A simplified real-business-cycle model with additive technology shocks.
(This follows Blanchard and Fischer, 1989, pp. 329–331.) Consider an economy consisting of a constant population of infinitely lived individuals. The
representative individual maximizes the expected value of P\infty t=0 u(Ct)/(1+\rho )t,
\rho > 0. The instantaneous utility function, u(Ct), is u(Ct) = Ct − \theta Ct2, \theta > 0.
Assume that C is always in the range where u′(C) is positive.
Output is linear in capital, plus an additive disturbance: Yt = AKt + et.
There is no depreciation; thus Kt +1 = Kt + Yt − Ct, and the interest rate is A.
Assume A = \rho . Finally, the disturbance follows a first-order autoregressive
process: et = \phi et −1 + \epsi t, where −1 < \phi < 1 and where the \epsi t’s are mean-zero,
i.i.d. shocks.
(a) Find the first-order condition (Euler equation) relating Ct and expectations
of Ct +1.
(b) Guess that consumption takes the form Ct = \alpha + \beta Kt + \gamma et. Given this
guess, what is Kt +1 as a function of Kt and et?
(c) What values must the parameters \alpha , \beta , and \gamma have for the first-order condition in part (a) to be satisfied for all values of Kt and et? (d) What are the effects of a one-time shock to \epsi on the paths of Y, K,
and C ?