Ch19

Chapter Nineteen
Profit-Maximization

Economic Profit
A firm uses inputs j = 1…,m to make
products i = 1,…n.
Output levels are y1,…,yn.
Input levels are x1,…,xm.
Product prices are p1,…,pn.
Input prices are w1,…,wm.

The Competitive Firm
The competitive firm takes all output
prices p1,…,pn and all input prices w1,
…,wm as given constants.

Economic Profit
The economic profit generated by
the production plan (x1,…,xm,y1,…,yn)
is
Π = p1y1 ++pnyn=− w 1x1 − w mxm .
Π

Economic Profit
Output and input levels are typically
flows.
E.g. x1 might be the number of labor
units used per hour.
And y3 might be the number of cars
produced per hour.
Consequently, profit is typically a
flow also; e.g. the number of dollars
of profit earned per hour.

Economic Profit
How do we value a firm?
Suppose the firm’s stream of
periodic economic profits is
Π 0 , Π 1 , Π 2 , … and r is the rate of
interest.
Then the present-value of the firm’s
economicΠ + Π1 + Πis + 
profit stream 2
PV =
0

1+r

(1 + r ) 2

Economic Profit
A competitive firm seeks to maximize
its present-value.
How?

Economic Profit
Suppose the firm is in a short-run
~ .
circumstance in which x 2 ≡ x 2
Its short-run production function is

~ ).
y = f ( x1 , x 2

Economic Profit
Suppose the firm is in a short-run
~ .
circumstance in which x 2 ≡ x 2
Its short-run production function is

~ ).
y = f ( x1 , x 2

~
The firm’s fixed cost is FC = w 2x 2
and its profit function is
~ .
Π = py − w 1x1 − w 2x 2

Short-Run Iso-Profit Lines
A $Π iso-profit line contains all the
production plans that provide a profit
level $Π .
A $Π iso-profit line’s equation is
~ .
Π ≡ py − w 1x1 − w 2x 2

A $Π iso-profit line contains all the
production plans that yield a profit
level of $Π .
The equation of a $Π iso-profit line is
~ .
Π ≡ py − w 1x1 − w 2x 2
I.e.
~
w1
Π + w 2x 2
y=
x1 +
.

p

p

~
w1
Π + w 2x 2
y=
x1 +
p
p

has a slope of

w1
+
p

and a vertical intercept of

~
Π + w 2x 2
.
p

y

ing
s
ea t
r
nc rofi
I
p

Π ≡ Π′′′
Π ≡ Π′′

Π ≡ Π′

w1
Slopes = +
p

x1

Short-Run Profit-Maximization
The firm’s problem is to locate the
production plan that attains the
highest possible iso-profit line, given
the firm’s constraint on choices of
production plans.
Q: What is this constraint?

The firm’s problem is to locate the
production plan that attains the
highest possible iso-profit line, given
the firm’s constraint on choices of
production plans.
Q: What is this constraint?
A: The production function.

y

The short-run production function and
~
x2 ≡ x2 .
technology set for

~
y = f ( x1 , x 2 )

Technically
inefficient
plans
x1

y

ing
s
ea t
r
nc rofi
I
p

Π ≡ Π′′′
Π ≡ Π′′

Π ≡ Π′

~
y = f ( x1 , x 2 )

w1
Slopes = +
p

x1

y

Π ≡ Π′′′
Π ≡ Π′′

Π ≡ Π′

w1
Slopes = +
p

y*

*
x1

x1

y

~
Given p, w1 and x 2 ≡ x 2 , the short-run
* ~
*
profit-maximizing plan is ( x1 , x 2 , y ).
Π ≡ Π′′

w1
Slopes = +
p

y*

*
x1

x1

y

y*

~
Given p, w1 and x 2 ≡ x 2 , the short-run
* ~
*
profit-maximizing plan is ( x1 , x 2 , y ).
And the maximum
Π ≡ Π′′
possible profit
is Π′′ .
w1
Slopes = +
p

