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Chapter Twenty
Cost Minimization
Cost Minimization
A firm is a cost-minimizer if it
produces any given output level y ≥
0 at smallest possible total cost.
c(y) denotes the firm’s smallest
possible total cost for producing y
units of output.
c(y) is the firm’s total cost function.
Cost Minimization
When the firm faces given input
prices w = (w1,w2,…,wn) the total cost
function will be written as
c(w1,…,wn,y).
The Cost-Minimization Problem
Consider a firm using two inputs to
make one output.
The production function is
y = f(x1,x2).
Take the output level y ≥ 0 as given.
Given the input prices w 1 and w2, the
cost of an input bundle (x1,x2) is
w 1 x1 + w 2 x 2 .
The Cost-Minimization Problem
For given w1, w2 and y, the firm’s
cost-minimization problem is to
min w 1x1 + w 2x 2
solve
x1 ,x 2 ≥ 0
subject to f ( x1 , x 2 ) = y.
The Cost-Minimization Problem
The levels x1*(w1,w2,y) and x1*(w1,w2,y) in
the least-costly input bundle are the
firm’s conditional demands for inputs
1 and 2.
The (smallest possible) total cost for
producing y output units is therefore
*
c( w 1 , w 2 , y ) = w 1x1 ( w 1 , w 2 , y )
*
+ w 2x 2 ( w 1 , w 2 , y ).
Conditional Input Demands
Given w1, w2 and y, how is the least
costly input bundle located?
And how is the total cost function
computed?
Iso-cost Lines
A curve that contains all of the input
bundles that cost the same amount
is an iso-cost curve.
E.g., given w1 and w2, the $100 isocost line has the equation

w 1x1 + w 2x 2 = 100.
Iso-cost Lines
Generally, given w1 and w2, the
equation of the $c iso-cost line is
w 1x1 + w 2x 2 = c
i.e.
w1
c
x2 = −
x1 +
.
w2
w2
Slope is - w1/w2.
Iso-cost Lines
x2

c” ≡ w1x1+w2x2
c’ ≡ w1x1+w2x2
c’ < c”
x1
Iso-cost Lines
x2

Slopes = -w1/w2.

c” ≡ w1x1+w2x2
c’ ≡ w1x1+w2x2
c’ < c”
x1
The y’-Output Unit Isoquant
x2

All input bundles yielding y’ units
of output. Which is the cheapest?

f(x1,x2) ≡ y’
x1
The Cost-Minimization Problem
x2

All input bundles yielding y’ units
of output. Which is the cheapest?

f(x1,x2) ≡ y’
x1
The Cost-Minimization Problem
x2

All input bundles yielding y’ units
of output. Which is the cheapest?

f(x1,x2) ≡ y’
x1
The Cost-Minimization Problem
x2

All input bundles yielding y’ units
of output. Which is the cheapest?

f(x1,x2) ≡ y’
x1
The Cost-Minimization Problem
x2

All input bundles yielding y’ units
of output. Which is the cheapest?

x2*
f(x1,x2) ≡ y’
x1*

x1
The Cost-Minimization Problem
x2

At an interior cost-min input bundle:
* *
(a) f ( x1 , x 2 ) = y ′

x2*
f(x1,x2) ≡ y’
x1*

x1
The Cost-Minimization Problem
x2

At an interior cost-min input bundle:
* *
(a) f ( x1 , x 2 ) = y ′ and
(b) slope of isocost = slope of
isoquant

x2*
f(x1,x2) ≡ y’
x1*

x1
The Cost-Minimization Problem
x2

At an interior cost-min input bundle:
* *
(a) f ( x1 , x 2 ) = y ′ and
(b) slope of isocost = slope of
isoquant; i.e.
w1
MP1
−
= TRS = −
at ( x* , x* ).
1 2
w2
MP2

x2*
f(x1,x2) ≡ y’
x1*

x1
A Cobb-Douglas Example of Cost
Minimization
A firm’s Cobb-Douglas production
function is
1/ 3 2/ 3
y = f ( x1 , x 2 ) = x1 x 2 .
Input prices are w1 and w2.
What are the firm’s conditional input
demand functions?
A Cobb-Douglas Example of Cost
Minimization
At the input bundle (x1*,x2*) which minimizes
the cost of producing y output units:
(a)
y = ( x* )1/ 3 ( x* ) 2/ 3
and
1
2
(b)
w1
∂ y / ∂ x1
(1 / 3)( x* ) −2/ 3 ( x* ) 2/ 3
1
2
−
=−
=−
w2
∂ y / ∂ x2
( 2 / 3)( x* )1/ 3 ( x* ) −1/ 3
1
2
*
x2
=−
.
*
2x1
A Cobb-Douglas Example of Cost
Minimization
* 1/ 3 * 2/ 3
(a) y = ( x1 ) ( x 2 )

w 1 x*
= 2 .
(b)
w 2 2x*
1
A Cobb-Douglas Example of Cost
Minimization
* 1/ 3 * 2/ 3
(a) y = ( x1 ) ( x 2 )

2w 1 *
*
x1 .
From (b), x 2 =
w2

w 1 x*
= 2 .
(b)
w 2 2x*
1
A Cobb-Douglas Example of Cost
Minimization
* 1/ 3 * 2/ 3
(a) y = ( x1 ) ( x 2 )

2w 1 *
*
x1 .
From (b), x 2 =
w2

w 1 x*
= 2 .
(b)
w 2 2x*
1

Now substitute into (a) to get
2/ 3
* 1/ 3  2w 1 * 
y = ( x1 ) 
x1 
 w2 
A Cobb-Douglas Example of Cost
Minimization
* 1/ 3 * 2/ 3
(a) y = ( x1 ) ( x 2 )

2w 1 *
*
x1 .
From (b), x 2 =
w2

w 1 x*
= 2 .
(b)
w 2 2x*
1

Now substitute into (a) to get
2/ 3
2/ 3
 2w 1 
* 1/ 3  2w 1 * 
*
y = ( x1 ) 
x1 
=
x1 .

