chapter 4 in managerial economic

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In chapter three, we looked at the demand function model. A demand function model

shows the relationship between the factors that influence the demand of a product and

the quantity demand.

Managers of a firm need to estimate the values of the coefficient of the demand function

to reduce uncertainty in decision-making and to achieve the objective of the firm that is to

maximise its value.

Large automobile manufacturers such as General Motors, Daimler Chrysler use

empirical estimates of demand in making decisions about how many units of each model

to produce and what price to charge for different car models.

Empirical demand functions are demand equations derived from actual market data.

From these functions managers get quantitative estimates of the effect on sales of

changes in the price, consumer income and other determinants of demand.

In this chapter we will see how the firm estimates the demand function of the product it

sells.

4DEMAND ESTIMATION


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n

Se = (Yt - Y)2

t = 1

n - k

Key terms for review:

Regression analysis

Coefficient

Parameters

Independent variable

Dependent variableCoefficient of determination

Scatter diagram

Standard error of estimation

Degrees of freedom

F - statistic

t - statistics

t - test

Linear model

Standard error of coefficient

Time series

Cross section

Market experiments

Consumer interviewMultiplier regression

Single regression

Identification

Multi-collinearity

Heteroskedasticity

Auto correlation

Parameters


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CHAPTER OVERVIEW


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Learning Objectives

After reading this chapter, the students should be able to:

1. Estimate a demand function or any related function.

2. Interpret and evaluate the function.

3. Make the necessary adjustment to improve the estimated model.


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4.0 INTRODUCTION TO DEMAND ESTIMATION

A demand function model shows the relationship between the factors that influence the

demand of a product and the quantity demand. In order to estimate a demand function

for a product, it is necessary to use a specific functional form For example; a linear

demand function and determinant of demand can be simplified:

Qs = f (Ps, A, I)

Qs = - 1 Ps + 2 A + 3 I ------- eqn 4a

Where Qs is the quantity demand for shoes, Ps is the price. A is the advertising

expenditure and I refers to household income. 1 measures the change in quantity

demanded when the price of the product change by one unit. 2 measures the change

in quantity demanded when advertising expenditure change by one unit 3 measures the

change in quantity demanded when income change by one unit. In this chapter we will

see how firms estimates and analyses its demand function of its product. That is, how

the intercept and coefficients 1 to 3 are determined and tested.

Managers of a firm need to estimate the values of the coefficient of the demand function

to reduce uncertainty in decision-making and to achieve the objective of the firm.

Managers are faced with various forms of uncertainty whenever decisions are made. The

form of uncertainty that is of concern to corporate economists is economic uncertainty.

Economic uncertainty includes recession, fluctuation in exchange rate, inflation andgovernment regulation.

The other reason for estimating the demand function is to achieve the objective of the

firm, to maximise its value. When the values of the coefficients have been estimated,

they can be used to make decision on optimality. For example, it will help to explain how

much will the revenue of a firm change after increasing the price of its product by a


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certain amount or by how much the enrolment decline, if the fees of in college increase,

let say by 10%.

The estimated demand function can also be used for forecasting. Forecasting involves

predicting future economic conditions affecting firms operation that is, for planning

production, introducing new products and investment decisions.

4.1 DEMAND ESTIMATION

Demand estimation and forecasting requires a good set of data. The data collected

could be time-series or cross-sectional data. Time series data provide information on

one variable over a period of time. Cross-section data provide information on a variable

at a given period time. The values represent a cross-section of observations taken from

different entities. A disadvantage of time-series data is the influences of uncontrollable

variables on the results of the observations. Though cross-section analysis eliminates

the problem of uncontrollable variables that change over time, it introduces new sets of

problems such as differences that may exist between and among entities at a particular

point of time.

There are three analytical tools available for doing empirical demand analysis. They

are direct method that is, marketing research approaches such as consumer survey,

observational research, expert opinion, consumer clinics; market experiments and

indirect method that uses econometric techniques.

The survey research is the most direct and simple way of estimation. It involves

interviewing customers or potential customers directly. Though, seemingly simple, this

approach is fraught with problems such as randomness of the sample, interviewer bias,confusing, misinterpreting or unknown responses and best of intentions problem.

Simulated market situations are synonymous with consumer clinics; whereby participants

are given play money and asked on spend this money in a artificially created

environment. The results may not be representative of the entire markets reaction

because clinics have to be kept small under pressures of monetary and time constraints.


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Moreover, participants could react differently and they might behave as to conform to the

desire of the experiments.

In direct market experiments, one or more cities or regions would be chosen and

experiments conducted in these test markets to gauge customers acceptance of the

product or to identify impacts of changes in one or more controllable variables. Cost and

time could be a constraint and experiment has to be conducted on a small scale over a

short time period. Results may be bias by extraneous occurrences.

Given the limitation of these direct methods, most economists have turned to a more

practical approach that is econometrics. The objective of econometrics is to provide

empirical contend to economic theory. Econometric methods integrate economics,

mathematics and econometrics to measure the relationship between variables.

Econometric modelling involves four distinct steps: namely model specifications,

coefficient estimation, validity and policy simulation

The econometric technique discussed here will be regression analysis as the others are

too complex. Besides this, regression analysis is said to be the most useful and used

method of estimating demand. Although this chapter use examples that are based on

demand analysis, econometric techniques can also in other economic indicators of

interest such as inflation, production and interest rates.

4.2 REGRESSION ANALYSIS

Regression analysis describes the way in which one variable is related to another. It

derives an equation that can be used to estimate the unknown values of one variable on

the basis of the known values of another variable. There are two types of regressionanalysis; they are simple regression analysis and multiple regression analysis.

