household demand for fresh potatoes: a disaggregated cross...

Household Demand for FreshPotatoes: A Disaggregated

Cross-Sectional Analysis

Thomas L. Cox, Rod F. Ziemer and Jean-Paul Chavas

A model of household fresh potato consumption incorporating prices, income, family sizeand other socioeconomic effects is estimated by maximum likelihood Tobit procedures. Theeffects of truncation bias due to non-purchasing households are evaluated and decompositionsof the Tobit elasticities are performed for various sub-groups of the data. The market devel-opment implications of this type of disaggregated commodity analysis are explored.

The use of socioeconomic variables toaugment the more traditional money in-come specification of household food ex-penditure functions from cross-sectiondata has been increasingly accepted. Priceet al. and others have focused upon ex-penditure function analysis for broad foodaggregates incorporating socioeconomicand demographic factors. Adrian andDaniel, Allen and Gadson and others havefocused upon the impact of household so-cioeconomic characteristics on selectedfood nutrients. In more commodity spe-cific frameworks, Price et al. (fruits andvegetables), Huang et al. (whole and low-fat fresh milk), and Capps and Love (freshvegetables) have demonstrated the ex-planatory usefulness of socioeconomicvariables for disaggregated cross-sectionalanalysis.

Most food consumption analysis of cross-section data derives from traditional con-

The authors are respectively Research Associate andAssistant Professor, Department of Agricultural Eco-nomics, Texas A&M University, College Station,Texas, and Associate Professor of Agricultural Eco-nomics, University of Wisconsin, Madison, Wiscon-sin.

The authors acknowledge the helpful comments ofJohn P. Nichols, Michael K. Wohlgenant, the editorsand anonymous Journal reviewers.

Texas Agricultural Experiment Station, Texas A&MUniversity, Technical Article No. 19264.

Western Journal of Agricultural Economics, 9(1): 41-57© 1984 by the Western Agricultural Economics Association

sumer demand theory, where demand isa function of own price, the prices of sub-stitutes and complements, income, andhousehold size. This traditional specifica-tion is then commonly augmented withsocioeconomic variables as proxies forhousehold taste characteristics. Unfortu-nately, much cross-section data, particu-larly from Bureau of Labor Statistics (BLS)sources, contain only household expendi-tures and socioeconomic information.Given the lack of price information, pricesare generally not included in cross-sec-tional analysis (e.g., West and Price; Buseand Salathe).

Recent cross-section data sources suchas the USDA's 1977-78 Nationwide FoodConsumption Survey (NFCS), contain de-tailed information on household socioeco-nomic characteristics as well as food ex-penditures and their correspondingquantities consumed for the survey week.The inclusion of both quantities con-sumed and expenditures can then be usedto derive commodity price information. Ifone considers disaggregated commodityprices from the 1977-78 NFCS it is ob-served that prices are generally anythingbut constant in this cross-section demanddata. Assuming that the structure of com-modity demand is constant over the sur-vey period and that regional and quarter-ly differences in cross-sectional prices

Western Journal of Agricultural Economics

reflect commodity supply forces, thenthese prices can be hypothesized to iden-tify a commodity demand curve as intime-series data. Therefore, it appears thatthe NFCS can provide a basis for inves-tigating the role of prices and substitutionrelationships in disaggregated, cross-sec-tional demand analysis.

The primary objective of this paper isto investigate price effects in disaggregat-ed cross-sectional analysis of food con-sumption. This objective is accomplishedby developing a household consumptionmodel for the commodity fresh potatoes.The analysis is based on households sur-veyed over a week period in the 1977-78NFCS and located in the Western regionof the United States. By including theprices of commodities that are close sub-stitutes for fresh potatoes (canned or fro-zen, and dehydrated potatoes), the modelprovides useful information concerning is-sues such as substitutions between productform (e.g., fresh versus processed vegeta-bles). A secondary objective of the paperis to estimate the influence of a numberof socio-demographic variables on house-hold fresh potato consumption.

A problem with increased disaggrega-tion is that a number of households be-come less likely to consume particularcommodities during the one week surveyperiod, possibly leading to a large numberof zero valued observations on the depen-dent consumption variable. Tobin hasshown that traditional least squaresregression analysis based on a samplecharacterized by a truncated dependentvariable can lead to biased and inconsis-tent parameter estimates. In this paper,asymptotically efficient estimates of mod-el parameters and associated elasticities areobtained through maximum likelihood(ML) procedures. To provide some evi-dence on the importance of truncationbias, these estimates are compared andcontrasted to the traditional ordinary leastsquares (OLS) estimates obtained from the

full sample (i.e., all households) as well asfrom the truncated sample (i.e., includingonly purchasing households). The ML es-timates are then discussed and interpretedin the context of a disaggregated demandanalysis. Implications of the results formarket development strategies are ex-plored.

The plan of the paper is as follows. Sec-tion 2 discusses theoretical considerationsfrom previous research and the empiricalmodel. Section 3 then describes the dataand estimation procedures. Section 4 pre-sents the results from the alternativeestimation procedures and their impliedelasticities. Conclusions are offered in Sec-tion 5.

