Introduction to Modern Economic Growth almost all growth models–are more naturally formulated as infinite-horizon problems This is obvious in the context of economic growth, but is also the case in repeated games, political economy or industrial organization, where even if individuals may have finite expected lifes, the end date of the game or of their lives may be uncertain For this reason, the canonical model of optimization and economic problems is the infinite-horizon one 7.3.1 The Basic Problem: Necessary and Sufficient Conditions Let us focus on infinite-horizon control with a single control and a single state variable Using the same notation as above, the problem is (7.28) max W (x (t) , y (t)) ≡ x(t),y(t) subject to (7.29) Z ∞ f (t, x (t) , y (t)) dt x˙ (t) = g (t, x (t) , y (t)) , and (7.30) y (t) ∈ R for all t, x (0) = x0 and lim x (t) ≥ x1 t→∞ The main difference is that now time runs to infinity Note also that this problem allows for an implicit choice over the endpoint x1 , since there is no terminal date The last part of (7.30) imposes a lower bound on this endpoint In addition, we have further simplified the problem by removing the feasibility requirement that the control y (t) should always belong to the set Y, instead simply requiring this function to be real-valued For this problem, we call a pair (x (t) , y (t)) admissible if y (t) is a piecewise continuous function of time, meaning that it has at most a finite number of discontinuities.4 Since x (t) is given by a continuous differential equation, the piecewise continuity of y (t) ensures that x (t) is piecewise smooth Allowing for piecewise continuous controls is a significant generalization of the above approach There are a number of technical difficulties when dealing with the infinite-horizon case, which are similar to those in the discrete time analysis Primary among those 4More generally, y (t) could be allowed to have a countable number of discontinuities, but this added generality is not necessary for any economic application 331