![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.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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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 − = = = ∂ ∂ = ⋅ ∂ ∂ = ∂ ∂ =
- 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