Definition and Properties of the Cost Function
Lecture XIII
I.
From Previous Lectures
A.
In the preceding lectures we first developed the production function as a
technological envelope demonstrating how inputs can be mapped into outputs.
Next, we showed how these functions could be used to derive input demand,
cost, and profit functions based on these functions and optimizing behavior.
B.
In this development, we stated that economist had little to say about the
characteristics of the production function. We were only interested in these
functions in the constraints that they imposed on optimizing behavior.
C.
Thus, the insight added by the “dual” approach is the fact that we could
simply work with the resulting optimizing behavior. In some cases, this
optimizing behavior can then be used to infer facts about the technology
underlying it.
D.
Gorman (1976) “Duality is about the choice of the independent variables in
terms of which one defines a theory.”
E.
Chambers (p. 49) “The essence of the dual approach is that technology (or in
the case of the consumer problem, preferences) constrains the optimizing
behavior of individuals. One should therefore be able to use an accurate
representation of optimizing behavior to study the technology.”
II.
The Cost Function Defined
A.
The cost function is defined as:
()
( )
0
,m
i
n
:
x
cwy
w xx V y
≥
⎡
⎤
=×
∈
⎣
⎦
1.
Literally, the cost function is the minimum cost of producing a given
level of output from a specific set of inputs.
2.
This definition depends on the production set
. In a specific
instant such as the CobbDouglas production function we can define
this production set analytically.
Vy
3.
Technology constrains the behavior or economic agents. For example,
we will impose the restriction on the technology so that at least some
input is used to produce any nonzero level of output.
B.
The goal is to place as few of restrictions on the behavior of economic agents
as possible to allow for the derivation of a fairly general behavioral response.
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 Fall '09
 Staff
 Economics, Operations Research, Optimization, Charles B. Moss

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