Consider the following planning economy problem: max ct,kt+1 such that an k+1 = L + (1-5) kt In addition, Above: c is aggregate consumption; k is the aggregate capital stock; Et is the ex- pectation operator conditional on the information available up and through period t inclusive; β E (0, 1) is the subjective parametric discount factor (β is the Greek letter “beta”); y is aggregate output; I is aggregate investment: δ > 0 is the depreciation rate of capital (δ is the Greek letter “delta”: is productivity; α E (0, 1) is a parame- ter (α is the Greek letter “alpha”); ρ E (0, 1) is the productivity process persistence parameter (ρ is the Greek letter “rho”); and ξz is disturbance term associated with the exogenous productivity process (ξ is the Greek letter “xi”). As usual, all non-price variables are normalized by the aggregate population, which consists of a unit mass
(a) State the Lagrangian associated with this problem using a single constraint
(b) State the first order conditions associated with this problem. Please show/explain your work thoroughly.
(c) Derive the steady state values of c, I, k, and z. Please show/explain your work thoroughly