 w2 
 w2 
A Cobb-Douglas Example of Cost
Minimization
* 1/ 3 * 2/ 3
(a) y = ( x1 ) ( x 2 )

2w 1 *
*
x1 .
From (b), x 2 =
w2

w 1 x*
= 2 .
(b)
w 2 2x*
1

Now substitute into (a) to get
2/ 3
2/ 3
 2w 1 
* 1/ 3  2w 1 * 
*
y = ( x1 ) 
x1 
=
x1 .

 w2 
 w2 
*  w2 
So x1 = 

 2w 1 

2/ 3

y is the firm’s conditional
demand for input 1.
A Cobb-Douglas Example of Cost
Minimization
2w 1 *
*  w2 
*
x1 and x1 = 
Since x 2 =

 2w 1 
w2
2/ 3
1/ 3
 2w 1 
2w 1  w 2 
*
x2 =
y=


 y
 w2 
w 2  2w 1 

2/ 3

y

is the firm’s conditional demand for input 2.
A Cobb-Douglas Example of Cost
Minimization
So the cheapest input bundle yielding y
output units is

(

*
*
x1 ( w 1 , w 2 , y ), x 2 ( w 1 , w 2 , y )

  w  2/ 3  2w  1/ 3 
1
=  2 
y, 
 y .
  2w 1 

 w2 



)
Conditional Input Demand Curves
x2

Fixed w1 and w2.

y′′′
y′′

y′

x1
Conditional Input Demand Curves
x2

y

Fixed w1 and w2.

y′
y x* ( y′ )
2
y′′′
y′′

x* ( y′ )
2

y′

x* ( y′ )
1

x*
2

y′

x1

x* ( y′ )
1

x*
1
Conditional Input Demand Curves
x2

y

Fixed w1 and w2.

y′′
y′

y x* ( y′ )
2

x* ( y′′ )
2

x* ( y′′ )
2
x* ( y′ )
2

y′′
y′

y′′′
y′′

y′

x* ( y′ )
1
x* ( y′′ )
1

x*
2

x1

x* ( y′ )
1
x* ( y′′ )
1

x*
1
Conditional Input Demand Curves
x2

y

Fixed w1 and w2.

y′′′

y′′
y′

y x* ( y′ ) x* ( y′′′ ) x*
2
2
2
*

x* ( y′′′ )
2
x* ( y′′ )
2
x* ( y′ )
2

y′′′

y′′
y′

y′′′
y′′

y′

x* ( y′ ) x* ( y′′′ )
1
1
x* ( y′′ )
1

x 2 ( y′′ )

x1

x* ( y′ ) x* ( y′′′ )
1
1
x* ( y′′ )
1

x*
1
Conditional Input Demand Curves
x2

y

Fixed w1 and w2.

y′′′

y′′
y′

output
expansion
path
x* ( y′′′ )
2
x* ( y′′ )
2
x* ( y′ )
2

y x* ( y′ ) x* ( y′′′ ) x*
2
2
2
*
y′′′

y′′
y′

y′′′
y′′

y′

x* ( y′ ) x* ( y′′′ )
1
1
x* ( y′′ )
1

x 2 ( y′′ )

x1

x* ( y′ ) x* ( y′′′ )
1
1
x* ( y′′ )
1

x*
1
Conditional Input Demand Curves
x2

y

Fixed w1 and w2.

y′′′

y′′
y′

output
expansion
path
x* ( y′′′ )
2
x* ( y′′ )
2
x* ( y′ )
2

y x* ( y′ ) x* ( y′′′ ) x*
2
2
2
*
y′′′

y′′′
y′′

y′

x* ( y′ ) x* ( y′′′ )
1
1
x* ( y′′ )
1

Cond. demand
for
input 2

x1

x 2 ( y′′ )

Cond.
demand
y′′
for
y′
input 1
x* ( y′ ) x* ( y′′′ ) x*
1
1
1
x* ( y′′ )
1
A Cobb-Douglas Example of Cost
Minimization
For the production function

y = f ( x1 , x 2 ) = x1 / 3x 2 / 3
1
2

the cheapest input bundle yielding y output
units is

(

x* ( w 1 , w 2 , y ), x* ( w 1 , w 2 , y )
1
2

  w  2/ 3  2w  1/ 3 
1
=  2 
y, 
 y .
  2w 1 

 w2 



)
A Cobb-Douglas Example of Cost
Minimization
So the firm’s total cost function is
c( w 1 , w 2 , y ) = w 1x* ( w 1 , w 2 , y ) + w 2x* ( w 1 , w 2 , y )
1
2
A Cobb-Douglas Example of Cost
Minimization
So the firm’s total cost function is
*
*
c( w 1 , w 2 , y ) = w 1x1 ( w 1 , w 2 , y ) + w 2x 2 ( w 1 , w 2 , y )
2/ 3
1/ 3
 w2 
 2w 1 
= w1 
y + w2

 y
 2w 1 
 w2 
A Cobb-Douglas Example of Cost
Minimization
So the firm’s total cost function is
c( w 1 , w 2 , y ) = w 1x* ( w 1 , w 2 , y ) + w 2x* ( w 1 , w 2 , y )
1
2
 w2 
= w1 

 2w 1 
 1
= 
 2

2/ 3

2/ 3

 2w 1 
y + w2

 w2 

1/ 3

y

w 1/ 3 w 2/ 3 y + 21/ 3 w 1/ 3 w 2/ 3 y
1
2
1
2
A Cobb-Douglas Example of Cost
Minimization
So the firm’s total cost function is
c( w 1 , w 2 , y ) = w 1x* ( w 1 , w 2 , y ) + w 2x* ( w 1 , w 2 , y )
1
2
 w2 
= w1 

 2w 1 
 1
= 
 2

2/ 3

2/ 3

 2w 1 
y + w2

 w2 

4

y

w 1/ 3 w 2/ 3 y + 21/ 3 w 1/ 3 w 2/ 3 y
1
2
1
2

2  1/ 3
 w 1w 2
 y.
= 3





1/ 3


A Perfect Complements Example of
Cost Minimization
The firm’s production function is

y = min{ 4x1 , x 2 }.
Input prices w1 and w2 are given.
What are the firm’s conditional
demands for inputs 1 and 2?
What is the firm’s total cost
function?
A Perfect Complements Example of
Cost Minimization
x2

