 1.74 1.38

x2 (f) 12.9 (5) 12.0 (5) 1.5 (5) 3.8 (5) 9.0 (5) 4.7 (5)

4: y = a + b (X

a 318.8 225.0 1014 98.77 134.5 60.73b10 -70.2 -16.9 -140 -31.6 -55.5 -11.4

10 3.261 3.721 4.023 6.179 6.658 4.719X

- 0.238 0.151 0.422 0.284 0.251- 0.310 0.199 0.501 0.438 0.308- 8.5 25 7.6 10 2.4- 5.1 13 3.0 4.1 1.1

R 0.50 0.42 0.79 0.78 0.86 0.79N 28 21 27 31 31 371 7 16 10 8 11 6

d 1.57 1.37 1.00 1.35 1.86 1.94F - 1.70 1.73 1.41 2.38 1.50

x2 (f) - 11.8 (5) 4.6 (5) 18.3 (5) 8.0 (5) 5.8 (5)

a -4.667 -3.831 -4.422 -5.905 -5.513 -4.451k 2290 592.6 5920 19230 12270 410.7

0.122 0.103 0.0656 0.183 0.123 0.1090.143 0.109 0.0587 0.169 0.165 0.1300.41 0.41 0.23 0.56 0.38 0.38

Provincial towns. (2.4.) 70 groups of 3 households.

groups

157

0.7333

61.781.72

0.823110

1.741.33

0.9032

51.311.22

0.913561.820.88

0.782912

1.95-

0.552711

1.254.39

0.822581.051.40

11.1 (5) 10.4 (5) 7.9 (5) 3.9 (5) - 52.5 (5) 1.8 (5)

43.22 49.65 57.45 30.38 113.6 44.56 208.6-9.90 -33.9 -14.3 -8.91 -57.8 -45.4 -56.7

5.271 7.420 5.527 5.928 4.991 5.665 3.261

0.311 0.509 0.235 0.314 0.385 0.5370.489 0.694 0.349 0.377 0.649 1.362.9 7.2 2.8 1.7 11 12

1.2 3.4 1.1 0.67 7.1 9.3

0.71 0.77 0.84 0.85 0.71 0.52 0.78

1.961.207.45 (7)

2.051.302.23 (7)

1.690.981.94

--

(7) -

- - 0.09- - 20.71- - 34.51 (7)

32 21 21 27 27 23 2110 11 17 10 10 11 141.492.48

31.9 (5)

1.501.86

19.6 (5)

0.792.21

12.2 (5)

1.131.446.6 (5)

1.362.84

25.2 (5)

1.106.38

88.7 (5)

1.94

0.69 0.74 0.88 - - - 0.6235 33 33 - - - 19

7 10 5 - - - 20

-4.858 -6.414 -5.394 - - - -9.571628.2 77220 2714 - - - 1080-10e

0.135 0.220 0.102 - - - 0.09080.148 0.251 0.101 - - - 0.4140.45 0.66 0.32 - - - 0.028

460 150000 1800 - - - 100.108

Washing & Durables excl. Personal Books, newspapers Sports, holidays, Transport incl. Union fees,cleaning vehicles hygiene etc. hobbies own car subscriptions etc.

158

Appendix A. Main results (continued). Lower public servants and

dF

1.820.92

1.591.25

1.571.12

2.041.60

2.210.95

2.190.79

x2 (f) 5.0 (7) 23.4 (7) 2.8 (7) 11.1 (7) 5.5 (7) 6.7 (7)

2: log y = a + b -a 2.477 2.411 3.018 2.197 2.512 1.911

-0.118 -0.0855 -0.0834 -0.162 -0.153 -0.09941.10+4 2.891 2.891 2.891 2.891 2.891 2.891X

si 0.146 0.143 0.0702 0.198 0.136 0.1180.148 0.156 0.0776 0.244 0.156 0.1150.015 0.015 0.0077 0.024 0.015 0.0110.011 0.011 0.0056 0.018 0.0011 0.0083

R 0.74 0.61 0.83 0.68 0.81 0.77N 45 43 43 55 39 491 8 9 9 6 15 10

d 1.66 1.69 1.45 2.18 1.58 1.85F 1.02 1.18 1.22 1.51 1.32 0.95x2 (f) 7.7 (7) 20.7 (7) 2.4 (7) 13.4 (7) 4.5 (7) 9.0 (7)

3:y= a + b(logx-logx)a 257.6 229.3 937.2 89.4 221.3 70.82b 484 380 1320 352 620 117

log x 3.429 3.444 3.467 3.280 3.342 3.4420.336 0.330 0.162 0.456 0.313 0.2720.343 0.284 0.173 0.502 0.370 0.2639.4 6.8 17 5.9 9.6 2.0

sa 43 32 79 27 42 9.1

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

log y = a + b (log x log x)

a 2.477 2.411 3.018 2.197 2.512 1.911b 0.835 0.550 0.577 1.061 1.117 0.712log x 3.585 3.585 3.585 3.585 3.585 3.585s1 0.146 0.143 0.0702 0.198 0.136 0.118

0.140 0.160 0.0743 0.250 0.133 0.1050.014 0.016 0.0074 0.025 0.013 0.0100.069 0.079 0.036 0.12 0.065 0.051

R 0.77 0.57 0.85 0.65 0.86 0.81N 44 42 42 47 55 51

10 9 lI 8 12 5

salaried employees. Rural districts. (3.2.) 102 groups of 3 households.

48 50 64 45 58 50 517 6 7 6 5 8 6

159

1.79 1.92 2.36 1.85 2.34 1.97 2.211.08 1.85 0.94 1.37 1.55 2.05 1.383.7 (7) 6.8 (7) 14.9 (7) 8.3 (7) 4.6 (7) 5.7 (7) 8.1 (7)

1.818 2.314 1.943 1.698 2.340 2.260 2.009-0.124 -0.167 -0.134 -0.131 -0.223 -0.207 -0.114

2.891 2.891 2.891 2.891 2.891 2.891 2.891

0.154 0.232 0.119 0.163 0.165 0.252 0.1020.180 0.311 0.129 0.197 0.228 0.382 0.1320.018 0.031 0.013 0.020 0.023 0.038 0.0130.013 0.022 0.0093 0.014 0.016 0.028 0.0095

0.69 0.60 0.82 0.68 0.81 0.60 0.7743

7

1.401.383.5 (7)

53.37115

5051.971.802.9 (7)

59.98551

4381.891.186.2

65.21157

(7)

377

1.751.478.1 (7)

37.3296.8

4912

1.911.926.7

314.21107

(7)

4991.762.305.5

358.71522

(7)

517

1.821.69

10.8 (7)

84.73165

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

Union fees,subscriptions etc.

1.818 2.314 1.943 1.698 2.340 2.260 2.0090.936 1.104 0.954 0.925 1.590 1.530 0.8193.585 3.585 3.585 3.585 3.585 3.585 3.5850.154 0.232 0.119 0.163 0.165 0.252 0.1020.160 0.316 0.115 0.191 0.205 0.361 0.1200.016 0.031 0.011 0.019 0.020 0.036 0.0120.078 0.15 0.056 0.093 0.10 0.18 0.059

0.77 0.58 0.86 0.70 0.85 0.65 0.81

3.393 3.166 3.378 3.356 3.585 3.585 3.4190.354 0.534 0.273 0.374 - - 0.2340.429 0.747 0.319 0.577 - - 0.3112.5 8.2 2,3 2.5 - - 2.8

11 49 10 11 - - 13

160

Appendix A. Main results (continued). Lower public servants and

5: log y = log k + log '1 (a + log x)

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

R 0.74 0.77 0.86 0.81 0.83 0.79N 43 47 42 52 29 491 8 9 9 6 15 10

d 1.60 1.60 1.57 2.08 1.46 1.96F 1.04 0.74 1.15 1.21 1.39 0.94

x2 (f) 9.5 (7) 8.1 (7) 8.9 (7) 12.5 (7) 19.4 (7) 13.9 (7)

4: y = a + b (

a 275.0 230.9 962.7 204.0 226.1 74.86bl0 -46.5 -38.3 -136 -64.1 -59.2 -11.9

ak

-5.0754481

-4.2741067

-4.3464738

-5.6798851

-9.573l573l0

-4.734660.3

s1 0.146 0.143 0.0702 0.198 0.136 0.1180.139 0.157 0.0726 0.247 0.369 0.1100.37 0.40 0.19 0.47 0.038 0.31

2900 510 1100 9500 220l0 310

R 0.76 0.74 0.86 0.76 0.61 0.81N 44 42 44 51 31 491 10 9 10 8 17 9d 1.90 1.63 1.65 2.09 0.30 2.20F 0.91 1.21 1.07 1.55 7.35 0.79

z2 (f) 2.89 (9) 20.85 (9) 8.98 (9) 11.00 (9) 34.47 (9) 4.92 (9)

l0-- 4.109 4.179 3.797 2.891 5.379 3.943X

s1 0.336 0.330 0.162 - 0.313 0.272s2 0.407 0.330 0.2 16 - 0.473 0.324sa 12 8.0 21 - 13 2.5

5.3 3.6 11 - 5.0 1.2

R 0.66 0.73 0.79 0.74 0.76 0.71N 35 33 33 53 23 431 14 19 23 8 24 14

d 1.22 1.41 1.08 2.14 0.91 1.41F 1.47 1.00 1.79 - 2.28 1.42

x2 (f) 14.1 (7) 14.7 (7) 14.8 (7) - 47.1 (7) 22.1 (7)

salaried employees. Rural districts. (3.2.) 102 groups of 3 households.

groups

161

0.99 1.82 1.45 2.01 1.23 1.78 1.542.15 2.14 - - - - -

36.8 (7) 47.0 (7) - - - -

1300 20000 1500 960 - 230.108 620

1.721.108.81 (9)

2.051.827.93 (9)

2.360.93

13.83 (9)

1.861.368.91 (9)

- 1.52- 2.64- 3.78 (9)

2.221.38

11.22 (9)

0.7241

7

1.421.47

14.0 (7)

56.40-11.5

0.764411

1.941.96

38.8 (7)

33.37-51.3

0.843715

1.791.36

18.6

100.6-28.9

(7)

0.683512

1.842.38

43.2 (7)

61.7-18.7

0.773916

1.91

--

314.2-137

0.4537142.18

--

358.7-174

0.7951

7

1.951.76

11.5

114.2-28.7

(7)

37 41 33 35 33 27 3910 11 15 13 15 14 14

0.73 0.58 0.86 0.71 - 0.30 0.8047 52 62 42 - 42 53

7 5 7 6 - 11 6

-5.340 -5.790 -5.395 -5.317 - -9.377 -5.0301688 15290 2538 1219 - 409.8.108 1397

0.154 0.232 0.119 0.163 - 0.252 0.1020.162 0.313 0.115 0.190 - 0.410 0.1200.38 0.55 0.29 0.40 - 0.13 0.26

4.578 7.814 2.891 2.891 2.891 2.891 2.891

0.354 0.534 - - -0.519 0.780 - .- -3.2 6.3 - - -1.3 4.3 - - -0.66 0.76 0.79 0.67 0.74 0.46 0.86

Washing & Durables excl. Personal Books, newspapers Sports, holidays, Transport incl. Union fees,cleaning vehicles hygiene etc. hobbies Own car subscriptions etc.

162

Appendix A. Main results (continued). Skilled workers.

ExpenditureParameter estimates

and test results

R 0.81 0.45 0.92 0.70 0.82 0.69N 25 19 29 21 22 28

7 10 6 6 5 5

Dwelling Fuel & light Food Tobacco Clothing Footwear

log y = a + b (log x log x)

a 2.409 2.336 2.982 2.168 2.366 1.769

b 0.926 0.322 0.649 0.933 1.021 0.617log x 3.488 3.488 3.488 3.488 3.488 3.488

s1 0.126 0.105 0.0481 0.185 0.134 0.129

s2 0.123 0.120 0.0520 0.166 0.134 0.1220.017 0.017 0.0073 0.023 0.019 0.0170.095 0.091 0.040 0.14 0.10 0.093

dFx2 (f)

1.650.950.6 (3)

1.431.302.3 (3)

2.031.179.4 (3)

2.120.815.9 (3)

1.950.993.0 (3)

1.940.898.6 (3)

-2.409 2.336 2.982 2.168 2.366 1.769

-0.126 -0.0565 -0.0870 -0.118 -0.141 -0.0843

3.523 3.523 3.523 3.523 3.523 3.523

0.126 0.105 0.0481 0.185 0.134 0.1290.120 0.111 0.0618 0.173 0.137 0.1240.017 0.016 0.0087 0.024 0.019 0.0170.013 0.012 0.0066 0.019 0.015 0.013

0.82 0.57 0.88 0.67 0.81 0.6825

5

1.720.914.1 (3)

2010

1.671.112.9 (3)

2561.321.662.2 (3)

2551.950.873.7 (3)

2541.841.056.2 (3)

2851.860.923.7 (3)

- log x)207.1 211.2 870.1 128.8 187.5 55.23555 254 1450 331 526 88.6

3.349 3.426 3.404 3.364 3.354 3.4070.291 0.243 0.111 0.425 0.309 0.2980.293 0.247 0.128 0.420 0.283 0.2639.3 7.5 16 8.3 8.1 2.1

57 45 100 53 51 13

log y = a + b (

ab101 . .X

s2

Sa

sylO4RN

dFX2 (f)

y = a + b (log xab

s,s2

Sa

Rural districts. (3.3.) 51 groups of 3 households.

163

304

239

217

284

295

305

206

2.30 1.50 1.47 1.55 2.37 2.23 1.870.99 2.41 1.48 1.20 1.51 1.91 1.051.1 (3) 3.7 (3) 11.5 (3) 1.5 (3) 7.1 (3) 4.9 (3) 6.6 (3)

1.747 2.114 1.826 1.598 2.169 2.247 2.182-0.116 -0.152 -0.108 -0.120 -0.208 -0.266 -0.123

3.523 3.523 3.523 3.523 3.523 3.523 3.523

0.132 0.215 0.106 0.136 0.164 0.252 0.09430.141 0.331 0.135 0.150 0.231 0.358 0.1140.020 0.046 0.019 0.021 0.032 0.050 0.0160.015 0.035 0.014 0.016 0.024 0.038 0.012

0.74 0.52 0.73 0.73 0.77 0.71 0.823041.961.13

2391.562.38

2251.311.64

2641.521.21

257

1.841.99

2952.112.03

21

61.311.47

3.2 (3) 2.6 (3) 10.6 (3) 1.3 (3) 2.3 (3) 2.0 (3) 2.2 (3)

47.21 208.9 58.53 34.82 224.3 49.40 128.6117 928 131 81.3 1170 751 310

3.364 3.494 3.378 3.375 3.494 3.181 3.3720.305 - 0.243 0.314 - 0.579 0.2170.357 - 0.282 0.369 - 0.812 0.2552.5 - 2.5 1.9 - 10 4.9

16 - 15 12 - 100 31

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

Union fees,subscriptions etc.

1.747 2.114 1.826 1.598 2.169 2.247 2.1820.875 1.068 0.806 0.862 1.606 1.964 0.9383.488 3.488 3.488 3.488 3.488 3.488 3.4880.132 0.215 0.106 0.136 0.164 0.252 0.09430.132 0.333 0.128 0.149 0.201 0.347 0.09640.018 0.047 0.018 0.021 0.028 0.049 0.0140.10 0.25 0.098 0.11 0.15 0.26 0.073

0.78 0.52 0.76 0.74 0.83 0.73 0.88

164

Appendix A. Main results (continued). Skilled workers.

ExpenditureParameter estimates

and test results

5: log y = log k + log D (a + log x)

Dwelling Fuel & light Food Tobacco Clothing Footwear

RN

dF

0.8123

7

1.701.02

0.632010

1.661.03

0.9026

61.601.35

0.6723

82.080.98

0.8323

41.710.84

0.7028

51.900.78

x2 (f) 5.4 (3) 3.3 (3) 3.4 (3) 5.4 (3) 0.5 (3) 4.0 (3)

4: y = a + b (

a 206.4 209.4 874.2 132.3 196.6 55.00b10-4 -52.2 -29.7 -15.5 -32.7 -52.0 -9.81

I 10+ 4.935 4.070 4.268 4.609 4.675 4.263X

si 0.291 0.243 0.111 0.425 0.309 0.2980.326 0.245 0.178 0.470 0.351 0.274

sa 10 7.3 23 9.6 11 2.2

SbiO4 5.7 4.9 14 5.8 6.2 1.4

R 0.79 0.66 0.84 0.64 0.77 0.71

N 17 24 17 25 19 251 12 9 13 9 9 6

d 1.31 1.82 0.81 1.81 1.19 1.70

F 1.26 1.02 2.60 1.22 1.29 0.85(f) 8.4 (3) 3.3 (3) 5.9 (3) 3.9( 3) 8.2 (3) 3.8 (3)

ak

-9.484134010

-3.527449.9

- -5.272- 4045

- -4.431- 341.6

s1 0.126 0.105 - 0.185 - 0.129s2 0.358 0.115 - 0.164 - 0.120

0.053 0.55 - 0.77 - 0.54Sk 260.102 190 - 6300 - 2500

R 0.27 0.59 - 0.66 - 0.68N 11 19 - 25 - 281 24 10 - 5 - 5

d 0.29 1.51 - 2.10 - 1.96F 8.07 1.21 - 0.79 - 0.87

z2 (f) 25.20 (5) 1.42 (5) - 2.80 (5) - 4.22 (5)

Rural districts. (3.3.) 51 groups of 3 households.

