Suppose there is an SDF m˜ with the property that for every function g there exists a portfolio θ (depending on g) such that n

i=1

θix˜i = g(m˜ ).

Consider an investor with no labor income y˜. Show that her optimal wealth is a function of m˜ . Hint: For any feasible w˜, define w˜ ∗ = E[ ˜w | ˜m], and show that w˜ ∗ is both budget feasible and at least as preferred as w˜, using the result of Section 1.5.

Note: The assumption in this exercise is a weak form of market completeness.

The exercise is inspired by Chamberlain (1988).