(8th edition) (the pearson series in economics) robert pindyck, daniel rubinfeld microecon 166

CHAPTER • Individual and Market Demand 141 Price 25 20 F IGURE 4.19 ESTIMATING DEMAND 15 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 d1, d2, and d3 that shift over time d1 10 d2 D d3 10 15 20 25 Quantity There is no reason to expect elasticities of demand to be constant Nevertheless, we often find it useful to work with the isoelastic demand curve, in which the price elasticity and the income elasticity are constant When written in its log-linear form, the isoelastic demand curve appears as follows: log(Q) = a - b log(P) + c log(I) (4.4) 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.11 Using the data in Table 4.5, for example, we obtained the regression line log(Q) = -0.23 - 0.34 log(P) + 1.33 log(I) This relationship tells us that the price elasticity of demand for raspberries is - 0.34 (that is, demand is inelastic), and that the income elasticity is 1.33 We have seen that it can be useful to distinguish between goods that are complements and goods that are substitutes Suppose that P2 represents the price of a second good—one which is believed to be related to the product we are studying We can then write the demand function in the following form: log(Q) = a - b log(P) + b log(P2) + c log(I) When b2, the cross-price elasticity, is positive, the two goods are substitutes; when b2 is negative, the two goods are complements 11 The natural logarithmic function with base e has the property that ⌬(log(Q)) = ⌬Q/Q for any change in log(Q) Similarly, ⌬(log(P)) = ⌬P/P for any change in log(P) It follows that ⌬(log(Q)) = ⌬Q/Q = - b[⌬(log(P))] = -b(⌬P/P) Therefore, (⌬Q/Q)/(⌬P/P) = - b, which is the price elasticity of demand By a similar argument, the income elasticity of demand c is given by (⌬Q/Q)/(⌬I/I)