Economic growth and economic development 274

Introduction to Modern Economic Growth
determining which value of x (t + 1) to choose for a given value of the state variable
x (t). In general, however, there will be two complications: first, a control reaching
the optimal value may not exist, which was the reason why we originally used the
notation sup; second, we may not have a policy function, but a policy correspondence, Π : X ⇒ X, because there may be more than one maximizer for a given
state variable. Let us ignore these complications for now and present a heuristic
exposition. These issues will be dealt with below.
Once the value function V is determined, the policy function is given straightforwardly. In particular, by definition it must be the case that if optimal policy is
given by a policy function π (x), then
V (x) = [U(x, π (x)) + βV (π (x))] , for all x ∈ X,
which is one way of determining the policy function. This equation simply follows
from the fact that π (x) is the optimal policy, so when y = π (x), the right hand side
of (6.1) reaches the maximal value V (x).
The usefulness of the recursive formulation in Problem A2 comes from the fact
that there are some powerful tools which not only establish existence of the solution,
but also some of its properties. These tools are not only employed in establishing the
existence of a solution to Problem A2, but they are also useful in a range of problems
in economic growth, macroeconomics and other areas of economic dynamics.
The next section states a number of results about the relationship between the
solution to the sequence problem, Problem A1, and the recursive formulation, Problem A2. These results will first be stated informally, without going into the technical
details. Section 6.3 will then present these results in greater formality and provide
their proofs.
6.2. Dynamic Programming Theorems
Let us start with a number of assumptions on Problem A1. Since these assumptions are only relevant for this section, we number them separately from the main
assumptions used throughout the book. Consider first a sequence {x∗ (t)}∞
t=0 which

attains the supremum in Problem A1. Our main purpose is to ensure that this
sequence will satisfy the recursive equation of dynamic programming, written here
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