107年 - 107 國立臺北大學_碩士班一般入學考試試題_經濟學系：個體經濟學#147947

科目：研究所、轉學考(插大)◆個體經濟學 | 年份：107年 | 選擇題數：0 | 申論題數：29

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所屬科目：研究所、轉學考(插大)◆個體經濟學

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申論題 (29)

一、(50%)

1. (5%) Consider a profit-maximizing monopolist faces the inverse demand function \( P(y) = \alpha - \beta y \) with \( \alpha, \beta > 0 \), and has the total cost function \( C(y) = cy \) with \( 0 < c < \alpha \), where \( y \) is the output amount. Suppose the government imposes a quantity tax of $4 per unit of output on this monopolist. Then, the equilibrium price will increase (1a) and the equilibrium quantity will decrease (1b).
1a
ab
2. (5%) Under the set-up of question 1 but with the monopolist operating at an output level with point price elasticity of demand equal to 3. If the government imposes a quantity tax of $4 per unit of output, then the price will increase (2).
3. (5%) Suppose that a representative firm in a perfectly competitive industry has the production function

$f(x_1, x_2) = \min\{x_1, x_2\}^n$ with $0 < \alpha < 1$, where $x_1$ and $x_2$ are the amounts of inputs 1 and 2, respectively. Denote $w_1$ and $w_2$ the prices of inputs 1 and 2, respectively. Suppose that this firm chooses the amounts of inputs 1 and 2 to minimize its costs for producing $y$ units of output. The long run cost function of this firm is (3a) and the long-run supply function of this firm is (3b).
3a
3b
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
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？
2. (16%)假設消費者對X與Y財的效用函數為 U(X, Y) = Y + 12X - X (2) / 2，X、Y財的價格分別為 P_X = 2、P_Y = 1，預算為 M = 30，

(1)求消費者均衡為何？
(2)求X财之普通需求函数 \(\mathrm{X}(P_X, P_Y, M)\)。
(3)求X财之Hicks受桶價需求函数。
(4)求 \(P_{X}\) 由2元降至1元時Slutsky定義的X財替代效果與所得效果。
(5)若所得M為30元， \(P_{Y} = 1\) ，求 \(P_{X}\) 由2元降至1元時之桶價變量、對等變量以及消費者剩餘的變動為多少？
3. (18%)某甲兩期消費的效用函數為 U(C_1, C_2) = C_1C_2，兩期商品價格均為1 (即 P_1 = P_2 = 1)，兩期所得原賦為 (Y_1, Y_2) = (700, 816)，名目利率為2%，

(1)請計算兩期的最適消費組合為多少？某甲於最適選擇點為貸出者還是賒借者？效用為多少？
(2)假設名目利率自 \(2\%\) 上升至 \(36\%\) ，請計算新的最適消費選擇點。某甲於新的最適選擇點為貸出者還是賒借者？某甲效用會增加還是減少？
(3)若兩期所得原賦為 \((Y_{1}, Y_{2}) = (800, 714)\)，名目利率自 \(2\%\) 上升至 \(36\%\) 時，請問某甲第一期消費會增加還是減少？某甲於新的最適選擇點為貸出者還是賒借者？某甲效用會增加還是減少？