Chapter 18
NAME
Technology
Introduction.
In this chapter you work with production functions, re-
lating output of a firm to the inputs it uses. This theory will look familiar
to you, because it closely parallels the theory of utility functions. In utility
theory, an
indifference curve
is a locus of commodity bundles, all of which
give a consumer the same utility. In production theory, an
isoquant
is a lo-
cus of input combinations, all of which give the same output. In consumer
theory, you found that the slope of an indifference curve at the bundle
(
x
1
, x
2
) is the ratio of marginal utilities,
MU
1
(
x
1
, x
2
)
/MU
2
(
x
1
, x
2
).
In
production theory, the slope of an isoquant at the input combination
(
x
1
, x
2
) is the ratio of the marginal products,
MP
1
(
x
1
, x
2
)
/MP
2
(
x
1
, x
2
).
Most of the functions that we gave as examples of utility functions can
also be used as examples of production functions.
There is one important difference between production functions and
utility functions. Remember that utility functions were only “unique up to
monotonic transformations.” In contrast, two different production func-
tions that are monotonic transformations of each other describe different
technologies.
Example:
If the utility function
U
(
x
1
, x
2
) =
x
1
+
x
2
represents a person’s
preferences, then so would the utility function
U
∗
(
x
1
, x
2
) = (
x
1
+
x
2
)
2
.
A person who had the utility function
U
∗
(
x
1
, x
2
) would have the same
indifference curves as a person with the utility function
U
(
x
1
, x
2
) and
would make the same choices from every budget. But suppose that one
firm has the production function
f
(
x
1
, x
2
) =
x
1
+
x
2
, and another has the
production function
f
∗
(
x
1
, x
2
) = (
x
1
+
x
2
)
2
. It is true that the two firms
will have the same isoquants, but they certainly do not have the same
technology.
If both firms have the input combination (
x
1
, x
2
) = (1
,
1),
then the first firm will have an output of 2 and the second firm will have
an output of 4.
Now we investigate “returns to scale.” Here we are concerned with
the change in output if the amount of every input is multiplied by a
number
t >
1. If multiplying inputs by
t
multiplies output by more than
t
, then there are increasing returns to scale.
If output is multiplied by
exactly
t
, there are constant returns to scale. If output is multiplied by
less than
t
, then there are decreasing returns to scale.
Example:
Consider the production function
f
(
x
1
, x
2
) =
x
1
/
2
1
x
3
/
4
2
. If we
multiply the amount of each input by
t
, then output will be
f
(
tx
1
, tx
2
) =
(
tx
1
)
1
/
2
(
tx
2
)
3
/
4
.
To compare
f
(
tx
1
, tx
2
) to
f
(
x
1
, x
2
), factor out the
expressions involving
t
from the last equation.
You get
f
(
tx
1
, tx
2
) =
t
5
/
4
x
1
/
2
1
x
3
/
4
2
=
t
5
/
4
f
(
x
1
, x
2
). Therefore when you multiply the amounts
of all inputs by
t
, you multiply the amount of output by
t
5
/
4
. This means
there are
increasing
returns to scale.