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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 px_1(x_2+x_3) stays fixed (positive), while the value \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.

Homogeneity is a notion whose usefulness contrasts with its simplicity. See homogeneity of means, of varianceof standard deviation, of correlationof conditional variance, and application to the Gauss-Markov theorem. And the Cobb-Douglas function is also in this club.

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