This article was downloaded by: [Ioan, Catalin Angelo]
On: 13 April 2011
Access details: Access Details: [subscription number 936329567]
Publisher Routledge
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 3741 Mortimer Street, London W1T 3JH, UK
Applied Economics Letters
Publication details, including instructions for authors and subscription information:
http://www.informaworld.com/smpp/title~content=t713684190
A generalization of a class of production functions
Catalin Angelo Ioana; Gina Ioana
a
Department of General Economics, Danubius University, Galati, Romania
First published on: 12 April 2011
To cite this Article Ioan, Catalin Angelo and Ioan, Gina(2011) 'A generalization of a class of production functions', Applied
Economics Letters,, First published on: 12 April 2011 (iFirst)
To link to this Article: DOI: 10.1080/13504851.2011.564117
URL: http://dx.doi.org/10.1080/13504851.2011.564117
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf
This article may be used for research, teaching and private study purposes. Any substantial or
systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or
distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representation that the contents
will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses
should be independently verified with primary sources. The publisher shall not be liable for any loss,
actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly
or indirectly in connection with or arising out of the use of this material.
Applied Economics Letters, 2011, 1–8, iFirst
A generalization of a class of
production functions
Catalin Angelo Ioan* and Gina Ioan
Downloaded By: [Ioan, Catalin Angelo] At: 05:19 13 April 2011
Department of General Economics, Danubius University, 800654
Galati, Romania
In this article we shall give a generalization of Cobb–Douglas, Constant
Elasticity of Substitution (CES), Lu–Fletcher, Liu–Hildebrand, Variable
Elasticity of Substitution (VES) and Kadiyala production functions. We
compute the principal indicators such as the marginal products, the marginal
rate of substitution, the elasticities of factors and the elasticity of substitution.
Finally, we formulate three theorems of characterization for the functions with
a proportional marginal rate of substitution, with constant elasticity of labour
and for those with constant elasticity of substitution (for n = 1).
Keywords: production functions; Cobb–Douglas; CES; Lu–Fletcher;
Liu–Hildebrand; VES; Kadiyala
JEL Classification: D24
I. Introduction
In the economical analysis, the production functions
had a long and interesting history (Kmenta, 1967;
Sato, 1974; Ioan, 2004, 2007; Mishra 2007).
A
production
function
is
defined
as
P:R+ · R+!R+, P = P(K,L), where P is the production, K the capital and L the labour such that
K ¼
@P
the marginal product of K;
@K
ð4Þ
L ¼
@P
the marginal product of L;
@L
ð5Þ
@P
RMS ¼ @L
the marginal rate of substitution; ð6Þ
@P
@K
@P
Pð0; 0Þ ¼ 0
the elasticity of K;
EK ¼ @K
P
ð1Þ
ð7Þ
K
P is differentiable of order 2 in any interior point of
the production set;
P is a homogenous function of degree 1, that is,
P(rK,rL) = rP(K,L) "rPR;
@P
@P
0;
0
@K
@L
ð2Þ
@2P
@2P
0;
0
@K2
@L2
ð3Þ
For any production function, we have a lot of indicators such as
@P
EL ¼ @L
the elasticity of L;
P
ð8Þ
L
s¼
@P @P
@L @K
@2 P
P @K@L
the elasticity of substitution:
ð9Þ
Cobb and Douglas (1928) formulated the well-known
production function: P(K,L) = aKpL1-p where
pP[0,1], which have many applications in various economical problems.
*Corresponding author. E-mail: catalin_angelo_ioan@univ_danubius.ro
Applied Economics Letters ISSN 1350–4851 print/ISSN 1466–4291 online # 2011 Taylor & Francis
http://www.informaworld.com
DOI: 10.1080/13504851.2011.564117
1
C. A. Ioan and G. Ioan
2
Arrow et al. (1961) generalized the preceding and
obtained the Constant Elasticity of Substitution (CES)
1
production function: PðK; LÞ ¼ aðbKr þ ð1 bÞLr Þr ,
which for r = 0 becomes Cobb–Douglas function.
The Lu–Fletcher production function also generalized the CES function into the form: PðK; LÞ
cð1bÞ b b1
¼ a dKb þ ð1 dÞ KL
L , which for c = 0,
Downloaded By: [Ioan, Catalin Angelo] At: 05:19 13 April 2011
= 1 becomes CES function.
