Managers can use the techniques of regression analysis outlined in Chapter 4 to obtain estimates of the demand for their firms’ products. The theoretical foundation for specifying and analyzing empirical demand functions is provided by the theory of consumer behavior, which was presented in Chapter 5. In this section, we will show you two possible specifications of the demand function to be estimated.
A General Empirical Demand Specification
To estimate a demand function for a product, it is necessary to use a specific functional form. Here we will consider both linear and nonlinear forms. Before proceeding, however, we must simplify the general demand relation. Recall that quantity demanded depends on the price of the product, consumer income, the price of related goods, consumer tastes or preferences, expected price, and the number of buyers. Given the difficulties inherent in quantifying taste and price expectations, we will ignore these variables—as is commonly done in many empirical demand studies—and write the general demand function as
Q 5 f (P, M, PR, N) where
Q 5 quantity purchased of a good or service P 5 price of the good or service
M 5 consumers’ income PR 5 price(s) of related good(s)
N 5 number of buyers
While this general demand specification seems rather simple and straightfor- ward, the task of defining and collecting the data for demand estimation requires careful consideration of numerous factors. For example, it is important to recognize the geographic boundaries of the product market. Suppose a firm sells its product only in California. In this case, the consumer income variable (M) should measure the buyers’ incomes in the state of California. Using average household income in the United States would be a mistake unless California’s household income level matches nationwide income levels and trends. It is also crucial to include the prices of all substitute and complement goods that affect sales of the firm’s prod- uct in California. Although we will illustrate empirical demand functions using just one related good (either a substitute or a complement), there are often nu- merous related goods whose prices should be included in the specification of an empirical demand function. Whether the market is growing (or shrinking) in size is another consideration. Researchers frequently include a measure of population in the demand specification as a proxy variable for the number of buyers. As you
can see from this brief discussion, defining and collecting data to estimate even a simple general demand function requires careful consideration.
A Linear Empirical Demand Specification
The simplest demand function is one that specifies a linear relation. In linear form, the empirical demand function is specified as
Q 5 a 1 bP 1 cM 1 dPR 1 eN
In this equation, the parameter b measures the change in quantity demanded that would result from a one-unit change in price. That is, b 5 DQ/DP, which is assumed to be negative. Also,
c 5 DQ/DM _ 0 if the good is Hnormal inferior
and
d 5 DQ/DPR _ 0 if commodity R is a H substitute complement
The parameter e measures the change in quantity demanded per one-unit change in the number of buyers; that is, e 5 DQ/DN, which is assumed to be positive.
Using the techniques of regression analysis, this linear demand function can be estimated to provide estimates of the parameters a, b, c, d, and e. Then t-tests are performed, or p-values examined, to determine whether these parameters are sta- tistically significant.
As stressed in Chapter 6, elasticities of demand are an important aspect of de- mand analysis. The elasticities of demand—with respect to price, income, and the prices of related commodities—can be calculated from a linear demand function without much difficulty. From our discussion in Chapter 6, it follows that, for a linear specification, the estimated price elasticity of demand is
ˆ E 5 ˆb 3 P __Q
As you know from the discussion of demand elasticity in Chapter 6, the price elasticity depends on where it is measured along the demand curve (note the P/Q term in the formula). The elasticity should be evaluated at the price and quantity values that correspond to the point on the demand curve being analyzed. In simi- lar manner, the income elasticity may be estimated as
ˆ E M 5 c ˆ3 M ___Q Likewise, the estimated cross-price elasticity is
ˆ E XR 5 dˆ 3 __PR Q
where the X in the subscript refers to the good for which demand is being esti- mated. Note that we denote values of variables and parameters that are statisti- cally estimated (i.e., empirically determined) by placing a “hat” over the variable or parameter. In this discussion, for example, the empirical elasticities are des- ignated as E ˆ, E ˆM , and E ˆXR, while the empirically estimated parameter values are designated by a , ˆ ˆ b , ˆ c , ˆ d , and e .ˆ
A Nonlinear Empirical Demand Specification
The most commonly employed nonlinear demand specification is the log-linear (or constant elasticity) form. A log-linear demand function is written as
Q 5 aPbMc P Rd N e
The obvious potential advantage of this form is that it provides a better estimate if the true demand function is indeed nonlinear. Furthermore, as you may recall from Chapter 4, this specification allows for the direct estimation of the elasticities.
Specifically, the value of parameter b measures the price elasticity of demand.
Likewise, c and d, respectively, measure the income elasticity and cross-price elasticity of demand.3
As you learned in Chapter 4, to obtain estimates from a log-linear demand function, you must convert it to natural logarithms. Thus, the function to be esti- mated is linear in the logarithms
ln Q 5 ln a 1 b ln P 1 c ln M 1 d ln PR 1 e ln N Choosing a Demand Specification
Although we have presented only two functional forms (linear and log-linear) as possible choices for specifying the empirical demand equation, there are many possible functional forms from which to choose. Unfortunately, the exact func- tional form of the demand equation generally is not known to the researcher. As noted in Chapter 4, choosing an incorrect functional form of the equation to be estimated results in biased estimates of the parameters of the equation. Selecting the appropriate functional form for the empirical demand equation warrants more than a toss of a coin on the part of the researcher.
