The first part of the result follows by the arbitrage arguments used previously. Moving on to the second statement, substituting the defining equation (1.2) for Z into (1.3) yields   The economy E is...

The first part of the result follows by the arbitrage arguments used previously. Moving on to the second statement, substituting the defining equation (1.2) for Z into (1.3) yields

The economy E is arbitrage-free if and only if there exists a pricing kernel, i.e. a strictly positive Z such that

Suppose such a Z exists. Then for any portfolio φ,

If φ is an arbitrage portfolio then the left-hand side of (1.4) is non-positive, and the right-hand side is non-negative. Hence (1.4) is identically zero. This contradicts φ being an arbitrage portfolio and so E is arbitrage-free. Conversely, suppose no such Z exists. We will in this case construct an arbitrage portfolio. Let C be the convex cone constructed from A1(·),