课程名称︰个体经济学一
课程性质︰必修
课程教师︰黄贞颖
开课学院：社会科学院
开课系所︰经济系大学部
考试日期（年月日）︰2016/1/11
考试时限（分钟）：180分钟
试题 :
Microeconomoics Final
January 11, 2016
1.
Tom is risk averse. He has an initial wealth of 100 but his car may be stolen
and so he runs a risk of a loss of 40 dollars. The probability of loss is 0.2
It is possible, however, for Tom to buy insurance. One unit of insurance costs
r dollars and pay 1 dollar if the loss occurs. Thus, if α units of insurance
are bought, the wealth of Tom will be 100-rα if there is no loss (a good
state). Tome is an expected utility maximizer with Bernoulli utility function
u(x)=ln(x) where x is his wealth in a state.
(a)(10pts)
Write down Tom's expected utility if he buys α units of insurance. Differet-
iate it with respect to α to derive the first order condition which will be
useful later.
(b)(10pts)
Suppose in 2015 insurance is acturially fair in the sense that the insurance
company breaks even on average. What should r be? How much insurance (α) will
Tom buy? What is his wealth in the good state? Does Tom face any risk after
being insured?
(c)(10pts)
In 2016 insurance company decides to increase the premium so now r becomes
2.5. How much insurance (α) will Tome buy now? What is his wealth in the good
state?
(d)(10pts)
Facing the premium increase from 2015 to 2016, what is the Slutsky substitu-
tion effect of Tom's wealth in the good state?
2.
Consider a mean-variance utility maximizer who call allocate his portfolio
between three different assets. The three assets have different expected re-
turns 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)(5pts)
Now assume the expected returns are 5%, 10% and 2%, respectively. Re-write
your answer to part (a) incorporating this information.
(d)(5pts)
Assume also that the variances of the returns are 4%, 4% and 0%, respectively.
Re-write your answer to part (b) incorporating this information.
(e)(10pts)
Write down the optimization problem that this consumer will try to solve, us-
ing the specific numbers fot 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)(5pts)
Solve for the optimal values of w1 and w2.
(g)(5pts)
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.
Have a great winter break!