Liu and Hildebrand (1965) made a new generalization of the CES function:
1
PðK; LÞ ¼ a ð1 dÞK þ dKm Lð1mÞ for m ¼ 0
Revankar (1971) introduced the Variable Elasticity of
Substitution (VES) function: PðK; LÞ ¼ aKrð1dmÞ
½L þ ðm 1ÞKrdm , which for m = 1, r = 1 is also
generalization of Cobb–Douglas production function.
Kadiyala (1972) made an important generalization
with
r
b þb
PðK; LÞ ¼ EðtÞ c11 Kb1 þb2 þ2c12 Kb1 Lb2 þ c22 Lb1 þb2 1 2
where c11 + 2c12 + c22 = 1, cij 0, b1(b1 + b2) . 0,
b2(b1 + b2) . 0
For c12 = 0, r = 1 Kadiyala (1972) obtained the
CES function, c22 = 0 generates directly the
Lu–Fletcher function, for c11 = 0, c22 = 0,
r = 1 – the Cobb–Douglas function and, finally, for
1
b1 ¼ dm
1, b2 = 1, c22 = 0 – the VES function.
In what follows, we shall make a new generalization, from another point of view, of these functions.
We have then the following cases:
pi1 ; pi2 ; pi3 2 ð1; 0Þ and pi3 ðpi1 þ pi2 Þ ¼ 1
ð14Þ
pi1 ; pi2 2 ð0; 1Þ; pi3 2 ½1; 1Þ and pi3 ðpi1 þ pi2 Þ ¼ 1
From Equation 13 we have that 9i ¼ 1; n such that
ci2 + ci1ci3 . 0, therefore if for such an i, we have
ci2 = 0 follows that ci1, ci3 . 0 and if ci2 . 0 follows
that ci1, ci3 are arbitrary (of course nonnegative).
If we note
K
¼w
L
ð15Þ
follows
P¼L
n
X
pi3
ai ðci1 wpi1 þpi2 þ ci2 wpi1 þ ci3 Þ
ð16Þ
i¼1
Because w 0 and for any i ¼ 1; n we have that ai . 0
and at least one of ci1, ci2 or ci3 is greater than 0 we
obtain P 0. Also from Equation 11: P(0,0) = 0 and
P is differentiable of order 2 in any interior point of the
production set.
We have now
PðrK; rLÞ ¼ rL
n
P
p
ai ðci1 wpi1 þpi2 þ ci2 wpi1 þ ci3 Þ i3 ¼ r1 PðK; LÞ;
i¼1
therefore P is homogenous of first degree.
Let now
Ai ðwÞ ¼ ci1 wpi1 þpi2 þ ci2 wpi1 þ ci3 >0; i ¼ 1; n
ð17Þ
From Equations 6 and 7 we have
II. The Sum Production Function
P¼L
Let the production function
PðK; LÞ ¼
n
X
pi1 þpi2 pi3
Þ ; n1
FðwÞ ¼
ð10Þ
where
n
P
ai Ai ðwÞpi3
ð19Þ
i¼1
From Equation 17 we obtain easily
ai >0 "i ¼ 1; n;
pi3 2 ð1; 0Þ¨½1; 1Þ; pi1 pi2 >0; pi3 ðpi1 þ pi2 Þ
n
P
ð18Þ
where
i¼1
¼ 1 "i ¼ 1; n
ai Ai ðwÞpi3 ¼ LFðwÞ
i¼1
ai ðci1 Kpi1 þpi2 þ ci2 Kpi1 Lpi2
þci3 L
n
P
ð11Þ
ð12Þ
ðci2 þ ci1 ci3 Þ>0; cij 0 "i ¼ 1; n "j ¼ 1; 3 ð13Þ
A0i ðwÞ ¼ wpi1 1 ½ci1 ðpi1 þ pi2 Þwpi2 þ ci2 pi1 ; i ¼ 1; n ð20Þ
A00i ðwÞ ¼ wpi1 2 ½ci1 ðpi1 þ pi2 Þðpi1 þ pi2 1Þwpi2
þci2 pi1 ðpi1 1Þ; i ¼ 1; n
ð21Þ
i¼1
From Equation 15 we obtain after partial derivation
From Equation 12 follows that if pi3 , 0 then pi1, pi2
, 0. If pi3 1 then pi1, pi2 . 0 and pi1 þ pi2 ¼ p1i3 1,
therefore 1 - pi1 pi2 . 0, 1 - pi2 pi1 . 0.
