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︰经济学系
考试日期（年月日）︰2018/01/08
考试时限（分钟）：180
试题 :
1. Tom has an initial wealth of 100. His car may be stolen so he runs a risk
of a loss of 20 dollars. The probability of loss is 0.2. It is also
possible that someone breaks in Toms home. In this case, he runs a risk of
a loss of 80 dollars. The probability of loss is 0.1. These two
probabilities (events) are independent. It is possible, however, for Tom to
buy insurance. One unit of car insurance cost γ dollars and pays 1 dollar
if the loss occurs. On the other hand, one unit of home insurance cost γ'
dollars and pays 1 dollar if the loss occurs. Thus if α units of car
insurance are bought and α' units of home insurance are bought, the wealth
of Tom will be 100－γα－γ'α' if there is no loss. In the case that only
his car is stolen, his wealth is 100－γα－γ'α'－20＋α. When only his
home is broken in, his wealth is 100－γα－γ'α'－80＋α'. In the worst
scenario that both his car is stolen and his home is broken in, his wealth
is 100－γα－γ'α'－20－80＋α＋α'. Tom is an expected utility
maximizer with Bernoulli utility function u(x)＝In(x) where x is his
wealth in a state.
(a) (15 pts) Write down Tom's expected utility if he buys α units of car
insurance and α' units of home insurance. Differentiate it with
respect to α and α' to derive the first order conditions which will
be useful later.
(b) (10 pts) Suppose insurance is actuarially fair in the sense that both
car and home insurance companies break even on average. What should γ
be? What should γ' be?
(c) (15 pts) Continue from (b). How much car insurance (α) will Tom buy?
How much home insurance (α') will Tom buy? Does Tom face any risk
after being insured?
2. Consider a mean-variance utility maximizer who can allocate his portfolio
between three different assets. The three assets have differnt expected
returns and different variance of returns. The returns of different assets
are all uncorrelated with each other.
(a) (10 pts) If μ1, μ2, and μ3 are the expected returns on the three
assts, and w1, w2 are the shars 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) (10 pts) If σ1^2, σ2^2, and σ3^2 are the variance of the returns on
the three assts, and w1, w2 are the shars of the portfolio allocated to
the first and second assets, respectively, write down the formula for the
variace of the return on the consumer's portfolio.
(c) (5 pts) Now assume the expected returns are 5%, 10%, and 2%,
respectively. Re-write your answer to part (a) incorporating this
information.
(d) (5 pts) Assume also that the variance of the returns are 4%, 4%, and
0%, respectively. Re-write your answer to part (b) incorporating this
information.
(e) (10 pts) 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) (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) (10 pts) Explain why the consumer chooses to hold asset 1 given that it
has the same variance but a lower expected return than asset 2.