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Introduction to Modern Economic Growth At first sight, the recursive formulation might appear not as a great advance over the sequence formulation After all, functions might be trickier to work with than sequences Nevertheless, it turns out that the functional equation of dynamic programming is easy to work with in many instances In applied mathematics and engineering, it is favored because it is computationally convenient In economics, perhaps the major advantage of the recursive formulation is that it often gives better economic insights, similar to the logic of comparing today to tomorrow In particular, U (x, y) is the“ return for today” and βV (y) is the continuation return from tomorrow onwards, equivalent to the “return for tomorrow” Consequently, in many applications we can use our intuitions from two-period maximization or economic problems Finally, in some special but important cases, the solution to Problem A2 is simpler to characterize analytically than the corresponding solution of the sequence problem, Problem A1 In fact, the form of Problem A2 suggests itself naturally from the formulation Problem A1 Suppose Problem A1 has a maximum starting at x (0) attained by the ∗ optimal sequence {x∗ (t)}∞ t=0 with x (0) = x (0) Then under some relatively weak technical conditions, we have that ∗ V (x (0)) = ∞ X β t U (x∗ (t) , x∗ (t + 1)) t=0 = U(x (0) , x∗ (1)) + β ∞ X β s U(x∗ (s + 1) , x∗ (s + 2)) s=0 ∗ ∗ = U(x (0) , x (1)) + βV (x∗ (1)) This equation encapsulates the basic idea of dynamic programming: the Principle of Optimality, and it is stated more formally in Theorem 6.2 Essentially, an optimal plan can be broken into two parts, what is optimal to today, and the optimal continuation path Dynamic programming exploits this principle and provides us with a set of powerful tools to analyze optimization in discrete-time infinite-horizon problems As noted above, the particular advantage of this formulation is that the solution can be represented by a time invariant policy function (or policy mapping), π : X → X, 259