Economic growth and economic development 377

Introduction to Modern Economic Growth
models, and also imposes an additional technical assumption that is not easy to verify; instead the results here are stated under the assumption of weak monotonicity,
which is satisfied in almost all economic applications.
The original economic interpretation of the Maximum Principle appeared in
Dorfman (1969). The interpretation here builds on the discussion by Dorfman, but
extends this based on the no-arbitrage interpretation of asset values in the HamiltonJacobi-Bellman equation. This interpretation of Hamilton-Jacobi-Bellman is well
known in many areas of macroeconomics and labor economics, but is not often used
to provide a general economic interpretation for the Maximum Principle. Weitzman
(2003) also provides an economic interpretation for the Maximum Principle related
to the Hamilton-Jacobi-Bellman equation.
The classic reference for exploitation of a non-renewable resource is Hotelling
(1931). Weitzman (2003) provides a detailed treatment and a very insightful discussion. Dasgupta and Heal (1979) and Conrad (1999) are also useful references for
applications of similar ideas to sustainability and environmental economics. Classic
references on investment with costs of adjustment and the q-theory of investment
include Eisner and Strotz (1963), Lucas (1967), Tobin (1969) and Hayashi (1982).
Detailed treatments of the q-theory of investment can be found in any graduate-level
economics textbook, for example, Blanchard and Fisher (1989) or Romer (1996), as
well as in Dixit and Pindyck’s (1994) book on investment under uncertainty and
Caballero’s (1999) survey. Caballero (1999) also includes a critique of the q-theory.
7.10. Exercises
Exercise 7.1. Consider the problem of maximizing (7.1) subject to (7.2) and (7.3)
as in Section 7.1. Suppose that for the pair (ˆ
x (t) , yˆ (t)) there exists a time interval
(t0 , t00 ) with t0 < t00 such that
λ˙ (t) 6= − [fx (t, xˆ (t) , yˆ (t)) + λ (t) gx (t, xˆ (t) , yˆ (t))] for all t ∈ (t0 , t00 ) .
Prove that the pair (ˆ
x (t) , yˆ (t)) could not attain the optimal value of (7.1).
Exercise 7.2. * Prove that, given in optimal solution xˆ (t) , yˆ (t) to (7.1), the maximized Hamiltonian defined in (7.16) and evaluated at xˆ (t), M (t, xˆ (t) , λ (t)), is
differentiable in x and satisfies λ˙ (t) = −Mx (t, xˆ (t) , λ (t)) for all t ∈ [0, t1 ] .
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