and the sum of observed characteristics of the competitors (which the literature assumes as exogenous and provides a sense of how crowded is the product space) as our instruments. Both sets of instruments will shift markups through changes in the competitive environment, and therefore, will be correlated with price, but uncorrelated with the unobserved characteristics.
The price index is built based on the changes in compensating variation derived from the estimation of the model described above. The compensating variation provides a measure of how much income could be taken away from (or given to) an individual and leave him indifferent between facing the old choice set and the new improved (inferior) choice set. Given the logit functional form assumption,
the compensating variation is calculated as
C ut V t =
where ut is the unconditional indirect utility ut = maxj ujt
marginal utility of income. Small and Rosen (1981) show that ut can be computed as
u t = l n X e x p ( δ j )
Trajtenberg (1990) shows that if the price change takes the form of a shift by a factor of (1 µt) in the distribution of prices but the variance remains the same, then the price index can be obtained as
P I t = ( 1
µt)P It 1
C V t C V t + ¯ p t
and p¯t is the average price in period t.
Suppose now that the indirect utility of consumer i when he chooses product j in market t is represented by: