Raising the bar

Solution to exercise 6.1: how to use homogeneity

Suppose a firm produces just one output $latex y$ using three inputs $latex x_1$, $latex x_2$, and $latex x_{3}$ according to the production function: $latex y=x_1(x_2+x_3)$.  The prices of goods $latex y,x_1,x_2,x_3$ are $latex p,w_1,w_2,w_3>0$, respectively. We assume that the firm can neither produce negative quantities of $latex y$ nor use negative quantities of the inputs, so that $latex y,x_1,x_2,x_3\geq 0$.

This is a good opportunity to learn using the homogeneity notion. Assuming, for simplicity, that a function $latex f$ has positive arguments, we say that it is homogeneous of degree $latex \alpha$ if $latex f(tx_1,...,tx_n)=t^{\alpha}f(x_1,...,x_n)$ for all $latex t>0.$ For example, our production function $latex f(x_1,x_2,x_3) = x_1(x_2+x_3)$ is homogeneous of degree 2 and the cost function $latex g(x_1,x_2,x_3)=w_1x_1+w_2x_2+w_3x_3$ is homogeneous of degree 1. Let's see how this affects the properties of the profit function

$latex \pi(x_1,x_2,x_3)=px_1(x_2+x_3)-(w_1x_1+w_2x_2+w_3x_3).$

Suppose we scale the inputs by $latex t>0,$ then

$latex \pi(tx_1,tx_2,tx_3)=t^2px_1(x_2+x_3)-t(w_1x_1+w_2x_2+w_3x_3)$

$latex =t^2[px_1(x_2+x_3)-\frac{1}{t}(w_1x_1+w_2x_2+w_3x_3)].$

This means the following: if we start with any bundle of positive inputs $latex (x_1,x_2,x_3)$ and move along the ray $latex (tx_1,tx_2,tx_3)$ to infinity, the value of the output $latex px_1(x_2+x_3)$ stays fixed (positive), while the value of the inputs $latex \frac{1}{t}(w_1x_1+w_2x_2+w_3x_3)$ tends to zero. For $latex t$ sufficiently large, the value in the brackets $latex [px_1(x_2+x_3)-\frac{1}{t}(w_1x_1+w_2x_2+w_3x_3)]$ becomes close to $latex px_{1}(x_{2}+x_{3}).$ As there is the factor $latex t^2$ in front of the brackets, the profit function tends to infinity along such a ray. Since the initial bundle $latex (x_1,x_2,x_3)$ is arbitrary, such rays cover the whole quadrant $latex {(x_1,x_2,x_3):x_1,x_2,x_3>0}.$ So the profit tends to infinity along any ray and not only along the "diagonal" $latex (x,x,x),$ 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.