1
Definition and Properties of the Production Function:
Lecture II
I.
Overview of the Production Function – Chambers
A.
“The production function (and indeed all representations of technology) is a
purely technical relationship that is void of economic content.
Since
economists are usually interested in studying economic phenomena, the
technical aspects of production are interesting to economists only insofar as
they impinge upon the behavior of economic agents.” (Chambers p. 7).
B.
“Because the economist has no inherent interest in the production function, if
it is possible to portray and to predict economic behavior accurately without
direct examination of the production function, so much the better.
This
principle, which sets the tone for much of the following discussion, underlies
the intense interest that recent developments in duality have aroused.”
(Chambers p. 7).
1.
The point of these two statements is that economists are not engineers
and have no insights into why technologies take on any particular
shape.
We are only interested in those properties that make the
production function useful in economic analysis, or those properties
that make the system solvable.
2.
There are several interpretations of the dual.
Let use briefly discuss
one concept.
Assume that we are interested in analyzing production of
some crop (say cotton).
a.
One approach would be to estimate a production function, say
a CobbDouglas production function in two relevant inputs:
12
yx
x
α
β
=
b.
Given this production function, we could derive a cost function
by minimizing the cost of the two inputs subject to some level
of production:
11
2 2
,
min
..
xx
wx
st y x x
αβ
+
=
Forming the Lagrangian of this optimization problem, we have
( )
1 2
1
2
22
0
0
0
Lw
x w
x
x
L
w
L
w
L
x
λ
=+
+−
∂
=−
=
∂
∂
=
∂
∂
=
∂
Taking the first two firstorder conditions together we have
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View Full DocumentAEB 6184 – Production Economics
Lecture II
Professor Charles B. Moss
Fall 2005
2
11
2
2
12
21
1
2
L
xw
x
w
x
x
L
wx
w
x
∂
∂
⇒=
⇒
=
∂
∂
Substituting this relationship into the final firstorder condition
yields
()
1
*
22
2
1
2
0,
,
ww
L
yx
x
x
w
w
y
y
α
β
αβ
λ
+
+
∂
⇒−
=⇒
=
∂
By substituting this relationship back into the previous
condition with respect that solves
1
x
as a function of
2
x
, we
have
1
*
2
112
1
,,
w
xwwy y
w
+
+
=
Substituting both of these optimal relationships (output
conditional input demand curves) back into the cost function
yields
1
2
1
1
2
Cww y w y
w y
y
w
w
++
+
+
+
=+
c.
Thus, in the end, we are left with a cost function that relates
input prices and output levels to the cost of production based
on the economic assumption of optimizing behavior.
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 Fall '09
 Staff
 Economics, Production Economics, Professor Charles B. Moss

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