Chapter Nineteen
Profit-Maximization
利润最大化
Structure
?Economic profit
?Short-run profit maximization
–Comparative statics
?Long-run profit maximization
?Profit maximization and returns to
scale
?Revealed profit maximization
Economic Profit
?A firm uses inputs j = 1…,m to make
products i = 1,…n.
?Output levels are y1,…,y n.
?Input levels are x1,…,x m.
?Product prices are p1,…,p n.
?Input prices are w1,…,w m.
The Competitive Firm
?The competitive firm takes all output
prices p1,…,p n and all input prices
w1,…,w m as given constants.
Economic Profit
?The economic profit generated by the
production plan (x1,…,x m,y1,…,y n) 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.
Opportunity Costs (机会成本 )
?All inputs must be valued at their
market value.
?Labor
?Capital
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 isPV
r r? ? ? ? ? ??
? ?
0 1 2 21 1( ) ?
Profit Maximization
?A competitive firm seeks to maximize
its present-value.
?How?
?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 Profit Maximization
Short-Run Iso-Profit Lines
?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 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
~
.
Short-Run Iso-Profit Lines
? ?? ?? ?? ??
? ?? ???
y
x1
Slo p es wp? ? 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?
?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
Slo p es wp? ? 1
y
y f x x? (,~ )1 2? ?? ?
? ?? ??? ?? ???
Short-Run Profit-Maximization
x1
y
? ?? ?? ?? ??
? ?? ???
Slo p es wp? ? 1
x1*
y*
Short-Run Profit-Maximization
x1
y
Slo p es wp? ? 1
Given p,w1 and the short-run
profit-maximizing plan is
? ?? ??
x1*
y*
x x2 2? ~,(,~,).* *x x y
1 2
Short-Run Profit-Maximization
x1
y
Slo p es wp? ? 1
Given p,w1 and the short-run
profit-maximizing plan is
And the maximum
possible profit
is
x x2 2? ~,(,~,).* *x x y
1 2
???.
? ?? ??
x1*
y*
Short-Run Profit-Maximization
x1
y
Slo p es wp? ? 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
Slo p es wp? ? 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? 1is the marginal revenue product(边际
收益产量 ) of input 1,the rate at which revenue
Increases with the amount used of input 1.
If then profit increases with x1.
If then 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? 11/ 3 21/ 3~,
The marginal product of the variable
input 1 is MP yx x x1
1
1
2 3
2
1/ 31
3? ?
??
?
/ ~,
The profit-maximizing condition is
M R P p MP p x x w1 1 1 2 3 21/ 3 13? ? ? ??( ) ~,* /
Short-Run Profit-Maximization; A
Cobb-Douglas Examplep
x x w3 1 2 3 21/ 3 1( ) ~* /? ?Solving for x1 gives
( ) ~,* /x w
p x1
2 3 1
2
1/ 3
3? ?
Short-Run Profit-Maximization; A
Cobb-Douglas Examplep
x x w3 1 2 3 21/ 3 1( ) ~* /? ?Solving for x1 gives
( ) ~,* /x w
p x1
2 3 1
2
1/ 3
3? ?
That is,( )
~* /
x p x
w1
2 3 21/ 3
13
?
Short-Run Profit-Maximization; A
Cobb-Douglas Examplep
x x w3 1 2 3 21/ 3 1( ) ~* /? ?Solving for x1 gives
( ) ~,* /x w
p x1
2 3 1
2
1/ 3
3? ?
That is,( )
~* /
x p x
w1
2 3 21/ 3
13
?
so
x p x
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?
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
? ?? ?? ?? ??
? ?? ???
Slo p es wp? ? 1
y
y f x x? (,~ )1 2
x1*
y*
Comparative Statics of Short-Run
Profit-Maximization
x1
Slo p es wp? ? 1
y
y f x x? (,~ )1 2
x1*
y*
Comparative Statics of Short-Run
Profit-Maximization
x1
Slo p es wp? ? 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? 11/ 3 21/ 3~
y p
w
x* ~,? ?
??
?
??3 1
1/ 2
2
1/ 2and its short-runsupply is
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? 11/ 3 21/ 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? 11/ 3 21/ 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
? ?? ?? ?? ??
? ?? ???
Slo p es wp? ? 1
y
y f x x? (,~ )1 2
x1*
y*
Comparative Statics of Short-Run
Profit-Maximization
x1
Slo p es wp? ? 1
y
y f x x? (,~ )1 2
x1*
y*
? ?? ?? ?? ??
? ?? ???
Comparative Statics of Short-Run
Profit-Maximization
x1
Slo p es wp? ? 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? 11/ 3 21/ 3~
y p
w
x* ~,? ?
??
?
??3 1
1/ 2
2
1/ 2and its short-runsupply 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? 11/ 3 21/ 3~
x1* decreases as w1 increases.
y p
w
x* ~,? ?
??
?
??3 1
1/ 2
2
1/ 2and its short-runsupply 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? 11/ 3 21/ 3~
x1* decreases as w1 increases.
y* decreases as w1 increases.
y p
w
x* ~,? ?
??
?
??3 1
1/ 2
2
1/ 2and its short-runsupply is
Long-Run Profit-Maximization
?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.
?The profit-maximization problem is
?FOCs are:
Long-Run Profit-Maximization
12
1 2 1 1 2 2,m a x (,),xx p f x x w x w x??
12
1
1
( *,* ) 0.f x xpw
x
? ??
?
12
2
2
( *,* ) 0.f x xpw
x
? ??
?
Factor Demand Functions
?Demand for inputs 1 and 2 can be
solved as,
1 1 1 2
2 2 1 2
(,,)
(,,)
x x w w p
x x w w p
?
?
Inverse Factor Demand Functions
?For a given optimal demand for x2,
inverse demand function for x1 is
?For a given optimal demand for x1
inverse demand function for x2 is
1 1 1 2(,* )w p M P x x?
2 2 1 2( *,)w p M P x x?
Inverse Factor Demand Curves
x1
w1
1 1 2(,* )p M P x x
Example,C-D Production Function
The production function is
1 / 3 1 / 3
12y x x?
First-order conditions are:
2 / 3 1 / 31
1 2 13
1 / 3 2 / 31
1 2 23
p x x w
p x x w
?
?
?
?
?Solving for x1 and x2,
x p
w w2
3
1 2
227
* ?
3
*
1 2
12
.
27
px
ww
?
y p
w
p
w w
p
w w
*,? ?
??
?
??
?
?
??
?
?
?? ?
3 27 91
1/ 2 3
1 2
2
1/ 2 2
1 2
Plug-in production function to get:
Example,C-D Production Function
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.
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”
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 is chosen at prices sois profit-maximizing at these prices.
Slop e wp? ??
?x
?y
(,)? ?x y (,)? ?w p(,)? ?x y
Revealed Profitability
x
y is chosen at prices sois profit-maximizing at these prices.
Slop e wp? ??
?x
?y
(,)? ?x y (,)? ?w p(,)? ?x y
??x
??y
(,)?? ??x ywould give lower
profits,so
? ? ? ? ? ? ? ?? ? ? ??p y w x p y w x,
Revealed Profitability
x
y
??y
??x ?x
?y
(,)?? ??w p
is chosen at prices
so
(,)?? ??x y
(,)?? ??w p
?? ?? ? ?? ?? ? ?? ? ? ?? ?p y w x p y w x,
(x”,y”) is chosen at prices (w”,p”) so
(x”,y”) is profit-maximizing at these
prices.
Weak Axiom of Profit Maximization
x
y
??y
??x ?x
?y
(,)? ?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
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