Question

6. Entry Deterrence 2: Consider the Cournot duopoly game with demand p= 100 - (qı+q2) and variable costs c;(q;) = 0 for i € {1, 2}. The twist is that there is now a fixed cost of production k > 0 that is the same for both firms. Assume first that both firms choose their quantities simultaneously. Model this as a normal-form game. b. Write down the firm's best-response function for k = 1000 and solve for a pure-strategy Nash equilibrium. Is it unique? C. Now assume that firm 1 is a "Stackelberg leader” in the sense that it moves first and chooses 91. Then after observing 9 firm 2 chooses 92. Also assume that if firm 2 cannot make strictly positive profits then it will not produce at all. Model this as an extensive-form game tree as best you can and find a subgame-perfect equilibrium of this game for k = 25. Is it unique? d. How does your answer in (c) change for k = 225?

Answer 1

(a) Suppose firms choose qi simultaneously. Write down the firms' best-response functions for k = 1000 and solve for a pure-strategy Nash equilibrium. Is it unique? Firm l's payoff is 10001-91-9291-k if qı >0 and is (if qı = 0. The first-order condition for qı > 0 is 100 - 241 - 92 = 0, whence q1 = 100/92, exactly as in the usual Cournot duopoly; for this game, however, this is only the best response if it gives a payoff that is at least 0. Plugging (100 – 22)/2 in for 91 (and doing some algebra) gives a payoff of (100 – 22)/4 – k; in particular, for k = 1000, if q2 > 100 – 20/10 36.75, the best response is 0; it's only below that point that (100 – 22)/2 is the best response. The Nash equilibrium we find with k = 0, in which qı = 42 = 33], is still a Nash equilibrium; now, however, qı = 50,92 = 0 and qı = 0,92 = 50 are also Nash equilibria.

(b) Now assume that firm 1 is a "Stackelberg leader”, viz. it moves first, choosing 91, and then, after observing 91, firm 2 chooses 92. Also assume that if firm 2 is indifferent between producing a positive quantity and producing 92 = 0, it will choose 92 = 0. Model this as an extensive-form game tree as best you can and find a subgame-perfect equilibrium for k = 25. Is this the only subgame perfect equilibrium? This is a little bit hard to draw in a game tree because each firm has an infinite number of possible actions at each deci- sion node — and there are, accordingly, an infinite number of decision nodes for player 2! Sequential rationality requires that player 2 here best-respond to player l's action, where

again 42 (100 - 91)/2 if qı < 100 - 2Vk but q2 = 0 if 91 > 100 – 2Vk. This describes the strategy that firm 2 plays in any SPNE of the game. Firm l's payoff is 10001 - 41 - k if qı > 100 – 2VK, 10001 - 91 - (100 - 9141/2-k= 5091 - 1/2 - k if 100 – 2Vk > 41 >0, and 0 if qı = 0. Both cases for qı > 0 have q1 50 satisfy- ing the first-order condition, so you might think that 91 50 should be the answer. However, as long as 5001 - 97/2 > 0 at 41 = 100 – 2Vk, there's a jump in profit at q1 = 100 – 2Vk. If Vk = 5, this is qı = 90, where the payoff jumps to 100 x 90 – 902 – 25 = 875; the payoff at qı = 50 is 50 x 50-(50)2/2-25 = 1225, so 91 50 is indeed firm l's best response to firm 2's se- quentially rational strategy. 92 = (100 – 50)/2 = 25 is firm 2's action.