Managerial Economics (chapter 4)

Chapter 4: Demand Estimation Managerial Economics Instructor: Maharouf Oyolola

Outline of the lecture • -Introduction • Statistical estimation of the demand function Model OLS estimation technique Interpretation of the results Testing

• The preceding chapter developed the theory of demand, including the concepts of price elasticity, income elasticity, and cross-elasticity of demand. • A manager who is contemplating an increase in the price of one of the firm’s products needs to know the impact of this increase on: • (1) quantity demanded • (2) total revenue • (3) profits

What questions should the manager answer? • - Is the demand elastic, inelastic, or unit elastic with respect to price over the range of contemplated price increase? • -What will happen to demand if consumer incomes increase or decrease as a result of an economic expansion or contraction. • Managers face these types of problems everyday whether in a profit-seeking enterprises, not-for-profit organizations or governments.

Example • What will be the impact of cigarette taxes on my quantity demanded of my product? • What effect will a tuition increase have on local state university revenues? • These are the types of questions the empirical investigation attempt to answer

Statistical Estimation of the Demand Function • Econometrics is a collection of statistical techniques available for testing economic theories by empirically measuring relationships among economic variables. • The measurement of economic relationships is a necessary step in using economic theories and models to obtain estimates of the numerical values of variables that are of interest to the decision maker.

The estimation of a demand function using econometric techniques involves the following steps • • • • •

- Identification of the variables - Collection of the data - Formulation of the demand model -Estimation of the parameters - Development of forecasts (estimates) based on the model

Identification of the variables • As discussed in the previous chapter, the demand function may be viewed as the relationship between the quantity demanded (the dependent variable) and several independent variables. • The first task in developing a statistical demand model is to identify the independent variables that are likely to influence quantity demanded. • These might include factors such as the price of the good in question, price of competing or substitute goods, population, per capita income, and advertising and promotional expenditures.

Collection of the data • Once the variables have been identified, the next step is to collect data on the variables. Data can be obtained from a number of different sources.

Formulation of the model • The next step is to specify the form of the equation, that indicates the relationship between the independent variables and the dependent variable.

Example • The linear model, which is the most common form of estimation equation:

Y = α + β 1 X1 + β 2 X 2 + β 3 X 3 + ε

Linear Model • α, β1, β2, β3, ε are the parameters of the model and ε is the error term. • The error term is included in the model to reflect the fact that the relationship is not an exact one, i.e., the observed demand value may not always be equal to the theoretical value.

Interpretation of the value of β • The value of each β coefficient provides an estimate of the change in quantity demanded associated with a one-unit change in the given independent variable, holding constant all other independent variables.

Interpretation of the value of β The β coefficients are equivalent to the partial derivatives of the demand function:

∆Y β1 = ∆X 1

∆Y β2 = ∆X 2 ∆Y β3 = ∆X 3

Interpretation of the value of β

ED ED

∆Y X 2 = . ∆X Y X2 =β2 . Y

Simple Linear Regression Model • The analysis in this section is limited to the case of one independent and one dependent variable (two-variable case), where the form of the relationship between the two variables is linear.

Estimating the simple linear regression coefficients n

(X ∑

1 b =i =

−

i

− X )(Yi − Y) − 2

n

(X ∑ i= 1

n

X Y ∑

1 b =i = n

i

i

X ∑ i= 1

i

−

2

i

− X)

− −

− n X Y − 2

− nX

Example 1 • Sherwin-Williams company is attempting to develop a demand model for its line of exterior house paints. The company’s chief economist feels that the most important variables affecting paint sales (Y) (measured in gallons) are: • (1) promotional expenditures(X1) (measured in dollars). These include expenditures on advertising (radio, TV, and newspaper), in-store displays and literature, and customer rebate programs.

Example1 • (2) Selling price (X2) (measured in dollars per gallon). • (3) Disposable income per household (X3) (measured in dollars) • The chief economist decides to collect data on the variables in a sample of ten company sales regions that are roughly equal in population. Data on paint sales, promotional expenditures, and selling prices were obtained from the company’s marketing department. Data on disposable income (per capita) was obtained from the Bureau of Labor Statistics

Answer • Y= a + b.X • Y= 120.75 + 0.434X • Interpretation: The coefficient of X (0.434) indicates that for one-unit increase in X ($1,000 in additional promotional expenditures), expected sales (Y) will increase by 0.434 (X1,000)= 434 gallons in a given sales region.

Using the regression equation to make predictions • A regression equation can be used to make predictions concerning the value of Y, given any particular value of X. This is done by substituting the particular value of X, namely Xp, into the sample regression equation

Example • Suppose one is interested in estimating Sherwin-William’s paint sales for a metropolitan area with promotional expenditures equal to $185,000 (i.e., Xp=185). • Y=120.75 + 0.434(185) • Y=201.045 gallons or 201,045 gallons

Example • Xp=300 or $300,000 • Remark: Xp falls outside of the series of observations for which the regression line was calculated. • Thus, because of the above remark, we cannot be certain that the prediction of paint sales based on the regression model would reasonable.

