課程名稱︰ 個體經濟學二 課程性質︰ 必修 課程教師︰ 古慧雯 開課學院： 社科院 開課系所︰ 經濟系 考試日期（年月日）︰ 2017/6/20 考試時限（分鐘）： 100 + 30 min (大家寫不完 所以加了時間) 試題 : 個體經濟學期末考(2017.6) 古慧雯 總分41分。答題皆須附說明，未作解釋的答案概不計分。 1. In a city of 60 resident, there are 3 aquariums. Residents visit aquariums every weekend without any other recreational alterna- tives. Each person will decide which aquarium to visit this weekend and they all share the same preference. The value of the visit de- pends on how many residents crowd in an aquarium. For i = a,b,c, when there are ni persons in aquarium i, the value of the visit per person in monetary term is: AVa = 120 - 2na AVb = 30 - nb AVc = 20 - nc (a) (2 points) In equilibrium, how many people will visit aquarium a if all aquariums are free of charge? (b) (2 points) If wr want to maximize total value of visits, how many people should be allowed to visit aquarium a? (c) (2 points) Our first goal is to maximize total value of visits. The second goal is to minimize entry fee(s) charged. Under these considerations, how should we charge entry fees at these three aquariums? 2. A firm has the following production function: q = (kl)^(1/2) (根號kl) where q is the output and k and l are quantities of captial and labor. The wage rate is w, the rental rate on capital is r and the product price is p. We shall consider a long-run problem in which the firm couls adjust both k and l. (a) (1 point) Please derive the marginal revenue product of captial in terms of k, l and p. (b) (1 point) Are captial and labor complements or substitutes in production? (c) (2 points) If the firm wishes to produce q units, how many units of labor will he hire? (Answer in terms of q,w and r) (d) (1 point) Please derive the average cost function AC(q) in terms of r and w. (e) (1 point) What is this firm's break-even price in terms of r and w ? (f) In the following, we shall consider a perfectly competitive in- dustry with many firms identical to the above one. We shall assume that this industry plays a very small role in the market of capital, and r is fixed at $1. On the other hand, an increase of labor employment by this industry will drive up the wage rate. Labor supplied to this industry is: L = w, where L is the units of labor hired by the whole industry. Let Q denote the total output supplied by the industry. i. (1 point) Please derive L as function of Q and w. ii. (1 point) Is this industry a constant-cost industry? iii. (2 points) Please derive the industry supply function Q(p). 3. (3 points) We shall consider a two-peroid model in which people only produce and consume apples. You do not have to consume exactly the amount you harvest in a period. You could borrow or lend apples at a market interest rate. People have the usual convex time preference and apples consumed in either period are normal goods. Apples of the first period were harvested a short while ago. An unexpected drought will severely lower the harvest in the seconf period. Use a model of the reprensentative agent to figure out how this drought affects the market interest rate? 4. X lives for periods, and has an income of $100 in both periods. His utility depends on consumption of both periods in the following manner: u = 12 ln c1 + 10 ln c2, where c1 and c2 denote the consumption in the first period and the consumption in the second perios. If X saves in the first period, he will earn an interest rate of 5%. On the other hand, if he borrows to consume in the first periof, the credit card company will charge him an interest of 25%. (a) (1 point) Put c1 on the horizontal axis and c2 on the vertical axis and draw X's budget line. (b) (1 point) Please prove that the marginal rate of substitution is 6c2/5c1. (c) (2 points) How much will X spend in the first period? (d) (2 points) What happens if the bank increases the interest rate of saving account from 5% to 9% and lowers the interest rate of consumption loan from 25% to 10%? How will X change his spenfing in the first period? Show the substitution effect and income effect in a graph. 5. Y, who is risk averse, enter a casino with $100 in his pocket. He has an opportunity to gamble at 1-to-2 odds on the toss of an unbaiased coin. For every dollar he bets on heads (tails), he will win (lose) $1 if heads comes up and he will lose (win) $2 if tails comes up. Y is thinking how to gamble. (a) (1 point) Put Y's wealth when heads come up on the horizontal axis and his wealth when tails come up on the vertical axis. Derive and draw Y's budget line in a figure. (b) (2 points) Draw Y's indifferent curve that passes his endow- ment point E, i.e. the situation if he does not gamble. Please caculate the marginal rate of subsitution at E. (c) (2 points) Will Y gamble? If so, will he bet on heads or tails? And how much will he bet? 6. Z's utility depends on his consumption c, and leisure l: u = c + 2 l^(1/2) (a) (1 point) Please prove that MRS = |dc/dl| = 1/(l)^(1/2). (b) (1 point) Does Z have a convex preference? (c) (1 point) The daily wage is $1, and Z has no other sourve of income. His weekly consumption completely comes from his labor income: c = $1 * (7 - l) where 7 - l is his working day(s) per week. How many days does Z work in a week? (d) (1 point) Z's parents decide to give Z $5 weekly from now on. How many days will Z work in a week when he becomes richer in this case ? (e) (2 points) Is Z's labor supply curve backward-bending? 7. There are only two stocks in the market: a and b. Let ra and rb denote the rate of return of stock a and the rate of return of b, respectively. The expected value of ra is 12% and the standard deviation of ra is 4%. The expected value of rb is 8% and the standard deviation of rb is 3%. The covariance of ra and rb is 0. (a) (2 points) An investor consider how to divide $100 between these two stocks so that the variance of the rate of return of this portfolio will be minimized. How much will hw invest in stock a? And what is the expected rate of return of his portfolio? (b) (3 points) Consider the captial asset pricing model which de- scribes the equilibrium situation of the finacial markets. Be- sides stocks a and b mentioned above, and investor could bor- row/lend at a risk-free rate 2%. Please compare the market value of all stocks of a and the market value of all stocks of b. --

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