*
x1

x1

At the short-run profit-maximizing plan,
y the slopes of the short-run production
function and the maximal Π ≡ Π′′
iso-profit line are
equal.
w1
Slopes = +
y*
p

*
x1

x1

At the short-run profit-maximizing plan,
y the slopes of the short-run production
function and the maximal Π ≡ Π′′
iso-profit line are
equal.
w1
Slopes = +
y*
p
w1
MP1 =
p
~
at ( x* , x , y* )
1

*
x1

2

x1

w1
MP1 =
p

⇔ p × MP1 = w 1

p × MP1 is the marginal revenue product of

input 1, the rate at which revenue increases
with the amount used of input 1.
If p × MP1 > w1 then profit increases with x 1.
If p × MP1 < w1 then profit decreases with x 1.

Short-Run Profit-Maximization; A
Cobb-Douglas Example
Suppose the short-run production
~
function is y = x1/ 3x1/3 .
1
2
The marginal product of the variable
∂ y 1 − 2/ 3~ 1/ 3
input 1 is
MP1 =
= x1 x 2 .
∂ x1 3
The profit-maximizing condition is
p * − 2/ 3 ~ 1/ 3
MRP1 = p × MP1 = ( x1 )
x2 = w1 .
3

p * − 2/ 3 ~ 1/ 3
x 2 = w 1 for x1 gives
Solving ( x1 )
3
3w 1
* − 2/ 3
( x1 )
=
.
~
px1/ 3
2

p * − 2/ 3 ~ 1/ 3
Solving ( x1 )
3
3w 1
* − 2/ 3
( x1 )
=
.
~
px1/ 3
2

That is,

~
px1/ 3
2
( x* ) 2/ 3 =
1
3w 1

p * − 2/ 3 ~ 1/ 3
Solving ( x1 )
3
3w 1
* − 2/ 3
( x1 )
=
.
~
px1/ 3
2

That is,

so

~
px1/ 3
2
( x* ) 2/ 3 =
1
3w 1

~ 1/ 3  3/ 2  p  3/ 2
*  px 2 
~ 1/ 2 .

x1 = 
=
x2


 3w 1 
 3w 1 

*  p 
x1 = 

 3w 1 

3/ 2

~ 1/ 2
x2

is the firm’s
short-run demand
for input 1 when the level of input 2 is
~
fixed at x 2 units.

*  p 
x1 = 

 3w 1 

3/ 2

~ 1/ 2
x2

is the firm’s
short-run demand
for input 1 when the level of input 2 is
~
fixed at x 2 units.
The firm’s short-run output level is thus
* 1/ 3 ~ 1/ 3  p 
y = ( x1 ) x 2 = 

 3w 1 
*

1/ 2

~ 1/2 .
x2

Comparative Statics of Short-Run
Profit-Maximization
What happens to the short-run profitmaximizing production plan as the
output price p changes?

Profit-Maximization
The equation of a short-run iso-profit line
~
is
w1
Π + w 2x 2
y=
x1 +

p

p

so an increase in p causes
-- a reduction in the slope, and
-- a reduction in the vertical intercept.

Profit-Maximization Π ≡ Π′′′
y

Π ≡ Π′′
Π ≡ Π′

~
y = f ( x1 , x 2 )
y*

w1
Slopes = +
p

*
x1

x1

Profit-Maximization
y

~
y = f ( x1 , x 2 )
y*

w1
Slopes = +
p
*
x1

x1

Profit-Maximization
y

~
y = f ( x1 , x 2 )

y*

w1
Slopes = +
p
*
x1

x1

Profit-Maximization
An increase in p, the price of the
firm’s output, causes
– an increase in the firm’s output
level (the firm’s supply curve
slopes upward), and
– an increase in the level of the
firm’s variable input (the firm’s
demand curve for its variable input
shifts outward).