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.Own car

Union fees,subscriptions etc.

165

0.76 0.59 0.78 0.21 0.56 - 0.8630 23 21 15 27 - 224 9 7 14 7 - 62.290.992.51 (5)

1.522.333.16 (5)

1.441.445.95 (5)

0.278.37

23.25 (5)

1.462.588.84 (5)

- 1.83- 1.06- 2.36 (5)

0.73264

0.2117

9

0.7822

5

0.6924

8

0.4113

16

0.73239

0.82196

1.94 2.54 1.24 1.53 2.35 1.45 1.221.37 - 1.34 1.39 - 1.96 1.387.0 (3) - 4.5 (3) 4.8 (3) - 4.1 (3) 1.5 (3)

50.09 109.5 59.15 35.22 224.3 52.37 129.9-12.1 -37.1 -13.5 -7.60 -133 -72.4 -29.6

4.496 5.628 4.530 4.622 3.479 6.782 4.675

0.305 0.494 0.243 0.314 - 0.579 0.2170.452 1.21 0.343 0.401 - 1.02 0.3513.5 20 3.0 2.1 - 15 6.82.1 11 1.8 1.2 - 12 4.0

0.65 0.42 0.73 0.66 0.50 0.65 0.7217 15 21 24 15 21 15

7 11 6 8 16 12 15

-5.151 -5.664 -4.963 -9.510 -9.384 - -5.3181161 8798 959.2 220.3.108 548.108 - 4529

0.132 0.215 0.106 0.136 0.164 - 0.09430.131 0.327 0.127 0.394 0.263 - 0.09710.51 0.80 0.41 0.054 0.099 - 0.36

1100 16000 700 42.108 230.108 - 3300

1.51 1.17 0.85 1.35 2.19 0.91 0.722.20 6.03 1.99 1.63 - 3.10 2.61

19.3 (3) 52.0 (3) 6.4 (3) 10.0 (3) - 5.5 (3) 11.0 (3)

166

Appendix A. Main results (continued). Unskilled workers.

ExpenditureParameter estimates

and test resultsDwelling Fuel & light Food Tobacco Clothing Footwear

log y = a + b (log x - log x)a.blog x

2

Sb

RN 44 391 5 10

d 1.91 1.54F 1.31 1.04

x2 (f) 3.8 (7) 17.7 (7)

2: log y = a + b (X X

a 2.321 2.336b104 -0.112 -0.0795

3: y = a + b (log x - log x)a 163.1 184.7b 416 350

1 . 10 3.788 3.788X

s1 0.133 0.114s2 0.164 0.110

0.017 0.011

5b 10 0.0097 0.0065

R 0.77 0.79N 35 401 11 14

d 1.66 1.71F 1.51 0.93x2 (f) 7.4 (7) 8.2 (7)

2.321 2.336 2.984 2.121 2.366 1.7430.981 0.642 0.665 1.275 0.993 0.4983.468 3.468 3.468 3.468 3.468 3.4680.133 0.114 0.0565 0.169 0.112 0.1120.152 0.116 0.0520 0.187 0.136 0.1250.016 0.012 0.0054 0.019 0.014 0.0130.076 0.058 0.026 0.093 0.068 0.063

0.80 0.76 0.94 0.82 0.84 0.6446

649

646

6

478

1.74 2.28 2.43 2.170.85 1.22 1.47 1.259.0 (7) 11.7 (7) 3.6 (7) 8.9 (7)

2.984 2.121 2.366 1.743-0.0768 -0.145 -0.111 -0.0565

3.788 3.788 3.788 3.788

0.0565 0.169 0.112 0.1120.0619 0.202 0.153 0.1290.0064 0.021 0.016 0.0130.0037 0.012 0.0091 0.0077

0.91 0.78 0.79 0.613591.231.20

51

8

1.951.43

457

1.911.85

478

2.031.33

2.9 (7) 10.6 (7) 1.5 (7) 11.7 (7)

827.9 172.9 173.7 52.661420 531 461 65.7

3.336 3.468 3.269 3.3700.130 - 0.259 0.2580.126 - 0.367 0.260

11 - 7.5 1.558 - 37 7.4

s1

3.2810.306

3.3240.262

s2 0.385 0.2497.3 5.1

Sb 36 26

Rural districts. (3.4.) 93 groups of 3 households.

groups

167

469

456

51

847

651

649

9366

1.93 1.76 1.96 1.79 2.02 1.85 1.860.91 1.75 0.94 0.96 1.23 1.54 0.799.0 (7) 7.3 (7) 3.8 (7) 5.2 (7) 5.2 (7) 4.0 (7) 10.6 (7)

1.696 2.068 1.775 1.556 2.073 2.057 2.180-0.0993 -0.104 -0.0990 -0.107 -0.179 -0.174 -0.108

3.788 3.788 3.788 3.788 3.788 3.788 3.788

0.136 0.207 0.108 0.146 0.173 0.236 0.08570.144 0.264 0.118 0.146 0.214 0.336 0.09110.015 0.027 0.012 0.015 0.022 0.035 0.00950.0086 0.016 0.0070 0.0087 0.013 0.020 0.0054

0.77 0.57 0.83 0.79 0.83 0.67 0.9039

944

7

4010

406

456

3910

376

1.55 1.90 1.49 1.69 1.60 1.42 1.341.13 1.61 1.20 1.01 1.53 2.02 1.13

10.5 (7) 7.8 (7) 14.1 (7) 8.8 (7) 4.9 (7) 4.5 (7) 9.4 (7)

39.87 89.28 46.16 27.16 33.27 219.7 114.592.5 279 110 71.5 322 1060 282

3.295 3.246 3.290 3.274 3.090 3.468 3.2870.313 0.477 0.248 0.335 0.398 - 0.1970.344 0.636 0.268 0.358 0.634 - 0.2 181.6 7.0 1.4 1.1 3.8 - 2.97.8 35 7.1 5.7 26 - 14

Washing & Durables excl. Personalcleaning vehicles hygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

Union fees,subscriptions etc.

1.696 2.068 1.775 1.556 2.073 2.057 2.1800.888 0.791 0.872 0.914 1.571 1.659 0.9353.468 3.468 3.468 3.468 3.468 3.468 3.4680.136 0.207 0.108 0.146 0.173 0.236 0.08570.129 0.275 0.104 0.143 0.192 0.294 0.07600.013 0.028 0.011 0.015 0.020 0.030 0.00790.065 0.14 0.052 0.071 0.096 0.15 0.038

0.82 0.52 0.87 0.80 0.86 0.76 0.93

168

Appendix A. Main results (continued). Unskilled workers.

log y = log k + log 1) (a + log x)

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

R 0.77 0.82 0.93 0.68 0.79 0.68N 35 44 43 39 49 491 11 14 6 8 8 5

d 1.67 1.69 1.49 2.23 1.65 2.08F 1.58 0.91 0.94 - 2.02 1.02

x2 (f) 11.3 (7) 17.7 (7) 5.9 (7) - 15.2 (7) 5.1 (7)

4: y = a + b ( -a 248.0 182.8 834.5 172.9 276.3 52.94bl04 -60.3 -29.7 -124 -54.0 -72.5 -6.10

I 10 3.788 5.460 5.206 3.788 3.788 4.771X

s1 - 0.262 0.130 - - 0.258- 0.287 0.194 - - 0.284- 5.9 18 - - 1.6

s 10 - 2.3 7.3 - - 0.72

R 0.67 0.80 0.87 0.73 0.69 0.66N 31 40 27 39 37 451 11 15 17 10 9 9

d 1.73 1.35 0.70 1.37 2.31 1.78

F - 1.20 2.22 - - 1.21

x2 (f) - 20.7 (7) 33.7 (7) - - 3.6 (7)

RN1

dFx2 (f)

0.7544

51.921.31

4.29 (9)

0.804111

1.640.97

14.49 (9)

0.9446

61.740.84

14.75 (9)

0.473318

0.753.83

13.96 (9)

0.8246

6

2.381.494.29 (9)

0.6847

82.181.249.18 (9)

a -5.369 -4.449 -4.504 -9.373 -5.397 -3.998k. 7442 1350 6526 484.7.108 8801 188.6

1 0.133 0.114 0.0565 0.169 0.112 0.112

0.152 0.112 0.0518 0.330 0.137 0.1250.33 0.31 0.15 0.58 0.28 0.33

5200 590 1400 11.108 5200 68

Rural districts. (3.4.) 93 groups of 3 households.

groups

670 1400 580 620 11010

0.79 0.59 0.86 0.82 0.58

169

1.07 1.78 0.90 1.28 1.38 2.27 1.661.82 - 2.09 1.70 - - -

42.1 (7) - 30.6 (7) 28.1 (7) -

1.880.918.19 (9)

1.801.707.93 (9)

1.950.945.01 (9)

1.820.94

11.25 (9)

0.982.52

11.29 (9)

0.7835

91.581.21

0.6642

7

1.971.77

0.864481.511.17

0.8242

7

1.641.14

0.8131

14

0.972.53

0.1617

201.62

-

0.903561.291.22

31.4 (7) 26.2 (7) 27.4 (7) 13.9 (7) 35.7 (7) - 12.7 (7)

41.69 152.7 45.17 26.93 175.9 219.7 169.9-7.63 -33.7 -9.03 -5.62 -68.7 -99.9 -40.1

5.619 3.788 6.025 6.204 3.788 3.788 3.788

0.313 - 0.248 0.3350.422 - 0.358 0.4362.0 - 1.9 1.40.78 - 0.71 0.51

0.73 0.68 0.80 0.75 0.52 0.45 0.8831 42 31 35 27 19 2913 7 12 14 14 22 15

48 45 47 49 339 6 8 6 19

-5.118 -4.876 -5.077 -5.195 -9.3061022 1497 1127 868.9 327.710

0.136 0.207 0.108 0.146 0.1730.130 0.270 0.105 0.141 0.2750.34 0.54 0.27 0.37 0.075

Washing & Durables excl. Personal Books, newspapers Sports, holidays, Transport incl. Union fees,cleaning vehicles hygiene etc. hobbies own car subscriptions etc.

170

Appendix A. Main results (continued). Agricultural workers.

ExpenditureParameter estimates

and test resultsDwelling Fuel & light Food Tobacco Clothing Footwear

log y = a + b (log x - i)ablog x

s2

s

RN1

dFx2 (f)

2: log y = a + b (

ab1041 . 10X

s

RN1

dFx2 (f)

3: y = a + b (log xablog xsis2

s

21

6276

277

268

285

225

1.57 1.95 1.58 2.26 2.05 1.831.31 1.36 1.21 1.56 1.54 0.876.6 (3) 8.6 (3) 5.8 (3) 1.6 (3) 3.2 (3) 2.4 (3)

2.173 2.276 2.937 1.997 2.195 1.609-0.0816 -0.0655 -0.0592 -0.0969 -0.112 -0.0439

4.803 4.803 4.803 4.803 4.803 4.803

0.170 0.105 0.0618 0.177 0.133 0.1170.206 0.132 0.0670 0.233 0.166 0.1090.028 0.018 0.0092 0.032 0.023 0.0150.013 0.0085 0.0043 0.015 0.011 0.0070

0.65 0.74 0.89 0.67 0.83 0.66

2.173 2.276 2.937 1.997 2.195 1.6090.962 0.764 0.651 1.132 1.236 0.4873.361 3.361 3.361 3.361 3.361 3.3610.170 0.105 0.0618 0.177 0.133 0.1170.194 0.123 0.0680 0.221 0.165 0.1090.027 0.017 0.0093 0.030 0.023 0.0150.14 0.087 0.048 0.16 0.12 0.077

0.70 0.78 0.88 0.71 0.83 0.66

216

19

1429

5

256

294

236

1.40 1.67 1.65 2.02 2.04 1.831.47 1.58 1.17 1.74 1.55 0.882.6 (3) 4.7 (3) 1.8 (3) 0.6 (3) 6.3 (3) 2.9 (3)

- log x)130.3 168.0 756.6 75.22 103.4 38.77291 298 1270 242 360 46.8

3.205 3.234 3.240 3.158 3.135 3.2730.391 0.242 0.142 0.407 0.306 0.2680.500 0.3 17 0.148 0.534 0.409 0.2569.7 7.7 16 6.5 6.9 1.4

51 40 84 34 38 7.3

Rural districts. (3.5.) 53 groups of 3 households.

171

238

305

326

314

227

256

226

1.43 2.33 1.78 1.52 1.34 1.83 1.852.07 1.39 0.94 1.54 1.54 2.20 1.278.8 (3) 1.3 (3) 5.6 (3) 6.1 (3) 1.8 (3) 9.7 (3) 3.5 (3)

1.660 1.953 1.655 1.464 1.900 1.846 2.080-0.0634 -0.0823 -0.0656 -0.106 -0.118 -0.0677 -0.0782

4.803 4.803 4.803 4.803 4.803 4.803 4.803

0.132 0.220 0.109 0.153 0.191 0.208 0.1150.214 0.252 0.106 0.172 0.265 0.327 0.1330.029 0.035 0.015 0.024 0.036 0.045 0.0180.014 0.016 0.0068 0.011 0.017 0.021 0.0085

0.54 0.58 0.80 0.80 0.70 0.41 0.7923

6

354

247

286

247

238

254

1.15 2.50 1.72 1.90 1.07 1.66 1.752.65 1.31 0.95 1.25 1.91 2.47 1.331.7(3) 1.1 (3) 5.3 (3) 3.3 (3) 6.6 (3) 8.1 (3) 3.4 (3)

43.69 72.85 38.63 14.84 37.47 76.44 93.5883.5 210 74.6 74.7 244 136 2373.237 3.169 3.224 3.092 3.062 3.214 3.1880.303 0.507 0.251 0.354 0.441 0.479 0.2650.546 0.564 0.251 0.380 0.776 0.894 0.3003.5 6.4 1.4 1.0 5.7 9.7 4.3

18 34 7.4 6.0 36 51 22

groups

Washing &cleaning

Durables excl.vehicles

Personalhygiene

Books, newspapersetc.

Sports, holidays,hobbies

Transport incl.own car

Union fees,subscriptions etc.

1.660 1.953 1.655 1.464 1.900 1.846 2.0800.865 0.847 0.727 1.096 1.429 0.922 0.8753.361 3.361 3.361 3.361 3.361 3.361 3.3610.132 0.220 0.109 0.153 0.191 0.208 0.1150.189 0.260 0.105 0.191 0.237 0.309 0.1300.026 0.036 0.014 0.026 0.033 0.042 0.0180.13 0.18 0.075 0.13 0.17 0.22 0.092

0.67 0.54 0.81 0.75 0.77 0.51 0.80

172

Appendix A. Main results (continued). Agricultural workers.

