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Consider an economy with a continuum of infinitely lived individ- uals each of who work one of two work shifts or not at all. The shifts correspond to working only a straight time shift (h) or straight time plus over time (h+h2). Let nt be the fraction of individuals that work only straight time and not be the fraction that work straight time plus overtime. Each individual has preferences given by E3 [log & + Alog (1 h)] t=0 where he {0, h1, h1 + h}. Output, which can be used for consumption or investment, is produced using a Cobb- Douglas technology that is a function of capital, labor and a technology shock. That is, 1-0 Yt et K(H + H) where K, is the tock of capital, Ht is total straight time hours worked and H2t is total overtime hours worked. Output can be used as consumption or investment (it). Investment in period t become productive capital in period t+1, and the stock of capital depreciates at the rate 6. Finally, the technology shock,zt, evolves over time according to a first order autoregression, Zt+1= pzt + Et+1, Et ~ ~ N (0,0) A) [12 points] Write down the social planners problem for this economy as a dynamic program. The planner should give equal weight to the utility of all individuals in its objective function. B) [12 points] Derive a set of first order necessary conditions that characterize a solu- tion to this problem. In particular, find equations that determine the following variables: Yt, Ct, nit, n2t, Ht, it, Kt+1. Here, H, is total hours worked. Also, characterize the steady state for a nonstochastic version of this problem. C) [12 points] Suppose that you are given the following statistics computed from U. S. data: (1) average labors share; (2) the average capital to output ratio (annual); (3) the average investment to output ratio; and (4) the average fraction of time that individuals spend working in the market sector. Suppose that a period is one quarter of a year. Your job is to calibrate the economy so that the steady state matches the U.S. averages in these four respects. Show how these facts can be used to find values for A, B, 0, and 6. That is, give a set of equations that can be solved to obtain values for these parameters given values for h and h. D) [12 points] Write the equations from part B in terms of log deviations from steady state. Derive a linear approximation of this set of equations. E) [12 points] From these equations, describe the response of hit and het to an increase in z holding other variables constant. Provide economic intuition for what you find. (6 points)