In practice, choosing the functional form to use is, to a large degree, a mat- ter of judgment and experience. Nevertheless, there are some things a manager can do to suggest the best choice of functional form. When possible, a manager should consider the functional form used in similar empirical studies of demand.
If a linear specification has worked well in the past or has worked well for other products that are similar, specifying a linear demand function may be justified. In some cases a manager may have information or experience that indicates whether the demand function is either linear or curvilinear, and this functional form is then used to estimate the demand equation.
Now try Technical Problem 2.
Now try Technical Problem 3.
3The appendix to this chapter shows the derivation of the elasticities associated with the log-linear demand specification.
I L L U S T R AT I O N 7. 1 Demand for Imported Goods in Trinidad
and Tobago:
A Log-Linear Estimation
Trinidad and Tobago, two small developing countries in the Caribbean, rely heavily on imports from other nations to provide their citizens with consumer and capital goods. Policymakers in these two countries need estimates of the demand for various imported goods to aid them in their trade-related negotiations and to make forecasts of trade balances in Trinidad and Tobago. The price elasticities and income elasticities of demand are of particular interest.
Using data on prices and income, John S. Gafar estimated the demand for imported goods in the two countries, using a log-linear specification of demand.a According to Gafar, the two most common functional forms used to estimate import demand are the linear and log-linear forms. As we noted, the choice of functional form is often based on the past experience of experts in a particular area of empirical research. Gafar chose to use the log-linear specification because a number of other import stud- ies “have shown that the log-linear specification is preferable to the linear specification.”b Gafar noted that he experimented with both the linear and log- linear forms and found the log-linear model had the higher R2.
In his study, Gafar estimated the demand for imports of eight groups of commodities. The demand for any particular group of imported goods is specified as
Qd 5 aPbMc
where Qd is the quantity of the imported good demanded by Trinidad and Tobago, P is the price of the imported good (relative to the price of a bundle of domestically produced goods), and M is an income variable. Taking natural logarithms of the demand
equation results in the following demand equation to be estimated:
ln Qd 5 ln a 1 b ln P 1 c ln M
Recall from the discussion in the text that b is the price elasticity of demand and c is the income elasticity of demand. The sign of ˆ b is expected to be negative and the sign of cˆ can be either positive or negative.
The results of estimation are presented in the accompanying table.
Estimated Price and Income Elasticities in Trinidad and Tobago
Product group
Price elasticity estimates ( b )ˆ
Income elasticity estimates ( c )ˆ
Food 20.6553 1.6411
Beverages and tobacco 20.0537n 1.8718 Crude materials (except fuel) 21.3879 4.9619 Animal and vegetable oils and fats 20.3992 1.8688
Chemicals 20.7211 2.2711
Manufactured goods 20.2774n 3.2085
Machinery and transport equipment 20.6159 2.9452 Miscellaneous manufactured articles 21.4585 4.1997
Only two of the estimated parameters are not statis- tically significant at the 5 percent level of significance (denoted by “n” in the table). Note that all the product groups have the expected sign for ˆ b , except manufac- tured goods, for which the parameter estimate is not statistically significant. The estimates of c suggest that ˆ all eight product groups are normal goods ( ˆc . 0).
As you can see, with the log-linear specification, it is much easier to estimate demand elasticities than it is with a linear specification.
aJohn S. Gafar, “The Determinants of Import Demand in Trinidad and Tobago: 1967–84,” Applied Economics 20 (1988).
bIbid.
Sometimes researchers employ a series of regressions to settle on a suitable specification of demand. If the estimated coefficients of the first regression speci- fication have the wrong signs, or if they are not statistically significant, the speci- fication of the model may be wrong. Researchers may then estimate some new specifications, using the same data, to search for a specification that gives signifi- cant coefficients with the expected signs.4
For the two specifications we have discussed here, a choice between them should consider whether the sample data to be used for estimating demand are best repre- sented by a demand function with varying elasticities (linear demand) or by one with constant elasticity (log-linear demand). When price and quantity observations are spread over a wide range of values, elasticities are more likely to vary, and a linear specification with its varying elasticities is usually a more appropriate specification of demand. Alternatively, if the sample data are clustered over a narrow price (and quantity) range, a constant-elasticity specification of demand, such as a log-linear model, may be a better choice than a linear model. Again we stress that experience in estimating demand functions and additional training in econometric techniques are needed to become skilled at specifying the empirical demand function. We now discuss how to estimate the parameters of the empirical demand function.