Consider a continuous-time economy with complete markets and a representative individual having an ‘external habit formation’ or ‘keeping up with the Joneses’ utility function so that, at any time t, the individual wants to maximize Et  T t e−δ(s−t)u(cs, Xs) ds
, where u

(c, X) = 1 1−γ (c − X)1−γ for c > X ≥ 0.
Define Yt = − ln (
1 − Xt ct )
.

(a) Argue that Yt is positive. Would you call a situation where Yt is high a ‘good state’ or a ‘bad state’? Explain!

(b) Argue that the unique state-price deflator is given by ζt = e −δt c −γ
t eγ Yt c −γ
0 eγ Y0 .
First, write the dynamics of consumption and the variable Yt in the general way:
dct = ct[μct dt + σ
ct dzt], dYt = μYt dt + σ
Yt dzt, where z = (zt) is a multidimensional standard Brownian motion.

(c) Find the dynamics of the state-price deflator and identify the continuously compounded short-term risk-free interest rate r f t and the market price of risk λt.
An asset i pays an uncertain terminal dividend but no intermediate dividends. The price dynamics is of the form dPit = Pit 
μit dt + σ
it dzt
.

(d) Explain why μit − r f t = βictηct + βiYtηYt, where βict = (σ
itσ ct)/σ ct2 and βiYt = (σ
itσ Yt)/σ Yt2. Express ηct and ηYt in terms of previously introduced parameters and variables.
Next, consider the specific model:
dct = ct[μc dt + σc dz1t], dYt = κ[Y¯ − Yt] dt + σY Yt 
ρ dz1t +
'
1 − ρ2 dz2t 
, where (z1, z2) is a two-dimensional standard Brownian motion, μc, σc, κ, Y¯ , and σY are positive constants, and ρ ∈ (−1, 1).

(e) What is the short-term risk-free interest rate and the market price of risk in the specific model?
Assume that the price dynamics of asset i is dPit = Pit 
μit dt + σit 
ψ dz1t +
'
1 − ψ2 dz2t  , where σit > 0 and ψ ∈ (−1, 1).

(f) What is the Sharpe ratio of asset i in the specific model? Can the specific model generate counter-cyclical variation in Sharpe ratios (if necessary, provide parameter conditions ensuring this)?