Microeconomics
Economics Department
National Chi Nan University
Spring 2022
Lecture19
Profit maximization
1
Microeconomics, Spring 2022
Outline
Economic profit
Short-run profit maximization
Comparative statics
Long-run profit maximization
Profit maximization and returns to scale
Revealed profitability






2
Microeconomics, Spring 2022
Economic profit





3
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.
Microeconomics, Spring 2022
The economic profit for competitive firms

The competitive firm takes all output prices p1,…,pn and
all input prices w1,…,wm as given constants.

The economic profit generated by the production plan
(x1,…,xm,y1,…,yn) is
  p1 y1    pn yn  w1x1  wm xm.
4
Microeconomics, Spring 2022
Economic profit is a flow

Output and input levels are typically flows. 流量

E.g. x1 might be the number of labor units used per
hour.

And y 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.
5
Microeconomics, Spring 2022
Outline
Economic profit
Short-run profit maximization
Comparative statics
Long-run profit maximization
Profit maximization and returns to scale
Revealed profitability






6
Microeconomics, Spring 2022
Short-run profit maximization

Suppose the firm is in a short-run circumstance in
~
x

x
which 2
2.

Its short-run production function is
~ ).
y  f ( x1 , x
2
7
Microeconomics, Spring 2022
Short-run profit maximization

Suppose the firm is in a short-run circumstance in
~ .
x2  x
which
2
Its short-run production function is

The firm’s fixed cost is

~ ).
y  f ( x1 , x
2
~
FC  w 2x 2
and its profit function is
~ .
  py  w 1x1  w 2x
2
8
Microeconomics, Spring 2022
Short run iso-proft lines 等利潤線

A $ iso-profit line contains all the production plans that
yield a profit level of $ .
The equation of a $ iso-profit line is

I.e.

~
  py  w 1x1  w 2x 2 .
~
w1
  w 2x
2.
y
x1 
p
p
9
Microeconomics, Spring 2022
Short run iso-proft lines
~
w1
  w 2x 2
y
x1 
p
p
has a slope of
w1

p
and a vertical intercept of
~
  w 2x 2
.
p
10
Microeconomics, Spring 2022
Short run iso-proft lines
y
   
   
  
w1
Slopes  
p
x1
11
Microeconomics, Spring 2022
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?

A: The production function.
12
Microeconomics, Spring 2022
Profit-maximization
The short-run production function and
y
~
x

x
technology set for 2
2.
~ )
y  f ( x1 , x
2
Technically
inefficient
plans
x1
13
Microeconomics, Spring 2022
Profit-maximization
y
   
   
  
~ )
y  f ( x1 , x
2
w1
Slopes  
p
x1
14
Microeconomics, Spring 2022
Profit-maximization
y
   
   
  
w1
Slopes  
p
y*
*
x1
15
x1
Microeconomics, Spring 2022
Profit-maximization
~ , the short-run
Given p, w1 and x 2  x
2
* ~
*
y
profit-maximizing plan is ( x1 , x 2 , y ).
   
w1
Slopes  
p
y*
*
x1
16
x1
Microeconomics, Spring 2022
Profit-maximization
~ , the short-run
Given p, w1 and x 2  x
2
* ~
*
y
profit-maximizing plan is ( x1 , x 2 , y ).
And the maximum
   
possible profit
is   .
w
1
*
Slopes


y
p
*
x1
17
x1
Microeconomics, Spring 2022
Profit-maximization
y
At the short-run profit-maximizing plan, the
slopes of the short-run production function and
the maximal
   
iso-profit line are
equal.
w1
Slopes  
p
y*
*
x1
18
x1
Microeconomics, Spring 2022
Profit-maximization
At the short-run profit-maximizing plan,
y the slopes of the short-run production
function and the maximal    
iso-profit line are
equal.
w
1
*
Slopes


y
p
w1
MP1 
p
~ , y* )
at ( x* , x
1
2
*
x1
19
x1
Microeconomics, Spring 2022
Profit-maximization
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. 邊際收益產出
If p  MP1  w 1 then profit increases with x1.
Ifp  MP1  w 1 then profit decreases with x1.
20
Microeconomics, Spring 2022
A Cobb-Douglas example
Suppose the short-run production
~ 1/ 3 .
function is y  x1/3
x
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 )
x 2  w1 .
3
21
Microeconomics, Spring 2022
A Cobb-Douglas example
p *  2 / 3 ~ 1/ 3
Solving ( x1 )
x 2  w 1 for x1 gives
3
3w 1
*  2/ 3
( x1 )