Profit-Maximization
The Cobb-Douglas example: When
1/ 3 ~ 1/ 3
y = x1 x 2
then the firm’s short-run
demand for its variable input 1 is
*  p 
x1 = 

 3w 1 

 p 
y =

 3w 1 
*

3/ 2

~ 1/ 2
x2

1/ 2

~
x1/ 2 .
2

and its short-run
supply is

Profit-Maximization
1/ 3 ~ 1/ 3
y = x1 x 2
*  p 
x1 = 

 3w 1 

 p 
y =

 3w 1 
*

3/ 2

~ 1/ 2
x2

1/ 2

~
x1/ 2 .
2

and its short-run
supply is

x* increases as p increases.
1

Profit-Maximization
1/ 3 ~ 1/ 3
y = x1 x 2
*  p 
x1 = 

 3w 1 

 p 
y =

 3w 1 
*

3/ 2

~ 1/ 2
x2

1/ 2

~
x1/ 2 .
2

and its short-run
supply is

x* increases as p increases.
1
*
y increases as p increases.

Profit-Maximization
What happens to the short-run profitmaximizing production plan as the
variable input price w1 changes?

Profit-Maximization
The equation of a short-run iso-profit line
~
is
w1
Π + w 2x 2
y=
x1 +

p

p

so an increase in w1 causes
-- an increase in the slope, and
-- no change to the vertical intercept.

Profit-Maximization
Π ≡ Π′′′
Π ≡ Π′′

y

Π ≡ Π′

~
y = f ( x1 , x 2 )
y*

w1
Slopes = +
p

*
x1

x1

Profit-Maximization
Π ≡ Π′′′
Π ≡ Π′′

y

Π ≡ Π′

~
y = f ( x1 , x 2 )
w1
Slopes = +
p

y*

*
x1

x1

Profit-Maximization
An increase in w1, the price of the
firm’s variable input, causes
– a decrease in the firm’s output
level (the firm’s supply curve shifts
inward), and
– a decrease in the level of the firm’s
variable input (the firm’s demand
curve for its variable input slopes
downward).

Profit-Maximization
1/ 3 ~ 1/ 3
y = x1 x 2
*  p 
x1 = 

 3w 1 

 p 
y =

 3w 1 
*

3/ 2

~ 1/ 2
x2
and its short-run

1/ 2

~
x1/ 2 .
2

supply is

Profit-Maximization
1/ 3 ~ 1/ 3
y = x1 x 2
*  p 
x1 = 

 3w 1 

 p 
y =

 3w 1 
*

3/ 2

~ 1/ 2
x2
and its short-run

1/ 2

~
x1/ 2 .
2

supply is

x* decreases as w1 increases.
1

Profit-Maximization
1/ 3 ~ 1/ 3
y = x1 x 2
*  p 
x1 = 

 3w 1 

 p 
y =

 3w 1 
*

3/ 2

~ 1/ 2
x2
and its short-run

1/ 2

~
x1/ 2 .
2

supply is

x* decreases as w1 increases.
1
y* decreases as w1 increases.

Long-Run Profit-Maximization
Now allow the firm to vary both input
levels.
Since no input level is fixed, there
are no fixed costs.

Both x1 and x2 are variable.
Think of the firm as choosing the
production plan that maximizes
profits for a given value of x2, and
then varying x2 to find the largest
possible profit level.

The equation of a long-run iso-profit line
is
w1
Π + w 2x 2
y=
x1 +

p

p

so an increase in x2 causes
-- no change to the slope, and
-- an increase in the vertical intercept.

y

y = f ( x1 , x ′ )
2

x1

y
y = f ( x1 , 3x ′ )
2
y = f ( x1 , 2x′ )
2

y = f ( x1 , x ′ )
2

Larger levels of input 2 increase the
productivity of input 1.

x1

y
y = f ( x1 , 3x ′ )
2
y = f ( x1 , 2x′ )
2

y = f ( x1 , x ′ )
2

The marginal product
of input 2 is
diminishing.
Larger levels of input 2 increase the
productivity of input 1.

x1

y

p × MP1 − w 1 = 0 for each short-run

production plan.

