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 aij; let A be the mxn matrix whose (i, j) entry is aij. Let be the probability mass of State i; > 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 m maxCER,TER", bER uo(c) + (bi); s.t. b = Ax; i=1 c+px w; c 0; bi 0, for i = 1, ..., m. Here up and u 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.