Economic growth and economic development 292

Introduction to Modern Economic Growth
defines the set of maximizers Π (x) for Problem A2. Let x∗ = (x (0) , x∗ (1) , ...) with
x∗ (t + 1) ∈ Π (x∗ (t)) for all t ≥ 0. Then from Theorems 6.1 and 6.2, x is also and

Ô

optimal plan for Problem A1.

These two proofs illustrate how different approaches can be used to reach the
same conclusion, once the equivalences in Theorems 6.1 and 6.2 have been established.
An additional result that follows from the second version of the theorem (which
can also be derived from version 1, but would require more work), concerns the
properties of the correspondence of maximizing values
Π : X ⇒ X.
An immediate application of the Theorem of the Maximum (see Mathematical Appendix) implies that Π is a upper hemi-continuous and compact-valued correspondence. This observation will be used in the proof of Corollary 6.1. Before turning
to this corollary, we provide a proof of Theorem 6.4, which shows how Theorem 6.8
can be useful in establishing a range of results in dynamic optimization problems.
Proof of Theorem 6.4. Recall that C (X) is the set of continuous (and
bounded) functions over the compact set X. Let C0 (X) ⊂ C(X) be the set of

bounded, continuous, (weakly) concave functions on X, and let C00 (X) ⊂ C0 (X)

be the set of strictly concave functions. Clearly, C0 (X) is a closed subset of the

complete metric space C(X), but C00 (X) is not a closed subset. Let T be as defined
in (6.15). Since it is a contraction, it has a unique fixed point in C (X). By Theorem
6.8, proving that T [C0 (X)] ⊂ C00 (X) ⊂ C0 (X) would be sufficient to establish that

this unique fixed point is in C00 (X) and hence the value function is strictly concave.
Let V ∈ C0 (X) and for x0 6= x00 and α ∈ (0, 1), construct,
xα = αx0 + (1 − α)x00 .

Let y 0 ∈ G(x0 ) and y 00 ∈ G(x00 ) be solutions to Problem A2 with state vectors x0 and

x00 . This implies that

T V (x0 ) = U (x0 , y 0 ) + βV (y 0 ) and
(6.16)

T V (x00 ) = U (x00 , y 00 ) + βV (y 00 )
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