4x1 = x2

min{4x1,x2} ≡ y’

x1
A Perfect Complements Example of
Cost Minimization
x2

4x1 = x2

min{4x1,x2} ≡ y’

x1
A Perfect Complements Example of
Cost Minimization
x2

4x1 = x2 Where is the least costly

input bundle yielding
y’ output units?

min{4x1,x2} ≡ y’

x1
A Perfect Complements Example of
Cost Minimization
x2

4x1 = x2 Where is the least costly

input bundle yielding
y’ output units?

min{4x1,x2} ≡ y’

x2* = y

x1*
= y/4

x1
A Perfect Complements Example of
Cost Minimization
The firm’s production function is

y = min{ 4x1 , x 2 }

and the conditional input demands are
y
*
*
x1 ( w 1 , w 2 , y ) =
and x 2 ( w 1 , w 2 , y ) = y.

4
A Perfect Complements Example of
Cost Minimization
The firm’s production function is

y = min{ 4x1 , x 2 }

and the conditional input demands are
y
*
*
x1 ( w 1 , w 2 , y ) =
and x 2 ( w 1 , w 2 , y ) = y.

4

So the firm’s total cost function is
*
c( w 1 , w 2 , y ) = w 1x1 ( w 1 , w 2 , y )

*
+ w 2x 2 ( w 1 , w 2 , y )
A Perfect Complements Example of
Cost Minimization
The firm’s production function is

y = min{ 4x1 , x 2 }

and the conditional input demands are
y
*
*
x1 ( w 1 , w 2 , y ) =
and x 2 ( w 1 , w 2 , y ) = y.

4

So the firm’s total cost function is
*
c( w 1 , w 2 , y ) = w 1x1 ( w 1 , w 2 , y )

+ w 2x* ( w 1 , w 2 , y )
2
y
 w1

= w1 + w 2y = 
+ w 2  y.
 4

4
Average Total Production Costs
For positive output levels y, a firm’s
average total cost of producing y
units is
c( w 1 , w 2 , y )
AC( w 1 , w 2 , y ) =
.

y
Returns-to-Scale and Av. Total Costs
The returns-to-scale properties of a
firm’s technology determine how
average production costs change with
output level.
Our firm is presently producing y’
output units.
How does the firm’s average
production cost change if it instead
produces 2y’ units of output?
Constant Returns-to-Scale and Average
Total Costs
If a firm’s technology exhibits
constant returns-to-scale then
doubling its output level from y’ to
2y’ requires doubling all input levels.
Constant Returns-to-Scale and Average
Total Costs
If a firm’s technology exhibits
constant returns-to-scale then
doubling its output level from y’ to
2y’ requires doubling all input levels.
Total production cost doubles.
Constant Returns-to-Scale and Average
Total Costs
If a firm’s technology exhibits
constant returns-to-scale then
doubling its output level from y’ to
2y’ requires doubling all input levels.
Total production cost doubles.
Average production cost does not
change.
Decreasing Returns-to-Scale and
Average Total Costs
If a firm’s technology exhibits
decreasing returns-to-scale then
doubling its output level from y’ to
2y’ requires more than doubling all
input levels.
Decreasing Returns-to-Scale and
Average Total Costs
If a firm’s technology exhibits
decreasing returns-to-scale then
doubling its output level from y’ to
2y’ requires more than doubling all
input levels.
Total production cost more than
doubles.
Decreasing Returns-to-Scale and
Average Total Costs
If a firm’s technology exhibits
decreasing returns-to-scale then
doubling its output level from y’ to
2y’ requires more than doubling all
input levels.
Total production cost more than
doubles.
Average production cost increases.
Increasing Returns-to-Scale and
Average Total Costs
If a firm’s technology exhibits
increasing returns-to-scale then
doubling its output level from y’ to
2y’ requires less than doubling all
input levels.
Increasing Returns-to-Scale and
Average Total Costs
If a firm’s technology exhibits
increasing returns-to-scale then
doubling its output level from y’ to
2y’ requires less than doubling all
input levels.
Total production cost less than
doubles.
Increasing Returns-to-Scale and
Average Total Costs
If a firm’s technology exhibits
increasing returns-to-scale then
doubling its output level from y’ to
2y’ requires less than doubling all
input levels.
Total production cost less than
doubles.
Average production cost decreases.
Returns-to-Scale and Av. Total Costs
$/output unit
AC(y)

decreasing r.t.s.
constant r.t.s.
increasing r.t.s.
y
Returns-to-Scale and Total Costs
What does this imply for the shapes
of total cost functions?
Returns-to-Scale and Total Costs
$

Av. cost increases with y if the firm’s
technology exhibits decreasing r.t.s.

c(2y’)

Slope = c(2y’)/2y’
= AC(2y’).
Slope = c(y’)/y’
= AC(y’).

c(y’)
y’

2y’

y
Returns-to-Scale and Total Costs
$

Av. cost increases with y if the firm’s
technology exhibits decreasing r.t.s.
c(y)

c(2y’)

Slope = c(2y’)/2y’
= AC(2y’).
Slope = c(y’)/y’
= AC(y’).

c(y’)
y’

2y’

y
Returns-to-Scale and Total Costs
$
c(2y’)

Av. cost decreases with y if the firm’s
technology exhibits increasing r.t.s.
Slope = c(2y’)/2y’
= AC(2y’).
Slope = c(y’)/y’
= AC(y’).

c(y’)

y’

2y’

y
Returns-to-Scale and Total Costs
$
c(2y’)

Av. cost decreases with y if the firm’s
technology exhibits increasing r.t.s.
c(y)
Slope = c(2y’)/2y’
= AC(2y’).
Slope = c(y’)/y’
= AC(y’).

c(y’)

y’

2y’

y
Returns-to-Scale and Total Costs
c(2y’)
=2c(y’)

