Solving a real-business-cycle model by finding the social optimum.34 Consider the model of Section 5.5. Assume for simplicity that n = g =  =  = 0. Let V(Kt,At), the value function, be the expected present value from the current period forward of lifetime utility of the representative individual as a function of the capital stock and technology.

(a) Explain intuitively why V (•) must satisfy

This condition is known as the Bellman equation. Given the log-linear structure of the model, let us guess that V(•) takes the form V(Kt,At) = β0 + βK ln Kt + βA ln At, where the values of the β’s are to be determined. Substituting this conjectured form and the facts that Kt +1 = Yt − Ct and Et [ln At +1] = ρA ln At into the Bellman equation yields

(b) Find the first-order condition for Ct. Show that it implies that Ct/Yt does not depend on Kt or At.

(c) Find the first-order condition for lt. Use this condition and the result in part (b) to show that lt does not depend on Kt or At

(d) Substitute the production function and the results in parts (b) and (c) for the optimal Ct and t into the equation above for V(•), and show that the resulting expression has the form V(Kt,At) = β’0 + β’K ln Kt + β’A ln At.

(e) What must βK and βA be so that β’K = βK and β’A = βA?35

(f) What are the implied values of C/Y and l? Are they the same as those found in Section 5.5 for the case of n = g = 0?