Profit-Maximization
利润最大化





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 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
  p1y1  pnyn  w1x1  wmxm .

Profit: Revenues minus economic costs.


Accounting cost: a firm’s actual cash payments for
its inputs (explicit costs)
Economic cost: the sum of explicit cost and
opportunity cost (机会成本) (implicit cost).

All inputs must be valued at their market value.
Labor
 Capital


Opportunity cost: The next (second) best
alternative use of resources sacrificed by making a
choice.
Item
Wages (w)
Interest paid
Accounting
cost
$ 40 000
Economic
cost
$ 40 000
10 000
10 000
w of owner
0
3000
w of owner’s
wife
Rent
0
1000
0
5000
Total cost
50 000
59 000




Output and input levels are typically flows(流量
).
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.



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   0 
1
1r

2
(1  r )
2



A competitive firm seeks to maximize its presentvalue.
How?



Suppose the firm is in a short-run circumstance
~
in which
x 2  x2 .
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

A $ iso-profit line (等利润线) contains all the
production plans that yield a profit level of $ .
The equation of a $ iso-profit line is

I.e.

~
  py  w1x1  w 2x 2 .
~
w1
  w 2x 2
y
x1 
.
p
p
y
w1
p
~
  w 2x 2
x1 
p
has a slope of

w1
p
and a vertical intercept of
~
  w 2x 2
p
.
y
   
   
  
Slopes  
w1
x1
p



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
~ .
x

x
technology set for 2
2
~ )
y  f ( x1 , x
2
Technically
inefficient
plans
x1
y
   
   
  
~ )
y  f ( x1 , x
2
Slopes  
w1
p
x1
y
   
   
  
Slopes  
*
y
*
x1
x1
w1
p
y
~ , the short-run
Given p, w1 and x 2  x
2
* ~
*
profit-maximizing plan is ( x1 , x 2 , y ).
   
Slopes  
*
y
*
x1
x1
w1
p
y
~ , the short-run
Given p, w1 and x 2  x
2
* ~
*
profit-maximizing plan is ( x1 , x 2 , y ).
And the maximum
   
possible profit
is   .
Slopes  
*
y
*
x1
x1
w1
p
y
At the short-run profit-maximizing plan,
the slopes of the short-run production
function and the maximal    
iso-profit line are
equal.
Slopes  
*
y
*
x1
x1
w1
p
y
At the short-run profit-maximizing plan,
the slopes of the short-run production
function and the maximal    
iso-profit line are
equal.
Slopes  
*
y
MP1 
w1
p
* ~
*
at ( x1 , x
,
y
)
2
*
x1
x1
w1
p
MP1 
w1
p

p  MP1  w1
is the value of marginal product of
(边际产品价值) of input 1,
p  MP1
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.
Suppose the short-run production
3 ~ 1/ 3
function is y  x1/
1 x2 .
The marginal product of the variable
y
1  2/ 3 ~ 1/ 3
input 1 is
MP1 
 x1

3
x1
x2 .
The profit-maximizing condition is
MRP1  p  MP1 
p
3
*  2 / 3 ~ 1/ 3
x 2  w1 .
( x1 )
Solving
p
3
*  2 / 3 ~ 1/ 3
( x1 )
x 2  w1
for x1 gives
3w 1
*  2/ 3
( x1 )

.
~ 1/ 3
px
2
That is,
* 2/ 3
( x1 )
so

~ 1/ 3 

px
*
2 
x1  

3
w

1 
1/ 3
~
px 2
3w 1
3/ 2
 p 


 3w 1 
3/ 2
1/ 2
~
x2 .