Standard Error of the estimate • The error term ei is defined as the difference between the observed and predicted value of the dependent variable. • The standard deviation of the error terms is calculated as: n

Se =

∑ei i =1

n −2

n

=

2 ( y − a − bx ) ∑ i i i =1

n −2

• The standard error of the estimate (Se) can be used to construct prediction intervals for Y. An approximate 95 percent prediction interval is equal to

Y ±2 S e

Returning to our previous example

Y ± 2S e = 201.045 ± 2(22.7) 155.447 to 246.643 (that is, from 155,447 gallons to 246,643 gallons).

Testing • H0: β=0 ( No relationship between X and Y) • Ha: β≠0 ( linear relationship between X and Y)

Testing •

There are two ways of doing the testing:

3) Calculate the t statistic and compare it to the critical value 4) Use the p-value technique

Correlation Coefficient n

r=

∑( x

i

i =1 n

∑( x i =1

i

−

−

− x)( y − y ) −

−

− x ) ( yi − y ) 2

2

Correlation Coefficient • The correlation coefficient measures the degree to which two variables tend to vary together. • Correlation analysis is useful in explanatory studies of the relationship among economic variables. The information obtained in the correlation analysis can then be used as a guide in building descriptive models of economic phenomena that can serve as a basis for prediction and decision making.

Correlation Coefficient • The value of the correlation coefficient ® ranges from +1 for the two variables with perfect positive correlation to -1 for two variables with perfect negative correlation.

The Coefficient of Determination ^

r

2

=

−

( y −y)

2

−

( yi − y )

2

The Coefficient of Determination • It measures the proportion of the variation in the dependent variable that is explained by the regression line (the independent variable). • The coefficient of determination ranges from 0 ( when none of the variation in Y is explained by the regression) to 1( when all the variation in Y is explained by the regression.

Example • If r2=0.519 ( from the Sherwin-William’s company example). • Interpretation: the regression equation, with promotional expenditures as the independent variable, explains about 52 percent of the variation in paint sales in the sample. • Remark: In the two-variable linear regression model, the coefficient of determination is equal to the square of the correlation coefficient, i.e., r2=0.519=( r)2=(0.72059)2.

F-ratio • It is used to test whether the estimated regression equation explains a significant proportion of the variation in the dependent variable. • The decision is to reject the null hypothesis of no relationship between X and Y ( that is, no explanatory power) at the k level of significance if the calculated F-ratio is greater than the Fk,1,n-2 value obtained from the F-distribution.

Example • If F=8.641 • The value of F0.05,1,8 from the F-distribution (from the table) is 5.32. • We reject, at the 0.05 level of significance, the null hypothesis that there is no relationship between promotional expenditures and paint sales. In other words, we conclude that the regression models does explain a significant proportion of the variation in paint sales in the sample.

Association and Causation • The presence of association (correlation) does not necessarily imply causation.

Multiple Linear Regression • A functional relationship containing two or more independent variables is known as a multiple linear regression model.

Y = α + β1 X 1 + β 2 X 2 + ...... + β m X m + ε

Regression Techniques

Regression techniques • Consider a simple demand equation Q= a + bP. The law of demand implies that the coefficient b should be negative, indicating that less of the product is demanded at higher prices.

Estimating Coefficients • Consider a small restaurant chain specializing in fresh lobster dinner. The business has collected information on prices and the average number of meals served per day for a random sample of eight restaurants in the chain. These data are shown below. Use regression analysis to estimate the coefficients of the demand function Qd= a +bP. Based on the estimated equation, calculate the point price elasticity of demand at the mean values of the variables.

The Least-Squares regression estimation −

(X ∑ b=

i

−X )(Y −Y )

∑( X −

a=

−

− i

−X

−

Y −b X

)

2

Estimating the demand for lobsters dinners using the OLS Qi

Pi

Qi-Q(bar)

Pi-P(bar)

(Pi-P(bar))2

(Pi-P(bar))(Qi-Q(bar))

100

15

0

-1

1

0

90

18

-10

2

4

-20

85

19

-15

3

9

-45

110

14

10

-2

4

-20

120

13

20

-3

9

-60

90

19

-10

3

9

-30

105

16

5

0

0

0

100

14

0

-2

4

0

40

-175

b

-4.375

a

170

Estimating the demand for lobster dinners • Using the ordinary Least Squares regression, we find the estimates of the demand for lobsters. • We can now use our results to determine the point elasticity of demand for lobsters.

Written assignment • Problem #6 page 180