Definition and Properties of the Cost Function

Lecture XIII

From Previous Lectures

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.

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.

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.

– 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.”

The Cost Function Defined

The cost function is defined as:

– Literally, the cost function is the minimum cost of producing a given level of output from a specific set of inputs.

– This definition depends on the production set V(y). In a specific instant such as the Cobb-Douglas production function we can define this production set analytically.

0

, min :x

c w y w x x V y

– 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-zero level of output.

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.

Not to loose sight of the goal, we are interested in be able to specify the cost function based on input prices and output prices:

Is a standard form of the quadratic cost function that we use in empirical research. We are interested in developing the properties under which this function represents optimizing behavior.

01 1, ' ' ' ' '2 2c w y w w Aw y y By w y

In addition, we will demonstrate Shephard’s lemma which states that

Or, that the derivative of the cost function with respect to the input price yields the demand equation for each input.

*,,i i i i

i

c w yx w y A w y

w

Properties of the Cost Function

– c(w,y)>0 for w>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 w ', ,c w y c w y

'y y , , 'c w y c w y

– 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 to w.

In order to develop these costs, we begin with the basic notion that technology set is closed and nonempty. Thus V(y) implies . Thus,

0

min : ' 0;x

w x w x x x V y

V y

'x

'wx

Discussion of Properties

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.

Property 2B.2 likewise seems obvious, if one of the input prices increases, then the cost of production increases.

A

BC

iw2iw1

iw

,c w y1 1w x

– First, if we constrain our discussion to the original input bundle, x1, it is clear that w1x1 < w2 x1 if w2 > w1. Next, we have to establish that the change does not yield change in inputs such that the second price is lower than the first. This conclusions follows from the previous equation:

In other words, it is impossible for w1x2 < w1x1.

0

min : ' 0;x

w x w x x x V y

– Taken together this results yields the fundamental inequality of cost minimization:

If we focus on one price,

1 2 1 2 0w w x x

1 2 1 2 0i i i iw w x x

Continuous and concave in w. – This fact is depicted in the above graph.

• Note that A, B, and C lie on a straight line that is tangent to the cost function at B.

• Movement from B to C would 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 they produce a lower cost.

– To develop a more rigorous proof, let w0, w1, and w11 be vectors of prices, and x1 and x11 be associated input bundles such that

Thus, w1 is one vector of input prices, and w11 is another vector of input prices. w0 is then a linear combination of input prices. We then want to show that

0 1 111 ; 0 1w w w

0 1 11, , 1 ,c w y c w y c w y

– Let x0 be the cost minimizing bundles associated with w0. By cost minimization,

1 0 1 1 11 0 11 11 and w x w x w x w x

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

Positive Linear Homogeneity

No fixed costs.

0

0

, min :

min :

,

x

x

c tw y tw x x V y

t w x x V y

t c w y

Shephard’s lemma

In general, Shephard’s 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

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 maximize f(x,) through choosing x, but assume that is fixed. To do this, we form the first-order conditions conditional on :

1 2, , ,nf x x x

*1 2

* * *1 2

, , , 0

, , ,

i n i i

n

f x x x x x

y x x x

– The question is then: How does the solution change with respect to a change in . To see this we differentiate the optimum objective function value with respect to to obtain:

* *

1

*

. . .

.given 0

. .

ni

i i

i

y f x f

x

fi

x

y f

– Similarly, in the case of the constrained optimum:

1 2

1 2

*

*

* * * *1 2

max , , ,

. . , , ,

0

, , ,

n

n

i ii ii

n

f x x x

s t g x x x

Lf g

x xxL f g

Lg

y x x x

– Again, differentiating the optimum with respect to , we get

To work this out, we also differentiate the cost function with respect to :

* *

1

. .

. . .but 0

ni

i i

i i i

y f x x f

x

f f g

x x x

* * *1 2

*

1

, , ,

. . .

n

ni

i i

g x x x

g g x g

x

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

Thus, following the envelope theorem:

1 1 2 2

1 2

1 1 2 2 0 1 2

* **

1 11 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 Lx x w y

w w

– More explicitly,

* * ** 1 21 1 2

1 1 1

11

22

* * ** * 1 21

1 1 1 2 1

,

However, by first-order conditions

,

c w y x xx w w

w w w

fw

x

fw

x

c w y f x f xx

w x w x w

– However, differentiating the constraint of the minimization problem, we see

Thus, the second term in the preceding equation is zero and we have demonstrated Shephard’s lemma.

* ** * 0 1 2

0 1 21 1 1 2 2

. ., 0

f fy x xy f x x

w x w x w