Cost Functions
R. Pope
Probably no concept in Econ. 380 is as useful as cost or expenditure. In Economics 110,
you learned about opportunity cost and sunk costs both powerful concepts. Here we
formalized costs a bit further and greater understanding is obtained. Modern
microeconomics emphasizes a duality between the production function and costs. That is,
the cost function can be used to recover the economically relevant portion (where a cost
minimizer would be) of an isoquant. In this lecture, we develop fewer examples because
cost minimization and expenditure minimization (which you have already studied) are
formally identical. See page 342 of the text for worked examples.
As you know, we distinguish costs in the short-run where one or more inputs are fixed
from costs in the long run where all inputs are considered variable. Continuing to work
with two inputs, capital and labor as our workhorse, cost,
(1)
C =
vk
wl
+
.
So cost is simple, we just sum expenditures on each of the inputs where v is the rental
price of capital and w is the wage rate. The rental price of capital
v
contains depreciation
and the opportunity cost of capital (more in class). When
k
, capital, is fixed we will say
we are in the short fun. When
and
k
l
are both freely chosen, this is the long run.
Though the equation in (1) is cost, it isn’t the cost function. The cost function embodies
the least cost quantity of
and
k
l
to produce a given level of output. The economists
view will specify the cost equation as a minimization problem whereas (1) would be
more like an accounting identity.
The economists state cost minimization as:
(2)
,
( , ; )
min {
:
( , )}
k l
C v w q
vk
wl q
f k l
=
+
=
i.e., choose the level of capital and labor so as to minimize cost given a particular
isoquant. One then thinks of doing this for all isoquants. Thus, the cost function in the
long run depends in general on input prices and ouput. The factor demands from (2) are
called variously “cost minimizing factor (input) demands” or “conditional factor (input)
demands”.
Costs in the Short-Run
When plant and equipment,
k
is fixed, the problem becomes:
(3)
(
, ; )
min {
:
( , )}
l
SC vk w q
FC
VC
vk
wl q
f k l
=
+
=
+
=
where
vk
=total fixed cost or TFC and the
c
wl
is total variable cost, TVC where
c
l
is the
cost minimizing level of labor. Fortunately, it is incredibly easy with two inputs to
explain what
c
l
is. With k fixed at
k
, there is only one level of l that generates a given q: