Economic growth and economic development 298

Introduction to Modern Economic Growth
x ∈Φ (x (0)), define
∆x ≡ lim

T →∞

T
X
t=0

β t [U(x∗ (t) , x∗ (t + 1)) − U (x (t) , x (t + 1))]

as the difference of the objective function between the feasible sequences x∗ and x.
From Assumptions 6.2 and 6.5, U is continuous, concave, and differentiable. By
definition of a concave function, we have
∆x ≥

lim

T →∞

T
X
t=0

β t [∇Ux (x∗ (t) , x∗ (t + 1)) · (x∗ (t) − x (t))

+∇Uy (x∗ (t) , x∗ (t + 1)) · (x∗ (t + 1) − x (t + 1))]
for any x ∈Φ (x (0)). Using the fact that x∗ (0) = x (0) and rearranging terms, we

obtain

∆x ≥
( T
X
β t [∇Uy (x∗ (t) , x∗ (t + 1)) + β∇Ux (x∗ (t + 1) , x∗ (t + 2))] · (x∗ (t + 1) − x (t + 1))
lim
T →∞

t=0

ª
+ β ∇Uy (x∗ (T ) , x∗ (T + 1)) · (x∗ (T + 1) − x (T + 1)) .
T

Since x∗ satisfies (6.21), the terms in first line are all equal to zero.

Therefore,

substituting from (6.21), we obtain
∆x ≥ − lim β T ∇Ux (x∗ (T ) , x∗ (T + 1)) · (x∗ (T ) − x (T ))
T →∞

≥ − lim β T ∇Ux (x∗ (T ) , x∗ (T + 1)) · x∗ (T )
≥ 0

T →∞

where the second inequality uses the fact that from Assumption 6.4, U is increasing
in x, i.e., ∇x U ≥ 0 and x ≥ 0, and the last inequality follows from (6.25). This

implies that ∆x ≥ 0 for any x ∈Φ (x (0)). Consequently, x∗ yields higher value than
any feasible x (x (0)), and is therefore optimal.

Ô

We now illustrate how the tools that will so far can be used in the context of
the problem of optimal growth, which will be further discussed in Section 6.6.
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