As noted earlier, estimating the parameters of the empirical demand function can be accomplished using regression analysis. However, the method of estimating demand depends on whether the demand to be estimated is a single price-setting firm’s demand or is a competitive price-taking industry’s demand. As previously mentioned, estimating industry demand requires more advanced statistical meth- ods than estimating the demand for a single firm;5 thus we will limit our discus- sion in this text to estimating demand for a price-setting firm. In this section, we will show you how to use ordinary regression, as set forth in Chapter 4, to esti- mate a single price-setting firm’s demand.
Before discussing an example of how to estimate the demand for a price-setting firm, we give you the following step-by-step guide.
4In a strict statistical sense, it is incorrect to estimate more than one model specification with the same set of data. This practice is common, however, given the high costs often associated with collecting sample data.
5For the least-squares method of estimating the parameters of a regression equation to yield unbiased estimates of the regression parameters, the explanatory variables cannot be correlated with the random error term of the equation. Virtually all the applications covered in this book involve explanatory variables that are not likely to be correlated with the random error term in the equation.
There is one important exception, however, and it involves the estimation of industry demand, because the price of the product—an explanatory variable in all demand functions—varies with shifts in both demand and supply. As it turns out, correctly estimating the industry demand for price-taking firms requires the use of a special technique called two-stage least-squares (2SLS). Online Topic 1, which can be found through McGraw-Hill Connect®, shows you how to use 2SLS to estimate both demand and supply equations for competitive industries and to forecast future industry prices and quantities.
Step 1: Specify the price-setting firm’s demand function As discussed previ- ously in Section 7.2, the demand function for the firm is specified by choosing a linear or curvilinear functional form and by deciding which demand-shifting variables to include in the empirical demand equation along with the price of the good or service.
Step 2: Collect data on the variables in the firm’s demand function Data must be collected for quantity and price as well as for the demand-shifting variables specified in Step 1.
Step 3: Estimate the price-setting firm’s demand The parameters of the firm’s demand function can be estimated using the linear regression procedure set forth in Chapter 4. Demand elasticities can then be computed as discussed previously in Section 7.2.
Estimating the Demand for a Pizza Firm: An Example
We will now illustrate how a firm with price-setting power can estimate the de- mand equation for its output. Consider Checkers Pizza, one of only two home delivery pizza firms serving the Westbury neighborhood of Houston. The man- ager and owner of Checkers Pizza, Ann Chovie, knows that her customers are rather price-conscious. Pizza buyers in Westbury pay close attention to the price she charges for a home-delivered pizza and the price her competitor, Al’s Pizza Oven, charges for a similar home-delivered pizza.
Ann decides to estimate the empirical demand function for her firm’s pizza.
She collects data on the last 24 months of pizza sales from her own company records. She knows the price she charged for her pizza during that time period, and she also has kept a record of the prices charged at Al’s Pizza Oven. Ann is able to obtain average household income figures from the Westbury Small Business Development Center. The only other competitor in the neighborhood is the local branch of McDonald’s. Ann is able to find the price of a Big Mac for the last 24 months from advertisements in old newspapers. She adjusts her price and income data for the effects of inflation by deflating the dollar figures, using a deflator she obtained from the Survey of Current Business.6 To measure the number of buyers in the market area (N), Ann collected data on the number of residents in Westbury. As it turned out, the number of residents had not changed during the last 24 months, so Ann dropped N from her specification of demand. The data she collected are presented in the appendix at the end of this chapter.
6The Survey of Current Business can be found at the website for the U.S. Department of Commerce, Bureau of Economic Analysis: www.bea.gov/scb. Implicit price deflators are presented quarterly from 1959 to the present in Table C.1, “GDP and Other Major NIPA Aggregates.”
I L L U S T R AT I O N 7. 2 Estimating the Demand for Corporate Jets
Given the success that many of our former students have experienced in their careers and the success we predict for you, we thought it might be valuable for you to examine the market for corporate aircraft.
Rather than dwell on the lackluster piston-driven and turboprop aircraft, we instead focus this illustration on the demand for corporate jets. In a recent empiri- cal study of general aviation aircraft, McDougall and Cho estimated the demand using techniques that are similar to the ones in this chapter.a Let’s now look at how regression analysis can be used to estimate the demand for corporate jets.