$

Av. cost is constant when the firm’s
technology exhibits constant r.t.s.
c(y)
Slope = c(2y’)/2y’
= 2c(y’)/2y’
= c(y’)/y’
so
AC(y’) = AC(2y’).

c(y’)

y’

2y’

y
Short-Run & Long-Run Total Costs
In the long-run a firm can vary all of
its input levels.
Consider a firm that cannot change
its input 2 level from x2’ units.
How does the short-run total cost of
producing y output units compare to
the long-run total cost of producing y
units of output?
Short-Run & Long-Run Total Costs
The long-run cost-minimization
problem is min w 1x1 + w 2x 2
x1 ,x 2 ≥ 0
subject to f ( x1 , x 2 ) = y.
The short-run cost-minimization
problem is min w 1x1 + w 2x ′
2
x1 ≥ 0
subject to f ( x1 , x ′ ) = y.
2
Short-Run & Long-Run Total Costs
The short-run cost-min. problem is the
long-run problem subject to the extra
constraint that x2 = x2’.
If the long-run choice for x2 was x2’
then the extra constraint x2 = x2’ is not
really a constraint at all and so the
long-run and short-run total costs of
producing y output units are the same.
Short-Run & Long-Run Total Costs
The short-run cost-min. problem is
therefore the long-run problem subject
to the extra constraint that x2 = x2”.
But, if the long-run choice for x2 ≠ x2”
then the extra constraint x2 = x2”
prevents the firm in this short-run from
achieving its long-run production cost,
causing the short-run total cost to
exceed the long-run total cost of
producing y output units.
Short-Run & Long-Run Total Costs
x2

y ′′′

Consider three output levels.

y ′′
y′

x1
Short-Run & Long-Run Total Costs
x2

y ′′′
y ′′
y′

In the long-run when the firm
is free to choose both x1 and
x2, the least-costly input
bundles are ...

x1
Short-Run & Long-Run Total Costs
x2

y ′′′
y ′′
y′

Long-run
output
expansion
path

x ′′′
2
x ′′
2
x′
2

x1 x1 x1
′ ′′ ′′′

x1
Short-Run & Long-Run Total Costs
x2

Long-run costs are:

y ′′′
y ′′
y′

Long-run
output
expansion
path

c( y ′ ) = w 1x1 + w 2x ′
′
2
c( y ′′ ) = w 1x1 + w 2x ′′′
′′
2
c( y ′′′ ) = w 1x1 w 2x ′′′
′′′+
2

x ′′′
2
x ′′
2
x′
2

x1 x1 x1
′ ′′ ′′′

x1
Short-Run & Long-Run Total Costs
Now suppose the firm becomes
subject to the short-run constraint
that x2 = x2”.
Short-Run & Long-Run Total Costs
x2

y ′′′ Short-run
y ′′
y′

output
expansion
path

Long-run costs are:
c( y ′ ) = w 1x1 + w 2x ′
′
2
c( y ′′ ) = w 1x1 + w 2x ′′′
′′
2
c( y ′′′ ) = w 1x1 w 2x ′′′
′′′+
2

x ′′′
2
x ′′
2
x′
2

x1 x1 x1
′ ′′ ′′′

x1
Short-Run & Long-Run Total Costs
x2

y ′′′ Short-run
y ′′
y′

output
expansion
path

Long-run costs are:
c( y ′ ) = w 1x1 + w 2x ′
′
2
c( y ′′ ) = w 1x1 + w 2x ′′′
′′
2
c( y ′′′ ) = w 1x1 w 2x ′′′
′′′+
2

x ′′′
2
x ′′
2
x′
2

x1 x1 x1
′ ′′ ′′′

x1
Short-Run & Long-Run Total Costs
x2

y ′′′ Short-run
y ′′
y′

output
expansion
path

Long-run costs are:
c( y ′ ) = w 1x1 + w 2x ′
′
2
c( y ′′ ) = w 1x1 + w 2x ′′′
′′
2
c( y ′′′ ) = w 1x1 w 2x ′′′
′′′+
2

Short-run costs are:
c s ( y ′ ) > c( y ′ )

x ′′′
2
x ′′
2
x′
2

x1 x1 x1
′ ′′ ′′′

x1
Short-Run & Long-Run Total Costs
x2

y ′′′ Short-run
y ′′
y′

output
expansion
path

Long-run costs are:
c( y ′ ) = w 1x1 + w 2x ′
′
2
c( y ′′ ) = w 1x1 + w 2x ′′′
′′
2
c( y ′′′ ) = w 1x1 w 2x ′′′
′′′+
2

Short-run costs are:
c s ( y ′ ) > c( y ′ )
cs ( y ′′ ) = c( y ′′ )

x ′′′
2
x ′′
2
x′
2

x1 x1 x1
′ ′′ ′′′

x1
Short-Run & Long-Run Total Costs
x2

y ′′′ Short-run
y ′′
y′

output
expansion
path

Long-run costs are:
c( y ′ ) = w 1x1 + w 2x ′
′
2
c( y ′′ ) = w 1x1 + w 2x ′′′
′′
2
c( y ′′′ ) = w 1x1 w 2x ′′′
′′′+
2

Short-run costs are:
c s ( y ′ ) > c( y ′ )
cs ( y ′′ ) = c( y ′′ )

x ′′′
2
x ′′
2
x′
2

x1 x1 x1
′ ′′ ′′′

x1
Short-Run & Long-Run Total Costs
x2

y ′′′ Short-run
y ′′
y′

output
expansion
path

Long-run costs are:
c( y ′ ) = w 1x1 + w 2x ′
′
2
c( y ′′ ) = w 1x1 + w 2x ′′′
′′
2
c( y ′′′ ) = w 1x1 w 2x ′′′
′′′+
2

