Economic growth and economic development 278

Introduction to Modern Economic Growth
Moreover, if any x∗ ∈ Φ (x (0)) satisfies (6.3), then it attains the optimal value

in Problem A1.

This theorem is the major conceptual result in the theory of dynamic programming. It states that the returns from an optimal plan (sequence) x∗ ∈ Φ (x (0))

can be broken into two parts; the current return, U(x∗ (t) , x∗ (t + 1)), and the con-

tinuation return βV ∗ (x∗ (t + 1)), where the continuation return is identically given
by the discounted value of a problem starting from the state vector from tomorrow
onwards, x∗ (t + 1). In view of the fact that V ∗ in Problem A1 and V in Problem
A2 are identical from Theorem 6.1, (6.3) also implies
V (x∗ (t)) = U(x∗ (t) , x∗ (t + 1)) + βV (x∗ (t + 1)).
Notice also that the second part of Theorem 6.2 is equally important. It states
that if any feasible plan, starting with x (0), x∗ ∈ Φ (x (0)), satisfies (6.3), then x∗
attains V ∗ (x (0)).

Therefore, this theorem states that we can go from the solution of the recursive
problem to the solution of the original problem and vice versa. Consequently, under Assumptions 6.1 and 6.2, there is no risk of excluding solutions in writing the
problem recursively.
The next results summarize certain important features of the value function V
in Problem A2. These results will be useful in characterizing qualitative features
of optimal plans in dynamic optimization problems without explicitly finding the
solutions.
Theorem 6.3. (Existence of Solutions) Suppose that Assumptions 6.1 and
6.2 hold. Then there exists a unique continuous and bounded function V : X → R

that satisfies (6.1). Moreover, an optimal plan x∗ ∈ Φ (x (0)) exists for any x (0) ∈
X.

This theorem establishes two major results. The first is the uniqueness of the
value function (and hence of the Bellman equation) in dynamic programming problems. Combined with Theorem 6.1, this result naturally implies the existence and
uniqueness of V ∗ in Problem A1. The second result is that an optimal solution also
exists. However, as we will see below, this optimal solution may not be unique (even
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