A firm's revenues as a function of its capital stock k are given by R(kt)=aktkt2/2, where a is a sufficiently high, positive constant. Depreciation rate is >0 and interest rate is constant at r. The total cost of investing I is given by C(I)=I+I2/2, where I2/2 is the adjustment costs. a. Write down the firm's problem and set-up the Hamiltonian. b. Derive k=0 and q=0 loci and present them in the (k,q) space. Is the long-run level of capital, k, unique? Characterize the optimal path for capital accumulation. c. Suppose that the interest rate will decrease at some future date. Describe graphically how does this change affect the firm's investment behavior assuming that the firm is initially at its long-run equilibrium. d. Suppose that the government is considering a policy that will increase coefficient a. The proposal will be voted on at some future date T, and it has a 50 percent chance of passing. How does this policy affect the firm's investment behavior assuming that the firm is initially at its long-run equilibrium. A firm's revenues as a function of its capital stock k are given by R(kt)=aktkt2/2, where a is a sufficiently high, positive constant. Depreciation rate is >0 and interest rate is constant at r. The total cost of investing I is given by C(I)=I+I2/2, where I2/2 is the adjustment costs. a. Write down the firm's problem and set-up the Hamiltonian. b. Derive k=0 and q=0 loci and present them in the (k,q) space. Is the long-run level of capital, k, unique? Characterize the optimal path for capital accumulation. c. Suppose that the interest rate will decrease at some future date. Describe graphically how does this change affect the firm's investment behavior assuming that the firm is initially at its long-run equilibrium. d. Suppose that the government is considering a policy that will increase coefficient a. The proposal will be voted on at some future date T, and it has a 50 percent chance of passing. How does this policy affect the firm's investment behavior assuming that the firm is initially at its long-run equilibrium