Question

Two firms compete in a market with inverse demand P(Q) = a − Q, where the aggregate quantity is Q = q1 + q2. The profit of firm i ∈ {1, 2} is πi(q1, q2) = P(Q)qi − cqi , where c is the constant marginal cost, with a > c > 0. The timing of the game is: (1) firm 1 chooses its quantity q1 ≥ 0; (2) firm 2 observes q1 and then chooses its quantity q2 ≥ 0; (3) the firms receive payoffs πi(q1, q2). (a) (20 points) Find the subgame perfect Nash equilibrium of this game. (b) (15 points) Would the equilibrium aggregate quantity be smaller, larger, or the same, if the choice was simultaneous instead of sequential?

Answer 1

For solving part a sequential game equilibrium, we used the method of backward induction by finding the best respónse of firm 2 in 2nd stage and then solving nash equilibrium with of firm 1 in stage 1.

For solving part b simultaneous equilibrium, we solve the best responses of firm 1 and firm 2 simultaneously to obtain nash equilibrium of both firms.

After solving both sequential and simultaneous equilibrium we founded the aggregate quantity in each case and concluded that the aggregate quantity is higher when the choice is simultaneous.