(This problem is based on Longstaff and Piazzesi (2004).) Consider a continuous-time model of an economy with a representative agent and a single non-durable good. The objective of the agent at any time t is to maximize the expected time-additive CRRA utility, Et

∞

t e

−δ(s−t) C1−γ s 1 − γ

ds

, where γ > 0 and Cs denotes the consumption rate at time s. The agent can invest in a bank account, that is borrow and lend at a short-term interest rate of rt. The bank account is in zero net supply. A single stock with a net supply of one share is available for trade. The agent is initially endowed with this share. The stock pays a continuous dividend at the rate Dt. The agent receives an exogenously given labour income at the rate It.

(a) Explain why the equilibrium consumption rate must equal the sum of the dividend rate and the labour income rate, that is Ct = It + Dt.

Let Ft denote the dividend-consumption ratio, that is Ft = Dt/Ct. Assume that Ft =

exp{−Xt}, where X = (Xt) is the diffusion process dXt = (μ − κXt) dt − η

Xt dz1t.

Here μ, κ, and η are positive constants and z1 = (z1t) is a standard Brownian motion.

(b) Explain why Ft is always between zero and one.

Assume that the aggregate consumption process is given by the dynamics dCt = Ct



α dt + σ

Xtρ dz1t + σ

Xt

'

1 − ρ2 dz2t



, where α, σ, and ρ are constants and z2 = (z2t) is another standard Brownian motion, independent of z1.

(c) What is the equilibrium short-term interest rate in this economy?

(d) Use Itô’s Lemma to derive the dynamics of Ft and of Dt.

(e) Show that the volatility of the dividend rate (the standard deviation of the relative changes in the dividend rate) is greater than the volatility of the consumption rate

(the standard deviation of the relative changes in the consumption rate) if and only if σρ > −η/2.

Let Pt denote the price of the stock, that is the present value of all the future dividends.

(f) Show that the stock price can be written as Pt = Cγ
t Et 
∞
t e −δ(s−t)
C1−γ s Fs ds
.
It can be shown that Pt can be written as a function of t, Ct, and Ft:
Pt = Ct ∞
t e −δ(s−t)
A(t,s)F−B(t,s)
t ds.
Here A(t,s) and B(t,s) are some deterministic functions of time that we will leave unspecified.
(g) Use Itô’s Lemma to show that dPt = Pt 
... dt + (ρσ + ηHt)
Xt dz1t + σ
'
1 − ρ2 Xt dz2t 
, where the drift term is left out (you do not have to compute the drift term!) and where Ht = − ∞
t e−δ(s−t)A(t,s)B(t,s)F−B(t,s)
t ds ∞
t e−δ(s−t)A(t,s)F−B(t,s)
t ds .
(h) Show that the expected excess rate of return on the stock at time t can be written as ψt = γXt (
σ 2 + σρηHt )
and as ψt = γ σ 2 Ct + γρHtσCtσFt, where σCt and σFt denote the percentage volatility of Ct and Ft, respectively.
(i) What would the expected excess rate of return on the stock be if the dividendconsumption ratio was deterministic? Explain why the model with stochastic dividend-consumption ratio has the potential to resolve (at least partially) the equity premium puzzle.