8.8. Suppose that the costs of adjustment exhibit constant returns in and K. Specifically, suppose they are given by C(k/k)k, where C(0) = 0, C'(0) = 0, C" () > 0. In addition, suppose capital depreciates at rate &; thus k(t) = I(t)- Sk(t). Consider the representative firm's maximization problem.

(a) What is the current-value Hamiltonian?

(b) Find the three conditions that characterize optimal behavior analogous to equations (8.18), (8.19), and (8.20), in Section 8.2.

(c) Show that the condition analogous to (8.18) implies that the growth rate of each firm's capital stock, and thus the growth rate of the aggregate capital stock, is determined by q. In (K, q) space, what is the K = 0 locus?

(d) Substitute your result in part

(c) into the condition analogous to (8.19) to express in terms of K and q.

(e) In (K,q) space, what is the slope of the = 0 locus at the point where 9=1?