Economic growth and economic development 276

Introduction to Modern Economic Growth
This assumption is also natural. We need to impose that G (x) is compactvalued, since optimization problems with choices from non—compact sets are not
well behaved (see the Mathematical Appendix). In addition, the assumption that U
is continuous leads to little loss of generality for most economic applications. In all
the models we will encounter in this book, U will be continuous. The most restrictive
assumption here is that X is compact. This assumption will not allow us to study
endogenous growth models where the state variable, the capital stock, can grow
without bounds. Nevertheless, everything stated in this chapter can be generalized
to the case in which X is not compact, though this requires additional notation and
more advanced mathematical tools. For this reason, we limit the discussion in this
chapter to the case in which X is compact.
Note also that since X is compact, G (x) is continuous and compact-valued,
XG is also compact. Since a continuous function from a compact domain is also
bounded, Assumption 6.2 also implies that U is bounded, which will be important
for some of the results below.
Assumptions 6.1 and 6.2 together ensure that in both Problems A1 and A2, the
supremum (the maximal value) is attained for some feasible plan x. We state all
the relevant theorems incorporating this fact.
To obtain sharper results, we will also impose:

Assumption 6.3. U is strictly concave, in the sense that for any α ∈ (0, 1) and

any (x, y), (x0 , y 0 ) ∈ XG , we have

U [α(x, y) + (1 − α)(x0 , y 0 )] ≥ αU(x, y) + (1 − α)U(x0 , y 0 ),
and if x 6= x0 ,
U[α(x, y) + (1 − α)(x0 , y 0 )] > αU(x, y) + (1 − α)U(x0 , y 0 ).
Moreover, G is convex in the sense that for any α ∈ [0, 1], and x, x0 ∈ X, whenever
y ∈ G(x) and y 0 ∈ G(x0 ), then we have

αy + (1 − α)y 0 ∈ G[αx + (1 − α)x0 ].

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