Introduction to Modern Economic Growth the state variable x is free Below, we will see that in the context of finite-horizon economic problems, the formulation where x1 is not a choice variable may be simpler (see Example 7.1), but the development in this section is more natural when the terminal value x1 is free In addition, to simplify the exposition, throughout we assume that f and g are continuously differentiable functions The difficulty in characterizing the optimal solution to this problem lies in two features: (1) We are choosing a function y : [0, t1 ] → Y rather than a vector or a finite dimensional object (2) The constraint takes the form of a differential equation, rather than a set of inequalities or equalities These features make it difficult for us to know what type of optimal policy to look for For example, y may be a highly discontinuous function It may also hit the boundary of the feasible set–thus corresponding to a “corner solution” Fortunately, in most economic problems there will be enough structure to make optimal solutions continuous functions Moreover, in most macroeconomic and growth applications, the Inada conditions make sure that the optimal solutions to the relevant dynamic optimization problems lie in the interior of the feasible set These features considerably simplify the characterization of the optimal solution In fact, when y is a continuous function of time and lies in the interior of the feasible set, it can be characterized by using the variational arguments similar to those developed by Euler, Lagrange and others in the context of the theory of calculus of variations Since these tools are not only simpler but also more intuitive, we start our treatment with these variational arguments The variational principle of the calculus of variations simplifies the above maximization problem by first assuming that a continuous solution (function) yˆ that lies everywhere in the interior of the set Y exists, and then characterizes what features this solution must have in order to reach an optimum (for the relationship of the results here to the calculus of variations, see Exercise 7.3) 315