Economic growth and economic development 297

Introduction to Modern Economic Growth
However, this Euler equation is not sufficient for optimality. In addition we need
the transversality condition. The transversality condition is essential in infinitedimensional problems, because it makes sure that there are no beneficial simultaneous changes in an infinite number of choice variables. In contrast, in finitedimensional problems, there is no need for such a condition, since the first-order
conditions are sufficient to rule out possible gains when we change many or all of
the control variables at the same time. The role that the transversality condition
plays in infinite-dimensional optimization problems will become more apparent after
we see Theorem 6.10 and after the discussion in the next subsection.
In the general case, the transversality condition takes the form:
(6.25)

lim β t ∇x(t) U (x∗ (t) , x∗ (t + 1)) · x∗ (t) = 0,

t→∞

where “·” denotes the inner product operator. In the one-dimensional case, we have
the simpler transversality condition:
(6.26)

lim β t

t→∞

∂U(x∗ (t) , x∗ (t + 1)) ∗
· x (t) = 0.
∂x (t)

In words, this condition requires that the product of the marginal return from the
state variable x times the value of this state variable does not increase asymptotically
at a rate faster than 1/β.
The next theorem shows that the transversality condition together with the
transformed Euler equations in (6.21) are sufficient to characterize an optimal solution to Problem A1 and therefore to Problem A2.

Theorem 6.10. (Euler Equations and the Transversality Condition)
Let X ⊂ RK
+ , and suppose that Assumptions 6.1-6.5 hold. Then the sequence

∗
∗
{x∗ (t + 1)}∞
t=0 , with x (t + 1) ∈IntG(xt ), t = 0, 1, . . . , is optimal for Problem A1

given x (0), if it satisfies (6.21) and (6.25).

Proof. Consider an arbitrary x (0) and x∗ ≡ (x (0) , x∗ (1) , ...) ∈ Φ (x (0)) be

a feasible (nonnegative) sequence satisfying (6.21) and (6.25). We first show that

x∗ yields higher value than any other x ≡ (x (0) , x (1) , ...) ∈ Φ (x (0)). For any
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