Reference no: EM132413290

In class, we studied the Stakelberg and Cournot models of oligopoly, which di?er in terms of whether ?rms move sequentially or simultaneously. Below we explore a model that combines main features of the two. Firms 1, 2, and 3 compete in quantity in a market with the inverse demand p(y) = a-by, where y ≥ 0 is the total quantity in the market and a,b > 0. For each i ∈{1,2,3}, let yi ≥ 0 be the quantity supplied by ?rm i and assume that ?rm i produces at a constant marginal cost c > 0, with no ?xed cost. Assume that a > c. Firm 1 is the leader and ?rms 2 and 3 are "simultaneous followers". Firm 1 ?rst chooses its quantity y1. Then observing y1, ?rms 2 and 3 choose their respective quantities y2 and y3 at the same time. Note once again that ?rms 2 and 3 move simultaneously after ?rm 1 moves. Now applying backward induction, we search for the oligopolistic outcome generated by this kind of competition.

(a) Suppose that ?rm 1 has chosen y1, which ?rms 2 and 3 observe. Given y1, what is ?rm 2's best response b2(y3) to y3?

(b) Using your answer to part (a), ?nd all Nash equilibria of the subgame that begins with ?rm 1's choice of y1. In other words, state what ?rm 1 should expect from ?rms 2 and 3 when it chooses y1.

(c) The output choices y2 and y3 by ?rms 2 and 3 in part (b) can be viewed as a function of y1: y2 = y2(y1) and y3 = y3(y1). Using this, ?nd the optimal (pro?t-maximizing) choice of y1 for ?rm 1 and the resulting price and total quantity in the market. Hint: It may be useful to let λ ≡ a-c b .

(d) Based on your answer to part (c), ?nd the deadweight loss of oligopoly. Hint: If the market were perfectly competitive, the market outcome would be e?cient. You may compare the outcome in part (c) to the perfectly competitive outcome.