Imagine an N firm oligopoly for "nominally differentiated" goods. That is, each of the N firms produces a product that "looks" different from the products of its competitors, but that "really" isn't any different. However, each firm is able to fool some of the buying public. Specifically, each of the N firms (which are identical and have zero marginal cost of production) has a captive market -consumers who will buy only from that firm. The demand generated by each of these captive markets is given by the demand function Pn A- Xn , where Xn is the amount supplied to this captive market and Pn is the price of the production of firm n. There is also a group of intelligent consumers who realize that the products are really undifferentiated. These consumers are represented by the demand curve P = A- X / B ,where P is the price of goods sold to these consumers and X is their demand. (If X n > A or X / B > A, then the prices in the respective markets are zero. Prices do not become negative.)
Firms compete Cournot style. Each firm n supplies a total quantity Xn , which is divided between its loyal customers and the customers who are willing to buy from any firm. If we let Xn be the part of Xn that goes to loyal customers, then the price of good n is necessarily 1:n = A - Xn·
The price that goes to the "shoppers" is P A -( En(Xn -Xn)) /B . In an equilibrium, Pn P , and if Xn > Xn , then Pn P . (That is, a firm can
charge a "higher than market" price for its own output, but then it will sell only to its own loyal customers.) By Coumot competition, we mean that each firm n assumes that the other firms will hold fixed their total amounts of output.
(a) For a given vector of output quantities by the firms, (X1, • • • ,XN ), is there a unique set of equilibrium prices (and a division of the output of each firm) that meets the conditions given above for equilibrium? If so, how do you find those prices? If not, can you characterize all the market equilibria?
(b) Find as many Cournot equilibria as you can for this model, as a function of the parameters N and B . (WARNING: Do not assume that solutions of first-order conditions are necessarily global maxima.)