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)

0

subject to

k˙ (t) = i (t) − δk (t)

(7.59)

and k (t) ≥ 0, with k (0) > 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) = 0
H

ˆ k (k, i, q) = f 0 (k (t)) − δq (t) = rq (t) − q˙ (t)
H
lim exp (−rt) q (t) k (t) = 0.

t→∞

The first necessary condition implies that
(7.60)

q (t) = 1 + φ0 (i (t)) for all t.
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