Chapter Eighteen 9/27/2015 1 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....

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Chapter Eighteen
9/27/2015
1
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
      p y p y w x w xn n m m1 1 1 1  .
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 0, 1,
2, … and r is the rate of interest.
Then the present-value of the firm’s
economic profit stream is
PV
r r
 




 
0
1 2
21 1( )

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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
 Its short-run production function is
y f x x ( , ~ ).1 2
x x2 2
~ .
Economic Profit
Suppose the firm is in a short-run
circumstance in which
 Its short-run production function is
The firm’s fixed cost is
and its profit function is
y f x x ( , ~ ).1 2
   py w x w x1 1 2 2
~ .
x x2 2
~ .
FC w x 2 2
~
Short-Run Iso-Profit Lines
A $ iso-profit line contains all the
production plans that provide a profit
level $ .
A $ iso-profit line’s equation is
   py w x w x1 1 2 2
~ .
Short-Run Iso-Profit Lines
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 x w x1 1 2 2
~ .
y
w
p
x
w x
p
 
1
1
2 2
~
.
Short-Run Iso-Profit Lines
y
w
p
x
w x
p
 
1
1
2 2
~
has a slope of

w
p
1
and a vertical intercept of
  w x
p
2 2
~
.
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Short-Run Iso-Profit Lines
  
  
  
y
x1
Slopes
w
p
  1
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?
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?
A: The production function.
Short-Run Profit-Maximization
x1
Technically
inefficient
plans
y The short-run production function and
technology set for x x2 2
~ .
y f x x ( , ~ )1 2
Short-Run Profit-Maximization
x1
Slopes
w
p
  1
y
y f x x ( , ~ )1 2
  
  
  
Short-Run Profit-Maximization
x1
y
  
  
  
Slopes
w
p
  1
x1
*
y*
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Short-Run Profit-Maximization
x1
y
Slopes
w
p
  1
Given p, w1 and XXXXXXXXXXthe short-run
profit-maximizing plan is
  
x1
*
y*
x x2 2
~ ,
( , ~ , ).
* *x x y1 2
Short-Run Profit-Maximization
x1
y
Slopes
w
p
  1
Given p, w1 and XXXXXXXXXXthe short-run
profit-maximizing plan is
And the maximum
possible profit
is
x x2 2
~ ,
( , ~ , ).
* *x x y1 2
 .
  
x1
*
y*
Short-Run Profit-Maximization
x1
y
Slopes
w
p
  1
At the short-run profit-maximizing plan,
the slopes of the short-run production
function and the maximal
iso-profit line are
equal.
  
x1
*
y*
Short-Run Profit-Maximization
x1
y
Slopes
w
p
  1
At the short-run profit-maximizing plan,
the slopes of the short-run production
function and the maximal
iso-profit line are
equal.
MP
w
p
at x x y
1
1
1 2

( , ~ , )
* *
  
x1
*
y*
Short-Run Profit-Maximization
MP
w
p
p MP w1
1
1 1   
p MP 1 is the marginal revenue product of
input 1, the rate at which revenue increases
with the amount used of input 1.
If XXXXXXXXXXthen profit increases with x1.
If XXXXXXXXXXthen profit decreases with x1.
p MP w 1 1
p MP w 1 1
Short-Run Profit-Maximization; A
Cobb-Douglas Example
Suppose the short-run production
function is y x x 1
1/3
2
1/3~ .
The marginal product of the variable
input 1 is
MP
y
x
x x1
1
1
2 3
2
1/31
3
  


/ ~ .
The profit-maximizing condition is
MRP p MP
p
x x w1 1 1
2 3
2
1/3
1
3
   ( ) ~ .* /
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Short-Run Profit-Maximization; A
Cobb-Douglas Example
p
x x w
3
1
2 3
2
1/3
1( )
~* / Solving for x1 gives
( )
~
.
* /x
w
px
1
2 3 1
2
1/3
3 
Short-Run Profit-Maximization; A
Cobb-Douglas Example
p
x x w
3
1
2 3
2
1/3
1( )
~* / Solving for x1 gives
( )
~
.
* /x
w
px
1
2 3 1
2
1/3
3 
That is,
( )
~
* /x
px
w
1
2 3 2
1/3
13

Short-Run Profit-Maximization; A
Cobb-Douglas Example
p
x x w
3
1
2 3
2
1/3
1( )
~* / Solving for x1 gives
( )
~
.
* /x
w
px
1
2 3 1
2
1/3
3 
That is,
( )
~
* /x
px
w
1
2 3 2
1/3
13

so x
px
w
p
w
x1
2
1/3
1
3 2
1
3 2
2
1/2
3 3
*
/ /~
~ .







