Research Article Some Characterizations of the...
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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 761832, 6 pages http://dx.doi.org/10.1155/2013/761832 Research Article Some Characterizations of the Cobb-Douglas and CES Production Functions in Microeconomics Xiaoshu Wang 1 and Yu Fu 2 1 School of Investment and Construction Management, Dongbei University of Finance and Economics, Dalian 116025, China 2 School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China Correspondence should be addressed to Yu Fu; [email protected] Received 5 November 2013; Accepted 10 December 2013 Academic Editor: Grzegorz Lukaszewicz Copyright © 2013 X. Wang and Y. Fu. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. It is well known that the study of the shape and the properties of the production possibility frontier is a subject of great interest in economic analysis. Vˆ ılcu (Vˆ ılcu, 2011) proved that the generalized Cobb-Douglas production function has constant return to scale if and only if the corresponding hypersurface is developable. Later on, the authors A. D. Vˆ ılcu and G. E. Vˆ ılcu, 2011 extended this result to the case of CES production function. Both results establish an interesting link between some fundamental notions in the theory of production functions and the differential geometry of hypersurfaces in Euclidean spaces. In this paper, we give some characterizations of minimal generalized Cobb-Douglas and CES production hypersurfaces in Euclidean spaces. 1. Introduction In microeconomics and macroeconomics, the production functions are positive nonconstant functions that specify the output of a firm, an industry, or an entire economy for all combinations of inputs. Almost all economic theories presuppose a production function, either on the firm level or the aggregate level. Hence, the production function is one of the most key concepts of mainstream neoclassical theories. By assuming that the maximum output technologically possible from a given set of inputs is achieved, economists always using a production function in analysis are abstracting from the engineering and managerial problems inherently associated with a particular production process; see [1]. In 1928, Cobb and Douglas [1] introduced a famous two- factor production function, nowadays called Cobb-Douglas production function, in order to describe the distribution of the national income by help of production functions. Two- factor Cobb-Douglas production function is given by = 1− , (1) where denotes the labor input, is the capital input, is the total factor productivity, and is the total production. e Cobb-Douglas (CD) production function is espe- cially notable for being the first time an aggregate or economy-wide production function was developed, esti- mated, and then presented to the profession for analysis. It gave a landmark change in how economists approached macroeconomics. e Cobb-Douglas function has also been applied to many other issues besides production. e generalized form of the Cobb-Douglas production function is written as ( 1 ,..., ) = 1 1 ⋅⋅⋅ , (2) where > 0( = 1,...,), is a positive constant, and 1 ,..., are nonzero constants. In 1961, Arrow et al. [2] introduced another two-input production function given by = ( + (1 − ) ) 1/ , (3) where is the output, the factor productivity, the share parameter, and the primary production factors, = ( − 1)/, and = 1/(1−) the elasticity of substitution. Hence this is called constant elasticity of substitution (CES) production function [3, 4].
Transcript of Research Article Some Characterizations of the...
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 761832 6 pageshttpdxdoiorg1011552013761832
Research ArticleSome Characterizations of the Cobb-Douglas and CESProduction Functions in Microeconomics
Xiaoshu Wang1 and Yu Fu2
1 School of Investment and Construction Management Dongbei University of Finance and Economics Dalian 116025 China2 School of Mathematics and Quantitative Economics Dongbei University of Finance and Economics Dalian 116025 China
Correspondence should be addressed to Yu Fu yufudufegmailcom
Received 5 November 2013 Accepted 10 December 2013
Academic Editor Grzegorz Lukaszewicz
Copyright copy 2013 X Wang and Y Fu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
It is well known that the study of the shape and the properties of the production possibility frontier is a subject of great interestin economic analysis Vılcu (Vılcu 2011) proved that the generalized Cobb-Douglas production function has constant return toscale if and only if the corresponding hypersurface is developable Later on the authors A D Vılcu and G E Vılcu 2011 extendedthis result to the case of CES production function Both results establish an interesting link between some fundamental notions inthe theory of production functions and the differential geometry of hypersurfaces in Euclidean spaces In this paper we give somecharacterizations of minimal generalized Cobb-Douglas and CES production hypersurfaces in Euclidean spaces
1 Introduction
In microeconomics and macroeconomics the productionfunctions are positive nonconstant functions that specifythe output of a firm an industry or an entire economy forall combinations of inputs Almost all economic theoriespresuppose a production function either on the firm level orthe aggregate level Hence the production function is one ofthemost key concepts ofmainstreamneoclassical theories Byassuming that the maximum output technologically possiblefrom a given set of inputs is achieved economists alwaysusing a production function in analysis are abstractingfrom the engineering and managerial problems inherentlyassociated with a particular production process see [1]
In 1928 Cobb and Douglas [1] introduced a famous two-factor production function nowadays called Cobb-Douglasproduction function in order to describe the distribution ofthe national income by help of production functions Two-factor Cobb-Douglas production function is given by
119884 = 1198871198711198961198621minus119896
(1)
where 119871 denotes the labor input119862 is the capital input 119887 is thetotal factor productivity and 119884 is the total production
The Cobb-Douglas (CD) production function is espe-cially notable for being the first time an aggregate or
economy-wide production function was developed esti-mated and then presented to the profession for analysisIt gave a landmark change in how economists approachedmacroeconomics The Cobb-Douglas function has also beenapplied to many other issues besides production
The generalized form of the Cobb-Douglas productionfunction is written as
119865 (1199091 119909119899) = 1198601199091205721
1sdot sdot sdot 119909120572119899
119899 (2)
where 119909119894 gt 0 (119894 = 1 119899) 119860 is a positive constant and1205721 120572119899 are nonzero constants
In 1961 Arrow et al [2] introduced another two-inputproduction function given by
119876 = 119887(119886119870119903+ (1 minus 119886) 119871
119903)1119903
(3)
where 119876 is the output 119887 the factor productivity 119886 the shareparameter 119870 and 119871 the primary production factors 119903 = (119904 minus
1)119904 and 119904 = 1(1minus119903) the elasticity of substitution Hence thisis called constant elasticity of substitution (CES) productionfunction [3 4]
2 Abstract and Applied Analysis
The generalized formof CES production function is givenby
119865 (1199091 119909119899) = 119860(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588
(4)
where 119886119894 120574 119860 and 120588 are nonzero constants and 119860 119886119894 gt
0 The CES production functions are of great interest ineconomy because of their invariant characteristic namelythat the elasticity of substitution between the parameters isconstant on their domains
It is easy to see that the CES production function is ageneralization of Cobb-Douglas production function andboth of them are homogeneous production functions
Note that the production function 119865 can be identifiedwith a production hypersurface in the (119899 + 1)-dimensionalEuclidean space E119899+1 through the map 119891 R119899
+rarr R119899+1+
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (5)
In the case of 119899 inputs we have a hypersurface and an analysisof the generalized CD and CES production functions fromthe point of view of differential geometry
In [5] Vılcu established an interesting link betweensome fundamental notions in the theory of productionfunctions and the differential geometry of hypersurfacesTheauthor proved that the generalizedCobb-Douglas productionfunction has constant return to scale if and only if the cor-responding hypersurface is developable The author jointlywith A D Vılcu further [6] generalized this result to the caseof the generalized CES production function with 119899-inputsFor further study of production hypersurfaces we refer thereader to Chenrsquos series of interesting papers on homogeneousproduction functions quasi-sum production models andhomothetic production functions [7ndash13] and G E Vılcu andA D Vılcursquos paper [14]
The theory of minimal surfaces (hypersurfaces) is veryimportant in the field of differential geometry In this paperwe give further study of the generalized Cobb-Douglas andCES production functions as hypersurfaces in Euclideanspace E119899+1 In particular we give some characterizations ofthe generalized Cobb-Douglas and CES production hyper-surfaces under the minimality condition in Euclidean spaces
2 Some Basic Concepts in Theory ofHypersurfaces
Each production function119865(1199091 119909119899) can be identifiedwitha graph of a nonparametric hypersurface of an Euclidean (119899+
1)-space E119899+1 given by
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (6)
We recall some basic concepts in the theory of hypersurfacesin Euclidean spaces
Let119872119899 be a hypersurface in an 119899+1-dimension Euclideanspace The Gauss map V is defined by
V 119872119899997888rarr 119878119899sub E119899+1
(7)
