Comparative Statics and Duality of the Cost Function

Lecture VII

Comparative Statics Comparative statics with respect to changes in input prices.

The most common results of the comparative statics with respect to input prices involve intuition about derived demand functions.

From the primal approach, we expect the demand functions for each input to be downward sloping with respect to input prices.

Starting from the cost function:

By Shephard’s lemma. In addition, we know that if i=j then by the concavity of the

cost function in input prices:

2 *, ,i

i j j

c w y x w yw w w

2 *, ,0i

i i i

c w y x w yw w w

In addition, we know that by Young’s theorem, the Hessian matrix for the cost function is symmetric:

The Hessian matrix for the cost function is also singular.

2 2, ,

i j j i

c w y c w yw w w w

Euler’s Theorem: Euler’s theorem is based on the definition of homogeneity:

Differentiating both sides with respect to t and applying the chain rule yields:

rf tx t f x

1

1

1

1

Ni r

i i i

Nr

ii i

f tx txrt f x

tx x

f txx rt f x

tx

Letting t=1 then yields

Coupling this result with the observation that if a function is homogeneous of degree r, then its derivative is homogeneous of degree r-1. We know that the input demand functions are homogeneous of degree zero in prices. Thus,

1

N

ii i

f xx rf x

x

* 2

1 1

, ,0

N Ni

j jj jj i j

x w y c w yw w

w w w

Multiplying this expression by xi*(w,y) yields

Given that we know that ii< 0, this result imposes restrictions on the cross-price elasticities.

*

*1 1

,0

,

N Nji

iji ij i

wx w yw x w y

Briefly, let us prove the homogeneity of the marginal cost function. Starting with the marginal cost, differentiate with respect to each price:

2

1 1

*

1

*

1

2

1

, , ,

,

, ,

,,

N N

i ii ii i

N

i ii

N

i ii

N

ii i

c w y c w y c w yw w

y w y y w

x w y wy

c w y x w y w

c w yw c w y

w y y

Comparative statics with respect to output levels.

Following from the restrictions on the cross price elasticities above, the comparative statics with respect to output levels imply that not all inputs can be inferior or regressive.

An inferior input is an input whose use declines as production increases while the use of a normal input increases as production increases.

Like the cross-price results above, we start with the sum of the differences of individual demand functions with respect to the level of output:

* 2

1 1

, ,

,

N Ni

i ii i i

x w y c w yw w

y w y

c w yy

In order to develop the effect of output on total cost, we start with the original Lagrangian from the primal problem:

Solving for the output using the first-order condition on the Lagrange multiplier and differentiating the solution then yields:

0

0

x ii

f xL w

xL w x y f xL y f x

1

N

ii i

f xy f x dy dx

x

From the first set of first-order conditions, we see

i

i

f x wx

1 1 1

N N Ni i

i i i ii i i

w dxdy dx dy w dx wdy

Which proves the definition of consistent with the envelope theory. Therefore,

Given this optimum , we can then sum over the initial first-order conditions:

**

1

,,

Ni

ii

dx w yw y w

dy

*

1 1

,, ,

N N

i i ii i i

f x w yw x w y x w y

x

Introducing the notion of an elasticity of scale:

*

1

*

1

1

*

,, ,

,

, ( , ),

,,,

,

,

Ni i

i

N

i ii

w x w yy x w y

w y y

w x w y c w yc w y

c w yc w y y w yy y

n y w

y w

Chambers defines

as the cost flexibility (the ratio between the marginal and average costs).

,ln ,

,, ln

c w y yy c w yn y w

c w y y

Remember the elasticity of scale in the production function:

which is the ratio between the marginal physical and average physical products.

1 1

1

ln .ln

N Ni

ii ii

f x f xx y

What is developed here is not quite the same, but is actually the elasticity of size.

It answers the question: Do I build one large plant or several small ones?

Defining y*=y/m

is related to the homogeneity of the cost function in terms of scale. Taking

*, ,m y m

*, ,* *, , ,

m y wc w y c w my m c w y

* *

1lim , , ,m

m y w n y w

If n(y*,w) > 1 then (y,x(w,y)) < 1 there are no efficiencies to centralization {diseconomies of scale}.

If n(y*,w) < 1 then (y,x(w,y)) > 1 there are efficiencies to centralization {economies of scale}.

( , )c w y

y

Just like the MPP-APP comparison, we can envision the ratio be marginal cost and average cost. It is clear that

Also, it is apparent that average cost equals marginal cost at the minimum of the average cost curve:

,, ,

, 1 1,

c w yc w y c w yyn y w

c w y y yy

,, , , ,1 1 1 0

c w yc w y c w y c w y c w yy

y y y y y y y y

Duality between the Cost and Production Functions

In our discussion of the primal we demonstrated how the production function placed restrictions on economic behavior. The question posed in duality is whether the optimizing behavior can be used to recover or reconstruct the properties of the production function.

Minkowski’s Theorem: A closed, convex set is the intersection of half-spaces that support it:

The half space H(m,k) is defined as

*X H

, :H m k x m x k

Thus, based on the definition of a cost function we have:

This definition actually recovers the original production set:

, : ,N w y x w x c w y

* : , 0V y x w x c w y w

If V*(y)=V(y), the original technology can be recovered.