Economic growth and economic development 275

Introduction to Modern Economic Growth
as
(6.2)

V (x∗ (t)) = U(x∗ (t) , x∗ (t + 1)) + βV (x∗ (t + 1)), for all t = 0, 1, 2, ...,

and that any solution to (6.2) will also be a solution to Problem A1, in the sense
that it will attain its supremum. In other words, we are interested in in establishing
equivalence results between the solutions to Problem A1 and Problem A2.
To prepare for these results, let us define the set of feasible sequences or plans
starting with an initial value x (t) as:
Φ(x (t)) = {{x (s)}∞
s=t : x (s + 1) ∈ G(x (s)), for s = t, t + 1, ...}.
Intuitively, Φ(x (t)) is the set of feasible choices of vectors starting from x (t). Let
us denote a typical element of the set Φ(x (0)) by x = (x (0) , x (1) , ...) ∈ Φ(x (0)).
Our first assumption is:

Assumption 6.1. G (x) is nonempty for all x ∈ X; and for all x (0) ∈ X and
P
x ∈ Φ(x (0)), limn→∞ nt=0 β t U(x (t) , x (t + 1)) exists and is finite.
This assumption is stronger than what is necessary to establish the results that

will follow. In particular, for much of the theory of dynamic programming, it is
sufficient that the limit in Assumption 6.1 exists. However, in economic applications,
we are not interested in optimization problems where households or firms achieve
infinite value. This is for two obvious reasons. First, when some agents can achieve
infinite value, the mathematical problems are typically not well defined. Second,
the essence of economics, trade-offs in the face of scarcity, would be absent in these
cases. In cases, where households can achieve infinite value, economic analysis is still
possible, by using methods sometimes called “overtaking criteria,” whereby different
sequences that give infinite utility are compared by looking at whether one of them

gives higher utility than the other one at each date after some finite threshold.
None of the models we study in this book require us to consider these more general
optimality concepts.
Assumption 6.2. X is a compact subset of RK , G is nonempty, compact-valued
and continuous. Moreover, let XG = {(x, y) ∈ X × X : y ∈ G(x)} and assume that

U : XG → R is continuous.

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