Introduction to Modern Economic Growth where for each t, x (t) and y (t) are finite-dimensional vectors (i.e., x (t) ∈ RKx and y (t) ∈ RKy , where Kx and Ky are integers) We refer to x as the state variable Its behavior is governed by a vector-valued differential equation (i.e., a set of differential equations) given the behavior of the control variables y (t) The end of the planning horizon t1 can be equal to infinity The function W (x (t) , y (t)) denotes the value of the objective function when controls are given by y (t) and the resulting behavior of the state variable is summarized by x (t) We also refer to f as the objective function (or the payoff function) and to g as the constraint function The problem formulation is general enough to incorporate discounting, since both the instantaneous payoff function f and the constraint function g depend directly on time in an arbitrary fashion We will start with the finite-horizon case and then treat the infinite-horizon maximization problem, focusing particularly on the case where there is exponential discounting 7.1 Variational Arguments Consider the following finite-horizon continuous time problem Z t1 f (t, x (t) , y (t)) dt (7.1) max W (x (t) , y (t)) ≡ x(t),y(t),x1 subject to (7.2) x˙ (t) = g (t, x (t) , y (t)) and (7.3) y (t) ∈ Y (t) for all t, x (0) = x0 and x (t1 ) = x1 Here the state variable x (t) ∈ R is one-dimensional and its behavior is governed by the differential equation (7.2) The control variable y (t) must belong to the set Y (t) ⊂ R Throughout, we assume that Y (t) is nonempty and convex We refer to a pair of functions (x (t) , y (t)) that jointly satisfy (7.2) and (7.3) as an admissible pair Throughout, as in the previous chapter, we assume the value of the objective function is finite, that is, W (x (t) , y (t)) < ∞ for any admissible pair (x (t) , y (t)) Let us first suppose that t1 < ∞, so that we have a finite-horizon optimization problem Notice that there is also a terminal value constraint x (t1 ) = x1 , but x1 is included as an additional choice variable This implies that the terminal value of 314