1
Ch 20. Cost Minimization
This chapter examines how to minimize the cost of producing any given level of output,
and then we will study how to choose the most profitable level of output. Combining both
these problems enables the firm to solve its profit-maximization problem in a competitive
environment.
20.1
Cost Minimization
A firm purchases two factors (
x
1
, x
2
) at prices (
w
1
, w
2
) to produce a given level of output
y
. The cheapest way to produce is to choose (
*
2
*
1
,
x
x
) that solves the following problem:
()
=
+
=
y
x
x
f
t
s
x
w
x
w
C
x
x
2
1
2
2
1
1
,
,
___
:
.
.
_
min
2
1
The optimal choices of inputs that minimize costs for the firm are called the
conditional
factor demand functions
(or
derived factor demands
), which measure the relation
between the demand for factors and their prices conditional on output
y
to be produced.
That is,
()
y
w
w
x
x
i
i
,
,
2
1
*
=
. By contrast, the profit-maximizing
factor demands
take the
form of
()
p
w
w
x
x
i
i
,
,
2
1
*
=
, as shown in Ch 19.
The minimum cost necessary to achieve the given level of output is known as the
cost
function
:
()
()
()
(
)
y
C
y
w
w
C
y
w
w
x
w
y
w
w
x
w
C
=
=
+
=
,
,
,
,
,
,
2
1
2
1
2
2
2
1
1
1
*
We now use a graph to illustrate the problem-solving. First, define an
isocost line
that
contains all the combinations of inputs that have some given level of cost C:
C
x
w
x
w
=
+
2
2
1
1
. Then, change this equation to have a function for the isocost line:
1
2
1
2
2
x
w
w
w
C
x
−
=
The line has a slope of
2
1
/
w
w
−
and a vertical intercept of
C / w
2
. As changing C, we get
a family of isocost lines. Higher isocost lines are associated with greater costs.