5: log y = log k + log I (a + log x)

Parameter estimatesand test results

Expenditure

Dwelling Fuel & light Food Tobacco Clothing Footwear

R 0.63 0.72 0.90 0.71 0.80 0.67N 23 21 31 25 27 221 7 19 5 6 7 6d 1.60 1.82 1.68 2.02 1.90 1.88F 1.63 1.72 1.09 1.72 1.78 0.91

(f) 4.9 (3) 20.2 (3) 0.8 (3) 3.2 (3) 2.1 (3) 8.5 (3)

4:y= a+ b(_)a 128.8 169.3 759.5 75.24 192.6 39.22b10 -20.6 -21.3 -90.3 -15.6 -41.8 -3.39

ak

-5.1934530

-4.6571965

-4.3235381

-5.6358802

-5.90328920

-3.849131.7

si 0.170 0.105 0.0618 0.177 0.133 0.1170.193 0.122 0.0651 0.220 0.162 0.1080.60 0.39 0.24 0.60 0.45 0.49

5600 1200 1800 13000 34000 68

R 0.59 0.71 0.90 0.70 0.82 0.67N 21 27 29 28 28 221 6 6 5 4 5 5

d 1.56 1.94 1.69 2.25 2.15 1.89

F 12.9 1.35 1.11 1.54 1.49 0.85

z2 (f) 2.90 (5) 10.64 (5) 3.69 (5) 0.05 (5) 3.16 (5) 4.00 (5)

I 10+ 7.083 6.497 6.400 7.905 4.803 5.840X

s1 0.391 0.242 0.142 0.407 - 0.268s2 0.529 0.344 0.180 0.611 - 0.269sa 10 8.5 20 7.4 - 1.5

3.5 3.0 7.1 2.5 - 0.57

R 0.64 0.71 0.87 0.66 0.72 0.64N 23 17 23 21 23 21

1 9 19 10 10 7 6d 1.39 1.47 1.20 1.65 1.94 1.73

F 1.83 2.02 1.60 2.25 - 1.00

x2 (f) 3.3 (3) 3.2 (3) 8.3 (3) 2.9 (3) - 7.0 (3)

Rural districts. (3.5.) 53 groups of 3 households.

groups

173

680 2100 250 2500 - 2400 1700

45.74 72.83 39.47 12.30 28.90 84.90 94.73-579 -14.0 -5.15 4.84 -16.6 -10.3 -16.1

6.339 7.524 6.558 9.269 9.861 6.176 7.203

0.303 0.507 0.251 0.354 0.441 0.479 0.2650.646 0.580 0.299 0.400 0.986 0.998 0.3664.3 6.7 1.7 1.0 6.6 12 5.31.6 2.2 0.61 0.37 2.8 4.6 1.8

0.46 0.66 0.76 0.88 0.64 0.29 0.78

0.5223

81.372.144.91 (5)

0.6030

5

23.71.331.98 (5)

0.8332

61.830.90

13.66 (5)

0.8431

41.691.396.66 (5)

- 0.34- 25- 6- 1.88- 2.21- 10.29 (5)

0.822441.861.243.03 (5)

21 33 16 27 17 17 238 4 19 8 7 12 8

0.53 0.66 0.82 0.90 0.71 0.37 0.8323 31 26 28 18 21 25

8 4 7 8 7 9 41.17 2.23 1.73 1.99 0.76 1.67 1.81

3.24 1.24 1.00 1.15 3.10 3.49 1.28

3.2 (3) 2.7 (3) 3.6 (3) 4.0 (3) 29.5 (3) 34.4 (3) 3.2 (3)

1.00 2.12 1.37 1.85 0.49 1.47 1.444.53 1.31 1.43 1.28 4.99 4.34 1.90

10.0 (3) 5.1 (3) 9.3 (3) 2.4 (3) 45.4 (3) 44.3 (3) 13.1 (3)

-4.918 -4.898 -4.562 -5.549 - -5.072 -4.964778.2 1467 399.4 2066 - 1638 2244

0.132 0.220 0.109 0.153 - 0.208 0.1150.193 0.254 0.103 0.180 - 0.309 0.1280.47 0.79 0.41 0.53 - 0.74 0.41

Washing & Durables excl. Personal Books, newspapers Sports, holidays, Transport incl. Union fees,cleaning vehicles hygiene etc. hobbies own car subscriptions etc.

174

Appendix B.

The tables show the correlation coefficient') between the residuals of the differentexpenditure items, where the calculated expenditures are calculated from the functionlog y = a + b (log x - log x). The tables are shown separately for each of the twelvegroups of wage and salary earners cf. the heads of the tables.

The numbers 1-13 in the head and the front column of the tables denote the differentexpenditure items according to the following code:

Dwelling.Full and light.Food.Tobacco.Clothing.Footwear.Washing and cleaning.Durables (excl, own car).Personal hygiene.Books, newspapers etc.Sports, holidays, hobbies.Transport incl, own car.Union fees, subscriptions etc.

) Cf. chapter VI, p. 105.

1.1 Higher public servants and salaried employees. The capital.

1 2 3 4 5 6 7 8 9 10 11 12 13

- 0.309 0.076 0.045 -0.093 -0.043 0.161 0.176 0.067 -0.096 0.023 0.057 -0.1002 - - 0.017 -0.053 -0.240 -0.081 -0.002 0.099 -0.145 -0.181 -0.385 -0.060 -0.1563 - - - 0.288 0.290 0.102 0.365 -0.045 0.145 -0.032 0.176 0.004 0.0334 - - - - 0.163 0.055 0.176 -0.116 0.233 0.160 0.122 0.032 -0.1565 - - - - - 0.578 0.319 -0.052 0.485 0.141 0.359 0.003 0.1006 - - - - - - 0.191 0.060 0.432 0.116 0.127 -0.114 -0.0227 - - - - - - - -0.011 0.338 0.020 0.258 -0.027 0.0778 - - - - - - - - 0.017 -0.139 -0.030 -0.043 0.041g - - - - - - - - - 0.381 0.342 -0.028 0.044

10 - - - - - - - - - - 0.378 0.053 0.08111 - - - - - - - - - - - 0.065 0.14112 - - - - - - - - - - - - 0.08013 - - - - - - - - - - - - -

1.2 Lower public servants and salaried employees. The capital.

1 2 3 4 5 6 7 8 9 10 11 12 13

1 - 0.395 0.082 -0.055 -0.046 0.122 0.154 0.039 0.02 1 0 077 0.074 -0.028 0.0212 - - 0.076 -0.055 -0.125 -0.075 0.073 -0.016 -0.139 0.014 -0.144 -0.070 0.0063 - - - 0.262 0.013 0.004 0.204 0.048 0.053 0.166 0.073 -0.061 0.1524 - - - - 0.013 -0.064 0.128 -0.011 0.042 0.158 0.067 0.102 0.1875 - - - - - 0.469 0.199 0.099 0.365 0.124 0.302 -0.046 0.0216 - - - - - - 0.132 0.148 0.370 0.078 0.172 -0.073 -0.0067 - - - - - - - 0.043 0.235 0.114 0.185 0.095 0.0558 - - - - - - - - 0.094 0.146 -0.024 -0.054 0.1229 - - - - - - - - - 0.222 0.207 -0.041 0.061

10 - - - - - - - - - - 0.109 -0.116 0.34711 - - - - - - - - - - - 0.051 0.00412 - - - - - - - - - - - - 0.08213 - - - - - - - - - - - - -

1.3 Skilled workers. The capital.

1 2 3 4 5 6 7 8 9 10 11 12 13

1 - 0.369 -0.117 0.003 -0.324 -0.283 -0.046 0.054 -0.047 0.155 0.025 0.017 0.162

2 - - 0.110 -0.057 -0.185 -0.146 -0.126 0.015 -0.038 0.205 -0.066 -0.124 0.2253 - - - -0.017 -0.193 -0.037 0.267 -0.165 0.065 0.079 0.048 -0.340 0.1974 - - - - -0.127 -0.089 0.244 0.085 0.120 0.034 0.171 -0.185 -0.0315 - - - - - 0.480 -0.176 0.112 0.265 0.061 0.110 0.010 -0.2076 - - - - - - -0.132 0.056 0.120 -0.077 0.074 -0.077 -0.1227 - - - - - - - -0.055 0.018 0.099 0.204 -0.035 0.0868 - - - - - - - - 0.076 0.152 0.109 -0.182 -0.0399 - - - - - - - - - 0.173 0.116 0.018 -0.055

10 - - - - - - - - - - 0.165 -0.110 0.06411 - - - - - - - - - - - -0.219 -0.22712 - - - - - - - - - - - - -0.09713 - - - - - - - - - - - - -

1.4 Unskilled workers. The capital.

1 2 3 4 5 6 7 8 9 10 11 12 13

1 - 0.336 -0.097 0.024 0.161 0.122 0.151 0.067 0.169 0.074 0.034 -0.167 -0.0162 - - -0.166 -0.034 -0.171 -0.123 -0.073 0.133 0.079 -0.031 -0.049 0.020 0.127

3 - - - 0.133 0.025 -0.086 0.265 -0.143 0.014 0.114 0.049 -0.032 0.186

4 - - - - -0.100 -0.030 0.186 -0.104 0.025 0.069 0.024 0.041 0.129

5 - - - - - 0.532 0.172 0.196 0.338 0.126 0.182 -0.187 0.034

6 - - - - - - 0.265 0.099 0.300 0.053 0.092 -0.032 -0.0667 - - - - - - - -0.034 0.121 0.180 -0.049 -0.026 -0.0298 - - - - - - - - 0.111 -0.023 -0.053 -0.161 0.040

9 - - - - - - - - - 0.110 0.083 -0.012 0.114

10 - - - - - - - - - - 0.102 -0.171 0.190

11 - - - - - - - - - - - -0.164 0.186

12 - - - - - - - - - - - - -0.01613 - - - - - - - - - - - - -

2.1 Higher public servants and salaried employees. Provincial towns.

1 2 3 4 5 6 7 8 9 10 11 12 13

- 0.294 0.129 0.076 -0.052 -0.102 0.119 0.115 0.054 0.114 -0.053 -0.122 0.2402 - - 0.338 -0.026 -0.192 -0.131 0.050 -0.075 -0.182 0.044 -0.078 -0.086 0.0473 - - - 0.016 -0.235 -0.200 0.087 -0.108 -0.071 -0.039 -0.206 -0.093 0.1354 - - - - 0.282 0.088 0.307 -0.095 0.368 0.072 0.288 -0.070 0.2435 - - - - - 0.469 0.252 0.130 0.507 0.121 0.237 -0.062 0.0876 - - - - - - 0.060 0.001 0.423 0.078 0.268 -0.115 -0.0507 - - - - - - - -0.048 0.366 0.2 14 0.032 -0.093 0.37 1

8 - - - - - - - - -0.024 -0.096 -0.130 -0.122 -0.0729 - - - - - - - - - 0.210 0.323 -0.114 0.311

10 - - - - - - - - - - 0.282 -0.162 0.06611 - - - - - - - - - - - -0.182 0.06612 - - - - - - - - - - - --0.04913

2.2 Lower public servants and salaried employees. Provincial towns.

1 2 3 4 5 6 7 8 9 10 11 12 13

1 - 0.250 0.001 0.023 -0.066 -0.109 0.124 -0.010 -0.101 -0.130 -0.160 -0.054 -0.0072 - - 0.127 -0.040 -0.252 -0.223 0.029 -0.075 -0.252 -0.045 -0.165 -0.167 0.1133 - - - 0.009 -0.045 0.073 0.028 -0.132 -0.090 -0.076 -0.117 -0.112 0.1594 - - - - 0.018 -0.053 0.204 -0.094 -0.026 0.084 0.191 -0.055 0.1295 - - - - - 0.510 0.080 0.020 0.390 0.065 0.111 -0.075 0.0266 - - - - - - 0.024 -0.008 0.397 0.167 0.138 -0.042 0.0287 - - - - - - - -0.044 0.098 0.157 0.104 0.014 0.0198 - - - - - - - - 0.052 0.002 - 0.116 -0.042 -0.1079 - - - - - - - - - 0.162 0.172 -0.029 -0.050

10 - - - - - - - - - - 0.182 -0.032 0.07811 - - - - - - - - - - - -0.148 0.05812 - - - - - - - - - - - - -0.13913 - - - - - - - - - - - - -

2.3 Skilled workers. Provincial towns.

2.4 Unskilled workers. Provincial towns.

2 3 4 5 6 7 8 9 10 11 12 13

- 0.278 0.026 -0.032 0.046 -0.104 -0.013 -0.064 0.000 -0.089 -0.013 0.062 -0.0342 - - 0.149 -0.135 -0.125 -0.109 -0.202 0.097 0.128 0.216 -0.274 -0.129 0.1353 - - - -0.077 0.061 0.024 -0.034 -0.236 -0.035 0.239 -0.101 -0.138 0.2884 - - - - -0.034 0.097 0.293 -0.038 -0.009 0.085 0.255 -0.158 0.0935 - - - - - 0.596 0 179 0.191 0.136 0.000 0.191 -0.396 -0.1236 - - - - - - 0.122 0.174 0.148 0.144 0.127 -0.336 -0.0487 - - - - - - - -0.098 0.156 0.061 0.467 -0.070 0.1308 - - - - - - - - 0.164 0.004 -0.141 -0.224 -0.2429 - - - - - - - - - 0.309 0.086 -0.045 0.028

10 - - - - - - - - - - 0.014 -0.223 0.25111 - - - - - - - - - - - -0.186 0.26912 - - - - - - - - - - - - -0.09113

2 3 4 5 6 7 8 9 10 11 12 13

- 0.292 -0.065 -0.104 -0.066 -0.119 0.120 -0.122 0.012 -0.075 -0.035 -0.232 0.0362 - - 0.217 0.009 -00l3 -0.059 0.099 -0.040 -0.128 0.105 -0.174 -0.288 0.1083 - - - 0.042 -0.388 -0.327 -0.111 -0.183 -0.326 0.193 -0.209 -0.208 0.2604 - - - - -0.032 -0.078 0.073 -0.135 0.017 0.211 -0.063 0.009 0.061

5 - - - - - 0.590 0.445 0.086 0.373 0.028 -0.006 -0.135 -0.1986 - - - - - - 0.307 0,088 0.394 0.019 0.149 -0.109 -0.1157 - - - - - - - -0.088 0.482 0.137 -0.096 -0.196 0.0678 - - - - - - - - 0.015 -0.174 -0.101 -0.191 -0.0179 - - - - - - - - - 0.073 0.101 -0.081 -0.200

10 - - - - - - - - - - 0.085 -0.151 0.09911 - - - - - - - - - - - 0.411 -0.04312 - - - - - - - - - - - - -0.01113

3.2 Lower public servants and salaried employees. Rural districts.

3.3 Skilled workers. Rural districts.

1 2 3 4 5 6 7 8 9 10 11 12 13

- 0.183 0.036 -0.005 -0.112 -0.079 -0.044 -0.199 -0.037 -0.112 -0.114 -0.061 -0.0112 - - -0.024 -0.099 -0.222 -0.227 -0.123 -0.087 -0.248 0.011 -0.263 -0.186 -0.0243 - - - 0.046 0.001 -0.042 -0.016 -0.014 -0.047 0.085 0.124 0.037 -0.1724 - - - - 0.157 0.028 0.195 0.010 0.192 0.017 0.315 -0.066 0.0605 - - - - - 0.428 0.152 0.251 0.359 0.042 0.090 -0.068 0.0696 - - - - - - 0.220 0.092 0.360 0.041 0.032 -0.032 -0.0567 - - - - - - - -0.044 0.291 0.171 -0.152 0.038 0.1108 - - - - - - - - 0.097 -0.076 -0.004 -0.149 -0.0119 - - - - - - - - - 0.085 0.178 -0.071 -0.111

10 - - - - - - - - - - -0.081 0.130 0.05211 - - - - - - - - - - - -0.022 -0.18012 - - - - - - - - - - - - -0.02113 - - - - - - - - - - - - -

1 2 3 4 5 6 7 8 9 10 11 12 13

I - 0.341 0.005 0.021 -0.097 -0.205 0.109 -0.102 -0.004 0.062 -0.149 -0.167 0.0352 - - 0.093 0.254 0.036 -0.144 0.087 -0.197 -0.154 0.251 -0.224 0.002 0.0143 - - - 0.231 0.145 -0.118 0.196 -0.327 -0.035 0.009 -0.096 -0.301 0.3144 - - - - 0.092 0.040 0.251 -0.234 -0.015 0.230 -0.010 -0.207 0.1655 - - - - - 0.496 0.429 -0.292 0.082 0.057 0.084 -0.124 -0.0256 - - - - - - 0.052 -0.098 0.255 0.104 0.216 0.068 -0.0257 - - - - - - - -0.274 0.384 0.058 0.198 -0.305 0.0278 - - - - - - - - -0.229 -0.188 -0.054 -0.265 -0.2709 - - - - - - - - - 0.276 0.199 0.024 0.103

10. - - - - - - - - - - 0.087 -0.017 0.07611 - - - - - - - - - - - 0.136 0.04112 - - - - - - - - - - - - -0.09513 - - - - - - - - - - - - -

3.4 Unskilled workers. Rural districts.

1 2 3 4 5 6 7 8 9 10 11 12 13

- 0.191 -0.174 -0.067 -0.109 0.013 -0.125 -0.083 0.075 0.066 -0.132 0.122 0.036

2 - - 0.108 -0.253 -0.097 -0.027 0.080 0.095 -0.146 0.296 -0.447 -0.033 -0.0113 - - - -0.018 0.040 -0.101 0.165 -0.134 0.017 -0.058 -0.048 -0.173 0.060

4 - - - - -0.020 0.003 0.163 -0.045 0.228 0.098 0.052 0.020 0.083

5 - - - - - 0.447 0.043 0.096 0.398 -0.006 0.224 -0.062 -0.2306 - - - - - - 0.147 0.065 0.402 0.066 0.254 0.036 -0.0667 - - - - - - - 0.026 0.152 0.090 0.193 -0.004 0.210

8 - - - - - - - - 0.240 0.103 -0.057 -0.158 -0.1759 - - - - - - - - - 0.087 0.216 -0.074 -0.083

10 - - - - - - - - - - -0.091 0.096 0.149

11 - - - - - - - - - - - -0.003 0.007

12 - - - - - - - - - - - - 0.092

13 - - - - - - - - - - - - -

3.5 Agricultural workers. Rural districts.

1 2 3 4 5 6 7 8 9 10 11 12 13

1 - 0.491 -0.190 0.031 0.051 0.015 0.116 -0.076 0.076 -0.062 -0.233 -0.079 -0.0842 - - 0.056 -0.050 0.014 -0.146 0.127 -0.123 -0.090 0.250 -0.228 -0.092 0.136

3 - - - -0.060 0.046 -0.136 0.003 -0.186 -0.153 0.220 0.120 -0.165 0.045

4 - - - - 0.055 -0.052 0.222 -0.124 0.220 0.060 0.015 0.073 -0.0135 - - - - - 0.458 -0.061 0.170 0.299 -0.041 0.025 -0.195 -0.1626 - - - - - - 0.035 0.134 0.149 -0.070 0.165 -0.117 -0.1247 - - - - - - - -0.214 -0.013 0.262 0.333 0.322 0.990

8 - - - - - - - - 0.290 -0.115 -0.245 0.070 -0.0969 - - - - - - - - - -0.080 -0.048 -0.104 -0.026

10 - - - - - - - - - - -0.101 -0.030 0.240

il - - - - - - - - - - - -0.008 0.123

12 - - - - - - - - - - - - -0.00913 - - - - - - - - - - - - -

181

Appendix C. The basic material.

