Now consider an environment, with potentially heterogeneous firms. Firms only use labour as a production factor and operate a technology given by thetha_t*sqrt(kt) where theta_t is firm level productivity that follows a persistent process with 2 values theta_h = 6 and theta_l = 2 with a transition matrix pi = [0.75 0.25. 0.25 0.75]. Assume no depreciation of capital. The firm maximises life-time profits and discount the future at the rate r. The firm manager has to spend a convex cost on investment gamma(k_t+1 -kt)^2 when they invest k_t+1 -kt >0 and linear cost of disinvestment mu when they want to downsize k_t+1 -kt <0. (1) Set up the dynamic programming problem of a single firm in this environment. Define carefully the state and decision variables. (2) Calculate the optimal levels of capital next period for both shocks (denote them as k^h and k^l) in the frictionless economy ( gamma = mu = 0). (3) Assum gamma > 0 and mu >0. Use the optimality condition for capital and economic intuition to show that we will have k_t >k^l when productivity is low and k_t < k^h when productivity is high for the current capital such tha k^l < k_t < k^h. (4) Explain intuitively the dynamics of capital in this environment assuming that the initial capita k_0 satisfies k^l < k_0 < k^h. Consider two cases. First assume that the firm experiences along sequence of high shocks. Second, assume that the firm experiences a long sequence of low shocks. (5) Extend this environment to general equilibrium