whichmaps119872119899 to the unit hypersphere 119878119899 ofE119899+1TheGaussmap is always defined locally that is on a small piece of thehypersurface It can be defined globally if the hypersurface isorientableTheGaussmap is a continuousmap such that V(119901)is a unit normal vector 120585(119901) of119872119899 at point 119901
The differential 119889V of the Gauss map V can be used todefine an extrinsic curvature known as the shape operatoror Weingarten map Since at each point 119901 isin 119872
119899 the tangentspace 119879119901119872 is an inner product space the shape operator 119878119901can be defined as a linear operator on this space by the relation
119892 (119878119901119906 119908) = 119892 (119889V (119906) 119908) (8)
where 119906 119908 isin 119879119901119872 and 119892 is the metric tensor on119872119899
Moreover the second fundamental form ℎ is related to theshape operator 119878119901 by
119892 (119878119901119906 119908) = 119892 (ℎ (119906 119908) 120585 (119901)) (9)
for 119906 119908 isin 119879119901119872It is well known that the determinant of the shape
operator 119878119901 is called the Gauss-Kronecker curvature Denoteit by 119866(119901) When 119899 = 2 the Gauss-Kronecker curvatureis simply called the Gauss curvature which is intrinsicdue to famous Gaussrsquos Theorema Egregium The trace ofthe shape operator 119878119901 is called the mean curvature of thehypersurfaces In contrast to the Gauss curvature the meancurvature is extrinsic which depends on the immersion ofthe hypersurface A hypersurface is calledminimal if itsmeancurvature vanishes identically
Denote the partial derivatives 120597119865120597119909119894 1205972119865120597119909119894119909119895 and
so forth by 119865119894 119865119894119895 and so forth Put
119882 = radic1 +
119899
sum
119894=1
1198652
119894 (10)
Recall some well-known results for a graph of hypersurface(6) in E119899+1 from [7 15 16]
Proposition 1 For a production hypersurface of E119899+1 definedby
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (11)
One has the following
(1) The unit normal 120585 is given by
120585 =1
119882(minus1198651 minus119865119899 1) (12)
(2) The coefficient 119892119894119895 = 119892(120597120597119909119894 120597120597119909119895) of the metrictensor is given by
119892119894119895 = 120575119894119895 + 119865119894119865119895 (13)
where 120575119894119895 = 1 if 119894 = 119895 otherwise 0(3) The volume element is
119889119881 = radic1198921198941198951198891199091 and sdot sdot sdot and 119889119909119899 (14)
Abstract and Applied Analysis 3
(4) The inverse matrix (119892119894119895) of (119892119894119895) is
119892119894119895= 120575119894119895 minus
119865119894119865119895
1198822 (15)
(5) The matrix of the second fundamental form ℎ is
ℎ119894119895 =
119865119894119895
119882 (16)
(6) The matrix of the shape operator 119878119901 is
119886119895
119894= sum
119896
ℎ119894119896119892119896119895
=1
119882(119865119894119895 minussum
119896
119865119894119896119865119896119865119895
1198822) (17)
(7) The mean curvature119867 is given by
119867 =1
119899119882(sum
119894
119865119894119894 minus1
1198822sum
119894119895
119865119894119865119895119865119894119895) (18)
(8) The Gauss-Kronecker curvature 119866 is
119866 =
det ℎ119894119895det119892119894119895
=
det119865119894119895119882119899+2
(19)
(9) The sectional curvature119870119894119895 of the plane section spannedby 120597120597119909119894 and 120597120597119909119895 is
119870119894119895 =
119865119894119894119865119895119895 minus 1198652
119894119895
1198822 (1 + 1198652
119894+ 1198652
119895)
(20)
(10) The Riemann curvature tensor 119877 satisfies
119892(119877(120597
120597119909119894
120597
120597119909119895
)120597
120597119909119896
120597
120597119909119897
) =
119865119894119897119865119895119896 minus 119865119894119896119865119895119897
1198824 (21)
3 Generalized Cobb-Douglas ProductionHypersurfaces
In this section we give a nonexistence result of minimalgeneralizedCobb-Douglas production hypersurfaces inE119899+1
Theorem 2 There does not exist a minimal generalized Cobb-Douglas production hypersurface in E119899+1
Proof Let 119872 be a generalized Cobb-Douglas productionhypersurface in E119899+1 given by a graph
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (22)
where 119865 is the generalized Cobb-Douglas production func-tion
119865 (1199091 119909119899) = 1198601199091205721
1sdot sdot sdot 119909120572119899
119899 (23)
Note that the generalizedCobb-Douglas production function(23) is homogeneous of degree ℎ = sum
119899
119894=1120572119894 It follows from
(23) that
119865119894 =120572119894
119909119894
119865 119894 = 1 119899
119865119894119894 =120572119894 (120572119894 minus 1)
1199092
119894
119865 119894 = 1 119899
119865119894119895 =
120572119894120572119895
119909119894119909119895
119865 119894 = 119895 119894 119895 = 1 119899
(24)
By assumption the production hypersurface is minimal thatis119867 = 0 Thus by (18) and (10) we have
(
119899
sum
119894=1
119865119894119894)(1 +
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (25)
which reduces to119899
sum
119894=1
119865119894119894 +sum
119894119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (26)
Note that sum119894119895(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 for 119894 = 119895 Hence (26)
becomes119899
sum
119894=1
119865119894119894 + sum
119894 = 119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (27)
Substituting (24) into (27) we obtain
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
minus 1198652sum
119894 = 119895
1205722
119894120572119895
1199092
1198941199092
119895
= 0 (28)
Differentiating (28) with respect to 119909119896 one has
minus2120572119896 (120572119896 minus 1)
1199093
119896
minus2120572119896
119909119896
1198652sum
119894 = 119895
1205722
119894120572119895
1199092
1198941199092
119895
+2120572119896
119909119896
1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0
(29)
which reduces to
120572119896 minus 1
1199092
119896
+ 1198652sum
119894 = 119895
1205722
119894120572119895
1199092
1198941199092
119895
minus 1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0 (30)
Substituting (28) into (30) we have
120572119896 minus 1
1199092
119896
+
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
minus 1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0 (31)
Moreover differentiating (31) with respect to 119909119897 (119897 = 119896) we get
minus2120572119897 (120572119897 minus 1)
1199093
119897
+2120572119897120572119896 + 2120572
2
119897
1199093
1198971199092
119896
1198652minus
2120572119897
119909119897
1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0
(32)
4 Abstract and Applied Analysis
Combining (32) with (31) gives
120572119897 minus 1
1199092
119897
minus120572119896 + 120572119897
1199092
1198971199092
119896
1198652+
120572119896 minus 1
1199092
119896
+
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
= 0 (33)
Differentiating (33) with respect to 119909119897 again one has
(120572119896 + 120572119897) (120572119897 minus 1)
1199093
1198971199092
119896
1198652+
(120572119897 minus 1) (1205721 + 1)
1199093
119897
= 0 (34)
which yields either 120572119897 = 1 (119897 = 119896) or
(120572119896 + 120572119897) 1198652= (120572119897 + 1) 119909
2
119896 (35)
Since 119865 is defined by (23) with nonzero constants 120572119894 for 119894 =
1 119899 it follows that (35) is impossible Hence we have 120572119897 =1
Without loss of generality we may choose 119896 = 1 In thiscase 120572119897 = 1 for 119897 = 1 reduces to
1205722 = sdot sdot sdot = 120572119899 = 1 (36)
Therefore (33) becomes
minus1205721 + 1
1199092
11199092
119897
1198652+
1205722
1minus 1
1199092
1
= 0 (37)
The above equation implies that either 1205721 = minus1 or
1198652= (1205721 minus 1) 119909
2
119897 (38)
It is easy to see that the latter case is impossible At thismoment we obtain the function 119865 as follows
119865 (1199091 119909119899) = 119860119909minus1
11199092 sdot sdot sdot 119909119899 (39)
Consider (31) for 119896 = 1 which turns into
1
1199092
1
minus1198652
1199092
119896
(1
1199092
2
+ sdot sdot sdot +1
1199092119899
) = 0 (40)
Substituting (39) into (40) we have
11986021199092
2sdot sdot sdot 1199092
119899(
1
1199092
2
+ sdot sdot sdot +1
1199092119899
) = 1199092
119896 (41)
In view of (41) by comparing the degrees of the left and theright of the equality we have 119899 = 3 Hence (41) becomes
1198602(1199092
2+ 1199092
3) = 1199092
119896 119896 = 2 3 (42)
which is impossible since 119860 = 0Therefore there is not a minimal generalized Cobb-
Douglas production hypersurface in E119899+1This completes theproof of Theorem 2
4 Generalized CES Production Hypersurfaces
In economics goods that are completely substitutable witheach other are called perfect substitutes They may becharacterized as goods having a constant marginal rate ofsubstitution Mathematically a production function is calleda perfect substitute if it is of the form
119865 (1199091 119909119899) =
119899
sum
119894=1
119886119894119909119894 (43)
for some nonzero constants 1198861 119886119899In the following we deal with the case of minimal
generalized CES production hypersurfaces in E119899+1
Theorem 3 An 119899-factor generalized CES production hyper-surface in E119899+1 is minimal if and only if the generalized CESproduction function is a perfect substitute
Proof Assume that 119872 is a generalized CES productionhypersurface in E119899+1 given by a graph
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (44)
where 119865 is the generalized CES production function
119865 (1199091 119909119899) = 119860(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588
(45)
with 119860 119886119894 gt 0 and 119909119894 gt 0 The generalized CES productionfunction (45) is homogeneous of degree 120574 By assumptionthat the production hypersurface is minimal from (18) theminimality condition is equivalent to
(
119899
sum
119894=1
119865119894119894)(1 +
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (46)
We define another minimal production hypersurface as
