1. Investment with Adjustment costs and taxes- + Consider a representative firm that uses capital, K,, to produce output. The production function for the firm is linear given by F(K,) = AK,. The firm purchases investment good, I,, at price p.. Investment has a second cost, namely it is expensive to install. Specifically, the cost of installation is given by C(1,) = = 1,". Capital does not depreciate and hence evolves as follows: K. = 1.. We assume that the initial capital stock and initial investment price are given as Ko and Po, respectively. The output price is normalized as 1. The firm maximizes the present value of profits discounted by the interest rate 7.+ (a) Write down the first order conditions and the transversality condition for this problem. ~ (b) Solve for the optimal investment as a function of investment prices and parameters only.+ (c) Derive the Tobin's q at time t, q,. (d) Using a phase diagram for (K., q, ) show that there is a unique solution. What is the steady state value for q,?+ (e) Suppose that the productivity A drops to A/2. If the investment price remains the same, explain, using a phase diagram, what happens to K, and q.. Discuss which variables jump and which adjust slowly, and why. ~