Homogeneous Functions

In this chapter we consider homogeneous functions. Functions with this property play an important role in economic theory. Thus, in standard economic models, profit functions and cost functions derived from production functions are naturally homogeneous functions, and the same is true for demand functions derived from utility functions. Furthermore, while differential calculus and the notion of the partial derivative allow us to establish a connection between the change in only one independent variable and the resulting change in the functional value, with all the other variables unchanged, for homogeneous functions we can estimate what happens to the functional value if values of all variables simultaneously change by the same percentage. In the case of homogeneous production functions, this leads to the concepts of increasing, decreasing and constant returns to scale. We also present some basic properties of homogeneous functions as well as present the very important Euler’s theorem. Finally, we introduce the notion of partial elasticity, which shows how strongly a function of n variables y = f ( x 1 , x 2 … , x n ) $$y=f(x_{1}, x_{2}\ldots ,x_{n})$$ reacts to a change in the variable x i $$x_{i}$$ , assuming that the value of all the other variables is held constant. We show how elasticity is applied to a demand function to obtain the (own) price and cross price elasticity of demand, which enable us to determine if goods are substitutes or complements.