Suppose you had access to the following lottery: if you spend one dollar, it pays a quantity of money a1 >0 with probability and a (potentially different) quantity of money a2 >0 with probability . A gambler can buy any non-negative quantity z of the lottery at a monetary cost of pr where p> 0. Specifically, if the gambler with initial wealth w > 0 buys 1 I units of the lottery, his or her wealth after the gamble is w + (a - p)x with probability and w+ (a2 p)r with the remaining probability. Assume that aj < p < az and a1 + a2 > 2p. Suppose the investor is an expected utility maximizer with a differentiable, strictly concave, and strictly increasing Bernoulli utility u: (0, o0) R that satisfies lim0 e du(e) du(c) 1. (10 minutes) Show that if u(-) is homogeneous of degree A > 1 then is homogeneous of degree A - 1. 2. (5 minutes) State formally the investor's utility maximization problem. Derive the first order conditions. 3. (20 mimutes) Show that the optimal value of a is proportional to w when u() is logarithmic, and when u(-) is homogeneous of degree X > 1.