Profit-Maximization 利润最大化. A firm uses inputs j = 1…,m to make products i = 1,…n. Output levels are y 1,…,y n. Input levels are x 1,…,x m. Product

Chapter Nineteen

Profit-Maximization利润最大化

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

Profit: Revenues minus economic costs.

p y p y w x w xn n m m1 1 1 1 .

Economic cost vs. accounting cost

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).

Opportunity Costs ( 机会成本 )

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.

An Example: accounting cost and economic cost

Item Accounting cost

Economic cost

Wages (w) $ 40 000 $ 40 000

Interest paid

10 000 10 000

w of owner 0 3000

w of owner’s wife

0 1000

Rent 0 5000

Total cost 50 000 59 000

Economic Profit 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.

Economic Profit How do we value a firm? Suppose the firm’s stream of periodic

economic profits is … and r is the rate of interest.

Then the present-value of the firm’s economic profit stream is

PVr r

0

1 221 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 isand 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

~ .

ywp

xw xp

11

2 2 ~.

Short-Run Iso-Profit Linesy

wp

xw xp

11

2 2 ~

has a slope of

wp1

and a vertical intercept of

w xp2 2~.

Short-Run Iso-Profit Lines

Increasing

profit

y

x1

Slopeswp

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.


x1

Technicallyinefficientplans

y The short-run production function andtechnology set for x x2 2~ .

y f x x ( , ~ )1 2


x1

Increasing

profit

Slopeswp

1

y

y f x x ( , ~ )1 2


x1

y

Slopeswp

1

x1*

y*


x1

y

Slopeswp

1

Given p, w1 and the short-runprofit-maximizing plan is

x1*

y*

x x2 2~ ,( , ~ , ).* *x x y1 2


x1

y

Slopeswp

1

Given p, w1 and the short-runprofit-maximizing plan is And the maximumpossible profitis

x x2 2~ ,( , ~ , ).* *x x y1 2

.

x1*

y*


x1

y

Slopeswp

1

At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximaliso-profit line areequal.

x1*

y*


x1

y

Slopeswp

1

At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximaliso-profit line areequal.

MPwp

at x x y

11

1 2

( , ~ , )* *

x1*

y*


MPwp

p MP w11

1 1

p MP 1 is the value of marginal product of (边际产品价值 ) of input 1,

p MP w 1 1

p MP w 1 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.

Short-Run Profit-Maximization; A Cobb-

Douglas ExampleSuppose the short-run productionfunction is y x x 1

1/321/3~ .

The marginal product of the variableinput 1 is MP

yx

x x11

12 3

21/31

3

/ ~ .

The profit-maximizing condition is

MRP p MPp

x x w1 1 12 3

21/3

13 ( ) ~ .* /


Douglas Examplepx x w

3 12 3

21/3

1( ) ~* / Solving for x1 gives

( )~

.* /xw

px1

2 3 1

21/3

3

That is,( )

~* /x

pxw1

2 3 21/3

13

so xpx

wpw

x121/3

1

3 2

1

3 2

21/2

3 3*

/ /~~ .


Douglas Examplex

pw

x11

3 2

21/2

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 xpw

x* *( ) ~ ~ .

1

1/321/3

1

1/2

21/2

3

Comparative Statics of Short-Run Profit-

Maximization What happens to the short-run profit-

maximizing production plan as the output price p changes?


Maximization

ywp

xw xp

11

2 2 ~The equation of a short-run iso-profit lineis

so an increase in p causes -- a reduction in the slope, and -- a reduction in the vertical intercept.


Maximization

x1

Slopeswp

1

y

y f x x ( , ~ )1 2

x1*

y*


Maximization

x1

Slopeswp

1

y

y f x x ( , ~ )1 2

x1*

y*


Maximization

x1

Slopeswp

1

y

y f x x ( , ~ )1 2

x1*

y*


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).


MaximizationThe Cobb-Douglas example: When

then the firm’s short-rundemand for its variable input 1 is

y x x 11/3

21/3~

y* increases as p increases.

and its short-runsupply is

x1* increases as p increases.

xpw

x11

3 2

21/2

3*

/~

ypw

x* ~ .

3 1

1/2

21/2


Maximization What happens to the short-run profit-

maximizing production plan as the variable input price w1 changes?


Maximization

ywp

xw xp

11

2 2 ~The equation of a short-run iso-profit lineis

so an increase in w1 causes -- an increase in the slope, and -- no change to the vertical intercept.


Maximization

x1

Slopeswp

1

y

y f x x ( , ~ )1 2

x1*

y*


Maximization

x1

Slopeswp

1

y

y f x x ( , ~ )1 2

x1*

y*


Maximization

x1

Slopeswp

1

y

y f x x ( , ~ )1 2

x1*

y*


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).


Maximization

xpw

x11

3 2

21/2

3*

/~

The Cobb-Douglas example: When

then the firm’s short-rundemand for its variable input 1 isy x x 1

1/321/3~

x1* decreases as w1 increases.

y* decreases as w1 increases.

ypw

x* ~ .

3 1

1/2

21/2

and its short-runsupply is

Long-Run Profit-Maximization( 长期利润最大

化 ) 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

1 21 2 1 1 2 2

,max ( , ) .x x

pf x x w x w x

1 21

1

( *, *)0.

f x xp w

x

1 22

2

( *, *)0.

f x xp w

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 （反要素需求函数

） 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 pMP x x

2 2 1 2( *, )w pMP x x

Inverse Factor Demand Curves

x1

w1

1 1 2( , *)pMP x x

Example: C-D Production Function

The production function is1/3 1/31 2y x x

First-order conditions are:

2/3 1/311 2 13

1/3 2/311 2 23

px x w

px x w

Solving for x1 and x2 :

xp

w w2

3

1 2227

*

3*1 2

1 2

.27

px

w w

ypw

p

w w

pw w

* .

3 27 91

1/2 3

1 22

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 （规模报酬递减），

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*

Decreasingreturns-to-scale


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.


x

y

y f x ( )

y”

x’

Increasingreturns-to-scale

y’

x”

Increasing

profit


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） ?


x

y

y f x ( )

y”

x’

Constantreturns-to-scaley’

x”

Increasing

profit


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.


x

y

y f x ( )

y”

x’

Constantreturns-to-scaley’

x”

= 0

Structure

Economic profit Short-run profit maximization

Comparative statics Long-run profit maximization Profit maximization and returns to scale

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Profit-Maximization 利润最大化. A firm uses inputs j = 1…,m to make products i = 1,…n. Output levels are y 1,…,y n. Input levels are x 1,…,x m. Product