Disposable income, expenditures on 13 items, savings, assets and some other informationseparately for each of 3098 households of wage and salary earners in the year 1955.

All amounts are in Danish kroner, (respectively 100 Danish kroner). The amountsgiven in columns 3-16 are in kroner per person, all other informations are per household.The columns l-22 contains the following information separately for each household.

Column Informationno.

1 Household number within the socialgroup in question.

2 Size of household measured in number of persons.

3 Disposable income per person of the household (i.e. total money income less paid personaltaxes).

4 Expenditure per person on dwelling.

5 Expenditure per person on fuel and light.

6 Expenditure per person on food.

7 Expenditure per person on tobacco.

8 Expenditure per person on clothing.

9 Expenditure per person on footwear.

10 Expenditure per person on washing and cleaning.

Il Expenditure per person on durables (excl. own car).

12 Expenditure per person on personal hygiene.

13 Expenditure per person on books, newspapers, etc.

14 Expenditure per person on sports, holidays, hobbies.

15 Expenditure per person on transport (incl, own car).

16 Expenditure per person on union fees, subscriptions, etc.

17 Savings (net changes in assests and debts).

18 Income changes in the period 1953-55 according to the following code:Rising through the whole period 1953 / 1954 / 1955.Constant - - - - 1953 = 1954 = 1955.Decreasing - - - 1953 \ 1954 \ 1955.Unknown 1953-1954, rising 1954-1955.Unknown 1953-1954, constant 1954-1955.Unknown 1953-1954, decreasing 1954-1955.

9. No information.

19 Type of household according to the following code:Single men.Single women.

182

Couples without children.Couples with one child.Couples with two children.Couples with three children.Couples with four or more children.Single men with one child or more.Single women with one child or more.

O. Other types of household.

20 Net assets in 100 kr.

21 Wage- and salary income in per cent of total income.

22 Year of establishment of the household (60 denotes no information).

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1.3

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Skilled workers. The capital.

9 10 11 12 13 114 15 16 17 18 19 20 21 22

181 1 8990 5140 1147 1685 1062 1208 167 158 376 312 55 1327 880 336 530 1 1 1 100 52182 2 9067 660 572 2378 1001 414Q 1140 130 332 281 112 592 1190 318 610 6 3 1 914 143

183 1 9066 968 562 2118 1100 190 107 360 20 73 75 755 220 318 1160 1 1 143 95 181814 2 9120 353 192 3021 973 361 101 138 917 3146 170 10814 321 287 430 1 3 -11 99 53185 2 9280 512 187 1560 210 1330 199 135 565 290 97 1292 3510 2914 -2630 1 3 325 93141

186 2 9350 1466 21414 1993 659 8314 2149 66 811 210 108 833 230 379 1990 1 3 72 99 31187 2 9353 2017 620 1560 1457 515 55 147 668 160 87 503 42 593 5970 14 3 182 9267188 1 93514 9148 560 21466 6214 797 122 97 714 161 236 863 370 399 160 3 1 7 9927189 2 9361 766 233 1865 372 1288 270 1140 672 1402 169 817 180 303 510 1 3 140 99 51190 2 91425 1423 228 2780 310 1432 127 81 31 120 1147 506 272 392 5560 1 3 110 99 38

191 2 9588 522 161 2217 5314 6142 60 208 1114 231 125 976 229 600 14380 9 3 2314 97 32192 2 9876 612 232 25142 786 768 2214 121 110 210 78 1008 173 1427 520 1 3 33 991414

193 2 9937 778 230 1998 181 697 77 101 33 2914 106 3149 3519 336 6Go 1 3 -12 99 511914 1 10076 780 120 3032 1266 272 143 230 885 200 1146 906 212 377 980 1 1 78 99 146195 2 1014148 9814 148 1668 279 675 66 131 1177 125 256 357 3722 273 2140 1 3 _143 99148

196 1 10825 1020 63 2791 1250 1078 152 252 860 157 288 2028 388 1466 -1770 14 1 -7 90 145

197 2 11123 710 85 1905 6514 757 11414 321 16143 155 23 11140 155 337 230 1 5 9 10055198 1 11220 375 17 850 1480 501 86 190 390 87 72 985 2585 336 1350 1 2 -17 100 514199 1 11260 800 0 2600 11448 668 8 609 1001 273 0 1039 1330 5114 -850 1 1 9 100514200 1 11273 660 146t 14200 566 367 86 319 12 124 126 12114 210 5114 80 1 1 2 99314

201 1 12262 830 162 23914 321 987 291 1142 9514 186 120 11420 1112 569 1130 1 1 2 97 60202 1 12928 1287 3148 2695 3140 1499 115 158 190 62 1014 553 143 599 6320 1 1 26 82 35203 1 13116 906 1493 3570 637 751 152 585 579 305 14283116 178 1423 360 14 1 14 99 52

2014 1 114178 1512 566 3168 11450 902 1140 390 1097 125 21626214 508 676 -60 14 1 210 92 20205 1 114220 1320 15 2290 1000 1457 115 1470 85 1514 120 1686 14871 788 -1300 3 1 -65 100 52

206 1 15952 810 53 23140 1235 1508 207 298 12141 379 132 5698 2180 1149 1280 1 1 11410038

1.4 Unskilled workers. The capital.

1 2 3 14 5 6 7 8 9 10 11 12 13 114 15 16 17 18 19 20 21 22

1 6 1268 68 107 607 126 82 26 23 115 13 22 3 0 56 0 14 7 -9 100492 £ 15147 235 100 673 149 101 147 78 32 16 2 37 4l 814 200 1 7 0 98 1443 3 1715 210 350 38 133 35 22 344 14 33 13 72 33 3 200 1 14 0 87 5314 5 1735 3141 1614 832 0 6 30 90 35 23 16 1 80 145 144o 6 9 1 114 30571752 1614737579315614638 85213995 132 10 -5060 -182606 2 1773 0 6 1452 135 259 69 43 1145 1114 21 251 110 148 -1400 9 9 214 71 507 6 1897 76 6 867 171 191 67 32 17 55 59 1440 37 68 -880 9 7 -1 84 408 5 1957 120 133 912 220 181 83 143 314 61 38 149 56 61 160 14 6 0 851439 5 1988 131 115 828 166 392 137 73 63 76 19 142 9 914 370 14 6 1 97 445

10 5 20144 100 115 823 186 83 57 544 111 80 66 107 146 109 1450 6 6 -1 93 144

11 14 2076 12 158 8014 309 69 30 53 131 146 61 148 61. 158 2560 1 5 -146 9814812 3 21514 232 193 1250 81 392 106 50 237 120 69 395 155 66 -14960 3 9 293 81 3613 44 2167 207 168 650 317 158 96 85 11 714 65 31 397 7 120 9 5 0100143114 14 2167 209 150 669 200 141 67 141 2214 142 58 152 75 107 290 3 5 0 71 44915 14 2212 1114 1.147 10140 63 171 144 82 125 71 142 149 11 1141 0 6 5 010039

16 7 2238 99 113 960 107 338 83 25 1 59 15 31 1.140 70 160 9 7 0 99 14117 5 22143 312 178 10140 0 62 50 57 165 148 0 8 3 6 130 9 0 -10 27 14518 7 2251 151 131 631 176 1141 27 77 143 68 142 102 171 95 300 1 0 -1 59 1919 5 2283 113 1614 800 34414 175 147 83 101 89 14.8 58 26 172 -130 1 6 -1 99 14620 5 2293 105 86 1212 2144 141 38 56 1614 36 22 214 314 93 1420 1 6 1 69 148

21 5 21409 i14 177 811 126 160 142 129 98 193 36 113 122 88 380 1 6 2 98 14722 3 214.29 307 201 867 187 186 100 1141 11 1214. 143 59 52 130 500 1 9 0 76 14223 14 21487 392 1.99 8145 301 40 30 29 14 105 0 27 1914 108 1110 1 5 -115 95 1421452507 11431291236263753957 5613662 144 21 80-23026 14993625 14 2515 180 116 1000 150 110 70 96 60 67 115 1514 114 138 200 6 5 -37 90 53

26 5 2537 156 173 603 267 182 31. 122 111 118 140 175 111 89 0 14 6 2 96 5227 1 2572 611 332 1040 60 445 19 32 13 99 12 56 36 130 0 9 2 0 100 3228 5 25914 1814 128 1062 272 202 25 148 55 85 86 205 85 120 380 14 6 142 914 14229 14 2655 287 1.37 10214 i4 1145 81 86 166 714 38 139 61 157 1420 1 5 13 100 14630 14 26614 214.0 126 11148 237 1514. 1014 60 152 143 147 27 145 169 120 1 5 -6 149146

31 14 2680 300 193 1009 146 263 77 62 160 69 15 300 140 97 1400 14 5 0 88 5232 5 2705 59 169 10140 14o 266 77 144 236 23 114 77 0 109 4o 1 6 9 89 21433 14 2721 581 161 10140 12 257 147 41 62 80 63 160 2214 1143 -1580 1 5 50 96 36344 14 27146 256 155 814 160 1714 72 63 251 130 82 81 252 75 -80 6 5 0 93 4935 5 2791 216 119 936 88 195 81 29 21 81 129 2014 68 119 390 6 6 -5 98144

36 5 2814.14 351 131 1092 250 119 66 31 221 99 37 110 714 140 0 1 6 _51#3 100 3337 14 2905 126 128 8014 1149 230 88 96 103 61 58 158 76 83 1130 14 5 -2 99 5338 14 2905 200 i14 9714 177 270 122 95 2143 79 91 185 39 1145 610 1 5 1 98 14939 44 29145 293 276 975 57 173 56 38 96 40 30 369 327 1147 _9140 6 0 3914 82 14040 5 2960 4114 157 6214 192 369 92 47 1142 103 85 326 145 61 570 2 6 10 98 145

141. 5 3023 212 117 850 3714 376 112 31 263 78 65 277 78 72 160 14 6 -23 99 60142 3 30449 0 310 867 151 1469 614 162 351 93 118 48 18 215 130 6 14 -6 100 147143 5 3061 1714 237 1196 2914 326 116 68 159 103 86 51 148 80 1400 1 6 -1 91433144 14 3072 303 116 675 0 276 83 57 118 1014 51 715 33 181 1150 1 5 0 99147145 3 3101 292 279 867 223 268 714 60 118 69 39 395 265 126 0 1 14 0 100 51

146 2 3135 1457 268 780 93 332 1514 78 0 128 56 285 50 1214 160 14 9 2 89 211#7 6 3172 1143 138 8914 1409 320 100 49 93 70 85 198 136 176 250 9 7 11 97 33148 1 3215 525 87 750 180 535 131 144 5 121 20 70 170 110 -180 9 2 28 96 142149 3 32214. 200 207 1200 520 60 114 100 8 153 60 205 275 56 120 1 14 0 99 5050 2 3285 4114 182 1560 306 145 1 106 6 65 39 146 12 208 0 14 9 2 86 44

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2 3 14 5 6 7 8 9 10 11 12 13 114 15 16 17 1819 20 2122

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21 5 2116 132 168 1060 1142 89 147 19 16 14o 114 101 20 101 14140 9 6 19 9814622 14 2161 216 223 715 219 121 3 614 162 78 28 95 62 110 -9140 6 5 11 974.823 6 2217 258 183 800 107 1214 65 28 95 55 39 191 141 914 14o 1 7 -18 99 39214 14 2220 150 220 600 150 296 147 50 171 95 8 70 19 814 510 6 5 17 97 51425 7 2251 39 129 9143 73 14o 66 714 68 88 9 119 6 143 14o 1 7 59 99 39

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36 5 2513 310 215 690 150 377 66 314 50 73 25 514 258 50 600 1 0 18 914142

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56 14 2825 309 2146 8145 120 14214 122 52 11 128 50 2149 214 102 330 1 5 12 99 1414

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1 2 3

2.2

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Lower public servants and salaried employees. Provincial towns.

5 6 7 8 9 10 11 12 13 114 15 16 17 1819 20 21 22

121 1 5605 1480 33 720 0 758 139 0 430 119 30 110 50 75 0 1 2 010055122 14 3615 1145 211 958 326 313 73 72 53 98 58 190 21 223 1980 1 5 -38 99 48123 3 3626 871 257 11114 1455 1147 141 66 814 77 33 0 42 169 1770 1 14 89 9262126 3 3662 632 301 1167 0 497 126 60 390 112 27 266 165 151 -1610 14 14 141 98 50125 3 3666 382 176 900 169 663 75 62 260 56 68 11148 12 119 100 1 14 15 99 52

126 3 3655 680 166 867 55 3149 88 67 63 914 13 19 1766 80 -1960 1 14 33 99 51127 3 3656 0 191 1250 667 352 135 78 156 137 147 298 362 82 600 1 14 -25 9167128 14 3673 779 382 1050 39 168 76 22 262 36 58 99 609 121 2350 1 0 212 88 67129 5 3682 328 186 1600 105 616 169 145 665 128 117 150 33 85 -1520 9 4 0 92 56130 2 3696 360 2144 1200 620 360 136 65 0 181 37 221 350 36 80 2 0 5 714 50

131 14 3696 605 196 16814 191 1446 115 106 21 106 70 110 298 76 180 1 5 -125 9832132 3 3697 283 128 851 7 3145 105 56 72 132 52 507 6i 137 1880 1 14 1145 93 l4133 3 3702 322 266 1200 0 6314 101 70 286 118 52 165 78 88 720 1 14 129 67 1431314 3 3703 308 166 1217 92 6614 79 52 1657 125 143 669 50 95 -5160 1 14 227 91 26135 14 3709 325 196 975 100 582 914 55 607 61 60 186 17 1146 570 1 5 19 914 50

136 14 3710 392 300 900 120 225 1414 914 137 914 38 236 102 82 -90 2 5 -98 99 52137 4 3715 327 592 1000 257 102 58 141 15 814 59 3614 6 73 2630 1 5 60 71439138 14 3732 3014 185 975 600 626 65 97 336 75 60 190 27 99 80 9 0 -6 914 33139 3 3756 112 168 1037 1141 107 6 35 32 114 25 79 10 91 5130 1 14 3149 86 37i6o 14 3771 155 193 1250 1148 358 89 145 1314 90 32 370 57 128 1360 1 0 14 87 21

1141 3 3783 230 300 1033 8 615 95 38 199 120 1142 308 75 185 250 1 14 11 96681142 3 3791 183 393 10140 120 5314 63 85 2314 101 83 280 217 79 600 6 14 10 98 531143 2 3860 900 31414 600 387 17 50 120 9 27 514 14o 125 83 6140 14 3 -27 96271144 14 3865 239 237 1265 96 631 914 87 1140 101 26 69 67 113 360 9 5 1 100 32i6 5 3918 336 153 1020 216 1443 86 53 26 96 67 115 16 101 2850 9 0 l4 961414