119891 (1205821199091 120582119909119899) = (1205821199091 120582119909119899 119865 (1205821199091 120582119909119899))
120582 isin R+
(47)
Since the CES production function is homogeneous of degree120574 we conclude that the hypersurface given by
119891 (1199091 119909119899) = (1199091 119909119899 120582120574minus1
119865 (1199091 119909119899)) (48)
is also minimal In this case the minimality condition (46)becomes
(
119899
sum
119894=1
119865119894119894)(1205822minus2120574
+
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (49)
Comparing (49) with (46) we obtain that either 120574 = 1 or 120574 = 1
and119899
sum
119894=1
119865119894119894 = 0 (50)
sum
119894119895
119865119894119865119895119865119894119895 = 0 (51)
Abstract and Applied Analysis 5
Moreover it follows from (45) that
119865119894 = 119860120574119886120588
119894119909120588minus1
119894(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus1
(52)
119865119894119894 = 119860120574119886120588
119894119909120588minus2
119894((120588 minus 1) sum
119896 = 119894
119886120588
119896119909120588
119896+ (120574 minus 1) 119886
120588
119894119909120588
119894)
times(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
(53)
119865119894119895 = 119860120574 (120574 minus 120588) 119886120588
119894119886120588
119895119909120588minus1
119894119909120588minus1
119895(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
119894 = 119895
(54)
We divide it into two cases
Case A (120574 = 1) Substituting (52) and (54) into (51) we get
11986031205743(120574 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120574120588minus4
timessum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(55)
If 120588 = 120574 then the above equation reduces to
sum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0 (56)
which is equivalent to
(
119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894)
2
= 0 (57)
Hence119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894= 0 (58)
Since 120588 = 1 this case is impossibleTherefore we conclude that 120588 = 120574 Substituting (53) into
(50) we have
119860120574 (120574 minus 1)(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
times (
119899
sum
119894=1
119886120588
119894119909120588
119894)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
= 0
(59)
Noting 120574 = 1 it is easy to see that (59) is impossible
Case B (120574 = 1) We note that the minimality conditionreduces to
119899
sum
119894=1
119865119894119894 + sum
119894 = 119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (60)
Substituting (52)ndash(54) into (60) we have
119860 (120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
1120588minus2
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198603(120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
minus 1198603(1 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(61)
Note that120588 = 1 fulfills (61) Hence in this case the generalizedCES production function 119865 is a perfect substitute
In the following we will show that the case 120588 = 1 isimpossible In fact if 120588 = 1 (61) reduces to
(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
2minus2120588
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198602(sum
119896 = 119894
119886120588
119896119909120588
119896) sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
+ 1198602sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(62)
Since 119909119894 gt 0 and 119886119894 gt 0 for 119894 = 1 119899 it follows that the leftof equality (62) is positive Therefore it is a contradiction
Conversely it is easy to verify that if the generalized CESproduction function is a perfect substitute then the 119899-factorgeneralized CES production hypersurface in E119899+1 is minimalThis completes the proof of Theorem 3
Remark 4 Remark that Chen proved in [7] a more generalresult for 2-factor ℎ-homogeneous production function is aperfect substitute if and only if the production surface is aminimal surface
Acknowledgments
The authors would like to thank the referees for theirvery valuable comments and suggestions to improve thispaper This work is supported by the Mathematical TianyuanYouth Fund of China (no 11326068) the general FinancialGrant from the China Postdoctoral Science Foundation (no2012M510818) and theGeneral Project for Scientific Researchof Liaoning Educational Committee (no W2012186)
References
[1] C W Cobb and P H Douglas ldquoA theory of productionrdquoAmerican Economic Review vol 18 pp 139ndash165 1928
6 Abstract and Applied Analysis
[2] K J Arrow H B Chenery B S Minhas and R M SolowldquoCapital-labor substitution and economic efficiencyrdquo Review ofEconomics and Statistics vol 43 no 3 pp 225ndash250 1961
[3] L Losonczi ldquoProduction functions having the CES propertyrdquoAcademiae Paedagogicae Nyıregyhaziensis vol 26 no 1 pp 113ndash125 2010
[4] D McFadden ldquoConstant elasticity of substitution productionfunctionsrdquo Review of Economic Studies vol 30 no 2 pp 73ndash831963
[5] G E Vılcu ldquoA geometric perspective on the generalized Cobb-Douglas production functionsrdquo Applied Mathematics Lettersvol 24 no 5 pp 777ndash783 2011
[6] A D Vılcu and G E Vılcu ldquoOn some geometric properties ofthe generalized CES production functionsrdquo Applied Mathemat-ics and Computation vol 218 no 1 pp 124ndash129 2011
[7] B-Y Chen ldquoOn some geometric properties of ℎ-homogeneousproduction functions in microeconomicsrdquo Kragujevac Journalof Mathematics vol 35 no 3 pp 343ndash357 2011
[8] B-Y Chen ldquoOn some geometric properties of quasi-sumproduction modelsrdquo Journal of Mathematical Analysis andApplications vol 392 no 2 pp 192ndash199 2012
[9] B-Y Chen ldquoA note on homogeneous production modelsrdquoKragujevac Journal ofMathematics vol 36 no 1 pp 41ndash43 2012
[10] B-Y Chen ldquoGeometry of quasi-sum production functions withconstant elasticity of substitution propertyrdquo Journal of AdvancedMathematical Studies vol 5 no 2 pp 90ndash97 2012
[11] B-Y Chen ldquoClassification of homothetic functions with con-stant elasticity of substitution and its geometric applicationsrdquoInternational Electronic Journal of Geometry vol 5 no 2 pp67ndash78 2012
[12] B-Y Chen ldquoSolutions to homogeneousMongemdashAmpere equa-tions of homothetic functions and their applications to produc-tion models in economicsrdquo Journal of Mathematical Analysisand Applications vol 411 no 1 pp 223ndash229 2014
[13] B-Y Chen and G E Vılcu ldquoGeometric classifications ofhomogeneous production functionsrdquo Applied Mathematics andComputation vol 225 pp 345ndash351 2013
[14] A D Vilcu and G E Vilcu ldquoOn homogeneous productionfunctions with proportional marginal rate of substitutionrdquoMathematical Problems in Engineering vol 2013 Article ID732643 5 pages 2013
[15] B-Y Chen Pseudo-Riemannian Geometry 120575-Invariants andApplications World Scientific Hackensack NJ USA 2011
[16] B-y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
The generalized formof CES production function is givenby
119865 (1199091 119909119899) = 119860(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588
(4)
where 119886119894 120574 119860 and 120588 are nonzero constants and 119860 119886119894 gt
0 The CES production functions are of great interest ineconomy because of their invariant characteristic namelythat the elasticity of substitution between the parameters isconstant on their domains
It is easy to see that the CES production function is ageneralization of Cobb-Douglas production function andboth of them are homogeneous production functions
Note that the production function 119865 can be identifiedwith a production hypersurface in the (119899 + 1)-dimensionalEuclidean space E119899+1 through the map 119891 R119899
+rarr R119899+1+
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (5)
In the case of 119899 inputs we have a hypersurface and an analysisof the generalized CD and CES production functions fromthe point of view of differential geometry
In [5] Vılcu established an interesting link betweensome fundamental notions in the theory of productionfunctions and the differential geometry of hypersurfacesTheauthor proved that the generalizedCobb-Douglas productionfunction has constant return to scale if and only if the cor-responding hypersurface is developable The author jointlywith A D Vılcu further [6] generalized this result to the caseof the generalized CES production function with 119899-inputsFor further study of production hypersurfaces we refer thereader to Chenrsquos series of interesting papers on homogeneousproduction functions quasi-sum production models andhomothetic production functions [7ndash13] and G E Vılcu andA D Vılcursquos paper [14]
The theory of minimal surfaces (hypersurfaces) is veryimportant in the field of differential geometry In this paperwe give further study of the generalized Cobb-Douglas andCES production functions as hypersurfaces in Euclideanspace E119899+1 In particular we give some characterizations ofthe generalized Cobb-Douglas and CES production hyper-surfaces under the minimality condition in Euclidean spaces
2 Some Basic Concepts in Theory ofHypersurfaces
Each production function119865(1199091 119909119899) can be identifiedwitha graph of a nonparametric hypersurface of an Euclidean (119899+
1)-space E119899+1 given by
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (6)
We recall some basic concepts in the theory of hypersurfacesin Euclidean spaces
Let119872119899 be a hypersurface in an 119899+1-dimension Euclideanspace The Gauss map V is defined by
V 119872119899997888rarr 119878119899sub E119899+1
(7)
whichmaps119872119899 to the unit hypersphere 119878119899 ofE119899+1TheGaussmap is always defined locally that is on a small piece of thehypersurface It can be defined globally if the hypersurface isorientableTheGaussmap is a continuousmap such that V(119901)is a unit normal vector 120585(119901) of119872119899 at point 119901
The differential 119889V of the Gauss map V can be used todefine an extrinsic curvature known as the shape operatoror Weingarten map Since at each point 119901 isin 119872
