Consider a discrete-time economy with L individuals with identical preferences so that agent l = 1, ... , L at time 0 wants to maximize E

⎡

⎣



T t=0

βt 1 1 − γ c 1−γ

l,t

⎤

⎦

where cl,t denotes the consumption rate of individual l at time t. Let c∗

l,t be the optimal consumption rate of individual l at time t.

(a) Argue why

ζt+1

ζt

= β

L



L l=1

c∗

l,t+1 c∗

l,t

−γ

(*)

is a state-price deflator between time t and time t + 1.

(b) If the market is complete, explain why the next-period state-price deflator in (*) can be written as

ζt+1

ζt

= β

L l=1 c∗

l,t+1

L l=1 c∗

l,t

−γ

.