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4) Consider the model of investment with adjustment costs from lecture. Production profits to shareholders are given by: "(K,)k - 1-C(It) where IT'(K,) 0 and C" > 0 That is, the first and second derivatives of C(*) with respect to I, are both positive. The firm maximizes the present discounted value of profits: E=[(1+r)]' [II(K;)k, -1, -C(15] The capital accumulation equation for each period t is: Ki+1 = It + (1-d)k, + sk; where 0 > > 0. a) Set up the Lagrangian to this problem. b) Take the FOC with respect to It. Explain intuitively. c) Take the FOC with respect to ke+1. Explain intuitively. d) Solve for the steady state of the model. Write down an equation that k must solve in steady state and impose any conditions that are necessary so that a steady state exists and is unique. e) Suppose the industry is at its steady state but then the natural rate of growth, s, increases? Graphically show what happens to q, and k, in both the short run and the long run. () Suppose s > d. Explain why a steady state would not exist in this optimization