Research Article Some Characterizations of the Cobb-Douglas and CES Production Functions in Microeconomics

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1 Hdaw Publshg Corporato Abstract ad Appled Aalyss Volume 2013, Artcle ID , 6 pages Research Artcle Some Characterzatos of the Cobb-Douglas ad CES Producto Fuctos Mcroecoomcs Xaoshu Wag 1 ad Yu Fu 2 1 School of Ivestmet ad Costructo Maagemet, Dogbe Uversty of Face ad Ecoomcs, Dala , Cha 2 School of Mathematcs ad Quattatve Ecoomcs, Dogbe Uversty of Face ad Ecoomcs, Dala , Cha Correspodece should be addressed to Yu Fu; yufudufe@gmal.com Receved 5 November 2013; Accepted 10 December 2013 Academc Edtor: Grzegorz Lukaszewcz Copyrght 2013 X. Wag ad Y. Fu. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. It s well kow that the study of the shape ad the propertes of the producto possblty froter s a subect of great terest ecoomc aalyss. Vîlcu Vîlcu, 2011 proved that the geeralzed Cobb-Douglas producto fucto has costat retur to scale f ad oly f the correspodg hypersurface s developable. Later o, the authors A. D. Vîlcu ad G. E. Vîlcu, 2011 exteded ths result to the case of CES producto fucto. Both results establsh a terestg lk betwee some fudametal otos the theory of producto fuctos ad the dfferetal geometry of hypersurfaces Eucldea spaces. I ths paper, we gve some characterzatos of mmal geeralzed Cobb-Douglas ad CES producto hypersurfaces Eucldea spaces. 1. Itroducto I mcroecoomcs ad macroecoomcs, the producto fuctos are postve ocostat fuctos that specfy theoutputofafrm,adustry,oraetreecoomyfor all combatos of puts. Almost all ecoomc theores presuppose a producto fucto, ether o the frm level or the aggregate level. Hece, the producto fucto s oe of the most key cocepts of mastream eoclasscal theores. By assumg that the maxmum output techologcally possble from a gve set of puts s acheved, ecoomsts always usg a producto fucto aalyss are abstractg from the egeerg ad maageral problems heretly assocated wth a partcular producto process; see [1]. I 1928, Cobb ad Douglas [1]troducedafamoustwofactor producto fucto, owadays called Cobb-Douglas producto fucto, order to descrbe the dstrbuto of the atoal come by help of producto fuctos. Twofactor Cobb-Douglas producto fucto s gve by Y=bL k C 1 k, 1 where L deotes the labor put, C s the captal put, b s the total factor productvty, ad Y s the total producto. The Cobb-Douglas CD producto fucto s especally otable for beg the frst tme a aggregate or ecoomy-wde producto fucto was developed, estmated,adthepresetedtotheprofessoforaalyss. It gave a ladmark chage how ecoomsts approached macroecoomcs. The Cobb-Douglas fucto has also bee appled to may other ssues besdes producto. The geeralzed form of the Cobb-Douglas producto fucto s wrtte as Fx 1,...,x =Ax α 1 1 xα, 2 where x > 0 = 1,...,, A s a postve costat, ad α 1,...,α are ozero costats. I1961,Arrowetal.[2] troducedaothertwo-put producto fucto gve by Q=baK r + 1 a L r 1/r, 3 where Q s the output, b the factor productvty, a the share parameter, K ad L the prmary producto factors, r = s 1/s, ad s = 1/1 r the elastcty of substtuto. Hece ths s called costat elastcty of substtuto CES producto fucto [3, 4].

