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課程名稱︰個體經濟學一
課程性質︰經濟系必修
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）︰2007/01/15
考試時限（分鐘）：兩小時(最後有延長)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Consider a consumer who lives for two periods. The consumer gets utility
from consumption in each period. The consumer also gets an endowment of time
_
in each period, L, which the consumer can use to work or consumer as leisure.
The consumer gets NO utility from leisure, however.
(a) (10pts) There is no borrowing or lending. Let w1 and w2 be the wage rates
in periods 1 and 2 respectively. Write down the constraints on consumption
that the consumer faces in period 1 and 2. What levels of consumption will
the consumer choose in each period?
(b) (10 pts) Now assume that in period 1, the consumer can spend some of the
time endowment of obtaining education. Call this amount of the time h.
This time provides neither utility nor disutility, but it reduces the time
available for working. The benefit of spending time on education is that
this raises the wage the consumer can earn in period 2. In particular,
assume
w2 = h*w1
Under these assumptions, write down the budget constraints the consumer
faces in period 1 and 2.
(c) (10pts) Assume the consumer's utility function is
u(c1,c2) = ln(c1) + βln(c2)
where c1 is the consumption of period 1, c2 that of period 2 and β>0 a
constant. Write down the optimization problem this consumer faces.
(d) (10pts) Solve for the optimal h of this consumer.
(e) (10pts) What effect does an increase in β have on the decision to invest
in education? Interpret.
2. Consider a mean-variance utility maximizer who can allocate his portfolio
between three different assets. The three assets have different expected
returns and different variance of returns. The returns of different assets
are all uncorrelated with each other.
^^^^^^^^^^^^
(a) (10pts) If μ1, μ2, and μ3 are the expected returns on the three assets.
and w1, w2 are the shares of the portfolio 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.
(b) (10pts) If (σ1)^2, (σ2)^2, and (σ3)^2 are the variance of the returns
on the three assets, and w1, w2 are the shares of the portfolio allocated
to the first and second assets, respectively, write down the formula for
the variance of the return on the consumer's portfolio.
(c) (10pts) Now assume the expected returns are 8%, 10%, and 2%, repectively.
Re-write your answer to part (a) incorporating this information.
(d) (10pts) Assume also that the variance of the returns are 5%, 5%, and 0%,
repectively. Re-write your answer to part (b) incorporating this
information.
(e) (10pts) Write down the optimization problem that this consumer will try to
solve, using the specific numbers for means and variances of the returns
and assuming the utility function is
u(μ, σ^2) = μ-σ^2
where μ is the expected return of the portfolio and σ^2 is its variance.
(f) (10pts) Solve for the optimal values of w1 and w2.
(g) (10pts) Interpret your solution for the demand for the third asset.
(h) (10pts) Explain why the consumer chooses to hold asset 1 given that it has
the same variance but a lower expected return than asset 2.
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