Solution to exercise 6.1: how to use homogeneity
Suppose a firm produces just one output using three inputs , , and according to the production function: . The prices of goods are , respectively. We assume that the firm can neither produce negative quantities of nor use negative quantities of the inputs, so that .
This is a good opportunity to learn using the homogeneity notion. Assuming, for simplicity, that a function has positive arguments, we say that it is homogeneous of degree if for all For example, our production function is homogeneous of degree 2 and the cost function is homogeneous of degree 1. Let's see how this affects the properties of the profit function
Suppose we scale the inputs by then
This means the following: if we start with any bundle of positive inputs and move along the ray to infinity, the value of the output stays fixed (positive), while the value of the inputs tends to zero. For sufficiently large, the value in the brackets becomes close to As there is the factor in front of the brackets, the profit function tends to infinity along such a ray. Since the initial bundle is arbitrary, such rays cover the whole quadrant So the profit tends to infinity along any ray and not only along the "diagonal" as the guide says.
Homogeneity is a notion whose usefulness contrasts with its simplicity. See homogeneity of means, of variance, of standard deviation, of correlation, of conditional variance, and application to the Gauss-Markov theorem. And the Cobb-Douglas function is also in this club.