4. (5%) Consider a representative firm in a perfectly competitive market using three inputs to produce its output: labor \((L)\), capital \((K)\), and material \((M)\). The production function is given by \(f(L, K, M) = L^2 K^2 M^2\). Suppose the unit price of labor, capital, and material are \(w = 1\), \(r = 1\), and \(m = 1\), respectively. Each firm is assumed to solve the cost minimization problem given the capital level \(K = 1\). The short-run cost function of each firm is (4a) with efficient scale (4b).
4a

4b

5. (5%) As in question 4, the short-run supply function of each firm is (5a) and the market supply function is (5b) if there are 100 identical firms in this industry. 5a

5b

6. (5%) Consider the following two-player game. Player 1 can choose \( T \), \( M \), or \( B \), while player 2 can choose \( L \), \( C \), or \( R \). The payoffs under players' various action profiles are given below. All the Nash equilibria including the mixed-strategy ones are (6).

7. (5%) There is an industry with three firms, each having zero marginal cost and zero fixed cost. The inverse demand function faced by these three firms is $P(q_1, q_2, q_3) = 60 - (q_1 + q_2 + q_3)$. Suppose that these three are Cournot competitors. Then, the Cournot-Nash equilibrium is (7a) with equilibrium profits of three firms (7b). 7a

7b

8. (5%) Consider the three firms and inverse demand function as assumed in question 7. But now suppose that firm 1 can commit to a certain level of output in advance. Afterwards, firms 2 and 3 will choose their outputs to maximize their profits independently and simultaneously. The subgame perfect Nash equilibrium is (8a) with equilibrium profits of three firms (8b). 8a

8b

9. (5%) Mary and Danny are risk averse and von Neumann-Morgenstern expected utility maximizers with utility function \( U(w) = w^{\frac{1}{2}} \) at all wealth level \( w > 0 \). Suppose that Mary's wealth is $10,000 and she may face a 40% chance of losing $7,500. Danny' wealth is $1,000,000 and he also faces a 40% chance of losing $7,500. The certainty equivalent of wealth for Mary, which is the wealth level with certainty providing Mary the same expected utility as the risky bundle, is (9a). The certainty equivalent of wealth for Danny is (9b). 9a

9b

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