.
~ 1/ 3
px
2
That is,
so
22
~ 1/ 3
px
2
( x*1 ) 2/ 3 
3w 1
3/ 2
3/ 2
1/
3
~


 p 
*  px 2 
1/2
~
x1  

x2 .


 3w 1 
 3w 1 
Microeconomics, Spring 2022
A Cobb-Douglas example
*  p 
x1  

 3w 1 
3/ 2
1/ 2
~
x2
The firm’s short-run demand for input
~
1 when the level of input 2 is fixed at x
2
units.
23
Microeconomics, Spring 2022
A Cobb-Douglas example
The firm’s short-run output level is thus
* 1/ 3 ~ 1/3  p 
y  ( x1 ) x 2  

 3w 1 
*
24
1/ 2
1/ 2
~
x2 .
Microeconomics, Spring 2022
Outline






Economic profit
Short-run profit maximization
Comparative statics
Long-run profit maximization
Profit maximization and returns to scale
Revealed profitability
25
Microeconomics, Spring 2022
Comparative statics of short-run profitmaximization

What happens to the short-run profit-maximizing
production plan as the output price p changes?
26
Microeconomics, Spring 2022
Comparative statics of short-run profitmaximization
The equation of a short-run iso-profit line
~
is
w1
  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.
27
Microeconomics, Spring 2022
Comparative statics of short-run profitmaximization
   
   
y
  
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
*
x1
28
x1
Microeconomics, Spring 2022
Comparative statics of short-run profitmaximization
y
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
*
x1
29
x1
Microeconomics, Spring 2022
Comparative statics of short-run profitmaximization
y
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
*
x1
30
Microeconomics, Spring 2022
x1
The change of the output price

31
An increase in p, the price of the firm’s output, causes

an increase in the firm’s output level y (the firm’s supply
curve slopes upward)

an increase in the level of the firm’s variable input x (the
firm’s demand curve for its variable input shifts outward).
Microeconomics, Spring 2022
A Cobb-Douglas example
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 
*
32
3/ 2
1/ 2
~
x2
1/ 2
and its short-run
supply is
~ 1/ 2 .
x
2
Microeconomics, Spring 2022
A Cobb-Douglas example
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 
3/ 2
1/ 2
~
x2
1/ 2
and its short-run
supply is
 p  ~ 1/ 2
y 
 x2 .
 3w 1 
x*1 increases as p increases.
*
y increases asMicroeconomics,
p increases.
33
Spring 2022
*
The change of the input price

What happens to the short-run profit-maximizing
production plan as the variable input price w1 changes?
34
Microeconomics, Spring 2022
The change of the input price
The equation of a short-run iso-profit line
~
is
w1
  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.
35
Microeconomics, Spring 2022
The change of the input price
y
   
   
  
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
*
x1
36
x1
Microeconomics, Spring 2022
The change of the input price
   
   
y
  
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
*
x1
37
x1
Microeconomics, Spring 2022
The change of the input price
   
   
y
  
~ )
y  f ( x1 , x
2
w1
Slopes  
p
y*
*
x1
38
x1
Microeconomics, Spring 2022
The change of the input price

39
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).
Microeconomics, Spring 2022
A Cobb-Douglas example
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 
*
40
3/ 2
1/ 2
~
x2
and its short-run
1/ 2
supply is
~ 1/ 2 .
x
2
Microeconomics, Spring 2022
A Cobb-Douglas example
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 
*
y
x*1
3/ 2
1/ 2
~
x2
and its short-run
1/ 2
supply is
 p  ~ 1/ 2

 x2 .
 3w 1 
decreases as w1 increases.
*
y
decreases asMicroeconomics,
w1 increases.
41
Spring 2022
Outline






