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

## Question:

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.

## Step by Step Answer:

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