第19章讲义利润最大化

equal. y*
MP1

w1 p
Slopes w1 p
at (x*1, x~2, y* )
x*1
x1
Short-Run Profit-Maximization
MP1

w1 p

p MP1 w1
p MP1 is 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.
Q: What is this constraint? A: The production function.
Short-Run Profit-Maximization
y
Ttehcehsnhoolorgt-yrusnetpfroorduxc2tionx~
function
2.
and
y f(x1, x~2 )
Cobb-Douglas Example
Solving
p 3
(x*1
)

2/
3
x~12/
3

w1
for
x1
gives
(x*1) 2/3

3w1 px~12/ 3
.
Short-Run Profit-Maximization; A
Cobb-Douglas Example
Solving
p 3
(x*1
)
y f(x1, x~2 ).
fixed cost is FC

w 2x~2
and its profit function is py w1x1 w 2x~2.
Short-Run Iso-Profit Lines
A $ iso-profit line (等利润线) contains all the production plans that yield a profit level of $ .
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.
The equation of a $ iso-profit line is
py w1x1 w 2x~2.
I.e.
y

w1 p
x1



w 2x~ 2 p
.
Short-Run Iso-Profit Lines
y

w1 p
x1



w 2x~ 2 p
has a slope of
is . y*
Slopes w1 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
p1y1pnyn w1x1wmxm.
Economic Profit
Output and input levels are typically flows.
E.g. x1 might be the number of labor units used per hour.
Technically inefficient plans
x1
Short-Run Profit-Maximization
y




y
f(x1
,
x~ 2
)
Slopes w1 p
x1
Short-Run Profit-Maximization
y

Chapter Nineteen
Profit-Maximization
Structure
Economic profit Short-run profit maximization
– Comparative statics Long-run profit maximization Profit maximization and returns to
x*1


p 3w 1
3/2
x~12/ 2
is the firm’s short-run demand
for input 1 when the level of input 2 is fixed at x~2 units.
Short-Run Profit-Maximization; A Cobb-Douglas Example
The marginal product of the variable
input 1 is
MP1

y x1

1 3
x1
2/
3x~12/
3
.
The profit-maximizing condition is
MRP1

p

MP1

p 3
( x*1
)

2/
3
x~12/
3

w1.
Short-Run Profit-Maximization; A
equal. y*
Slopes w1 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
Comparative Statics of Short-Run Profit-Maximization
The equation of a short-run iso-profit line
is
y

2x~ 2 p
so an increase in p causes -- a reduction in the slope, and -- a reduction in the vertical intercept.
x*1
x1
Short-Run Profit-Maximization
y
Given p, w1 and x2 x~2,
profit-maximizing plan is
the short-run
(x*1, x~2, y* ).
And the maximum

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

0

1 1r

2 (1 r)2


Profit Maximization
A competitive firm seeks to maximize its present-value.
x*1


p 3w 1
3/2
x~12/ 2
is the firm’s short-run demand
for input 1 when the level of input 2 is fixed at x~2 units.
The firm’s short-run output level is thus
How?
Short-Run Profit Maximization
Suppose the firm is in a short-run
circumstance in which x2 x~2.
Its short-run production function is
The
firm’s
3
x~12/
3

w1
for
x1
gives
(x*1) 2/3

3w1 px~12/ 3
.
That is,
( x*1 ) 2/ 3

px~12/ 3 3w1
so
x*1


px~12/ 3 3w1

3/2


p 3w 1
3/2
x~12/ 2 .
Short-Run Profit-Maximization; A Cobb-Douglas Example
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
w1 p
and a vertical intercept of
w2x~2 . p
Short-Run Iso-Profit Lines
y

Slopes w1
p
x1
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.
If p MP1 w1 then profit increases with x1. If p MP1 w1 then profit decreases with x1.
Short-Run Profit-Maximization; A Cobb-Douglas Example
Suppose the short-run production function is y x11/3x~12/3.
scale Revealed 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.
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.

2/
3
x~12/
3

w1
for
x1
gives
(x*1) 2/3

3w1 px~12/ 3
.
That is,
( x*1 ) 2/ 3

px~12/ 3 3w1
Short-Run Profit-Maximization; A
Cobb-Douglas Example
Solving
p 3
(x*1
)

2/
y*

(x*1)1/3 x~12/3


p 3w1

1/
2
x~12/
2
.
Comparative Statics of Short-Run Profit-Maximization
What happens to the short-run profitmaximizing production plan as the output price p changes?

y*
Slopes w1 p
x*1
x1
Short-Run Profit-Maximization
y
Given p, w1 and x2 x~2,
profit-maximizing plan is
the short-run
(x*1, x~2, y* ).

y*
Slopes w1 p