 






Short-Run Profit-Maximization; A
Cobb-Douglas Example
x
p
w
x1
1
3 2
2
1/2
3
*
/
~





 is the firm’s
short-run demand
for input 1 when the level of input 2 is
fixed at units. ~x2
Short-Run Profit-Maximization; A
Cobb-Douglas Example
x
p
w
x1
1
3 2
2
1/2
3
*
/
~





 is the firm’s
short-run demand
for input 1 when the level of input 2 is
fixed at units. ~x2
The firm’s short-run output level is thus
y x x
p
w
x* *( ) ~ ~ . 





1
1/3
2
1/3
1
1/2
2
1/2
3
Comparative Statics of Short-Run
Profit-Maximization
What happens to the short-run profit-
maximizing production plan as the
output price p changes?
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Comparative Statics of Short-Run
Profit-Maximization
y
w
p
x
w x
p
 
1
1
2 2
~
The equation of a short-run iso-profit line
is
so an increase in p causes
-- a reduction in the slope, and
-- a reduction in the vertical intercept.
Comparative Statics of Short-Run
Profit-Maximization
x1
  
  
  
Slopes
w
p
  1
y
y f x x ( , ~ )1 2
x1
*
y*
Comparative Statics of Short-Run
Profit-Maximization
x1
Slopes
w
p
  1
y
y f x x ( , ~ )1 2
x1
*
y*
Comparative Statics of Short-Run
Profit-Maximization
x1
Slopes
w
p
  1
y
y f x x ( , ~ )1 2
x1
*
y*
Comparative Statics of Short-Run
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).
Comparative Statics of Short-Run
Profit-Maximization
x
p
w
x1
1
3 2
2
1/2
3
*
/
~






The Cobb-Douglas example: When
then the firm’s short-run
demand for its variable input 1 is
y x x 1
1/3
2
1/3~
y
p
w
x* ~ .






3 1
1/2
2
1/2
and its short-run
supply is
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Comparative Statics of Short-Run
Profit-Maximization
The Cobb-Douglas example: When
then the firm’s short-run
demand for its variable input 1 is
y x x 1
1/3
2
1/3~
x1
* increases as p increases.
and its short-run
supply is
x
p
w
x1
1
3 2
2
1/2
3
*
/
~






y
p
w
x* ~ .






3 1
1/2
2
1/2
Comparative Statics of Short-Run
Profit-Maximization
The Cobb-Douglas example: When
then the firm’s short-run
demand for its variable input 1 is
y x x 1
1/3
2
1/3~
y* increases as p increases.
and its short-run
supply is
x1
* increases as p increases.
x
p
w
x1
1
3 2
2
1/2
3
*
/
~






y
p
w
x* ~ .






3 1
1/2
2
1/2
Comparative Statics of Short-Run
Profit-Maximization
What happens to the short-run profit-
maximizing production plan as the
variable input price w1 changes?
Comparative Statics of Short-Run
Profit-Maximization
y
w
p
x
w x
p
 
1
1
2 2
~
The equation of a short-run iso-profit line
is
so an increase in w1 causes
-- an increase in the slope, and
-- no change to the vertical intercept.
Comparative Statics of Short-Run
Profit-Maximization
x1
  
  
  
Slopes
w
p
  1
y
y f x x ( , ~ )1 2
x1
*
y*
Comparative Statics of Short-Run
Profit-Maximization
x1
Slopes
w
p
  1
y
y f x x ( , ~ )1 2
x1
*
y*
  
  
  
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Comparative Statics of Short-Run
Profit-Maximization
x1
Slopes
w
p
  1
y
y f x x ( , ~ )1 2
x1
*
y*
  
  
  
Comparative Statics of Short-Run
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).
Comparative Statics of Short-Run
Profit-Maximization
x
p
w
x1
1
3 2
2
1/2
3
*
/
~






The Cobb-Douglas example: When
then the firm’s short-run
demand for its variable input 1 is
y x x 1
1/3
2
1/3~
y
p
w
x* ~ .






3 1
1/2
2
1/2
and its short-run
supply is
Comparative Statics of Short-Run
Profit-Maximization
x
p
w
x1
1
3 2
2
1/2
3
*
/
~






The Cobb-Douglas example: When
then the firm’s short-run
demand for its variable input 1 is
y x x 1
1/3
2
1/3~
x1
* decreases as w1 increases.
y
p
w
x* ~ .






