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.
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