Economic growth and economic development 367

Introduction to Modern Economic Growth structure given by the convex function φ (i) In some models, the adjustment cost is taken to be a function of investment relative to capital, i.e., φ (i/k) instead of φ (i), but this makes no difference for our main focus We also assume that installed capital depreciates at an exponential rate δ and that the firm maximizes its net present discounted earnings with a discount rate equal to the interest rate r, which is assumed to be constant The firm’s problem can be written as Z ∞ exp (−rt) [f (k (t)) − i (t) − φ (i (t))] dt max k(t),i(t) subject to k˙ (t) = i (t) − δk (t) (7.59) and k (t) ≥ 0, with k (0) > given Clearly, both the objective function and the constraint function are weakly monotone, thus we can apply Theorem 7.14 Notice that φ (i) does not contribute to capital accumulation; it is simply a cost Moreover, since φ is strictly convex, it implies that it is not optimal for the firm to make “large” adjustments Therefore it will act as a force towards a smoother time path of investment To characterize the optimal investment plan of the firm, let us write the currentvalue Hamiltonian: ˆ (k, i, q) ≡ [f (k (t)) − i (t) − φ (i (t))] + q (t) [i (t) − δk (t)] , H where we used q (t) instead of the familiar µ (t) for the costate variable, for reasons that will be apparent soon The necessary conditions for this problem are standard (suppressing the “ˆ” to denote the optimal values in order to reduce notation): ˆ i (k, i, q) = −1 − φ0 (i (t)) + q (t) = H ˆ k (k, i, q) = f (k (t)) − δq (t) = rq (t) − q˙ (t) H lim exp (−rt) q (t) k (t) = t→∞ The first necessary condition implies that (7.60) q (t) = + φ0 (i (t)) for all t 353