Economic growth and economic development 343

Introduction to Modern Economic Growth
to express, and since we will make no use of the constrained maximization problems
in this book, we will not state these theorems.
The vector-valued theorems are direct generalizations of the ones presented above
and are useful in growth models with multiple capital goods. In particular, let
Z t1
f (t, x (t) , y (t)) dt
(7.21)
max W (x (t) , y (t)) ≡
x(t),y(t)

0

subject to
(7.22)

x˙ (t) = g (t, x (t) , y (t)) ,

and
(7.23)

y (t) ∈ Y (t) for all t, x (0) = x0 and x (t1 ) = x1 .

Here x (t) ∈ RK for some K ≥ 1 is the state variable and again y (t) ∈ Y (t) ⊂ RN

for some N ≥ 1 is the control variable. In addition, we again assume that f and g
are continuously differentiable functions. We then have:

Theorem 7.6. (Maximum Principle for Multivariate Problems) Consider the problem of maximizing (7.21) subject to (7.22) and (7.23), with f and g
continuously differentiable, has an interior continuous solution y
ˆ (t) ∈IntY (t) with

corresponding path of state variable x
ˆ (t). Let H (t, x, y, λ) be given by
(7.24)

H (t, x, y, λ) ≡ f (t, x (t) , y (t)) + λ (t) g (t, x (t) , y (t)) ,

ˆ (t) and the corresponding path of the
where λ (t) ∈ RK . Then the optimal control y

state variable x (t) satisfy the following necessary conditions:
(7.25)

∇y H (t, x
ˆ (t) , y
ˆ (t) , λ (t)) = 0 for all t ∈ [0, t1 ] .

(7.26)

λ˙ (t) = −∇x H (t, x
ˆ (t) , y
ˆ (t) , λ (t)) for all t ∈ [0, t1 ] .

(7.27)
ˆ (t) , y
ˆ (t) , λ (t)) for all t ∈ [0, t1 ] , x (0) = x0 and x (1) = x1 .
x˙ (t) = H (t, x
Ô

Proof. See Exercise 7.10.

Moreover, we have straightforward generalizations of the sufficiency conditions.
The proofs of these theorems are very similar to those of Theorems 7.4 and 7.5 and
are thus omitted.
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