Consider a firm which has profits as a function of its capital stock as: Π(K). Suppose there is a cost to investment of the form C(I) = b|I|.
Depreciation occurs at rate δ, and the discount rate is r.

(a) Derive the conditions for the optimal level of investment.

(b) Suppose the government suddenly decides to reward investment by paying firms τ for each unit of investment they do (where τ b). Assuming we are initially in the steady state, show the path of in- vestment, profits and the capital stock over time.

(c) Answer the previous question assuming that the government an- nounces that the reward will be made at some specified time in the future.