Economic profit
Short-run profit maximization
Comparative statics
Long-run profit maximization
Profit maximization and returns to scale
Revealed profitability
42
Microeconomics, Spring 2022
Long-run profit-maximization


43
Now allow the firm to vary both input levels.
Since no input level is fixed, there are no fixed costs.
Microeconomics, Spring 2022
Long-run profit-maximization

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.
44
Microeconomics, Spring 2022
Long-run profit-maximization
The equation of a long-run iso-profit line
is
w1
  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.
45
Microeconomics, Spring 2022
Long-run profit-maximization
y
y  f ( x1 , x2 )
x1
46
Microeconomics, Spring 2022
Long-run profit-maximization
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.
47
Microeconomics, Spring 2022
x1
Long-run profit-maximization
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.
48
Microeconomics, Spring 2022
x1
Long-run profit-maximization
p  MP1  w 1  0 for each short-run
y
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 )
49
x*1 ( x2 )
x*1 ( 3x2 )
x*1 ( 2x2Microeconomics,
)
Spring 2022
x1
Long-run profit-maximization
p  MP1  w 1  0 for each short-run
y
production plan.
y  f ( x1 , 3x2 )
y  f ( x1 , 2x 2 )
y* ( 3x 2 )
y* ( 2x 2 )
*
y ( x 2 )
50
y  f ( x1 , x2 )
The marginal product
of input 2 is
diminishing so ...
x*1 ( x2 )
x*1 ( 3x2 )
x*1 ( 2x2Microeconomics,
)
Spring 2022
x1
Long-run profit-maximization
p  MP1  w 1  0 for each short-run
y
production plan.
y  f ( x1 , 3x2 )
y  f ( x1 , 2x 2 )
y* ( 3x 2 )
y* ( 2x 2 )
*
y ( x 2 )
51
y  f ( x1 , x2 )
the marginal profit
of input 2 is
diminishing.
x*1 ( x2 )
x*1 ( 3x2 )
x*1 ( 2x2Microeconomics,
)
Spring 2022
x1
The conditions for long-run profit
maximization

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.
52
Microeconomics, Spring 2022
The conditions for long-run profit
maximization

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.

53
And p  MP1  w 1  0 is satisfied in any short-run,
so ...
Microeconomics, Spring 2022
The conditions for long-run profit
maximization

The input levels of the long-run profit-maximizing plan
satisfy
p  MP1  w 1  0
p  MP2  w 2  0.

That is, marginal revenue equals marginal cost for all
inputs.
54
Microeconomics, Spring 2022
A Cobb-Douglas example
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 
3/ 2
1/ 2
~
x2
and its short-run
1/ 2
supply is
 p  ~ 1/ 2
y 
 x2 .
 3w 1 
Short-run profit is therefore …
*
55
Microeconomics, Spring 2022
A Cobb-Douglas example
  py*  w1 x1*  w2 x2
1/ 2
 p 
 p

 3w1 
x
1/2
2
1/ 2
 p 
 p

 3w1 
 p 
 w1 

 3w1 
3/ 2

x1/2
2  w2 x2
1/2
x
1/2
2
p  p 
 w1


3w1  3w1 
x1/2 2  w2 x2
1/ 2
2p  p 



3  3w1 
1/ 2
 4p 


27
w
1 

3
56

x1/2
2  w2 x2

x1/2
2  w2 x2 .
Microeconomics, Spring 2022
A Cobb-Douglas example
1/ 2
3
 4p 
~ 1/ 2  w x
~ .

  
x
2
2 2
 27w 1 
What is the long-run profit-maximizing
level of input 2? Solve
1/ 2
3
  1  4p 
~  1/ 2  w

0  ~  
x
2
2
 x 2 2  27w 1 
to get
57
~  x* 
x
2
2
p3
27w 1w 2
2
.
Microeconomics, Spring 2022
A Cobb-Douglas example
What is the long-run profit-maximizing
input 1 level? Substitute
x*2 
p
3
27w 1w 2
2
into
*  p 
x1  

