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

10. (5%) Assume that Mary and Danny have the utility function and wealth levels as specified in question 9, and Mary will face the same risky bundle stated in question 9 as well. However, Danny now faces no risk at all. Suppose that Mary offers to pay Danny \( R > 0 \) to bear her risk of losing $7,500. Mary will pay Danny \( R \) whether or not she suffers the loss. If Mary loses $7,500, Danny must pay her $7,500. Under the circumstance, the lower bound of \( R \) is (10a) and the upper bound of \( R \) is (10b) so that Mary and Danny will agree this arrangement. 10a

10b

二、(50%) 1. (16%)甲有6間房子(X)和2筆土地(Y)，乙有現金10000萬元，房子市價為1000萬，土地市價為1000萬，甲和乙的效用函數皆為 U(X, Y) = X (1/2) Y (1/2)，求 (1)甲和乙二人對房子的最適消費量與效用水準分別為多少？

(2)若房子市價下跌為500萬，甲的效用會提高或降低？乙的效用會提高或降低？

(3)對乙而言，房子市價下跌為500萬時，Hicks定義的X財所得效果和替代效果各為多少？

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