Consider a one-period, finite-state economy where an individual has utility from the consumption of two goods. The consumption of the first good is denoted by c, and the consumption of the second good is denoted by q. The first good is the numeraire good, and Pq denotes the price of the second good. The individual is assumed to have timeadditive expected utility and therefore faces the problem max u(c0, q0) + e

−δ 

S

ω=1 pωu(cω, qω), s.t. c0 + Pq 0q0 ≤ e0 −

I i=1

θiPi, cω + Pq

ωqω ≤ eω +

I i=1 Diωθi, ω = 1, ... , S, where we assume the separable utility function u

(c, q) = 1 1 − γ c 1−γ + b 1 1 − γ

q1−γ

for some constants b > 0 and γ > 1. The choice variables are the portfolio θ =

(θ1, ... , θI), time 0 consumption (c0, q0), and state-dependent time 1 consumption

(c1, q1), ... ,(cS, qS).

(a) Set up the Lagrangian associated with the problem and show that the first-order conditions with respect to c0 and q0 imply that c0 = b−1/γ (Pq 0)

1/γ q0.

Similarly, show that the first-order conditions with respect to cω and qω imply that cω = b−1/γ (Pq

ω)

1/γ qω, ω = 1, ... , S.

Explain why these relations make economic sense.

(b) Show that the first-order condition with respect to θi implies that

ζ = e

−δ

 c c0

−γ

is a valid state-price deflator.

(c) Discuss the potential of a model with the above preferences for explaining the standard asset pricing puzzles.