euler's theorem

elers theorem

1. Euler’s theorem

2. Homogeneous Function
),,,(
0 wherenumber any for
if, degree of shomogeneou is function A
21
21
n
k
n
sxsxsxfYs
ss
k),x,,xf(xy


=
>
=
[Euler’s Theorem]
Homogeneity of degree 1 is often
called linear homogeneity.
An important property of
homogeneous functions is given by
Euler’s Theorem.

3. Euler’s Theorem
argument.ith its
respect toith function w theof derivative partial theis
where, valuesofset any for
, degree of shomogeneou isthat
function temultivariaany For
2121
212111
21
),x,,x(xf),x,,x(x
),x,,x(xfx),x,,x(xfxky
k
),x,,xf(xy
nin
nnnn
n



++=
=

4. Proof Euler’s Theorem
. degree of shomogeneou isfunction original then the
true,is above theIf holds. theorem thisof converse The
),,,(),,,(
Theorem sEuler'get we, Letting
),,,(),,,(
respect to with above theof derivative partial theTake
),,,(function shomogeneou Definition
212111
212111
1
21
k
xxxfxxxxfxky
1s
sxsxsxfxsxsxsxfxyks
s
sxsxsxfys
nnnn
nnnn
k
n
K



++=
=
++=
=
-

5. Division of National Income
( )[ ] ( )
[ ]
( )YYwLrKYHence
YKLKK
K
Y
rKand
YLLKL
L
Y
wL
L
L
Y
K
K
Y
LKY
bb
bba
bab
a
bb
bb
bb
-+=+=
==
¶
¶
=
-=-=
¶
¶
=
¶
¶
+
¶
¶
=
=
--
-
-
1,
.
11
implies This wage.real andreturn really their respective
paid arelabor and capital n,competitioperfect under Now
Y
therefore1, degree of shomogeneou is which
isfunction production national that theSuppose
11
1

6. Properties of Marginal
Products
( )
( ) ( )
b
b
ba
ab
÷
ø
ö
ç
è
æ
-=-=
¶
¶
÷
ø
ö
ç
è
æ
==
¶
¶
-=
¶
¶
=
¶
¶
-
-
--
-
--
L
K
LαKβ
L
Y
and
K
L
LβαK
K
Y
LαKβ
L
Y
LβαK
K
Y
ββ
ββ
ββ
ββ
11
as products marginal thecan write We.1
Labor, ofproduct marginal for the Likewise
zero. degree of shomogeneou is which
function, production accounting income nationalour For
1
11
11

7. Arguments of Functions that are
Homogeneous degree zero
QED
x
x
x
x
x
x
f),x,,x,,xf(x
then
x
sLet
),sx,,sx,,sxf(sx),x,,x,,xf(xs
nianyfor
x
x
x
x
x
x
f
),x,,x,,xf(x
i
n
ii
ni
i
nini
i
n
ii
ni
÷
÷
ø
ö
ç
ç
è
æ
=
=
=
=
÷
÷
ø
ö
ç
ç
è
æ
,,1,,,
,
1
0, degree of shomogeneou isfunction theSince :Proof
.,...,2,1,,1,,,
as written becan zero degree of
shomogeneou is that function Any
21
21
2121
0
21
21





8. First Partial Derivatives of
Homogeneous Functions
( )
( )
. degree of
shomogeneou is n,,1,2,iany for
,,,
sderivative partialfirst ists ofeach then
, degree of shomogeneou is ,,, function, theIf
21
21
k-1
x
xxxf
f
kxxxf
i
n
i
n



=
¶
¶
=

9. Proof of previous slide
( ) ( )
( ) ( )
()
()
( )
( )
( )
( ) ( )
( ) ( )
. degree of shomogeneou is derivative theimpliesWhich
,,,,,,
,,,,,,
equal two thesetting ,,,
,,,
,,,
,,,,,,
,,,,,,
21
1
21
2121
21
21
21
2121
2121
k-1
xxxfssxsxsxf
orxxxfssxsxsxsf
xxxfs
x
xxxfs
andsxsxsxsf
dx
sxd
sx
sxsxsxf
x
sxsxsxf
xxxfssxsxsxfknowWe
ni
k
ni
ni
k
ni
ni
k
i
n
k
ni
i
i
i
n
i
n
n
k
n







-
=
=
=
¶
¶
=
×
¶
¶
=
¶
¶
=

10. Homothetic function
. allfor 0 ifor allfor 0 is
thatmonotonic,strictly is function theiffunction
homothetic a is then function, shomogeneou
a is if This functin. shomogeneou a
ofation transformmontonic a isfunction homotheticA
21
yg'(y)yg'(y)
g(y)
g(y)z
),x,,xf(xy
n
<>
=
= 

11. Example homothetic function
( )
( )
s.homogeneou are functions homothetic allnot
,homothetic are functions shomogeneou whileTherefore,
?. degree
of shomogeneou isfunction original aren except whe
)ln(
)ln()ln()ln()ln()ln(
)ln()ln(lnLet
. degree of shomogeneou is which
wssw
szxszsxnow
zx(y)w
zxylet
k
¹++=
+++=+
+==
+=
ba
bababa
ba
ba
ba