Suppose each investor h has a concave utility function, and suppose an allocation (w˜ 1,...,w˜ m) of market wealth w˜ m satisfies the first-order condition u

h(w˜ h) = γhm˜

for each investor h, where m˜ is an SDF and is the same for each investor. Show that the allocation solves the social planner’s problem (4.2) with weights λh =

1/γh. Note: The first-order condition holds withthe SDF beingthe same for each investor in a competitive equilibrium of a complete market, because there is a unique SDF in a complete market. Recall that γh in the first-order condition is the marginal value of beginning-of-period wealth (Section 3.1). Thus, the weights in the social planner’s problem can be taken to be the reciprocals of the marginal values of wealth. Otherthings equal, investors with high wealth have low marginal values of wealth and hence have high weights in the social planner’s problem.