An individual with a utility function u = lnw has the opportunity to invest in a risky asset whose final payoff is distributed as (1 + h, 1 2;1 −

d, 1 2) with h>d.The risk-free return on bonds is zero. The initial price of the risky asset is normalized to unity.

(a) Write the first order condition for α∗ and solve it explicitly. Are you surprised by the result that α∗ is a linear function of initial wealth?

(b) Which property of the utility function generates the result that α∗ is increasing in initial wealth?

(c) Show that when d increases, α∗ falls. Is this surprising? What about the impact on α∗ of an increase in h?

(d) Apply equation (4.4) to find the approximation for the proportion of wealth invested in the risky asset. Compare with the exact value you obtained under (a).