(4) Ks+1 = Ks + /,, K, is given . As defined, vik gives the optimal discounted profits -- discounted back to year t -- given that the firm begins in year t with K, as its capital stock. The function v,[K ) is known as the optimal-value function or the Bellman function of the firm . Let (/]),, be the solution of the problem of maximizing (1) subject to (2), and K, be the capital stock in year t under the optimal investment program , ie ., (5) KN = K, '+ 1, K' = Ka for t = 0. (a) Let q, = v,[K,"] be the shadow price of capital (or the Tobin's q) in period t along the optimal trajec tory. Find the difference equations that link (9,], K,+,) to (q, ", K,"). Assume that the productivity parameter is constant over time, say A, = A = constant. Draw a phase diagram with K on the horizontal axis and q on the vertical axis , then explain how the system converges to the steady state . (b) Suppose that the system is in steady state. Suddenly, there is an unanticipated permanent increase in A. Show the new steady state and the transition to the new steady state . Answer to question (a) (a) Let the firm begins at time t with K, as stock of capital. It has to find an investment program (IJ), to maximize discounted profits . Generally , it solves the following maximization problem : (1) v.[KJ =maxor (2;_, (1) (A, FIK, ] -1, - }x1,3)) subject to K,. = K, + 1,, s = t, t+ 1, ..., K, is given . To solve this problem, we use dynamic programming. Applying the principle of optimality to equation 1, we obtain the following Bellman equation : (2) v.[KJ =max, (A, FIK] - 1, - } X12 + 1 VICK, + 1,1)