Consider the Dixit capacity investment model when the inverse market demand curve is P(Q) = 200 − Q, w = 20, r = 10, and K = 200. a. Derive a new firm’s post-entry best reply function. b. Given initial capacity of x, derive the incumbent firm’s post-entry best reply function. c. Given x, derive the post-entry equilibrium profit of a new firm. d. Derive the minimum capacity that makes entry unprofitable. e. Derive the optimal capacity choice of the incumbent firm.

3. Using the Cournot model, consider a market where demand for the good is given by the equation, P = 16 - Q. Assume that marginal cost is constant at 4. There are two, equal-sized firms whose outputs are 9₁ and 92, and there is no further entry. Total market output is q₁ + q2 = Q. Marginal revenue for each of the identical firms is given by MR₁ 16-291-92 and MR₂ = 16-292-91. a. What is the best-response function for each firm? = b. What is each firm's output? c. What is the market price? 91 = 92 = P = d. Suppose Firm 1 asserts itself as the leader in this market, and acts as such. How will the output of the two firms change? 91 = 92 = e. Now, assume that there are no antitrust laws and the two firms can collude. What will be the market output? What will the price be in the market? Q= P =

M8

1. A firm that sells in two separate markets has the profit function where q1 and q2 sales in the two markets. Find the profit maximizing sales in each market, using the Hessian to check second-order conditions for a maximum. 2. Find the values of q1 and q2 that will maximizing the profit function and check that second-order conditions are met using the Hessian matrix.

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