# For all products to have positive market share the following condition has to be satisfied

δ_{J }

δ_{J 1 }

< ··· <

2 δ

1 δ

p_{J }

p_{J 1 }

p_{2 }

p_{1 }

<

δ 1

p_{1 }

δ 0

and therefore the market shares are computed as

s_{1}(δ, p; Ω, F ) = F (

δ 1 δ p_{1 }0 | θ )

F(

δ p_{2 }2

δ p_{1 }1

|θ)

s_{j}(δ, p; Ω, F ) = F (

δ j p j

δ_{j 1 }p j 1

|θ)

F(

δ_{j 1 }p j 1

δ j p j | θ ) ,

for

1<j <J

s_{J }(δ, p; Ω, F ) = F (

δ_{J }p_{J }

δ_{J 1 }p J 1

|θ)

The market share of a product is therefore determined by its price-adjusted quality distance from its neighboring products.

The market share equations are inverted recursively to obtain the estimated values of δ as follows:

s_{0t }= 1

F(

δ_{1t }

p_{1t }

δ 0 t | θ )

I f p 0 a n d δ 0 a r e a s s u m e d t o b e 0 , t h e n w i t h d a t a o n p r i c e s a n d m a r k e t s h a r e and the log normal assumption for the distribution of _{i}, we can solve for the δ’s recursively ,

δ_{1t }= p_{1t}F ^{1}(1

s_{0t}|θ)

δ j t

=

δ j

1t

+

(p

_{jt }

p j

_{1 t})F

1

(1

s_{0t }s

1t

···

s_{j 1 }_{t }

|θ)

# In the estimation stage of this model we estimate all the parameters Ω = ( , β, θ) by means of a GMM procedure where we exploit the moment conditions

o f t h e k i n d E ( Z 0

ξ(Ω)) = 0, where Z contains functions of observable attributes

uncorrelated with

ξ(Ω)

The price index is computed in a similar way to the previous quality-adjusted price index, with the difference that the unconditional indirect utility is computed

by simulating 1000 consumers drawing from the distribution of

_{i}, and

then

obtaining CV _{t }

as an average of the individuals’ compensating variations.

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