Theoretical Considerations

Traditional consumer theory assumesthat consumption units (households) at-tempt to maximize utility from theservices of goods purchased in the mar-ketplace subject to a money income con-straint. This motivates the inclusion ofprices and income in demand specifica-tions. However, numerous non-market so-cioeconomic factors such as family size,age/sex composition, education, occupa-tional and life-cycle variables have beenshown to influence consumption decisions(e.g., West and Price; Buse and Salathe).Life-cycle concepts (Ferber) and the "newhousehold economic theory" (Becker;Lancaster) have extended the applicabil-ity of traditional consumer theory andhave motivated the incorporation ofhousehold socioeconomic characteristicsvia the household production framework.In particular, household production the-ory can help explain the allocation ofhousehold resources to competing non-market factors and improve the predic-tive ability of consumption models (Da-vis).

As an approximation to the underlyingbehavior suggested by traditional and

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July 1984

Household Demand for Fresh Potatoes

household production theory', a reducedform household demand specification ishypothesized as:

Qij = f(Pij, PSij, Ij, FS, SE) i = 1, ... , N, (1)

where Qii and Pij represent the quantityand price, respectively of the jth commod-ity consumed by the ith household; PSijrepresents the prices of commodities hy-pothesized to be substitutes for the jth com-modity; Ii is total household income; FS,represents the age/sex composition of theith household; and SE, represents other rel-evant socioeconomic characteristics of thehousehold. Previous research suggests thatrelevant socioeconomic variables includeoccupation of the household head (Priceet al.), degree of urbanization (Adrian andDaniel), region (Burk), number of mealsconsumed (Allen and Gadson), season (Be-loian), life-cycle proxies such as age ofhousehold head (Ferber; Allen and Gad-son), sex and education of the meal plan-ner (Allen and Gadson). In addition,equivalence scale research suggests thatthe age/sex composition of the family isrelevant to household consumption deci-sions (Price; Buse and Salathe).

Since a number of households did notconsume a given disaggregated commod-ity during the survey period, the demandrelation (1) is specified for estimation pur-poses as a truncated dependent variableor Tobit model:

Qi = Xijj- + uij if Xijj- + uii > 0

= 0 if Xijfj + uij 0 (2)

i = 1, 2, . . . N,

where Xij is a (1 X K) vector of relevantexogenous variable values, fj is a (K X 1)

'The lack of a computationally tractable simulta-neous equation Tobit estimator makes it difficult totest Tobit demand systems for the adding up andsymmetry restrictions implied by demand theory.Therefore the estimated equation(s) should be con-sidered approximations to the underlying behaviorsuggested by demand theory.

parameter vector, and uij is a random vari-able assumed to be distributed as normalwith mean zero and variance a2.

A number of approaches are availablefor estimating the parameters in (2). Wewill distinguish two OLS estimators: (1)truncated OLS, resulting from the use ofjust the non-limit observations (i.e., thosehouseholds where Qij > 0); and (2) fullsample OLS, resulting from the use of allavailable observations (i.e., including thosehouseholds where Qij = 0). Both OLS es-timators have been shown to yield biasedand inconsistent estimates of fj and a2 forthe Tobit model in (2) (Tobin; Greene).

Procedures to correct for truncation biasinclude: a consistent method proposed byHeckman involving a probit instrumentalvariable, two-step procedure; Amemiya'smaximum likelihood procedure derivedfrom the truncated normal likelihoodfunction and the method of Newton witha consistent initial estimator; an iterativemaximum likelihood procedure proposedby Fair; and, a relatively simple, methodof moments approximation proposed byGreene. Since the truncated normal like-lihood function has a unique maximum(Olsen), Fair's iterative procedure is uti-lized as it is computationally easier thanAmemiya's maximum likelihood proce-dure.2

The interpretation of the coefficientsand the elasticities which result from To-bit analysis differ from those of the tra-ditional normal linear model due to thecorrection for possible truncation bias. Theexpected value of consumption from (2)has been shown by Tobin to be:

E(Q) = XS(Z) + ao(Z), (3)

2 The maximum likelihood procedures of both Ame-miya and Fair were evaluated. Since the results areequivalent, there is no loss in using the computa-tionally easier Fair procedure. The covariance ma-trix for the Fair estimates, however, is calculatedby the asymptotic covariance matrix proposed byAmemiya (p. 1006).

43

Cox, Ziemer and Chavas


where E(Q) is the expected value of con-sumption, Z = XP/3/, ¢(Z) is the standardnormal density function, $(Z) is the stan-dard normal distribution function, and ais the standard deviation of the normalerror term from (2). Amemiya has shownthat the expected value of conditionalconsumption (i.e., conditional upon beingnon-zero), is simply X/ plus the expectedvalue of the truncated normal, conditionalerror term:

E(Q*) = E(QIQ > 0)= E(Qu > -X/)= Xf + a0(Z)/1(Z). (4)

Therefore, expected consumption is di-rectly related to the expected conditionalconsumption via b(Z), the probability ofnon-zero consumption as follows:

E(Q) = k(Z)E(Q*). (5)

Notice that these predicted values, andhence their derivatives with respect to thevariables in the design X, can be consid-erably different than those from the tra-ditional linear model in which E(Q) = Xf.

McDonald and Moffitt suggest the use-ful decomposition of the marginal effectson (5) due to a change in the kth variableof X:

dE(Q)/aXk = b(z)(dE(Q*)/axk)

+ E(Q*)(0d(Z)/aXk). (6)

This decomposition of the Tobit total pre-dicted response indicates two effects: thechange in quantity consumed of the pur-chasing households weighted by the prob-ability of being a purchasing household(the first component on the right hand sideof (6)); and, the change in the probabilityof being a purchasing household weightedby the expected value of consumption forsuch a household (the second term on theright hand side of (6)). Thus, expression(6) decomposes the total effect of a changein Xk on expected consumption E(Q) intotwo additive terms: the conditional effect

(given Q > 0) plus the probability or par-ticipation effect. The conditional andprobability effect components of (6) areuseful for interpretation of Tobit esti-mates.