Short-run costs are:
c s ( y ′ ) > c( y ′ )
c s ( y ′′ ) = c( y ′′ )
c s ( y ′′′ ) > c( y ′′′ )

x ′′′
2
x ′′
2
x′
2

x1 x1 x1
′ ′′ ′′′

x1
Short-Run & Long-Run Total Costs
Short-run total cost exceeds longrun total cost except for the output
level where the short-run input level
restriction is the long-run input level
choice.
This says that the long-run total cost
curve always has one point in
common with any particular shortrun total cost curve.
Short-Run & Long-Run Total Costs
A short-run total cost curve always has
$
one point in common with the long-run
total cost curve, and is elsewhere higher
than the long-run total cost curve.
cs(y)
c(y)

F=
w 2x ′′
2

y′

y ′′

y ′′′ y

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Ch20

  • 2. Cost Minimization A firm is a cost-minimizer if it produces any given output level y ≥ 0 at smallest possible total cost. c(y) denotes the firm’s smallest possible total cost for producing y units of output. c(y) is the firm’s total cost function.
  • 3. Cost Minimization When the firm faces given input prices w = (w1,w2,…,wn) the total cost function will be written as c(w1,…,wn,y).
  • 4. The Cost-Minimization Problem Consider a firm using two inputs to make one output. The production function is y = f(x1,x2). Take the output level y ≥ 0 as given. Given the input prices w 1 and w2, the cost of an input bundle (x1,x2) is w 1 x1 + w 2 x 2 .
  • 5. The Cost-Minimization Problem For given w1, w2 and y, the firm’s cost-minimization problem is to min w 1x1 + w 2x 2 solve x1 ,x 2 ≥ 0 subject to f ( x1 , x 2 ) = y.
  • 6. The Cost-Minimization Problem The levels x1*(w1,w2,y) and x1*(w1,w2,y) in the least-costly input bundle are the firm’s conditional demands for inputs 1 and 2. The (smallest possible) total cost for producing y output units is therefore * c( w 1 , w 2 , y ) = w 1x1 ( w 1 , w 2 , y ) * + w 2x 2 ( w 1 , w 2 , y ).
  • 7. Conditional Input Demands Given w1, w2 and y, how is the least costly input bundle located? And how is the total cost function computed?
  • 8. Iso-cost Lines A curve that contains all of the input bundles that cost the same amount is an iso-cost curve. E.g., given w1 and w2, the $100 isocost line has the equation w 1x1 + w 2x 2 = 100.
  • 9. Iso-cost Lines Generally, given w1 and w2, the equation of the $c iso-cost line is w 1x1 + w 2x 2 = c i.e. w1 c x2 = − x1 + . w2 w2 Slope is - w1/w2.
  • 10. Iso-cost Lines x2 c” ≡ w1x1+w2x2 c’ ≡ w1x1+w2x2 c’ < c” x1
  • 11. Iso-cost Lines x2 Slopes = -w1/w2. c” ≡ w1x1+w2x2 c’ ≡ w1x1+w2x2 c’ < c” x1
  • 12. The y’-Output Unit Isoquant x2 All input bundles yielding y’ units of output. Which is the cheapest? f(x1,x2) ≡ y’ x1
  • 13. The Cost-Minimization Problem x2 All input bundles yielding y’ units of output. Which is the cheapest? f(x1,x2) ≡ y’ x1
  • 14. The Cost-Minimization Problem x2 All input bundles yielding y’ units of output. Which is the cheapest? f(x1,x2) ≡ y’ x1
  • 15. The Cost-Minimization Problem x2 All input bundles yielding y’ units of output. Which is the cheapest? f(x1,x2) ≡ y’ x1
  • 16. The Cost-Minimization Problem x2 All input bundles yielding y’ units of output. Which is the cheapest? x2* f(x1,x2) ≡ y’ x1* x1
  • 17. The Cost-Minimization Problem x2 At an interior cost-min input bundle: * * (a) f ( x1 , x 2 ) = y ′ x2* f(x1,x2) ≡ y’ x1* x1
  • 18. The Cost-Minimization Problem x2 At an interior cost-min input bundle: * * (a) f ( x1 , x 2 ) = y ′ and (b) slope of isocost = slope of isoquant x2* f(x1,x2) ≡ y’ x1* x1
  • 19. The Cost-Minimization Problem x2 At an interior cost-min input bundle: * * (a) f ( x1 , x 2 ) = y ′ and (b) slope of isocost = slope of isoquant; i.e. w1 MP1 − = TRS = − at ( x* , x* ). 1 2 w2 MP2 x2* f(x1,x2) ≡ y’ x1* x1
  • 20. A Cobb-Douglas Example of Cost Minimization A firm’s Cobb-Douglas production function is 1/ 3 2/ 3 y = f ( x1 , x 2 ) = x1 x 2 . Input prices are w1 and w2. What are the firm’s conditional input demand functions?
  • 21. A Cobb-Douglas Example of Cost Minimization At the input bundle (x1*,x2*) which minimizes the cost of producing y output units: (a) y = ( x* )1/ 3 ( x* ) 2/ 3 and 1 2 (b) w1 ∂ y / ∂ x1 (1 / 3)( x* ) −2/ 3 ( x* ) 2/ 3 1 2 − =− =− w2 ∂ y / ∂ x2 ( 2 / 3)( x* )1/ 3 ( x* ) −1/ 3 1 2 * x2 =− . * 2x1
  • 22. A Cobb-Douglas Example of Cost Minimization * 1/ 3 * 2/ 3 (a) y = ( x1 ) ( x 2 ) w 1 x* = 2 . (b) w 2 2x* 1
  • 23. A Cobb-Douglas Example of Cost Minimization * 1/ 3 * 2/ 3 (a) y = ( x1 ) ( x 2 ) 2w 1 * * x1 . From (b), x 2 = w2 w 1 x* = 2 . (b) w 2 2x* 1
