research article on homogeneous production functions with...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 732643 5 pageshttpdxdoiorg1011552013732643

Research ArticleOn Homogeneous Production Functions withProportional Marginal Rate of Substitution

Alina Daniela Vicirclcu1 and Gabriel Eduard Vicirclcu23

1 Department of Information Technology Mathematics and Physics Petroleum-Gas University of PloiestiBulevardul Bucuresti No 39 100680 Ploiesti Romania

2 Faculty of Mathematics and Computer Science Research Center in Geometry Topology and AlgebraUniversity of Bucharest Street Academiei No 14 Sector 1 70109 Bucharest Romania

3 Department of Mathematical Modelling Economic Analysis and Statistics Petroleum-Gas University of PloiestiBulevardul Bucuresti No 39 100680 Ploiesti Romania

Correspondence should be addressed to Alina Daniela Vılcu danielavilcuupg-ploiestiro

Received 11 December 2012 Accepted 10 February 2013

Academic Editor Gradimir Milovanovic

Copyright copy 2013 A D Vılcu and G E Vılcu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We completely classify homogeneous production functions with proportional marginal rate of substitution and with constantelasticity of labor and capital respectively These classifications generalize some recent results of C A Ioan and G Ioan (2011)concerning the sum production function

1 Introduction

It is well known that the production function is one of thekey concepts of mainstream neoclassical theories with a lotof applications not only in microeconomics and macroeco-nomics but also in various fields like biology [1 2] educa-tional management [3 4] and engineering [5ndash8] Roughlyspeaking the production functions are the mathematicalformalization of the relationship between the output of afirmindustryeconomy and the inputs that have been usedin obtaining it In fact a production function is a map 119891 ofclass 119862infin 119891 R119899

+rarr R

+ 119891 = 119891(119909

1 1199092 119909

119899) where 119891 is

the quantity of output 119899 is the number of the inputs and1199091 1199092 119909

119899are the factor inputs (such as labor capital land

and raw materials) In order for these functions to model aswell the economic reality they are required to be homoge-neous that is there exists a real number119901 such that119876(120582sdot119909) =120582119901119876(119909) for all 119909 isin R119899

+and 120582 isin R

+ that means if the inputs

are multiplied by same factor then the output is multipliedby some power of this factor If 120582 = 1 then the functionis said to have a constant return to scale if 120582 gt 1 then wehave an increased return to scale and if 120582 lt 1 then we saythat the function has a decreased return to scale Among the

family of production functions themost famous is the Cobb-Douglas (CD) production function introduced in 1928 byCobb and Douglas [9] in order to describe the distributionof the national income of the USA In its most standardform for production of a single good with two factors theCobb-Douglas production function is given by

119891 = 119862119870120572119871120573 (1)

where 119891 is the total production119870 is the capital input 119871 is thelabor input and 119862 is a positive constant which signifies thetotal factor productivity We note that in the original defini-tion of Cobb and Douglas we have 120572 + 120573 = 1 so the produc-tion function had a constant return to scale but this conditionhas been later relaxed and CD production functions weregeneralized (see [10 11]) Some very interesting informationabout other production functions of great interest in eco-nomic analysis like Leontief Lu-Fletcher Liu-HildebrandKadiyala Arrow-Chenery-Minhas-Solow (ACMS) constantelasticity of substitution (CES) and variable elasticity ofsubstitution (VES) production functions can be found in[12] Recently C A Ioan and G Ioan [13] introduced a new

2 Mathematical Problems in Engineering

class of production functions called sum production func-tion as a two-factor production function defined by

119891 (119870 119871) =119899

sum119894=1

120572119894(11988811989411198701199011198941+1199011198942 + 119888

119894211987011990111989411198711199011198942 + 119888

11989431198711199011198941+1199011198942)

