Consider an infinitely-lived firm that chooses investment and capital to maximize lifetime profits. The firm has the following production function: f(kx) = k Output and investment have unit prices, but non-zero investment also incurs a cost C(1) = (1-6k,). Capital depreciates at rate and so has the following law of motion: k+1=It + (1-6) k Assume future profits are discounted using the fixed interest rate r. (a) Setup the firm's Lagrangian function. Derive and interpret the two first-order conditions. (b) Find expressions for the steady state levels of q and k. Plot these expressions in {k, q} space. (c) Show the effect on your plot from part (b) of an increase in the depreciation rate, 6. (d) Now suppose that 6 increases from 0.1 to 0.2. If r = 0.02 and a = 0.3, solve for the steady state levels of k and q for both values of 6. (e) Will the economy transition to the new steady state in one period? How do you know? (5)