  • 24. A Cobb-Douglas Example of Cost Minimization * 1/ 3 * 2/ 3 (a) y = ( x1 ) ( x 2 ) 2w 1 * * x1 . From (b), x 2 = w2 w 1 x* = 2 . (b) w 2 2x* 1 Now substitute into (a) to get 2/ 3 * 1/ 3  2w 1 *  y = ( x1 )  x1   w2 
  • 25. A Cobb-Douglas Example of Cost Minimization * 1/ 3 * 2/ 3 (a) y = ( x1 ) ( x 2 ) 2w 1 * * x1 . From (b), x 2 = w2 w 1 x* = 2 . (b) w 2 2x* 1 Now substitute into (a) to get 2/ 3 2/ 3  2w 1  * 1/ 3  2w 1 *  * y = ( x1 )  x1  = x1 .   w2   w2 
  • 26. A Cobb-Douglas Example of Cost Minimization * 1/ 3 * 2/ 3 (a) y = ( x1 ) ( x 2 ) 2w 1 * * x1 . From (b), x 2 = w2 w 1 x* = 2 . (b) w 2 2x* 1 Now substitute into (a) to get 2/ 3 2/ 3  2w 1  * 1/ 3  2w 1 *  * y = ( x1 )  x1  = x1 .   w2   w2  *  w2  So x1 =    2w 1  2/ 3 y is the firm’s conditional demand for input 1.
  • 27. A Cobb-Douglas Example of Cost Minimization 2w 1 * *  w2  * x1 and x1 =  Since x 2 =   2w 1  w2 2/ 3 1/ 3  2w 1  2w 1  w 2  * x2 = y=    y  w2  w 2  2w 1  2/ 3 y is the firm’s conditional demand for input 2.
  • 28. A Cobb-Douglas Example of Cost Minimization So the cheapest input bundle yielding y output units is ( * * x1 ( w 1 , w 2 , y ), x 2 ( w 1 , w 2 , y )   w  2/ 3  2w  1/ 3  1 =  2  y,   y .   2w 1    w2    )
  • 29. Conditional Input Demand Curves x2 Fixed w1 and w2. y′′′ y′′ y′ x1
  • 30. Conditional Input Demand Curves x2 y Fixed w1 and w2. y′ y x* ( y′ ) 2 y′′′ y′′ x* ( y′ ) 2 y′ x* ( y′ ) 1 x* 2 y′ x1 x* ( y′ ) 1 x* 1
  • 31. Conditional Input Demand Curves x2 y Fixed w1 and w2. y′′ y′ y x* ( y′ ) 2 x* ( y′′ ) 2 x* ( y′′ ) 2 x* ( y′ ) 2 y′′ y′ y′′′ y′′ y′ x* ( y′ ) 1 x* ( y′′ ) 1 x* 2 x1 x* ( y′ ) 1 x* ( y′′ ) 1 x* 1
  • 32. Conditional Input Demand Curves x2 y Fixed w1 and w2. y′′′ y′′ y′ y x* ( y′ ) x* ( y′′′ ) x* 2 2 2 * x* ( y′′′ ) 2 x* ( y′′ ) 2 x* ( y′ ) 2 y′′′ y′′ y′ y′′′ y′′ y′ x* ( y′ ) x* ( y′′′ ) 1 1 x* ( y′′ ) 1 x 2 ( y′′ ) x1 x* ( y′ ) x* ( y′′′ ) 1 1 x* ( y′′ ) 1 x* 1
  • 33. Conditional Input Demand Curves x2 y Fixed w1 and w2. y′′′ y′′ y′ output expansion path x* ( y′′′ ) 2 x* ( y′′ ) 2 x* ( y′ ) 2 y x* ( y′ ) x* ( y′′′ ) x* 2 2 2 * y′′′ y′′ y′ y′′′ y′′ y′ x* ( y′ ) x* ( y′′′ ) 1 1 x* ( y′′ ) 1 x 2 ( y′′ ) x1 x* ( y′ ) x* ( y′′′ ) 1 1 x* ( y′′ ) 1 x* 1
  • 34. Conditional Input Demand Curves x2 y Fixed w1 and w2. y′′′ y′′ y′ output expansion path x* ( y′′′ ) 2 x* ( y′′ ) 2 x* ( y′ ) 2 y x* ( y′ ) x* ( y′′′ ) x* 2 2 2 * y′′′ y′′′ y′′ y′ x* ( y′ ) x* ( y′′′ ) 1 1 x* ( y′′ ) 1 Cond. demand for input 2 x1 x 2 ( y′′ ) Cond. demand y′′ for y′ input 1 x* ( y′ ) x* ( y′′′ ) x* 1 1 1 x* ( y′′ ) 1
  • 35. A Cobb-Douglas Example of Cost Minimization For the production function y = f ( x1 , x 2 ) = x1 / 3x 2 / 3 1 2 the cheapest input bundle yielding y output units is ( x* ( w 1 , w 2 , y ), x* ( w 1 , w 2 , y ) 1 2   w  2/ 3  2w  1/ 3  1 =  2  y,   y .   2w 1    w2    )
  • 36. A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is c( w 1 , w 2 , y ) = w 1x* ( w 1 , w 2 , y ) + w 2x* ( w 1 , w 2 , y ) 1 2
  • 37. A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is * * c( w 1 , w 2 , y ) = w 1x1 ( w 1 , w 2 , y ) + w 2x 2 ( w 1 , w 2 , y ) 2/ 3 1/ 3  w2   2w 1  = w1  y + w2   y  2w 1   w2 
  • 38. A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is c( w 1 , w 2 , y ) = w 1x* ( w 1 , w 2 , y ) + w 2x* ( w 1 , w 2 , y ) 1 2  w2  = w1    2w 1   1 =   2 2/ 3 2/ 3  2w 1  y + w2   w2  1/ 3 y w 1/ 3 w 2/ 3 y + 21/ 3 w 1/ 3 w 2/ 3 y 1 2 1 2
  • 39. A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is c( w 1 , w 2 , y ) = w 1x* ( w 1 , w 2 , y ) + w 2x* ( w 1 , w 2 , y ) 1 2  w2  = w1    2w 1   1 =   2 2/ 3 2/ 3  2w 1  y + w2   w2  4 y w 1/ 3 w 2/ 3 y + 21/ 3 w 1/ 3 w 2/ 3 y 1 2 1 2 2  1/ 3  w 1w 2  y. = 3    1/ 3 
  • 40. A Perfect Complements Example of Cost Minimization The firm’s production function is y = min{ 4x1 , x 2 }. Input prices w1 and w2 are given. What are the firm’s conditional demands for inputs 1 and 2? What is the firm’s total cost function?
  • 41. A Perfect Complements Example of Cost Minimization x2 4x1 = x2 min{4x1,x2} ≡ y’ x1
  • 42. A Perfect Complements Example of Cost Minimization x2 4x1 = x2 min{4x1,x2} ≡ y’ x1
  • 43. A Perfect Complements Example of Cost Minimization x2 4x1 = x2 Where is the least costly input bundle yielding y’ output units? min{4x1,x2} ≡ y’ x1
  • 44. A Perfect Complements Example of Cost Minimization x2 4x1 = x2 Where is the least costly input bundle yielding y’ output units? min{4x1,x2} ≡ y’ x2* = y x1* = y/4 x1