@w
1 @w
K
w
¼ ;
¼ 2 ¼
@K L @L
L
L
ð22Þ
A generalization of a class of production functions
From Equation 18 we have
@P
@P
¼ FðwÞ wF0 ðwÞ;
¼ F0 ðwÞ
@L
@K
ð23Þ
3
Lemma 1: Let qi P R*, i ¼ 1; m, m 1, qi qj
"i; j ¼ 1; m, i j. Therefore the functions
wqi ; i ¼ 1; m and the constant function 1 are linear indem
P
pendent, that is from the equality:
bi wqi þ bm þ1 ¼ 0
i¼ 1
therefore
follows bi = 0, i ¼ 1; m þ 1.
@P P
@P
¼ w
@L L
@K
ð24Þ
which can also be derived from Euler’s formula for
homogenous functions.
By derivation with L and after with K in
Equation 24, we obtain
@P
@ P @LL P w @P
@2P
@2P
w
¼
w
¼
þ
L2
@L2
L @K
@L@K
@L@K
2
@ P
1 @P 1 @P
@2P
@2P
¼
w 2 ¼ w 2
@L@K L @K L @K
@K
@K
Downloaded By: [Ioan, Catalin Angelo] At: 05:19 13 April 2011
2
therefore
@2P
@2P
¼ w
2
@L
@L@K
2
ð25Þ
2
@ P
1 @ P
¼
@K2
w @L@K
ð26Þ
2
@2P
2@ P
¼
w
@L2
@K2
ð27Þ
@2P
@2P
have the
From Equation 27 we have that 2 and
@L
@K2
same sign.
We obtain from Equation 18
@P
¼
K ¼ @K
n
P
ai Ai ðwÞpi3 1 wpi1 1 ðci1 wpi2 þ ci2 pi1 pi3 Þ ð28Þ
i¼ 1
@P
L ¼ @K
¼
n
P
i¼ 1
ai Ai ðwÞpi3 1 ðAi ðwÞ wpi3 Ai ðwÞÞ
0
ð29Þ
Because
Ai ðwÞ wpi3 A0i ðwÞ ¼ ðci1 wpi1 þpi2 þ ci2 wpi1 þ ci3 Þ
wpi3 ci1 ðpi1 þ pi2 Þwpi1 þpi2 1 þ ci2 pi1 wpi1 1
¼ ci2 pi2 pi3 w
pi1
þ ci3
We obtain from Equation 29
L ¼ @P
@L ¼
n
P
ai Ai ðwÞpi3 1 ðci2 pi2 pi3 wpi1 þ ci3 Þ
ð30Þ
i¼1
From Equations 11–14 we can see easily that
@P=@K 0.
We have now the following lemma, which will be
useful in all what follows:
Pm
Proof 1: Differentiating the equality
þbm þ1 ¼ 0 m-times, we obtain
m
P
bi
i ¼1
where
qi
k
¼
i¼ 1
bi w qi
qi qi k
¼ 0; k ¼ 1; m
w
k
qi ðqi 1Þ . . . ðqi k þ 1Þ
; k ¼ 1; m :
k!
Computing the determinant of the system, we obtain
q1 q 1
1
1 w
q1 q 2
1
2 w
..
.
q1 q m
1
m w
wq2 1
1
q2 q2 2
w
2
..
. q2 q2 m
w
m
q2
wqm 1 1
qm qm 2 w
2
..
.
qm qm m w
m
qm
¼ wq1 m . . . wqm m
q1 m1
1 w
qm m2
2 w
..
.
q1
m
wm1
1
q2 m2
w
2
..
. q2
q1
1
q1
2
..
.
q1
m
q2
1
q2
2
..
. q2
m
m
¼ wq1 m . . . wqm m w
ðm1Þm
2
1
qm 2
..
