Raising the bar Solution to exercise 6.1: how to use homogeneity and common sense

Jan 18

Solution to exercise 6.1: how to use homogeneity

Solution to exercise 6.1: how to use homogeneity

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

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

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

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

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

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

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