119899 the tangentspace 119879119901119872 is an inner product space the shape operator 119878119901can be defined as a linear operator on this space by the relation
119892 (119878119901119906 119908) = 119892 (119889V (119906) 119908) (8)
where 119906 119908 isin 119879119901119872 and 119892 is the metric tensor on119872119899
Moreover the second fundamental form ℎ is related to theshape operator 119878119901 by
119892 (119878119901119906 119908) = 119892 (ℎ (119906 119908) 120585 (119901)) (9)
for 119906 119908 isin 119879119901119872It is well known that the determinant of the shape
operator 119878119901 is called the Gauss-Kronecker curvature Denoteit by 119866(119901) When 119899 = 2 the Gauss-Kronecker curvatureis simply called the Gauss curvature which is intrinsicdue to famous Gaussrsquos Theorema Egregium The trace ofthe shape operator 119878119901 is called the mean curvature of thehypersurfaces In contrast to the Gauss curvature the meancurvature is extrinsic which depends on the immersion ofthe hypersurface A hypersurface is calledminimal if itsmeancurvature vanishes identically
Denote the partial derivatives 120597119865120597119909119894 1205972119865120597119909119894119909119895 and
so forth by 119865119894 119865119894119895 and so forth Put
119882 = radic1 +
119899
sum
119894=1
1198652
119894 (10)
Recall some well-known results for a graph of hypersurface(6) in E119899+1 from [7 15 16]
Proposition 1 For a production hypersurface of E119899+1 definedby
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (11)
One has the following
(1) The unit normal 120585 is given by
120585 =1
119882(minus1198651 minus119865119899 1) (12)
(2) The coefficient 119892119894119895 = 119892(120597120597119909119894 120597120597119909119895) of the metrictensor is given by
119892119894119895 = 120575119894119895 + 119865119894119865119895 (13)
where 120575119894119895 = 1 if 119894 = 119895 otherwise 0(3) The volume element is
119889119881 = radic1198921198941198951198891199091 and sdot sdot sdot and 119889119909119899 (14)
Abstract and Applied Analysis 3
(4) The inverse matrix (119892119894119895) of (119892119894119895) is
119892119894119895= 120575119894119895 minus
119865119894119865119895
1198822 (15)
(5) The matrix of the second fundamental form ℎ is
ℎ119894119895 =
119865119894119895
119882 (16)
(6) The matrix of the shape operator 119878119901 is
119886119895
119894= sum
119896
ℎ119894119896119892119896119895
=1
119882(119865119894119895 minussum
119896
119865119894119896119865119896119865119895
1198822) (17)
(7) The mean curvature119867 is given by
119867 =1
119899119882(sum
119894
119865119894119894 minus1
1198822sum
119894119895
119865119894119865119895119865119894119895) (18)
(8) The Gauss-Kronecker curvature 119866 is
119866 =
det ℎ119894119895det119892119894119895
=
det119865119894119895119882119899+2
(19)
(9) The sectional curvature119870119894119895 of the plane section spannedby 120597120597119909119894 and 120597120597119909119895 is
119870119894119895 =
119865119894119894119865119895119895 minus 1198652
119894119895
1198822 (1 + 1198652
119894+ 1198652
119895)
(20)
(10) The Riemann curvature tensor 119877 satisfies
119892(119877(120597
120597119909119894
120597
120597119909119895
)120597
120597119909119896
120597
120597119909119897
) =
119865119894119897119865119895119896 minus 119865119894119896119865119895119897
1198824 (21)
3 Generalized Cobb-Douglas ProductionHypersurfaces
In this section we give a nonexistence result of minimalgeneralizedCobb-Douglas production hypersurfaces inE119899+1
Theorem 2 There does not exist a minimal generalized Cobb-Douglas production hypersurface in E119899+1
Proof Let 119872 be a generalized Cobb-Douglas productionhypersurface in E119899+1 given by a graph
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (22)
where 119865 is the generalized Cobb-Douglas production func-tion
119865 (1199091 119909119899) = 1198601199091205721
1sdot sdot sdot 119909120572119899
119899 (23)
Note that the generalizedCobb-Douglas production function(23) is homogeneous of degree ℎ = sum
119899
119894=1120572119894 It follows from
(23) that
119865119894 =120572119894
119909119894
119865 119894 = 1 119899
119865119894119894 =120572119894 (120572119894 minus 1)
1199092
119894
119865 119894 = 1 119899
119865119894119895 =
120572119894120572119895
119909119894119909119895
119865 119894 = 119895 119894 119895 = 1 119899
(24)
By assumption the production hypersurface is minimal thatis119867 = 0 Thus by (18) and (10) we have
(
119899
sum
119894=1
119865119894119894)(1 +
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (25)
which reduces to119899
sum
119894=1
119865119894119894 +sum
119894119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (26)
Note that sum119894119895(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 for 119894 = 119895 Hence (26)
becomes119899
sum
119894=1
119865119894119894 + sum
119894 = 119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (27)
Substituting (24) into (27) we obtain
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
minus 1198652sum
119894 = 119895
1205722
119894120572119895
1199092
1198941199092
119895
= 0 (28)
Differentiating (28) with respect to 119909119896 one has
minus2120572119896 (120572119896 minus 1)
1199093
119896
minus2120572119896
119909119896
1198652sum
119894 = 119895
1205722
119894120572119895
1199092
1198941199092
119895
+2120572119896
119909119896
1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0
(29)
which reduces to
120572119896 minus 1
1199092
119896
+ 1198652sum
119894 = 119895
1205722
119894120572119895
1199092
1198941199092
119895
minus 1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0 (30)
Substituting (28) into (30) we have
120572119896 minus 1
1199092
119896
+
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
minus 1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0 (31)
Moreover differentiating (31) with respect to 119909119897 (119897 = 119896) we get
minus2120572119897 (120572119897 minus 1)
1199093
119897
+2120572119897120572119896 + 2120572
2
119897
1199093
1198971199092
119896
1198652minus
2120572119897
119909119897
1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0
(32)
4 Abstract and Applied Analysis
Combining (32) with (31) gives
120572119897 minus 1
1199092
119897
minus120572119896 + 120572119897
1199092
1198971199092
119896
1198652+
120572119896 minus 1
1199092
119896
+
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
= 0 (33)
Differentiating (33) with respect to 119909119897 again one has
(120572119896 + 120572119897) (120572119897 minus 1)
1199093
1198971199092
119896
1198652+
(120572119897 minus 1) (1205721 + 1)
1199093
119897
= 0 (34)
which yields either 120572119897 = 1 (119897 = 119896) or
(120572119896 + 120572119897) 1198652= (120572119897 + 1) 119909
2
119896 (35)
Since 119865 is defined by (23) with nonzero constants 120572119894 for 119894 =
1 119899 it follows that (35) is impossible Hence we have 120572119897 =1
Without loss of generality we may choose 119896 = 1 In thiscase 120572119897 = 1 for 119897 = 1 reduces to
1205722 = sdot sdot sdot = 120572119899 = 1 (36)
Therefore (33) becomes
minus1205721 + 1
1199092
11199092
119897
1198652+
1205722
1minus 1
1199092
1
= 0 (37)
The above equation implies that either 1205721 = minus1 or
1198652= (1205721 minus 1) 119909
2
119897 (38)
It is easy to see that the latter case is impossible At thismoment we obtain the function 119865 as follows
119865 (1199091 119909119899) = 119860119909minus1
11199092 sdot sdot sdot 119909119899 (39)
Consider (31) for 119896 = 1 which turns into
1
1199092
1
minus1198652
1199092
119896
(1
1199092
2
+ sdot sdot sdot +1
1199092119899
) = 0 (40)
Substituting (39) into (40) we have
11986021199092
2sdot sdot sdot 1199092
119899(
1
1199092
2
+ sdot sdot sdot +1
1199092119899
) = 1199092
119896 (41)
In view of (41) by comparing the degrees of the left and theright of the equality we have 119899 = 3 Hence (41) becomes
1198602(1199092
2+ 1199092
3) = 1199092
119896 119896 = 2 3 (42)
which is impossible since 119860 = 0Therefore there is not a minimal generalized Cobb-
Douglas production hypersurface in E119899+1This completes theproof of Theorem 2
4 Generalized CES Production Hypersurfaces
In economics goods that are completely substitutable witheach other are called perfect substitutes They may becharacterized as goods having a constant marginal rate ofsubstitution Mathematically a production function is calleda perfect substitute if it is of the form
119865 (1199091 119909119899) =
119899
sum
119894=1
119886119894119909119894 (43)
for some nonzero constants 1198861 119886119899In the following we deal with the case of minimal
generalized CES production hypersurfaces in E119899+1
Theorem 3 An 119899-factor generalized CES production hyper-surface in E119899+1 is minimal if and only if the generalized CESproduction function is a perfect substitute
Proof Assume that 119872 is a generalized CES productionhypersurface in E119899+1 given by a graph
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (44)
where 119865 is the generalized CES production function
119865 (1199091 119909119899) = 119860(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588
(45)
with 119860 119886119894 gt 0 and 119909119894 gt 0 The generalized CES productionfunction (45) is homogeneous of degree 120574 By assumptionthat the production hypersurface is minimal from (18) theminimality condition is equivalent to
(
119899
sum
119894=1
119865119894119894)(1 +
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (46)
We define another minimal production hypersurface as
119891 (1205821199091 120582119909119899) = (1205821199091 120582119909119899 119865 (1205821199091 120582119909119899))
120582 isin R+
(47)
Since the CES production function is homogeneous of degree120574 we conclude that the hypersurface given by
119891 (1199091 119909119899) = (1199091 119909119899 120582120574minus1
119865 (1199091 119909119899)) (48)
is also minimal In this case the minimality condition (46)becomes
(
119899
sum
119894=1
119865119894119894)(1205822minus2120574
+
