課程名稱︰個體經濟學上 課程性質︰必修 課程教師︰黃貞穎 開課學院：社會科學院 開課系所︰經濟系 考試日期（年月日）︰2010/12/27 考試時限（分鐘）：120 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. Roger's utility function is U = min{a1,a2} ×min{b1,b2}, where a1 and a2 are the number of piano lessons he consumes this year and next year and b1 and b2 are the number of ice skating lessons he consumes this year and next year. The price of piano lessons is $10 each and the price of ice skating lessons is $4 each. The price won't change, but the interest rate is 7%. Roger consumes 20 piano lessons this year. (a) (10%) Argue that Roger must consume the same number of piano lessons this year and next year. Similarly, he must consume the same number of ice skating lessons this year and next year. (b) (10%) Calculate the present value of Roger's income over this year and next year. (c) (10%) How many ice-skating lessons will Roger consume next year? 2. Forecasters predict there is a 50 percent probability that the upcoming growing season will be a drought. Assume that the farmer Jane is an expected utility maximizer with Bernoulli utility function u(w) = ln(w) where w is the wealth of Jane. Her initial wealth is $0. Jane initially has the choice between two crops with payoffs: Normal Rain Drought Potatoes $5000 $40000 Strawberries $20000 $12000 (a) (10%) If she can only plant one crop, which crop should she plant? (b) (10%) Assume she can instead plant half her land with each crop. Which crop mix (all potatoes, all stawberries or half of each) gives the highest expected income? Will Jane choose the crop mix giving her the highest expected income? Explain. (c) (10%) Assume Jane can choose any combination of potatoes and strawberries, provided that their total sums to 100 percent. What mix of crops maximizes Jane's expected utility? (d) (10%) Assume Jane decide to plant half her land with each crop. She is offered strawberry insurance. This insurance cost $5,000 and pays $10,000 in the case of a drought. Hence the policy is actuarially fair. Would Jane buy it? Explain. 3. If you invest $100 now in firm A, in one year you will get back $(30+T), where T is the average temperature during the next summer. If you invest $100 now in firm B, in one year you will get back $(180-T). The expected value of T is 70 and the standard deviation of T is 10. (a) (10%) Suppose you invest $50 in A and $50 in B. Calculate the expected value and the standard deviation of this investment. (b) (10%) Draw a graph showing the combinations of expected value and standard deviation that you can have by dividing $100 between stock in A and stock in B. (Hint: Think about your answer in (a).) (c) (10%) There is another firm, C. If you invest $100 now in firm C, in one year you will get back $(50+X), where X is the average number of earthquakes during the next summer. The expected value of X is 70 and the standard deviation of X is 20. T and X are statistically independent so that the covariance between them is zero. Stocks A, B and C are the only assets in the world. For an investor having mean-variance preference, show that he will always invest (weakly) more in stock B than in stock A. (Hint: Think about your answer in (b).) 助教表示：難易度順序：3>1>2 --

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