Reference no: EM133063947

Finite horizon economy with habit formation

Consider the usual two-period endowment economy studied in class (Topic 1, slide 15...). Assume that the representative agent features habit formation in consumption that is, the utility function is U(Ct , Xt) = β (Ct - Xt)^ 1-γ /1 - γ where Xt is the reference level of consumption and γ the relative risk aversion.

(a) Derive the F.O.C. for the the optimal investment in the risky and risk-free asset

(b) Assume that gt+1 = log ( Ct+1 /Ct) = µc + σcεc t+1, εc t+1 ∼ N(0, 1) , rt+1 = log Rt+1 = µr + σrε r t+1, εs t+1 ∼ N(0, 1) , ft+1 = log St+1 St = µs + σsε f t+1, ε f t+1 ∼ N(0, 1) where St = (Ct-Xt)/Ct is the surplus consumption ratio. Random variables above have the following covariances: Cov(gt+1, rt+1) = σcr, Cov(ft+1, rt+1) = σfr and Cov(gt+1, ft+1) = σcf . Compute the equilibrium risk-free rate and the equity premium. (Hint: need to compute the expected value of log-normal random variables...just more parameters than usual...) How does µs, σs and σcf affect he equilibrium risk-free rate? How does σcr and σfr affect the equity premium? It is important you explain the economic intuition behind your results.

(c) Assume now that ft+1 = log (St+1)/ St = µs + ρft + σsε s t+1, εc t+1 ∼ N(0, 1) where 0 < ρ < 1. How does the realized value of ft affect the risk free rate in this case? how does ft affect the equity premium. It is important you explain the economic intuition behind your results.