To estimate the demand for corporate jets, McDougall and Cho specified the variables that affect jet aircraft sales and the general demand relation as follows:
QJ 5 f (P, PR, M, D) when
QJ 5 number of new corporate jets purchased P 5 price of a new corporate jet
PR 5 price of a used corporate jet M 5 income of the buyers
D 5 so-called dummy variable to account for seasonality of jet sales
Since the market for used jets is extensively used by corporations, the price of used jets (a substitute) is included in the demand equation. We should note that P and PR are not the actual prices paid for the air- craft but are instead the user costs of an aircraft. A jet aircraft provides many miles of transportation, not all of which are consumed during the first period of ownership. The user cost of a jet measures the cost per mile (or per hour) of operating the jet by spreading the initial purchase price over the lifetime of jet transpor- tation services and adjusting for depreciation in the value of the aircraft.
The income of the buyers (M) is approximated by corporate profit because most buyers of small jet aircraft are corporations. The data used to estimate
the demand equation are quarterly observations (1981I–1985III). Many corporations purchase jets at year-end for tax purposes. Consequently, jet sales tend to be higher in the fourth quarter, all else constant, than in the other three quarters of any given year. Ad- justing for this pattern of seasonality is accomplished by adding a variable called a “dummy variable,”
which takes on values of 1 for observations in the fourth quarter and 0 for observations in the other three quarters. In effect, the dummy variable shifts the esti- mated demand equation rightward during the fourth quarter. A complete explanation of the use of dummy variables to adjust for seasonality in data is presented later in this chapter (Section 7.5).
The following linear model of demand for corpo- rate jets is estimated:
QJ 5 a 1 bP 1 cPR 1 dM 1 eD4
McDougall and Cho estimated this demand equation using least-squares estimation rather than two-stage least-squares because they noted that the supply curve for aircraft is almost perfectly elastic or horizontal.
If the supply of jets is horizontal, the supply price of new jets is constant no matter what the level of out- put. Because the market price of jets is determined by the position of the (horizontal) jet supply curve, and the position of supply is fixed by the exogenous determinants of supply, the price of jets is itself exog- enous. If jet price (P) is exogenous, least-squares re- gression is appropriate. The computer output obtained by McDougall and Cho from estimating this equation is shown on the next page.
Theoretically, the predicted signs of the esti- mated coefficients are (1) ˆ b , 0 because demand for corporate jets is expected to be downward-sloping;
(2) cˆ . 0 because new and used jets are substitutes;
(3) dˆ. 0 because corporate jets are expected to be normal goods; and (4) e ˆ . 0 because the tax effect at year-end should cause jet demand to increase (shift rightward) during the fourth quarter. All the esti- mates, except dˆ, match the expected signs.
The p-values for the individual parameter esti- mates indicate that all the variables in the model play
Since the price of pizza at Checkers Pizza is set by Ann—she possesses a degree of market power—she can estimate the empirical demand equation using linear regression. Ann first estimates the following linear specification of demand using the 24 monthly observations she collected:
Q 5 a 1 bP 1 cM 1 dPAl 1 ePBMac where
Q 5 sales of pizza at Checkers Pizza P 5 price of a pizza at Checkers Pizza
M 5 average annual household income in Westbury PAl 5 price of a pizza at Al’s Pizza Oven
PBMac 5 price of a Big Mac at McDonald’s a statistically significant role in determining sales of jet
aircraft, except corporate profits. The model as a whole does a good job of explaining the variation in sales of corporate jets: 86 percent of this variation is explained by the model (R2 5 0.8623). The F-ratio indicates that the model as a whole is significant at the 0.01 per- cent level.
McDougall and Cho estimated the price and cross- price elasticities of demand using the values of QJ, P, and PR in the third quarter of 1985:
E 5 ˆ b______P1985III Q1985III5 23.95
ENU5 c ˆ_______PR’1985III Q1985III 5 6.41
where ENU is the cross-price elasticity between new- jet sales and the price of used jets. The price elasticity estimate suggests that the quantity demanded of new corporate jets is quite responsive to changes in the price of new jets (|E| . 1). Furthermore, a 10 percent decrease in the price of used jets is estimated to cause a 64.1 percent decrease in sales of new corporate jets.
Given this rather large cross-price elasticity, used jets appear to be viewed by corporations as extremely close substitutes for new jets, and for this reason, we advise all of our managerial economics students to look closely at the used-jet market before buying a new corporate jet.
aThe empirical results in this illustration are taken from Gerald S. McDougall and Dong W. Cho, “Demand Estimates for New General Aviation Aircraft: A User-Cost Approach,”
Applied Economics 20 (1988).
