The following is a general result from matrix theory: Let $\mathbf{A}$ be an $m \times n$ matrix. Suppose that the equation $\mathbf{A x}=\mathbf{p}$ can achieve no $\mathbf{p} \geq \mathbf{0}$ except $\mathbf{p}=\mathbf{0}$. Then there is a vector $\mathbf{y}>\mathbf{0}$ with $\mathbf{A}^{T} \mathbf{y}=\mathbf{0}$. Use this result to show that if there is no arbitrage, there are positive state prices; that is, prove the positive state price theorem in Section 11.9. [If there are $S$ states and $N$ securities, let $\mathbf{A}$ be an appropriate $(S+1) \times N$ matrix.]

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Positive state prices theorem A set of positive state prices exists if and only if there are no arbitrage opportunities. Proof: Suppose first that there are positive state prices. Then it is clear that no arbitrage is possible. To see this, suppose a security d can be constructed with d0. We have d= (d',d2,...,d) with d" >0 for each s=1,2,...,S. The price of d is P=V,d", which since , > 0 for all s, gives P>0. Indeed P > 0 if d0 and P=0 if d = 0. Hence there is no arbitrage possibility. To prove the converse, we assume that there are no arbitrage opportunities, and we make use of the result on the portfolio choice problem of Section 11.7. This proof requires some additional assumptions. (A more general proof is outlined in Exercise 14.) We assume there is a portfolio 60 such that 10d, >0. We assign positive probabilities p.,s = 1,2,...,S, to the states arbitrarily, with Ps=1, and we select a strictly increasing utility function U. Since there is no arbitrage, there is, by the portfolio choice theorem of Section 11.7, a solution to the optimal portfolio choice problem. We assume that the optimal payoff has x* > 0. The necessary conditions (11.4) show that for any security d with price P, E[U'(x)d] =P, (11.14) where x* is the (random) payoff of the optimal portfolio and >> O is the Lagrange multiplier.