New York University
Department of Economics
V31.0006
C. Wilson
Mathematics for Economists
May 7, 2008
Homogeneous Functions
For any
α
∈
R
, a function
f
:
R
n
++
→
R
is
homogeneous of degree
α
if
f
(
λx
)=
λ
α
f
(
x
)
for all
λ>
0
and
x
∈
R
n
++
. A function is
homogeneous
if it is homogeneous of degree
α
for some
α
∈
R
.
Afunct
ion
f
is
linearly homogenous
if it is homogeneous of degree 1.
•
Along any ray from the origin, a homogeneous function de
f
nes a power function. If
f
is linearly
homogeneous, then the function de
f
ned along any ray from the origin is a linear function.
Example:
Consider a CobbDouglas production,
f
(
x
Q
n
j
=1
x
α
j
j
,
where each
α
j
>
0
,
and let
β
=
P
n
j
=1
α
j
.
Then
f
(
λx
n
Y
j
=1
(
λx
j
)
α
j
=
λ
β
n
Y
j
=1
(
x
j
)
α
j
=
λ
β
f
(
x
)
.
So
f
is homogeneous of degree
β.
Example:
Consider a CES production function
f
(
x
)=(
P
n
i
=1
α
i
x
ρ
i
)
β
,
where
β,ρ >
0
.
Then
f
(
λx
Ã
n
X
i
=1
α
i
(
λx
i
)
ρ
!
β
=
λ
ρβ
Ã
n
X
i
=1
α
i
x
ρ
i
!
β
=
λ
ρβ
f
(
x
)
.
So
f
is homogeneous of degree
ρβ.
•
If
f
is homogeneous of degree
0
,
then
f
(
λx
f
(
x
)
.
Why?
•
If
f
is homogeneous of degree
α
6
=0
,
then
f
1
α
is homogenenous of degree 1. Why?
•
Let
f
(
x
³
Q
n
j
=1
x
α
j
j
´
1
β
,
where each
α
j
>
0
and
β
=
P
n
j
=1
α
j
.
Then
f
is linearly homogeneous.
Why?
•
f
(
x
)=min
{
x
i
:
i
=1
,...,n
}
is linearly homogeneous. Why?
The following theorem relates the value of a homogeneous function to its derivative.
Theorem 1:
If
f
:
R
n
++
→
R
is continuously di
f
erentiable and homogeneous of degree
α,
then
Df
(
x
)
·
x
=
n
X
i
=1
f
i
(
x
)
x
i
=
αf
(
x
)
.
(Euler’s theorem)
Proof.
If
f
is homogeneous of degree
α,
then for any
x
∈
R
n
++
and any
0
,
we have
f
(
λx
λ
α
f
(
x
)
.
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May 7, 2008
Page 2
Then holding
x
f
xed and di
f
erentiating both sides with repect to
λ,
we obtain
df
(
λx
)
dλ
=
Df
(
λx
)
·
x
=
n
X
i
=1
f
i
(
λx
)
x
i
=
d
(
λ
α
f
(
x
))
dλ
=
αλ
α
−
1
f
(
x
)
Letting
λ
=1
,
yields the statement to be proved.
The next theorem relates the homogeneity of a function to the homogeneity of its partial derivatives.
Theorem 2:
If
f
:
R
n
++
→
R
is continuously di
f
erentiable and homogeneous of degree
α,
then
each partial derivative
f
i
is homogeneous of degree
α
−
1
.
Proof.
For
f
xed
x
∈
R
n
++
and
λ>
0
,
de
f
ne each
g
i
,h
i
:(
−
x
i
,
∞
)
→
R
by
g
i
(
t
)=
f
(
λ
(
x
+
e
i
t
))
and
h
i
(
t
λ
α
f
(
x
+
e
i
t
)
Then the homogeneity of
f
implies
g
i
(
t
f
(
λ
(
x
+
te
i
)) =
λ
α
f
(
x
+
te
i
h
i
(
t
)
and therefore
g
0
i
(
t
h
0
i
(
t
)
for all
t
∈
(
−
x
i
,
∞
)
But
g
0
i
(0)
=
Df
(
λx
)
·
λe
i
=
λf
i
(
λx
)
h
0
i
(0)
=
λ
α
Df
(
x
)
·
e
i
=
λ
α
f
i
(
x
)
So
f
i
(
λx
λ
α
−
1
f
i
(
x
)
.
Example:
In the example above, we showed that
f
(
x
Q
n
i
=1
x
α
i
i
is homogeneous of degree
β
=
P
n
i
=1
α
i
.
To verify Euler’s theorem, observe that, for each
j
,...,n,
we have
f
j
(
x
α
j
x
α
j
−
1
j
Y
i
6
=
j
x
α
i
i
=
α
j
x
j
n
Y
i
=1
x
α
i
i
=
α
j
f
(
x
)
x
j
.
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 Spring '11
 PP
 Derivative, λ, Monotonic function, Convex function, homogeneous functions

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