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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