The economy E is arbitrage-free if and only if there exists a strictly positive random variable Z such that   Extending Definition 1.3, we call Z a pricing kernel for the economy E. Suppose, further,...

The economy E is arbitrage-free if and only if there exists a strictly positive random variable Z such that

Extending Definition 1.3, we call Z a pricing kernel for the economy E. Suppose, further, that
 for some i. Then the economy E is arbitrage-free if and only if there exists a strictly positive random variable κ with E[κ]=1 such that

Proof: The first result follows immediately from Theorem 1.7 by setting
 To prove the second part of the theorem we show that (1.9) and (1.10) are equivalent. This follows since, given either of Z or κ, we can define the other via