1199011198943

(2)

where 119899 ge 1 120572119894ge 0 119901

1198943isin (minusinfin 0) cup [1infin) 119901

11989411199011198942

gt 01199011198943(1199011198941+ 1199011198942) = 1 sum119899

119894=1(1198881198942+ 11988811989411198881198943) gt 0 and 119888

119894119895ge 0 for all 119894 isin

1 2 119899 for all 119895 isin 1 2 3It is easy to see that this production function is homoge-

neous of degree 1 and integrates in an unitary expression var-ious production functions including CD CES and VES In[13] C A Ioan and G Ioan compute the principal indicatorsof the sum production function and prove three theoremsof characterization for the functions with a proportionalmarginal rate of substitution with constant elasticity of laborand for those with constant elasticity of substitution asfollows

Theorem 1 (see [13]) The sum production function has aproportional marginal rate of substitution if and only if itreduces to the Cobb-Douglas function

Theorem 2 (see [13]) The sum production function has aconstant elasticity of labor if and only if it reduces to the Cobb-Douglas function

Theorem 3 (see [13]) If 119899 = 1 then the sum productionfunction has constant elasticity of substitution if and only if itreduces to the Cobb-Douglas or CES function

We recall that for a production function 119891 with two fac-tors (119870-capital and 119871-labor) the marginal rate of substitution(between capital and labor) is given by

MRS =120597119891120597119871

120597119891120597119870 (3)

where the elasticities of 119871 and119870 are defined as

119864119871=120597119891120597119871

119891119871 119864

119870=120597119891120597119870

119891119870 (4)

while the elasticity of substitution is given by

120590 = ((1 (119870 (120597119891120597119870))) + (1 (119871 (120597119891120597119871))))

times (minus ((12059721198911205971198702) (120597119891120597119870)2

)

+ ((2 (1205972119891120597119870120597119871)) ((120597119891120597119870) (120597119891120597119871)))

minus ((12059721198911205971198712) (120597119891120597119871)2

) )minus1

(5)

It is easy to verify that in the case of constant returnto scale Eulerrsquos theorem implies the following more simpleexpression for the elasticity of substitution

120590 =(120597119891120597119871) (120597119891120597119870)

119891 (1205972119891120597119870120597119871) (6)

We note that it was proved by Losonczi [14] that twicedifferentiable two-input homogeneous production functionswith constant elasticity of substitution (CES) property areCobb-Douglas and ACMS production functions which isobviously a more general result than Theorem 3 This resultwas recently generalized by Chen for an arbitrary number ofinputs [15] In the next section we prove the following resultwhich is a generalization of Theorems 1 and 2

Theorem 4 Let 119891 be a twice differentiable homogeneous ofdegree 119903 nonconstant real valued production function withtwo inputs (119870-capital and 119871-labor) Then one has the follow-ing

(i) 119891 has a constant elasticity of labor 119896 if and only if it isa Cobb-Douglas production function given by

119891 (119870 119871) = 119862119870119903minus119896119871119896 (7)

where 119862 is a positive constant(ii) 119891 has a constant elasticity of capital 119896 if and only if it is

a Cobb-Douglas production function given by

119891 (119870 119871) = 119862119870119896119871119903minus119896 (8)

where 119862 is a positive constant(iii) 119891 satisfies the proportional rate of substitution property

between capital and labor (ie MRS = 119896(119870119871) where119896 is a positive constant) if and only if it is a Cobb-Douglas production function given by

119891 (119870 119871) = 119862119870119903(119896+1)119871119903119896(119896+1) (9)

where 119862 is a positive constant

In the last section of the paper we generalize the abovetheorem for an arbitrary number of inputs 119899 ge 3We note thatother classification results concerning production functionswere proved recently in [16ndash20]

2 Proof of Theorem 4

Proof Consider the following

(i) We first suppose that 119891 has a constant elasticity oflabor 119896 Then we have

120597119891

120597119871= 119896

119891

119871 (10)

But with 119891 being homogeneous of degree 119903 it followsthat it can be written in the form

119891 (119870 119871) = 119870119903ℎ (119906) (11)

or

119891 (119870 119871) = 119871119903ℎ (119906) (12)