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (49)
Comparing (49) with (46) we obtain that either 120574 = 1 or 120574 = 1
and119899
sum
119894=1
119865119894119894 = 0 (50)
sum
119894119895
119865119894119865119895119865119894119895 = 0 (51)
Abstract and Applied Analysis 5
Moreover it follows from (45) that
119865119894 = 119860120574119886120588
119894119909120588minus1
119894(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus1
(52)
119865119894119894 = 119860120574119886120588
119894119909120588minus2
119894((120588 minus 1) sum
119896 = 119894
119886120588
119896119909120588
119896+ (120574 minus 1) 119886
120588
119894119909120588
119894)
times(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
(53)
119865119894119895 = 119860120574 (120574 minus 120588) 119886120588
119894119886120588
119895119909120588minus1
119894119909120588minus1
119895(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
119894 = 119895
(54)
We divide it into two cases
Case A (120574 = 1) Substituting (52) and (54) into (51) we get
11986031205743(120574 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120574120588minus4
timessum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(55)
If 120588 = 120574 then the above equation reduces to
sum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0 (56)
which is equivalent to
(
119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894)
2
= 0 (57)
Hence119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894= 0 (58)
Since 120588 = 1 this case is impossibleTherefore we conclude that 120588 = 120574 Substituting (53) into
(50) we have
119860120574 (120574 minus 1)(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
times (
119899
sum
119894=1
119886120588
119894119909120588
119894)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
= 0
(59)
Noting 120574 = 1 it is easy to see that (59) is impossible
Case B (120574 = 1) We note that the minimality conditionreduces to
119899
sum
119894=1
119865119894119894 + sum
119894 = 119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (60)
Substituting (52)ndash(54) into (60) we have
119860 (120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
1120588minus2
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198603(120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
minus 1198603(1 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(61)
Note that120588 = 1 fulfills (61) Hence in this case the generalizedCES production function 119865 is a perfect substitute
In the following we will show that the case 120588 = 1 isimpossible In fact if 120588 = 1 (61) reduces to
(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
2minus2120588
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198602(sum
119896 = 119894
119886120588
119896119909120588
119896) sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
+ 1198602sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(62)
Since 119909119894 gt 0 and 119886119894 gt 0 for 119894 = 1 119899 it follows that the leftof equality (62) is positive Therefore it is a contradiction
Conversely it is easy to verify that if the generalized CESproduction function is a perfect substitute then the 119899-factorgeneralized CES production hypersurface in E119899+1 is minimalThis completes the proof of Theorem 3
Remark 4 Remark that Chen proved in [7] a more generalresult for 2-factor ℎ-homogeneous production function is aperfect substitute if and only if the production surface is aminimal surface
Acknowledgments
The authors would like to thank the referees for theirvery valuable comments and suggestions to improve thispaper This work is supported by the Mathematical TianyuanYouth Fund of China (no 11326068) the general FinancialGrant from the China Postdoctoral Science Foundation (no2012M510818) and theGeneral Project for Scientific Researchof Liaoning Educational Committee (no W2012186)
References
[1] C W Cobb and P H Douglas ldquoA theory of productionrdquoAmerican Economic Review vol 18 pp 139ndash165 1928
6 Abstract and Applied Analysis
[2] K J Arrow H B Chenery B S Minhas and R M SolowldquoCapital-labor substitution and economic efficiencyrdquo Review ofEconomics and Statistics vol 43 no 3 pp 225ndash250 1961
[3] L Losonczi ldquoProduction functions having the CES propertyrdquoAcademiae Paedagogicae Nyıregyhaziensis vol 26 no 1 pp 113ndash125 2010
[4] D McFadden ldquoConstant elasticity of substitution productionfunctionsrdquo Review of Economic Studies vol 30 no 2 pp 73ndash831963
[5] G E Vılcu ldquoA geometric perspective on the generalized Cobb-Douglas production functionsrdquo Applied Mathematics Lettersvol 24 no 5 pp 777ndash783 2011
[6] A D Vılcu and G E Vılcu ldquoOn some geometric properties ofthe generalized CES production functionsrdquo Applied Mathemat-ics and Computation vol 218 no 1 pp 124ndash129 2011
[7] B-Y Chen ldquoOn some geometric properties of ℎ-homogeneousproduction functions in microeconomicsrdquo Kragujevac Journalof Mathematics vol 35 no 3 pp 343ndash357 2011
[8] B-Y Chen ldquoOn some geometric properties of quasi-sumproduction modelsrdquo Journal of Mathematical Analysis andApplications vol 392 no 2 pp 192ndash199 2012
[9] B-Y Chen ldquoA note on homogeneous production modelsrdquoKragujevac Journal ofMathematics vol 36 no 1 pp 41ndash43 2012
[10] B-Y Chen ldquoGeometry of quasi-sum production functions withconstant elasticity of substitution propertyrdquo Journal of AdvancedMathematical Studies vol 5 no 2 pp 90ndash97 2012
[11] B-Y Chen ldquoClassification of homothetic functions with con-stant elasticity of substitution and its geometric applicationsrdquoInternational Electronic Journal of Geometry vol 5 no 2 pp67ndash78 2012
[12] B-Y Chen ldquoSolutions to homogeneousMongemdashAmpere equa-tions of homothetic functions and their applications to produc-tion models in economicsrdquo Journal of Mathematical Analysisand Applications vol 411 no 1 pp 223ndash229 2014
[13] B-Y Chen and G E Vılcu ldquoGeometric classifications ofhomogeneous production functionsrdquo Applied Mathematics andComputation vol 225 pp 345ndash351 2013
[14] A D Vilcu and G E Vilcu ldquoOn homogeneous productionfunctions with proportional marginal rate of substitutionrdquoMathematical Problems in Engineering vol 2013 Article ID732643 5 pages 2013
[15] B-Y Chen Pseudo-Riemannian Geometry 120575-Invariants andApplications World Scientific Hackensack NJ USA 2011
[16] B-y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
(4) The inverse matrix (119892119894119895) of (119892119894119895) is
119892119894119895= 120575119894119895 minus
119865119894119865119895
1198822 (15)
(5) The matrix of the second fundamental form ℎ is
ℎ119894119895 =
119865119894119895
119882 (16)
(6) The matrix of the shape operator 119878119901 is
119886119895
119894= sum
119896
ℎ119894119896119892119896119895
=1
119882(119865119894119895 minussum
119896
119865119894119896119865119896119865119895
1198822) (17)
(7) The mean curvature119867 is given by
119867 =1
119899119882(sum
119894
119865119894119894 minus1
1198822sum
119894119895
119865119894119865119895119865119894119895) (18)
(8) The Gauss-Kronecker curvature 119866 is
119866 =
det ℎ119894119895det119892119894119895
=
det119865119894119895119882119899+2
(19)
(9) The sectional curvature119870119894119895 of the plane section spannedby 120597120597119909119894 and 120597120597119909119895 is
119870119894119895 =
119865119894119894119865119895119895 minus 1198652
119894119895
1198822 (1 + 1198652
119894+ 1198652
119895)
(20)
(10) The Riemann curvature tensor 119877 satisfies
119892(119877(120597
120597119909119894
120597
120597119909119895
)120597
120597119909119896
120597
120597119909119897
) =
119865119894119897119865119895119896 minus 119865119894119896119865119895119897
1198824 (21)
3 Generalized Cobb-Douglas ProductionHypersurfaces
In this section we give a nonexistence result of minimalgeneralizedCobb-Douglas production hypersurfaces inE119899+1
Theorem 2 There does not exist a minimal generalized Cobb-Douglas production hypersurface in E119899+1
Proof Let 119872 be a generalized Cobb-Douglas productionhypersurface in E119899+1 given by a graph
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (22)
where 119865 is the generalized Cobb-Douglas production func-tion
119865 (1199091 119909119899) = 1198601199091205721
1sdot sdot sdot 119909120572119899
119899 (23)
Note that the generalizedCobb-Douglas production function(23) is homogeneous of degree ℎ = sum
119899
119894=1120572119894 It follows from
(23) that
119865119894 =120572119894
119909119894
119865 119894 = 1 119899
119865119894119894 =120572119894 (120572119894 minus 1)
1199092
119894
119865 119894 = 1 119899
119865119894119895 =
120572119894120572119895
119909119894119909119895
119865 119894 = 119895 119894 119895 = 1 119899
(24)
By assumption the production hypersurface is minimal thatis119867 = 0 Thus by (18) and (10) we have
(
119899
sum
119894=1
119865119894119894)(1 +
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (25)
which reduces to119899
sum
119894=1
119865119894119894 +sum
119894119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (26)
Note that sum119894119895(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 for 119894 = 119895 Hence (26)
becomes119899
sum
119894=1
119865119894119894 + sum
119894 = 119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (27)
Substituting (24) into (27) we obtain
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
minus 1198652sum
119894 = 119895
1205722
119894120572119895
1199092
1198941199092
119895
= 0 (28)
Differentiating (28) with respect to 119909119896 one has
minus2120572119896 (120572119896 minus 1)
1199093
119896
minus2120572119896
119909119896
1198652sum
119894 = 119895
1205722
119894120572119895
1199092
1198941199092
119895
+2120572119896
119909119896
1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0
(29)
which reduces to
120572119896 minus 1
1199092
119896
+ 1198652sum
119894 = 119895
1205722
119894120572119895
1199092
1198941199092
119895
minus 1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0 (30)
Substituting (28) into (30) we have
120572119896 minus 1
1199092
