Economics 2301

Lecture 28

Multivariate Calculus

Homogeneous Function

),,,(

0 wherenumber any for if, degree of shomogeneou is function A

21

21

nk

n

sxsxsxfYs

ssk),x,,xf(xy

Cobb-Douglas Function

scale. toreturns increasing have we,1 Ifscale. toreturnsconstant have we,1 If

scale. toreturns decreasing have we,1 If. degree of shomogeneou isfunction Douglas-CobbOur

factor aby inputs theIncrease

Function Production Douglas-Cobb thehave We

1

YsLδKssLsKδY

sLKY

βαβαβαβα

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

xxxfxxxxfxky1s

sxsxsxfxsxsxsxfxyks

ssxsxsxfys

nnnn

nnnnk

nK

Division of National Income

YYwLrKYHence

YKLKKKYrKand

YLLKLLYwL

LLYK

KYLKY

1,

.

11

implies This wage.real andreturn really their respectivepaid 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

LKLαKβ

LY

andKLLβαK

KY

LαKβLY

LβαKKY

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

xx

xx

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

thenx

sLet

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

nianyforxx

xx

xxf

),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-1x

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-1xxxfssxsxsxf

orxxxfssxsxsxsf

xxxfsx

xxxfs

andsxsxsxsfdxsxd

sxsxsxsxf

xsxsxsxf

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, shomogeneoua is if This functin. shomogeneou aofation 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,

?. degreeof shomogeneou isfunction original aren except whe

)ln(

)ln()ln()ln()ln()ln()ln()ln(lnLet

. degree of shomogeneou is which

wssw

szxszsxnowzx(y)w

zxylet

k