Mathematical Problems in Engineering 3

where 119906 = 119871119870 (with 119870 = 0) respectively 119906 = 119870119871(with 119871 = 0) and ℎ is a real valued function of 119906 ofclass 1198622 on its domain of definition We can supposewithout loss of generality that the first situationoccurs so 119891(119870 119871) = 119870119903ℎ(119906) with 119906 = 119871119870 Thenwe have

120597119891

120597119871= 119870119903minus1ℎ1015840 (119906) (13)

From (10) and (13) we obtain

119870119903minus1ℎ1015840 (119906) = 119896119870119903ℎ (119906)

119871 (14)

and therefore we deduce that the constant elasticity oflabor property implies the following differential equa-tion

ℎ1015840 (119906) = 119896ℎ (119906)

119906 (15)

Solving the above separable differential equation weobtain

ℎ (119906) = 119862119906119896 (16)

where 119862 is a positive constant Finally from (11) and(16) we derive that 119891 is a Cobb-Douglas productionfunction given by

119891 (119870 119871) = 119862 sdot 119870119903minus119896119871119896 (17)

The converse is easy to verify(ii) The proof follows similarly as in (i)(iii) Since the production function satisfies the pro-portional rate of substitution property it follows that

120597119891

120597119871= 119896

119870

119871

120597119891

120597119870 (18)

On the other hand from Eulerrsquos homogeneous func-tion theorem we have

119870120597119891

120597119870+ 119871

120597119891

120597119871= 119903119891 (119870 119871) (19)

Combining now (18) and (19) we obtain

120597119891

120597119870=

119903

119896 + 1

119891

119870 (20)

From (20) we deduce that

119891 (119870 119871) = 119862119870119903(119896+1)119906 (119871) (21)

where 119862 is a real constant But with 119891 being a homo-geneous function of degree 119903 it follows from (21) that

119906 (119871) = 119871119903119896(119896+1) (22)

Therefore from (21) and (22) we derive that

119891 (119870 119871) = 119862119870119903(119896+1)119871119903119896(119896+1) (23)

where 119862 is a real constant Finally since 119891 is a non-constant production function it follows that 119891 gt 0and therefore we deduce that 119862 is in fact a positiveconstant So 119891 is a Cobb-Douglas production func-tionThe converse is easy to check and the proof is nowcomplete

3 Generalization to an ArbitraryNumber of Inputs

Let 119891 be a homogeneous production function with 119899 inputs1199091 1199092 119909

119899 119899 gt 2 Then the elasticity of production with

respect to a certain factor of production 119909119894is defined as

119864119909119894=120597119891120597119909

119894

119891119909119894

(24)

while themarginal rate of technical substitution of input 119895 forinput 119894 is given by

MRS119894119895=120597119891120597119909

119895

120597119891120597119909119894

(25)

A production function is said to satisfy the proportionalmarginal rate of substitution property if and only if MRS

119894119895=

119909119894119909119895 for all 1 le 119894 = 119895 le 119899 Now we are able to prove the

following result which generalizesTheorem 4 for an arbitrarynumber of inputs

Theorem 5 Let 119891 be a twice differentiable homogeneousof degree 119903 nonconstant real valued function of 119899 variables(1199091 1199092 119909

119899) defined on 119863 = R119899

+ where 119899 gt 2 Then one

has the following

(i) The elasticity of production is a constant 119896119894with

respect to a certain factor of production 119909119894if and only if

119891 (1199091 1199092 119909

119899) = 119909119896119894

119894119909119903minus119896119894

119895119865 (1199061 119906

119899minus2) (26)

where 119895 is any element settled from the set 1 119899119894and 119865 is a twice differentiable real valued function of119899 minus 2 variables

1199061 119906

119899minus2 =

119909119896

119909119895

| 119896 isin 1 119899 119894 119895 (27)