  • 45. A Perfect Complements Example of Cost Minimization The firm’s production function is y = min{ 4x1 , x 2 } and the conditional input demands are y * * x1 ( w 1 , w 2 , y ) = and x 2 ( w 1 , w 2 , y ) = y. 4
  • 46. A Perfect Complements Example of Cost Minimization The firm’s production function is y = min{ 4x1 , x 2 } and the conditional input demands are y * * x1 ( w 1 , w 2 , y ) = and x 2 ( w 1 , w 2 , y ) = y. 4 So the firm’s total cost function is * c( w 1 , w 2 , y ) = w 1x1 ( w 1 , w 2 , y ) * + w 2x 2 ( w 1 , w 2 , y )
  • 47. A Perfect Complements Example of Cost Minimization The firm’s production function is y = min{ 4x1 , x 2 } and the conditional input demands are y * * x1 ( w 1 , w 2 , y ) = and x 2 ( w 1 , w 2 , y ) = y. 4 So the firm’s total cost function is * c( w 1 , w 2 , y ) = w 1x1 ( w 1 , w 2 , y ) + w 2x* ( w 1 , w 2 , y ) 2 y  w1  = w1 + w 2y =  + w 2  y.  4  4
  • 48. Average Total Production Costs For positive output levels y, a firm’s average total cost of producing y units is c( w 1 , w 2 , y ) AC( w 1 , w 2 , y ) = . y
  • 49. Returns-to-Scale and Av. Total Costs The returns-to-scale properties of a firm’s technology determine how average production costs change with output level. Our firm is presently producing y’ output units. How does the firm’s average production cost change if it instead produces 2y’ units of output?
  • 50. Constant Returns-to-Scale and Average Total Costs If a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’ to 2y’ requires doubling all input levels.
  • 51. Constant Returns-to-Scale and Average Total Costs If a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’ to 2y’ requires doubling all input levels. Total production cost doubles.
  • 52. Constant Returns-to-Scale and Average Total Costs If a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’ to 2y’ requires doubling all input levels. Total production cost doubles. Average production cost does not change.
  • 53. Decreasing Returns-to-Scale and Average Total Costs If a firm’s technology exhibits decreasing returns-to-scale then doubling its output level from y’ to 2y’ requires more than doubling all input levels.
  • 54. Decreasing Returns-to-Scale and Average Total Costs If a firm’s technology exhibits decreasing returns-to-scale then doubling its output level from y’ to 2y’ requires more than doubling all input levels. Total production cost more than doubles.
  • 55. Decreasing Returns-to-Scale and Average Total Costs If a firm’s technology exhibits decreasing returns-to-scale then doubling its output level from y’ to 2y’ requires more than doubling all input levels. Total production cost more than doubles. Average production cost increases.
  • 56. Increasing Returns-to-Scale and Average Total Costs If a firm’s technology exhibits increasing returns-to-scale then doubling its output level from y’ to 2y’ requires less than doubling all input levels.
  • 57. Increasing Returns-to-Scale and Average Total Costs If a firm’s technology exhibits increasing returns-to-scale then doubling its output level from y’ to 2y’ requires less than doubling all input levels. Total production cost less than doubles.
  • 58. Increasing Returns-to-Scale and Average Total Costs If a firm’s technology exhibits increasing returns-to-scale then doubling its output level from y’ to 2y’ requires less than doubling all input levels. Total production cost less than doubles. Average production cost decreases.
  • 59. Returns-to-Scale and Av. Total Costs $/output unit AC(y) decreasing r.t.s. constant r.t.s. increasing r.t.s. y
  • 60. Returns-to-Scale and Total Costs What does this imply for the shapes of total cost functions?
  • 61. Returns-to-Scale and Total Costs $ Av. cost increases with y if the firm’s technology exhibits decreasing r.t.s. c(2y’) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). c(y’) y’ 2y’ y
  • 62. Returns-to-Scale and Total Costs $ Av. cost increases with y if the firm’s technology exhibits decreasing r.t.s. c(y) c(2y’) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). c(y’) y’ 2y’ y
  • 63. Returns-to-Scale and Total Costs $ c(2y’) Av. cost decreases with y if the firm’s technology exhibits increasing r.t.s. Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). c(y’) y’ 2y’ y
  • 64. Returns-to-Scale and Total Costs $ c(2y’) Av. cost decreases with y if the firm’s technology exhibits increasing r.t.s. c(y) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). c(y’) y’ 2y’ y
