In a year from now, there may be two states of the economy - "good" and "bad." Assume that the probability of "good" state is 1/3, and the probability of "bad" state is 2/3. There are two investment opportunities:

(i) risky asset will return you $1.32 in a year from now on each $1 you invest "today" if the economy is in "good" state, and $0.90, if the economy is in "bad" state;

(ii) safe asset will return you $1.02 in a year from now on each $1 you invest "today" in each state of the economy.

You current wealth is w0 > 0. You are an expected utility maximizer, and your utility of wealth is u(w) = ln(w). You want to invest your wealth in the optimal way.

Suppose you invest the fraction of your wealth in the risky asset, and the fraction 1 in the safe asset.

(a) What wealth will you have in a year from now in each state of the economy?

(b) What will be your expected wealth?

(c) What will be your expected utility of wealth?

(d) Which will maximize your expected utility?

(e) How does the optimal fraction of your wealthinvested in the risky asset depend on your current wealth?

(f) How does the optimal amount of moneyw0invested in the risky asset depend on your current wealth?