![Page 1: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/1.jpg)
1
Profit Maximization
Molly W. DahlGeorgetown UniversityEcon 101 – Spring 2009
![Page 2: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/2.jpg)
2
Economic Profit Suppose the firm is in a short-run
circumstance in which Its short-run production function is
The firm’s profit function is
y f x x ( , ~ ).1 2
py w x w x1 1 2 2~ .
x x2 2~ .
![Page 3: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/3.jpg)
3
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~ .
![Page 4: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/4.jpg)
4
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
Rearranging
py w x w x1 1 2 2~ .
ywp
xw xp
11
2 2 ~.
![Page 5: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/5.jpg)
5
Short-Run Iso-Profit Lines
ywp
xw xp
11
2 2 ~
has a slope of
wp1
and a vertical intercept of
w xp2 2~.
![Page 6: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/6.jpg)
6
Short-Run Iso-Profit Lines
Increasing
profit
y
x1
Slopeswp
1
![Page 7: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/7.jpg)
7
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.
![Page 8: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/8.jpg)
8
Short-Run Profit-Maximization
x1
Increasing
profit
Slopeswp
1
y
y f x x ( , ~ )1 2
![Page 9: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/9.jpg)
9
Short-Run Profit-Maximization
x1
y
Slopeswp
1
x1*
y*
![Page 10: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/10.jpg)
10
Short-Run Profit-Maximization
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*
![Page 11: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/11.jpg)
11
Short-Run Profit-Maximization
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*
![Page 12: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/12.jpg)
12
Short-Run Profit-Maximization
MPwp
p MP w11
1 1
p MP 1 is the marginal revenue product ofinput 1, the rate at which revenue increaseswith the amount used of input 1.
If then profit increases with x1.If then profit decreases with x1.
p MP w 1 1p MP w 1 1
![Page 13: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/13.jpg)
13
Short-Run Profit-Max: A Cobb-Douglas Example
In class
![Page 14: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/14.jpg)
14
Comparative Statics of SR Profit-Max
What happens to the short-run profit-maximizing production plan as the variable input price w1 changes?
![Page 15: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/15.jpg)
15
Comparative Statics of SR Profit-Max
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.
![Page 16: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/16.jpg)
16
Comparative Statics of SR Profit-Max
x1
Slopeswp
1
y
y f x x ( , ~ )1 2
x1*
y*
![Page 17: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/17.jpg)
17
Comparative Statics of SR Profit-Max
x1
Slopeswp
1
y
y f x x ( , ~ )1 2
x1*
y*
![Page 18: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/18.jpg)
18
Comparative Statics of SR Profit-Max
x1
Slopeswp
1
y
y f x x ( , ~ )1 2
x1*
y*
![Page 19: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/19.jpg)
19
Comparative Statics of SR Profit-Max
An increase in w1, the price of the firm’s variable input, causesa decrease in the firm’s output level, anda decrease in the level of the firm’s
variable input.
![Page 20: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/20.jpg)
20
Comparative Statics of SR Profit-Max
What happens to the short-run profit-maximizing production plan as the output price p changes?
![Page 21: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/21.jpg)
21
Comparative Statics of SR Profit-Max
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.
![Page 22: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/22.jpg)
22
Comparative Statics of SR Profit-Max
x1
Slopeswp
1
y
y f x x ( , ~ )1 2
x1*
y*
![Page 23: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/23.jpg)
23
Comparative Statics of SR Profit-Max
x1
Slopeswp
1
y
y f x x ( , ~ )1 2
x1*
y*
![Page 24: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/24.jpg)
24
Comparative Statics of SR Profit-Max
x1
Slopeswp
1
y
y f x x ( , ~ )1 2
x1*
y*
![Page 25: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/25.jpg)
25
Comparative Statics of SR Profit-Max
An increase in p, the price of the firm’s output, causesan increase in the firm’s output level, andan increase in the level of the firm’s
variable input.
![Page 26: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/26.jpg)
26
Long-Run Profit-Maximization
Now allow the firm to vary both input levels (both x1 and x2 are variable).
Since no input level is fixed, there are no fixed costs.
For any given level of x2, the profit-maximizing condition for x1 must still hold.
![Page 27: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/27.jpg)
27
Long-Run Profit-Maximization
The input levels of the long-run profit-maximizing plan satisfy
That is, marginal revenue equals marginal cost for all inputs.
Solve the two equations simultaneously for the factor demands x1(p, w1, w2) and x2(p, w1, w2)
p MP w 2 2 0.p MP w 1 1 0 and
![Page 28: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/28.jpg)
28
Returns-to-Scale and Profit-Max
If a competitive firm’s technology exhibits decreasing returns-to-scale then the firm has a single long-run profit-maximizing production plan.
![Page 29: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/29.jpg)
29
Returns-to Scale and Profit-Max
x
y
y f x ( )
y*
x*
Decreasingreturns-to-scale
![Page 30: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/30.jpg)
30
Returns-to-Scale and Profit-Max
If a competitive firm’s technology exhibits exhibits increasing returns-to-scale then the firm does not have a profit-maximizing plan.
![Page 31: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/31.jpg)
31
Returns-to Scale and Profit-Max
x
y
y f x ( )
y”
x’
Increasingreturns-to-scale
y’
x”
Increasing
profit
![Page 32: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/32.jpg)
32
Returns-to-Scale and Profit-Max
So an increasing returns-to-scale technology is inconsistent with firms being perfectly competitive.
![Page 33: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/33.jpg)
33
Returns-to-Scale and Profit-Max
What if the competitive firm’s technology exhibits constant returns-to-scale?
![Page 34: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/34.jpg)
34
Returns-to Scale and Profit-Max
x
y
y f x ( )
y”
x’
Constantreturns-to-scaley’
x”
Increasing
profit
![Page 35: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/35.jpg)
35
Returns-to Scale and Profit-Max
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.
![Page 36: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/36.jpg)
36
Returns-to Scale and Profit-Max
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.
![Page 37: 1 Profit Maximization Molly W. Dahl Georgetown University Econ 101 – Spring 2009](https://reader035.vdocument.in/reader035/viewer/2022070401/56649f1d5503460f94c344c2/html5/thumbnails/37.jpg)
37
Returns-to Scale and Profit-Max
x
y
y f x ( )
y”
x’
Constantreturns-to-scaley’
x”
= 0