119896
+
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
minus 1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0 (31)
Moreover differentiating (31) with respect to 119909119897 (119897 = 119896) we get
minus2120572119897 (120572119897 minus 1)
1199093
119897
+2120572119897120572119896 + 2120572
2
119897
1199093
1198971199092
119896
1198652minus
2120572119897
119909119897
1198652
119899
sum
119894=1
120572119894120572119896 + 1205722
119894
1199092
1198941199092
119896
= 0
(32)
4 Abstract and Applied Analysis
Combining (32) with (31) gives
120572119897 minus 1
1199092
119897
minus120572119896 + 120572119897
1199092
1198971199092
119896
1198652+
120572119896 minus 1
1199092
119896
+
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
= 0 (33)
Differentiating (33) with respect to 119909119897 again one has
(120572119896 + 120572119897) (120572119897 minus 1)
1199093
1198971199092
119896
1198652+
(120572119897 minus 1) (1205721 + 1)
1199093
119897
= 0 (34)
which yields either 120572119897 = 1 (119897 = 119896) or
(120572119896 + 120572119897) 1198652= (120572119897 + 1) 119909
2
119896 (35)
Since 119865 is defined by (23) with nonzero constants 120572119894 for 119894 =
1 119899 it follows that (35) is impossible Hence we have 120572119897 =1
Without loss of generality we may choose 119896 = 1 In thiscase 120572119897 = 1 for 119897 = 1 reduces to
1205722 = sdot sdot sdot = 120572119899 = 1 (36)
Therefore (33) becomes
minus1205721 + 1
1199092
11199092
119897
1198652+
1205722
1minus 1
1199092
1
= 0 (37)
The above equation implies that either 1205721 = minus1 or
1198652= (1205721 minus 1) 119909
2
119897 (38)
It is easy to see that the latter case is impossible At thismoment we obtain the function 119865 as follows
119865 (1199091 119909119899) = 119860119909minus1
11199092 sdot sdot sdot 119909119899 (39)
Consider (31) for 119896 = 1 which turns into
1
1199092
1
minus1198652
1199092
119896
(1
1199092
2
+ sdot sdot sdot +1
1199092119899
) = 0 (40)
Substituting (39) into (40) we have
11986021199092
2sdot sdot sdot 1199092
119899(
1
1199092
2
+ sdot sdot sdot +1
1199092119899
) = 1199092
119896 (41)
In view of (41) by comparing the degrees of the left and theright of the equality we have 119899 = 3 Hence (41) becomes
1198602(1199092
2+ 1199092
3) = 1199092
119896 119896 = 2 3 (42)
which is impossible since 119860 = 0Therefore there is not a minimal generalized Cobb-
Douglas production hypersurface in E119899+1This completes theproof of Theorem 2
4 Generalized CES Production Hypersurfaces
In economics goods that are completely substitutable witheach other are called perfect substitutes They may becharacterized as goods having a constant marginal rate ofsubstitution Mathematically a production function is calleda perfect substitute if it is of the form
119865 (1199091 119909119899) =
119899
sum
119894=1
119886119894119909119894 (43)
for some nonzero constants 1198861 119886119899In the following we deal with the case of minimal
generalized CES production hypersurfaces in E119899+1
Theorem 3 An 119899-factor generalized CES production hyper-surface in E119899+1 is minimal if and only if the generalized CESproduction function is a perfect substitute
Proof Assume that 119872 is a generalized CES productionhypersurface in E119899+1 given by a graph
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (44)
where 119865 is the generalized CES production function
119865 (1199091 119909119899) = 119860(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588
(45)
with 119860 119886119894 gt 0 and 119909119894 gt 0 The generalized CES productionfunction (45) is homogeneous of degree 120574 By assumptionthat the production hypersurface is minimal from (18) theminimality condition is equivalent to
(
119899
sum
119894=1
119865119894119894)(1 +
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (46)
We define another minimal production hypersurface as
119891 (1205821199091 120582119909119899) = (1205821199091 120582119909119899 119865 (1205821199091 120582119909119899))
120582 isin R+
(47)
Since the CES production function is homogeneous of degree120574 we conclude that the hypersurface given by
119891 (1199091 119909119899) = (1199091 119909119899 120582120574minus1
119865 (1199091 119909119899)) (48)
is also minimal In this case the minimality condition (46)becomes
(
119899
sum
119894=1
119865119894119894)(1205822minus2120574
+
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (49)
Comparing (49) with (46) we obtain that either 120574 = 1 or 120574 = 1
and119899
sum
119894=1
119865119894119894 = 0 (50)
sum
119894119895
119865119894119865119895119865119894119895 = 0 (51)
Abstract and Applied Analysis 5
Moreover it follows from (45) that
119865119894 = 119860120574119886120588
119894119909120588minus1
119894(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus1
(52)
119865119894119894 = 119860120574119886120588
119894119909120588minus2
119894((120588 minus 1) sum
119896 = 119894
119886120588
119896119909120588
119896+ (120574 minus 1) 119886
120588
119894119909120588
119894)
times(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
(53)
119865119894119895 = 119860120574 (120574 minus 120588) 119886120588
119894119886120588
119895119909120588minus1
119894119909120588minus1
119895(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
119894 = 119895
(54)
We divide it into two cases
Case A (120574 = 1) Substituting (52) and (54) into (51) we get
11986031205743(120574 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120574120588minus4
timessum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(55)
If 120588 = 120574 then the above equation reduces to
sum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0 (56)
which is equivalent to
(
119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894)
2
= 0 (57)
Hence119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894= 0 (58)
Since 120588 = 1 this case is impossibleTherefore we conclude that 120588 = 120574 Substituting (53) into
(50) we have
119860120574 (120574 minus 1)(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
times (
119899
sum
119894=1
119886120588
119894119909120588
119894)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
= 0
(59)
Noting 120574 = 1 it is easy to see that (59) is impossible
Case B (120574 = 1) We note that the minimality conditionreduces to
119899
sum
119894=1
119865119894119894 + sum
119894 = 119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (60)
Substituting (52)ndash(54) into (60) we have
119860 (120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
1120588minus2
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198603(120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
minus 1198603(1 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(61)
Note that120588 = 1 fulfills (61) Hence in this case the generalizedCES production function 119865 is a perfect substitute
In the following we will show that the case 120588 = 1 isimpossible In fact if 120588 = 1 (61) reduces to
(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
2minus2120588
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198602(sum
119896 = 119894
119886120588
119896119909120588
119896) sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
+ 1198602sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(62)
Since 119909119894 gt 0 and 119886119894 gt 0 for 119894 = 1 119899 it follows that the leftof equality (62) is positive Therefore it is a contradiction
Conversely it is easy to verify that if the generalized CESproduction function is a perfect substitute then the 119899-factorgeneralized CES production hypersurface in E119899+1 is minimalThis completes the proof of Theorem 3
Remark 4 Remark that Chen proved in [7] a more generalresult for 2-factor ℎ-homogeneous production function is aperfect substitute if and only if the production surface is aminimal surface
Acknowledgments
The authors would like to thank the referees for theirvery valuable comments and suggestions to improve thispaper This work is supported by the Mathematical TianyuanYouth Fund of China (no 11326068) the general FinancialGrant from the China Postdoctoral Science Foundation (no2012M510818) and theGeneral Project for Scientific Researchof Liaoning Educational Committee (no W2012186)
References
[1] C W Cobb and P H Douglas ldquoA theory of productionrdquoAmerican Economic Review vol 18 pp 139ndash165 1928
6 Abstract and Applied Analysis
[2] K J Arrow H B Chenery B S Minhas and R M SolowldquoCapital-labor substitution and economic efficiencyrdquo Review ofEconomics and Statistics vol 43 no 3 pp 225ndash250 1961
[3] L Losonczi ldquoProduction functions having the CES propertyrdquoAcademiae Paedagogicae Nyıregyhaziensis vol 26 no 1 pp 113ndash125 2010
[4] D McFadden ldquoConstant elasticity of substitution productionfunctionsrdquo Review of Economic Studies vol 30 no 2 pp 73ndash831963
[5] G E Vılcu ldquoA geometric perspective on the generalized Cobb-Douglas production functionsrdquo Applied Mathematics Lettersvol 24 no 5 pp 777ndash783 2011
[6] A D Vılcu and G E Vılcu ldquoOn some geometric properties ofthe generalized CES production functionsrdquo Applied Mathemat-ics and Computation vol 218 no 1 pp 124ndash129 2011
[7] B-Y Chen ldquoOn some geometric properties of ℎ-homogeneousproduction functions in microeconomicsrdquo Kragujevac Journalof Mathematics vol 35 no 3 pp 343ndash357 2011
[8] B-Y Chen ldquoOn some geometric properties of quasi-sumproduction modelsrdquo Journal of Mathematical Analysis andApplications vol 392 no 2 pp 192ndash199 2012
[9] B-Y Chen ldquoA note on homogeneous production modelsrdquoKragujevac Journal ofMathematics vol 36 no 1 pp 41ndash43 2012
[10] B-Y Chen ldquoGeometry of quasi-sum production functions withconstant elasticity of substitution propertyrdquo Journal of AdvancedMathematical Studies vol 5 no 2 pp 90ndash97 2012
[11] B-Y Chen ldquoClassification of homothetic functions with con-stant elasticity of substitution and its geometric applicationsrdquoInternational Electronic Journal of Geometry vol 5 no 2 pp67ndash78 2012
