(This problem is based on Lochstoer (2009).) Suppose a representative individual has preferences of the form E

⎡

⎣



T t=0 e

−δt u(Bt, Lt; Xt)

⎤

⎦ , u(B, L; X) = 1 1 − γ

(

Lα[B − X]

1−α

)1−γ

, where α ∈ (0, 1) and γ > 1. Here Bt denotes the time t consumption of basic goods and Lt the consumption of luxury goods. X is an external benchmark for basic consumption and works as a time-varying subsistence level of basic consumption similar to the Campbell–

Cochrane model of Section 9.2.3. Let the basic good be the numeraire and let PL t denote the time t price of the luxury good (in units of the basic good).

(a) Argue why the optimal consumption of the two goods must be so that Bt − Xt = 1 − α

α PL t Lt.

The state-price deflator process is in this case given by

ζt = e

−δt uB(Bt, Lt; Xt)

uB(B0, L0; X0)

.

Let Mt+1 = ζt+1/ζt denote the next-period state-price deflator.

(b) Show that Mt+1 = e

−δ

Lt+1 Lt

−γ 

PL t+1 PL t

−γ +α(γ −1)

.

Let t = ln Lt, pL t = ln PL t , t+1 = t+1 − t, and pL t+1 = pL t+1 − pL t . Suppose that

t+1 = a + bσ 2

,t + σ,tε,t+1, ln σ 2

,t = ω + β1,ε,t + β2, ln σ 2

,t−1, pL t+1 = ap + bpσ 2 p,t + σ 2 p,tεp,t+1, ln σ 2 p,t = ωp + β1,pεp,t + β2,p ln σ 2 p,t−1, where ε,t+1, εp,t+1 ∼ N(0, 1) with Corr[ε,t+1, εp,t+1] = ρ.

(c) Compute the continuously compounded one-period risk-free interest rate, r f

t .

(d) Compute the maximal conditional Sharpe ratio based on log-returns, that is the maximal possible value of Et[ri,t+1] − r f

t + 1 2 σ 2 i,t

σi,t over all risky assets i, compare Eq. (4.25). Here σ 2 i,t = Vart[ri,t+1].

(e) Discuss the potential of this model in explaining asset pricing puzzles.