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ECON5020

Microeconomic Theory

Problem Set 6

1.   In  the  portfolio  optimal  choice  problem,  assume  that  a  consumer  has  preferences represented by an expected utility function, and that moreover u(w) > 0 and u′′ (w) < 0. The consumer invests an amount a ∈ [0, w] in the risky asset with uncertain return  , Ri in state i, while the safe asset has a return of r. Prove that if Ri ≥ r for all i and that Ri > r for at least one i, then a = w.

2.   For the same type of problem, but with r = 0, if E() > 0 and assuming that a < w, prove that d(a/w)/dw  > 0 if the consumer’s coefficient of relative risk aversion is decreasing in w.

3.   A consumer’s preferences are represented by an expected utility function with the sub- utility u(w) = 2w0.5 . There is a safe asset with a rate of return r = 0.1 and a risky asset that gives a return of R1 = 0.4 with probability 几1 = 0.75 and R2 = −0.2 with 几2 = 0.25. The initial wealth is w = 810,000.

a.   What is the expected rate of return of the risky asset?

b.   What is the amount a ∈ [0, ∞) that the consumer invests in the risky asset? What is the consumer’s position in the safe asset?

c.   Would the consumer invest more or less in the risky asset when his wealth increases?

d.   Would the consumer invest more or less in the risky asset as a fraction of his wealth when his wealth increases?