y = f ( x1 , 3x ′ )
2
y = f ( x1 , 2x′ )
2

y* ( 3x ′ )
2
y* ( 2x ′ )
2

y = f ( x1 , x ′ )
2

*

y ( x′ )
2
x* ( x ′ )
x* ( 3x′ )
1 2
1
2
*
x1 ( 2x′ )
2

x1

y


production plan.

y = f ( x1 , 3x ′ )
2
y = f ( x1 , 2x′ )
2

y* ( 3x ′ )
2
y* ( 2x ′ )
2

y = f ( x1 , x ′ )
2

The marginal product
of input 2 is
diminishing so ...

*

y ( x′ )
2
x* ( x ′ )
x* ( 3x′ )
1 2
1
2
*
x1 ( 2x′ )
2

x1

y


production plan.

y = f ( x1 , 3x ′ )
2
y = f ( x1 , 2x′ )
2

y* ( 3x ′ )
2
y* ( 2x ′ )
2

y = f ( x1 , x ′ )
2

the marginal profit
of input 2 is
diminishing.

*

y ( x′ )
2
x* ( x ′ )
x* ( 3x′ )
1 2
1
2
*
x1 ( 2x′ )
2

x1

Profit will increase as x2 increases so
long as the marginal profit of input 2
p × MP2 − w 2 > 0.

The profit-maximizing level of input 2
therefore satisfies
p × MP2 − w 2 = 0.

Profit will increase as x2 increases so
long as the marginal profit of input 2
p × MP2 − w 2 > 0.

The profit-maximizing level of input 2
therefore satisfies
p × MP2 − w 2 = 0.

And p × MP1 − w1 = 0 is satisfied in
any short-run, so ...

The input levels of the long-run
profit-maximizing plan satisfy
p × MP1 − w 1 = 0 and p × MP2 − w 2 = 0.

That is, marginal revenue equals
marginal cost for all inputs.

1/ 3 ~ 1/ 3
y = x1 x 2 then the firm’s short-run
*  p 
x1 = 

 3w 1 

 p 
y =

 3w 1 
*

3/ 2

~ 1/ 2
x2

1/ 2

~
x1/ 2 .
2

and its short-run
supply is

Short-run profit is therefore …

~
Π = py* − w 1x* − w 2x 2
1
 p 
= p

 3w 1 

1/ 2

~ 1/ 2 − w  p 
x2

1
 3w 1 

3/ 2

~
~
x1/ 2 − w 2x 2
2

~
Π = py* − w 1x* − w 2x 2
1
 p 
= p

 3w 1 

1/ 2

 p 
= p

 3w 1 

1/ 2

~ 1/ 2 − w  p 
x2

1
 3w 1 

3/ 2

~
~
x1/ 2 − w 2x 2
2

~ 1/ 2 − w p  p 
x2


1
3w 1  3w 1 

1/ 2

~
− w 2x 2

~
Π = py* − w 1x* − w 2x 2
1
 p 
= p

 3w 1 

1/ 2

 p 
= p

 3w 1 

1/ 2

2p  p 
=


3  3w 1 

~ 1/ 2 − w  p 
x2

1
 3w 1 

3/ 2

~
~
x1/ 2 − w 2x 2
2

~ 1/ 2 − w p  p 
x2


1
3w 1  3w 1 

1/ 2

~
~
x1/ 2 − w 2x 2
2

1/ 2

~
− w 2x 2

~
Π = py* − w 1x* − w 2x 2
1
 p 
= p

 3w 1 

1/ 2

 p 
= p

 3w 1 

1/ 2

2p  p 
=


3  3w 1 

~ 1/ 2 − w  p 
x2

1
 3w 1 

3/ 2

~
~
x1/ 2 − w 2x 2
2

~ 1/ 2 − w p  p 
x2


1
3w 1  3w 1 

1/ 2

~
~
x1/ 2 − w 2x 2
2

3  1/ 2
 4p
~
~

=
x1/ 2 − w 2x 2 .
2
 27w 

1

1/ 2

~
− w 2x 2

3  1/ 2
 4p
~
~

Π=
x1/ 2 − w 2x 2 .
2
 27w 

1

What is the long-run profit-maximizing
level of input 2? Solve
3  1/ 2
∂ Π 1  4p
~−