SIMPLE REGRESSION ANALYSIS

In a simple regression analysis the dependent variable (Y) is a function of only one

independent variable (X). The function is written as:


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Y = a + b X ------------ eqn 4b

To estimate the relationship between these two variables we need to gather and analyse

their historical data. Once this is done we can analyse it in two ways. The easiest and

most common way of analysing the data is to plot and visually study the data. That is to

get a scatter diagram.

From the scatter diagram we can determine the relationship between two variables. To

get the linear relationship we can eye-ball or draw a straight line that bests fits between

the data points (so that the date points are equally a part of both side of the line). By

extending the line to the vertical axis, we will get the value of the intercept (a). The slope

(b) is derived by dividing the change in y by the change in x.

For example, to determine the relationship between sales and advertising expenditure,

we shall use the data on sales and advertising from Table 4.1.

Year Sales

(million dollars)

Advertising Expenditure

(million dollars)

1997

1998

1999

2000

2001

2002

20032004

2005

2006

44

58

48

46

42

60

5254

56

40

10

13

11

12

11

15

1213

14

9

Table 4.1The sales and advertising expenditure of Syarikat Mohd Rizal


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20

25

30

35

40

45

50

55

60

65

5 7 9 11 13 15 17

Adver ti s ing Expend itur e X

Sales

Y

4.1 Scatter diagram

A relationship between these two variables can be seen by plotting the data

points on a scatter diagram as in Diagram 4.1, and then by drawing a straight line

that best fits the data points and extending it to the vertical axis we can get the

values of the intercept (a). And by dividing the change in sales by the change in

advertising expenditure we can get the slope (b) of this line. The equation is:

------------ eqn 4c

the hat (^) above the variables and coefficient show it is an estimated

value.

One of the disadvantages of this method is that different researchers will fit a somewhat

different line to the same data point and obtain somewhat different results. Another

Y = a + b X

y = a + b x


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major problem is the impossibility of drawing a line if there is more than one independent

variable (multi-regression).

Besides this, note that for any line drawn through the points, there will be some

discrepancy between the actual points and the line. The distance of the dashed line

gives the deviations (error term e) between the actual points and the line. Since the line

represents the expected relation between Y and X, these deviations are analogous to the

deviations from the mean used to calculate the variance of a random variable

Econometrician uses regression analysis, which is a statistical technique for obtaining

line that minimizes the sum of the squared vertical deviations of each point from the

regression line. The method used is called ordinary least-squares method (OLS) and the

regression line estimated is the line that best fits data points. (You may refer to

econometric textbooks for further details)

There are two approaches to get the estimated regression line using OLS, the first

approach is to use formula to calculate the value for constant a and coefficient b. It is

quite handful and feasible if there is only one independent variable, when there is more

than one independent variable the calculation becomes tedious.

The other approach to determine the values of the constant and coefficient is to use

Software packages such as Excel, SPSS and TSP. These packages make regression

analysis easy to use.

In our discussion we use SPSS. Based on the data given in Table 4.1 the results are

shown in Table 4.2. The regression line is

Y = 7.60 + 3.53 X

You will notice both approaches have the same results.

Where 7.60 is the intercept that explains when advertising expenditure is zero the sales

will be $7.60 million, whereas 3.53 is value of the slope or the coefficient. It shows that

when advertising expenditure increase by $1 million, sales will increase by $3.53 million.


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Y denotes the estimated sales in millions given the value of X. For example, using the

last observation from Table 4.1 if the value of X is $9 million the sales will be $39.37

million. Diagram 4.2 shows the estimated regression line.

Diagram 4.2 The Regression Line

y = 7.60 + 3.53 x

Y-Y [

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12 14 16

Sales

Y

Advertising Expenditure X


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LS // Dependent variable is SAL

SMPL range 1986 - 1995

Number of observation 10

Variable Coefficient Std. Error T-Stat 2-Tail Sig.

C

ADV

7.6000000

3.5333333

6.3323245

0.5222813

1.2001912

6.751919

0.264

0.000

R-squared 0.851212 Mean of dependent var

50.00000

Adjusted R-squared 0.832614 S.D of dependent var 6.992059

S.E. of regression 2.860653 Sum of squared resid 65.46667

Durbin-Watson stat 1.224915 F-statistic 45.76782

Log likelihood -23.58417

Table 4.2 Computer printout for single variable regression

The equation can be written out as

Y = 7.6 + 3.53 X .. eqn 4e(revisited)

After estimating the regression line using the available data, the next step is to evaluate

and interpret the results, this is explain in part 4.3

MULTIPLE REGRESSION ANALYSIS

A multiple regression analysis involves more than one independent variable. For

example, we want to determine how sales is influence by advertising expenditure and

price, the regression equation will be:


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Y = a1 + b1X1 + b2X2

Where: Y is sales volume

X1is advertising expenditures

X2is price of the product

a is the vertical intercept

b1 is Y/ X1, the marginal effect of advertising

expenditure on salesb2 is Y/ X2, the marginal effect of price on sales.

The process of estimating a multiple regression is the same as in simple regression

When we regress sales using data in Table 4.3, the results are as in Table 4.4.

Year Sales

(Y)

Advertising

expenditure(X1

)

Price

(X2)

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

44

58

48

46

42

60

52

54

56

40

10

13

11

12

11

15

12

13

14

9

1

1.2

2

1.8

2.1

0.8

1.4

2.0

1.5

1.0

Table 4.3 Sales, advertising expenditure and price


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LS // dependent variable is SAL (Y)

SMPL range : 1986 - 1995

Number of observation : 10

Variable Coefficient Std. Error T-Stat 2-Tail Sig.

