Exercise . Suppose the market is complete and = (t) is the unique stateprice deflator. Then

Exercise . Suppose the market is complete and ζ = (ζt) is the unique stateprice deflator. Then the present value (the costs) of any consumption process c = (ct)t∈[,T] is E[

T

 ζtct dt]. For an agent with time-additive preferences, an initial wealth of W and no future income except from her financial transactions, the utility-maximization problem can then be formulated as max c=(ct)t∈[,T]

E

( T



e

−δt u(ct) dt)

s.t. E ( T



ζtct dt)

≤ W.

Use the Lagrangian technique for constrained optimization to show that the optimal consumption process must satisfy e

−δt u

(ct) = αζt, t ∈ [, T], where α is a Lagrange multiplier. Explain why you can conclude that

ζt = e−δt u

(ct)/u

(c)