Chapter 20
NAME
Cost Minimization
Introduction.
In the chapter on consumer choice, you studied a con-
sumer who tries to maximize his utility subject to the constraint that he
has a fixed amount of money to spend.
In this chapter you study the
behavior of a firm that is trying to produce a fixed amount of output
in the cheapest possible way.
In both theories, you look for a point of
tangency between a curved line and a straight line. In consumer theory,
there is an “indifference curve” and a “budget line.” In producer theory,
there is a “production isoquant” and an “isocost line.”
As you recall,
in consumer theory, finding a tangency gives you only one of the two
equations you need to locate the consumer’s chosen point.
The second
equation you used was the budget equation. In cost-minimization theory,
again the tangency condition gives you one equation. This time you don’t
know in advance how much the producer is spending; instead you are told
how much output he wants to produce and must find the cheapest way
to produce it. So your second equation is the equation that tells you that
the desired amount is being produced.
Example.
A firm has the production function
f
(
x
1
, x
2
) = (
√
x
1
+
3
√
x
2
)
2
.
The price of factor 1 is
w
1
= 1 and the price of factor 2
is
w
2
= 1.
Let us find the cheapest way to produce 16 units of out-
put.
We will be looking for a point where the technical rate of sub-
stitution equals
−
w
1
/w
2
.
If you calculate the technical rate of sub-
stitution (or look it up from the warm up exercise in Chapter 18),
you find
TRS
(
x
1
, x
2
) =
−
(1
/
3)(
x
2
/x
1
)
1
/
2
.
Therefore we must have
−
(1
/
3)(
x
2
/x
1
)
1
/
2
=
−
w
1
/w
2
=
−
1.
This equation can be simplified
to
x
2
= 9
x
1
. So we know that the combination of inputs chosen has to
lie somewhere on the line
x
2
= 9
x
1
. We are looking for the cheapest way
to produce 16 units of output. So the point we are looking for must sat-
isfy the equation (
√
x
1
+ 3
√
x
2
)
2
= 16, or equivalently
√
x
1
+ 3
√
x
2
= 4.
Since
x
2
= 9
x
1
, we can substitute for
x
2
in the previous equation to get
√
x
1
+3
√
9
x
1
= 4. This equation simplifies further to 10
√
x
1
= 4. Solving
this for
x
1
, we have
x
1
= 16
/
100. Then
x
2
= 9
x
1
= 144
/
100.
The amounts
x
1
and
x
2
that we solved for in the previous para-
graph are known as the
conditional factor demands for factors 1 and 2
,
conditional on the wages
w
1
= 1,
w
2
= 1, and output
y
= 16.
We ex-
press this by saying
x
1
(1
,
1
,
16) = 16
/
100 and
x
2
(1
,
1
,
16) = 144
/
100.
Since we know the amount of each factor that will be used to pro-
duce 16 units of output and since we know the price of each factor,
we can now calculate the cost of producing 16 units.
This cost is
c
(
w
1
, w
2
,
16) =
w
1
x
1
(
w
1
, w
2
,
16)+
w
2
x
2
(
w
1
, w
2
,
16). In this instance since
w
1
=
w
2
= 1, we have
c
(1
,
1
,
16) =
x
1
(1
,
1
,
16) +
x
2
(1
,
1
,
16) = 160
/
100.