lectures in macroeconomics- charles w. upton the cobb-douglas production function y = ak l (1- )
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Lectures in Macroeconomics- Charles W. Upton
The Cobb-Douglas Production Function
Y = AKL(1-)
The Cobb-Douglas Production Function
History
• Developed by Paul Douglas and C. W. Cobb in the 1930’s
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 - ??
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?
The Cobb-Douglas Production Function
The General Form
1tttt LKAY
The Cobb-Douglas Production Function
Increasing Capital
1
1
)2( oo
ooo
LKAY
LAKY
The Cobb-Douglas Production Function
Increasing Capital
ooo
oo
ooo
YLAK
LKAY
LAKY
22
)2(1
1
1
The Cobb-Douglas Production Function
Increasing Capital
ooo
oo
ooo
YLAK
LKAY
LAKY
22
)2(1
1
1
Diminishing returns to proportion
The Cobb-Douglas Production Function
Increasing Labor
1
1
)2( oo
ooo
LAKY
LAKY
The Cobb-Douglas Production Function
Increasing Labor
ooo
oo
ooo
YLAK
LAKY
LAKY
111
1
1
22
)2(
The Cobb-Douglas Production Function
Increasing Labor
ooo
oo
ooo
YLAK
LAKY
LAKY
111
1
1
22
)2(
Diminishing returns to proportion
The Cobb-Douglas Production Function
Increasing Both
1
1
)2()2( oo
ooo
LKAY
LAKY
The Cobb-Douglas Production Function
Increasing Both
ooo
oo
ooo
YLAK
LKAY
LAKY
22
)2()2(1
1
1
The Cobb-Douglas Production Function
Increasing Both
ooo
oo
ooo
YLAK
LKAY
LAKY
22
)2()2(1
1
1
Constant returns to scale
The Cobb-Douglas Production Function
Substitution
ooo
ooo
YxLKAY
LAKY
1
1
)()2(
Capital and Labor Can be Substituted
The Cobb-Douglas Production Function
An Illustration
2/12/1
tttt LKAY
The Cobb-Douglas Production Function
An Illustration
2/12/1 LAKY
The Cobb-Douglas Production Function
An Illustration
KLAY
The Cobb-Douglas Production Function
An Illustration
KLAY A =3 L =10
K =10
The Cobb-Douglas Production Function
An Illustration
301003)10)(10(3 Y
The Cobb-Douglas Production Function
Doubling Capital
42230
2003)10)(20(3
Y
The Cobb-Douglas Production Function
Constant Returns to Scale
604003)20)(20(3 Y
The Cobb-Douglas Production Function
Substitution
5
30))(20(3
x
xY
The Cobb-Douglas Production Function
Estimation
)log()1(
)log()log()log(
1
t
ttt
tttt
L
KAY
LKAY
The Cobb-Douglas Production Function
Estimation
)log(
)log()log(
2
1
tt
tt
L
KtCY
The Cobb-Douglas Production Function
Estimation
)log(
)log()log(
2
1
tt
tt
L
KY
Statistical issues abound!
The Cobb-Douglas Production Function
Best Estimate
3
1
The Cobb-Douglas Production Function
Factor Payments
= % of Income going to owners of capital
• 1- = % of Income going to workers
The Cobb-Douglas Production Function
How well does it work?
tttt LKAY
You can’t beat something with nothing
The Cobb-Douglas Production Function
CES Production Function
rwLK
%
%
The Cobb-Douglas Production Function
CES Production Function
rwLK
%
%
Suppose 10% increase in wage rate leads to 10% increase in capital labor
ratio. = 1.
The Cobb-Douglas Production Function
CES Production Function
rwLK
%
%
Suppose 10% increase in wage rate leads to 5%
increase in capital labor ratio. = ½.
The Cobb-Douglas Production Function
CES Production Function
rwLK
%
%
In the Cobb-Douglas Production Function,
= 1.
The Cobb-Douglas Production Function
CES Production Function
• The CES allows for a different elasticity of substitution.
• Little gained.
The Cobb-Douglas Production Function
Leontief Production Function
K
L
The Cobb-Douglas Production Function
Leontief Production Function
• K = aY
• L = bY
K
L
The Cobb-Douglas Production Function
Leontief Production Function
• K = aY
• L = bY
K
L
The Cobb-Douglas Production Function
Leontief Production Function
• K = aY
• L = bY
K
L
= 0
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
The Cobb-Douglas Production Function
Other Factors?
1
ttttt LNDLKAY
The Cobb-Douglas Production Function
And in Conclusion…
1tttt LKAY
The Cobb-Douglas Production Function
End
©2003 Charles W. Upton. All rights reserved