Note that (6) can be alternatively ex-pressed in elasticity form by multiplyingby Xk/E(Q) and using (5). This gives thefollowing elasticity decomposition (seeHuang et al.)

jk = (aE(Q/)/aXk)(Xk/E(Q)))

+ (04(Zj)/OXk)(Xk/$(Zj)) (7)

where njk = (OE(Qj)/OXk)(Xk/E(Qj)). As in(6) above, expression (7) decomposes thetotal effect elasticity, 7jk, into the condi-tional elasticity associated with non-zeroconsumption (the first term on the righthand side of (7)) plus the probability ef-fect or participation elasticity as the per-centage change in the probability of be-coming a consuming household associatedwith a percentage change in Xj (the sec-ond term on the right hand side of (7)).This formulation will be useful in the dis-cussion of price and income elasticitiesfrom our model.3

Model Specification and Data

The Western region defined in theNFCS is utilized for this analysis. Totalhousehold income from wage and non-wage sources is aggregated over individ-ual household members. After deletingthose household observations where in-come was not reported (11.2 percent),

3 It is important to note that while the Huang, Car-ley, and Raunikar formulation enhances the inter-pretation of the elasticity decomposition's compo-nents, it suggests the unweighted or marginal effectsof a change in Xk on E(Q*) and $(Z) as the appro-priate conditional and probability slopes, respec-tively. Interest in the quantity effects of the decom-position, however, suggests that the components of(6) are more appropriate and so will be used as theslope components of the Tobit elasticities for thepurpose of this paper.

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July 1984


some 2,221 households were retained forthe analysis. The quantities consumed(pounds per household per week) of freshpotatoes (including home produced pota-toes) are used as dependent variables.From quantity and expenditure data, theprices for fresh potatoes (including sweetpotatoes and yams) and their close substi-tutes, commercially canned or frozen anddehydrated potatoes, were obtained. Theseprices (PFRESH, PFROZEN, and PDE-HYDRA for fresh, commercially frozen orcanned, and dehydrated potato prices ($/pound), respectively), were derived by di-viding household expenditures by thequantities consumed for each commodi-ty. 4

The degree of truncation (i.e., the per-centage of zero-valued quantity con-sumed observations) in the selected sam-ple is 29.8, 87.6, and 94.8 percent for thefresh, canned/frozen, and dehydratedcategories respectively. Because of thistruncation, only 6 households consumedall three commodities during the surveyweek, resulting in missing price values. Inorder to obtain price information for eachhousehold, the missing price values wereestimated using a mean price "grid" pro-cedure. Each missing price of a householdwas estimated as the average price forhouseholds from the same geographic sub-region (Mountain or Pacific) and for thesame quarter (spring, summer, fall, orwinter). Finer grid procedures were eval-uated but subsample cell size was deemedunacceptable.5

4 Quality dimensions to these "implicit" prices wereevaluated under the Prais and Houthakker hypoth-esis that quality effects would be manifest as price(hence, quality) increases with income. Price equa-tions as a function of region, urbanization, quarterand income were estimated to evaluate this hy-pothesis. The lack of significance of the estimatedincome coefficients of these equations supports theconclusion that quality effects in the prices used forthis analysis are not significant.

5 In estimating missing prices there is a tradeoff be-tween the significance of the variables in explainingthe actual prices and the size of the resulting cells

Various specifications with respect tofunctional form in family size and incomewere considered. Given the nonlinearityof the Tobit likelihood function, linear-in-parameters functional forms are generallyspecified to reduce analytical and com-putational difficulties. Prais and Houthak-ker suggest that semi-logarithmic incomespecifications are particularly appropriatefor food items as they allow commoditiesto appear as luxuries at low income levelsand as necessities at higher levels of in-come (Phlips, p. 111). Given this theoret-ical consideration and its empirical per-formance, the natural logarithm of income(LNINCOME) was retained in the finalmodel specification. 6

Demographic changes in family com-position and the equivalence scale litera-ture motivates the decomposition offamily size into age/sex categories. Thenon-significance of many of the tradition-al age/sex categories (particularly forchildren) and an interest in parsimony forcomputational reasons, led to the follow-ing specification of household composi-tion: the number of children ages 5 or less(CHILD < 5); the number of childrenages 6-15 (CHILD > 5); the number ofadult males (ADULTM); and, the num-ber of adult females (ADULT F). A qua-dratic term in family size (SQFAMSZ) wasretained in the final model to complementthe age/sex categories (which are a lineardecomposition of family size) and to eval-uate potential economies of scale in house-hold consumption. The 21MEALSZ vari-able has the standard definition, that is,

(that is, the number of observations from which theaverage grid price is computed). Using geographicsub-region and quarter represented what we feltwas the best tradeoff between cell size and explan-atory power. The USDA uses a similar average gridprice for households consuming a commodity butlacking price information when constructing thisdata.

6 Food stamp recipients comprised less than 5 per-cent of sample households. Therefore, no variablesrelating to food stamp expenditures or participationwere considered in the model.

45



the total number of meals eaten fromhousehold food supplies in the past week(including refreshment meal equivalents),divided by 21. The 21-meal-equivalentfamily size captures effects due to familysize and the proportion of meals con-sumed at home.

A number of specifications with respectto SE of (1) were considered. Occupationof the household head, season (quarter),and age of the household head were notfound to have significant impacts uponfresh potato consumption. Therefore thesevariables were dropped from the finalmodel in the interest of parsimony. Dis-crete zero-one variables for the sex andeducation of the meal planner (MALE-PLAN for male meal planners; ELEMEDfor meal planners with elementary schooleducation; and COLLED for college ed-ucated meal planners), urbanization(SUBURBAN; NONMETRO) and geo-graphic subregion (PACIFIC) were re-tained in the final model. Estimated pa-rameters on the own, cross-price andincome parameters were found to behighly stable with respect to the alterna-tive specifications of SE and FS in (1)which were evaluated.7

Results

Parameter Estimates

The parameter estimates from the fullsample OLS, the truncated sample OLS,and the ML Tobit results are presented inTable 1 along with the sample means forthe full sample (i.e., all households), thelimit sample (i.e., non-purchasing house-holds), and the non-limit sample (i.e., pur-

7 While simultaneity bias, errors in variables, multi-collinearity, and other standard econometric prob-lems can be argued to exist (as in most econometricmodels), we feel that these concerns do not seriouslydetract from the objectives or results of this paper.

chasing households). Note that prices,income, urbanization and geographic sub-region are fairly comparable for all threesample means, while considerably morevariation is evident in the family size/composition and sex/education of the mealplanner variables.

The maximum likelihood (ML) Tobitestimates closely parallel the full sampleOLS results with respect to sign. Notableexceptions are that ADULTF and theINTERCEPT coefficients are of oppositesigns (but statistically insignificant at thea = .10 level). The magnitudes of the coef-ficients are generally different. In 12 outof 18 coefficients, the ML Tobit results aregreater in relative value (14 out of 18 inabsolute value) than the full sample OLSresults. These results reflect the knowndownward asymptotic bias of OLS in alimited dependent variable model(Greene).

Comparison of the truncated OLS withthe ML Tobit results indicates more vari-ation with respect to signs of the coeffi-cients while the general downward biasof the OLS results is again evident.Coefficient signs are opposite in thetwo models for the INTERCEPT,SQFAMSIZ, MALEPLAN, SUBURBANand ADULTF. With respect to down-ward bias, in 11 of 18 coefficients the MLTobit results are larger in relative value(13 of 18 in absolute value) than the trun-cated OLS results.

The ML Tobit results generally appearreasonable with respect to signs and mag-nitudes. The own and cross price, as wellas income coefficients conform to a prioriexpectations. 8 Commercially canned/fro-zen and dehydrated potatoes are found tobe substitutes for fresh potatoes, which is

8 Comparison of the final model with and withoutprices yielded a log-likelihood ratio statistic of 20.59.Given a chi-square critical value of 7.81 for a = .05and 3 degrees of freedom, the model with prices ispreferred.

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TABLE 1. Comparison of the Full Sample and Truncated OLS Results with ML Tobit for House-hold Fresh Potato Consumption in the Western Region of the 1977-78 NFCS.

Sample Means Estimated Coefficientsa

Full Limit Non-Limit Full Sample Truncated MLVariable Data Data Data OLS OLS Tobit

INTERCEPT - -1.4245 2.0888 -. 4885(1.0090)

.145 .146 .144 -6.1755(.9088)***

.480 .484 .478 1.8265(.7505)**

.995 1.008 .990 .4410(.3675)

10.604 7.592 11.883 -. 0038(.0210)

2.537 1.889 2.813 1.0407(.1177)***

9.316 9.303 9.321 -. 1382(.0965)

.111 .210 .069 -. 2947(.2998)

.095 .057 .112 .6728(.2634)**

.405 .506 .362 -. 7007(.1586)***

.417 .429 .412 .0170(.1687)

.222 .147 .255 .6831(.2054)***

.727 .799 .697 -. 3332(.2022)*

.285 .215 .315 -. 6174(.2237)***

.498 .331 .569 .1379(.2159)

.967 .808 1.035 .3995(.2094)*

-. 2826(.2483)

11.6024

1.090 .955 1.148

(1.2668)

-5.9617(.9784)***

2.7370(.9297)***

.4071(.4759)

.0316(.0265)

.9630(.1554)***

-. 1711(.1213)

.4696(.4230)

.2892(.3164)

-. 7145(.2047)***

-. 0348(.2196)

.5453(.2549)**

-. 1444(.2544)

-. 7577(.2891)***

.0894(.2764)

.1694(.2799)

-. 2327(.3178)

13.1023

(1.3715)

-4.9268(1.3039)***

1.8279(.9954)**

.7679(.5007)

-. 0766(.0279)***

1.5317(.1576)***

-. 2546(.1310)**

-1.1880(.4212)***

1.1360(.3496)***

-. 9658(.2173)***.0588

(.2320)

.9172(.2761)***

-. 5795(.2737)***

-. 4294(.2998)

.4966(.2880)*

1.0104(.2839)***

.0533(.3333)

18.7369

a Asymptotic standard errors in parenthesis.Asterisks indicate: * significant at the a = .10 level; and ** significant at the a = .05 level; and *** significant atthe a = .01 level.

indicated as an inferior good (negative in-come effect). The family composition ef-fects appear quite reasonable; households

with younger children and female adultsare indicated to consume fewer fresh po-tatoes on average than households with

