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