3. (40pm) Investment-specic productivity shocks and endogenous utilization. Consider the social planner formulation of the following stochastic growth model: 133 2 WE: [Slogq + (1 _ 9)10g(1 htll S-t- (3) Ch (Janie: :0 y: = C: + is: + is: (4) yt = intket)": im"e (5) ks,t+l = (1 65)kst + is: (6) ke,t+l = (1 62(Ht))ke,t + grist (7) b (Mu) = 60 + Soa: > 1 (8) 103 9t = Pq 10g'1t1 + 5t55t N Na]: 0:) (9) c, 2 0,14,, 2 0,1,, 2 0 for all t (10) where Q is consumption, ht is labor supply, st is investment in structures and tag is investment in equipment. The variable a is the utilization rate of capital in equipment. Depreciation depends on utilization according to (8). The non-negativity constraint on each type of investment means that machines cannot be removed once they are installed. This economy features a shock to the efciency of new capital produced (aka investment- specic technological change). A rise in q is an improvement in the quality of new capital goods because the same raw units of investment goods have a larger impact on the capital stock used in production. In a competitive equilibrium, q is the ratio of the price of consumption and structures p,2 relative to the price of equipment pg, i.e. q = Pc/Pe- (a) (4pm) What are the (endogenous) state and control variables? What is the ex- ogenous state variable? (Hint: there are 5 control variables, 2 endogenous state variables, and 1 exogenous state variable)