(i) First find the full information optimal prices and compute profits. (ii) Find a c∗ such that for c>c∗, the full information optimal prices constitute a separating equilibrium. (iii) Assume now that c<>∗. What is the price that will be chosen by a low quality firm in a separating equilibrium? Find the lowest price charged by the high quality firm in a separating equilibrium. Does a separating equilibrium always exist?
(Prices as signals of quality).
Assume that a firm is a monopolist that produces a good of quality θ.
The firm knows the quality, the consumers do not.
If quality were observable, the firm would face a demand curve Q = θ−P.
Quality can be either high (θ = 2) or low (θ = 1). The marginal cost to the
firm of producing a low quality good is zero. The marginal cost of producing
a high quality good is c > 0. There is no fixed cost. The firm chooses
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prices as a function of θ. Denote by µ(P) be the probability that consumers
assign to the good being high quality given a price P. The demand curve is:
Q = 2µ(P) + (1 − µ(P)) − P
(i) First find the full information optimal prices and compute profits.
(ii) Find a c∗ such that for c>c∗, the full information optimal prices
constitute a separating equilibrium.
(iii) Assume now that c<>∗. What is the price that will be chosen by a
low quality firm in a separating equilibrium? Find the lowest price charged
by the high quality firm in a separating equilibrium. Does a separating
equilibrium always exist?
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