UCD Business Profit Maximization Current Value Hamiltonian Question

I’m working on a business question and need guidance to help me learn.

Consider an industry with An identical firms. A representative firm’s real profits at time t, neglecting any costs of acquiring and installing capital, are proportional to its capital stock, κ (t), and decreasing in the industry-wide capital stock, K(t ); thus, they take the form π(K(t ))κ(t ), where π’(•) < 0. The key assumption of the model is that firms face costs of adjusting their capital stocks. The adjustment costs are a convex function of the rate of change of the firm’s capital stock. The purchase price of capital goods is constant and equal to 1; thus, there are no external adjustment costs. Finally, for simplicity, the depreciation rate is assumed to be zero. These assumptions imply that the firm’s profits at a point in time are π(K)κ− I − C(I ). The firm maximizes the present value of these profits,

Π = ∫ e-rt[π(K(t ))κ(t )- I (t )- C(I (t ))]dt

t=0

where we assume for simplicity that the real interest rate is constant.

(a) Set up the current-value Hamiltonian for the firm’s problem:

(b) Derive the first-order conditions of the profit maximization by differentiating the Hamiltonian with respect the relevant variables and transversality condition.