*
x1  
p 

 3w 1 
3/ 2
1/ 2
~
x2
is the firm’s short-run demand for input
~
x
1 when the level of input 2 is fixed at 2
units.
The firm’s short-run output level is thus

* 1/ 3 ~ 1/ 3
y  ( x1 )
x2  
*
p 

 3w 1 
1/ 2
~ 1/ 2 .
x
2

What happens to the short-run profit-maximizing
production plan as the output price p changes?
The equation of a short-run iso-profit line
~
is
w
w x
y
1x 
1
p
2 2
p
so an increase in p causes
-- a reduction in the slope, and
-- a reduction in the vertical intercept.
   
   
y
  
~ )
y  f ( x1 , x
2
*
y
Slopes  
*
x1
w1
p
x1
y
~ )
y  f ( x1 , x
2
*
y
Slopes  
*
x1
w1
p
x1
y
~ )
y  f ( x1 , x
2
*
y
Slopes  
*
x1
w1
p
x1

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).

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

*
x1  
p 

 3w 1 
3/ 2
 p 
y 

 3w 1 
1/ 2
*
*
1/ 2
~
x2
and its short-run
supply is
~ 1/ 2 .
x
2
increases as p increases.
*
y increases as p increases.
x1

What happens to the short-run profit-maximizing
production plan as the variable input price w1
changes?
The equation of a short-run iso-profit line
~
is
w
w x
y
1x 
1
p
2 2
p
so an increase in w1 causes
-- an increase in the slope, and
-- no change to the vertical intercept.
   
   
y
  
~ )
y  f ( x1 , x
2
*
y
Slopes  
*
x1
w1
p
x1
   
   
y
  
~ )
y  f ( x1 , x
2
*
y
Slopes  
*
x1
w1
p
x1
   
   
y
  
~ )
y  f ( x1 , x
2
Slopes  
*
y
*
x1
w1
p
x1

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).

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

*
x1  
p 

 3w 1 
3/ 2
 p 
y 

 3w 1 
1/ 2
*
*
1/ 2
~
x2
and its short-run
supply is
~ 1/ 2 .
x
2
decreases as w1 increases.
*
y decreases as w1 increases.
x1


Now allow the firm to vary both input levels,
i.e., both x1 and x2 are variable.
Since no input level is fixed, there are no fixed
costs.
Long-Run Profit-Maximization

The profit-maximization problem is
max pf ( x1 , x2 )  w1 x1  w2 x2 .
x1 , x2

FOCs are:
p
f ( x1*, x2 *)
p
x1
f ( x1*, x2 *)
x2
 w1  0.
 w2  0.

Demand for inputs 1 and 2 can be solved as,
x1  x1 ( w1 , w2 , p)
x2  x2 ( w1 , w2 , p)


For a given optimal demand for x2, inverse demand
function for x1 is
w1  pMP1 ( x1 , x2 *)
For a given optimal demand for x1 inverse demand
function for x2 is
w2  pMP2 ( x1*, x2 )
w1
pMP1 ( x1 , x2 *)
x1
The production function is
yx
1/ 3
1
1/ 3
2
x
First-order conditions are:
1
3
1
3
2 / 3
1
1/ 3
2
x
 w1
1/ 3
1
2 / 3
2
 w2
px
px
x

Solving for x1 and x2 :
p
x 
*
1
.
2
1
27 w w2
p
*
x2 
3
3
2
27w 1w 2
Plug-in production function to get:
 p 
*
y 

 3w 1 
1/ 2


p



2
 27w 1w 2 
3
1/ 2

p
2
9 w 1w 2
.


If a competitive firm’s technology exhibits
decreasing returns-to-scale（规模报酬递减），
then the firm has a single long-run profitmaximizing production plan.
y
y  f(x)
y*
Decreasing
returns-to-scale
x*
x


If a competitive firm’s technology exhibits exhibits
increasing returns-to-scale（规模报酬递增），
then the firm does not have a profit-maximizing
plan.
y
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 （规模报酬不变）?
y
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)
=0
y”
Constant
returns-to-scale
y’
x’
x”
x


Economic profit
Short-run profit maximization



Comparative statics
Long-run profit maximization
Profit maximization and returns to scale