Empirical Estimation of Demand

Later in Lhis book, we discuss how demand information is used as an input to firms' economic decision making. For example. General Motors needs to understand automobile demand to dccidc whether to offer rebates or below-market-interest-rate loans for new cars. Knowledge about demand is also important for public policy decisions-understanding Lhe demand for oil can help Congress dccidc whether to pass an oil import tax. Here, wc briefly ex-

3 See Gregory Chow, 'Technological Change and the Demand for Computers," American Economic Review 57, No. 5 (Dec. 1967); 1117-113Q

See Robert J. Gordon, 'The Postwar Evolution of Computer Prices," in Technology and Capital Formation, DaleW. Jorgensonand Ralph I.andau,eds. (Cambridge, MA: M.I.T. Press, 1989).

amine some methods for evaluating and forecasting demand. The more basic statistical tools needed to estimate demand curves and demand elasticities are described' in the Appendix to the book.

Interview and Experimental Approaches to Demand Determination

The most direct way to obtain information about demand is through interviews in which consumers are asked how much of a product they might be willing to buy at a given price. Direct approaches such as these, however, are unlikely to succeed because people may lack information or interest, or may want to mislead the interviewer. Therefore, market researchers have designed more successful indirect interview approaches. Consumers might be asked, for example, what their current consumption behavior is and how they would respond if a certain product were available at a 10 percent discount. Or interviewees might be asked how they would expect others to behave. Although indirect survey approaches to demand estimation can be fruitful, the difficulties of the interview approach have forced economists and marketing specialists to look to alternative methods.

In direct marketing experiments, actual sales offers are posed to potential customers. An airline, for example, might offer a reduced price on certain flights for six months, partly to learn how this price change affects demand for its flights and how other firms will respond.

Direct experiments are real, not hypothetical, but substantial problems remain. The wrong experiment can be costly, and even if profits and sales rise, the firm cannot be sure that the increase was the result of the experimental change because other factors probably changed at the same time. Also the response to experiments-which consumers often recognize as short-lived-may differ from the response to a permanent change. Finally, a firm can afford to try only a limited number of experiments.

The Statistical Approach to Demand Estimation

Firms often rely on market data based on actual studies of demand. Properly applied, the statistical approach to demand estimation can enable one to sort out the effects of variables, such as income and the prices of other products, on the quantity of a product demanded. Here we outline some of the conceptual issues involved in the statistical approach.

The data in Table 4.5 describe the quantity of raspberries sold in a market each year. Information about the market demand for raspberries might be valuable to an organization representing growers; it would allow them to predict sales on the basis of their own estimates of price and other demand-determining variables. To focus our attention on demand, let's suppose that the quantity of raspberries produced is sensitive to weather conditions but not

Year Quantity (Q) Price (P) Income (I)

1980

4 "

24

10

1981

7

20

10

1982

8

17

10

1983

13

17

17

1984

16

10

17

1985

15

15

17

1986

, 19

12

20

1987 >

20

9

20

1988.

22

5

20

to the current price in the market (because farmers make their planting decisions based on last year's price).

The price and quantity data from Table 4.5 are graphed in Figure 4.16. If one believed that price alone determined demand, it would be plausible to describe the demand for the product by drawing a straight line (or other appropriate curve), Q- a - bP, which "fit" the points as shown by demand curve D. The "least-squares" method of curve-fitting is described in the Appendix to the book.

FIGURE 4.16 Estimating Demand. Price and quantity data can be used to determine the form of a demand relationship. But the same data could describe a single demand curve D, or three demand curves di di, and d3 that shift over time.

Does curve D (given by the equation Q = 28.7 - 0.98P) really represent the demand for the product? The answer is yes, but only if no important factors other than product price affect demand. But in Table 4.5, we have included data for one omitted variable-the average income of purchasers of the product. Note that income (I) has increased twice during the study, suggesting that the demand for agricultural products has shifted twice. Thus, demand curves di, di, and ¿fe in Figure 4.16 give a more likely description of demand. This demand relationship would be described algebraically as

The income term in the demand equation allows the demand curve to shift in a parallel fashion as income changes. (The demand relationship, calculated using the least-squares method, is given by Q = 5.07 - 0.40P + 0.941.)

The Form of the Demand Relationship

The demand relationships discussed above are straight lines, so that the -effect of a change in price on quantity demanded is constant. However, the price elasticity of demand varies with the price level. For the demand equation Q = a - bP, for example, the price elasticity EP is:

Thus, the elasticity increases in magnitude as the price increases (and the quantity demanded falls).

There is no reason to expect elasticities of demand to be constant. Nevertheless, we often find the isoelastic demand curve, in which the price elasticity and the income elasticity are constant, useful to work with. When written in its log-linear form, it appears as follows:

where log () is the logarithmic function, and a, b, and c are the constants in the demand equation. The appeal of the log-linear demand relationship is that the slope of the line -b is the price elasticity of demand, and the constant c is the income elasticity.15 Using the data in Table 4.5, for example, we obtained the regression line log(0 = -0.81 - 0.24 log(P) + 1.46 log(7) This relation-

The logarithmic function has the property that A(log(0) = A Q!Q for any change in log(0. Similarly, A(log(P)) = AP/P for any change in log(P). It follows that A (log(Q)) = A2/g = -b[ A log (P))] = -b( &.P/P). Therefore, ( ¿\Q/Q)/(£iP/P) = -b, which is the price elasticity of demand. By a similar argument, the income elasticity of demand c is given by (A Q/Q)/( A I/I).

16 When reporting price and income elasticities of demand, we usually follow one of two procedures. Either we obtain their elasticities from constant elasticity demand equations, or we use other demand relationships and evaluate the price and elasticities-when each of the variables is evaluated at its mean. In the equation Q = a - bP, for example, we would calculate the price elasticity using the mean price P and the mean quantity sole Q , so that Ep = b (P/Q).

ship tells us that the price elasticity of demand for raspberries is -0.24 (that is, demand is inelastic), and the income elasticity is 1.46.

The constant elasticity form can also be useful for distinguishing between goods that are complements and goods that are substitutes. Suppose that Pi represents the price of a second good, which is believed to be related to the product we are studying. Then, we can write the demand function in the following form:

When ¿>2, the cross-price elasticity, is positive, the two goods are substitutes, and when b2 is negative the two goods are complements.

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