C

ADV(X1)

PR(X2)

11.60403

3.4936051

-2.3836921

6.9633945

0.5078770

1.9495316

1.6665152

6.8788413

-1.2226999

0.140

0.000

0.261

R-squared 0.877397 Mean of dependent var 50.00000

Adjusted R-squared 0.842367 S.D of dependent var 6.992059

S.E. of regression 2.776058 Sum of squared resid 53.94549

Durbin-Watson stat 1.414870 F-statistic 25.04734

Log likelihood -22.61633

Table 4.4 Multi-regression analysis

From the results shown in table 4.4 the regression equation is:

Y = 11.604 + 3.493 X1 - 2.383 X2 --------- -eqn 4f

4.3 EVALUATION OF RESULTS

Once the results have been derived either by using the software package or manually

using formula, the next step will be to interpret the results carefully.

Besides analysing the coefficients itself as seen in chapter three, there are a number of

statistics that are of importance to evaluate the results. In our study we shall discuss the

following test


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a) Testing the overall Explanatory Power [Coefficient of determination (R2) ]

b) Test of Significant of Coefficient Estimates [t-stats]

c) Use the equation to predict the value of the dependent variable given the

values of the independent variables.

d) F- Statistics

We shall evaluate results from computer printout Table 4.2 based on simple regression

analysis. Nevertheless it can be extended to the multi-variable equations.

a) Testing the overall Explanatory Power - Coefficient of Determination (R2)

The coefficient of determination (R2) is used to determine how well the regression line fits

the data. It is a test of goodness of fit R2 measures the proportion of total variation in the

dependent variable that is explained by the regression equation.

The value of coefficient of determination ranges from 0 to 1. If the value is 0 it shows

that none of the independent variables explain the changes in the dependent variable, If

the value is 1 it shows that all the changes in the dependent variable is explained by the

variation in independent variable used in the regression. Therefore a value closer to 1 is

preferred.

Diagram 4.3 shows regression line with different coefficient of determination, that is R2 =

0.2, R2 = 0.6, R2 = 0.8 and R2 = 1. When R2 = 1 the regression line is a perfect fit.

(Rough sketch!).

X X

Y = f (x) Y = f(x)

X X

Y YR2 = 0.2 R2 = 0.6

X

X

X

XX

X

X

X

X

X

X

X

X

X

XX

X

X


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X X

Y = f(x) Y= f(x)

R2 = 0.8 Y R2 = 1 Y

Diagram 4.3 Regression line with different R2.

From the computer printout (Table 4.2) we can get the value of R2 for the above data as

0.85. It shows that 85% of changes in the dependent variable sales can be explained by

the independent variables, advertising expenditure. The other 15% cannot be explained

by the regression analysis. This may due to the omission of some important

independent variables.

As a rule of thumb, the higher the value of R2, the higher the explanatory power of the

estimated equation and the more accurate for forecasting purposes.

b) Test of Significant of Coefficient Estimates (T Stats)

The t - test is used to determine if there is a significant relationship between the

dependent and each independent variable. To carry out this test we need the standard

error of coefficient (Sb) to calculate thet statistics.

To calculate the t statistics we divide the estimated coefficient (b) by the standard error

coefficient (Sb).

t statistics = b

Sb

X

X

XX

X

X

XX

X

X

X

X

X

X

X

X

X

X

XX

PERFECT FIT


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Next, we get the critical value that is, t n-k-1.0.05 value from the studentt distribution table.

Where n is number of observation, k represents the number of independent variables

estimated and 0.05 is the significant level .Sometimes Rule of thumb can be also be

applied for t n k1, 0.05, where its the critical t value will be 2

Finally, to determine if there is a significant relationship between the dependent and

each independent variable, the calculatedt statistics is compare with the value from the

student t distribution table. If the value is greater than the value from the student t

distribution, than the independent variable is said to be statistically significant

otherwise it is not statistically significant in explaining the dependent variable.

From the computer printout (Table 4.2) the standard error of coefficient of price is 0.52.

Thet statistic is calculate by :

Calculated t = b = 3.53 = 6.79

0.52

Sb

The critical value from the studentt distribution table with t n k1,0.05 = t 10 1,.0.05 = t8 at95% confidence interval the critical t value is 2.306. Since the value oft statistics is

6.76 and greater than the critical t value 2.306, we say advertising expenditure is

statistically significant in explaining the variations is sales at 95% confidence interval.

c) Predicting the Value of the Dependent Variable

The standard error of estimation is a measure of the dispersion of the data points from

the line of best fit (regression line). Actual data points usually do not lie on theregression line but are dispersed above and below the line. This means that a value

predicted by the regression line will be subjected to error. The standard error of

estimation measures the probable error in the predicted value. For example using the

data from 4.1, when the advertising expenditure is $9 the sales is $40. If we use the

regression results the sale is $39.37. Therefore the value predicted will have an error.


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The standard error of estimation is useful in determining prediction interval that is, the

range within which the dependent variable will lie at a specified probability. At 95%

probability the dependent variable will lie in the predicted interval of

Y tn k-1,0.05 SE

Where Y is the predicted value of the dependent value based on the regression, which

can also be written as

Y 2 SE, when the rule of thumb is used.

When the range of Y is determine, it will show that 95 percent of the time the actual

value of Y will fall within the range calculated.

The graph in diagram 4.4 shows regression line with different standard error of

estimation. The smaller the standard error of estimation the closer the data points are to

the regressed line. The standard error of estimation is used to indicate the accuracy of aregression model.

Y

Y = f(x)

Y= f(x)

X

SE = 1.5 SE = 1.0

X

XX

X X

XX

X

X

X

X

X

X

X


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Y

Y= f(x)

XSE = 0.25

Diagram 4.4 Regression line with different SEs

From the computer printout (Table 4.2 ) the SE is 2.8. The range in which the sales will

fall at 95% confidence interval when advertising expenditure is $9 is given by:

Y t n k1 SE

Y = 7.6 + 3.53 (9) = 39.37

Where n is 10 and K is 1, With t 10-1-1,0.05, the critical t value from the student t distribution

table is 2.306.