47

PFRESH

PFROZEN

PDEHYDRA

SQFAMSIZ

21MEALSZ

LNINCOME

MALEPLAN

ELEMED

COLLED

SUBURBAN

NONMETRO

PACIFIC

CHILD < 5

CHILD > 5

ADULT-M

ADULT-F

VARIANCE



adult males and older children. 9 Also, thesign of SQFAMSIZ indicates that, withinsome range of the data, fresh potato con-sumption increases at a decreasing ratewith respect to family size. 10

Households with a male meal plannerform a larger proportion of the limit (non-purchasing) sample. Furthermore, they arealmost entirely households without a fe-male household head present, tend to havehigher educations and incomes, live in ur-ban or suburban areas, and tend to havesmaller family sizes (fewer children andadult females) than the full and non-limitsamples. In general, households with malemeal planners tend to reflect the charac-teristics of the non-purchasing sample ofhouseholds. These notions are supportedby the negative sign of the statistically sig-nificant coefficient for MALEPLAN.

Predicted Consumption

Given the differences between the al-ternative parameter estimates, the generaldownward bias of the OLS results, and the

9 The following alternative decomposition of familysize was considered: children less than 1, 1-5, 6-10,and 11-15 years of age with ADULT M andADULTF. This specification yielded the ML pa-rameter estimates -4.9575 (PFRESH), 1.8454(PFROZEN), 0.8153 (PDEHYDRA), -0.0770(SQFAMSZ), 1.5311 (21MEALSZ), and -0.2608(LNINCOME) in contrast to the results presentedin Table 1. Given the robustness of these coefficientsas well as those of the dummy variables (not pre-sented) to alternative decompositions of family size,the 6-10 and 11-15 year old children categorieswere aggregated in the interests of parsimony inthe final model.

10 Utilizing the sum of the household compositioncoefficients (CHILD < 5, CHILD > 5,ADULTM, ADULTF) as the linear term ofthe quadratic family size relationship, fresh potatoconsumption increases with family size up to about7.4 household members. Given a full sample meanhousehold size of 2.85 members, with a standarddeviation of 1.59, the positive linear component ofthe quadratic family size relationship dominatesthe negative quadratic component for most of thefull sample households.

statistical significance of many of the dis-crete zero-one variables, one would expectsimilar differences with respect to pre-dicted values and elasticity measures.These differences should be manifest bothbetween models and within model resultsevaluated at various subsample means. Ingeneral, one would expect the OLS pre-dicted values to be lower than those fromML Tobit due to the downward bias ofOLS. With respect to the elasticities theissue is less certain due to the generallylower OLS slope values being offset bylarger weights used in the computation ofthe elasticities (i.e., the lower predictedconsumption E(Q)). The variation of theelasticity results due to the difference ofsubsample mean evaluation points (that is,the specific values of X at which the elas-ticity is evaluated) suggests similar expec-tations. Based on the ML Tobit results, ur-ban households in the Pacific region withhigher incomes, more educated mealplanners, fewer older children and feweradult males would generally be expectedto consume fewer fresh potatoes.

Table 2 compares the predicted freshpotato consumption (pounds per house-hold per week) for the ML Tobit, the fullsample and truncated OLS results evalu-ated at various subsample means (evalu-ation points). Components used in theMcDonald and Moffitt decompositions arealso included. As expected, there is fre-quently a large difference between the re-sults at different evaluation points. Someof the evaluation points considered in-clude the Pacific sub-region, urban or non-metro, female only or both male and fe-male headed households, and high schoolor college educated meal planners. Thesubsample data headings indicate: Region(P = Pacific); Urbanization (U = urban,N = non-metro); Household Head Status(F = only female head, B = both male andfemale head); and Education of the MealPlanner (H = high school, C = college).Those subsamples designated MEAN areevaluated at their respective subsample

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TABLE 2. Comparison of ML Tobit Predicted Consumption Values with the OLS Results andComponents of the McDonald and Moffitt Decompositions.

Truncated Full Fraction ofML Tobit OLS Sample OLS Mean TotalExpected ML Tobit Expected Expected Probability Response

Conditional Expected Value: Value: Non-Zero Due ToConsumption: Consumption: Xfl- X:- Consumption: Conditional

Sub-Samplea E(Q*) E(Q) Truncated Full Sample ,(Z) Response

FULLMEAN 4.1822 2.7555 3.6585 2.7914 0.6589 0.4620PUFCMEAN 3.0146 1.1452 1.6490 0.6130 0.3799 0.3021PUFHMEAN 3.6332 1.9762 2.9097 1.8618 0.5439 0.3881PUBCMEAN 3.8366 2.2632 3.0037 2.1213 0.5899 0.4159PUBHMEAN 4.3933 3.0561 3.9141 3.0268 0.6956 0.4894PNBCMEAN 4.6214 3.3792 4.2038 3.4198 0.7312 0.5183PNBHMEAN 5.2881 4.3043 5.0040 4.2546 0.8140 0.5980PUBCH10 3.6290 1.9704 2.4331 1.6327 0.5430 0.3876PUBCH01 4.0115 2.5120 3.2802 2.3879 0.6262 0.4394PUBCL10 3.7692 2.1677 2.6703 1.8243 0.5751 0.4067PUBCL01 4.2302 2.8239 3.6786 2.7228 0.6676 0.4683Note: Predicted consumption values are measured in pounds per household per week.a P = Pacific region; U = urban, N = non-metro; F = female head only, B = both male and female head; H =

high school education, C = college educated; MEAN = evaluated at sample means for other explanatory vari-ables; income level: H = $20,000, L = 5,000; 10 = younger child, 01 = older child.

