4. [10 marks] Consider the one-period investment problem. There are n securities available on Date 0 and their prices are given by the 1 x n matrix p. There are m states on Date 1. The price of Security j in State i is any; let A be the m x n matrix whose (i, j) entry is ay. Let a; be the probability mass of State i; n, > 0 for every i and _," = 1. The investor has wealth w > 0 at the beginning of Date 0 for consumption and investment. Her problem is the following IIIAX ER,IER" BERm "O(c)+ ) (b,); s.t. b = Ar; ct pr s w; c 20; b; 2 0, for i = 1, .... m. Here up and w are strictly increasing and continuous functions. Assume that the market is complete and the Law of One Price holds. In addition, assume that the state price of every state is positive. The goal is to show that the problem always has a solution. (a) [3 marks] Before we attack the original problem, consider the following simpler problem with q e R "being the unique state price vector satisfying the condition that q' A = p: i=1 s.t. ctq'bew; c 20; b; 2 0, for i = 1, .... m. By assumption, q; > 0 for i = 1, .... m. Show that this new problem always has a solution. (b) [3 marks] Assume that (c*, b*) is a solution to the new problem. Show that we can always find an I' E R" such that b' = Ar*. (c) [4 marks] To show that the (c', c*, b* ) solves the original investor's problem by contradiction, sup- pose that there exists some (c, 2, b) satisfying all constraints of the original problem and vo(c) + Ein mul(b;) > uo(c")+ , Tu (b;). Show that this contradicts the optimality of (c*, b* ) in the new