39.37 t 10 - 11 (2.8)

39.37 2.306 (2.8)

39.37 6.457

Based on this statistical analysis, when the advertising expenditure is $9 million, 95percent of the time the sales will range from $32.913 million to $45.827 million.

e) F statist ics

The F statistic is used in a multiple regression analysis. It is another test of overall

explanatory power of the regression is the analysis of variance, which uses the F statistic

X

X X

X

X

X X

X

X

X X

X


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or the F ratio. The F statistic is used to test the hypothesis that the variation in the

independent variable explains a significant portion of the variation in the dependent

variable. The F statistics can be calculated as shown below.

F = explained variation / (k - 1)

unexplained variation / (n - k)

F = R2 / (k - 1)

(1 - R2) / (n - k)

To conduct the F test, we compare the calculated F values with the critical value from tabled F

values. If the calculated F value is higher, we can conclude that there is a significant relationship

between the independent variables and the dependent variable.

From the computer printout (Table 4.4) the f-value is 25.04. To determine the critical value of the

F distributions, we use the F distribution table which uses 5 percent significant level. The f

distribution for each level of statistical significant is defined in terms of 2 degree of freedom that

is the numerator and the denominator. The degree of freedom for the numerator is k - 1 which

is 2 - 1 = 1 and for the denominator is n - k which is 10 - 2 = 8. Using these two values the

critical tabled F value is 5.32. Since the calculated (printout) result 25.04 is greater than the

critical f value, we say there is a statistically significant relationship between the independent and

dependent variable.

4.4 PROBLEMS IN REGRESSION ANALYSIS

Regression analysis may face some serious problems, among them are multicollinearity,

heteroscedasticity, auto-correlation, specification errors, measurement errors and identification

problem.


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MULTICOLLINEARITY

One of the assumptions in the regression analysis is that independent variables are not related

to each other. If this assumption is not observed, then the estimated coefficient may give a

distorted result of the impact of the change in the independent variables. Multicollinearity arises

because if two variables are closely related, it is difficult to separate out the effect that each has

on the dependent variables.

There are two ways of detecting multicollinearity problems :

(i) if the regression result pass the F test but fails the t test.

(ii) by looking at the correlation coefficient between pairs of independent variables (as a

thumb of rule correlation coefficient of 0.7 or more would indicate the existence of

multicollinearity - most software produce a correlation coefficient matrix).

Multicollinearity will introduce a upward bias to the standard error of the coefficient and hence,

reduce the t values and the variable to be insignificant. Multicollinearity would not pose a

problem if estimated regression results are used for forecasting purposes but if researcher

wishes to understand more about the underlying structure of the demand function, then the

problem has to be resolved. The problem can be resolved by

(i) increasing the sample size data

(ii) transform the functional relationship

(iii) drop one of the highly collinear variables

HETEROSCEDASTICITY

Regression analysis presume homoscedasticity of the error terms (deviations from the

line of best fit is constant for all values of the independent variables). Heteroscedasticity

causes a systematic relationship (the residual of each X becomes larger as value of X

becomes larger). This problem of often occurs in cross-sectional data.


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Heteroscedasticity causes the standard error of the coefficient to biased and the R2 to be

high. The detect this problem, simply plot the values of the resisuals against the values

of the independent variables. Most software will produce graphs for visual inspection.

Heteroscedasticity can be overcomed by respecifying the independent variables, by

changing the functional form of the relationship, and by transformation of the data.

AUTOCORRELATION

It is indicated by a sequential pattern is the error term (i.e. the size of the error term

becomes progressively larger or smaller, or exhibits a cyclical or any other pattern with

respect to the X observations - meaning some other variables is changing systematically

and influencing the dependent variable). Autocorrelation problem usually appears when

time-series data are used. It can also arise due to the existence of trends and cycles in

economic variables, when important variables are excluded from the function or from

nonlinearities in the data.

Autocorrelation gives a downward bias to the standard error of the estimated regression

coefficient (t values are exaggerated) and hence the estimated coefficient are concluded

to be statistically significant when in reality they are not.

A standard test for identifying the presence of autocorrelation is the Durbin-Watson test

(it is presented automatically in the computer printout). As a rule of thumb, a value of d

= 2 indicates the absence of autocorrelation. To overcome autocorrelation problem a

researcher can include time as an additional variable to take into account trend patterns,

re-estimate the regression in a non-linear form or introduce lag data in the time-series.

SPECIFICATION ERROR

This problem arises if a wrong functional form of the relationship is used. For example,

the relationship is stated in linear form when in fact it should be nonlinear and vice versa.

To determine which functional form best explains the variance, all functional forms

should be tried and a comparison of the R2 be made. The second specification error

involves the omission of an important explanatory variable. This leads to unreliability of

the regression coefficient.


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MEASUREMENT ERROR

A pitfall to be avoided is the improper measurement of the variables. The most notorious

being the price variable. In many instances the actual price paid has not being

accurately depicted.

IDENTIFICATION PROBLEM

Regression analysis is conducted on the assumption that a single equation explains the

entire relationship. In the case of the demand estimation, price is the result of

simultaneous equations of both demand and supply. There is insufficient data in the

regression analysis to identify the shifts of the demand curves. In the case of time-series

demand estimation, the demand function cannot be expected to remain stable for long

extended period of time.

4.5 STEPS IN THE REGRESSION ANALYSIS

The most common method of demand estimation is the regression analysis because it is

more objective, inexpensive and provides more information. Generally the four steps to

follow are :

Model specification The first step is to identify the factors important in determining the

demand. This can be scouted from in depth knowledge of market conditions or

economic theory. Important variables should not be omitted, otherwise the results

become bias. On the other hand, variables should not be too many, otherwise therewould be econometric difficulties.

Data collection The next step involves collecting data of the variables used in the

model. Data can be in time-series or cross-section. Where no data are available, proxy

can be used. Data can be of primary or secondary source.