means (derived from the full data) for allother independent variables. Thus, for ex-ample, PUBCMEAN indicates an averagehousehold in the Pacific region, in an ur-ban area, both male and female householdheads present, and a college educated mealplanner. In addition, those subsamples notdesignated MEAN, indicate the influenceof income level (H = $20,000, L = $5,000)and contrast a younger child (... 10) ver-sus an older child (... 01). Thus, for ex-ample, PUBCH10 represents a PUBChousehold as above with the exceptionsthat a high income level ($20,000), oneyounger and no older children have re-placed the sample means for these vari-ables in the evaluation point.

Since the full sample OLS model utiliz-es the full sample of data, it's predictedvalue should reflect the expected uncon-ditional consumption of all observations.As expected, the full sample OLS pre-dicted values are generally less than thosefrom the ML Tobit E(Q) due to the down-ward bias of the OLS coefficients. The ex-ceptions are the FULLMEAN andPUFCMEAN evaluation points. Similarly,

since the truncated OLS model utilizesonly the non-limit observations (i.e., thoseconditional on non-zero consumption), it'spredicted value should reflect expectedconditional consumption comparable tothe ML Tobit E(Q*) results. The ML To-bit E(Q*) are again larger than the trun-cated OLS predicted values, as expected.

Note that expected conditional con-sumption is always larger than the uncon-ditional due to the probability aspects of(5). Furthermore, as expected, urbanhouseholds with only female householdheads, and college educated meal plan-ners are all predicted to consume less freshpotatoes than their counterparts, i.e., non-urban households with both male and fe-male household heads and high school ed-ucated meal planners, ceteris paribus. Thelast four subsamples reflect the influenceof income level and age of children. Asexpected, higher income households andthose with a young child are predicted toconsume fewer fresh potatoes.

It is also interesting to note that the dif-ferences between the full sample OLSpredicted consumption and the ML Tobit

49



E(Q) decrease with increased expectedconsumption. The same relationship is truefor the differences between the truncatedOLS predicted consumption and the MLTobit E(Q*). Thus, for example, the fullsample and truncated OLS predicted con-sumption levels are 54.7 and 53.5 percentof their ML Tobit equivalents (E(Q) andE(Q*), respectively) for the lowest ex-pected consumption subsample, PUFC-MEAN. In contrast, for the highestexpected consumption subsample, PNBH-MEAN, the full sample and truncated OLSpredicted consumption levels are 98.8 and94.7 percent of the ML Tobit E(Q) andE(Q*), respectively. In general, the fullsample OLS predicted consumption levelsare closer to the appropriate ML Tobitpredicted consumption levels than thetruncated OLS results. This is not surpris-ing given that truncated OLS discards theinformation contained in the limit (non-purchasing) households.

Tobit Decompositions

Table 2 also presents two componentsused in the McDonald and Moffitt decom-positions. The probability of non-zeroconsumption as evaluated by the cumu-lative distribution function b(Z), reflectsthe ordering of the ML Tobit estimates forE(Q) and E(Q*) (see (5)). Thus, higher ex-pected consumption reflects a higherprobability of non-zero consumption. Thefraction of mean total response due toconditional response (i.e., due to the re-sponse of actual consuming households),follows a similar pattern." At the two ex-treme subsample evaluation points,PUFCMEAN households have a predict-ed probability of non-zero consumption of38 versus 81 percent for PNBHMEANhouseholds. The percentage of the aver-

11 This component, part of the dE(Q*)/dXk term ofthe conditional quantity response in (6), is deriva-ble as [1 - ZO(Z)/$(Z) - k(Z)2

/$(Z)2] (McDonald

and Moffitt).

age total response due to the response ofnon-limit households is 30 percent for thePUFCMEAN households. In contrast, 60percent of the average total response forPNBHMEAN households is due to the re-sponse above the limit. This type of resultindicates that the Tobit model can pro-vide useful information on the factors in-fluencing the probability of consumptionduring a particular period.

Table 3 compares the Tobit decompo-sitions as elasticity and slope (quantity) re-sponses for selected independent vari-ables. These comparisons are evaluated atthe full sample means (FULLMEAN) andthe subsample extremes with respect toexpected consumption, PUFCMEAN andPNBHMEAN. Approximate asymptoticstandard errors computed using the gen-eral procedures in Silvey, allow inferencesconcerning the significance of these eval-uation points and their differences. Aswould be expected, the tests of elasticityor slope differences from zero generallyfollow the significance of the associatedML Tobit parameter estimates. An eval-uation of significant differences betweenthe PUFCMEAN and PNBHMEAN To-bit decompositions is summarized in Ta-ble 4.12

Several characteristics of the results inTables 3 and 4 are worthy of note. First,the subsample evaluation points generallydiffer from the FULLMEAN and fromeach other. Table 4 indicates that thesedifferences are statistically significant forall responses except the PDEHYDRA es-timates. The significantly different sub-

12 Treating the PUFCMEAN and PNBCMEAN To-bit decompositions as asymptotically normal ran-dom variables, their difference is thus a normalrandom variable with variance equal to the sum oftheir respective variances minus twice their co-variance (Mood et al., p.178-79). Thus, the teststatistic (PUFCMEAN - PNBHMEAN)/(var-(PUFCMEAN) + var(PNBHMEAN) - 2*cov(PUFCMEAN, PNBHMEAN))½ will be asymptot-ically distributed as N(O, 1) under the null hypoth-esis that the two subsample results are equal.

