Economic growth and economic development 365

Introduction to Modern Economic Growth
Therefore, the initial value of the costate variable µ (0) must be chosen so as to
satisfy this equation.
Notice also that in this problem both the objective function, u (y (t)), and the
constraint function, −y (t), are weakly monotone in the state and the control vari-

ables, so the stronger form of the transversality condition, (7.55), holds. You are
asked to verify that this condition is satisfied in Exercise 7.20.
7.6. A First Look at Optimal Growth in Continuous Time
In this section, we briefly show that the main theorems developed so far apply
to the problem of optimal growth, which was introduced in Chapter 5 and then analyzed in discrete time in the previous chapter. We will not provide a full treatment
of this model here, since this is the topic of the next chapter.
Consider the neoclassical economy without any population growth and without
any technological progress. In this case, the optimal growth problem in continuous
time can be written as:
max∞

[k(t),c(t)]t=0

Z

∞

exp (−ρt) u (c (t)) dt,

0

subject to
k˙ (t) = f (k (t)) − δk (t) − c (t)
and k (0) > 0. Recall that u : R+ → R is strictly increasing, continuously differen-

tiable and strictly concave, while f (·) satisfies our basic assumptions, Assumptions

1 and 2. Clearly, the objective function u (c) is weakly monotone. The constraint
function, f (k) − δk − c, is decreasing in c, but may be nonmonotone in k. HowÊ Ô
ever, without loss of any generality we can restrict attention to k (t) ∈ 0, k¯ , where
¡ ¢
k¯ is defined such that f 0 k¯ = δ. Increasing the capital stock above this level

would reduce output and thus consumption both today and in the future. When
Ê Ô
k (t) ∈ 0, k¯ , the constraint function is also weakly monotone in k and we can apply

Theorem 7.14.

Let us first set up the current-value Hamiltonian, which, in this case, takes the
form
(7.58)

ˆ (k, c, µ) = u (c (t)) + µ (t) [f (k (t)) − δk (t) − c (t)] ,
H
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