Transcript Nov. 3
Economics 2301
Lecture 28
Multivariate Calculus
Homogeneous Function
A function y f(x1,x2 , ,xn ) is homogeneous of degree k if,
for any number s where s 0
s k Y f ( sx1 , sx2 ,, sxn )
Cobb-Douglas Function
We have theCobb - Douglas P roductionFunction
Y K L
Increasetheinputsby a factors
Y1 δ sK sL s α β δK α Lβ s α βY
α
β
Our Cobb - Douglas functionis homogeneous of degree .
If 1, we havedecreasingreturnsto scale.
If 1, we haveconstantreturnsto scale.
If 1, we haveincreasingreturnsto scale.
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
For any multivariat e function y f(x1,x2 , ,xn )
thatis homogeneous of degree k ,
ky x1 f1(x1,x2 , ,xn ) xn f n(x1,x2 , ,xn )
for any set of values(x1,x2 , ,xn ), where f i(x1,x2 , ,xn )
is thepartialderivativeof thefunction with respect to
its ith argument.
Proof Euler’s Theorem
Definit ionhomogeneous funct ions K y f ( sx1 , sx2 , , sxn )
T aket hepart ialderivat iveof t heabove wit h respect t os
ks k 1 y x1 f1 ( sx1 , sx2 , , sxn ) xn f n ( sx1 , sx2 , , sxn )
Let t ings 1, we get Euler's T heorem
ky x1 f1 ( x1 , x2 , , xn ) xn f n ( x1 , x2 , , xn )
T heconverseof t his t heoremholds. If t heaboveis t rue,
t hen t heoriginalfunct ionis homogeneous of degree k .
Division of National Income
Suppose that thenationalproductionfunctionis
Y K L1 which is homogeneous of degree1, therefore
Y
Y
Y
K
L
K
L
Now under perfectcompetition, capitaland labor are paid
respect ively theirreal returnand real wage. T hisimplies
Y
wL
L 1 K L L 1 Y
L
Y
and rK
K K 1 L1 K Y .
K
Hence, Y rK wL Y 1 Y
Properties of Marginal
Products
For our nat ionalincomeaccountingproductionfunct ion,
Y
βαK β 1 L1 β which is homogeneous of degree zero.
K
Likewise for t hemarginalproduct of Labor,
Y
1 β αK β L β . Wecan writ ethemarginalproductsas
L
1
Y
L
β 1 1 β
βαK L
K
K
and
Y
K
1 β αK β L β 1
L
L
Arguments of Functions that are
Homogeneous degree zero
Any funct ion f(x1,x2 , ,xi , ,xn ) t hatis homogeneous
of degree zero can be writ t enas
x1 x2
xn
f , , ,1, , for any i 1,2,...,n.
xi
xi xi
P roof: Since t hefunct ionis homogeneous of degree 0,
s 0 f(x1,x2 , ,xi , ,xn ) f(sx1,sx2 , ,sxi , ,sxn )
Let s
1
, then
xi
x x
x
f(x1,x2 , ,xi , ,xn ) f 1 , 2 , ,1, , n
xi
xi xi
QED
First Partial Derivatives of
Homogeneous Functions
If thefunction,f x1 , x2 ,, xn is homogeneous of degree k ,
theneach of ists first partialderivatives
fi
f x1 , x2 ,, xn
for any i 1,2,, n, is homogeneous
xi
of degree k-1.
Proof of previous slide
We know f sx1 , sx2 , , sxn s k f x1 , x2 , , xn
f sx1 , sx2 , , sxn f sx1 , sx2 , , sxn d sxi
xi
sxi
dxi
sf i sx1 , sx2 , , sxn and
s k f x1 , x2 , , xn
s k f i x1 , x2 , , xn set t ingt he t wo equal
xi
sf i sx1 , sx2 , , sxn s k f i x1 , x2 , , xn or
f i sx1 , sx2 , , sxn s k 1 f i x1 , x2 , , xn
W hichimplies t hederivat iveis homogeneous of degree k-1.
Homothetic function
A homotheticfunctionis a montonictransformationof
a homogeneous functin. T hisif y f(x1,x2 , ,xn ) is a
homogeneous function,then z g(y)is a homothetic
functionif thefunctiong(y)is strictlymonotonic,that
is g'(y) 0 for all y or if g'(y) 0 for all y.
Example homothetic function
let y x z which is homogeneous of degree .
Let w ln (y) ln(x) ln(z )
now ln(sx ) ln(sz ) ln(x) ln(z ) ln(s )
w ln(s) s k w
except when are originalfunctionis homogeneous of
degree ?.
T herefore,while homogeneous functionsare homothetic
,
not all homotheticfunctionsare homogeneous.