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Specifying Functional Form An appropriate functional form to be estimated has to be

determined. The simplest model is the linear model, example

Qx = a + b1Px + b2Py = b3I + .... + e

Model can also be non linear, for example in power form

Qx = A Pxa Py

b

Usually both forms are estimated and the one that gives the best result is reported.

Testing and evaluating econometric results The final step is to evaluate the regression

results. This involves

(i) checking the signs of the coefficients to see if they conforms to the

economic theory.

(ii) conduct t test on the estimated parameters to determined if they are

statistically significant

(iii) evaluate the R2

(iv) run other econometric test to ensure that problems such as

multicollinearity, heteroscedasticity and autocorrelation do not exists.


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QUESTIONS

1. You are given the following data of a company selling T-shirts.

Year $ MillionAdver tising Exp

$ Million QualityControl

$ Million Salesrevenue

1234

5678910

109

1112111213131415

3433456778

44404246485254585660

a) Use multiple regression analysis to estimate sales as a linear function of

advertising and Quality Control. Write the equation, t - statistics, and

coefficient of determination.

b) Using the results show which independent variable is significant at 95% in

explaining the dependent variable.

c) What can you say about the coefficient of determination?

d) Given the values for advertising as $13 million, and quality control as $7

million, at 95% confidence interval in what range will actual sales fall?

2. Assuming this results are taken from a computer printout.

Qx = 12.5 - I.SPx + 3.2 Px + 2.81

(0.8) (1.0) (1.9)

Se = 6.4

R2 = 0.8


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where Q is the quality demanded in thousands for product x, P, is the price in RM

, Py is the price of related product in RM and I 'is the income level in RM. The

figures in parenthesis are standard error of coefficient.

Answer the following questions.

a) At 95% confidence interval state which variable is significant in explaining

the dependent variables.

b) What does the R2 in the question implies?

c) Given the value of Px = 2.3, Py = 3.5, 1 = 3.5, calculate at 95% confidence

interval the range in which the quantity demanded will fall.

d) Using the information in (c) derive the

(i) Demand curve function

(ii) Total revenue function.

(iii) Marginal revenue function.

3. The results from a computer-printout for a linear demand -function is as follows.

Variable Constant Price Income Price of'

other good

Coefficient 0.248 -2.243 1.374 1.203

Standard error 0.018 0.93 0.501 0.814

t - statistics 1 3.778 2.412 2.742 1.478

Number of observation = 20 R2 = 0.5

Se = 4.89


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a) Write out the equation based on the above information.

b) What will be the quantity demanded if the values of the independent variables

are

Price = $10

Income = $90

Price of other goods = $12

c) Derive the demand curve function, and calculate the price, income and cross

elasticity.

d) Given marginal cost $20, what is the profit maximizing quantity and price.

Give the data given in (b)

e) At the profit-maximizing price, calculate the range in which the quantity will fall

at 95% confidence interval.

f) What can you say about the results? Will you accept the results?

4. Given the following data on advertisement, price and quantity demanded for

product X,

Observation Adver tisement Expenses (Million S) Sales (units) Price (S)1 1 4 02 2 6 13 4 8 24 8 14 35 0 12 4

6 5 10 57 8 16 68 9 16 79 7 12 8

a) Using the SPSS package, estimate sales as a linear function of

advertisement. Write down the equation, coefficient of determination, t -

statistics.


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b) Explain whether advertising expenditure is significant at 95% confidence

in explaining the variation in sales.

c) Now estimate sales as a linear function of advertisement and price, Write

the equation coefficient of determination, standard error of estimation and

t - statistics.

d) Is there any difference in the results between (a) and (c). If there is,

explain.

SUGGESTED SOLUTIONS

1 (a) Sal = 17.943 + 1.873 Adv + 1.915 Qu

(2.66) (2.81)

F - statistic = 46.61 Se=2.09

R 2 = 0.93

Sal : Sales

Adv : Advertising expenditure

QU : Quality control

(The values in parenthesis are t - statistics)

(b) To determine the significance, we have to calculate the 't' value for each

variable and compare it with the critical 't' value.

Advertising Expenditure

Calculate 't' = 1.87 = 2.66

0.703


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Quality control

Calculated 't' = 1.915 = 2.812

0.681

The critical 't' value

t10-2, 0 05 = 2.306

Since both the calculate 't' values for advertising expenditure and quality

control is above the critical 't' value (2.306), we can say that at 95%

confidence interval advertising expenditure and quality control Is

significant in explain sales.

(c) The R 2 is 0.93, showing that 93% of the changes in sales is

explained by the changes in advertising expenditure and quality control.

The f - statistics is 46.6 whereas the critical 'F' value at 95% confidence

interval is 4.74. Therefore we say there is statically significant relationship

between dependent and independent variable.

(d) Sales = 17.9 + 1.873 (13) + 1.915 (7)

= 55.654

To calculate the range we use the following formula

^

Y t n-k, 0.05 (SE) , tn -k, 0.08 = - 2.306

55.654 2.306 (2.09)

= 50.834, 60.473

At 95% confidence interval with the given values for advertising

expenditure and quality control the sales will range between

RM50.834 million and RM60.471 million.


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2. (a) To test the significance of the variable we get the 't' statistics for all

the variables.

Variable : Px (Price)

't' statistics = 1.8 = 2.6

0.8

Variable: Py (Price of related product)

't' statistics = 3.2 = 2

1.0

Variable: I (Income)

't' statistics = 2.8 = 1.47

1.9

In this case, since the number of observation is not known, we use rule

of thumb. Where tn-k, 0.05 = 2

By comparing the value of 't' statistics with this value, we see that price of

the product and price Of related product is significant. In explaining the

variation in quantity demanded at 95% confidence interval.

(b) The R 2 = 0, 8. It shows 80% of the variation in quantity demanded

is explain by the chances in price, price of related product and income.