. qm m ¼ wq1 m . . . wqm m w
qm
ðm1Þm
2
w
1
qm m2 w
2
..
.
qm
m
q2
D
qm
m1
C. A. Ioan and G. Ioan
4
The degree of the determinant-like function of q1,
q2 . . ., qm is
mðm þ 1Þ
1 þ 2 þ ... þ m ¼
2
If qi = qj, i j we have that columns i and j are equals
then D = 0. Also, if qi = 0 for an i ¼ 1; m follows that
m
m
i¼1
i; j¼1
ij
D = 0. From this, follows that D ¼ a qi ðqi qj Þ
with a a constant because the degree of the right side is
¼ mðmþ1Þ
. For m = 2 we have that D =
m þ mðm1Þ
2
2
q1q2(q2-q1), therefore a = 1.
We now have the determinant of the system:
wq1 m . . . wqm m w
ðm1Þm
2
m
Y
qi
Downloaded By: [Ioan, Catalin Angelo] At: 05:19 13 April 2011
i¼1
m
Y
ðqi qj Þ0
i; j¼1
ij
From the system we obtain that bi = 0, i ¼ 1; m and
&
from the initial equality follows that bm+1 = 0.
Returning to the production functions we have
from Equation 28 that if @P/@K = 0 then it follows
that ci1 wpi2 þ ¼ ci2 pi1 pi3 ¼ 0 "i ¼ 1; n, therefore from
Lemma 1, ci1 ¼ ci2 ¼ 0 "i ¼ 1; n, which is a contradiction with Equation 15. We have finally that
@P/@K . 0.
From Equation 30 we have that @P/@L 0. If
@P/@K = 0 we have ci2 pi2 pi3 wpi1 þ ci3 ¼ 0, therefore
ci2 ¼ ci3 ¼ 0 "i ¼ 1; n, which is a contradiction with
Equation 13. We have finally that @P/@L . 0.
Let us now compute the second derivatives.
n
@2P X
ai pi3 ðpi3 1ÞAi ðwÞpi3 2 Ai02 ðwÞ
L 2¼
@K
i¼1
þ
n
X
ai pi3 Ai ðwÞpi3 1 A00i ðwÞ
þci1 ci3 ð1 pi1 pi2 Þwpi2
þci1 ci2 pi2 pi3 ð1 pi2 Þwpi2 þpi1 ð31Þ
From Equations 11–14 follows that L@ 2P/@K2 0.
If @ 2P/@K2 = 0 we have that
c2i2 pi1 pi2 p2i3 wpi1 þ ci3 ci2 pi1 pi3 ð1 pi1 Þ
þ ci1 ci3 ð1 pi1 pi2 Þwpi2
þ ci1 ci2 pi2 pi3 ð1 pi2 Þwpi2 þpi1 ¼ 0
and from Lemma 1 we have ci2 = 0, ci1ci3 = 0, which is
a contradiction with Equation 13. We have therefore
@ 2P/@K2 , 0. From Equation 27 we obtain that @ 2P/
@L2 , 0 and from Equation 26 that @ 2P/@L@K . 0.
