Consider the quadratic utility function 2 U(W ) = aW bW where a and b are positive parameters.

a. What constraints need to be placed on the parameters a and b to ensure that the marginal utility of wealth is positive?

b. Does this function exhibit increasing or decreasing risk aversion?

c. Suppose that this person is considering investing in a risky asset. One dollar invested in this asset will yield 1 + r dollars at the end of one period, where r is a random variable with E(r) = r > 0 and variance given by 2 r . Any funds not invested in this asset will have an unchanged value at the end of this one period (that is, the risk-free rate of return is zero). How will this person's next period's wealth depend on the fraction ( k ) of current wealth he or she invests in the risky asset?

d. How should this person choose k to maximize the expected utility of his or her next period's wealth? (Note: to answer this part you will have to know the mathematical identities that for any random variable, x, ( ) ( ), ( ) ( ), 2 E kx = kE x Var kx = k Var x and 2 2 2 E(x ) [E(x)] = x + -- see chapter 2 and Problem 2.14). e. Explain explicitly how your optimal value for k here depends on this person's initial level of wealth. Does this seem consistent with real world observations?