A decision maker must decide how much of his wealth to invest in a risk-free asset and a risky asset. The consumer has utility-for-money function u(x) = Ln(x). There are two states of the world, G and B. The risk-free asset returns 15% in either state (i.e., $1 invested in the risk-free asset is worth $1.15 in either state of the world). The risky asset returns 32% if the state is G and 10% if the state is B (i. e., $1 invested in y is worth $1.32 in state G and $1.1 in state B). Let p denote the probability of state G occurring.

The decision maker's initial wealth is w = 1000. Let a denote the fraction of his/her wealth the decision maker invests in the risky asset.

a. If the investor invests fraction a of their wealth in the risky asset, what is the dollar investment in the risky asset?

b. If the decision maker chooses portfolio weight a for the risky asset, what is their final wealth in states G and B?

c. Suppose that the probability of the state G is p = 0.25. How much of their wealth does the decision maker invest in the risky asset?

d. Repeat part c when the probability of state G is p = 0.3. How much of their wealth does the decision maker invest in the risky asset in this case?