The marginal rate of substitution is
P
@P
RMS ¼ @L
¼L
@P
@K
n
P
¼
@P
w @K
@P
@K
¼
P
w
@P
L @K
ai Ai ðwÞpi3 1 ðci2 pi2 pi3 wpi1 þ ci3 Þ
ð32Þ
i¼1
n
P
ai Ai ðwÞpi3 1 wpi1 1 ðci1 wpi2 þ ci2 pi1 pi3 Þ
i¼1
The elasticities of L and K are
@P
EL ¼ @L
¼1w
P
L
n
P
¼
@P
@K
w KP
ai Ai ðwÞpi3 1 ðci2 pi2 pi3 wpi1 þ ci3 Þ
ð33Þ
i¼1
n
P
ai Ai ðwÞpi3
i¼1
i¼1
¼
n
X
ai Ai ðwÞpi3 2 wpi1 2 c2i2 pi1 pi2 p2i3 wpi1
EK ¼ 1 EL
ð34Þ
i¼1
þci3 ci2 pi1 pi3 ð1 pi1 Þ
s¼
The elasticity of substitution:
@P @P
@L @K
@2 P
P @K@L
n
P
ai aj Ai ðwÞpi3 1 Aj ðwÞpj3 1 wpj1 1 ðci2 pi2 pi3 wpi1 þ ci3 Þðcj1 wpj2 þ cj2 pj1 pj3 Þ
i;j¼1
¼
w
n
P
i; j¼1
h
i
ai aj Ai ðwÞpi3 Aj ðwÞpj3 2 wpj1 2 c2j2 pj1 wpj1 pj2 p2j3 þ cj3 cj2 pj1 pj3 ð1 pj1 Þ þ cj1 cj3 ð1 pj1 pj2 Þwpj2 þ cj1 cj2 wpj2 þpj1 pj2 pj3 ð1 pj2 Þ
ð35Þ
A generalization of a class of production functions
5
III. Particular Cases
For n = 1 we have
PðK; LÞ ¼ aðc1 Kp1 þp2 þ c2 Kp1 Lp2 þ c3 Lp1 þp2 Þ
p3
ð36Þ
For n = 1, p1 = 1 - , p2 = , P (0,1), c1 = 0, c2 = 1,
c3 = 0 we have P(K,L) = aK1 - L .
where the conditions 11–13 become
ð37Þ
a>0
The Cobb–Douglas production function
p3 2 ð1; 0Þ ¨ ½1; 1Þ; p1 p2 >0; p3 ðp1 þ p2 Þ ¼ 1 ð38Þ
The CES production function
For n = 1, p1 ¼ 2, p2 ¼ 2, c1 = , c2 = 0, c3 = 1 -1 ,
P(0,1) we have PðK; LÞ ¼ aðdK þ ð1 dÞL Þ :
The Lu–Fletcher production function
c2 þ c1 c3 >0; c1 ; c2 ; c3 0
ð39Þ
For n = 1, p1 = -(1-b), p2 = (1-b)+b, c1 = ,
c2 = 1-, c3 = 0, P(0,1) we obtain PðK; LÞ ¼
1
a dKb þ ð1 dÞKð1bÞ Lð1bÞþb b .
ð40Þ
The Liu–Hildebrand production function
From Equations 28 and 30–35 we obtain
Downloaded By: [Ioan, Catalin Angelo] At: 05:19 13 April 2011
@P
¼ aAðwÞp3 1 wp1 1 ðc1 wp2 þ c2 p1 p3 Þ
K ¼ @K
p3 1
L ¼ @P
ðc2 p2 p3 wp1 þ c3 Þ
@L ¼ aAðwÞ
ð41Þ
p3 2 p1 2 2
@2P
L @K
w
c2 p1 wp1 p2 p23
2 ¼ aAðwÞ
þ c1 c2 w
RMS ¼
¼
p2 p3 ð1 p2 Þ
ð42Þ
ð43Þ
EL ¼
EK ¼
s¼
wp1 ðc1 wp2 þ c2 p1 p3 Þ
c1 wp1 þp2 þ c2 wp1 þ c3
1
1
þK dm1 LÞdm ¼ aK1dm ððm 1ÞK þ LÞdm a particular
case of VES production function.
The Kadiyala production function
aAðwÞp3 1 wp1 1 ðc1 wp2 þ c2 p1 p3 Þ
aAðwÞp3 1 ðc2 p2 p3 wp1 þ c3 Þ
aAðwÞp3
c2 p2 p3 wp1 þ c3
¼
c1 wp1 þp2 þ c2 wp1 þ c3
1
For n = 1, p1 ¼ dm
1, p2 = 1, c1 = m-1, c2= 1, c3 = 0,
mP (0,1), m 1 we have PðK; LÞ ¼ a ðm 1ÞK dm
aAðwÞp3 1 ðc2 p2 p3 wp1 þ c3 Þ
c2 p2 p3 wp1 þ c3
wp1 1 ðc1 wp2 þ c2 p1 p3 Þ
1
þbKd Lð1dÞ Þ .
The VES production function
þ c3 c2 p1 p3 ð1 p1 Þ þ c1 c3 ð1 p1 p2 Þwp2
p2 þp1
For n = 1, p1 = , p2 = (1-), c1 = 1-b, c2 = b,
c3 = 0, P (0,1) we have PðK; LÞ ¼ aðð1 bÞK
ð44Þ
For c1+c2+c3 = 1 and c2 0 or c2 = 0, but c1, c3 . 0,
we obtain a particular case of Kadiyala production
function.
