Final answer:
The subgame perfect Nash equilibrium for the asymmetric game between two firms is found using backward induction. Firm two's optimal choice is based on firm one's output, which is found by equalizing marginal revenue to marginal cost. Then, firm one chooses its output to maximize its profit, considering firm two's optimal reaction.
Step-by-step explanation:
The subgame perfect Nash equilibrium of the asymmetric game is determined by backward induction since firm two reacts to firm one's output choice. First, we would calculate firm two's best response function by setting its marginal revenue equal to its marginal cost, which means solving dπ2/dq2 = 0. Given the payoff function π2(q1,q2)=80q2−q2²−q1q2, the best response for firm two can be found by taking the derivative with respect to q2 and setting it to zero.
Next, firm one anticipates firm two's best response and chooses its output to maximize its own payoff, taking into account the reaction of firm two. This involves solving dπ1/dq1 = 0, with π1(q1,q2)=100q1−q1²−q1q2. Firm one will use firm two's response function to find the optimal q1.
The final result is the quantity pair (q1, q2) that constitutes the equilibrium quantities for both firms. This outcome is the subgame perfect Nash equilibrium, where neither firm has an incentive to deviate from their chosen quantities given the other firm's decision.