[10 marks] Consider the one-period investment problem. There are n securities available on Date 0 and their prices are given by the 1n matrix p. There are m states on Date 1 . The price of Security j in State i is aij; let A be the mn matrix whose (i,j) entry is aij. Let i be the probability mass of State i;i>0 for every i and i=1mi=1. The investor has wealth w>0 at the beginning of Date 0 for consumption and investment. Her problem is the following cR,xRn,bRmmaxs.t.u0(c)+i=1miu1(bi);b=Ax;c+pxw;c0;bi0,fori=1,,m. Here u0 and u1 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 qRm being the unique state price vector satisfying the condition that qTA=p : maxcR,bRms.t.u0(c)+i=1miu1(bi)c+qTbw;c0bi0,fori=1,,m. By assumption, qi>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 xRn such that b=Ax. c) [4 marks] To show that the (c,x,b) solves the original investor's problem by contradiction, suppose that there exists some (c~,x~,b~) satisfying all constraints of the original problem and u0(c~)+ i=1miu1(b~i)>u0(c)+i=1miu1(bi). Show that this contradicts the optimality of (c,b) in the new problem. [10 marks] Consider the one-period investment problem. There are n securities available on Date 0 and their prices are given by the 1n matrix p. There are m states on Date 1 . The price of Security j in State i is aij; let A be the mn matrix whose (i,j) entry is aij. Let i be the probability mass of State i;i>0 for every i and i=1mi=1. The investor has wealth w>0 at the beginning of Date 0 for consumption and investment. Her problem is the following cR,xRn,bRmmaxs.t.u0(c)+i=1miu1(bi);b=Ax;c+pxw;c0;bi0,fori=1,,m. Here u0 and u1 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 qRm being the unique state price vector satisfying the condition that qTA=p : maxcR,bRms.t.u0(c)+i=1miu1(bi)c+qTbw;c0bi0,fori=1,,m. By assumption, qi>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 xRn such that b=Ax. c) [4 marks] To show that the (c,x,b) solves the original investor's problem by contradiction, suppose that there exists some (c~,x~,b~) satisfying all constraints of the original problem and u0(c~)+ i=1miu1(b~i)>u0(c)+i=1miu1(bi). Show that this contradicts the optimality of (c,b) in the new