Micro Prelim Question - Summer 2008

Consider the classic Cournot model of quantity competition, with asymmetric marginal costs. Two firms i ∈ {1, 2} simultaneously choose production levels qi ∈ R⁺; each firm i produces those units at a constant marginal cost of c_i per unit, and sells them at the market price, which is determined by the inverse demand function

P = max {0, 100 - q₁ - q₂}

1. Calculate firm i's best-response given a production level q_j of his opponent. (Be sure to account for the cases where it is optimal to set q_i = 0.)

2. Let c₁ = 25 and c₂ = 55. Find the Nash equilibrium in which both players produce, and calculate both firms' profits.

3. If firm 1 produced 45 units, firm 2's best-response would be not to produce at all. Calculate firm 1's profits in this event. Is it higher or lower than your answer to part 2? Is (q₁, q₂) = (45, 0) an equilibrium? Why or why not?

Now let G be any two-player simultaneous-move game, with strategy spaces A_i = R⁺ and payoff functions u_i: A_i × A_j → R which are continuous and differentiable. Let G₁ be a variation on the game G where player 1 moves first, then player 2 observes 1's action and moves second. (When G is the Cournot game, G₁ is known as the Stackelberg game.)

4. Let BR_i: A_j ⇒ A_i be player i's best-response correspondence for the game G, that is, BR_i(a_j) = arg max_ai∈Ai u_i(a_i, a_j). Show that if BR₂ is single-valued (BR₂(a₁) is a singleton for every a₁), then player 1's payoff in any subgame-perfect equilibrium of G₁ is at least as high as his payoff in any pure-strategy Nash equilibrium of G.

5. Now suppose BR₁ and BR₂ are both single-valued. Show that if BR₂ is weakly decreasing and u₁ is weakly decreasing in a₂, then player 1's strategy in any subgame-perfect equilibrium of G₁ is at least as high as his strategy in any pure-strategy Nash equilibrium of G.

6. Let G be the Cournot game described above, with marginal costs c1 and c2, and suppose G has a unique equilibrium in which both firms produce strictly positive quantities. Show that the statements in parts 4 and 5 hold strictly: firm 1 produces strictly more in the Stackelberg game than in the simultaneous-move Cournot game, and earns strictly higher profits.