课程名称︰个体经济学 课程性质︰必修 课程教师︰黄贞颖 开课学院：社科院 开课系所︰经济系 考试日期（年月日）︰2012/1/9 考试时限（分钟）：180 min 是否需发放奖励金：yes （如未明确表示，则不予发放） 试题 : 1. Consider a mean-variance utility maximizer who can allocate his portfolio between three different assets. The three assets have different expected return and different variance of returns. The return of different assets are all uncorrelated with each other. μσ (a)(10 points) If μ1 μ2 μ3 are the expected returns on the three assets and w1, w2 and w3 are the shares of the profolio allocated to the first and second assets(so 1-w1-w2 is the share allocated to the third asset). respectively, write down the formula for the expected return on this consumer's portfolio. 2 2 2 (b)(10 pts)If σ1, σ2 and σ3 are the variances of the returns of the three assets.Write down the formula for the variance of the consumer's portfolio. (c)(5 pts)Now assume the expected returns are 5%, 10% and 2%. Re-write your answer to part (a) incorporating this information. (d)(5 pts)Assume also that the variances of the returns are 4%, 4% and 0%. Re-write your answer to part (b) incorporating this information. (e)(10 pts)Write sown the optimization problem that this consumer will try to solve, using specific numbers for means and variances of the returns and assuming the utility function is 2 2 u(μ, σ) = μ-σ 2 Where μis the expected return of the portfolio and σ is its variance. (f)(5 pts)Solve for the optimal values of w1 and w2. (g)(5 pts)Interpret your solution for the demand for the third asset. (h)(5 pts)Explain why the consumer chooses to hold asset 1 while it has the same variance but a lower expected retyrn than asset 2. 2. T or F: You need to explain briefly your answers. (a)(5 pts)In the CAPM model, if a stock lies strictly below the budget line(recall that we label the standard deviation of a stock's return on the x-axis and the mean of that on the y-axis), then it is for sure that no one will want to hold this stock in his optimal choice since there is always an alternative with less risk or greater expected return (b)(5 pts)If apples are a normal good for a net supplier of apples, then it's possible that when the price of apples increases, his demand for apples increases as well. (c)(5 pts)Suppose there are two states of nature. State 1 occurs with probability p1 and state 2 with p2=1-p1. A consumer has the utility of the following p1 p2 U(c1,c2,p1,p2)=c1 c2 where c1 is his consumption at state 1 and c2 is that at state 2. The consumer's utility function takes the expected utility form. (d)(5 pts)Continue from above. Facing a risky stream and the expected value of the same income stream, the consumer prefers the latter than the latter. 1/3 (y+√x) 3.A comsumer has the utility function U(x,y)=e where x is the good in concern and y is the money that can be spent on all other goods(so the price of y is normalized to be 1). The income of this consumer is 100. (a)(10 pts)Derive the demand function of x for this consumer. (b)(5 pts)Calculate the price elasticity of the demand function in (a). Is it true that the absolute value if the elasticity of the demand decreases as the amount of x increases? (c)(10 pts)Suppose price if x decreases from 1/2 to 1/4. Calculate the compensating variation of this price change. --

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