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.