(c) Qx = 12.5 - 1.8 (2.3) + 3.2 (3.5) + 2.8 (3.5)

= 29.36

^Y + 2 Se

29.36 2 (6.5)

= 11, 31.36

At 95% confidence interval given the figures where Px = 2.3, Py : 3.5 and I = 3.2,

the quantity demanded will range between RM11 and $31.36.


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(d) (i) Demand curve

Qd = A - bPy

A = 12.5 + 3.2 (3.5) + 2.8 (3.5)

= 335

Qd = 33.5 - 1.8Px

Px = 18.6 - 0.55 Qx

(ii) Total revenue = Px Qx

= ( 1 8.6- 0.55 Qx) Q

= 18.6Q 0.55Q2

(iii) Marginal revenue

TR =18.6 -1.111 Q

Q

3. (a) Qx = 0.248 - 2.243 (Px) +1.375 (I) - 1.203(12)

(b) Qx = 0.248 - 2.243 (10) + I.375 (90)+ 1.203 ( 12)

=116.004

(c) Demand curve function

Qd = A - bPy

= 0.248 - 2.243 (10) + I.375 (90) )- 1 203 (12)

= 138.434

Qd = 138.434- 2.243 Px

Px = 61.71 - 0.445 Qx


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Price elasticity

Q. P = -2.243 x 10 = 0.19

P Q 116-004

Income elasticity

Q. I = 1.375 x 90 = 1,060

I P 116.004

Cross elasticity

Q. Py = 1.203 x 12 = 0.12Py Q 116.004

(d) Total Revenue

P = 61.71 - 0.445Qx

TR = 61.71Q - 0.445Q2

MR = 61.71 - 0.89Q

Profit maximizing quantity

MR = MC

61.71 - 0.89Q = 20

41.71 = 0.89 Q

Q = 46.86

p = 61.71 - 0.445 (46.86)

= 40.85

(e) The range is calculated by using the standard error of estimation

46-86+ tn-k, 0.05 (4.89)


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46.86 t 20-2, 0.05(4.89)

46.86 + 2.110(4.89)

36.54 ; 57.17

At 95% confidence interval at profit maximizing price the quantity

demanded will range from $36.54 to $57.17

(f) To reject or accept the regression we shall look at the 't' statistics and

coefficient of determination.

To measure the strength of the relationship we calculate the 't'

statistics for each variable, and compare with the critical @t' value

from the student 't' distribution table.

Price

t - statistics = - 2.243 = 2.41

0.93

Income

t-statistics = 1.374 = 2.7

0.501

Price of other good

t - statistics = 1.203 = 1.477

0.814

tn-k,0.05 = t20-3,0.05 = 2.110

From the findings we see that only price and income are statistically

significant at 95% confidence interval.

In terms of coefficient of determination only 50% of the variations in the

dependent variable is explain by the changes in the independent variable, price,

income and price of related goods.


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The results can be accepted with changes that is, more variables should be

added.

4. (a) Sal = 2.5522 + 1.4820 Adv

Standard error of coefficient (0. 1007)

't' statistics (14.71 14)

R2 = 0.964

Se = 0.806

F = 216.4261

(b) The 't' statistics is 14.7 1 14, whereas tile critical t' value is

t9-2,0.05 = 2.365

It is seen that the calculated 't' value is greater than the tabled 't' value.

Therefore at 95% confidence interval advertisement expenditure is

significant in explaining variation in sales.

(c) Sal = 2.4763) + 1.6097 Adv - 0. 1 448 (Pr

Standard error of coefficient: (0. 15939) (0,1405)

't' statistics = (10.0991) (1.0310)

R2 = 0.969

Se = 0.80

F = 109.5951

(d) Yes, the estimated value of b, (Adv Exp) changed from 1.48 to 1.61 that is

when Adv Exp changes by RM1 million the sales increases by RM1.61 million.

This is because in the second regression, we held price constant whereas the

earlier estimate we did not.

Overall there is a slight improvement in the results.


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SUMMARY

Demand estimation is important in order to help reduce uncertainty associated with

decision making and to achieve the objective of the firm.

There are basically two types of regression analysis, they are linear and multiple

regression. The econometric method used to estimate the coefficients of a function

ordinary is least square method (OLS).

Once the regression coefficients are estimated, we can use the standard error of

estimation, coefficient of determinationt statistic and f statistic to evaluate the

strength of variable in explaining the dependent variable and the forecasting power of

the estimated model.

PRACTICE QUESTIONS

Q1. Cyber Corporation Sdn Bhd has estimated the demand function for its Banana

computer using regression analysis to be:

Qx = 251 20.3 Px + 24.1Pz + 2.5Ax + 1.3 Y

(6.3) (-2.9) (1.56) (2.3) (5.1)

R2 = 0.93

Std error of estimate = 1260

DW stats =.2.1

F stat + 46.2

Where Qx is the quantity of computers demanded per month


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Px is the price of the computer

Pz is the price of another competitors computer

Ax is the advertising expenditures per month

Y is the consumers average monthly income.

The figures in parentheses are the t-statistic of standard error of coefficients.

a) Interpret the above regression result and state whether the result can be

accepted or rejected.

b) Given the current values of independent variables Px = RM2500 Pz =

RM2300 Ax = RM1500 Y = RM 7000, calculate the price and cross

elasticity of demand for Banana computers. What do these values

indicate?

c) Suppose that the marginal cost of Banana computer is constant at

RM1800, at what price and output should the corporation charge to

maximize profit?

d) At the profit maximizing price, what range of sales volume can be

expected at the 95% confidence interval?