3 1
1/2
2
1/2
and its short-run
supply is
Comparative Statics of Short-Run
Profit-Maximization
x
p
w
x1
1
3 2
2
1/2
3
*
/
~






The Cobb-Douglas example: When
then the firm’s short-run
demand for its variable input 1 is
y x x 1
1/3
2
1/3~
x1
* decreases as w1 increases.
y* decreases as w1 increases.
y
p
w
x* ~ .






3 1
1/2
2
1/2
and its short-run
supply is
Long-Run Profit-Maximization
Now allow the firm to vary both input
levels.
Since no input level is fixed, there
are no fixed costs.
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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.
Long-Run Profit-Maximization
y
w
p
x
w x
p
 
1
1
2 2
The equation of a long-run iso-profit line
is
so an increase in x2 causes
-- no change to the slope, and
-- an increase in the vertical intercept.
Long-Run Profit-Maximization
x1
y
y f x x ( , )1 2
Long-Run Profit-Maximization
x1
y
y f x x ( , )1 22
y f x x ( , )1 2
y f x x ( , )1 23
Larger levels of input 2 increase the
productivity of input 1.
Long-Run Profit-Maximization
x1
y
y f x x ( , )1 22
y f x x ( , )1 2
y f x x ( , )1 23
Larger levels of input 2 increase the
productivity of input 1.
The marginal product
of input 2 is
diminishing.
Long-Run Profit-Maximization
x1
y
y f x x ( , )1 22
y f x x ( , )1 2
y f x x ( , )1 23
Larger levels of input 2 increase the
productivity of input 1.
The marginal product
of input 2 is
diminishing.
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Long-Run Profit-Maximization
x1
y
y f x x ( , )1 22
y f x x ( , )1 2
y f x x ( , )1 23
y x*( )2
x x1 2
*
( )
x x1 22
*
( )
x x1 23
*
( )
y x*( )2 2
y x*( )3 2
p MP w  1 1 0 for each short-run
production plan.
Long-Run Profit-Maximization
x1
y
y f x x ( , )1 22
y f x x ( , )1 2
y f x x ( , )1 23
The marginal product
of input 2 is
diminishing so ...
y x*( )2
x x1 2
*
( )
x x1 22
*
( )
x x1 23
*
( )
y x*( )2 2
y x*( )3 2
for each short-run
production plan.
p MP w  1 1 0
Long-Run Profit-Maximization
x1
y
y f x x ( , )1 22
y f x x ( , )1 2
y f x x ( , )1 23
the marginal profit
of input 2 is
diminishing.
y x*( )2
x x1 2
*
( )
x x1 22
*
( )
x x1 23
*
( )
y x*( )2 2
y x*( )3 2
for each short-run
production plan.
p MP w  1 1 0
Long-Run Profit-Maximization
Profit will increase as x2 increases so
long as the marginal profit of input 2
The profit-maximizing level of input 2
therefore satisfies
p MP w  2 2 0.
p MP w  2 2 0.
Long-Run Profit-Maximization
Profit will increase as x2 increases so
long as the marginal profit of input 2
The profit-maximizing level of input 2
therefore satisfies
And XXXXXXXXXXis satisfied in
any short-run, so ...
p MP w  1 1 0
p MP w  2 2 0.
p MP w  2 2 0.
Long-Run Profit-Maximization
The input levels of the long-run
profit-maximizing plan satisfy
That is, marginal revenue equals
marginal cost for all inputs.
p MP w  2 2 0.p MP w  1 1 0 and
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Long-Run Profit-Maximization
x
p
w
x1
1
3 2
2
1/2
3
*
/
~






The Cobb-Douglas example: When
then the firm’s short-run
demand for its variable input 1 is
y x x 1
1/3
2
1/3~
y
p
w
x* ~ .






3 1
1/2
2
1/2
and its short-run
supply is
Short-run profit is therefore …
Long-Run Profit-Maximization
   






 





 
py w x w x
p
p
w
x w
p
w
x w x
* *
/
~
~ ~ ~
1 1 2 2
1
1/2
2
1/2
1
1
3 2
2
1/2
2 2
3 3
Long-Run Profit-Maximization
   






 





 






 





 
py w x w x
p
p
w
x w
p
w
x w x
p
p
w
x w
p
w
p
w
w x
* *
/
~
~ ~ ~
~ ~
1 1 2 2
1
1/2
2
1/2
1
1
3 2
2
1/2
2 2
1
1/2
2
1/2
1
1 1
1/2
2 2
3 3
3 3 3
Long-Run Profit-Maximization
   






 





 






 