DEPENDENT VARIABLE: QJ R-SQUARE F-RATIO P-VALUE ON F OBSERVATIONS: 18 0.8623 20.35 0.0001 PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE INTERCEPT 17.33 43.3250 0.40 0.6956 P 20.00016 0.000041 23.90 0.0018 PR 0.00050 0.000104 4.81 0.0003 M 20.85010 0.7266 21.17 0.2630 D4 31.99 8.7428 3.66 0.0030
The following computer printout shows the results of her least-squares regression:
DEPENDENT VARIABLE: Q R-SQUARE F-RATIO P-VALUE ON F OBSERVATIONS: 24 0.9555 101.90 0.0001
PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE INTERCEPT 1183.80 506.298 2.34 0.0305 P 2213.422 13.4863 215.83 0.0001 M 0.09109 0.01241 7.34 0.0001 PAL 101.303 38.7478 2.61 0.0171 PBMAC 71.8448 27.0997 2.65 0.0158
Ann tests the four estimated slope parameters ( ˆb , c , ˆ d , and ˆ e ) for statistical signifi-ˆ cance at the 2 percent level of significance. The critical t-value for 19 degrees of freedom (n − k 5 24 − 5) at the 2 percent significance level is 2.539. The t-ratios for all four slope parameters exceed 2.539, and thus the coefficients are all statistically significant. She is pleased to see that the model explains about 95.5 percent of the variation in her pizza sales (R2 5 0.9555) and that the model as a whole is highly significant, as indicated by the p-value on the F-statistic of 0.0001.
Ann decides to calculate estimated demand elasticities at values of P, M, PAl, and PBMac that she feels “typify” the pizza market in Westbury for the past 24 months. These values are P 5 9.05, M 5 26,614, PAl 5 10.12, and PBMac 5 1.15.
At this “typical” point on the estimated demand curve, the quantity of pizza demanded is
Q 5 1,183.80 2 213.422(9.05) 1 0.09109(26,614) 1 101.303(10.12) 1 71.8448(1.15) 5 2,784.4
The elasticities for the linear demand specification are estimated in the now familiar fashion
E ˆ 5 ˆb (P/Q) 5 2213.422(9.05/2,784.4) 5 20.694 E ˆM 5 c (M/Q) 5 0.09109(26,614/2,784.4) 5 0.871ˆ E ˆXAl 5 d ˆ(PAl /Q)5 101.303(10.12/2,784.4) 5 0.368 E ˆXBMac 5 e (Pˆ BMac /Q) 5 71.8448(1.15/2,784.4) 5 0.030
Ann’s estimated elasticities show that she prices her pizzas at a price where demand is inelastic (| ˆ E | , 1). A 10 percent increase in average household income will cause sales to rise by 8.71 percent—pizzas are a normal good in Westbury. The estimated cross-price elasticity E ˆXAl suggests that if Al’s Pizza Oven raises its pizza price by 10 percent, sales of Checkers’ pizzas will increase by 3.68 percent. While the price of a Big Mac does play a statistically significant role in determining sales of Check- ers’ pizzas, the effect is estimated to be quite small. Indeed, a 10 percent decrease
in the price of a Big Mac will decrease sales of Checkers’ pizzas only by about one-third of 1 percent (0.30 percent). Apparently families in Westbury aren’t very willing to substitute a Big Mac for a home-delivered pizza from Checkers Pizza.
While Ann is satisfied that a linear specification of demand for her firm’s pizza does an outstanding job of explaining the variation in her pizza sales, she decides to estimate a nonlinear model just for comparison. Ann chooses a log-linear de- mand specification of the form
Q 5 aPbMc P Ald P BMace
which can be transformed (by taking natural logarithms) into the following esti- mable form:
ln Q 5 ln a 1 b ln P 1 c ln M 1 d ln PAl 1 e ln PBMac The regression results from the computer are presented here:
DEPENDENT VARIABLE: LNQ R-SQUARE F-RATIO P-VALUE ON F OBSERVATIONS: 24 0.9492 88.72 0.0001
PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE INTERCEPT 20.72517 1.31437 20.55 0.5876 LNP 20.66269 0.04477 214.80 0.0001 LNM 0.87705 0.12943 6.78 0.0001 LNPAL 0.50676 0.14901 3.40 0.0030 LNPBMAC 0.02843 0.01073 2.65 0.0158
While the F-ratio and R2 for the log-linear model are just slightly smaller than those for the linear specification and the intercept estimate is not statistically significant at any generally used level of significance, the log-linear specification certainly performs well. Recall that the slope parameter estimates in a log-linear model are elasticities. Although the elasticity estimates from the log-linear model come close to the elasticity estimates from the linear model, linear and log-linear specifications may not always produce such similar elasticity estimates. In general, the linear demand is appropriate when elasticities are likely to vary, and a log-linear specification is appropriate when elastiticies are constant. In this case, Ann could use either one of the empirical demand functions for business decision making.