0= ~ = 
x 2 1/ 2 − w 2
∂ x 2 2  27w 1 



to get

~
x 2 = x* =
2

p3
27w 1w 2
2

.

input 1 level? Substitute
x* =
2

p

3

27w 1w 2
2

to get

into

*  p 
x1 = 

 3w 1 

3/ 2

~
x1/ 2
2

input 1 level? Substitute
x* =
2

p

3

into

27w 1w 2
2

*  p 
x1 = 

 3w 1 

3/ 2

~
x1/ 2
2

to get
*  p 
x1 = 

 3w 1 

3 / 2


p



2
 27w 1w 2 
3

1/ 2

=

p

3

2
27w 1 w 2

.

output level? Substitute
x* =
2

p

3

27w 1w 2
2

to get

 p 
into y = 

 3w 1 
*

1/ 2

~
x1/ 2
2

output level? Substitute
x* =
2

p

 p 
into y = 

 3w 1 

3

*

27w 1w 2
2

1/ 2

~
x1/ 2
2

to get
*  p 
y =

 3w 1 

1/ 2





2
 27w 1w 2 
p

3

1/ 2

p2
=
.
9 w 1w 2

So given the prices p, w 1 and w2, and
the production function y = x1/ 3x1/ 3
1
2
the long-run profit-maximizing production
plan is

p3
p3
p2 
.
( x* , x* , y* ) = 
,
,
1 2


2
 27w 1 w 2 27w 1w 2 9 w 1w 2 
2

Returns-to-Scale and ProfitMaximization
If a competitive firm’s technology
exhibits decreasing returns-to-scale
then the firm has a single long-run
profit-maximizing production plan.

Returns-to Scale and ProfitMaximization
y

y = f(x)
y*

Decreasing
returns-to-scale
x*

x

If a competitive firm’s technology
exhibits exhibits increasing returnsto-scale then the firm does not have
a profit-maximizing plan.

y
g
asin
ncre it
I
prof

y = f(x)

y”
y’

Increasing
returns-to-scale
x’

x”

x

So an increasing returns-to-scale
technology is inconsistent with firms
being perfectly competitive.

What if the competitive firm’s
technology exhibits constant
returns-to-scale?

y
g
asin
ncre it
I
prof

y = f(x)

y”

Constant
returns-to-scale

y’
x’

x”

x

So if any production plan earns a
positive profit, the firm can double
up all inputs to produce twice the
original output and earn twice the
original profit.

Therefore, when a firm’s technology
exhibits constant returns-to-scale,
earning a positive economic profit is
inconsistent with firms being
perfectly competitive.
Hence constant returns-to-scale
requires that competitive firms earn
economic profits of zero.

y

y = f(x)

Π =0
y”

Constant
returns-to-scale

y’
x’

x”

x

Revealed Profitability
Consider a competitive firm with a
technology that exhibits decreasing
returns-to-scale.
For a variety of output and input
prices we observe the firm’s choices
of production plans.
What can we learn from our
observations?

If a production plan (x’,y’) is chosen
at prices (w’,p’) we deduce that the
plan (x’,y’) is revealed to be profitmaximizing for the prices (w’,p’).

y

( x′ , y′ ) is chosen at prices ( w ′ , p′ )

w′
Slope =
p′
y′

x′

x

y

( x′ , y′ ) is chosen at prices ( w ′ , p′ ) so

( x′ , y′ ) is profit-maximizing at these prices.

w′
Slope =
p′
y′

x′

x

y



w′
Slope =
p′
y′
y′′

( x′′ , y′′ ) would give higher

profits, so why is it not
chosen?