 






 
py w x w x
p
p
w
x w
p
w
x w x
p
p
w
x w
p
w
p
w
w x
p p
w
x w x
* *
/
~
~ ~ ~
~ ~
~ ~
1 1 2 2
1
1/2
2
1/2
1
1
3 2
2
1/2
2 2
1
1/2
2
1/2
1
1 1
1/2
2 2
1
1/2
2
1/2
2 2
3 3
3 3 3
2
3 3
Long-Run Profit-Maximization
   






 





 






 





 






 









py w x w x
p
p
w
x w
p
w
x w x
p
p
w
x w
p
w
p
w
w x
p p
w
x w x
p
w
x
* *
/
~
~ ~ ~
~ ~
~ ~
~
1 1 2 2
1
1/2
2
1/2
1
1
3 2
2
1/2
2 2
1
1/2
2
1/2
1
1 1
1/2
2 2
1
1/2
2
1/2
2 2
3
1
1/2
2
3 3
3 3 3
2
3 3
4
27
1/2
2 2 w x
~ .
Long-Run Profit-Maximization
 







 
4
27
3
1
1/2
2
1/2
2 2
p
w
x w x~ ~ .
What is the long-run profit-maximizing
level of input 2? Solve
0
1
2
4
272
3
1
1/2
2
1/2
2 







 



~
~
x
p
w
x w
to get ~ .*x x
p
w w
2 2
3
1 2
2
27
 
9/27/2015
12
Long-Run Profit-Maximization
What is the long-run profit-maximizing
input 1 level? Substitute
x
p
w
x1
1
3 2
2
1/2
3
*
/
~





x
p
w w
2
3
1 2
2
27
*  into
to get
Long-Run Profit-Maximization
What is the long-run profit-maximizing
input 1 level? Substitute
x
p
w
x1
1
3 2
2
1/2
3
*
/
~





x
p
w w
2
3
1 2
2
27
*  into
to get
x
p
w
p
w w
p
w w
1
1
3 2 3
1 2
2
1/2
3
1
2
2
3 27 27
*
/
.













 
Long-Run Profit-Maximization
What is the long-run profit-maximizing
output level? Substitute
x
p
w w
2
3
1 2
2
27
*  into
to get
y
p
w
x* ~






3 1
1/2
2
1/2
Long-Run Profit-Maximization
What is the long-run profit-maximizing
output level? Substitute
x
p
w w
2
3
1 2
2
27
*  into
to get
y
p
w
p
w w
p
w w
*
.













 3 27 91
1/2 3
1 2
2
1/2
2
1 2
y
p
w
x* ~






3 1
1/2
2
1/2
Long-Run Profit-Maximization
So given the prices p, w1 and w2, and
the production function y x x 1
1/3
2
1/3
the long-run profit-maximizing production
plan is
( , , ) , , .
* * *x x y
p
w w
p
w w
p
w w
1 2
3
1
2
2
3
1 2
2
2
XXXXXXXXXX









Returns-to-Scale and Profit-
Maximization
 If a competitive firm’s technology
exhibits decreasing returns-to-scale
then the firm has a single long-run
profit-maximizing production plan.
9/27/2015
13
Returns-to Scale and Profit-
Maximization
x
y
y f x ( )
y*
x*
Decreasing
returns-to-scale
Returns-to-Scale and Profit-
Maximization
 If a competitive firm’s technology
exhibits exhibits increasing returns-
to-scale then the firm does not have
a profit-maximizing plan.
Returns-to Scale and Profit-
Maximization
x
y
y f x ( )
y”
x’
Increasing
returns-to-scale
y’
x”
Returns-to-Scale and Profit-
Maximization
So an increasing returns-to-scale
technology is inconsistent with firms
being perfectly competitive.
Returns-to-Scale and Profit-
Maximization
What if the competitive firm’s
technology exhibits constant
returns-to-scale?
Returns-to Scale and Profit-
Maximization
x
y
y f x ( )
y”
x’
Constant
returns-to-scaley’
x”
9/27/2015
14
Returns-to Scale and Profit-
Maximization
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.
Returns-to Scale and Profit-
Maximization
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.
Returns-to Scale and Profit-
Maximization
x
y
y f x ( )
y”
x’
Constant
returns-to-scaley’
x”
 = 0
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?
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 profit-
maximizing for the prices (w’,p’).
Revealed Profitability
x
y
Slope
w
p



x
y
( , ) x y is chosen at prices ( , ) w p
9/27/2015
15
Revealed Profitability
x
y
is chosen at prices XXXXXXXXXXso
is profit-maximizing at these prices.
Slope
w
p