(ii) The elasticity of production is a constant 119896119894with respect

to all factors of production 119909119894 119894 isin 1 2 119899 if and

only if

1198961+ 1198962+ sdot sdot sdot + 119896

119899= 119903 (28)


and 119891 reduces to the Cobb-Douglas production func-tion given by

119891 (1199091 1199092 119909

119899) = 119862119909

1198961

11199091198962

2sdot sdot sdot 119909119896119899119899 (29)

where 119862 is a positive constant(iii) The production function satisfies the proportional mar-

ginal rate of substitution property if and only if itreduces to the Cobb-Douglas production function givenby

119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(30)


Proof Consider the following(i) The if part of the statement is easy to verify Nextwe prove the only if part Since the elasticity of pro-duction with respect to a certain factor of production119909119894is a constant 119896

119894 we have

120597119891

120597119909119894

= 119896119894

119891

119909119894

(31)

On the other hand since 119891 is a homogeneous ofdegree 119903 it follows that it can be expressed in the form

119891 (1199091 119909

119899) = 119909119903119895ℎ (1199061 119906

119899minus1) (32)

where 119895 can be settled in the set 1 119899 and

119906119896=

119909119896

119909119895

1 le 119896 le 119895 minus 1

119909119896+1

119909119895

119895 le 119896 le 119899 minus 1

(33)

If we settle 119895 such that 119895 = 119894 then we derive from (32)

120597119891

120597119909119894

=

119909119903minus1119895

120597ℎ

120597119906119894

if 119894 lt 119895

119909119903minus1119895

120597ℎ

120597119906119894minus1

if 119894 gt 119895

(34)

Replacing now (34) in (31) we obtain

119896119894ℎ =

119906119894

120597ℎ

120597119906119894

if 119894 lt 119895

119906119894minus1

120597ℎ

120597119906119894minus1

if 119894 gt 119895

(35)

and solving the partial differential equations in (35)we derive

ℎ (1199061 119906

119899minus1)

=

119862119906119896119894

119894119865 (1199061 119906

119894 119906

119899minus1) if 119894 lt 119895

119862119906119896119894

119894minus1119865 (1199061 119906

119894minus1 119906

119899minus1) if 119894 gt 119895

(36)

where 119862 is a positive constant 119865 is a twice differ-entiable real valued function of 119899 minus 2 variables andthe symbol ldquordquo means that the corresponding term isomittedThe conclusion follows now easily from (32) and (36)taking into account (33)(ii) This assertion follows immediately from (i)(iii) It is easy to show that if 119891 is a Cobb-Douglasproduction function given by

119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(37)

then 119891 satisfies the proportional marginal rate of sub-stitution property We prove now the converse Since119891 satisfies the proportional marginal rate of substitu-tion property it follows that

1199091

120597119891

1205971199091

= 1199092

120597119891

1205971199092

= sdot sdot sdot = 119909119899

120597119891

120597119909119899

(38)

On the other hand since 119891 is a homogeneous ofdegree 119903 the Euler homogeneous function theoremimplies that

1199091

120597119891

1205971199091

+ 1199092

120597119891

1205971199092

+ sdot sdot sdot + 119909119899

120597119891

120597119909119899

= 119903119891 (39)


119909119894

120597119891

120597119909119894

=119903

119899119891 119894 isin 1 2 119899 (40)

Finally from the above system of partial differentialequations we obtain the solution

119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(41)

where 119862 is a positive constant and the conclusionfollows

Acknowledgments

The authors would like to thank the referees for carefullyreading the paper and making valuable comments andsuggestions The second author was supported by CNCS-UEFISCDI Project no PN-II-ID-PCE-2011-3-0118

References

[1] I B Adinya B O Offem and G U Ikpi ldquoApplication of astochastic frontier production function for measurement andcomparision of technical efficiency of mandarin fish and clownfish production in lowlands reservoirs ponds anddams ofCrossRiver State Nigeriardquo Journal of Animal and Plant Sciences vol21 no 3 pp 595ndash600 2011


[2] E Chassot D Gascuel and A Colomb ldquoImpact of trophicinteractions on production functions and on the ecosystemresponse to fishing a simulation approachrdquo Aquatic LivingResources vol 18 no 1 pp 1ndash13 2005