 3w 1 
to get
58
Microeconomics, Spring 2022
3/ 2
~ 1/ 2
x
2
A Cobb-Douglas example
What is the long-run profit-maximizing
input 1 level? Substitute
x*2 
p
3
into
27w 1w 2
2
*  p 
x1  

 3w 1 
3/ 2
~ 1/ 2
x
2
to get
*  p 
x1  

 3w 1 
59
3 / 2

p


2
 27w 1w 2 
3
1/ 2

p
3
2
27w 1 w 2
Microeconomics, Spring 2022
.
A Cobb-Douglas example
What is the long-run profit-maximizing
output level? Substitute
x*2 
p
3
27w 1w 2
2
 p 
into y  

 3w 1 
*
to get
60
Microeconomics, Spring 2022
1/ 2
~ 1/2
x
2
A Cobb-Douglas example
What is the long-run profit-maximizing
output level? Substitute
x*2 
p
 p 
into y  

 3w 1 
3
*
27w 1w 2
2
1/ 2
~ 1/2
x
2
to get
*  p 
y 

 3w 1 
61
1/ 2



 27w 1w 22 
p
3
1/ 2
p2

.
9 w 1w 2
Microeconomics, Spring 2022
A Cobb-Douglas example
So given the prices p, w1 and w2, and
the production function y  x1/ 3x1/ 3
1
2
the long-run profit-maximizing production
plan is
3
3
2 

p
p
p
 .
( x*1 , x*2 , y* )  
,
,
 27w 12 w 2 27w 1w 22 9 w 1w 2 
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Microeconomics, Spring 2022
Outline






Economic profit
Short-run profit maximization
Comparative statics
Long-run profit maximization
Profit maximization and returns to scale
Revealed profitability
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Microeconomics, Spring 2022
Profit-maximization and returns to scale

If a competitive firm’s technology exhibits decreasing
returns-to-scale then the firm has a single long-run profitmaximizing production plan.
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Microeconomics, Spring 2022
Decreasing return to scale
y
y  f(x)
y*
Decreasing
returns-to-scale
x*
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Microeconomics, Spring 2022
x
Profit-maximization and returns to scale

If a competitive firm’s technology exhibits exhibits
increasing returns-to-scale then the firm does not have a
profit-maximizing plan.
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Microeconomics, Spring 2022
Increasing returns to scale
y
y  f(x)
y”
y’
Increasing
returns-to-scale
x’
67
x”
Microeconomics, Spring 2022
x
Increasing returns to scale

So an increasing returns-to-scale technology is
inconsistent with firms being perfectly competitive.
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Microeconomics, Spring 2022
Constant returns to scale

What if the competitive firm’s technology exhibits
constant returns-to-scale?
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Microeconomics, Spring 2022
Constant returns to scale
y
y  f(x)
y”
Constant
returns-to-scale
y’
x’
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x”
Microeconomics, Spring 2022
x
Constant returns to scale

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.
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Microeconomics, Spring 2022
Constant returns to scale

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.
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Microeconomics, Spring 2022
Constant returns to scale
y
y  f(x)
=0
y”
Constant
returns-to-scale
y’
x’
73
x”
Microeconomics, Spring 2022
x
Outline






Economic profit
Short-run profit maximization
Comparative statics
Long-run profit maximization
Profit maximization and returns to scale
Revealed profitability
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Microeconomics, Spring 2022
Revealed profitability 顯示性獲利

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?
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Microeconomics, Spring 2022
Revealed profitability

