Profit Maximization
Molly W. Dahl
Georgetown University
Econ 101 – Spring 2009
1
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 profit function is
~
  py  w1x1  w 2x 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  w1x1  w 2x2 .
3
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

~
  py  w1x1  w 2x2 .

Rearranging
~
w1
  w 2x
2
y
x1 
.
p
p
4
Short-Run Iso-Profit Lines
~
w1
  w 2x 2
y
x1 
p
p
has a slope of
w1

p
and a vertical intercept of
~
  w 2x 2
.
p
5
Short-Run Iso-Profit Lines
y
   
   
  
w1
Slopes  
p
x1
6
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.
7
Short-Run Profit-Maximization
y
   
   
  
~ )
y  f ( x1 , x
2
w1
Slopes  
p
x1
8
Short-Run Profit-Maximization
y
   
   
  
w1
Slopes  
p
y*
x*1
x1
9
Short-Run Profit-Maximization
~
y
y*
Given p, w1 and x 2  x 2 , the short-run
* ~
*
profit-maximizing plan is ( x1 , x 2 , y ).
And the maximum
   
possible profit
is   .
w1
Slopes  
p
x*1
x1
10
Short-Run 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
x*1
2
x1
11
Short-Run Profit-Maximization
w1
MP1 
p
 p  MP1  w1
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 x1.
If p  MP1  w1 then profit decreases with x1.
12
Short-Run Profit-Max: A Cobb-Douglas
Example

In class
13
Comparative Statics of SR Profit-Max

What happens to the short-run profitmaximizing production plan as the variable
input price w1 changes?
14
Comparative Statics of SR Profit-Max
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.
15
Comparative Statics of SR Profit-Max
   
   
y
  
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
x*1
x1
16
Comparative Statics of SR Profit-Max
   
   
y
  
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
x*1
x1
17
Comparative Statics of SR Profit-Max
   
   
y
  
~ )
y  f ( x1 , x
2
w1
Slopes  
p
y*
x*1
x1
18
Comparative Statics of SR Profit-Max

An increase in w1, the price of the firm’s
variable input, causes
a
decrease in the firm’s output level, and
 a decrease in the level of the firm’s variable
input.
19
Comparative Statics of SR Profit-Max

What happens to the short-run profitmaximizing production plan as the output
price p changes?
20
Comparative Statics of SR Profit-Max
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.
21
Comparative Statics of SR Profit-Max
   
   
y
  
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
x*1
x1
22
Comparative Statics of SR Profit-Max
y
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
x*1
x1
23
Comparative Statics of SR Profit-Max
y
~ )
y  f ( x1 , x
2
y*
w1
Slopes  
p
x*1
x1
24
Comparative Statics of SR Profit-Max

An increase in p, the price of the firm’s
output, causes
 an
increase in the firm’s output level, and
 an increase in the level of the firm’s
variable input.
25
Long-Run Profit-Maximization
Now allow the firm to vary both input
levels (both x1 and x2 are variable).
 Since no input level is fixed, there are
no fixed costs.
 For any given level of x2, the profitmaximizing condition for x1 must still
hold.

26
Long-Run Profit-Maximization

The input levels of the long-run profit-maximizing
plan satisfy
p  MP1  w1  0 and p  MP2  w 2  0.


That is, marginal revenue equals marginal cost
for all inputs.
Solve the two equations simultaneously for the
factor demands x1(p, w1, w2) and x2(p, w1, w2)
27
Returns-to-Scale and Profit-Max

If a competitive firm’s technology exhibits
decreasing returns-to-scale then the firm
has a single long-run profit-maximizing
production plan.
28
Returns-to Scale and Profit-Max
y
y  f(x)
y*
Decreasing
returns-to-scale
x*
x
29
Returns-to-Scale and Profit-Max

If a competitive firm’s technology exhibits
exhibits increasing returns-to-scale then
the firm does not have a profit-maximizing
plan.
30
Returns-to Scale and Profit-Max
y
y  f(x)
y”
y’
Increasing
returns-to-scale
x’
x”
x
31
Returns-to-Scale and Profit-Max

So an increasing returns-to-scale
technology is inconsistent with firms being
perfectly competitive.
32
Returns-to-Scale and Profit-Max

What if the competitive firm’s technology
exhibits constant returns-to-scale?
33
Returns-to Scale and Profit-Max
y
y  f(x)
y”
Constant
returns-to-scale
y’
x’
x”
x
34
Returns-to Scale and Profit-Max

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.
35
Returns-to Scale and Profit-Max
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.

36
Returns-to Scale and Profit-Max
y
y  f(x)
=0
y”
Constant
returns-to-scale
y’
x’
x”
x
37