(This follows Hansen and Singleton, 1983.) Suppose instantaneous utility is of the constant-relative-risk-aversion form,  Assume that the real interest rate, r, is constant but not necessarily equal to the discount rate, ρ

 

(a) Find the Euler equation relating Ct to expectations concerning C_{t +1}.

 

(b) Suppose that the log of income is distributed normally, and that as a result the log of C_t+1 is distributed normally; let σ² denote its variance conditional on information available at time t. Rewrite the expression in part (a) in terms of ln Ct, Et [ln C_t+1], σ², and the parameters r, ρ, and θ. (Hint: If a variable x is distributed normally with mean μ and variance V, E [ex] = eμeV/2.)

 

(c) Show that if r and σ² are constant over time, the result in part (b) implies that the log of consumption follows a random walk with drift: ln C_t+1 = a + ln Ct + u_t+1, where u is white noise. (d) How do changes in each of r and σ² affect expected consumption growth, Et [ln C_t+1 − ln Ct]? Interpret the effect of σ² on expected consumption growth considering the discussion of precautionary saving in Section 8.6.