Euler’s theorem

Homogeneous Function

),,,(

0 wherenumber any for

if, degree of shomogeneou is function A

21

21

nk

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.

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

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

2121111

21

k

xxxfxxxxfxky

1s

sxsxsxfxsxsxsxfxyks

s

sxsxsxfys

nnnn

nnnnk

nK

Division of National Income

YYwLrKYHence

YKLKKK

YrKand

YLLKLL

YwL

LL

YK

K

Y

LKY

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

Properties of Marginal Products

L

KLαKβ

L

Y

andK

LLβαK

K

Y

LαKβL

Y

LβαKK

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

111

11

Arguments of Functions that are Homogeneous degree zero

QED

x

x

x

x

x

xf),x,,x,,xf(x

thenx

sLet

),sx,,sx,,sxf(sx),x,,x,,xf(xs

nianyforx

x

x

x

x

xf

),x,,x,,xf(x

i

n

iini

i

nini

i

n

ii

ni

,,1,,,

,1

0, degree of shomogeneou isfunction theSince :Proof

.,...,2,1,,1,,,

as written becan zero degree of

shomogeneou is that function Any

2121

21210

21

21

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

xxxff

kxxxf

i

ni

n

Proof of previous slide

. degree of shomogeneou is derivative theimpliesWhich

,,,,,,

,,,,,,

equal two thesetting ,,,,,,

,,,

,,,,,,

,,,,,,

211

21

2121

2121

21

2121

2121

k-1

xxxfssxsxsxf

orxxxfssxsxsxsf

xxxfsx

xxxfs

andsxsxsxsf

dx

sxd

sx

sxsxsxf

x

sxsxsxf

xxxfssxsxsxfknowWe

nik

ni

nik

ni

nik

i

nk

ni

i

i

i

n

i

n

nk

n

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

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