Question

Two Cournot duopolists compete in a market with inverse demand given by p=32.00−2Q, where p is the per-unit price, qi is the output for firm i (either firm 1 or firm 2), and Q=q₁+q₂

Both firms face constant marginal costs of $2 per unit. Assume no fixed costs. What is the optimal output for firm 1 ? (Round to two decimals if necessary.) What is the optimal output for firm 2? (Round to two decimals if necessary.)

Answer 1

The optimal output for firm 1 in the Cournot duopoly model is q1 = 1, while the optimal output for firm 2 is Q - q1 = 1.

Step-by-step explanation:

The optimal output for firm 1 can be determined by using the Cournot duopoly model. In this model, each firm maximizes its profit by setting its output level where its marginal revenue (MR) is equal to its marginal cost (MC). Given that both firms have constant marginal costs of $2 per unit, the equilibrium output for firm 1 can be found by setting its MR equal to its MC and solving for its output. Since the inverse demand function is given by p=32.00−2Q, the total output in the market (Q) is equal to the sum of the outputs of both firms (q1 + q2).

To find the optimal output for firm 1, we need to substitute the total output in the market into the inverse demand function. Let's assume that the output of firm 1 is q1, then the output of firm 2 would be q2 = Q - q1. Therefore, the inverse demand function for firm 1 becomes: p = 32.00 - 2(Q - q1) = 32.00 - 2Q + 2q1. To find the optimal output, we set MR = MC, which gives: 2 = 2q1. Solving for q1, we find that the optimal output for firm 1 is q1 = 1. The optimal output for firm 2 can be found using the same process. Since q2 = Q - q1, we substitute the value of q1 into this equation, which gives q2 = Q - 1. Therefore, the inverse demand function for firm 2 becomes: p = 32.00 - 2(Q - (Q - 1)) = 32.00 - 2 + 2Q. Setting MR = MC, we get: 2 = 2Q. Solving for Q, we find that the optimal output for firm 2 is Q = 1.