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Definition and Properties of
the Cost Function
Lecture XIII
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From Previous Lectures
j In the preceding lectures we first developed
the production function as a technological
envelope demonstrating how inputs can bemapped into outputs.
jNext, we showed how these functions could
be used to derive input demand, cost, and
profit functions based on these functions
and optimizing behavior.
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j In this development, we stated that
economist had little to say about the
characteristics of the production function.j We were only interested in these functions
in the constraints that they imposed on
optimizing behavior.
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j Thus, the insight added by the dual
approach is the fact that we could simply
work with the resulting optimizingbehavior.
In some cases, this optimizing behavior can
then be used to infer facts about the technology
underlying it.
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Gorman (1976) Duality is about the choice of
the independent variables in terms of which one
defines a theory.
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.
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The Cost Function Defined
j The cost function is defined as:
Literally, the cost function is the minimum costof producing a given level of output from aspecific set of inputs.
This definition depends on the production setV(y). In a specific instant such as the Cobb-Douglas production function we can define thisproduction set analytically.
0
, min :x
c w y w x x V yu
! v
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Technology constrains the behavior or
economic agents. For example, we will impose
the restriction on the technology so that at least
some input be used to produce any non-zerolevel of output.
j The goal is to place as few of restrictions on
the behavior of economic agents as possibleto allow for the derivation of a fairly general
behavioral response.
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jNot to loose sight of the goal, we are interested inbe able to specify the cost function based on inputprices and output prices:
Is a standard form of the quadratic cost functionthat we use in empirical research. We are
interested in developing the properties underwhich this function represents optimizingbehavior.
0 1 1, ' ' ' ' '2 2c w y w w Aw y y By w yE E F! +
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j In addition, we will demonstrate Shephards
lemma which states that
Or, that the derivative of the cost function
with respect to the input price yields thedemand equation for each input.
*, ,i i i ii
c w yx w y A w y
wE x ! ! +x
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Properties of the Cost Function
c(w,y)>0 forw>0 and y>0 (nonnegativity);
If , then
(nondecreasing in w);
concave and continuous in w;
c(tw,y)=tc(w,y), t>0 (positively linearly
homogeneous);
If , then(nondecreasing in y); and
c(w,0)=0 (no fixed costs).
'w wu ', ,c w y c w yu
'y yu , , 'c y c yu
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If the cost function is differentiable in w, then
there exists a vector of costs minimizing
demand functions for each input formed from
the gradient of the cost function with respect tow.
j In order to develop these costs, we begin
with the basic notion that technology set isclosed and nonempty. Thus V(y) implies .
Thus,
_ a0min : ' 0;
x x x x x V y
u
v v e
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V y
'x
'wx
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Discussion of Properties
j Property 2B.1 simply states that it is
impossible to produce a positive output at
zero cost. Going back to the production
function, it was impossible to produce
output without inputs. Thus, given positive
prices, it is impossible to produce outputs
without a positive cost.
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j Property 2B.2 likewise seems obvious, if
one of the input prices increases, then the
cost of production increases.
A
BC
iw
2i
w1i
w
,c w y1 1
w xg
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First, if we constrain our discussion to theoriginal input bundle,x1, it is clear thatw1x1 < w2x1 ifw2 > w1. Next, we have to
establish that the change does not yield changein inputs such that the second price is lowerthan the first. This conclusions follows fromthe previous equation:
In other words, it is impossible forw1x2 < w1x1.
_ a0min : ' 0;x w x w x x x V yu v v e
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Taken together this results yields the
fundamentalinequality ofcostminimization:
If we focus on one price,
1 2 1 2 0w w x x e
1 2 1 2 0i i i iw w x x e
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j Continuous and concave in w.
This fact is depicted in the above graph.
Note thatA, B, and Clie on a straight line that is
tangent to the cost function at B.
Movement from B to Cwould assume that input
bundle optimal at B is also optimal at C.
If, however, are opportunities to substitute one input
for another, such opportunities will be used if theyproduce a lower cost.