x′′

x′

x

y



w′
Slope =
p′
y′
y′′


chosen? Because it is
not a feasible plan.
x′′

x′

x

y



w′
Slope =
p′
y′
y′′


chosen? Because it is
not a feasible plan.
x
x′
So the firm’s technology set must lie under the
iso-profit line.
x′′

y



w′
Slope =
p′
y′
y′′

The technology
set is somewhere
in here
x
x′
So the firm’s technology set must lie under the
iso-profit line.
x′′

y

( x′′′ , y′′′ ) is chosen at prices ( w ′′′ , p′′′ ) so
( x′′′ , y′′′ ) maximizes profit at these prices.

w ′′′
Slope =
p′′′
( x ′′ , y′′ ) would provide higher

y′′

profit but it is not chosen
y′′′

x ′′′ x ′′

x

y


w ′′′
Slope =
p′′′
y′′


y′′′

because it is not feasible

x ′′′ x ′′

x

y


w ′′′
Slope =
p′′′
y′′


y′′′

because it is not feasible so
the technology set lies under
the iso-profit line.
x ′′′ x ′′

x

y


w ′′′
Slope =
p′′′
y′′

The technology set is
also somewhere in
here.

y′′′

x ′′′ x ′′

x

y

The firm’s technology set must lie under
both iso-profit lines

y′

y′′′

x ′′′

x′

x

y

both iso-profit lines

y′

The technology set
is somewhere
in this intersection

y′′′

x ′′′

x′

x

Observing more choices of
production plans by the firm in
response to different prices for its
input and its output gives more
information on the location of its
technology set.

y

all the iso-profit lines
( w ′ , p′ )
( w ′′′ , p′′′ )

y′′
y′

( w ′′ , p′′ )

y′′′

x ′′′

x′

x′′

x

y

( w ′ , p′ )
( w ′′′ , p′′′ )

y′′
y′

( w ′′ , p′′ )

y = f(x)

y′′′

x ′′′

x′

x′′

x

What else can be learned from the
firm’s choices of profit-maximizing
production plans?

y

( w ′ , p′ )
( w ′′ , p′′ )

( x′ , y′ ) is chosen at prices
( w ′ , p′ ) so

y′

p′ y′ − w ′x ′ ≥ p′ y′′ − w ′x′′ .
( x′′ , y′′ ) is chosen at prices

y′′

( w ′′ , p′′ ) so

p′′ y′′ − w ′′x′′ ≥ p′′ y′ − w ′′x′ .

x ′′

x′

x

p′y′ − w ′x ′ ≥ p′y′′ − w ′x ′′ and

p′′y′′ − w ′′x ′′ ≥ p′′y′ − w ′′x ′ so
p′y′ − w ′x ′ ≥ p′y′′ − w ′x ′′ and

− p′′y′ + w ′′x ′ ≥ − p′′y′′ + w ′′x ′′.

Adding gives
( p′ −p′ ) y′ −( w ′ −w ′ ) x ′ ≥
′
′
( p′ −p′ ) y′ −( w ′ −w ′ ) x ′ .
′
′
′
′

( p′ − p′′ ) y′ − ( w ′ − w ′′ ) x ′ ≥
( p′ − p′′ ) y′′ − ( w ′ − w ′′ ) x ′′
so

( p′ − p′′ )( y′ − y′′ ) ≥ ( w ′ − w ′′ )( x ′ − x ′′ )
That is,

∆p∆y ≥ ∆w∆x

is a necessary implication of profitmaximization.

∆ ∆ ≥∆ ∆
p y
w x

Suppose the input price does not change.
Then ∆w = 0 and profit-maximization
p y
implies ∆ ∆ ≥0; i.e., a competitive
firm’s output supply curve cannot slope
downward.

∆ ∆ ≥∆ ∆
p y
w x

Suppose the output price does not change.
Then ∆p = 0 and profit-maximization
w x
implies 0 ≥∆ ∆; i.e., a competitive
firm’s input demand curve cannot slope
upward.

Ch19

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