u_{ijt }

=

i p j t + β x j t + ξ j +

ξ j

Notice that in this model there is a different unobserved shock to preferences. This model

for each consumer i and there is no is the opposite case of the previous

logit model, where all the heterogeneity entered through the error term, while in this model the heterogeneity is present in the random coefficient for price, which is a permanent component. This one-random-coefficient model assumes that consumers agree on the ranking of product quality, and that they only differ

on

their

willingness

to

pay

for

it.^{7 }

In

this

model

there

is

also

an

outside

option

whose utility is normalized to zero. This model, as opposed to the logit one, potentially predict zero market share for a product.

could

# Ordering products by ascending price, a consumer will buy product j iff u_{ij }>

u _{i }_{k }∀ k =6

j, which implies

i

<

δ p

j

j

i

>

δ p_{k }

k

Therefore, consumers of type

i

,

if

j p > p_{k }

,

if

j p_{k }> p

δ p_{k }k

δ j p j

wil buy product j iff

i

< min

δ j

k < j p j

δ k

p_{k }

=

_{j }(δ, p)

and

i

> max k > j δ k p k

δ j p j

=

_{j }(δ, p)

Assuming product j as:

_{0}(δ, p) = ∞ and

_{J }(δ, p) = 0, we can write the market share for

s_{j}(δ, p; Ω, F ) = (F ( _{j}(δ, p)|θ) F ( _{j}(δ, p)|θ)) {

j

>

_{j}}

for

j = 0, . . . , J

where {} is an indicator function, Ω are the parameters, and F ( to the c.d.f. of i, which is assumed to be log-normal.

_{i}|θ) corresponds

^{7}Ongoing work extends this model to include more than one random coefficient, and therefore the product ranking is not a matter of consensus across consumers.

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