A Semigroups Approach to the Study of a Second Order Partial Diferential Equation Applied in Economics

In this paper we will study the well-known problem of production functions in an operator semigroup approach. In general, semigroups can be used to solve a large class of problems com-monly known as evolution equations. They are usually described by an initial value problem for a differential equation, also known as a Cauchy problem. After summarizing some of the major properties of semigroups theory, we will provide an application to the theory of production functions. In order to arrive to our main result, we consider that the production function F (L(t), K (t), t) is assumed to be homogenous of degree one. To simplify the computation procedure, we denote by x(t) = K (t)/ L(t), y = f ( x(t), t ), and we suppose that x(t) is the solution for a stochastic differential equation. Finally we present some concluding remarks.