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 261