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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 ) ∈ XG , we have U [α(x, y) + (1 − α)(x0 , y )] ≥ αU(x, y) + (1 − α)U(x0 , y ), and if x 6= x0 , U[α(x, y) + (1 − α)(x0 , y )] > αU(x, y) + (1 − α)U(x0 , y ) Moreover, G is convex in the sense that for any α ∈ [0, 1], and x, x0 ∈ X, whenever y ∈ G(x) and y ∈ G(x0 ), then we have αy + (1 − α)y ∈ G[αx + (1 − α)x0 ] 262