3. Consider the economic model described in the previous problem, and the equilibrium conditions derived in part a). Assume that = 1 and g = 0. In the general problem, we cannot obtain an analytic solution for the en- dogenous variables as functions of predetermined variables such as capital, and the exogenous variable technology. Hence we focused on determin- ing the steady-state. In this problem, you are asked to show that with full depreciation (4 = 1) we can obtain an analytic solution even out of steady-state. This problem asks yvou to derive this analytic solution and then consider what the solution implies for how the growth rate of capital and output will vary over time. The solution method here is to \"guess and verify\". In particular, you are asked to guess that the solution implies a constant savings rate at each point in time, along with a constant equilibrinm amount of labor. You then need to show that these guesses satisfy equilibrium conditions and provide algebraic solutions for the guesses. (a) Show that C: = (1-3s)Y; Ht+l = sY, N = N where () () is a solution to the equilibrium condi- tions provided in part a) problem 1 (note with = 1 these conditions hold even if the economy is not in the long-run balanced growth path). Provide alpebraic expressions for s and N*, ie. show that the savings rate and equilibrium amount of labor can be written as functions of the underlying parameters of the model o, 3, 0. (b L Now suppose that the (log of) technology follows a random walk: Inf, =Ind,_ + = where , is a mean-zero iid random variable. Show that, the equi- librium implies that the growth rate of the capital stock follows an autoregressive process of the form: &].I'.I H+'| = qh ].Ith + [1 = E':"}E: (e} Show that output growth also follows an autoresressive process of the form: AlnY; = AInY;_1 + (1 @)= for some coefficient : )