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 u_{t }V t =

u_{t 1 }

where u_{t }is the unconditional indirect utility u_{t }= max_{j }u_{jt }

and

is the

marginal utility of income. Small and Rosen (1981) show that u_{t }can be computed as

u t = l n X e x p ( δ j )

j

Trajtenberg (1990) shows that if the price change takes the form of a shift by a factor of (1 µ_{t}) in the distribution of prices but the variance remains the same, then the price index can be obtained as

P I t = ( 1

µ_{t})P I_{t }_{1 }

where

µ_{t }=

C V t C V t + ¯ p t

and p¯_{t }is the average price in period t.

3.3

# Vertical Model

Suppose now that the indirect utility of consumer i when he chooses product j in market t is represented by:

12