1
Cost, Learning Curves, and Scale Economies
Berndt Chapter 3
1.1
Economic Theory of Cost and Production
the relationship among inputs and output is summarized by a production function
y
=
f
(
x
1
; :::; x
n
;
A
)
y
:
output
f
(
°
) :
production function
x
i
; i
= 1
; :::n
:
input variables (capital, labor, ...)
A
:
technology
CobbDouglas production function with 3 input variables
y
=
f
(
x
1
; x
2
; x
3
;
A
) =
Ax
°
1
1
x
°
2
2
x
°
3
3
, where
°
1
; °
2
; °
3
are unknown parameters to be estimated
returns to scale:
what is the increase in output if all inputs are increased by 100%?
CobbDouglas:
r
=
°
1
+
°
2
+
°
3
economies of scale:
returns to scale minus 1
CobbDouglas:
r
±
1 =
°
1
+
°
2
+
°
3
±
1
cost minimizing behavior
a cost function: the minimum possible total cost of producing a given level of output
to the prices of the
n
inputs, the level of output
y
, and the state of technology
A
C
=
C
(
p
1
; :::; p
n
; y
;
A
)
, where
p
1
; :::; p
n
are input prices of
x
1
; :::; x
n
total cost:
C
=
C
(
p
1
; :::; p
n
; y
;
A
)
average cost:
C=y
marginal cost:
@[email protected]
cost minimization
minimize
P
p
i
x
i
subject to
y
=
f
(
x
1
; :::; x
n
;
A
)
min
x
i
L
=
P
p
i
x
i
+
±
[
y
±
f
(
x
1
; :::; x
n
;
A
)]
,
where
±
is the Lagrange multiplier
FOC:
@
L
[email protected]
i
=
p
i
±
±f
i
(
b
x
1
; :::;
b
x
n
;
A
) = 0
, for
i
= 1
; :::; n
@
L
[email protected]±
=
y
±
f
(
b
x
1
; :::;
b
x
n
;
A
) = 0
, where
f
i
is the °rst partial derivative of the production function w.r.t.
i
th input
1
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so we get
p
i
=p
n
=
f
i
(
b
x
1
; :::;
b
x
n
;
A
)
=f
n
(
b
x
1
; :::;
b
x
n
;
A
)
, for
i
= 1
; :::; n
±
1
then solve for each
b
x
i
and substitute it into the expression of total cost
C
=
P
p
i
b
x
i
cost minimization: CobbDouglas
minimize
P
p
i
x
i
subject to
y
=
Ax
°
1
1
x
°
2
2
x
°
3
3
min
x
i
L
=
P
p
i
x
i
+
±
[
y
±
Ax
°
1
1
x
°
2
2
x
°
3
3
]
,
where
±
is the Lagrange multiplier
FOC:
(1)
@
L
[email protected]
1
=
p
1
±
°
1
±Ax
°
1
°
1
1
x
°
2
2
x
°
3
3
=
p
1
±
°
1
±x
°
1
1
y
= 0
(2)
@
L
[email protected]
2
=
p
2
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 Spring '08
 SeikKim
 Microeconomics, Econometrics, log nt, log ct

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