[3] S T Cooper and E Cohn ldquoEstimation of a frontier productionfunction for the SouthCarolina educational processrdquoEconomicsof Education Review vol 16 no 3 pp 313ndash327 1997

[4] M E Da Silva Freire and J J R F Da Silva ldquoThe applicationof production functions to the higher education systemmdashsomeexamples from Portuguese universitiesrdquo Higher Education vol4 no 4 pp 447ndash460 1975

[5] J M Boussard ldquoBio physical models as detailed engineer-ing production functionsrdquo in Bio-Economic Models Appliedto Agricultural Systems pp 15ndash28 Springer Dordrecht TheNetherland 2011

[6] T G Gowing ldquoTechnical change and scale economies in anengineering production function The case of steam electricpowerrdquo Journal of Industrial Economics vol 23 no 2 pp 135ndash152 1974

[7] J Marsden D Pingry and A Whinston ldquoEngineering founda-tions of production functionsrdquo Journal of EconomicTheory vol9 no 2 pp 124ndash140 1974

[8] D L Martin D G Watts and J R Gilley ldquoModel andproduction function for irrigation managementrdquo Journal ofIrrigation and Drainage Engineering vol 110 no 2 pp 149ndash1641984

[9] C W Cobb and P H Douglas ldquoA theory of productionrdquoAmerican Economic Review vol 18 pp 139ndash165 1928

[10] 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

[11] G E Vılcu ldquoA geometric perspective on the generalized Cobb-Douglas production functionsrdquo Applied Mathematics Lettersvol 24 no 5 pp 777ndash783 2011

[12] S K Mishra ldquoA brief history of production functionsrdquoThe IUPJournal of Managerial Economics vol 8 no 4 pp 6ndash34 2010

[13] C A Ioan andG Ioan ldquoA generalization of a class of productionfunctionsrdquo Applied Economics Letters vol 18 pp 1777ndash17842011

[14] L Losonczi ldquoProduction functions having the CES propertyrdquoActa Mathematica Academiae Paedagogicae Nyıregyhaziensisvol 26 no 1 pp 113ndash125 2010

[15] B-Y Chen ldquoClassification of h-homogeneous production func-tions with constant elasticity of substitutionrdquo Tamkang Journalof Mathematics vol 43 no 2 pp 321ndash328 2012

[16] B-Y Chen ldquoOn some geometric properties of h-homogeneousproduction functions in microeconomicsrdquo Kragujevac Journalof Mathematics vol 35 no 3 pp 343ndash357 2011

[17] B-Y Chen ldquoOn some geometric properties of quasi-sumproduction modelsrdquo Journal of Mathematical Analysis andApplications vol 392 no 2 pp 192ndash199 2012

[18] B-Y Chen ldquoGeometry of quasy-sum production functionswith constant elasticity of substitution propertyrdquo Journal ofAdvanced Mathematical Studies vol 5 no 2 pp 90ndash97 2012

[19] 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

[20] B-Y Chen and G E V Vılcu ldquoGeometric classifications ofhomogeneous production functionsrdquo submitted

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Stochastic AnalysisInternational Journal of


class of production functions called sum production func-tion as a two-factor production function defined by

119891 (119870 119871) =119899

sum119894=1

120572119894(11988811989411198701199011198941+1199011198942 + 119888

119894211987011990111989411198711199011198942 + 119888

11989431198711199011198941+1199011198942)

1199011198943

(2)

where 119899 ge 1 120572119894ge 0 119901

1198943isin (minusinfin 0) cup [1infin) 119901

11989411199011198942

gt 01199011198943(1199011198941+ 1199011198942) = 1 sum119899

119894=1(1198881198942+ 11988811989411198881198943) gt 0 and 119888

119894119895ge 0 for all 119894 isin

1 2 119899 for all 119895 isin 1 2 3It is easy to see that this production function is homoge-

neous of degree 1 and integrates in an unitary expression var-ious production functions including CD CES and VES In[13] C A Ioan and G Ioan compute the principal indicatorsof the sum production function and prove three theoremsof characterization for the functions with a proportionalmarginal rate of substitution with constant elasticity of laborand for those with constant elasticity of substitution asfollows

Theorem 1 (see [13]) The sum production function has aproportional marginal rate of substitution if and only if itreduces to the Cobb-Douglas function

Theorem 2 (see [13]) The sum production function has aconstant elasticity of labor if and only if it reduces to the Cobb-Douglas function

Theorem 3 (see [13]) If 119899 = 1 then the sum productionfunction has constant elasticity of substitution if and only if itreduces to the Cobb-Douglas or CES function

We recall that for a production function 119891 with two fac-tors (119870-capital and 119871-labor) the marginal rate of substitution(between capital and labor) is given by

MRS =120597119891120597119871

120597119891120597119870 (3)

where the elasticities of 119871 and119870 are defined as

119864119871=120597119891120597119871

119891119871 119864

119870=120597119891120597119870

119891119870 (4)

while the elasticity of substitution is given by

120590 = ((1 (119870 (120597119891120597119870))) + (1 (119871 (120597119891120597119871))))

times (minus ((12059721198911205971198702) (120597119891120597119870)2

)

+ ((2 (1205972119891120597119870120597119871)) ((120597119891120597119870) (120597119891120597119871)))

minus ((12059721198911205971198712) (120597119891120597119871)2

) )minus1

(5)

It is easy to verify that in the case of constant returnto scale Eulerrsquos theorem implies the following more simpleexpression for the elasticity of substitution

120590 =(120597119891120597119871) (120597119891120597119870)

119891 (1205972119891120597119870120597119871) (6)

We note that it was proved by Losonczi [14] that twicedifferentiable two-input homogeneous production functionswith constant elasticity of substitution (CES) property areCobb-Douglas and ACMS production functions which isobviously a more general result than Theorem 3 This resultwas recently generalized by Chen for an arbitrary number ofinputs [15] In the next section we prove the following resultwhich is a generalization of Theorems 1 and 2

Theorem 4 Let 119891 be a twice differentiable homogeneous ofdegree 119903 nonconstant real valued production function withtwo inputs (119870-capital and 119871-labor) Then one has the follow-ing

(i) 119891 has a constant elasticity of labor 119896 if and only if it isa Cobb-Douglas production function given by

119891 (119870 119871) = 119862119870119903minus119896119871119896 (7)

where 119862 is a positive constant(ii) 119891 has a constant elasticity of capital 119896 if and only if it is

a Cobb-Douglas production function given by

119891 (119870 119871) = 119862119870119896119871119903minus119896 (8)

where 119862 is a positive constant(iii) 119891 satisfies the proportional rate of substitution property

between capital and labor (ie MRS = 119896(119870119871) where119896 is a positive constant) if and only if it is a Cobb-Douglas production function given by

119891 (119870 119871) = 119862119870119903(119896+1)119871119903119896(119896+1) (9)


In the last section of the paper we generalize the abovetheorem for an arbitrary number of inputs 119899 ge 3We note thatother classification results concerning production functionswere proved recently in [16ndash20]

2 Proof of Theorem 4

Proof Consider the following

(i) We first suppose that 119891 has a constant elasticity oflabor 119896 Then we have

120597119891

120597119871= 119896

119891

119871 (10)

But with 119891 being homogeneous of degree 119903 it followsthat it can be written in the form

119891 (119870 119871) = 119870119903ℎ (119906) (11)

or

119891 (119870 119871) = 119871119903ℎ (119906) (12)



120597119891

120597119871= 119870119903minus1ℎ1015840 (119906) (13)


119870119903minus1ℎ1015840 (119906) = 119896119870119903ℎ (119906)

119871 (14)


ℎ1015840 (119906) = 119896ℎ (119906)

119906 (15)


ℎ (119906) = 119862119906119896 (16)


119891 (119870 119871) = 119862 sdot 119870119903minus119896119871119896 (17)


120597119891

120597119871= 119896

119870

119871

120597119891

120597119870 (18)


