ECO 100B
Handout on CRS LongRun Costs
Suppose
f
is a CRS technology: i.e.,
f
(
&L;
) =
(
L;K
)
. Let
w
be the wage rate and
r
the rental rate of capital.
1
unit of output: that this
would involve
L
units of labor,
K
units of capital, and cost
C
=
wL
+
rK
(1)
So this would imply that
C
=
C
(1)
where
C
(
q
)
is the cost function that tells you the cheapest
way to produce
q
units of output.
Claim
:
C
(
q
) =
qC
for all
q
.
Proof
: We can argue this by contradiction.
Pick some arbitrary
q
, and for convenience, let
q >
1
. Suppose
C
q
)
6
qC
: then Case 1:
C
q
)
>
qC
or Case 2:
C
q
)
<
qC
. We will argue
that each of these cases will lead to a logical contradiction.
Case 1
:
Suppose
C
q
)
>
qC
.
Now,
f
(
L
;K
) = 1
, so
f
qL
;
qK
qf
(
L
;K
q
, because
f
is CRS. So one way to
produce
q
units of output is to scale up production and use
qL
units of labor and
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 Spring '07
 RAUCH
 Economics, Logic, rental rate, QC, cheapest way

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