50

July 1984


sample responses could provide useful in-formation for market segmentationanalysis and the targeting of promotionstrategies. As well, these results indicatethe inferential usefulness of standard errorsfor the Tobit decomposition components.This additional information is frequentlylacking in applied Tobit analysis. Second,the elasticity measures can be misleadingwhere larger predicted percentagechanges actually refer to significantlysmaller expected consumption, and hence,imply smaller predicted absolute quantityresponses. For example, contrast of the to-tal effect elasticities and their slope com-ponents generally reflects an oppositemovement with respect to increasing ex-pected consumption. Third, there is a con-siderably more detailed indication of con-sumption behavior through the use ofTobit analysis as reflected by the decom-positions and their probability compo-nents than with OLS in the traditional,normal linear model. Lastly, given thegeneral downward bias of OLS, one wouldexpect OLS derived results to frequentlyoverestimate consumption elasticities andunderestimate expected consumption,particularly as the degree of truncation(and, hence the potential bias of OLS) in-creases.

The variation in elasticity and slopecomponents due to the point of evaluationin Table 3 is-most noticeable in the totaleffect slopes and elasticities, the condi-tional effect slopes, and the probability ef-fect elasticities. Table 4 indicates that thesecomponents are statistically different be-tween the PUFCMEAN and PNBH-MEAN households for all estimated re-sponses except the PDEHYDRA measures.The point estimates of the statistically dif-ferent own price (PFRESH) total elastic-ities of Table 3 indicate that PUFCMEANis predicted to be twice as responsive asPNBHMEAN in percentage terms. Incontrast, however, the corresponding totaleffect PFRESH slopes (which are statisti-cally different from each other as well)

indicate that the implied quantity changein pounds of fresh potatoes due to unitchanges in PFRESH are exactly oppositeto the relationship of the elasticity results.Thus, for a $.10 change in the price offresh potatoes, PNBHMEAN householdsare predicted to change consumption by0.4 pounds (a 9.3 percent change) whereasPUFCMEAN households are predicted tochange fresh potato consumption by 0.2pounds (a 17.5 percent change).

A second example of the usefulness ofthe standard errors and slope componentsto augment the Tobit elasticities concernsthe cross price effects of the two substitutegoods, frozen and dehydrated potatoes.Note that the total, conditional and prob-ability effect elasticities for PFROZENand PDEHYDRA are fairly close in mag-nitude. The slope components of theseelasticities, however, again indicate thatthe quantities implied by approximatelyequal predicted percentage changes inconsumption, are frequently quite differ-ent. The predicted substitution responseof fresh potato consumption to PFROZENconsiderably dominates the PDEHYDRAresponse in quantity terms. For example,although the magnitude of their total elas-ticities are nearly identical, a $0.10 changein PFROZEN is predicted to induce a 0.07and 0.15 pound total change in fresh po-tato consumption for PUFCMEAN andPNBHMEAN households, respectively. Incontrast, a $0.10 change in PDEHYDRAis predicted to induce only a 0.03 and 0.06pound total change for PUFCMEAN andPNBCMEAN households, respectively.The standard errors of Table 3, however,indicate that the effects of PDEHYDRAon fresh potato consumption are not sta-tistically different from zero at the a =0.10 level of significance. Hence, compar-isons of substitution effects due toPFROZEN and PDEHRDRA estimatedhere, should be treated cautiously. Table4 likewise suggests that the estimated sub-stitution effects due to PDEHYDRA donot vary significantly among the subsam-

51



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TABLE 5. Comparison of the Full Sample and Truncated OLS Elasticities for Household FreshPotato Consumption in the Western Region of the 1977-78 NFCS.

Full Sample

Subsample PFRESH PFROZEN PDEHYDRA LNINCOME 21MEALSZ

FULLMEAN -0.3208 0.3141 0.1572 -0.4615 0.9459PUFCMEAN -1.5213 1.4422 0.7532 -2.0337 2.2327PUFHMEAN -0.4876 0.4768 0.2532 -0.6444 0.9749PUBCMEAN -0.4512 0.4202 0.2212 -0.6332 1.2957PUBHMEAN -0.3060 0.2927 0.1534 -0.4348 0.9497PNBCMEAN -0.2655 0.2494 0.1350 -0.3877 0.9446PNBHMEAN -0.1989 0.2104 0.1096 -0.3048 0.7700PUBCH10 -0.5863 0.5459 0.2874 -0.8388 1.6835PUBCH01 -0.4008 0.3733 0.1965 -0.5735 1.1510PUBCL10 -0.5247 0.4886 0.2572 -0.6456 1.5066PUBCL01 -0.3402 0.3253 0.1705 -0.4326 1.0557

Truncated

Subsample PFRESH PFROZEN PDEHYDRA LNINCOME 21MEALSZ

FULLMEAN -0.2363 0.3591 0.1107 -0.4356 0.6678PUFCMEAN -0.5459 0.8033 0.2585 -0.9352 0.7679PUFHMEAN -0.3012 0.4572 0.1496 -0.5101 0.5772PUBCMEAN -0.3076 0.4447 0.1442 -0.5532 0.8467PUBHMEAN -0.2285 0.3391 0.1095 -0.4159 0.6795PNBCMEAN -0.2085 0.3041 0.1014 -0.3901 0.7111PNBHMEAN -0.1632 0.2680 0.0860 -0.3206 0.6058PUBCH10 -0.3798 0.5490 0.1780 -0.6963 1.0453PUBCH01 -0.2817 0.4072 0.1320 -0.5165 0.7753PUBCL10 -0.3461 0.5002 0.1622 -0.5456 0.9524PUBCL01 -0.2431 0.3609 0.1165 -0.3961 0.7230

ples evaluated. Thus, the standard errorsfor the Tobit decompositions can provideuseful insight concerning inferences andinterpretation of Tobit model results.

Comparison of the conditional andprobability effects of the Tobit decom-position further indicates the informationcontained in the ML Tobit estimates fordisaggregated commodity analysis. Theconditional effect elasticities and slopesboth increase in predicted responsivenessas expected consumption increases. Incontrast, the probability effects parallel themagnitude relationships of the total ef-fects. Thus, the elasticities decrease andthe slopes increase in responsiveness as ex-pected consumption increases. Note thatthe conditional effects for the highest ex-pected consumption subsample (PNBH-MEAN) dominate the corresponding

probability effects of the decomposition.This relationship is reversed for the lowestexpected consumption subsample (PUFC-MEAN). This result is also indicated bythe fraction of mean total response due toresponse above the limit (see Table 2),suggesting that both percentage andquantity responses due to changes in theprobability of non-zero consumption aremore dominant than the conditional re-sponses among the lower expected con-sumption households.

OLS Elasticities

A comparison of the ML Tobit resultsof Table 3 with the full sample and trun-cated OLS elasticities of Table 5, sum-marizes their differences. Aside from theelasticity magnitude differences, where,

54

July 1984


as expected, the OLS results generallyoverstate responsiveness relative to the MLTobit results, one very important differ-ence between results is that the ML Tobitquantity effects (slopes) change with theevaluation point while the OLS slopes areconstant. Given the range of magnitudesof the ML Tobit quantity effects with re-spect to different household characteris-tics, ML Tobit results appear more infor-mative relative to OLS for marketsegmentation analysis. 13

A second major difference between theresults is the Tobit decomposition and itsprobability components. Following the ar-gument that truncated OLS reflects con-ditional consumption behavior, the pre-dicted elasticities from the truncated OLSare considerably larger in absolute valuethan either the estimated conditional ortotal ML Tobit elasticities, a result possi-bly due to their failure to adjust for theprobability response. It should be notedthat the apparent effects of truncation biasexhibited here reflect a sample that is notseverely truncated, with only 30 percentnon-purchasing households. In general,one would expect the discrepancies be-tween OLS and ML Tobit to increase withgreater sample truncation.

Conclusions

This paper has applied the McDonaldand Moffitt decomposition to ML Tobitresults, compared predicted consumptionand elasticities with full sample and trun-cated OLS results, and indicated the po-tential usefulness of Tobit analysis in dis-

13 Market segmentation seeks to identify market sub-groups with somewhat homogeneous characteris-tics and behavior that is distinct from other sub-groups. Based upon their distinctive responses andcharacteristics (if any), market promotion strate-gies can be devised to target specific sub-groups ofinterest. In this case, sub-group response measureswould be preferred to aggregate response measuresfor predicting potentially distinct sub-group be-havior.

aggregated, cross-sectional analysis ofregional fresh potato consumption. Theinclusion of price variables in the disag-gregated specification was found to yieldreasonable parameter and elasticity esti-mates, thus providing useful evidence onthe extent of substitution in fresh potatoconsumption among socioeconomic sub-groups in the Western region of the NFCS.Subsample evaluation revealed that ur-banization, household head status, and ed-ucation of the meal planner have a mea-surable influence on price and incomeelasticities. The estimation of the standarderrors for the Tobit decomposition indi-cated that the differences between the ex-tremes of the subsample evaluation points(PUFCMEAN and PNBHMEAN) werestatistically significant for all estimated re-sponse measures except the PDEHYDRAmeasures.

Our results suggest that decompositionof Tobit elasticity and slope estimates intoconditional and probability effect com-ponents can be useful for disaggregatedmarket development purposes. Socioeco-nomic variables in the analysis suggestedthat demand responsiveness to own priceand income may vary considerably amongmarket sub-groups. Thus, the approachproposed here could be useful in the iden-tification of target groups for market pro-motion strategies. In addition, given re-cent demographic trends such as regionalpopulation shifts, increasing numbers ofsingle parent households, urban to ruralmigration, etc., the procedure utilized inthis paper may contribute to a better un-derstanding of regional food consumptionpatterns and their future projection.

Comparison with full sample and trun-cated OLS results suggests that moderatetruncation may induce notable bias in OLSresults. ML Tobit estimates appear pref-erable for two reasons. First, they havedesirable asymptotic properties; in partic-ular they are consistent and asymptotic ef-ficient. Second, through the McDonald andMoffitt decomposition, they provide use-

55



ful information in the analysis of disag-gregated consumption behavior. Based onour results, evaluation of price effectsalong with those of socioeconomic vari-ables in other disaggregated, cross-sec-tional commodity models with differingdegrees of truncation and aggregation,appears to be an interesting avenue forfurther research.

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Fair, R. C. "Note on the Computation of the TobitEstimator." Econometrica, 45(1977): 7 1723-7.

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