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

δJ

δJ 1

< ··· <

2 δ

1 δ

pJ

pJ 1

p2

p1

<

δ 1

p1

δ 0

and therefore the market shares are computed as

s1(δ, p; Ω, F ) = F (

δ 1 δ p1 0 | θ )

F(

δ p2 2

δ p1 1

|θ)

sj(δ, p; Ω, F ) = F (

δ j p j

δj 1 p j 1

|θ)

F(

δj 1 p j 1

δ j p j | θ ) ,

for

1<j <J

sJ (δ, p; Ω, F ) = F (

δJ pJ

δJ 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:

s0t = 1

F(

δ1t

p1t

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 ,

δ1t = p1tF 1(1

s0t|θ)

δ j t

=

δ j

1t

  • +

    (pjt

p j

1 t)F

1

(1

s0t s

1t

···

sj 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.

14

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