Look at an individual with habit formation living in a continuous-time complete market economy. The individual wants to maximize his expected utility E

T 0

e

−δt u(ct, ht) dt

where the habit level ht is given by ht = h0e −αt + β
t 0 e −α(t−u)
cu du.
We can write the budget constraint as E T 0 ζtct dt
≤ W0, where ζ = (ζt) is the state-price deflator and W0 is the initial wealth of the agent (including the present value of any future non-financial income).

(a) Show that dht = (βct − αht) dt. What condition on α and β will ensure that the habit level declines, when current consumption equals the habit level?

(b) Show that the state-price deflator is linked to optimal consumption by the relation ζt = ke−δt !
uc(ct, ht) + β Et T t e −(δ+α)(s−t)
uh(cs, hs) ds" (*)
for some appropriate constant k. Hint: First consider what effect consumption at time t has on future habit levels.

(c) How does (*) look when u

(c, h) = 1 1−γ (c − h)1−γ ?