pi1´ = h1(zi2), zi2 ∈ C2´, (7a)
pi2´ = h2(zi1), zi1 ∈ C2´, (7b)
One can then compute a Laspeyres-like measure by taking a weighted average, using period 1 sales shares as weights, of the (actual and “predicted”) price ratios of the varieties available in period 1:
L12 = ∑C*wi(pi2/pi1) + ∑C1′ wi(pi2′/pi1) = [∑C*qi1pi2 + ∑C1′ qi1pi2′]/∑C1qi1pi1, (8a)
where, as usual, wi = qi1pi1/∑C1qi1pi1.
Similarly, using period 2 sales shares of the various varieties as weights and “predicting” the period 1 prices of varieties available only in period 2 yields a Paasche-like measure:
P12 = ∑C*wi(pi2/pi1) + ∑C2′ wi(pi2/pi1′) = [∑C2qi2pi2]/[∑C*qi2pi1 + ∑C2′ qi2pi1′], (8b)
where wi = qi2pi1/[∑C*qi2pi1 + ∑C2′ qi2pi1′] for zi2∈C*, and wi = qi2pi1′/[SC*qi2pi1 + SC2′ qi2pi1′] for zi2ŒC2′. One can combine these, as in (6c), to obtain a Fisher-like measure of the price relative for this product. Note again that if price ratios for all varieties are the same, as assumed by the time dummy method, all of these measures are equal.
To see the sense in which these two approaches give Laspeyres-like and Paasche-like measures, it is instructive to follow Pakes (2001) and consider a single consumer with income y in periods 1 and 2, with prices the same in both periods for all goods but widgets. In period 1, the consumer has available a set of varieties C1, the prices of which are given by the known hedonic function h1(z), and she purchases variety z1. In period 2, the consumer faces choice set C2 and known hedonic function h2(z), and she chooses variety z2.
Suppose this consumer is given h2(z1) − h1(z1) additional income in period 2. Is this greater or less than the compensating variation, the period 2 income increase that would leave her exactly as well off as in period 1? If z1∈C2, buying variety z1 in period 2 would leave her with y + h2(z1) − h1(z1) − h2(z1) = y − h1(z1) to spend on other goods, exactly as in period 1. So h2(z1) − h1(z1) is at least equal to the compensating variation. But because the two hedonic functions are different, it may be possible for the consumer to do even better by choosing some z2′ π z1 in C2. Thus h2(z1) − h1(z1) is greater than or equal to the compensating variation, depending on whether such a z1′ exists or not.
Similarly, suppose instead that the consumer’s period 1 income is reduced by h2(x2) − h1(x2). Is this greater or less than the equivalent variation, the period 1 income reduction that would leave her exactly as well off as in period 2? If z2∈C1, buying variety z2 in period 1 would leave her with y + h1(z2) − h2(z2) − h1(z2) = y − h2(z2) to spend on other goods, exactly as in period 1. So this income