Economics 202
Name:
_______________
Principles of Macroeconomics
Professor Melick
Problem Set #2
Due Wednesday January 30, 2008
This problem set is meant to provide you with a solid understanding of production
functions by using as an example the CobbDouglas production function  the most popular
functional form in economics and the cornerstone of most models of economic growth (a topic
we will cover in several weeks.)
The function is named after the two economists who
popularized it in the 1920s.
1
I.
Theory
Abel and Bernanke write a general production function as
(
)
N
K
F
A
Y
,
⋅
=
.
The Cobb
Douglas functional form is given by
Y
A K
N
=
⋅
⋅
α
β
where
output (GDP)
total factor productivity
capital stock
labor force
parameters, elasticity of output with respect
to capital and labor, respectively.
Y
A
K
N
=
=
=
=
=
α β
,
Most commonly, it is assumed that
α
β
+
=
1
and hence
Y
A K
N
=
⋅
⋅

α
α
1
We now want to convince ourselves of several properties of the CobbDouglas function.
We will
use mathematical derivations as well as a particular form of the CobbDouglas function where we
set
α
=
=
1
3
10
and A
giving
Y
K
N
=
⋅
⋅
10
1 3
2 3
A.
Constant Returns to Scale
1.
In the space below, use the properties of exponents
(
)
(
)
x y
x
y
and
x
x
x
z
z
z
a
b
a
b
⋅
=
⋅
⋅
=
+
,
to demonstrate that the following is true
(
)
(
)
Y
A
K
N
Y
'
=
⋅
⋅
⋅
⋅
=
⋅

λ
λ
λ
α
α
1
1
Charles W. Cobb; Paul H. Douglas, “A Theory of Production”,
The American Economic Review
, Vol. 18, No. 1,
Supplement, Papers and Proceedings of the Fortieth Annual Meeting of the American Economic Association. (Mar.,
1928), pp. 139165.
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Viola!!
Constant returns to scale.
You have demonstrated that increasing both
K
and
N
by
some factor
λ
, increases output by the same factor.
With Excel, we can confirm this for our
particular sample version of
Y
K
N
=
⋅
⋅
10
1 3
2 3
.
We are going to setup a spreadsheet that solves
for Y, given
K
and
N
. and the first several rows of the end result should look like this
0.333333 0.666667
10
K
N
Y
0
0
0
1
1
10
Put the parameters
1/3 , 2/3, and 10 in the top row, and put values for K in column A and the
values for N in column B starting in the second row.
The production function formula
Y
K
N
=
⋅
⋅
10
1 3
2 3
can be translated into an Excel formula as
=$C$1*(A3)^($A$1)*(B3)^($B$1)
in order to solve for Y in column C.
Fill in the rest of the
rows by having K and N increase by units of 1 until they reach 16 and answer the following
questions
2.
As K and N double from 1 to 2, output _____________ from ______ to __________.
3.
As K and N double from 2 to 4, output _____________ from ______ to __________.
4.
As K and N double from 4 to 8, output _____________ from ______ to __________.
Voila!
Constant returns to scale.
Now try changing the parameters so that both exponents
α
β
and
are equal to 2/3.
We find that if the coefficients sum to more than one, we get
increasing returns to scale.
Set them both equal to 1/3 to find a production function with
decreasing returns to scale.
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 Spring '10
 Jakes
 Economics, Macroeconomics, CobbDouglas production function, Paul H. Douglas, Charles W. Cobb

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