Consider a discrete-time representative individual economy equipped with preferences E[ t=0 et u(ct, qt)], where ct is

Consider a discrete-time representative individual economy equipped with preferences E[∞

t=0 e−δt u(ct, qt)], where ct is the consumption of good 1 and qt is the consumption of good 2. Assume a Cobb–Douglas type utility function, u

(c, q) = 1 1 − γ

(

c

αq1−α

)1−γ

where γ > 0 and α ∈ [0, 1].

(a) Determine the next-period state-price deflator ζt+1/ζt in terms ofct+1/ct, qt+1/qt, and the preference parameters.

Assume that lnct+1 ct



= μc + σcεc t+1, lnqt+1 qt



= μqt + σqt 

ρεc t+1 +

'

1 − ρ2ε

q t+1



, where σc > 0, ρ ∈ (−1, 1), σqt is a positive stochastic process, and εc t+1 and ε

q t+1 are independent N(0, 1)-distributed random variables.

(b) Determine the equilibrium one-period risk-free interest rate (with continuous compounding).

(c) What can you say about expected excess returns and Sharpe ratios on risky assets without additional assumptions? What if you assume that the log-return satisfies ln Ri,t+1 = μit + σit (

ξ1εc t+1 + ξ2ε

q t+1 + ξ3εi t+1

)

, where εi t+1 is also N(0, 1)-distributed and independent of εc t+1 and ε

q t+1 and ξ3 = '

1 − ξ 2 1 − ξ 2 2 ?

(d) Discuss the potential of such a model for explaining the equity premium puzzle, the risk-free rate puzzle, and the predictability puzzle.