Q2. Sports Masters Inc produces shuttle cocks. It conducted regression analysis to

estimate the demand for its product. Some of the results of the analysis are given

below:


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Variable Coefficient Standard error of

Coeffecient

Intercept

P

M

Pr

425120.0

-25930.6

1.024

-2478.0

220300.0

8774.0

0.251

785.0

R2 = 0.8435

Standard error of regression =26900

The dependent variable is Q which represents the number of boxes of shuttle

cocks sold quarterly. P is the wholesale price it charges for a box of shuttle costs

in (RM), M is the consumers annual income (in RM) and Pr is the average price

of badminton rackets (in RM)

a) Based on the above information, write down the estimated regression

equation.

b) Does each independent variable have a significant effect on the dependent

variable at 5% significance level/ give your reason.

c) Is the demand sign of the coefficient of each variable consistent with demand

theory?

d) Does the regression have a strong explanatory power? Why do you say so?

e) Are regression results acceptable? Support your answer.

f) Sports Masters plans to charge a wholesale price of RM1.65 per box. The

average price of a badminton racket is RM110 and consumers annual

average household income is RM24600. What is the estimated number of


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boxes of shuttle cocks demanded? Compute the 95% confidence interval

estimate for your answer.

Q3. A study was done to look at demand for Mar and Mas cheesecake. A

regression analysis was conducted using the following model.

Q = a + b1 P + b2 A + b3Y + b4 H + b5 Pc

Where Q = quantity demanded in hundreds

P = price in RM (55)

A = advertising expenditures in thousand (20)

Y = average household income in thousand (31)

H = total number of residential sales in thousand (10)

Pc = price of leading competitor in RM (50)

Values in parentheses are the assumed values.

Data were collected over 18 quarters. The results of the computer analysis are

shown below.

Variable Coefficient Std. Error t-statistic

Intercept

P

A

Y

H

Pc

40

-1.1

1.5

0.32

0.5

0.1

2.5

0.9

0.6

0.12

0.17

0.75

16

-1.22

2.5

2.67

2.94

0.13

Coefficient of determination = 0.91

Standard error of estimate = 2.8F-value = 311.43

a) Write down the estimated regression equation. Using the assumed values,

estimate quantity demanded.


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b) How concerned should this company be about price discounts by its leading

competitor? Explain.

c) Should this company consider discounting its price in order to increase total

revenue? Explain.

d) How effective do you think advertising is for this company?

e) What type of good is Mar and Mas cheese cake?

f) Describe the statistical significance of each individual independent variable

included in the model. Given the t-value at 95% confidence interval as 1.96.

g) Comment on the coefficient of determination and F values. Can this estimated

regression equation be accepted?

h) Indicate the 95 % confidence interval of the range o f forecasted demand and

the range of total revenue for cheese cake.

Q4. Sharp Edge Inc. (SEI) is a producer of Polo Knives, a set of kitchen cutlery, which

it markets on a nationwide basis. Their knife sees are either sold directly to the

public through national television marketing programs, or given away as

promotional items. Operating experience during the past year suggest the

following demand function for its knife sets

Q = 400 180P + 10N + 0.5Y + 0.4A

(101) (314) (0.22) (0.14)

R

2

= 0.83SEE = 283

The value in the parenthesis is the standard error of coefficient.

Where Q is the quantity, P is the price (RM). N is the average Neilson rating of

television programs during which the company advertise their knives. Y is


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average disposable income per household (RM ) and A is advertising

expenditures (RM)

The current values of the variables are P =20, N =18.5, Y= 2,200 and A = 5,000

a) Determine the demand curve equation faced by SEI in a typical market.

State the demand curve with quantity expressed as a function of price, and

with price expressed as a function of quantity.

b) Calculate the price necessary to sell 2,650 and 3,190 sets of knives.

c) Do you agree that a price increase will increase the total revenue of the

company.

d) Calculate the income elasticity. What is the impact on demand during

periods of recession?

e) Which of the independent variables are statistically significant at 95%

confidence interval in explaining quantity change? (use rule of thumb 2)

f) Is the regression acceptable? Explain your answer.

g) Based on 95% confidence interval, compute the range of quantity demanded

at the total revenue-maximising price.

Q5. Using a linear regression analysis Syarikat Juicy Sweet Sdn Bhd. estimated its

demand function for its orange juice and achieved the following results:

Q = 257.1 + 1.465A + 0.61Y - 121.47P

(80.35) (0.36) (0.77) ( 21.77)

where Q = quantity demanded (000 packets)

A = advertising budget (RM000)

Y = disposable income per household (RM000)


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P = price per packet of orange juice (RM)

Standard errors are in parenthesis

Standard error of estimate = 12

R- Square = 0.70

a) What would an R-squared of 0.70 indicate?

b) Using a 95% confidence interval criteria, identify the independent factors

which have influence on sales of Syarikat Juicy Sweet Sdn. Bhds orange

juice. (Use the rule of thumb = 2)

c) Given A = RM32, 500, Y = RM6, 250 and P = RM1.25, determine the

demand function for Syarikat Juicy Sweets orange juice.

d) As the marketing manager of Syarikat Juicy Sweet Sdn Bhd., would you

recommend a price increase for the orange juice if the firm wants to increase

total revenue? Why?

e) Using the income elasticity, how would you categorise the orange juice

necessity, inferior or luxury? Explain.

f) Compute the advertising elasticity of demand. Would you recommend an

increase in advertising budget? Explain.

g. Calculate the range within which you would expect to find the actual

quantity with 95% confidence interval.

h. If the marginal cost of the orange juice is constant at RM0.80 per packet,

calculate the profit maximizing price for the orange juice.