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To develop a more rigorous proof, let w0, w1,
and w11be vectors of prices, and x1 andx11be
associated input bundles such that
Thus, w1 is one vector of input prices, and w11
is another vector of input prices. w0
is then alinear combination of input prices. We then
want to show that
0 1 111 ; 0 1w w wU U U! e e
0 1 11, , 1 ,c w y c w y c w yU Uu
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Let x0be the cost minimizing bundles
associated with w0. By cost minimization,
1 0 1 1 11 0 11 11andw x w x w x w xu u
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0 0 0
1 11 0
1 0 11 0 1 11
1 0 1 1 1
11 0 11 11 11
,
1
1 , 1 ,
,
,
c w y w x
w w x
w x w x c w y c w y
w x c w y w x
w x c w y w x
U U
U U U U
!
!
! u
u !
u !
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j Positive Linear Homogeneity
jNo fixed costs.
_ a
_ a
0
0
, min :
min :
,
x
x
c tw y tw x x V y
t w x x V y
t c w y
u
u
! v
! v
!
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Shephards lemma
j In general, Shephards lemma holds that
1 *
1
*
2
2
*
,
,,
,,
,,
w
n
n
c w y
wx w y
c w yx w y
c w y w
x w yc w y
w
x
x x ! !x
x
x
M
M
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j At the most basic level, this proof is a
simple application of the envelope theorem:
First, assume that we want to maximize some
general function:
were we maximizef(x,E) through choosing x, but
assume that E is fixed. To do this, we form thefirst-order conditions conditional on E:
1 2, , ,nf x x x EL
*
1 2
* * *
1 2
, , , 0
, , ,
i n i i
n
f x x x x x
y x x x
E E
E E E E J E
! !
!
L
L
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The question is then: How does the solution
change with respect to a change in E. To see
this we differentiate the optimum objective
function value with respect to E to obtain:
* *
1
*
. . .
.given 0
. .
ni
i i
i
y f x f
x
f ix
y f
J
!
x x x x x! !
x x x x x
x ! x
x x !
x x
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Similarly, in the case of the constrained
optimum:
1 2
1 2
*
*
* * * *
1 2
max , , ,
. . , , ,
0
, , ,
n
n
i i
i ii
n
f x x x
s t g x x x
Lf g
x xxL f g
Lg
y x x x
E
E
PE
P P P E
P
E E E E
x! !x
! !x ! ! x
L
L
L
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Again, differentiating the optimum with respectto E, we get
To work this out, we also differentiate the cost
function with respect to E:
* *
1
. .
. . .but 0
ni
i i
i i i
y f x x f
x
f f g
x x x
J
P
!
x x x x x! !
x x x x x
x x x{ !
x x x
* * *
1 2
*
1
, , ,
. . .
n
ni
i i
g x x x
g g x g
x
E E E E
E
E E E!
x x x x!
x x x x
L
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j Putting the two halves together:
* * *
1 1
* *
1
*
. . . .
. . . .
. . . .0
n ni i
i ii i
ni
i i i
i i
y f x x f g x g
x x
y f x g x f g
x x
f x g y f gi
x x
E EP
E E E E EE
P PE E E E
P PE E E
! !
!
x x x x x x x!
x x x x x x x x x x x x x
! x x x x x x
x x x x x ! ! x x x x x
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j Thus, following the envelope theorem:
1 1 2 2
1 2
1 1 2 2 0 1 2
* * *
1 1
1 1
min ( , )
. . ,
,
, ,
c w y w x w x
s t f x x y
L w x w x y f x x
c w y L x x w yw w
P
!
!
!
x x ! ! !x x
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More explicitly,
* * ** 1 21 1 2
1 1 1
1
1
2
2
* * ** * 1 21
1 1 1 2 1
,
However, by first-order conditions
,
c w y x x x w w
w w wf
wx
fw x
c w y f x f xx
w x w x w
P
P
P
x x x!
x x xx
!x
x! x
x x x x x ! x x x x x
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However, differentiating the constraint of the
minimization problem, we see
Thus, the second term in the preceding equationis zero and we have demonstrated Shephards
lemma.
* ** * 0 1 2
0 1 2
1 1 1 2 2
. ., 0 f fy x xy f x xw x w x w
x xx x x! ! ! x x x x x
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