Lectures in Macroeconomics- Charles W. Upton The Cobb-Douglas Production Function Y = AK L (1- )
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Transcript of Lectures in Macroeconomics- Charles W. Upton The Cobb-Douglas Production Function Y = AK L (1- )
![Page 1: Lectures in Macroeconomics- Charles W. Upton The Cobb-Douglas Production Function Y = AK L (1- )](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649d545503460f94a31a70/html5/thumbnails/1.jpg)
Lectures in Macroeconomics- Charles W. Upton
The Cobb-Douglas Production Function
Y = AKL(1-)
![Page 2: Lectures in Macroeconomics- Charles W. Upton The Cobb-Douglas Production Function Y = AK L (1- )](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649d545503460f94a31a70/html5/thumbnails/2.jpg)
The Cobb-Douglas Production Function
History
• Developed by Paul Douglas and C. W. Cobb in the 1930’s
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The Cobb-Douglas Production Function
History
• Developed by Paul Douglas and C. W. Cobb in the 1930’s– Douglas went on to
be professor at Chicago and U.S. Senator
– Cobb - ??
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The Cobb-Douglas Production Function
The General Problem
• An increase in a nation’s capital stock or labor force means more output.
• Is there a mathematical formula that relates capital, labor and output?
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The Cobb-Douglas Production Function
The General Form
1tttt LKAY
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The Cobb-Douglas Production Function
Increasing Capital
1
1
)2( oo
ooo
LKAY
LAKY
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The Cobb-Douglas Production Function
Increasing Capital
ooo
oo
ooo
YLAK
LKAY
LAKY
22
)2(1
1
1
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The Cobb-Douglas Production Function
Increasing Capital
ooo
oo
ooo
YLAK
LKAY
LAKY
22
)2(1
1
1
Diminishing returns to proportion
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The Cobb-Douglas Production Function
Increasing Labor
1
1
)2( oo
ooo
LAKY
LAKY
![Page 10: Lectures in Macroeconomics- Charles W. Upton The Cobb-Douglas Production Function Y = AK L (1- )](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649d545503460f94a31a70/html5/thumbnails/10.jpg)
The Cobb-Douglas Production Function
Increasing Labor
ooo
oo
ooo
YLAK
LAKY
LAKY
111
1
1
22
)2(
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The Cobb-Douglas Production Function
Increasing Labor
ooo
oo
ooo
YLAK
LAKY
LAKY
111
1
1
22
)2(
Diminishing returns to proportion
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The Cobb-Douglas Production Function
Increasing Both
1
1
)2()2( oo
ooo
LKAY
LAKY
![Page 13: Lectures in Macroeconomics- Charles W. Upton The Cobb-Douglas Production Function Y = AK L (1- )](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649d545503460f94a31a70/html5/thumbnails/13.jpg)
The Cobb-Douglas Production Function
Increasing Both
ooo
oo
ooo
YLAK
LKAY
LAKY
22
)2()2(1
1
1
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The Cobb-Douglas Production Function
Increasing Both
ooo
oo
ooo
YLAK
LKAY
LAKY
22
)2()2(1
1
1
Constant returns to scale
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The Cobb-Douglas Production Function
Substitution
ooo
ooo
YxLKAY
LAKY
1
1
)()2(
Capital and Labor Can be Substituted
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The Cobb-Douglas Production Function
An Illustration
2/12/1
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The Cobb-Douglas Production Function
An Illustration
2/12/1 LAKY
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The Cobb-Douglas Production Function
An Illustration
KLAY
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The Cobb-Douglas Production Function
An Illustration
KLAY A =3 L =10
K =10
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The Cobb-Douglas Production Function
An Illustration
301003)10)(10(3 Y
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The Cobb-Douglas Production Function
Doubling Capital
42230
2003)10)(20(3
Y
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The Cobb-Douglas Production Function
Constant Returns to Scale
604003)20)(20(3 Y
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The Cobb-Douglas Production Function
Substitution
5
30))(20(3
x
xY
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The Cobb-Douglas Production Function
Estimation
)log()1(
)log()log()log(
1
t
ttt
tttt
L
KAY
LKAY
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The Cobb-Douglas Production Function
Estimation
)log(
)log()log(
2
1
tt
tt
L
KtCY
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The Cobb-Douglas Production Function
Estimation
)log(
)log()log(
2
1
tt
tt
L
KY
Statistical issues abound!
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The Cobb-Douglas Production Function
Best Estimate
3
1
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The Cobb-Douglas Production Function
Factor Payments
= % of Income going to owners of capital
• 1- = % of Income going to workers
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The Cobb-Douglas Production Function
How well does it work?
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You can’t beat something with nothing
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The Cobb-Douglas Production Function
CES Production Function
rwLK
%
%
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The Cobb-Douglas Production Function
CES Production Function
rwLK
%
%
Suppose 10% increase in wage rate leads to 10% increase in capital labor
ratio. = 1.
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The Cobb-Douglas Production Function
CES Production Function
rwLK
%
%
Suppose 10% increase in wage rate leads to 5%
increase in capital labor ratio. = ½.
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The Cobb-Douglas Production Function
CES Production Function
rwLK
%
%
In the Cobb-Douglas Production Function,
= 1.
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The Cobb-Douglas Production Function
CES Production Function
• The CES allows for a different elasticity of substitution.
• Little gained.
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The Cobb-Douglas Production Function
Leontief Production Function
K
L
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The Cobb-Douglas Production Function
Leontief Production Function
• K = aY
• L = bY
K
L
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The Cobb-Douglas Production Function
Leontief Production Function
• K = aY
• L = bY
K
L
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The Cobb-Douglas Production Function
Leontief Production Function
• K = aY
• L = bY
K
L
= 0
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The Cobb-Douglas Production Function
Leontief Production Function
• K = aY
• L = bY
K
L
Doesn’t work. We can and do
substitute labor for capital all the time
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The Cobb-Douglas Production Function
Other Factors?
1
ttttt LNDLKAY
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The Cobb-Douglas Production Function
And in Conclusion…
1tttt LKAY
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The Cobb-Douglas Production Function
End
©2003 Charles W. Upton. All rights reserved