Q6. Kintan cooler manufactures sells ice cubes. An in-house study for 3 years

revealed the following:


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Qc = 588 6.8Pc + 4.8Px + 18A + 88T + 18W

(318) (2.8) (2.8) (3.8) (18) (38)

R2 = 0.88

F5,30, = 0.01 = 3.7

Standard error of estimation = 18.8

n = 36 observations

Values in parentheses are standard errors

Where;

Qc = quantity of ice cube per plastic bag

Pc = Price of ice cube per plastic bag

Px = Price of competitors product

A = Advertising expenditures

T = Time

W = Weather

a) Fully evaluate and interpret the empirical results based on R2, F-statistic

and standard error of estimation.

b) Using a 95% confidence interval criterion, which independent variables

have influence on sales? (t30,=0.05=2.042)

c) In the 36th month, the average price is RM880, average competitors price

is RM780, advertising expenditures are RM880 and average monthly

temperature is 800F. Assuming this was a typical observation; derive the

relevant demand curve function for Kintan ice cube.

d) Given wholesale price is RM880; is there a probability for Kintan ice cube

to generate at least RM8, 888,000 in revenue?

e) Assuming preceding model and data given are relevant for the coming

period. Calculate the range within which you would expect to find actual

monthly sales revenue with 95% confidence. (t30, =0.05=2.042)


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Q7. The research department of Sutra Tiles Company wished to estimate the demand

function for its new product, Sutra XT200. A demand function had been

estimated on 120 respondents using regression analysis. The demand function is

as follows :

Q = 15,000 10P + 1500A + 4Pc + 2I

(5,234) (2.29) (525) (1.75) (1.5)

R2 = 0.65

F = 35.25

Standard error of estimate = 565

Values in parentheses are standard errors

Where

Q = Quantity demanded for Sutra XT200 tiles.

P = Price of Sutra XT200 tiles (RM7, 000)

A = Advertising expense, in thousands (RM54)

Pc

= Price of competitor's product (RM8, 000)

I = Average monthly income (RM4, 000)

a) Using a 95% confidence interval, interpret the regression result. Do you think the

equation can be accepted and used for forecasting purpose? Explain.

b) Derive an expression for the firms conventional demand curve for the new

product, Sutra XT200.

c) If the firms objective is to maximize total revenue from the sales of Sutra XT200tiles, at what price should the firm charge?

d) At that price, what range of sales volume can be expected at the 95% confidence

interval?


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e) Should Sutra Tiles Company consider reducing its price in order to increase its

total revenue? Explain.

f) Calculate income elasticity of demand. Is the Sutra XT200 a luxury, normal or

necessity good?

g) Calculate the cross elasticity of demand. Are Sutra XT200 tiles and

competitors tiles substitutes or complement?

STUDY NOTES


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APPENDIX


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ProbabilitiesDegree

ofFreedom .80 .60 .40 .20 .10 .05 .02 .01

1 0.325 0.727 1.376 3.078 6.314 12.706 31.821 63.6572 0.289 0.617 1.061 1.886 2.920 4.303 6.965 9.9253 0.277 0.584 0.978 1.638 2.353 .0182 4.541 5.8414 0.271 0.569 0.941 1.533 2.132 2.776 3.747 4.6045 0.267 0.559 0.920 1.476 2.015 2.571 3.365 4.0326 0.265 0.553 0.906 1.440 1.943 2.447 3.143 3.7077 0.263 0.549 0.896 1.415 1.895 2.365 2.998 3.4998 0.262 0.546 0.889 1.397 1.860 2.306 2.896 3.3559 0.261 0.543 0.883 1.383 1.833 2.262 2.821 3.25010 0.260 0.542 0.879 1.372 1.812 2.228 2.764 3.16911 0.260 0.540 0.876 1.363 1.796 2.201 2.718 3.10612 0.259 0.539 0.873 1.356 1.782 2.179 2.681 3.055

13 0.259 0.538 0.870 1.350 1.771 2.160 2.650 3.01214 0.258 0.537 0.868 1.345 1.761 2.145 2.624 2.97715 0.258 0.536 0.866 1.341 1.753 2.131 2.602 2.94716 0.258 0.535 0.865 1.337 1.746 2.120 2.583 2.92117 0.257 0.534 0.863 1.333 1.740 2.110 2.567 2.89818 0.257 0.534 0.862 1.330 1.734 2.101 2.552 2.87819 0.257 0.533 0.861 1.328 1.729 2.093 2.539 2.86120 0.257 0.533 0.860 1.325 1.725 2.086 2.528 2.84521 0.257 0.532 0.859 1.323 1.721 2.080 2.518 2.83122 0.256 0.532 0.858 1.321 1.717 2.074 2.508 2.81923 0.256 0.532 0.858 1.319 1.714 2.069 2.500 2.80724 0.256 0.531 0.857 1.318 1.711 2.064 2.492 2.79725 0.256 0.531 0.856 1.316 1.708 2.060 2.485 2.78726 0.256 0.531 0.856 1.315 1.706 2.056 2.479 2.77927 0.256 0.531 0.855 1.314 1.703 2.052 2.473 2.77128 0.256 0.530 0.855 3.313 1.701 2.084 2.467 2.763

29 0.256 0.530 0.854 1.311 1.699 2.045 2.462 2.75630 0.256 0.530 0.854 1.310 1.697 2.042 2.457 2.75040 0.255 0.529 0.851 1.303 1.684 2.021 2.423 2.70460 0.254 0.527 0.848 1.296 1.671 2.000 2.390 2.660120 0.254 0.526 0.845 1.289 1.658 1.980 2.358 2.617 0.253 0.524 0.842 1.282 1.645 1.960 2.326 2.576

Note: The probabilities given in the table are for two-tailed tests. Thus, a probability of 0.05 allows for 0.025 in each tail. For example, forthe probability of 0.05 and 21 df, t = 2.080. This means that 2.5 percent of the area under the distribution lies to the right of t = 2.080, and2.5 percent to the left oft = 22.080.

Source: From table III of Fisher and Yates, Statistical Tables for Biological, Agricultural and Medical Research, 6th

ed., 1974, published byLongman Group Ltd., London (previously by Oliver & Boyd, Edinburgh), by permission of the authors and publishers.

TABLE C-2

chapter 4 in managerial economic

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