x
y
( , ) x y ( , ) w p
( , ) x y
Revealed Profitability
x
y
is chosen at prices XXXXXXXXXXso
is profit-maximizing at these prices.
Slope
w
p



x
y
( , ) x y ( , ) w p
( , ) x y
x
y ( , ) x y would give higher
profits, so why is it not
chosen?
Revealed Profitability
x
y
is chosen at prices XXXXXXXXXXso
is profit-maximizing at these prices.
Slope
w
p



x
y
( , ) x y ( , ) w p
( , ) x y
x
y ( , ) x y would give higher
profits, so why is it not
chosen? Because it is
not a feasible plan.
Revealed Profitability
x
y
is chosen at prices XXXXXXXXXXso
is profit-maximizing at these prices.
Slope
w
p



x
y
( , ) x y ( , ) w p
( , ) x y
x
y ( , ) x y would give higher
profits, so why is it not
chosen? Because it is
not a feasible plan.
So the firm’s technology set must lie under the
iso-profit line.
Revealed Profitability
x
y
is chosen at prices XXXXXXXXXXso
is profit-maximizing at these prices.
Slope
w
p



x
y
( , ) x y ( , ) w p
( , ) x y
x
y
So the firm’s technology set must lie under the
iso-profit line.
The technology
set is somewhere
in here
Revealed Profitability
x
y
is chosen at prices XXXXXXXXXXso
maximizes profit at these prices.
( , ) x y ( , ) w p
y
x
Slope
w
p



x
y
( , ) x y
would provide higher
profit but it is not chosen
( , ) x y
9/27/2015
16
Revealed Profitability
x
y
is chosen at prices XXXXXXXXXXso
maximizes profit at these prices.
( , ) x y ( , ) w p
y
x x
y
( , ) x y
would provide higher
profit but it is not chosen
because it is not feasible
( , ) x y
Slope
w
p



Revealed Profitability
x
y
is chosen at prices XXXXXXXXXXso
maximizes profit at these prices.
( , ) x y ( , ) w p
y
x x
y
( , ) x y
would provide higher
profit but it is not chosen
because it is not feasible so
the technology set lies under
the iso-profit line.
( , ) x y
Slope
w
p



Revealed Profitability
x
y
is chosen at prices XXXXXXXXXXso
maximizes profit at these prices.
( , ) x y ( , ) w p
y
x x
y
( , ) x y
Slope
w
p



The technology set is
also somewhere in
here.
Revealed Profitability
x
y
y
x x
y
The firm’s technology set must lie under
both iso-profit lines
Revealed Profitability
x
y
y
x x
y
The firm’s technology set must lie under
both iso-profit lines
The technology set
is somewhere
in this intersection
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.
9/27/2015
17
Revealed Profitability
x
y
y
x x
y
The firm’s technology set must lie under
all the iso-profit lines
y
x
( , ) w p
( , ) w p
( , ) w p
Revealed Profitability
x
y
y
x x
y
The firm’s technology set must lie under
all the iso-profit lines
y
x
( , ) w p
( , ) w p
( , ) w p
Revealed Profitability
x
y
y
x x
y
The firm’s technology set must lie under
all the iso-profit lines
y
x
( , ) w p
( , ) w p
( , ) w p
y f x ( )
Revealed Profitability
What else can be learned from the
firm’s choices of profit-maximizing
production plans?
Revealed Profitability
x
y
y
x x
y
The firm’s technology set must lie under
all the iso-profit lines
( , ) w p
( , ) w p
is chosen at prices
so
( , ) x y
( , ) w p
          p y w x p y w x .
is chosen at prices
so
( , ) x y
( , ) w p
          p y w x p y w x .
Revealed Profitability
          p y w x p y w x
          p y w x p y w x
and
so
          p y w x p y w x
            p y w x p y w x .
and
Adding gives
( ) ( )
( ) ( ) .
         
        
p p y w w x
p p y w w x
9/27/2015
18
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 profit-
maximization.
Revealed Profitability
   p y w x
is a necessary implication of profit-
maximization.
Suppose the input price does not change.
Then w = 0 and profit-maximization
implies ; i.e., a competitive
firm’s output supply curve cannot slope
downward.
 p y  0
Revealed Profitability
   p y w x
is a necessary implication of profit-
maximization.
Suppose the output price does not change.
Then p = 0 and profit-maximization
implies ; i.e., a competitive
firm’s input demand curve cannot slope
upward.
0   w x
Answered 1 days AfterMay 10, 2021

Solution

Preeta answered on May 11 2021
22 Votes

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