econ
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 α of your wealth in the risky asset, and the fraction 1 − α 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 α will maximize your expected utility?
(e) How does the optimal fraction of your wealth α invested in the risky asset depend on your current wealth?
(f) How does the optimal amount of money αw0 invested in the risky asset depend on your current wealth?
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