[12] B-Y Chen ldquoSolutions to homogeneousMongemdashAmpere equa-tions of homothetic functions and their applications to produc-tion models in economicsrdquo Journal of Mathematical Analysisand Applications vol 411 no 1 pp 223ndash229 2014
[13] B-Y Chen and G E Vılcu ldquoGeometric classifications ofhomogeneous production functionsrdquo Applied Mathematics andComputation vol 225 pp 345ndash351 2013
[14] A D Vilcu and G E Vilcu ldquoOn homogeneous productionfunctions with proportional marginal rate of substitutionrdquoMathematical Problems in Engineering vol 2013 Article ID732643 5 pages 2013
[15] B-Y Chen Pseudo-Riemannian Geometry 120575-Invariants andApplications World Scientific Hackensack NJ USA 2011
[16] B-y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Combining (32) with (31) gives
120572119897 minus 1
1199092
119897
minus120572119896 + 120572119897
1199092
1198971199092
119896
1198652+
120572119896 minus 1
1199092
119896
+
119899
sum
119894=1
120572119894 (120572119894 minus 1)
1199092
119894
= 0 (33)
Differentiating (33) with respect to 119909119897 again one has
(120572119896 + 120572119897) (120572119897 minus 1)
1199093
1198971199092
119896
1198652+
(120572119897 minus 1) (1205721 + 1)
1199093
119897
= 0 (34)
which yields either 120572119897 = 1 (119897 = 119896) or
(120572119896 + 120572119897) 1198652= (120572119897 + 1) 119909
2
119896 (35)
Since 119865 is defined by (23) with nonzero constants 120572119894 for 119894 =
1 119899 it follows that (35) is impossible Hence we have 120572119897 =1
Without loss of generality we may choose 119896 = 1 In thiscase 120572119897 = 1 for 119897 = 1 reduces to
1205722 = sdot sdot sdot = 120572119899 = 1 (36)
Therefore (33) becomes
minus1205721 + 1
1199092
11199092
119897
1198652+
1205722
1minus 1
1199092
1
= 0 (37)
The above equation implies that either 1205721 = minus1 or
1198652= (1205721 minus 1) 119909
2
119897 (38)
It is easy to see that the latter case is impossible At thismoment we obtain the function 119865 as follows
119865 (1199091 119909119899) = 119860119909minus1
11199092 sdot sdot sdot 119909119899 (39)
Consider (31) for 119896 = 1 which turns into
1
1199092
1
minus1198652
1199092
119896
(1
1199092
2
+ sdot sdot sdot +1
1199092119899
) = 0 (40)
Substituting (39) into (40) we have
11986021199092
2sdot sdot sdot 1199092
119899(
1
1199092
2
+ sdot sdot sdot +1
1199092119899
) = 1199092
119896 (41)
In view of (41) by comparing the degrees of the left and theright of the equality we have 119899 = 3 Hence (41) becomes
1198602(1199092
2+ 1199092
3) = 1199092
119896 119896 = 2 3 (42)
which is impossible since 119860 = 0Therefore there is not a minimal generalized Cobb-
Douglas production hypersurface in E119899+1This completes theproof of Theorem 2
4 Generalized CES Production Hypersurfaces
In economics goods that are completely substitutable witheach other are called perfect substitutes They may becharacterized as goods having a constant marginal rate ofsubstitution Mathematically a production function is calleda perfect substitute if it is of the form
119865 (1199091 119909119899) =
119899
sum
119894=1
119886119894119909119894 (43)
for some nonzero constants 1198861 119886119899In the following we deal with the case of minimal
generalized CES production hypersurfaces in E119899+1
Theorem 3 An 119899-factor generalized CES production hyper-surface in E119899+1 is minimal if and only if the generalized CESproduction function is a perfect substitute
Proof Assume that 119872 is a generalized CES productionhypersurface in E119899+1 given by a graph
119891 (1199091 119909119899) = (1199091 119909119899 119865 (1199091 119909119899)) (44)
where 119865 is the generalized CES production function
119865 (1199091 119909119899) = 119860(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588
(45)
with 119860 119886119894 gt 0 and 119909119894 gt 0 The generalized CES productionfunction (45) is homogeneous of degree 120574 By assumptionthat the production hypersurface is minimal from (18) theminimality condition is equivalent to
(
119899
sum
119894=1
119865119894119894)(1 +
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (46)
We define another minimal production hypersurface as
119891 (1205821199091 120582119909119899) = (1205821199091 120582119909119899 119865 (1205821199091 120582119909119899))
120582 isin R+
(47)
Since the CES production function is homogeneous of degree120574 we conclude that the hypersurface given by
119891 (1199091 119909119899) = (1199091 119909119899 120582120574minus1
119865 (1199091 119909119899)) (48)
is also minimal In this case the minimality condition (46)becomes
(
119899
sum
119894=1
119865119894119894)(1205822minus2120574
+
119899
sum
119894=1
1198652
119894) = sum
119894119895
119865119894119865119895119865119894119895 (49)
Comparing (49) with (46) we obtain that either 120574 = 1 or 120574 = 1
and119899
sum
119894=1
119865119894119894 = 0 (50)
sum
119894119895
119865119894119865119895119865119894119895 = 0 (51)
Abstract and Applied Analysis 5
Moreover it follows from (45) that
119865119894 = 119860120574119886120588
119894119909120588minus1
119894(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus1
(52)
119865119894119894 = 119860120574119886120588
119894119909120588minus2
119894((120588 minus 1) sum
119896 = 119894
119886120588
119896119909120588
119896+ (120574 minus 1) 119886
120588
119894119909120588
119894)
times(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
(53)
119865119894119895 = 119860120574 (120574 minus 120588) 119886120588
119894119886120588
119895119909120588minus1
119894119909120588minus1
119895(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
119894 = 119895
(54)
We divide it into two cases
Case A (120574 = 1) Substituting (52) and (54) into (51) we get
11986031205743(120574 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120574120588minus4
timessum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(55)
If 120588 = 120574 then the above equation reduces to
sum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0 (56)
which is equivalent to
(
119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894)
2
= 0 (57)
Hence119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894= 0 (58)
Since 120588 = 1 this case is impossibleTherefore we conclude that 120588 = 120574 Substituting (53) into
(50) we have
119860120574 (120574 minus 1)(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
times (
119899
sum
119894=1
119886120588
119894119909120588
119894)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
= 0
(59)
Noting 120574 = 1 it is easy to see that (59) is impossible
Case B (120574 = 1) We note that the minimality conditionreduces to
119899
sum
119894=1
119865119894119894 + sum
119894 = 119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (60)
Substituting (52)ndash(54) into (60) we have
119860 (120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
1120588minus2
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198603(120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
minus 1198603(1 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(61)
Note that120588 = 1 fulfills (61) Hence in this case the generalizedCES production function 119865 is a perfect substitute
In the following we will show that the case 120588 = 1 isimpossible In fact if 120588 = 1 (61) reduces to
(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
2minus2120588
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198602(sum
119896 = 119894
119886120588
119896119909120588
119896) sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
+ 1198602sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(62)
Since 119909119894 gt 0 and 119886119894 gt 0 for 119894 = 1 119899 it follows that the leftof equality (62) is positive Therefore it is a contradiction
Conversely it is easy to verify that if the generalized CESproduction function is a perfect substitute then the 119899-factorgeneralized CES production hypersurface in E119899+1 is minimalThis completes the proof of Theorem 3
Remark 4 Remark that Chen proved in [7] a more generalresult for 2-factor ℎ-homogeneous production function is aperfect substitute if and only if the production surface is aminimal surface
Acknowledgments
The authors would like to thank the referees for theirvery valuable comments and suggestions to improve thispaper This work is supported by the Mathematical TianyuanYouth Fund of China (no 11326068) the general FinancialGrant from the China Postdoctoral Science Foundation (no2012M510818) and theGeneral Project for Scientific Researchof Liaoning Educational Committee (no W2012186)
References
[1] C W Cobb and P H Douglas ldquoA theory of productionrdquoAmerican Economic Review vol 18 pp 139ndash165 1928
6 Abstract and Applied Analysis
[2] K J Arrow H B Chenery B S Minhas and R M SolowldquoCapital-labor substitution and economic efficiencyrdquo Review ofEconomics and Statistics vol 43 no 3 pp 225ndash250 1961
[3] L Losonczi ldquoProduction functions having the CES propertyrdquoAcademiae Paedagogicae Nyıregyhaziensis vol 26 no 1 pp 113ndash125 2010
[4] D McFadden ldquoConstant elasticity of substitution productionfunctionsrdquo Review of Economic Studies vol 30 no 2 pp 73ndash831963
[5] G E Vılcu ldquoA geometric perspective on the generalized Cobb-Douglas production functionsrdquo Applied Mathematics Lettersvol 24 no 5 pp 777ndash783 2011
[6] A D Vılcu and G E Vılcu ldquoOn some geometric properties ofthe generalized CES production functionsrdquo Applied Mathemat-ics and Computation vol 218 no 1 pp 124ndash129 2011
[7] B-Y Chen ldquoOn some geometric properties of ℎ-homogeneousproduction functions in microeconomicsrdquo Kragujevac Journalof Mathematics vol 35 no 3 pp 343ndash357 2011
[8] B-Y Chen ldquoOn some geometric properties of quasi-sumproduction modelsrdquo Journal of Mathematical Analysis andApplications vol 392 no 2 pp 192ndash199 2012
[9] B-Y Chen ldquoA note on homogeneous production modelsrdquoKragujevac Journal ofMathematics vol 36 no 1 pp 41ndash43 2012
[10] B-Y Chen ldquoGeometry of quasi-sum production functions withconstant elasticity of substitution propertyrdquo Journal of AdvancedMathematical Studies vol 5 no 2 pp 90ndash97 2012