  • 65. Returns-to-Scale and Total Costs c(2y’) =2c(y’) $ Av. cost is constant when the firm’s technology exhibits constant r.t.s. c(y) Slope = c(2y’)/2y’ = 2c(y’)/2y’ = c(y’)/y’ so AC(y’) = AC(2y’). c(y’) y’ 2y’ y
  • 66. Short-Run & Long-Run Total Costs In the long-run a firm can vary all of its input levels. Consider a firm that cannot change its input 2 level from x2’ units. How does the short-run total cost of producing y output units compare to the long-run total cost of producing y units of output?
  • 67. Short-Run & Long-Run Total Costs The long-run cost-minimization problem is min w 1x1 + w 2x 2 x1 ,x 2 ≥ 0 subject to f ( x1 , x 2 ) = y. The short-run cost-minimization problem is min w 1x1 + w 2x ′ 2 x1 ≥ 0 subject to f ( x1 , x ′ ) = y. 2
  • 68. Short-Run & Long-Run Total Costs The short-run cost-min. problem is the long-run problem subject to the extra constraint that x2 = x2’. If the long-run choice for x2 was x2’ then the extra constraint x2 = x2’ is not really a constraint at all and so the long-run and short-run total costs of producing y output units are the same.
  • 69. Short-Run & Long-Run Total Costs The short-run cost-min. problem is therefore the long-run problem subject to the extra constraint that x2 = x2”. But, if the long-run choice for x2 ≠ x2” then the extra constraint x2 = x2” prevents the firm in this short-run from achieving its long-run production cost, causing the short-run total cost to exceed the long-run total cost of producing y output units.
  • 70. Short-Run & Long-Run Total Costs x2 y ′′′ Consider three output levels. y ′′ y′ x1
  • 71. Short-Run & Long-Run Total Costs x2 y ′′′ y ′′ y′ In the long-run when the firm is free to choose both x1 and x2, the least-costly input bundles are ... x1
  • 72. Short-Run & Long-Run Total Costs x2 y ′′′ y ′′ y′ Long-run output expansion path x ′′′ 2 x ′′ 2 x′ 2 x1 x1 x1 ′ ′′ ′′′ x1
  • 73. Short-Run & Long-Run Total Costs x2 Long-run costs are: y ′′′ y ′′ y′ Long-run output expansion path c( y ′ ) = w 1x1 + w 2x ′ ′ 2 c( y ′′ ) = w 1x1 + w 2x ′′′ ′′ 2 c( y ′′′ ) = w 1x1 w 2x ′′′ ′′′+ 2 x ′′′ 2 x ′′ 2 x′ 2 x1 x1 x1 ′ ′′ ′′′ x1
  • 74. Short-Run & Long-Run Total Costs Now suppose the firm becomes subject to the short-run constraint that x2 = x2”.
  • 75. Short-Run & Long-Run Total Costs x2 y ′′′ Short-run y ′′ y′ output expansion path Long-run costs are: c( y ′ ) = w 1x1 + w 2x ′ ′ 2 c( y ′′ ) = w 1x1 + w 2x ′′′ ′′ 2 c( y ′′′ ) = w 1x1 w 2x ′′′ ′′′+ 2 x ′′′ 2 x ′′ 2 x′ 2 x1 x1 x1 ′ ′′ ′′′ x1
  • 76. Short-Run & Long-Run Total Costs x2 y ′′′ Short-run y ′′ y′ output expansion path Long-run costs are: c( y ′ ) = w 1x1 + w 2x ′ ′ 2 c( y ′′ ) = w 1x1 + w 2x ′′′ ′′ 2 c( y ′′′ ) = w 1x1 w 2x ′′′ ′′′+ 2 x ′′′ 2 x ′′ 2 x′ 2 x1 x1 x1 ′ ′′ ′′′ x1
  • 77. Short-Run & Long-Run Total Costs x2 y ′′′ Short-run y ′′ y′ output expansion path Long-run costs are: c( y ′ ) = w 1x1 + w 2x ′ ′ 2 c( y ′′ ) = w 1x1 + w 2x ′′′ ′′ 2 c( y ′′′ ) = w 1x1 w 2x ′′′ ′′′+ 2 Short-run costs are: c s ( y ′ ) > c( y ′ ) x ′′′ 2 x ′′ 2 x′ 2 x1 x1 x1 ′ ′′ ′′′ x1
  • 78. Short-Run & Long-Run Total Costs x2 y ′′′ Short-run y ′′ y′ output expansion path Long-run costs are: c( y ′ ) = w 1x1 + w 2x ′ ′ 2 c( y ′′ ) = w 1x1 + w 2x ′′′ ′′ 2 c( y ′′′ ) = w 1x1 w 2x ′′′ ′′′+ 2 Short-run costs are: c s ( y ′ ) > c( y ′ ) cs ( y ′′ ) = c( y ′′ ) x ′′′ 2 x ′′ 2 x′ 2 x1 x1 x1 ′ ′′ ′′′ x1
  • 79. Short-Run & Long-Run Total Costs x2 y ′′′ Short-run y ′′ y′ output expansion path Long-run costs are: c( y ′ ) = w 1x1 + w 2x ′ ′ 2 c( y ′′ ) = w 1x1 + w 2x ′′′ ′′ 2 c( y ′′′ ) = w 1x1 w 2x ′′′ ′′′+ 2 Short-run costs are: c s ( y ′ ) > c( y ′ ) cs ( y ′′ ) = c( y ′′ ) x ′′′ 2 x ′′ 2 x′ 2 x1 x1 x1 ′ ′′ ′′′ x1
  • 80. Short-Run & Long-Run Total Costs x2 y ′′′ Short-run y ′′ y′ output expansion path Long-run costs are: c( y ′ ) = w 1x1 + w 2x ′ ′ 2 c( y ′′ ) = w 1x1 + w 2x ′′′ ′′ 2 c( y ′′′ ) = w 1x1 w 2x ′′′ ′′′+ 2 Short-run costs are: c s ( y ′ ) > c( y ′ ) c s ( y ′′ ) = c( y ′′ ) c s ( y ′′′ ) > c( y ′′′ ) x ′′′ 2 x ′′ 2 x′ 2 x1 x1 x1 ′ ′′ ′′′ x1
  • 81. Short-Run & Long-Run Total Costs Short-run total cost exceeds longrun total cost except for the output level where the short-run input level restriction is the long-run input level choice. This says that the long-run total cost curve always has one point in common with any particular shortrun total cost curve.
  • 82. Short-Run & Long-Run Total Costs A short-run total cost curve always has $ one point in common with the long-run total cost curve, and is elsewhere higher than the long-run total cost curve. cs(y) c(y) F= w 2x ′′ 2 y′ y ′′ y ′′′ y