Consider a two period firm model similar to the one you studied in Unit 1 modified to include uncertain productivity. In period 1, the firm starts with some capital K, = K that it owns. Within each period, the firm chooses labor L, at fixed wage rate w and produces output using the production function f(K,L) = 0, KL- where the productivity level, evolves randomly across periods. In each period, productivity can be one of three possible states: t = 1,2 0 = {0L.OM.OH} representing low, medium and high states respectively. The firm starts period 1 with low pro- ductivity. The evolution of productivity is independent through time. The probability of occurrence of each productivity state (in every period) is given by P(OL) = 1/5. P(0M)=3/5 and P(0)=1/5 The firm decides on investments I, in each period which becomes productive capital with one period delay. Capital depreciates every period at the rate of 5. The firm discounts future values with the discount factor 3. The firm is liquidated at the end of period 2 and the revenues from liquidation is distributed as dividends. Assume that all prices are constant and normalized to unity. Based on this information, answer the following two questions: What are the state and choice variables in this model? What is the value of the firm? Write down the sequential constrained optimization problem of the firm and derive first order conditions that characterize optimal investment policy. Interpret these equations. [20%] Numerically solve for investments using Newton's method if K = 30,3 = 0.96, a = 0.33,6 = 0.025, w = 0.25; and (1/2.1.3/2) are the low, medium and high state productivity values, respectively. Plot and interpret investment policy function for different values of initial capital K. Answer in not more than 250 words. [20%] You should include your Matlab code in the Appendix. The code should be commented (do not overdo this). Figures should be suitably labeled and titled. You can have them either in the main body or in the appendix.