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’).
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Microeconomics, Spring 2022
Revealed profitability
( x  , y ) is chosen at prices ( w  , p )
y
w
Slope 
p
y
x
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Microeconomics, Spring 2022
x
Revealed profitability
( x  , y ) is chosen at prices ( w  , p ) so
y
( x  , y ) is profit-maximizing at these prices.
w
Slope 
p
y
x
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Microeconomics, Spring 2022
x
Revealed profitability
( x  , y ) is chosen at prices ( w  , p ) so
y
( x  , y ) is profit-maximizing at these prices.
w
Slope 
p
y
y
( x  , y ) would give higher
profits, so why is it not
chosen?
x
79
x
Microeconomics, Spring 2022
x
Revealed profitability
( x  , y ) is chosen at prices ( w  , p ) so
y
( x  , y ) is profit-maximizing at these prices.
w
Slope 
p
y
y
( x  , y ) would give higher
profits, so why is it not
chosen? Because it is
not a feasible plan.
x
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x
Microeconomics, Spring 2022
x
Revealed profitability
( x  , y ) is chosen at prices ( w  , p ) so
y
( x  , y ) is profit-maximizing at these prices.
w
Slope 
p
y
y
( x  , y ) would give higher
profits, so why is it not
chosen? Because it is
not a feasible plan.
x
x
So the firm’s technology set must lie under the
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Microeconomics, Spring 2022
iso-profit line.
x
Revealed profitability
( x  , y ) is chosen at prices ( w  , p ) so
y
( x  , y ) is profit-maximizing at these prices.
w
Slope 
p
y
y
The technology
set is somewhere
in here
x
x
So the firm’s technology set must lie under the
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Microeconomics, Spring 2022
iso-profit line.
x
Revealed profitability
( x  , y ) is chosen at prices ( w  , p ) so
y ( 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
83
Microeconomics, Spring 2022
Revealed profitability
( x  , y ) is chosen at prices ( w  , p ) so
y ( x , y )
maximizes profit at these prices.
w 
Slope 
p
y
( x  , y ) would provide higher
y
profit but it is not chosen
because it is not feasible
x
x  x
84
Microeconomics, Spring 2022
Revealed profitability
( x  , y ) is chosen at prices ( w  , p ) so
y ( x , y )
maximizes profit at these prices.
w 
Slope 
p
y
( x  , y ) would provide higher
y
profit but it is not chosen
because it is not feasible so
the technology set lies under
the iso-profit line.
x
x  x
85
Microeconomics, Spring 2022
Revealed profitability
( x  , y ) is chosen at prices ( w  , p ) so
y ( x , y )
maximizes profit at these prices.
w 
Slope 
p
y
The technology set is
also somewhere in
here.
y
x
x  x
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Microeconomics, Spring 2022
Revealed profitability
The firm’s technology set must lie under
y
both iso-profit lines
y
y
x 
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x
Microeconomics, Spring 2022
x
Revealed profitability
The firm’s technology set must lie under
y
both iso-profit lines
y
The technology set
is somewhere
in this intersection
y
x 
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x
Microeconomics, Spring 2022
x
Revealed profitability

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.
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Microeconomics, Spring 2022
Revealed profitability
The firm’s technology set must lie under
y
all the iso-profit lines
( w  , p )
( w  , p )
y
y
( w  , p )
y
x 
90
x
x 
Microeconomics, Spring 2022
x
Revealed profitability
The firm’s technology set must lie under
y
all the iso-profit lines
( w  , p )
( w  , p )
y
y
( w  , p )
y
x 
91
x
x 
Microeconomics, Spring 2022
x
Revealed profitability
The firm’s technology set must lie under
y
all the iso-profit lines
( w  , p )
( w  , p )
y
y
( w  , p )
y  f(x)
y
x 
92
x
x 
Microeconomics, Spring 2022
x
Revealed profitability

What else can be learned from the firm’s choices of
profit-maximizing production plans?
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Microeconomics, Spring 2022
Revealed profitability
The firm’s technology set must lie under
y
all the iso-profit lines
( 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
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x
Microeconomics, Spring 2022
x
Revealed profitability
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  .
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Microeconomics, Spring 2022
Revealed profitability
( 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.
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Microeconomics, Spring 2022
Revealed profitability
 p y   w  x
is a necessary implication of profitmaximization.
Suppose the input price does not change.
Then w = 0 and profit-maximization
implies py  0; i.e., a competitive
firm’s output supply curve cannot slope
downward.
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Microeconomics, Spring 2022
Revealed profitability
 p y   w  x
is a necessary implication of profitmaximization.
Suppose the output price does not change.
Then p = 0 and profit-maximization
implies 0  w x; i.e., a competitive
firm’s input demand curve cannot slope
upward.
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Microeconomics, Spring 2022