[11] B-Y Chen ldquoClassification of homothetic functions with con-stant elasticity of substitution and its geometric applicationsrdquoInternational Electronic Journal of Geometry vol 5 no 2 pp67ndash78 2012
[12] B-Y Chen ldquoSolutions to homogeneousMongemdashAmpere equa-tions of homothetic functions and their applications to produc-tion models in economicsrdquo Journal of Mathematical Analysisand Applications vol 411 no 1 pp 223ndash229 2014
[13] B-Y Chen and G E Vılcu ldquoGeometric classifications ofhomogeneous production functionsrdquo Applied Mathematics andComputation vol 225 pp 345ndash351 2013
[14] A D Vilcu and G E Vilcu ldquoOn homogeneous productionfunctions with proportional marginal rate of substitutionrdquoMathematical Problems in Engineering vol 2013 Article ID732643 5 pages 2013
[15] B-Y Chen Pseudo-Riemannian Geometry 120575-Invariants andApplications World Scientific Hackensack NJ USA 2011
[16] B-y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Moreover it follows from (45) that
119865119894 = 119860120574119886120588
119894119909120588minus1
119894(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus1
(52)
119865119894119894 = 119860120574119886120588
119894119909120588minus2
119894((120588 minus 1) sum
119896 = 119894
119886120588
119896119909120588
119896+ (120574 minus 1) 119886
120588
119894119909120588
119894)
times(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
(53)
119865119894119895 = 119860120574 (120574 minus 120588) 119886120588
119894119886120588
119895119909120588minus1
119894119909120588minus1
119895(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
119894 = 119895
(54)
We divide it into two cases
Case A (120574 = 1) Substituting (52) and (54) into (51) we get
11986031205743(120574 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120574120588minus4
timessum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(55)
If 120588 = 120574 then the above equation reduces to
sum
119894119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0 (56)
which is equivalent to
(
119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894)
2
= 0 (57)
Hence119899
sum
119894=1
1198862120588
1198941199092120588minus2
119894= 0 (58)
Since 120588 = 1 this case is impossibleTherefore we conclude that 120588 = 120574 Substituting (53) into
(50) we have
119860120574 (120574 minus 1)(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
times (
119899
sum
119894=1
119886120588
119894119909120588
119894)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
120574120588minus2
= 0
(59)
Noting 120574 = 1 it is easy to see that (59) is impossible
Case B (120574 = 1) We note that the minimality conditionreduces to
119899
sum
119894=1
119865119894119894 + sum
119894 = 119895
(1198652
119894119865119895119895 minus 119865119894119865119895119865119894119895) = 0 (60)
Substituting (52)ndash(54) into (60) we have
119860 (120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
1120588minus2
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198603(120588 minus 1)(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
minus 1198603(1 minus 120588)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
3120588minus4
times sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(61)
Note that120588 = 1 fulfills (61) Hence in this case the generalizedCES production function 119865 is a perfect substitute
In the following we will show that the case 120588 = 1 isimpossible In fact if 120588 = 1 (61) reduces to
(sum
119896 = 119894
119886120588
119896119909120588
119896)(
119899
sum
119894=1
119886120588
119894119909120588
119894)
2minus2120588
(
119899
sum
119894=1
119886120588
119894119909120588minus2
119894)
+ 1198602(sum
119896 = 119894
119886120588
119896119909120588
119896) sum
119894 = 119895
(1198862120588
119894119886120588
1198951199092120588minus2
119894119909120588minus2
119895)
+ 1198602sum
119894 = 119895
(1198862120588
1198941198862120588
1198951199092120588minus2
1198941199092120588minus2
119895) = 0
(62)
Since 119909119894 gt 0 and 119886119894 gt 0 for 119894 = 1 119899 it follows that the leftof equality (62) is positive Therefore it is a contradiction
Conversely it is easy to verify that if the generalized CESproduction function is a perfect substitute then the 119899-factorgeneralized CES production hypersurface in E119899+1 is minimalThis completes the proof of Theorem 3
Remark 4 Remark that Chen proved in [7] a more generalresult for 2-factor ℎ-homogeneous production function is aperfect substitute if and only if the production surface is aminimal surface
Acknowledgments
The authors would like to thank the referees for theirvery valuable comments and suggestions to improve thispaper This work is supported by the Mathematical TianyuanYouth Fund of China (no 11326068) the general FinancialGrant from the China Postdoctoral Science Foundation (no2012M510818) and theGeneral Project for Scientific Researchof Liaoning Educational Committee (no W2012186)
References
[1] C W Cobb and P H Douglas ldquoA theory of productionrdquoAmerican Economic Review vol 18 pp 139ndash165 1928
6 Abstract and Applied Analysis
[2] K J Arrow H B Chenery B S Minhas and R M SolowldquoCapital-labor substitution and economic efficiencyrdquo Review ofEconomics and Statistics vol 43 no 3 pp 225ndash250 1961
[3] L Losonczi ldquoProduction functions having the CES propertyrdquoAcademiae Paedagogicae Nyıregyhaziensis vol 26 no 1 pp 113ndash125 2010
[4] D McFadden ldquoConstant elasticity of substitution productionfunctionsrdquo Review of Economic Studies vol 30 no 2 pp 73ndash831963
[5] G E Vılcu ldquoA geometric perspective on the generalized Cobb-Douglas production functionsrdquo Applied Mathematics Lettersvol 24 no 5 pp 777ndash783 2011
[6] A D Vılcu and G E Vılcu ldquoOn some geometric properties ofthe generalized CES production functionsrdquo Applied Mathemat-ics and Computation vol 218 no 1 pp 124ndash129 2011
[7] B-Y Chen ldquoOn some geometric properties of ℎ-homogeneousproduction functions in microeconomicsrdquo Kragujevac Journalof Mathematics vol 35 no 3 pp 343ndash357 2011
[8] B-Y Chen ldquoOn some geometric properties of quasi-sumproduction modelsrdquo Journal of Mathematical Analysis andApplications vol 392 no 2 pp 192ndash199 2012
[9] B-Y Chen ldquoA note on homogeneous production modelsrdquoKragujevac Journal ofMathematics vol 36 no 1 pp 41ndash43 2012
[10] B-Y Chen ldquoGeometry of quasi-sum production functions withconstant elasticity of substitution propertyrdquo Journal of AdvancedMathematical Studies vol 5 no 2 pp 90ndash97 2012
[11] B-Y Chen ldquoClassification of homothetic functions with con-stant elasticity of substitution and its geometric applicationsrdquoInternational Electronic Journal of Geometry vol 5 no 2 pp67ndash78 2012
[12] B-Y Chen ldquoSolutions to homogeneousMongemdashAmpere equa-tions of homothetic functions and their applications to produc-tion models in economicsrdquo Journal of Mathematical Analysisand Applications vol 411 no 1 pp 223ndash229 2014
[13] B-Y Chen and G E Vılcu ldquoGeometric classifications ofhomogeneous production functionsrdquo Applied Mathematics andComputation vol 225 pp 345ndash351 2013
[14] A D Vilcu and G E Vilcu ldquoOn homogeneous productionfunctions with proportional marginal rate of substitutionrdquoMathematical Problems in Engineering vol 2013 Article ID732643 5 pages 2013
[15] B-Y Chen Pseudo-Riemannian Geometry 120575-Invariants andApplications World Scientific Hackensack NJ USA 2011
[16] B-y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
[2] K J Arrow H B Chenery B S Minhas and R M SolowldquoCapital-labor substitution and economic efficiencyrdquo Review ofEconomics and Statistics vol 43 no 3 pp 225ndash250 1961
[3] L Losonczi ldquoProduction functions having the CES propertyrdquoAcademiae Paedagogicae Nyıregyhaziensis vol 26 no 1 pp 113ndash125 2010
[4] D McFadden ldquoConstant elasticity of substitution productionfunctionsrdquo Review of Economic Studies vol 30 no 2 pp 73ndash831963
[5] G E Vılcu ldquoA geometric perspective on the generalized Cobb-Douglas production functionsrdquo Applied Mathematics Lettersvol 24 no 5 pp 777ndash783 2011
[6] A D Vılcu and G E Vılcu ldquoOn some geometric properties ofthe generalized CES production functionsrdquo Applied Mathemat-ics and Computation vol 218 no 1 pp 124ndash129 2011
[7] B-Y Chen ldquoOn some geometric properties of ℎ-homogeneousproduction functions in microeconomicsrdquo Kragujevac Journalof Mathematics vol 35 no 3 pp 343ndash357 2011
[8] B-Y Chen ldquoOn some geometric properties of quasi-sumproduction modelsrdquo Journal of Mathematical Analysis andApplications vol 392 no 2 pp 192ndash199 2012
[9] B-Y Chen ldquoA note on homogeneous production modelsrdquoKragujevac Journal ofMathematics vol 36 no 1 pp 41ndash43 2012
[10] B-Y Chen ldquoGeometry of quasi-sum production functions withconstant elasticity of substitution propertyrdquo Journal of AdvancedMathematical Studies vol 5 no 2 pp 90ndash97 2012
[11] B-Y Chen ldquoClassification of homothetic functions with con-stant elasticity of substitution and its geometric applicationsrdquoInternational Electronic Journal of Geometry vol 5 no 2 pp67ndash78 2012
[12] B-Y Chen ldquoSolutions to homogeneousMongemdashAmpere equa-tions of homothetic functions and their applications to produc-tion models in economicsrdquo Journal of Mathematical Analysisand Applications vol 411 no 1 pp 223ndash229 2014
[13] B-Y Chen and G E Vılcu ldquoGeometric classifications ofhomogeneous production functionsrdquo Applied Mathematics andComputation vol 225 pp 345ndash351 2013
[14] A D Vilcu and G E Vilcu ldquoOn homogeneous productionfunctions with proportional marginal rate of substitutionrdquoMathematical Problems in Engineering vol 2013 Article ID732643 5 pages 2013
[15] B-Y Chen Pseudo-Riemannian Geometry 120575-Invariants andApplications World Scientific Hackensack NJ USA 2011
[16] B-y Chen Geometry of Submanifolds Marcel Dekker NewYork NY USA 1973
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
