Chapter Eighteen
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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
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
Co
-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
Co
-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
Co
-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
Co
-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
Co
-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
Co
-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 Co
-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 Co
-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 Co
-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