IV. Theorems
Theorem 1: The only case when RMS ¼ k KL where k
is a positive constant is the Cobb–Douglas function with
¼ 1=k þ 1.
ð45Þ
Proof 1: From Equation 32 we have
ðc2 p2 p3 wp1 þ c3 Þðc1 wp2 þ c2 p1 p3 Þ
c22 p1 wp1 p2 p23 þ c3 c2 p1 p3 ð1 p1 Þ þ c1 c3 ð1 p1 p2 Þwp2 þ c1 c2 wp2 þp1 p2 p3 ð1 p2 Þ
ð46Þ
C. A. Ioan and G. Ioan
6
n
X
From Lemma 1 we have
ai Ai ðwÞpi 31 ðci2 pi2 pi3 wpi1 þ ci3 Þ
i¼1
¼k
n
X
pi1 kpi2 ¼ 0 "i ¼ 1; n
i¼1
and with the notation pi2=p we have that pi1 = kp, pi2
1
1
= p, pi3 ¼ ðkþ1Þp
, p kþ1
. The production function
becomes
therefore
n
X
ð52Þ
ai Ai ðwÞpi 31 wpi1 ðci1 wpi2 þ ci2 pi1 pi3 Þ
ai Ai ðwÞpi3 1
PðK; LÞ ¼
ð47Þ
i¼1
ci1 wpi1 þpi2 þ ci2 pi3 ðpi1 kpi2 Þwpi1 þ ci3 ¼ 0
n
1
P
k
1
ai ci2 Kkp Lp ðkþ1Þp ¼ aKðkþ1Þ Lðkþ1Þ
after obvious notations.
Note that I ¼ fi ¼ 1; njpi1 ; pi2 ; pi3 <0g and J ¼ fj ¼
1; njpj1 ; pj2 ; pj3 >0g: Because Equation 47 holds for
every w, we have with Equation 14:
ð53Þ
i¼1
&
Theorem 2: The only case when EL = k where k is a
constant is the Cobb–Douglas function with = k.
Downloaded By: [Ioan, Catalin Angelo] At: 05:19 13 April 2011
Proof 2: From Equation 33 we have that
lim
n
X
w!1
ai Ai ðwÞ
X
w!1
i21
X
þ lim
w!1
X
w!0
½ci1 w
pi1
þ ci2 pi3 ðpi1 kpi2 Þw
þ ci3 pi3 1
pi1 þpi2
½ci1 w
pi1
þ ci2 pi3 ðpi1 kpi2 Þw
n
P
þ ci3 i2J
n
X
n
P
p
ð48Þ
aj cj3j3 1
therefore
n
X
i ¼1
w!0
þ lim
i2I
w!1
X
pj3 1 aj Aj ðwÞ
i2J
ai cpi1i3 1 þ
i2I
X
pj1 þ pj2
cj1 w
pj1
þ cj2 pj3 ðpj1 kpj2 Þw
p
þ cj3
þci3 ð1 kÞ kci1 wpi1 þpi2 Þ ¼ 0
ð49Þ
aj cj3j3
i2J
From Equations 48 and 49 we have that ci1 = ci3 = 0
"iPI and cj1 = cj3 = 0 "jPJ, therefore ci1 = ci3 = 0
"i ¼ 1; n.