1146 2 3930 198 3147 1300 130 395 179 125 62 178 65 165 77 614 120 1 9 0 99 351147 2 3933 0 3 1000 0 3143 314 80 652 195 72 53 0 1014 2180 1 3 3145 82 5141148 14 3967 3146 206 1087 225 512 133 35 209 85 28 150 986 1514 -1230 1 5 39 99 381149 3 3989 331 1437 1300 318 31414 87 214 68 116 35 91 151 111 550 1 14 97 98141150 14 993 305 175 910 209 812 217 35 375 151 72 271 12 1141 6140 9 5 -12 96 60

151 14 #011 360 1914 1080 89 5142 129 30 2614 99 51 1419 78 175 14140 9 5 22 99 60152 3 14018 77 13 1150 4140 1056 229 81 1499 1145 96 1091 58 117 -2620 1 14 81 981414153 14 4028 92 187 1312 303 6014 90 62 133 177 21 3714 19 165 530 9 5 17 99311514 3 14031 151 227 9145 12 201 82 52 148 72 27 6114 5 96 5110 1 14 277 72 145155 3 14055 1433 292 114140 59 3814 73 152 255 152 58 177 80 51 19140 1 9 -145 91 36

156 5 6061 269 2141 960 103 237 50 714 207 66 22 8 310 161 6160 6 0 78 98 68157 3 140714 6148 328 1027 399 268 101 82 122 70 145 1149 91 100 1580 1 14 -178 98 50158 3 141014 6214 322 1320 133 262 51 80 272 52 88 110 2142 138 _590 1 14 -90 95 14

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5 6 7 8 9 10 11 12 13 114 15 16 17 1819

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14 5 6 7 8 9 10 11 12 13 i'4 15 16 17 18 19

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187 14 14168 366 300 1205 210 336 91 814 289 92 36 317 99 76 930 1 5 -9 96 140

1882 14195 140 143 690 06147175 10 109 109 01096 766 140 9099 35960 189 1 14200 0 0 0 600 768 71 0 0 149 0 815 3750 97 -31470 14 1 145 99 55 190 2 14209 219 375 1200 118 153 97 73 166 91 66 193 145 88 1520 14 5 35 9825

191 2 142214 300 201 962 13 1428 75 14 2620 112 22 166 219 121 -3350 14 5 21 99 55 192 3 14228 1455 3511 1522 315 1479 92 92 911 90 15 14145 515 944 -1270 5 14 39 96 145

193 2 142140 3145 14i 1080 180 156 29 85 122 1044 95 338 831 51 170 14 3 2 91 514

1914 3 142144 0 3914 760 88 260 66 78 215 95 151 208 15 97 114140 1 0 109 70 514

195 14 142814 130 2114 1025 537 552 119 37 77 1614 63 501 101 150 2370 1 5 -28 71457

196 2 14289 2110 365 11460 793 1467 141 115 35 1140 62 75 87 61 -1720 6 5 -36 9823 197 3 14302 1458 1435 1560 1465 358 1014 77 178 120 97 814 16 128 -650 6 14 23 98 37 198 2 14i 202 558 960 130 2114 118 142 152 6k 91 327 92 158 370 1 5 51 99 52 199 3 143514 180 33 j67 579 1145 142 693 151 60 215 109 116 -500 1 14 86 96145 200 2 14576 585 216 1500 167 1498 105 86 711 116 7k 788 30 178 -11400 1 3 72 98 51

201 2 14386 1406 382 1560 270 5514 110 8k 3514 106 145 177 2614 112 1060 1 3 714 98214 202 1 141403 0 10 0 1035 571 108 180 0 103 79 8140 500 8k 690 14 1 15 99 k 203 2 1440k 3143 566 1260 156 259 814 90 701 59 112 158 20 126 1010 1 5 89 814 114

2014 2 141452 1476 ki 1500 129 koe 125 8k i14 85 21 128 70 107 500 1 5 77 9827 205 2 144433 202 536 1175 387 1233 161 1141 508 92 30 227 72 127 -1120 1 3 111 98 314

206 2 14456 1485 511 1150 221 kk 196 101 369 1148 50 257 1435 109 250 9 3 3 98 52 207 2 144141 2144 550 1275 k 593 62 65 215 50 67 158 30 1514 -120 1 3 21 98 18 208 2 1414614 256 1451 1281 23E 200 81 28 579 614 51 5114 111 115 30 14 3 73 9521 209 14466 167 520 1215 1145 205 107 56 70 96 27 358 59k 139 2520 1 14 -30 99 60 210 2 141487 1426 501 1821 90 393 121 71 559 100 149 117 112 99 300 1 3 58 97 21

211 1 141490 2k 63 250 1622 885 150 352 2140 55 1115 290 68 0 1 1 0 99148 212 2 141491 ko k6 1630 110 239 78 68 298 81 8k 50 113 130 ko 1 3 97 9622 213 1 14507 152 1481 0 1466 271 147 98 16 97 60 206 1422k 75 -13ko 6 1 66 98 60 2114 2 14517 195 365 1300 1143 131 76 59 2149 29 71 302 590 99 850 1 3 1146 98146 215 2 14522 159 366 20147 513 1426 1141 52 88 146 98 116 0 128 160 14 5 78 98 22

216 3 14571 klo 188 1267 205 633 120 93 256 110 50 298 108 57 1650 1 1 25 96148 217 5 14601 531 737 1167 1406 260 115 105 275 131 1114 1419 15 115 142ko 1 0 -53 87 k 218 5 14607 126 361 12142 265 566 99 171 91 65 139 91 1419 86 2280 1 0 314 95 56 219 2 14671 602 598 1820 555 125 72 56 239 92 68 196 66 121 580 1 5 -38 98 60

220 2 14680 1914 1460 1200 163 302 117 2k 1511 107 914 209 145 161 -260 9 5 107 99 18

221 2 14699 210 312 1560 2514 515 81 314 1414k 53 147 1142 5 158 9140 1 3 1414 97 19 222 2 14701 590 665 1275 126 505 126 59 157 102 105 158 225 121 klo 1 5 108 9823 223 2 1472k 1447 k6 1515 1142 83 21 514 53 117 69 14142 51498 115 -10500 1 5 251 96 50 2214 2 14755 300 228 1500 26 195 77 50 2509 50 148 276 80 161 -1160 1 3 25 79 55 225 2 14782 251 5145 9514 186 255 78 58 23k 142 32 205 514k iko 2650 6 3 1488 83 sk

226 1 14800 659 36 1762 1066 500 123 0 9 68 15 953 170 0 -703 1 1 0 100 60 227 2 14852 1461 147k 1500 221 875 155 78 912 79 72 386 326 96 -23140 1 5 71 9825 228 1 14857 555 397 1620 0 395 145 51 2140 150 32 180 0 95 890 1 2 50 9938 229 2 148814 281 s8 1500 288 k6 159 59 295 131 65 169 160 139 320 1 5 1 98 50 230 1 14920 0 0 0 555 1416 20 150 0 102 10 232 170 81 2500 9 1 66 96 60

231 2 14928 160 5149 765 50 135 93 61 597 75 148 14014 55 288 1160 1 5 275 93 60

232 3 149814 589 187 917 97 225 71 148 146e 68 70 116 505 82 14950 1 14 17 99 51 233 3 5012 362 1467 1587 210 503 51 65 259 125 78 32 68 87 1920 9 14 1314 10051

23141 5020 600 0200 0758115 75 15087 0140 50025021414011 810060 235 2 5021 150 1407 1100 686 570 117 514 195 96 72 i6k 62 115 1030 1 5 57 99 k

236 2 5056 567 610 1380 350 717 86 ko 127 139 105 112 39 350 350 9 3 -149 99 514

257 1 5065 0 6 0 821031 151 2k 269 257 197 1421 310 70 900 1 2 31000 238 3 5071 0 6147 10140 3114 5k 92 1142 789 191 57 376 10 103 780 1 14 1 99 514

239 2 5100 1485 1401 2010 103 295 18 61 690 99 27 165 222 119 160 14 3 76 9528 2140 2 51142 215 517 1200 192 208 105 66 k 52 55 556 57 118 51490 1 5 118 99 21

n /) f 3.2 Lower public servants and salaried employees. Rural districts.

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Lower public servants and salaried employees. Rural districts.

5 6 7 8 9 10 11 12 13 114 15 16 17 18 19 20 21 22

301 1 8105 600 18 1617 Ii-82 816 196 0 375 169 114 1257 1000 207 520 1 1 40 99 60302 1 8230 480 120 1540 0 737 93 34 52 87 0 859 9824 201 -6060 1 1 -29 98 54303 1 8318 571 777 2650 81 887 167 816 1475 29 160 309 2140 188 590 4 2 50 99 3143014 1 8368 1086 594 1700 21 837 121 183 1421 201 70 939 180 188 270 1 2 14 98 146305 2 8407 2141 732 1050 620 625 178 81 11 107 45 760 321 183 2810 1 3 -98 94 58

306 1 8565 0 0 2224 706 1778 149 64 2559 309 18 3177 1495 75 -4020 14 1 71 100 60307 1 8587 569 4814 1500 893 1188 151 628 70 314 186 400 1188 415 -1450 14 2 -62 100 47308 1 8992 7147 898 960 38 290 71 89 1014 98 192 503 95 150 4690 1 1 _314 78 60309 1 9077 420 250 2310 400 824 180 196 0 124 675 760 2703 522 -800 1 1 914 96 47310 1 9191 300 1458 2026 814 1231 196 122 595 283 472 993 483 139 1410 4 2 159 98 45

311 2 9216 369 310 1300 375 771 60 150 1858 1445 325 703 1830 169 1250 1 3 218 85 514312 1 9325 576 65 1815 530 878 190 120 28 183 50 20614 780 213 1310 9 1 20 86 60313 1 9326 588 121 2012 1261 902 111 295 0 168 8 1147 263 5314 790 1 1 20 99 603114 2 9335 425 856 1788 527 721 84 110 69 236 37 1910 0 232 1100 6 3 7 9932315 1 93140 14140 350 1800 10140 1975 210 350 8205 205 10 865 520 200 -2650 9 1 30 89 55

316 1 96814 723 1400 1092 20 688 65 152 285 40 66 355 21443 166 3050 9 2 3143 86 35317 1 10289 1037 8147 11440 1435 769 28 86 35 167 202 502 136 235 2020 1 2 137 92 38318 1 101447 360 0 2220 530 11498 228 277 325 193 555 1006 6298 200 -2190 1 1 50 100 514319 1 10913 1480 120 364 o 1498 187 1450 0 2114 58 398 1080 1436 2820 6 1 81 98 146520 1 11079 936 898 1960 7146 717 228 396 272 1814 62 148 5666 218 160 9 1 314 100 89

321 1 11782 602 1164 1200 260 375 181 296 1362 166 68 732 2140 220 3690 1 2 79 95 60322 1 13498 360 3614 1800 70 16149 2145 68 1026 209 196 1511 21425 1428 1810 1 2 208 89 142

2 2920 21 22

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149 90141

31 95 6013 983140 80 1474 98 31

2 914 145

25 9332614953565 98 60159950

2 87 50

12 91 6014 983129 89145277752

1 98 60

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59 99141148 97 35

13 814 143714 87 38

0 99 606 88 14914 99 149

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390 99 1441 98 5361 98 521 98 600 10060

53 971418 93147

23 9814827 92 14461 92 141

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3.3 Skilled workers. Rural districts.

1 2 3 14 5 6 7 8 9 10 11 12 13 114 15 16 17 18 19

1 11 1332 112 87 533 87 127 26 25 29 23 10 10 31 145 560 1 726 1332 814216 t#33 582812381693614237 18 1014_1280273 6 11459 75 1142 552 67 173 ti14 25 28 30 19 109 23 92 200 6 714 5 1562 250 78 826 89 113 148 22 514 57 144 68 12 203 1280 9 65 14 1603 150 126 690 16 250 514 31 142 140 15 12 21 38 110 1 5

6 5 1635 120 177 728 150 53 t#0 514 17 41 7 17 80 26 310 1 07 5 1639 99 158 1432 121 82 31 25 27 76 114 21 92 120 560 1 68 6 16147 128 130 520 37 112 31 t6 115 39 52 151 83 39 8140 1 79 5 1614-8 108 197 520 102 136 67 25 32 26 18 217 26 116 190 1 0101 1700 360 5 0 91131 35 0 0113 0851 1014 0 _it#Olil

11 5 17114 206 1146 62t# 36 t#2 21 30 118 25 20 28 226 133 950 5 612 1 1795 1i80 0 0 88 327 95 214 0 28 36 260 0 0 0 9 2135 1853 1201514735105131414337 12t# 14523514 181014 39014611451856 152182751511856528 2143326114 85 87 2001615 6 1890 136 180 730 88 112 514 32 90 69 27 83 140 112 180 1 7

16 5 1891 103 160 7114 79 168 142 314 187 63 20 160 92 100 6140 Ii 617 14 1923 83 152 968 101 60 20 25 150 18 22 25 35 111 1460 9 518 14 1970 120 138 715 275 95 144 56 60 63 23 97 79 129 170 14 519 14 1978 177 232 780 77 i148 61 23 20 314 26 62 57 85 130 9 520 6 1995 122 125 800 1148 1146 514 20 53 57 21 152 12 1146 0 1 7

21 6 2019 159 117 867 87 232 78 60 140 148 23 87 25 80 80 1 7226 2019 69808671331523133 7143722614116 81 72014723 5 2039 205 226 6214 112 2147 27 57 276 38 38 143 32 111 1140 9 6214142059 39698145321672531 1495613 6255 39 12014525 3 2089 210 189 907 1149 91 13 31 3 141 144 78 50 78 330 1 0

26 6 2133 1144 1142 650 37 296 61 314 12 57 6 1214 57 62 1970 6 727 3 21142 119 177 780 65 63 58 37 31 19 31 262 38 132 1430 9 14

28 5 2153 192 1145 832 95 196 146 144 53 25 20 91 30 120 590 1 629142190 181416281451146561751 25272537 5 60212014030 Ii 2198 273 171 750 158 97 53 37 159 li-s 6 53 10 126 650 1 5

31 5 2200 218 1814 7114 59 126 20 36 39 55 21 66 33 100 1180 1 632 14 2220 665 2146 780 914 25 7 58 36 57 25 26 514 140 700 9 533 6 2227 83 195 693 72 1146 58 21 300 149 25 59 32 101 610 1 7314 Ii 2228 70 169 910 80 165 70 53 314 31 10 20 62 182 560 1 535 14 2237 153 1314 1000 135 2140 73 25 1114 80 35 128 614 120 1#1O 1 5

36 5 22149 1014 226 7114 98 269 77 75 28 76 31 714 514 126 1000 1 037 5 2292 258 151 769 120 2146 73 140 55 72 17 138 55 122 870 1 638 5 22914 1514 155 6614 165 218 55 31 155 149 69 256 86 117 80 1 639 14 2300 71 191 10140 95 137 149 214 9 35 19 18 205 131 930 1 5140 5 2328 118 133 531# 172 3014 110 67 63 68 18 165 36 133 630 9 6

Iii 14 2336 387 268 830 183 714 77 142 514 106 144 26 0 i148 -130 1 5142 14 2337 192 219 750 6 132 56 31 814 78 17 109 1106 151 3120 9 5143 6 23140 1014 200 533 169 139 72 214 148 61 22 232 150 86 220 1 71i4 li 2357 212 1714 637 212 228 70 36 26 61 21 128 75 i14o 770 14 5145 14 2390 150 1314 o1 1414 78 9 32 55 62 35 113 170 112 250 6 5

146 6 21420 261 198 653 87 200 68 57 142 39 25 81 57 81 1370 9 7147 14 21472 269 155 975 135 112 36 109 131 52 27 151 5 1314 310 14 5148 14 21487 250 236 829 136 2814 37 17 26 143 27 269 1037 103 142140 9 5149 14 21491 185 171 9148 122 115 149 79 95 73 70 73 142 199 1430 6 650 3 21497 220 91 780 3 26E 142 58 512 56 35 131 71 128 2190 14 14

51 14 2508 195 211 780 229 62 31 28 2146 70 32 229 53 139 7140 1 552 5 25114 1142 232 780 173 255 78 50 ills 71 26 229 60 126 360 1 653 14 25114 202 si 750 714 360 62 25 78 78 36 87 25 86 1300 1 5514 5 25214 208 125 832 110 365 95 62 1614 96 19 57 55 115 580 9 655 14 25143 227 257 932 106 127 145 51 172 38 17 146 9 8 1330 6 0

