Economic growth and economic development 366

Introduction to Modern Economic Growth
with state variable k, control variable c and current-value costate variable µ.
From Theorem 7.14, the following are the necessary conditions:
ˆ c (k, c, µ) = u0 (c (t)) − µ (t) = 0
H

ˆ k (k, c, µ) = µ (t) (f 0 (k (t)) − δ) = ρµ (t) − µ˙ (t)
H
lim [exp (−ρt) µ (t) k (t)] = 0.

t→∞

Moreover, the first necessary condition immediately implies that µ (t) > 0 (since
u0 > 0 everywhere). Consequently, the current-value Hamiltonian given in (7.58)
consists of the sum of two strictly concave functions and is itself strictly concave and
thus satisfies the conditions of Theorem 7.15. Therefore, a solution that satisfies
these necessary conditions in fact gives a global maximum. Characterizing the
solution of these necessary conditions also establishes the existence of a solution in
this case.
Since an analysis of optimal growth in the neoclassical model is more relevant
in the context of the next chapter, we do not provide further details here.
7.7. The q-Theory of Investment
As another application of the methods developed in this chapter, we consider the
canonical model of investment under adjustment costs, also known as the q-theory
of investment. This problem is not only useful as an application of optimal control
techniques, but it is one of the basic models of standard macroeconomic theory.
The economic problem is that of a price-taking firm trying to maximize the
present discounted value of its profits. The only twist relative to the problems we
have studied so far is that this firm is subject to “adjustment” costs when it changes
its capital stock. In particular, let the capital stock of the firm be k (t) and suppose
that the firm has access to a production function f (k (t)) that satisfies Assumptions

1 and 2. For simplicity, let us normalize the price of the output of the firm to 1 in
terms of the final good at all dates. The firm is subject to adjustment costs captured
by the function φ (i), which is strictly increasing, continuously differentiable and
strictly convex, and satisfies φ (0) = φ0 (0) = 0. This implies that in addition to the
cost of purchasing investment goods (which given the normalization of price is equal
to i for an amount of investment i), the firm incurs a cost of adjusting its production
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