Problem
Consider a Lucas economy where the representative agent is equipped with exponential utility function of consumption, U(c) = - e -γC γ Assume that aggregate consumption follows Ct+1 = a + bCt + εt+1 where εt+1 ∼ N(0, σ), a > 0 and -1 < b < 1.
1. Compute the equilibrium risk-free rate at time t. What happens to the risk-free rate if γ increases? Explain.
2. The representative agent can also buy a risky asset which pays a random dividend in one period from now. The dividend D follows log Dt+1 = d0 + log Dt + εt+1 + ot+1 where ot+1 ∼ N(0, χ) and is uncorrelated with ε. Compute the price-dividend ratio of the one-period asset. What happens to the price-divided ratio if γ increases? Explain.
3. Compute the expected return of the one-period asset. What happens to expected returns if a increases? Explain.