119870120597119891

120597119870+ 119871

120597119891

120597119871= 119903119891 (119870 119871) (19)


120597119891

120597119870=

119903

119896 + 1

119891

119870 (20)


119891 (119870 119871) = 119862119870119903(119896+1)119906 (119871) (21)


119906 (119871) = 119871119903119896(119896+1) (22)


119891 (119870 119871) = 119862119870119903(119896+1)119871119903119896(119896+1) (23)






119864119909119894=120597119891120597119909

119894

119891119909119894

(24)


MRS119894119895=120597119891120597119909

119895

120597119891120597119909119894

(25)


119894119895=




119899) defined on 119863 = R119899


has the following



119891 (1199091 1199092 119909

119899) = 119909119896119894

119894119909119903minus119896119894

119895119865 (1199061 119906

119899minus2) (26)


1199061 119906

119899minus2 =

119909119896

119909119895

| 119896 isin 1 119899 119894 119895 (27)



only if

1198961+ 1198962+ sdot sdot sdot + 119896

119899= 119903 (28)



119891 (1199091 1199092 119909

119899) = 119862119909

1198961

11199091198962

2sdot sdot sdot 119909119896119899119899 (29)



119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(30)



119894 we have

120597119891

120597119909119894

= 119896119894

119891

119909119894

(31)


119891 (1199091 119909

119899) = 119909119903119895ℎ (1199061 119906

119899minus1) (32)


119906119896=

119909119896

119909119895

1 le 119896 le 119895 minus 1

119909119896+1

119909119895

119895 le 119896 le 119899 minus 1

(33)


120597119891

120597119909119894

=

119909119903minus1119895

120597ℎ

120597119906119894

if 119894 lt 119895

119909119903minus1119895

120597ℎ

120597119906119894minus1

if 119894 gt 119895

(34)


119896119894ℎ =

119906119894

120597ℎ

120597119906119894

if 119894 lt 119895

119906119894minus1

120597ℎ

120597119906119894minus1

if 119894 gt 119895

(35)


ℎ (1199061 119906

119899minus1)

=

119862119906119896119894

119894119865 (1199061 119906

119894 119906

119899minus1) if 119894 lt 119895

119862119906119896119894

119894minus1119865 (1199061 119906

119894minus1 119906

119899minus1) if 119894 gt 119895

(36)


119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(37)


1199091

120597119891

1205971199091

= 1199092

120597119891

1205971199092

= sdot sdot sdot = 119909119899

120597119891

120597119909119899

(38)


1199091

120597119891

1205971199091

+ 1199092

120597119891

1205971199092

+ sdot sdot sdot + 119909119899

120597119891

120597119909119899

= 119903119891 (39)


119909119894

120597119891

120597119909119894

=119903

119899119891 119894 isin 1 2 119899 (40)


119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(41)


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119891

120597119871= 119870119903minus1ℎ1015840 (119906) (13)


119870119903minus1ℎ1015840 (119906) = 119896119870119903ℎ (119906)

119871 (14)


ℎ1015840 (119906) = 119896ℎ (119906)

119906 (15)


ℎ (119906) = 119862119906119896 (16)


119891 (119870 119871) = 119862 sdot 119870119903minus119896119871119896 (17)


120597119891

120597119871= 119896

119870

119871

120597119891

120597119870 (18)


119870120597119891

120597119870+ 119871

120597119891

120597119871= 119903119891 (119870 119871) (19)


120597119891

120597119870=

119903

119896 + 1

119891

119870 (20)


119891 (119870 119871) = 119862119870119903(119896+1)119906 (119871) (21)


119906 (119871) = 119871119903119896(119896+1) (22)


119891 (119870 119871) = 119862119870119903(119896+1)119871119903119896(119896+1) (23)






119864119909119894=120597119891120597119909

119894

119891119909119894

(24)


MRS119894119895=120597119891120597119909

119895

120597119891120597119909119894

(25)


