A household has the vNM utility function over final wealth, u(x) = ln(x) and has $100,000 to invest. The household is deciding how to allocate this wealth over two types of assets:
- Asset 1: Risk-free. For every dollar invested, this asset will return 10%. For instance, if $100 is invested, the asset will be worth $110.
- Asset 2: Risky. With a 32% probability, the asset will return 0%. With a 68% probability, this asset will return 15%. For instance, if $100 is invested, the asset will worth $100 with 32% probability and $115 with 68% probability.
To model this decision, we will assume the household is selecting q , the percentage (in decimals) of its wealth to invest in Asset 1. Thus, if q = 0.20 , the household is investing $20,000 in the safe asset and $80,000 in the risky asset. Note if q is invested in the safe asset, then 1-q is invested in the risky asset.
a. With a given q , what is the household's expected utility? Hint: Be very careful setting this up. Specifically, there are only two possible outcomes. In both outcomes, for instance, the firm's wealth will be the value of their safe asset plus the value of their risky asset.
b. What percentage of its wealth will the household invest in the safe asset? In the risky asset?