56 14 2551 207 215 9141 80 132 18 21 166 13 20 51 8814 131 10s0 1 057 14 2556 357 180 830 0 59 21s 147 19 63 13 146 281 11 1550 1 558 1 2568 159 157 1000 162 101 28 22 30 32 35 97 914 116 1020 1 559 3 2607 182 179 7148 67 37 19 75 1314 27 114 131 387 11 230 5 ls603 2610 16026310201052773362 14365111221614 144 320 il#

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999Ç 6601T0T911?T?l44?941T69i1Ç4?69ÇÇj04hGt3011L?qL9466o90L60?TI9649901trtl?99LT??Ç0ilLELILTi96996 8-496cTL?TL419190Tth99L9To1ILÇILT069?t4

996O190?L910499Il094T999966 9LTO9L994?8999eÇi96c 9g4T069-?ITLL169T401ti469?49L4Loi6l4?0Ç39?li9ÇÇ 069?9906?T914?91T91ÇÇÇ964491001L9114119?9?94

04 66IlT-Io4T61110LT?tI199ilITÇ9T)OTl?T009??9ÇT9?i19

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S13J1S!P iJfl}J sJI3poM PIIP1S

09 ool 1111 T 6 oÇÇ- tT4 66 4611 oÇ OL z249 o 66 4L1 O 011T O 0116 0lTT T 441

ti 66 i4i- Ç 9 oÇ9Ç- Ç14 498L oi4T 601 L9T 001 91 191 119 0 041T 401i 1Ç9 60T 1 11 6 011 T 9 o44 1114 496 i9 9 Ç6 ÇÇ 091 1101 Ç8TI o 9Çi L o9s T8L6 ÇÇj

09 16 9 T 9 0Ç9- 19Ç 041 990T o4 Ll Ç6 ÇÇÇ Ç9 10111 119 94Ç1 L61 olL 9499 T 14T 09 cOT 91 I T 011 oL4 o61Ç 146 6L ÇÇ 11L 9 OL 91 t16Ç o4LI 001 0111 44t9 T 141

Ll Ç9 98 T T 049 609 411 i8 091 1I 001 1101 LI 091 T6 0061 L19 i6 4ti T 041 09 16 6T 1 T 09Ç1 Çi4 L9 949 46 icÇ O 9L1 19 ÇÇ o 4L61 09 0911 1119 T 6111

Ç4 96 49 T T OI11T T6 901 161 46 9L1 914 401 L9 494 L44 ffjll t19Ç 96 1-199L T 8111

09 66 L T 6 o9Ç 9611 T9Ç L9i 44 49T tl 91 ILT Çol 0 0491 011 oic TLLL T L11T

09 L9 91111 T Ç 098- tjI4 41I1 1901 111 IL 04 Loi L9L ooL 0911 911 L9Ç 190L T 9111

i4 001 0 T 6 o ILT 1911 9941 f101 81Ç o o4 4L4 0 01191 041 o Ç669 T 41I 09 66 91 1 T 019 lIlli 464 4111 091 9L1 tl filT Ill LÇ4 4t1t1 oLlT o l44 L849 T 11t11

66 04- 4 T o6 T91 6T ÇÇ1 IL ÇÇ g9Ç o4 i 4T 009 GL1T L91 961 0849 1 ÇflT

91 96 6 Ç 4 o46T 911 041 Ç411 14 T6 191 141 14T ÇÇ 0011 4911 81 LÇ9 T609 1 liT 4 96 Ç- Ç 1 096 081 OIl 949 101 Çp L91 6oT 18 9111 lÇÇ O4lT o4 Ç9Ç 919g 1 TIlT

ÇÇ 96 16 11 T 0011 491 Ç8 ILT L9 TIT 8Ç LÇ 011 911 O 094T l Ç4Ç 911 Ç 0111 04 49 0 T 9 oÇ 191 ÇÇ oÇ L ÇÇ o flOT 66 119 009 lÇTl 841 814 1414 T 6T 94 96 0 11 1 olfi- 8Çg 1811 o9Ç 4i ITT Ill 11 119 141 901 llTT 941 641 9914 Ç 8Ç1

Ç 66 01 i T 016 1ÇÇ 969 LT 1Ç 911 141 111 09 191 Lfll o9Çi 1181 091 1104 Ç LcT LI 6L tll4 Ç T 0191 Ill 991 6Ç oÇ L 191 81 1 9T9 94 0911 Ç9l 6Ll 949fl 1 9ÇT

09 Ç9 L9 0 11 OliOT L9Ç Ç9l 9L flfj 91 Lol 69 6 4t1 Ç 4191 91 9O 1T91 1 9196 11 Ç T 014 dl LTT L9r 19 111 L91 001 li OTT 464 011T LÇ 4L9 19411 1 t1ÇT

91 46 6 fi T cOL L91 Ç4 iLl 1 Ol L9 19 4L 6tt ÇÇ L411T oÇ 001 1L4 Ç ÇÇT

L 66 611 Ç 11 OT Ç111 Ill 4TÇ 091 LI dl 911 9L 19Ç 496 0041 ÇL 091 T o6 101 Ç T 004 ol 009 191 04 49 TOT i 19 911 061 dLlI fiLl 8L1 dLt 1 TÇT

6 66 14- I 09L1 641 fl1ï 99 94 041 991 611 116 T6 L4 OtIOT 6l 111 19111 Ç oT Ll 16 di Ç 6 064- 161 ldL 111 09 89 LÇ 9Ç 69 091 Lil 0191 L14 oIl 91Ç11 T 611 44 96 TT t1 oL6 Çol lfl 9Çi 14 0T 694 T4T dL 961 ÇTT 6flT1 ÇÇ 181 Ç1Ç11 Ç 811 i T6 641- 0 9 O1OÇ- 0fT 1111 9611 Ç1 64 666 19 09 66T 911 ÇLflT 9111 Lo9 Ç6T11 Ç L1T

44 46 0 11 T 04- 914 119 ILÇ Ll LOT 109 99 14 ÇÇ 191 TL8 4TT lT 611111 Ç 911

id 96 Ç 11 T 0991 94 TIT 8011 16 L4T 66 TIlT 19 041 111 096 Ill 0 611111 Ç 4TT

09 19 0 1 6 091 411 64 941 11i1 Ç9 64T 6Ç 99 014 691 41111 OLT 001 LTT1 Ç ill 04 19 II- 11 T 0TL 191 ToT Ç911 81 Ç6 611 44 19 49Ç Çoi ÇÇ8 941 lT9 0111 Ç lT

1166 ÇÇ Ç 6 09fr 191 L14T 06 44 OL oÇ dl 12 LL 041 O4LT do 141 00111 1 111 tid 66 TT Ç 11 0101- 9LT 461 104 L9 1141 06111 99 69 911 001 L8L 991 9ÇÇ LLO1 T T1T

11 Il Ol 61 9T LT 91 4T 11 T lT TT OT 6 9 L 9 4 Ç 1 T

1 2 3

3.4 Unskilled workers. Rural districts.

14- 5 6 7 8 9 10 11 12 13 114 15 16 17 1819 20 2122

1 8 981 100 78 390 32 69 18 18 146 9 14 8 82 69 1460 1 7 -9 149 1482 10 10143 68 57 390 1#9 714 28 12 62 20 14 15 32 58 620 4 7 -105 50 1413 7 1065 95 107 1486 32 103 23 24 10 20 15 12 58 72 1410 9 7 20 81 14514 6 1089 82 122 1425 33 814 38 18 35 27 10 39 15 72 110 6 7 -23 76 1455 8 1112 62 5 357 39 112 39 214 37 22 19 23 82 29 230 1 7 0 98 141

6 10 1217 214 138 516 68 156 63 1414 3 38 20 214 10 58 80 1 7 0 87317 3 1283 112 163 433 5 128 514 14i 81 55 32 140 22 35 0 2 9 0 601418 6 12814 105 914 433 102 100 214 21 72 32 0 32 75 76 1420 14 7 0 78 1489 7 1288 1514 98 571 68 125 36 23 17 15 13 114 16 65 310 9 7 _14 9943106 1326 116117 6071114132 2918 83140214114 21 80 -3097 -957238

ii 6 13140 87 1147 1433 17 138 51 145 79 30 314 69 37 63 430 1 7 .14 99 14712 7 1362 185 83 669 36 126 140 0 614 114 9 27 200 57 -1780 1 7 -15 10014913 8 1380 63 1214 377 14o 133 76 214 141 148 19 36 153 69 120 6 0 2 65 31i14-8 1396 8713146111411614 314 31 52214 721 5 56 78017 -102993915 7 114014 129 118 5914 35 96 140 21 614 35 25 65 27 81 1480 14 7 14 99 37

16811412 14711614221813714520 882011412207 70 310147 -17953517 6 11458 120 226 667 39 77 143 15 99 314 30 31 147 81 380 9 7 -200 9914318 3 11469 232 2814 700 0 30 114 36 3 30 18 23 0 37 50 14 9 0 1414 3519 5 1503 1141 101 780 12 105 143 30 146 30 25 38 30 92 -370 1 8 114 58 3720 7 1521 69 70 631 91 121 35 142 26 52 114 2145 38 37 370 1 7 141 9036

21 14 1537 180 121 650 107 71 28 27 91 36 13 714 12 63 930 9 5 11 10014622 6 1552 160 130 563 17 177 35 23 52 35 20 141 29 115 720 1 0 -58 69 141236 1559 621067801436123314 114 91281176 814 150147 3147628214 6 1561 96 136 737 18 2114 38 30 27 29 25 6 16 79 360 1 7 3 9014025 5 1606 103 133 6214 96 1146 141 142 22 38 10 81 33 100 370 1 6 0 79 50

26 6 1607 97 1143 601 92 103 22 32 614 145 18 oo 145 91 210 1 7 0 9814327 6 1621 97 131 693 66 107 143 15 214 30 11 53 10 89 1930 1 0 -50 9514528 5 1651 83 96 696 1614 1145 50 23 514 148 30 22 96 125 370 1 6 -14 91145295 1657 65162510211128131416030151143 5 714 1400146 142955130 5 1659 1814 231 676 714 76 29 15 141 28 18 8 2 113 230 1 6 0 99147

31 5 1666 108 1141 600 99 172 140 33 144 31 8 77 96 95 160 1 6 2 79 14i32 7 16814 159 93 793 105 105 149 18 26 35 9 140 35 71 7140 1 7 141 98 14133 5 1686 150 714 676 52 81 33 142 160 53 12 157 27 115 1140 14 6 30 100 147314 4 1717 50 2145 805 21 65 142 15 22 17 27 1314 1414 111 80 1 5 0 67 2135 5 17140 81 123 797 73 138 18 20 135 146 27 21 2 96 700 9 6 -16 100143

36 6 17146 83 102 693 61 155 68 27 27 314 22 110 25 108 590 14 7 143 993537 4 1752 128 166 806 145 1614 36 21 7 28 17 114 152 118 130 14 5 0 75 14938 5 1762 148 121 580 108 387 85 33 37 115 143 123 93 914 -250 9 6 -11 73 3039 7 17714 102 157 5143 32 63 20 30 714 15 12 214 1478 19 1130 1 7 25 99 144140 7 1778 31 100 7143 25 168 66 18 143 214 32 82 14 714 160 9 0 39 68 30

141 5 1783 155 135 728 125 59 18 52 188 214 13 8 16 914 660 1 6 -141 98 146142 14 1793 116 1145 650 12 152 65 16 62 66 26 87 75 133 14o 6 5 95 98 38143 5 1797 14 109 770 59 269 314 27 191 23 19 11 50 90 14o 1 6 35 781414144 6 1813 296 201 595 237 230 70 32 57 614 30 30 18 102 -1870 14 7 22 60 35145141816 11915991414292081414391081463512 35117-370145 79152146 6 1819 107 1514 10140 122 65 27 18 28 114 5 21 29 90 1140 9 0 9 61428147 14 1833 51 193 8145 10 175 37 30 514 146 15 146 1014 118 1140 1 5 17 73 39148 3 1833 893 189 638 28 93 21 39 50 52 30 37 37 30 -910 14 0 82 514 514

149 5 1837 72 93 728 83 160 70 37 101 31 20 158 23 129 230 1 6 0 9614550 3 1837 30 153 920 202 68 65 50 58 148 141 23 145 163 0 9 14 0 81 28

51 6 1839 53 171 693 165 78 2 79 38 26 6 38 159 81 16140 1 7 614 82 14852 5 1853 125 190 768 75 55 32 714 75 38 314 39 15 107 80 9 6 0 73 2053 14 1856 150 197 750 12 i14 72 1414 102 59 141 141 32 111k 230 1 5 72 8039514 5 1865 0 157 668 108 1147 50 7 301 67 8 29 75 92 -1140 6 6 0 9814755 14 1869 132 1814 650 127 130 26 214 38 38 22 60 32 1314 710 1 5 -63 85147

56 14 1888 175 158 585 56 96 28 82 32 62 214 814 51 133 320 1 5 -57 8614957 1 1909 2141 386 1000 10 116 140 31 514 62 60 95 10 111 0 9 2 161 61 14858 14 1928 276 210 975 187 177 76 35 265 146 143 25 62 1714 -2810 6 5 -25 90141459 14 1932 3140 205 600 76 163 59 33 181 142 23 28 147 130 -650 1 5 97 65 14660 5 1951 85 1114 616 132 117 36 29 28 23 38 114 13 95 2580 14 6 7 96149

1 2 3

3.4 Unskilled workers. Rural districts.

14 5 6 7 8 9 10 11 12 13 114 15 16 17 1.819

61111997 22710587532672027 147 5312 l4 19 137 12001562 6 2000 1614 130 693 116 231. 39 25 1.30 149 18 66 23 88 5110 1 763 2 2000 180 389 910 0 23 12 56 140 149 30 12 28 20 140 9 0614 5 2030 101 1.146 7114 77 202 148 23 143 514 12 102 75 107 380 1 665 5 2050 92 1118 728 714 186 118 33 131 142 141 82 32 82 260 1 6

66 5 2069 218 201. 720 125 1.32 l4 214 142 35 25 35 50 614 1390 14 067 5 2080 195 129 832 60 2611 119 28 51 56 20 37 59 92 360 1 6682 2087 901611170692061322 2310119112 59 80 1011369 14 2121 91 1511 10110 62 122 79 56 101 51 142 129 59 120 -870 14 570 14 2132 711 161 829 130 311 78 140 17 76 142 125 0 128 510 1 5

71 3 21117 162 338 913 0 210 26 71 7 52 38 144 147 210 170 1 072 14 2168 192 177 780 2111 1311 32 31 158 15 12 61 10 56 1110 14 573 5 2198 1.142 218 765 1014 215 62 27 176 58 12 81 56 109 2110 14 6711112201 18013187512919962 148 5311323211 31.121 117011575112217 1971399002112111639 0 602732157 21101 3301576 2 2233 255 1190 858 1214 131 70 35 37 50 35 214 20 69 210 9 377 14 2236 219 269 8115 169 131 37 30 214 28 8 37 12 129 -1250 9 578 5 22115 156 162 670 125 263 108 52 132 57 114 80 110 115 160 1 679 5 2257 68 122 728 914 2149 32 23 75 70 10 127 59 1014 960 1 0805 2259 35192720751898939 8950 7210 139 87 3601681112260 69185101101272297299 1117614460 37118 29011582 14 2277 215 155 753 33 198 22 59 113 31 21 39 1514 1141 1180 1 583 5 22814 111 196 10140 56 1611 67 35 107 70 31 119 26 125 1480 11 6814 5 2286 73 239 728 72 238 61 142 231 52 23 147 127 131 1010 1 685 14 2290 92 170 780 260 191 141 53 135 55 23 131 52 139 280 1 5

86 5 2292 113 1119 592 98 280 109 37 218 63 21 1711 86 112 1010 1 687 14 2298 133 2014 975 165 206 60 143 61 83 39 614 142 151 220 1 588112332 871358115196285772111861171397 51129 2601589 14 2339 292 153 756 1311 198 14o 18 176 52 30 105 67 166 230 14 590 14 23118 137 128 850 1140 159 75 1 106 914 141 13 96 128 200 6 5

91 5 2351 2711 172 810 82 217 35 36 91 59 314 148 183 137 930 1 692 2 2356 1211 290 918 150 227 714 1214 14i 85 30 388 70 90 -50 6 393 3 2357 148 271 1127 0 310 56 90 55 76 103 91 0 514 80 9 14

914 14 2370 205 179 8145 127 1911 65 115 223 85 148 82 62 1148 1410 14 595 14 2375 105 198 780 1114 1160 85 80 1445 97 29 56 119 101 6810 14 5

