Consider a closed economy with no population growth, no exogenous technological progress, and no physical capital. At any point in time, the representative household has one unit of time that can be divided between working (N) in the production sector and acquiring human capital. The law of motion for the workeris stock of human capital (H) is

Ht+1 = (1)Ht +A(1Nt)

where is the depreciation rate of human capital (satisfying 0 < < 1), and A is a

positive constant. The economy has the production function

Yt=(HtNt) ; 0< <1

for producing the consumption good. The good is perishable and consumption must be equal to output each period (Ct = Yt). A social planner maximizes the householdis utility

X+1 t log(Ct) t=0

given the above constraints and a positive initial stock of human capital. (a) Derive the Orst-order conditions for the planneris optimization problem.

(b) Does the planneris problem admit a steady-state equilibrium? If so, solve for the steady state. If not, explain why this problem does not have a steady-state solution.