From Equation 47 we now have
lim
n
X
w!1
ai Ai ðwÞpi 31 wpi1 ci2 pi3 ðpi1 kpi2 Þ ¼ 0
ð50Þ
X
¼ lim
w!1
¼
ai Ai ðwÞpi3 1 ðci2 ðpi2 pi3 kÞwpi1 þ ci3 ð1 kÞ kci1 wpi1 þpi2 Þ
i ¼1
X
ai Ai ðwÞpi3 1 ðci2 ðpi2 pi3 kÞwpi1 þ ci3 ð1 kÞ kci1 wpi1 þpi2 Þ
i2I
X
w!1
i ¼1
ai Ai ðwÞpi3 1 ðci2 ðpi2 pi3 kÞwpi1 þ ci3 ð1 kÞ kci1 wpi1 þpi2 Þ
i2J
ð1 kÞai cpi3i3 lim
w!1
i2I
where
lim
w!0
Ai ðwÞ ¼ ci2 wpi1
n
X
w!0
þ lim
w!0
n
P
i¼1
ai cpi2i3 wpi 1pi 3 pi3 ðpi1
kpi2 Þ ¼ 0
ð51Þ
X
kai cpi1i3 1
ð55Þ
i2J
ai Ai ðwÞpi3 1 ðci2 ðpi2 pi3 kÞwpi1 þ ci3 ð1 kÞ kci1 wpi1 þpi2 Þ
i¼1
¼ lim
that is
ð54Þ
Note again that I ¼ fi ¼ 1; njpi1 ; pi2 ; pi3 <0g and
J ¼ fj ¼ 1; njpj1 ; pj2 ; pj3 >0g: Because Equation 54
holds for every w, we have
þ lim
n
P
ai Ai ðwÞpi3 1 ðci2 ðpi2 pi3 kÞwpi1
i¼1
ai Ai ðwÞpi3 1 ½ci1 wpi1 þ pi2 þ ci2 pi3 ðpi1 kpi2 Þwpi1 þ ci3 X
¼ k0
ai Ai ðwÞpi3
i ¼1
X
ai Ai ðwÞpi3 1 ½ci1 wpi1 þ pi2 þ ci2 pi3 ðpi1 kpi2 Þwpi1 þ ci3 X
ai Ai ðwÞpi3 1 ðci2 pi2 pi3 wpi1 þ ci3 Þ
i¼1
aj Aj ðwÞpi3 1 cj1 wpj1 þpj2 þ cj2 pj3 ðpj1 kpj2 Þwpj1 þ cj3
i2J
¼ lim
¼
ai Ai ðwÞ
ai cpi3i3 þ
i2I
lim
pi2 þpi2
i ¼1
¼ lim
¼
pi3 1
X
ai Ai ðwÞpi3 1 ðci2 ðpi2 pi3 kÞwpi1 þ ci3 ð1 kÞ kci1 wpi1 þpi2 Þ
i2I
X
ai Ai ðwÞpi3 1 ðci2 ðpi2 pi3 kÞwpi1 þ ci3 ð1 kÞ kci1 wpi1 þpi2 Þ
i2J
¼ lim
w!0
X
i2I
kai cpi1i3 1 þ
X
i2J
ð1 kÞai cpi3i3
ð56Þ
A generalization of a class of production functions
From Equations 55 and 56 ci1 = 0 "iPJ, (1 - k)ci3 = 0
"iPI, ci1 = 0 "iPI and (1 - k)ci3 = 0 "iPJ, therefore
ci1 = 0 and (1 - k)ci3 = 0 "i ¼ 1; n.
If k = 1 we have only ci1 = 0 and Equation 54
becomes
n
P
ai Ai ðwÞpi3 1 ci2 ðpi2 pi3 1Þwpi1 ¼ 0
7
c1 c2 p2 p3 ðkð1 p2 Þ 1Þ ¼ 0
ð62Þ
c22 p1 p2 p23 ðk 1Þ ¼ 0
ð63Þ
c1 c3 ðkð1 p1 p2 Þ 1Þ ¼ 0
ð64Þ
c2 c3 p1 p3 ðkð1 p1 Þ 1Þ ¼ 0
ð65Þ
ð57Þ
i¼1
Because pi2pi3-1 = -pi1pi3 we can also write
n
P
ai Ai ðwÞpi3 1 ci2 pi1 pi3 wpi1 ¼ 0
ð58Þ
i¼1
Downloaded By: [Ioan, Catalin Angelo] At: 05:19 13 April 2011
From Lemma 1 we have now ci2 = 0, which is an
obvious contradiction. If k 1 then ci1 ¼ ci3 ¼ 0 "i
¼ 1; n and Equation 54 becomes
n
P
ai Ai ðwÞpi3 1 ci2 ðpi2 pi3 kÞwpi1 ¼ 0
ð59Þ
i¼1
or, from Lemma 1
ci2 ðpi2 pi3 kÞ ¼ 0 "i ¼ 1; n
ð60Þ
From Equation 60 we have now ci2 = 0, or ci2 0 but
pi2 ¼ k=pi3 , from where pi1 ¼ ð1 kÞ=pi3 .
