Profit-Maximization
Economic Profit
Profit maximization provides the rationale
for firms to choose the feasible production
plan.
 Profit is the difference between revenues
and costs.
 In the following analysis, we will assume
that firms operate within a competitive
market, which implies that the selling
price is exogenous to the firm.

Economic Profit
 When
computing profits we must
include all inputs used by the firm,
valued at their market price. In
particular, we must be aware of the
opportunity cost of using inputs in the
production of the firm rather than on
alternative uses. Example: labor of the
firm’s owner; capital of shareholders.
Economic Profit
 Thus,
the economic definition of
profits requires that we value all inputs
and outputs at their opportunity costs,
which means that the economic
definition will certainly differ from the
accountants definition.
Profits and the Stock Market Value
 Often
the production process that a
firm uses goes on for many years.
Inputs that are acquired today (such as
buildings or machines) last several
years and contribute to the future
production of output. Therefore, firms
have to value a flow of costs and
revenues over time. How should they
evaluate those flows?
Profits and the Stock Market Value
 Even
if we abstract from inflation,
receiving (or paying) today is different
from receiving (or paying) one year
from now. To measure future money
flows we need to calculate the present
value of these future flows, i.e. the
value of these flows from a today’s
perspective.
Profits and the Stock Market Value
The
present value is calculated with
the help of the (real) interest rate,
which can be thought of giving the
price of money in different
moments of time.
Profits and the Stock Market Value
 Therefore,
one can say that the present
value of the firm is the present value of
all future profits. In the case of
corporations where the capital consists
of many shares, the dividends
correspond to shares of profits. Thus,
the stock market value of the firm’s
share is, in a world of certainty, equal to
the correspondent fraction of the
present value of the firm’s future profits.
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
  p1y1 pnyn w1x1 wmxm .
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 euros of
profit earned per hour.
Economic Profit
 Fixed
Inputs are those the quantity of
which the firm cannot vary. As we
have seen, only in the short run can
we have this category.
 Variable Inputs are those the quantity
of which can be freely chosen by the
firm. In the long run, all inputs are
variable.
Economic Profit
 Quasi-Fixed
inputs are those that do
not depend on the quantity of output
being produced, but only apply as
long as the firm is producing a
positive amount of output. There can
easily be quasi-fixed factors in the
long run.
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
~ .
  py  w1x1  w 2x
2
Short-Run Profit-Maximization
 The
firm’s problem is to choose the
production plan that attains the
highest possible profit, 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

The profit maximization problem facing the firm can
be written as:
~
~
Max pf ( x1 , x2 )  w1 x1  w2 x2
x1
The first-order condition for this Max problem is:
*
f ( x1 , ~
x2 )
* ~
p
 w1  0  pMP1 ( x1 , x 2 )  w1
x1
It turns out that the production plan that maximizes
profits has a nice economic interpretation: at the
optimal production plan, the value of the marginal
product of an input must equal its price.
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.
Short-Run Iso-Profit Line
A € iso-profit line contains all the
production plans that provide a profit
level € .
A € iso-profit line’s equation is
~ .
  py  w1x1  w 2x
2
I.e.
~
w1
  w 2x
2.
y
x1 
p
p
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
Short-Run Iso-Profit Lines
y
   
   
  
w1
Slopes  
p
x1
Short-Run Profit-Maximization
y
The short-run production function and
~ .
x

x
technology set for 2
2
~ )
y  f ( x1 , x
2
Technically
inefficient
plans
x1
Short-Run Profit-Maximization
y
   
   
  
~ )
y  f ( x1 , x
2
w1
Slopes  
p
x1
Short-Run Profit-Maximization
y
y*
~ , the
Given p, w1, w2 and x 2  x
2
short-run
* ~
*
(
x
,
x
,
y
).
profit-maximizing plan is 1 2
   
And the maximum
possible profit
w1
is   .
Slopes  
p
x*1
x1
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
Short-Run Profit-Maximization; 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
Short-Run Profit-Maximization; A
Cobb-Douglas Example
p *  2/ 3 ~ 1/ 3
x 2  w1 for x1 gives
Solving ( x1 )
3
3w 1
*  2/ 3
( x1 )

.
~ 1/ 3
px
2
That is,
so
1/ 3
~
* 2 / 3 px 2
( x1 )

3w 1
3/ 2
3/ 2
1/
3
~


 p 
*  px 2 
1/ 2
~
x1  

x2 .


 3w 1 
 3w 1 
Short-Run Profit-Maximization; A
Cobb-Douglas Example
*  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
~ units.
fixed at x
2
Short-Run Profit-Maximization; A
Cobb-Douglas Example
*  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
~ units.
fixed at x
2
The firm’s short-run output level is thus
* 1/ 3 ~ 1/3  p 
y  ( x1 ) x 2  

 3w 1 
*
1/ 2
~ 1/ 2 .
x
2
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
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. The new
tangency point has higher y and higher
x1.
Comparative Statics of Short-Run
Profit-Maximization
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).
 An
Comparative Statics of Short-Run
Profit-Maximization
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, so the new
tangency point has lower y and lower x1.
Comparative Statics of Short-Run
Profit-Maximization
   
   
y
  
~ )
y  f ( x1 , x
2
w1
Slopes  
p
y*
x*1
x1
Long-Run Profit-Maximization
 Now
allow the firm to vary both input
levels.
 Since no input level is fixed, there
are no fixed costs.
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
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
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
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.
 And p  MP1  w1  0
any short-run, so ...
is satisfied in
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.