119894119895=




119899) defined on 119863 = R119899


has the following



119891 (1199091 1199092 119909

119899) = 119909119896119894

119894119909119903minus119896119894

119895119865 (1199061 119906

119899minus2) (26)


1199061 119906

119899minus2 =

119909119896

119909119895

| 119896 isin 1 119899 119894 119895 (27)



only if

1198961+ 1198962+ sdot sdot sdot + 119896

119899= 119903 (28)



119891 (1199091 1199092 119909

119899) = 119862119909

1198961

11199091198962

2sdot sdot sdot 119909119896119899119899 (29)



119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(30)



119894 we have

120597119891

120597119909119894

= 119896119894

119891

119909119894

(31)


119891 (1199091 119909

119899) = 119909119903119895ℎ (1199061 119906

119899minus1) (32)


119906119896=

119909119896

119909119895

1 le 119896 le 119895 minus 1

119909119896+1

119909119895

119895 le 119896 le 119899 minus 1

(33)


120597119891

120597119909119894

=

119909119903minus1119895

120597ℎ

120597119906119894

if 119894 lt 119895

119909119903minus1119895

120597ℎ

120597119906119894minus1

if 119894 gt 119895

(34)


119896119894ℎ =

119906119894

120597ℎ

120597119906119894

if 119894 lt 119895

119906119894minus1

120597ℎ

120597119906119894minus1

if 119894 gt 119895

(35)


ℎ (1199061 119906

119899minus1)

=

119862119906119896119894

119894119865 (1199061 119906

119894 119906

119899minus1) if 119894 lt 119895

119862119906119896119894

119894minus1119865 (1199061 119906

119894minus1 119906

119899minus1) if 119894 gt 119895

(36)


119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(37)


1199091

120597119891

1205971199091

= 1199092

120597119891

1205971199092

= sdot sdot sdot = 119909119899

120597119891

120597119909119899

(38)


1199091

120597119891

1205971199091

+ 1199092

120597119891

1205971199092

+ sdot sdot sdot + 119909119899

120597119891

120597119909119899

= 119903119891 (39)


119909119894

120597119891

120597119909119894

=119903

119899119891 119894 isin 1 2 119899 (40)


119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(41)


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 (1199091 1199092 119909

119899) = 119862119909

1198961

11199091198962

2sdot sdot sdot 119909119896119899119899 (29)



119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(30)



119894 we have

120597119891

120597119909119894

= 119896119894

119891

119909119894

(31)


119891 (1199091 119909

119899) = 119909119903119895ℎ (1199061 119906

119899minus1) (32)


119906119896=

119909119896

119909119895

1 le 119896 le 119895 minus 1

119909119896+1

119909119895

119895 le 119896 le 119899 minus 1

(33)


120597119891

120597119909119894

=

119909119903minus1119895

120597ℎ

120597119906119894

if 119894 lt 119895

119909119903minus1119895

120597ℎ

120597119906119894minus1

if 119894 gt 119895

(34)


119896119894ℎ =

119906119894

120597ℎ

120597119906119894

if 119894 lt 119895

119906119894minus1

120597ℎ

120597119906119894minus1

if 119894 gt 119895

(35)


ℎ (1199061 119906

119899minus1)

=

119862119906119896119894

119894119865 (1199061 119906

119894 119906

119899minus1) if 119894 lt 119895

119862119906119896119894

119894minus1119865 (1199061 119906

119894minus1 119906

119899minus1) if 119894 gt 119895

(36)


119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(37)


1199091

120597119891

1205971199091

= 1199092

120597119891

1205971199092

= sdot sdot sdot = 119909119899

120597119891

120597119909119899

(38)


1199091

120597119891

1205971199091

+ 1199092

120597119891

1205971199092

+ sdot sdot sdot + 119909119899

120597119891

120597119909119899

= 119903119891 (39)


119909119894

120597119891

120597119909119894

=119903

119899119891 119894 isin 1 2 119899 (40)


119891 (1199091 1199092 119909

119899) = 119862119909119903119899

11199091199031198992

sdot sdot sdot 119909119903119899119899

(41)


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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