Economic growth and economic development 277

Introduction to Modern Economic Growth
This assumption imposes conditions similar to those used in many economic
applications: the constraint set is assumed to be convex and the objective function
is concave or strictly concave.
Our next assumption puts some more structure on the objective function, in particular it ensures that the objective function is increasing in the state variables (its
first K arguments), and that greater levels of the state variables are also attractive
from the viewpoint of relaxing the constraints; i.e., a greater x means more choice.
Assumption 6.4. For each y ∈ X, U(·, y) is strictly increasing in each of its

first K arguments, and G is monotone in the sense that x ≤ x0 implies G(x) ⊂ G(x0 ).
The final assumption we will impose is that of differentiability and is also common in most economic models. This assumption will enable us to work with firstorder necessary conditions.

Assumption 6.5. G is continuously differentiable on the interior of its domain
XG .
Given these assumptions, the following sequence of results can be established.
The proofs for these results are provided in Section 6.4.
Theorem 6.1. (Equivalence of Values) Suppose Assumption 6.1 holds. Then
for any x ∈ X, Problem A2 has a unique value V (x), which is equal to V ∗ (x) defined

in Problem A1.

Therefore, both the sequence problem and the recursive formulation achieve
the same value. While important, this theorem is not of direct relevance in most
economic applications, since we do not care about the value but we care about the
optimal plans (actions). This is dealt with in the next theorem.
Theorem 6.2. (Principle of Optimality) Suppose Assumption 6.1 holds.
Let x∗ ∈ Φ (x (0)) be a feasible plan that attains V ∗ (x (0)) in Problem A1. Then we
have that
(6.3)

V ∗ (x∗ (t)) = U(x∗ (t) , x∗ (t + 1)) + βV ∗ (x∗ (t + 1))

for t = 0, 1, ... with x∗ (0) = x (0).
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