963 2375 132236953116150391191233113520 781611 180111197 3 2398 212 2117 10140 198 ii11 15 63 0 23 5 35 25 166 360 14 14

98 14 21408 203 157 780 62 212 514 70 2112 69 32 125 32 112 870 1 599 14 21420 121 135 650 252 1614 58 119 150 37 21 2014 178 129 6110 9 5

100 14 21428 2142 186 8145 156 53 35 19 377 35 65 318 29 119 -530 1 5

101 14 21430 175 227 9140 78 226 142 28 119 148 18 1142 105 189 380 14 5102 3 21137 1114 69 950 122 96 31 56 17 614 15 50 33 177 1800 6 11

103 5 21442 75 139 752 108 261 68 36 126 50 13 139 50 105 2160 1 61011 7 214144 55 611 5149 219 1475 614 15 338 140 614 190 122 73 210 1 7105 3 21445 218 236 867 0 38 314 67 133 60 36 81 17 180 10140 14 14

106 3 21465 270 196 607 53 714 23 88 147 58 30 37 77 162 1660 1 14

107 14 21192 18 192 8145 0 135 55 90 89 27 i14 337 562 1149 1100 1 0108 5 21498 118 138 780 1011 2149 6i 55 152 116 i14 117 66 110 970 9 6109 3 2500 108 189 1083 26 211 90 35 88 144 117 27 1148 178 180 1 14

110 5 2518 2714 219 612 135 205 86 68 i111 614 26 187 6 123 760 1 6

iii 14 2537 150 171 550 161 130 55 25 82 314 16 202 17 95 2930 1 5112 3 25145 266 212 867 52 230 59 26 111 214 35 166 63 157 1420 14 14

113 14 2560 371 161 1085 38 165 62 28 914 58 148 59 15 1145 8o 1 51111 14 2602 187 211 811 130 376 60 28 82 52 53 77 37 130 1270 9 5115 14 26044 306 1844 775 12 377 79 21 73 914 31 138 32 1146 1400 11 5

116 3 2611 223 171 1473 90 119 13 147 658 79 99 122 2148 1449 -100 11 11

117 14 2623 2147 233 650 60 572 101 51 102 97 28 28 i314 131 -300 9 0118 3 2626 311 2814 953 87 106 9 146 178 23 39 59 377 263 -20 1 1

1193 2628 2143178693814128140114 633 01011421161 250614120 5 2658 108 125 780 130 90 145 35 914 25 114 21 395 1143 2350 1 6

2 3

20 2122

3061451416 99 1420 1414 145

-102 714 35314 98 118

-39 81 39-20 97 1455179113114 98 3028 86 140

0 90 20-16 93 144-1 98 145096501 9952

103 99 22814 93 149

1 99 3851 5021.1789605149914533 77 3720 98 29

1711 914 37-5 99 51

-25 971411 99 149

-327930-28 96 149-27 100 37

10 9914783 97 1920 80211

_1214 98 1121 99 144

128531144 714 53

173 89 145-5 86 514

14 98 50

28 89 14262 1414 35111 61 142112 99 .29

-129 80 52

-39 86 514-8 944 29140 88 60

3 99 52-1514 81143

28 97 52-5 82 3950 89 314-7 96118_14 914 140

1 63 5522 85 141

2 87 35558218

-81 71 112

3.4 Unskilled workers. Rural districts.

114112818 16638799028831146050 0601214193 0117 60141 -699251142 3 2819 172 161 9814 120 75 114 90 528 t4 147 145 191 158 1470 1 14 -23 80 5141143 3 2823 131 3140 10140 258 338 514 150 81 50 1014 29 32 172 130 9 14 3 73 361114 1 2832 0 0 120 0 697 193 0 282 207 70 1421 279 81 180 14 2 0 100 601145 3 28141 263 1140 1261 1014 97 30 16 90 50 67 143 2142 183 1470 9 14 -9 98144

1146 14 28148 299 131 910 89 203 52 21 81 514 143 14314 149 132 1260 1 5 -140 95 1441147 3 28149 116 227 953 140 14c 20 87 1140 143 51 87 285 195 830 1 0 58 80 251148 3 2866 1037 328 867 1147 166 147 149 314 149 314 125 90 182 -2060 1 14 144 98 1461149 3 2867 1400 177 867 232 115 55 67 35 50 19 60 20 165 970 6 14 -9 8631150 2 2881 1914 366 980 203 14314 142 29 68 57 30 32 50 2214 170 1 3 147 12 36

151 3 2888 315 358 867 140 2214 97 66 145 77 57 195 22 63 770 1 14 7 99 53152 3 28914 169 222 10140 238 389 614 214 162 39 16 35 2147 1614 -80 1 14 36 62 30153 2 2901 102 275 1590 73 98 20 60 59 38 141 53 75 228 80 14 140 60 1141514 5 2902 1114 199 728 100 320 158 73 136 914 56 117 27 129 510 1 6 32 9931155 3 2907 100 231 1127 190 220 35 141 14 60 37 614 26 88 330 1 14 -6 99 53

156 14 2910 158 1148 975 1417 102 53 59 85 60 65 1148 65 201 670 1 5 -5 86 39157 3 2935 212 228 895 108 50 18 100 104-45 82 67 372 33 150 -2100 9 14 214 98 53158 3 29714 396 2140 717 2140 269 714 1414 101 93 35 1614 179 170 -70 1 14 14 9026159 5 2999 108 132 832 2148 1458 78 142 338 81 27 21414 5 138 570 1 6 62 96 36160 3 3010 28 2214 967 167 288 52 714 87 72 28 1514 215 1614 1440 9 14 11 99 52

161 14 3027 67 173 1035 27 229 60 50 152 146 38 236 70 131 1530 1 0 7 9860162 3 3027 3014 187 1O4C 100 3142 91 25 123 50 33 79 150 236 8140 6 14 32 814 1i4163 3 30314 1425 271 886 1014 79 50 59 112 6 26 26 0 172 1130 1 14 56 99 501614 3 3052 95 2314 i314o 1400 282 65 38 5 38 5 30 2114 177 70 14 14 149 89 60165 3 3070 210 282 1350 1140 286 83 167 65 63 145 227 231 178 -11410 3 14 73 9933

166 3 3073 251 229 1200 99 2514 36 56 226 147 12 142 168 173 180 14 14 -5 682616723081 209214511857141711336 165814698 144235 250143 1772625168 3 3109 97 282 10140 199 316 83 29 35 130 18 5 143 2140 980 9 14 123 99 314169 3 31145 222 175 10140 113 308 55 31 1514 149 22 270 1141 168 90 1 14 -3 14 38170 14 31147 250 233 650 8 351 58 314 37 26 37 1145 614 149 3760 1 5 70 14526

171 14 3166 256 2148 1170 1145 153 30 614 129 33 149 69 11 113 2550 1 5 3614 67 20172 2 3191 150 337 1112 119 300 142 60 102 514 59 130 326 2144 0 6 3 3 8629173 3 3198 185 2314 1277 282 1446 106 52 135 51 55 91 151 150 120 1 14 70 83 311714 14 3203 120 165 780 4-443 350 146 17 85 141 23 275 12 125 1060 1 5 _14 92 140175 3 32014 6149 236 971 279 377 68 26 107 71 55 131 33 175 -1050 1 14 110 98147

176 1 3210 0 0 0 350 666 281 314 332 350 10 350 76 714 290 1 2 -5 99 60177 2 3226 7 299 1560 269 57 28 25 2148 141 18 31 28 2140 1420 6 3 50 76 13178 L1 32112 202 219 935 14114 276 77 51 147 95 53 1114 1146 115 -4-490 1 5 -36 91 39179 5 3258 168 204-4 780 314 229 55 314 322 26 29 61 35 112 5860 1 6 -62 51 140180 3 32714 207 201 780 115 320 65 33 152 82 31 67 17 152 1760 1 14 60 991414

1 2 3 14 5 6 7 8 9 10 11 12 13 114 15 16 17 18 19 20 21 22

121 6 2660 111 87 985 213 373 71 35 169 145 37 216 209 172 -2140 9 7 29 86 314122 1 2963 84 215 1067 714 338 83 119 12 56 12 87 108 1114 830 1 5 214 98 42123 3 2663 158 233 1167 33 50 61 103 149 142 23 57 2145 176 -160 1 4 59 9921412143 2693 14009187318720362361686535146 381614 3701414 19053125 4 26914 315 196 814 103 150 148 143 275 60 31 99 1147 1314 520 1 5 -121 97 141

126 3 2719 152 166 10140 2140 180 68 56 29 97 41 166 7 50 11430 9 14 11 97147127 3 2730 179 232 733 100 2141 55 314 186 146 31 25 283 196 700 1 4 -25 99 31128 5 2736 229 178 663 186 1145 52 57 37 149 20 155 216 1140 -2140 1 6 0 93 53129 14 2736 158 218 592 93 129 141 50 126 26 314 111 3140 130 1260 1 5 30 85 42130 14 2759 235 225 980 232 576 75 89 135 72 55 1314 14o 129 910 1 5 -143 914 148

131 3 2762 215 2146 833 219 257 14i 214 2 72 22 39 27 161 960 1 14 26 81 147132 6 2763 256 190 867 125 285 72 37 147 61 66 60 295 36 670 1 0 -17 66142133 14 2777 216 151 910 121 163 29 140 260 38 27 146 206 122 1420 1 5 7 97 351314 14 2779 237 173 1150 117 297 61 141 1141 78 25 131 219 117 -10 1 5 61 89147135 3 2788 336 257 800 310 1147 55 50 60 144 31 39 115 186 680 1 14 -19 97 149

136 14 2796 393 173 10140 171 1148 27 27 14 140 17 514 0 131 1610 1 5 26 514 52137 3 2797 280 160 900 267 112 32 50 3714 51 63 198 7 157 250 1 14 -i6 98 514138 14 2801 321 233 912 100 180 31 146 62 79 19 69 92 123 2100 14 5 21 82 146139 3 2805 138 183 1333 140 250 53 0 77 23 14o 195 73 138 130 1 14 12 91 161140 1 2815 898 14141 1420 0 308 25 18 3 142 68 26 72 914 180 9 2 318 1414114

3.4 Unskilled workers. Rural districts.

1 2 3 14 5 6 7 8 9 10 11 12 13 114 15 16 17 18 19 20 21 22

1811 3280 14203591352 012614079111635141145 91123 0142 606322182 2 3297 80 292 900 10 320 142 0 15 30 65 197 170 35 1850 6 9 148 51 23183 3 3315 190 238 900 237 188 30 36 24. 77 13 73 3140 160 1780 14 14. 25 69 521814 5 3332 53 124 6814 22 296 76 149 806 37 814 75 2140 115 11460 14 0 158 9514.9

185 3 3335 182 166 1081 139 161 62 28 145 65 40 114.2 80 171 1770 1 14. 11 95 30

186 3 33112 2314. 261 1127 116 276 37 28 214.3 21 214 166 505 213 170 14 14 98 88 271873 3350 257252125336514.0887 514 10172 9116 55191 100914 14.96514188 14 3391 297 202 800 206 162 32 67 106 37 314. 250 659 214.2 1180 9 5 -72 91 261891431400 3112209751993088050111314145296 31135 50090 -398537190 3 3409 205 277 1020 130 325 90 36 112 39 37 967 10 1714 -400 1 0 119 714 214

191 3 314214 213 267 1020 1420 211 614 65 226 63 51 98 102 158 660 1 14 -73 99 41192 3 314214 196 315 1300 53 298 514 26 214 55 0 330 27 52 -630 14 14 512 91 35193 2 34140 311 291 8614 129 237 39 35 71 55 62 192 221 2014 520 14 3 36 91 261914 3 31446 311 277 1387 196 238 145 105 179 63 40 37 3 192 1110 t4 14 93 77 17195 3 31460 156 238 917 105 506 83 90 914 101 18 219 476 159 14io 1 14 -22 99 514

196 3 31469 1460 2146 1127 280 491 75 62 80 70 25 1144 75 167 200 1 14 -89 98 36197 3 3495 356 286 12514 83 1435 117 53 203 60 59 38 117 186 630 1 14 80 98 146198 2 3539 335 335 892 75 253 97 61 2149 80 87 302 85 14214 -50 1 3 36 90 25199 2 36514 3142 368 980 128 14th 67 149 336 82 29 173 98 270 160 9 3 1 99 52200 5 3689 118 218 1100 120 325 118 60 36 61 32 188 2057 1149 -5850 1 6 112 99 314

201 14 3705 1147 208 690 1149 340 714 145 152 47 39 293 958 153 -.500 1 5 805 514 32202 2 3750 252 285 io8 178 257 65 52 230 62 79 514 25 250 550 14 3 2 99 514203 14 3792 222 267 1147 193 1149 52 57 189 62 67 70 78 201 1300 1 5 31 98 14220141 37914 5140 0 8140700285113205 70103 52272 220 371 15091 26860205 2 3828 3148 260 1995 159 177 47 45 914 75 77 55 20 269 50 1 3 67 98 214

206 14 3927 1427 179 1270 814 2141 75 29 546 50 29 38 395 1147 870 1 5 166 99 14207 2 3968 i14 305 1625 98 1403 37 55 75 62 914. 31 71 287 580 1 3 18 98 22208 6 3975 165 1147 917 58 272 57 25 114.8 145 17 1147 768 88 5680 1 0 2 39 143209214025 30837912009827875691027895081 39279-50013 09960210 4 140142 363 281 910 262 180 28 53 61 76 29 59 66 197 3170 1 5 -814 74145

211 14 14069 102 118 780 121 599 107 146 1472 53 58 597 1141 170 1760 1 5 2 99 60212 4 14071 251 230 1200 329 335 86 36 326 173 55 300 75 197 1630 1 5 22 95 39213 3 1410 338 338 973 290 1489 93 147 422 95 40 185 1140 191 880 6 14 -65 94 382114 3 41114 56 3140 1000 176 171 140 70 1886 93 32 176 17 102 -710 1 4 61 85 55215 2 14129 462 446 1275 82 1418 57 57 38 106 65 515 68 286 320 1 3 113 79 18

216 3 14302 120 282 1387 8 439 87 149 2014 102 34 65 97 162 3330 14 14 10 98 51217 2 14308 631 329 1061 130 1400 50 88 35 95 52 176 1140 211 760 1 3 14o 62 25218 3 14310 967 300 800 83 355 7 23 165 73 77 338 118 1148 1550 1 14 -2148 99 52219 3 14311 371 291 867 2114 247 127 59 360 150 60 1422 225 165 1290 0 14 10 98 49220 2 4322 291 575 1620 389 3114 87 149 63 914 32 1214 86 99 1480 1 3 55 99 23

221 2 143147 389 309 1680 122 340 145 27 77 50 30 35 57 210 -210 6 3 -110 66 25222 3 14355 227 2014 1067 0 520 58 142 352 71 73 55 93 195 3850 1 4 1410 91 36223 2 4369 1140 363 1560 392 186 38 145 109 51 66 362 140 2149 780 14 3 -12 99 23224 14 14392 347 2114 1507 317 510 113 1314 101 151 81 245 390 177 1470 1 5 -109 914 35225 2 14412 398 1460 1632 400 226 17 8i 140 93 51 453 75 2149 60 6 3 20 80 38

226 3 14441 713 292 1345 14114 72 24 53 7 75 79 14o 105 179 25140 1 14 -87 90 30227 2 1414144 198 296 1300 125 110 32 37 27 25 33 53 0 163 14500 1 3 36 91 60228 14 4519 109 194 962 202 379 37 28 103 68 19 357 235 300 14920 9 5 229 87 15229 3 45514 292 262 1326 125 763 108 72 278 139 71 155 1427 206 630 1 14 148 90 38230 2 14703 240 3614 1881 200 1477 106 814 1403 1314 i14 117 315 2140 160 6 3 0 99 514

231 2 14723 398 523 1200 200 266 51 50 95 81 68 130 10 271 60 1 3 122 9821232 2 4726 2714 361 1770 57 1014 17 1214 287 14 53 53 521 213 1310 1 3 _141 96 11233 2 4788 288 11414 1620 1456 109 12 188 874 221 39 1114 30 326 120 6 3 -10 99 54234 2 4796 1141 388 io14o 150 1148 56 123 17 38 37 225 85 248 3670 9 3 320 76 27235 2 4882 183 1485 1717 338 792 76 75 162 73 55 266 708 129 -1320 1 3 116 98 22

236 2 4899 2142 269 1125 252 205 81 26 3140 95 16 2145 66 259 3120 1 3 634 67 141237 1 4920 299 769 1225 225 205 139 278 227 75 1148 565 60 14514 130 6 1 110 614 18238 2 14957 165 1451 1020 1014. 451 33 52 745 58 31 148 145 2140 2820 1 3 16 25 20239 2 4963 304 435 1820 476 325 814 118 675 108 97 3014 327 2714 -1860 1 3 132 90 302140 1 4992 83 366 1190 110 922 72 177 29 51 127 148 763 1421 120 1 1 66 69 19

Li g9 T i o9L1 49t1 o96T L?t1 411 94 T6 99 oL 11T LT6 L44 644 9LIiOT T 19?