Finally
PðK; LÞ ¼
n
P
i¼1
ai cpi2i3 K1k Lk ¼ aK1k Lk
Û
ð61Þ
Theorem 3: The only cases when for n = 1, s = k
where k is a positive constant are the Cobb–Douglas
function and CES function with ¼ k=1 k.
Proof 3: From Equation 35 we obtain
ðc2 p2 p3 wp1 þ c3 Þðc1 wp2 þ c2 p1 p3 Þ
¼ k c22 p1 wp1 p2 p23 þ c3 c2 p1 p3 ð1 p1 Þ
þc1 c3 ð1 p1 p2 Þwp2 þ c1 c2 wp2 þp1 p2 p3 ð1 p2 Þ
that is
ðkc1 c2 p2 p3 ð1 p2 Þ c1 c2 p2 p3 Þwp1 þp2
þ ðkc22 p1 p2 p23 c22 p1 p2 p23 Þwp1
þ ðkc1 c3 ð1 p1 p2 Þ c1 c3 Þwp2
þ kc3 c2 p1 p3 ð1 p1 Þ c2 c3 p1 p3 ¼ 0
From Lemma 1, we obtain
If c2 0 follows from Equation 63 that k = 1 and
from Equation 62 we have c1 = 0 and c3 = 0. The
function is PðK; LÞ ¼ aðc2 Kp1 Lp2 Þp3 ¼ bKp L1p with
obvious notations.
If c2 = 0, from Equation 64 we have k(1 - p1 - p2) 3
1 = 0, that is, k ¼ p13p1
and the function is
p
p p
PðK; LÞ ¼ aðc1 K þ c3 L Þ and k ¼ 1=1 p.
&
V. Conclusions
The sum production function, which we have introduced in this article, integrates in an unitary expression various production functions such as
Cobb–Douglas, CES, Lu–Fletcher, Liu–Hildebrand,
VES and Kadiyala. In a big enterprise with various
departments of production and peculiar production’s
processes, each of them has a production that follows
a different law. The total production is therefore the
sum of partial results that can be considered like the
function defined in this article.
The first two theorems presented here are very
important because they show that a condition
imposed upon the proportionality marginal rate of
substitution or upon the constancy of the elasticity of
labour can exist only when each component is
Cobb–Douglas with the same parameter. The third
theorem also shows that in the case of n = 1 the only
cases when the elasticity of substitution is constant are
the Cobb–Douglas function and CES function.
Finally, we can say that the sum production function can be considered like a class of functions with its
own indicators that generalize those of particular
cases.
References
Arrow, K. J., Chenery, H. B., Minhas, B. S. and Solow, R. M.
(1961) Capital labour substitution and economic efficiency, Review of Economics and Statistics, 63, 225–50.
Cobb, C. W. and Douglas, P. H. (1928) A theory of production, American Economic Review, 18, 139–65.
Ioan, C. A. (2004) Applications of Geometry at the Study of
Production Functions Annals of Danubius University,
Danubius University, Galati, pp. 27–39.
Ioan, C. A. (2007) Applications of the space differential
geometry at the study of production functions,
Euroeconomica, 18, 30–8.
8
Downloaded By: [Ioan, Catalin Angelo] At: 05:19 13 April 2011
Kadiyala, K. R. (1972) Production functions and elasticity of substitution, Southern Economic Journal, 38,
281–4.
Kmenta, J. (1967) On estimation of the CES production
function, International Economic Review, 8, 180–9.
Liu, T. C. and Hildebrand, G. H. (1965) Manufacturing
Production Functions in the United States, 1957,
Cornell University Press, Ithaca, NY.
C. A. Ioan and G. Ioan
Mishra, S. K. (2007) A Brief History of Production
Functions, North-Eastern Hill University, Shillong,
India.
Revankar, N. S. (1971) A class of variable elasticity of substitution production functions, Econometrica, 39,
61–71.
Sato, R. (1974) On the class of separable non-homothetic
CES functions, Economic Studies Quarterly, 15, 42–55.