Cost Minimization

Chapter Twenty

Given w1, w2 and y, how is the least costly

input bundle located?

How can the conditional factor demands be derived?

And how is the total cost function computed?

The Cost-Minimization Problem

The Cost-Minimization Problem

.MPMP

TRSww

2

1

2

1

The first-order condition for cost minimization is thus

Before we have had pMP1 = w1

pMP2 = w2

Eliminating p yields MP1/ MP2 = w1/ w2

Deriving Cost Functions Solving the problem of cost minimization

yields the cost functions

Cost functions will depend on technology

A Cobb-Douglas Example of Cost Minimization A firm’s Cobb-Douglas production function

is

Input prices are w1 and w2.

What are the firm’s conditional input demand functions?

y f x x x x ( , ) ./ /1 2 1

1 322 3

A Cobb-Douglas Example of Cost Minimization

At the input bundle (x1*,x2*) which minimizesthe cost of producing y output units:

(a)

(b) - w1/w2 = - MP1/MP2

y x x( ) ( )* / * /11 3

22 3 an

d

A Cobb-Douglas Example of Cost Minimization

At the input bundle (x1*,x2*) which minimizesthe cost of producing y output units:

(a)

(b) - w1/w2 = - MP1/MP2

y x x( ) ( )* / * /11 3

22 3 an

d

2/3*2

2/3*1

11 (x1/3(x

xy

MP ))

1/3*2

1/3*1

22 (x2/3(x

xy

MP ))

A Cobb-Douglas Example of Cost Minimization

At the input bundle (x1*,x2*) which minimizesthe cost of producing y output units:

(a)

(b) - w1/w2 = - MP1/MP2

y x x( ) ( )* / * /11 3

22 3 an

d

.2xx

)(x)(2/3)(x)(x)(1/3)(x

ww

*

1

*

2

1/3*

2

1/3*

1

2/3*

2

2/3*

1

2

1

A Cobb-Douglas Example of Cost Minimizationy x x( ) ( )* / * /

11 3

22 3 w

wx

x1

2

2

12

*

*.(a) (b)

From (b), xww

x21

21

2* * .

Now substitute into (a) to get

y xww

xww

x

( ) .* / */ /

*11 3 1

21

2 31

2

2 3

12 2

xww

y12

1

2 3

2*

/

So is the firm’s conditionaldemand for input 1.

A Cobb-Douglas Example of Cost Minimization

xww

x21

21

2* * xww

y12

1

2 3

2*

/

is the firm’s conditional demand for input 2.

Since and

xww

ww

yww

y21

2

2

1

2 31

2

1 32

22*

/ /

A Cobb-Douglas Example of Cost Minimization

So the cheapest input bundle yielding y output units is

x w w y x w w y

ww

yww

y

1 1 2 2 1 2

2

1

2 31

2

1 3

22

* *

/ /

( , , ), ( , , )

, .

A Cobb-Douglas Example of Cost Minimization

So the firm’s total cost function isc w w y w x w w y w x w w y

www

y www

y

w w y w w y

w wy

( , , ) ( , , ) ( , , )

.

* *

/ /

// / / / /

/

1 2 1 1 1 2 2 2 1 2

12

1

2 3

21

2

1 3

2 3

11 3

22 3 1 3

11 3

22 3

1 22 1 3

22

12

2

34