- Euler's theorem states that for a homogeneous function f(x) of degree k, the partial derivative of f with respect to x is equal to kf(x)/x. - Homogeneous functions have special properties related to their degree of homogeneity. One important property is described by Euler's theorem. - National income can be modeled as a homogeneous function of degree one, implying some key relationships between its arguments.

1. Euler’s theorem

2. Homogeneous Function
),,,(
0wherenumberanyfor
if,degreeofshomogeneouisfunctionA
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.ithits
respect toithfunction wtheofderivativepartialtheis
where,valuesofsetanyfor
,degreeofshomogeneouisthat
functiontemultivariaanyFor
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
.degreeofshomogeneouisfunctionoriginalthen the
true,isabovetheIfholds.theoremthisofconverseThe
),,,(),,,(
TheoremsEuler'getwe,Letting
),,,(),,,(
respect towithabovetheofderivativepartialtheTake
),,,(functionshomogeneouDefinition
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
ββ
ββα
βαβ
α
ββ
ββ
ββ
−+=+=
==
∂
∂
=
−=−=
∂
∂
=
∂
∂
+
∂
∂
=
=
−−
−
−
1,
.
11
impliesThiswage.realandreturnreally theirrespective
paidarelaborandcapitaln,competitioperfectunderNow
Y
therefore1,degreeofshomogeneouiswhich
isfunctionproductionnationalthat theSuppose
11
1

6. Properties of Marginal
Products
( )
( ) ( )
β
β
βα
αβ






−=−=
∂
∂






==
∂
∂
−=
∂
∂
=
∂
∂
−
−
−−
−
−−
L
K
LαKβ
L
Y
and
K
L
LβαK
K
Y
LαKβ
L
Y
LβαK
K
Y
ββ
ββ
ββ
ββ
11
asproductsmarginalthecan writeWe.1
Labor,ofproductmarginalfor theLikewise
zero.degreeofshomogeneouiswhich
function,productionaccountingincomenationalourFor
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






=
=
=
=





,,1,,,
,
1
0,degreeofshomogeneouisfunctiontheSince:Proof
.,...,2,1,,1,,,
aswrittenbecanzerodegreeof
shomogeneouisthatfunctionAny
21
21
2121
0
21
21





8. First Partial Derivatives of
Homogeneous Functions
( )
( )
.degreeof
shomogeneouisn,,1,2,ianyfor
,,,
sderivativepartialfirstistsofeachthen
,degreeofshomogeneouis,,,function,theIf
21
21
k-1
x
xxxf
f
kxxxf
i
n
i
n



=
∂
∂
=

9. Proof of previous slide
( ) ( )
( ) ( )
( )
( )
( )
( ) ( )
( ) ( )
( ) ( )
.degreeofshomogeneouisderivativetheimpliesWhich
,,,,,,
,,,,,,
equaltwothesetting,,,
,,,
,,,
,,,,,,
,,,,,,
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
.allfor0iforallfor0is
thatmonotonic,strictlyisfunctiontheiffunction
homotheticaisthenfunction,shomogeneou
aisifThisfunctin.shomogeneoua
ofationtransformmontonicaisfunctionhomotheticA
21
yg'(y)yg'(y)
g(y)
g(y)z
),x,,xf(xy n
<>
=
= 

11. Example homothetic function
( )
( )
s.homogeneouarefunctionshomotheticallnot
,homotheticarefunctionsshomogeneouwhileTherefore,
?.degree
ofshomogeneouisfunctionoriginalarenexcept whe
)ln(
)ln()ln()ln()ln()ln(
)ln()ln(lnLet
.degreeofshomogeneouiswhich
wssw
szxszsxnow
zx(y)w
zxylet
k
≠++=
+++=+
+==
+=
βα
βαβαβα
βα
βαβα