Economic growth and economic development 272

Introduction to Modern Economic Growth form that the constraint correspondence could take, since it does not depend on k (t)) Problem A1, also referred to as the sequence problem, is one of choosing an infinite sequence {x (t)}∞ t=0 from some (vector) space of infinite sequences (for example, ∞ ⊂ L∞ , where L∞ is the vector space of infinite sequences that {x (t)}∞ t=0 ∈ X are bounded with the k·k∞ norm, which we will denote throughout by the simpler notation k·k) Sequence problems sometimes have nice features, but their solutions are often difficult to characterize both analytically and numerically The basic idea of dynamic programming is to turn the sequence problem into a functional equation That is, it is to transform the problem into one of finding a function rather than a sequence The relevant functional equation can be written as follows: Problem A2 (6.1) : V (x) = sup [U(x, y) + βV (y)] , for all x ∈ X, y∈G(x) where V : X → R is a real-valued function Intuitively, instead of explicitly choosing the sequence {x (t)}∞ t=0 , in (6.1), we choose a policy, which determines what the control vector x (t + 1) should be for a given value of the state vector x (t) Since instantaneous payoff function U (·, ·) does not depend on time, there is no reason for this policy to be time-dependent either, and we denote the control vector by y and the state vector by x Then the problem can be written as making the right choice of y for any value of x Mathematically, this corresponds to maximizing V (x) for any x ∈ X The only subtlety in (6.1) is the presence of the V (·) on the right hand side, which will be explained below This is also the reason why (6.1) is also called the recursive formulation–the function V (x) appears both on the left and the right hand sides of equation (6.1) and is thus defined recursively The functional equation in Problem A2 is also called the Bellman equation, after Richard Bellman, who was the first to introduce the dynamic programming formulation, though this formulation was anticipated by the economist Lloyd Shapley in his study of stochastic games 258