96 9L 9111 1 T Olj 966 0T 94 t19 96 TT 461 6L t109 0941 64T o 661i6 i og T4 T6 6æ- I T 0LL? 9911 611 6TT o 6T f 091 110T 906 6 000? 9 1190T 9666 1 6L LT 4? t1i c 6 O11?- ffiT 0T11 6i L 4LL 6g 64T 600? L9? 009? 669 9?L 11611g ? 9L?

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09 96 401 T 11 C)0?T 0611 TI6T 4111? tIT OTT g ?16 611T 611T1 00? 9L4? 11 1104 1019 T 4L? 9 66 11?I T T OLTT 966 11LT gTÇ 64 8TT 4 61 11OT 4L6 9L 000 6o 4L9T 9608 1

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11 46 44T I T 0969- 9?4 49L 49 4T 64 411 ?T IITT ?T11 49 O1OT 09 L94T T46L I TL

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3.5 Agricultural workers. Rural districts. 2 R1 2 5 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

1 10 872 7 65 522 23 88 29 314 36 34 7 5 14 51 150 1 7 20 60 412 9 907 514 80 1404 15 82 53 16 140 21 5 6 42 75 5140 6 7 -119 53 1455 9 918 97 81 1404 7 148 34 24 51 17 5 17 194 18 420 5 7 0 98 814 9 967 130 88 549 19 20 22 26 7 15 0 6 91 55 220 2 7 21 75 575 5 985 37 1144 572 97 6 13 50 3 24 0 9 17 141 -160 5 6 -5 76 44

6 9 1000 28 91 289 21 85 50 28 414 19 8 90 22 143 220 14 7 1 99 417 5 1009 98 77 2140 24 71 34 51 53 42 0 50 73 83 250 1 6 67 67368 14 1075 18 108 440 156 85 26 27 5 50 14 86 10 5 70 14 5 -10 95 519 6 1097 170 105 435 15 58 18 22 6 10 17 50 24 26 710 1 7 22 55 40

10 2 1124 547 552 845 52 53 26 157 157 36 35 52 430 35 -3960 3 0 320 74 52

11 6 1127 191 129 14ss 3 26 13 47 100 39 15 9 5 92 520 1 7 0 76 4512 5 1142 73 106 624 144 25 14 15 17 15 10 6 10 75 0 14 6 ..5o 63 28156 1155 558452890572550 29412128 17 96 19017 34675514 6 1170 100 171 433 7 31 i14 57 41 22 56 15 62 76 260 1 7 -14 55 4815 7 1262 69 4 557 37 21 45 57 17 25 14 128 14 714 590 14 7 0 84 60

16 5 1288 96 102 510 87 91 51 21 25 59 20 63 40 1i4 80 9 6 0 74 42177 1295 77100624741515528 62201731 50 72-90047 -107232182 1317 2502877804382 757 22124129181 66-18504 5 517051419 6 1383 116 110 533 40 70 14 27 3 50 19 10 226 80 370 1 7 -8 98 5120 4 1401 0 90 s85 61 50 30 32 597 57 20 159 60 37 -630 1 5 9 100 60

21 5 1405 103 151 696 5 76 26 35 68 41 20 11 26 91 60 4 6 -14 65 4922 5 1408 117 214 552 55 62 50 44 70 26 8 38 21 102 120 1 6 17 52 4525 6 1415 0 162 6o 44 27 17 38 74 29 5 50 22 76 220 9 7 -7 99 4824 5 14114 212 252 515 76 i4 22 49 28 54 16 57 42 100 -420 1 6 -5 28 4255 1420 4810178042581425 26515250 19 95 8046 0712226 5 1477 118 1140 852 209 125 49 29 15 29 26 35 18 86 490 1 6 0 98 4727 5 1479 67 95 596 129 69 57 15 167 27 16 59 0 120 -2420 1 6 20 994528 2 1482 13 86 910 56 27 19 11 0 15 21 50 61 74 -150 4 3 12 51 5229 5 1488 914 124 572 52 55 24 16 42 33 12 61 10 97 470 9 6 0 98 4730 5 1515 80 1314 624 50 100 31 56 84 51 11 19 20 97 270 1 6 0 98 52

51 4 1545 81 175 910 50 169 56 45 84 45 54 65 70 95 -1580 2 5 80 76 14325 1575 012892811 6 5117 42216064 0 58 16036 0985155 6 1622 42 88 7814 117 150 48 56 41 50 26 52 70 56 -160 6 0 -15 965834 14 1630 63 166 780 80 164 12 51 105 38 7 12 57 116 60 4 5 -2 755355 6 1655 88 io4 695 131 98 42 514 74 42 26 66 11 78 560 1 7 -6 98 60

64 1682 127215 390 0136564717028 816 67 36154045 -4875537 6 1684 89 145 600 52 174 51 59 66 45 114 78 55 78 100 1 7 -36 75 38385 1686 173182867159731874 3281189 7169-65064 0945559 5 1688 60 166 6214 187 72 53 4 84 50 28 154 51 85 120 6 6 0 65 5040 4 1691 152 1814 530 0 2514 47 45 135 40 22 127 37 117 _40 1 5 42 97 54

4141709 186145650771051537 70391653 82144 35045 1975042 14 1714 116 150 1416 12 296 41 37 170 52 25 55 51 116 850 1 5 18 994643 2 1734 106 95 10140 59 15 15 70 5 10 56 22 0 196 160 14 3 50 50 2444 5 1745 145 64 520 58 512 65 6 79 26 20 25 56 105 620 1 6 11 96 4i4 5 1771 154 104 458 32 158 68 31 507 4o 20 151 24 31 570 1 6 -15 9948

46 5 1782 171 97 728 86 65 25 58 19 4 18 64 26 124 410 9 6 -55 7514t47 3 1799 2146 203 800 56 86 25 21 4 20 35 69 150 185 -400 14 4 L 62 604841800 123148845511556541 42642753 17110 13015 098514951808 18281468 016576362025218157141109-24046 355714150 5 1825 8 47 624 84 102 44 19 15 46 23 59 225 89 650 1 6 4 76 42

51418314 1551506251006233 9194312457 82 140 50095 -3975352 5 1861 87 162 7314 81 179 31 20 21 17 146 152 146 76 150 1 6 -3 98 5153 4 1883 1 i1 775 87 258 45 19 98 5 147 70 90 88 490 14 5 14 98 146514 4 1912 96 106 910 125 88 39 20 13 15 22 118 52 139 -90 14 5 328 84414553 1919 5517272816035 4 743375220214 30 56 160144 15995156141920 271826501321052175 80613011 42144123055 195054573 1930 215161867408425012 7255550 18118-540614 705852

14 19149 400 165 650 17 161 28 3 63 50 15 35 33 107 2720 1 5 0 98 4559 14 1968 156 187 910 96 62 35 22 172 36 62 40 36 122 100 9 5 0 773260 4 1992 223 182 1072 80 100 49 18 59 1444 46 46 14o 135 650 9 5 -3 84 42

20 21 22

-20 831438 90 52

15 86119-29938326752

119 47 14132 86 42138530

1 87 511-19918

3 90 148_3 975166 76 26

0 99 531978742

09831-28938814 11611534 33 113-8 49 30

1 98 4170 93 42-9 100 5135 69 44114 50 43

496441346127

88142408 100131600543124737165352752 40130311095 668741

119 2 30811 2114 307 10110 9 99 28 43 190 32 40 102 43 247 440 1 3 148 711 23120 2 3095 210 261 1423 284 71 40 6 17 4 66 66 235 256 120 6 3 0 74141

89 14 2414 232 136 1040 41 186 48 74 87 82 23 86 76 122 810 14 5 -58 783590 14 21444 150 187 861 323 129 24 44 97 4 41 37 186 114 320 1 5 -4 92 53

91 3 21490 133 193 1105 200 128 31 25 61 30 42 173 123 205 120 1 14 0 87 4592 14 2556 192 1144 1105 1144 180 30 36 105 41 1i4 114 354 30 -220 9 5 147 54 2093 3 2561 320 191 1127 33 914 25 25 108 67 12 77 45 160 180 6 14 0 99 3594 1 2563 180 13 549 161 266 117 158 33 19 614 159 1714 81 230 3 1 36 956095 2 2568 213 333 780 25 120 71 18 3 15 33 70 72 92 830 9 3 143 99 60

96 6 2582 233 1514 818 102 372 90 33 315 82 17 140 118 56 -1490 14 7 -152 96 3797 5 2585 167 142 936 41 149 68 25 1446 36 26 22 288 107 520 1 6 -83 87 4098 3 2587 219 195 1103 105 191 51 43 105 29 34 108 53 169 5110 1 14 6 92 4599 2 2627 220 252 1040 475 62 25 0 75 32 42 40 15 85 550 9 3 0 60 32

100 14 2636 19 170 765 175 338 57 38 261 106 4i 368 97 92 40 14 5 48 84 28

101 5 2638 142 155 864 45 221 50 18 72 32 0 32 43 53 2660 1 0 73 5641102 3 2645 100 146 1253 101 102 24 69 111 105 48 192 52 133 60 1 14 1 90 25103 3 2654 268 231 1127 73 219 41 62 138 60 36 67 37 171 -110 1 14 1 75 114104 1 2656 80 250 1105 138 142 20 56 81 69 50 711 55 121 770 1 5 52 99 27105 3 2685 178 1014 1387 106 501 63 67 554 121 17 159 85 169 -1650 9 14 20 97 36

1063 2694 90223133315175 738 82781012 23170 380411 -38152107 2 2711 378 362 1560 215 462 61 100 464 103 72 177 190 258 -3110 14 3 97 9921108 3 2719 198 252 953 94 142 52 61 69 8 36 98 24 191 11440 14 14 37 54 24109 3 2746 140 213 1127 226 192 17 52 71 81 32 23 41 166 800 1 14 41 97 52110 2 2825 180 259 1170 276 248 6o 37 88 42 25 58 45 266 120 6 3 0 98 48

111428314 45889547633260282045736130 92173103045 88335112 2 2871 166 26 845 373 352 50 22 1 87 37 42 1149 214 360 1 3 63 89 221132 2885 2025815602311205337 19748577 60 914 18023 099111114 14 29114 2 170 145o 87 388 71 614 98 81 33 45 134 59 1380 9 5 116 98 38115 2 2987 210 293 1300 190 153 29 59 176 56 30 35 799 243 -1620 2 3 42 72 20

1163 3017 16020713001333887047 79954567100174 26094 37435117 2 3024 135 1414 1139 250 506 58 28 6o 41 36 514 59 96 80 14 3 670 61 2411823080 156111113002732763124 69253415 24165 44043 389955

3.5 Agricultural workers. Rural districts.

1 2 3 14 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

61 3 1995 ill 186 867 67 164 28 18 81 41 33 45 27 154 350 14 14

62 5 1999 82 133 819 119 262 32 35 69 43 12 1149 6 25 120 3 0634 2000 105151 765 502022720 5521550 32 140 790156144 2017 0114787515613911878 9636 16104 35124 22015653 2020 149119104095943363 11511238118 76 55 606466 2 2026 80 168 825 77 78 21 26 156 30 48 25 33 254 200 1 367 5 2041 148 98 759 118 153 40 17 130 33 37 111 38 1014 11140 1 668142066 50181 650171223 12 12 119 314 3660220 87 -8809569 3 2094 100 191 953 76 105 25 27 33 35 22 90 21 173 0 1 14

703 2102 115218 907 01233331 71391028 52107 38011471 7 2103 7 76 929 96 461 63 16 88 52 19 61 67 45 250 1 572 11 2115 25 128 875 71 195 57 75 189 69 211 66 111 101 260 3 573 2 2116 311 211 800 100 86 31 31 1514 36 56 16 12 80 20 6 374 3 2134 0 175 833 1148 85 115 30 80 60 14 1611 42 51 810 1 14

75112138 7511378053112117111 1930 620 45 14262095763 2181 155188121390211111327 273916 0 3 46 i2011773 2193 2331146600261163637114935621482147173 260111478 3 2205 17 175 935 17 111 16 19 0 19 13 515 7 193 140 2 079 14 2239 307 181 325 52 167 92 57 143 55 18 26 316 1113 900 j 580 2 2240 180 225 805 132 191 44 62 231 714 25 35 100 253 -390 9 3

81 3 2273 240 131 693 121 206 28 33 129 27 24 11 139 175 160 6 14

82 14 2276 219 232 650 0 120 43 10 107 81 25 1140 18 180 1180 1 583 3 2296 294 200 867 0 159 48 24 11 39 25 12 10 167 1250 9 14

84 3 2305 181 341 693 59 398 127 10 2 30 57 28 6 190 430 14 085 2 2323 247 377 780 95 146 36 35 55 28 11 52 13 59 400 14 0

865 2392 781961040872033329163563197 36 71 65016872 2399 3423997501172104799 69272212136 68 12013

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240

Appendix D

Detailed account of the expenditure items in the questionnaire of the Danish consumption survey 1955.

Dwelling

Ordinary rentExpenditure on maintenance, etc.Mortgage paymentsTaxes on land and buildingsWater rates, etc.Glass insuranceOther insuranceExpenditure in conn. with purchases of real property

Fuel and lighting

Contribution to central heatingCoalCokeFuelwoodKindlingPeat and patent fuelLigniteOil for heatingTown gasBottled gasElectricityKeroseneElectric bulbsMatchesOther expenditure

Food')

Expenditure of foodstuffs bought (incl, beer, vine, and spirits for household use)Expenditure on regular eating out

Tobacco

Cigars, cigarillos, cherootsCigarettesCigarette tobacco and paperPipe tobacco

1) This item was taken up for special investigation in a separate food-survey, cf. Statistiske Efterretninger1958, No. 46.

Chewing tobacco and snuffPipesPipe cleanersTobacco purses, lighters, etc.

Clothing

27 different items of men's clothing25 different items of women's clothingRepair of clothing

Footwear

5 different items of men's footwear5 different items of women's footwearRepair of footwearShoe-laces, polish

Washing and cleaning

Outside washing, mangling, ironingOutside cleaning, pressing, proofingSoda and other softening agentsBrown soap, soft soapSoap flakes, soap powder, bar-soapDetergents, dish-washing preparationsSelf-acting washing preparationsStarch, bleaching solution, dye tabletsScouring powderMethylated spirits, hydrochloric acid, cleaning liquidsLacquer and varnishFloor polish and mop-oilOther expenditure

Durables

13 different items of furniture, lamps, and ornamental objects11 different items of bedclothes and table-linen10 different items of kitchen utensils and table wareWashing machinesWringing machinesKitchen ranges, cookers, and ovensRefrigerators and ice-boxesMixersVacuum-cleanersSewing machines

241

242

PerambulatorsBrushesBuckets, tubsIronsTools and implements (excl, those for professional use, hobby, and garden)Other acquisitions apart from transport equipmentRepair of durables

Personal hygiene

Bath, pedicure, ultra-violet ray treatmentHairdresser and beauty cultureHand soap, bathing soap, shaving soapHair-washing preparationsToothpasteNail brushes, sponges, face chothsCombs, hairbrushes, hairpinsRazor, shaversToothbrushes and tooth glassesHair-lotion, brilliantine, cream, perfume, lipstick, powder, nail-polish, and other

cosmetic articlesOther expenditure on personal hygiene

Books, newspapers, etc.

BooksNewspapersWeekly and monthly magazinesPeriodicals

Sports, hobbies

Consumption at restaurantsBeer, wine, and spirits outside the usual household consumptionRadio and televisionGramophoneMusical instrumentsTheatre, cinema, and concertsOther entertainment (sports games, etc.)Holiday dwellingOther holiday expenditure, incl, holiday transportGarden and domestic animalsSports, subscriptions, and accessoriesOther recreation

Transport

Public transport facilitiesTaxiAcquisition and maintenance of bicycleAcquisition and maintenance of motor-assisted bicycleAcquisition and maintenance of motor-car and motor-cyclePetrol and ailTaxes and insuranceOther transport expenditure

Union fees and subscriptions

Unemployment insuranceFire and burglary insuranceHealth insuranceOther insurance (excl, life and superannuation insurance)Fees and subscriptions to trade and professional associationsOther fees and subscriptions (excl, sports and motor associations)

243