We now consider a duopoly model where the firms offer products with different qualities and consumers differ from each other in how much they care for quality. Suppose the firms offer their products with quality si ∈ [0, 1] and consumers’ ‘location’ in terms of how much they care about quality is given by a parameter θ ∈ [0, 1] and consumers are uniformly distributed over this interval. Suppose the firms first choose their quality s1 and s2 and then set prices p1 and p2. A consumer of type θ derives utility vi = r − pi + θsi from consuming a unit from firm i and where reservation price r is high enough that everyone purchases from one of the two firms. Suppose the marginal cost of producing increases with quality. There are no fixed costs and the total variable cost is C(qi, si) = csiqi so that marginal cost is csi and increases with quality. Further, let c = 1 so marginal cost is si Consider the second stage where given a choice of qualities s1 and s2 with s1 < s2, the firms simultaneously choose their prices. (1) Compute θb, the location address of the indifferent consumer and derive demand for the two firms. Also, write down the profit function for each firm. (2) Compute the equilibrium prices, quantities, and profits for both firms.
We now consider a duopoly model where the firms offer products with different qualities
and consumers differ from each other in how much they care for quality.
Suppose the firms offer their products with quality si ∈ [0, 1] and consumers’ ‘location’ in terms of
how much they care about quality is given by a parameter θ ∈ [0, 1] and consumers are uniformly
distributed over this interval. Suppose the firms first choose their quality s1 and s2 and then set
prices p1 and p2.
A consumer of type θ derives utility
vi = r − pi + θsi
from consuming a unit from firm i and where reservation price r is high enough that everyone
purchases from one of the two firms. Suppose the marginal cost of producing increases with quality.
There are no fixed costs and the total variable cost is
C(qi, si) = csiqi
so that marginal cost is csi and increases with quality. Further, let c = 1 so marginal cost is si
Consider the second stage where given a choice of qualities s1 and s2 with s1 < s2, the firms
simultaneously choose their prices.
(1) Compute θb, the location address of the indifferent consumer and derive demand for the two
firms. Also, write down the profit function for each firm.
(2) Compute the
Step by step
Solved in 6 steps with 21 images
