Econ 302 Assignment 3 Solution. (a) The monopolist solves: The first order condition is max Π(Q) = Q(a bq) cq. Q a Q c = 0, or equivalently, Q = a c, which is the monopolist s optimal quantity; the associated price is ( ) a c P (Q) = a b = a + c 2. (b) Fix a symmetric strategy profile in which every firm produces q units. The profit of any firm (say firm ) when he produces q and the others each produce q is: Π (q ) = q P (q + (n )q) cq = q (a b(q + (n )q)) cq. Given q, firm wants to maximize Π (q ). respect to q is The first order condition of firm with a b(n )q q c = 0, which gives firm s best-response to q of others: q = a b(n )q c. At a symmetric Nash equilibrium, we must have q = q, i.e., q = a b(n )q c,
or equivalently, q = a c (n + )b. The market price given these quantities is P (nq) = a In summary, each firm produces q = n (a c) = n + n + a + n n + c. a c (n+)b is the symmetric pure-strategy Nash equilibrium. The equilibrium price is n+ a + n n+ c. Figure : Graph for Problem, part (c). (c) In the graph, P M and Q M are the monopolist s price and quantity, P C and Q C are the price and total quantity from the Nash equilibrium in the quantity competition game (Cournot competition), and Q P C = a c b is the quantity from perfect competition: P (Q P C ) = a bq P C = c. 2
The deadweight loss for monopoly is 2 (P M c)(q P C Q M ) = ( ) ( a + c a c c a c ) = 2 2 b (a c)2. 8b The deadweight loss for Cournot competition is 2 (P C c)(q P C Q C ) = ( 2 n + a + n ) ( a c n + c c b ) n(a c) = (n + )b (a c)2 2(n + ) 2 b. Notice that the deadweight loss for Cournot competition (n 2) is decreasing with n, and is always smaller than the deadweight loss for monopoly (and becomes the same when n = ). In other words, competition reduces the deadweight loss. (d) Assume Cournot/quantity competition. As the number n of firms tends to infinity, in the Nash equilibrium the price tends to c and the deadweight loss tends to 0, i.e., we get convergence to perfect competition. 2. The profit of firm when he produces q while firm 2 produces q 2 is: Π (q ) = q (0 (q + q 2 )) 2q. Given q 2, firm wants to maximize Π (q ). respect to q is The first order condition of firm with which gives firm s best-response to q 2 : 0 q 2 2q 2 = 0, q = 8 q 2. () 2 Likewise, the profit of firm 2 when he produces q 2 while firm produces q is: Π 2 (q 2 ) = q 2 (0 (q + q 2 )) (q 2 ) 2. Given q, firm wants to maximize Π 2 (q 2 ). respect to q 2 is The first order condition of firm 2 with 3
which gives firm 2 s best-response to q : 0 q 2q 2 2q 2 = 0, q 2 = 0 q. (2) 4 At a Nash equilibrium, we must have mutually best responses, i.e., a solution to Equations () and (2). It is easy to show that the solution, and hence the Nash equilibrium, is q = 22 7, q 2 = 2 7. The market price given these (Nash equilibrium) quantities is P (q + q 2 ) = 0 34 7 = 36 7. 3. There are many pure-strategy Nash equilibria in this game. Below I describe all possible equilibria. (a) For any p [2.0, 3], it is a Nash equilibrium for firm to set p as his price, and firm 2 to set p 2 = p + 0.0; in this equilibrium, firm captures the whole market and sells Q(p ) = 0 p units. In the Nash equilibrium (p, p 2 = p + 0.0) where p + 0.0 < 3, firm 2 sets his price below his marginal cost of 3, which seems irrational. However, this p 2 = p + 0.0 is a best response to p because firm 2 never gets a chance to actually sell at this price, so he gets a profit of 0 in any case. You should check that for firm i, any p i < c i is weakly dominated by p i = c i ; this is a good exercise in reviewing the concept of dominance. This means that a weakly dominated strategy can still be a part of a Nash equilibrium, in contrast to strict dominance. However, a Nash equilibrium with a weakly dominated strategy is not very plausible in practice. (b) For any p [2.0, 3], it is a Nash equilibrium for firm to set p as his price, firm j to set p j = p + 0.0 for some j between 2 and 5, and the rest of the firms to set p i > p, where i j and i ; in this equilibrium, firm captures the whole market and sells Q(p ) = 0 p units. 4
(c) For this case it makes sense to distinguish between two classes of Nash equilibria: Class. It is a Nash equilibrium for firm to set p = 6, and firm 2 to set any p 2 > 6; in this equilibrium, firm captures the whole market and sells Q(6) = 4 units. Class 2. For any p [2.0, 5.99], it is a Nash equilibrium for firm to set p as his price, and firm 2 to set p 2 = p + 0.0; in this equilibrium, firm captures the whole market and sells Q(p ) = 0 p units. Note that p = 6 is firm s optimal price when he is the monopolist: max q(0 q) 2q q = 4, P (q) = 6. q In part (a) and (b), firm also wants to have a price of 6, but he is forced to sell at a price of at most 3 because given p = 6 > 3 = c 2 firm 2 would undercut this price, so p = 6 is not a sustainable price. 5