2. Growth model with investment-specific productivity shocks. Consider the social planner formulation of the following stochastic growth model: V(ks, ke, q) = max Ologc + (1 -0) log(1 - h) + BE, [V (k,, k;, q')] s.t. chiside Ctis tie = kaskachl-as-de K, = (1 -6,)ks + is ke = (1 - be)ke + que c20, is 20, i. 20 log q = pq log q-1 + E, E ~ N(0. 02) (1) where c is consumption, h is labor supply, i, is investment in structures and i, is in- vestment in equipment. Here, the primes ' denote next-period value and the subscripts -1 denote past values. The non-negativity constraint on each type of investment means that machines cannot be removed once they are installed. System (1) is a recursive formulation. One can also write the problem as an infinitehorizon as follows: max BE [Olog at + (1 - 0) log(1 - ht)] s.t. Ct,ht,ist,fet 1=0 joel-as-de Ct + ist + let = hot hot he kat+1 = (1 - 6,)kat + ist kettl = (1 - 6)ket + qtlet C 2 0, ist 2 0, i t 2 0 for all t log qt = Pq log q1-1 + Et, Et ~ N(0, o?) This economy features a shock to the efficiency of new capital produced (aka investment- specific 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. relative to the price of equipment pe, i.e. q = Pc/Pe. You will not need to solve the business cycle version of the model[a] 1What are the state and control variables? (b) Characterize the equilibrium of the economy. The equilibrium conditions involve a resource constraint, labor leisure trade-off. and two {intertemporalj Euler equations. (Hint: take rst order conditions with respect to chat. k311+1 and Frey\" and rear range. You can follow the logic of the real business model closely but now note that there are two types of investment). {c} Carefully interpret the equilibrium conditions. in particular the intertemporal con dition with respect to investment in capital equipment. (d) Suppose there is a positive realization to 5;. Describe the transmission of the shock into the economy. ISompare the transmission to that of a total factor productivity shock in the standard real business cycle model. {e} 1Write the steadystate conditions of the model. (Take the equilibrium conditions and remove the time indices. You can simplifyr by using the rate of time preference r = [1 if] ,1\" Ii The steady state can be written as a function of c. km ha. and h alone. where output satises 3} = k:\"k:\"h1_*'\".) (f) Suppose equipment can be utilized at rate It. The law of motion for equipment is e;={1 arena + gig where 52m) = tin + ELF\". M :5- 1 Production 110\"! satises 3; = hp [ukgjmfilm'\". Take the rst order condition with respect to u in the household problem. You can do this by modifying the Lagrangian above. Ho'il.r does 11. depend on o and the marginal product of capital equipment nyfkg? (Hint: the first order condition is static; you don"t need to worry about multiple periods.) Provide intuition on the effects of an endogenous utilization rate on the transmission mechanism in the EJCGIIDHIF