2 2 Abstract ad Appled Aalyss by The geeralzed form of CES producto fucto s gve Fx 1,...,x =A a ρ xρ γ/ρ, 4 where a, γ, A,adρ are ozero costats ad A, a > 0. The CES producto fuctos are of great terest ecoomy because of ther varat characterstc, amely, that the elastcty of substtuto betwee the parameters s costat o ther domas. It s easy to see that the CES producto fucto s a geeralzato of Cobb-Douglas producto fucto, ad both of them are homogeeous producto fuctos. Note that the producto fucto F ca be detfed wth a producto hypersurface the + 1-dmesoal Eucldea space E +1 through the map f:r + R+1 +, f x 1,...,x = x 1,...,x,Fx 1,...,x. 5 Ithecaseof puts, we have a hypersurface ad a aalyss of the geeralzed CD ad CES producto fuctos from the pot of vew of dfferetal geometry. I [5], Vîlcu establshed a terestg lk betwee some fudametal otos the theory of producto fuctos ad the dfferetal geometry of hypersurfaces. The author proved that the geeralzed Cobb-Douglas producto fucto has costat retur to scale f ad oly f the correspodg hypersurface s developable. The author otly wth A. D.Vîlcu further [6] geeralzed ths result to the case of the geeralzed CES producto fucto wth -puts. For further study of producto hypersurfaces, we refer the reader to Che s seres of terestg papers o homogeeous producto fuctos, quas-sum producto models, ad homothetc producto fuctos [7 13]ad G.E.Vîlcu ad A. D. Vîlcu s paper [14]. The theory of mmal surfaces hypersurfaces s very mportat the feld of dfferetal geometry. I ths paper, we gve further study of the geeralzed Cobb-Douglas ad CES producto fuctos as hypersurfaces Eucldea space E +1. I partcular, we gve some characterzatos of the geeralzed Cobb-Douglas ad CES producto hypersurfaces uder the mmalty codto Eucldea spaces. 2. Some Basc Cocepts Theory of Hypersurfaces Each producto fucto Fx 1,...,x cabedetfedwth a graph of a oparametrc hypersurface of a Eucldea + 1-space E +1 gve by f x 1,...,x = x 1,...,x,Fx 1,...,x. 6 We recall some basc cocepts the theory of hypersurfaces Eucldea spaces. Let M beahypersurfacea+1-dmeso Eucldea space. The Gauss map V s defed by V :M S E +1, 7 whch maps M to the ut hypersphere S of E +1.TheGauss map s always defed locally, that s, o a small pece of the hypersurface. It ca be defed globally f the hypersurface s oretable. The Gauss map s a cotuous map such that Vp s a ut ormal vector ξp of M at pot p. The dfferetal dv of the Gauss map V ca be used to defe a extrsc curvature, kow as the shape operator or Wegarte map. Sce, at each pot p M,thetaget space T p M s a er product space, the shape operator S p ca be defed as a lear operator o ths space by the relato gs p u, w = g dv u,w, 8 where u, w T p M ad g s the metrc tesor o M. Moreover, the secod fudametal form h s related to the shape operator S p by gs p u, w = g h u, w,ξp 9 for u, w T p M. It s well kow that the determat of the shape operator S p s called the Gauss-Kroecker curvature. Deote t by Gp. Whe = 2,theGauss-Kroeckercurvature s smply called the Gauss curvature, whch s trsc due to famous Gauss s Theorema Egregum. The trace of the shape operator S p s called the mea curvature of the hypersurfaces. I cotrast to the Gauss curvature, the mea curvature s extrsc, whch depeds o the mmerso of the hypersurface. A hypersurface s called mmal f ts mea curvature vashes detcally. Deote the partal dervatves F/ x, 2 F/ x x,...,ad so forth by F,F,...,adsoforth.Put W= 1+ F Recall some well-kow results for a graph of hypersurface 6E +1 from [7, 15, 16]. Proposto 1. For a producto hypersurface of E +1 defed by f x 1,...,x = x 1,...,x,Fx 1,,x 11 Oe has the followg. 1 The ut ormal ξ s gve by ξ= 1 W F 1,..., F, The coeffcet g = g / x, / x of the metrc tesor s gve by where δ =1f =,otherwse0. 3 The volume elemet s g =δ +F F, 13 dv = g dx 1 dx. 14

3 Abstract ad Appled Aalyss 3 4 The verse matrx g of g s g =δ F F W The matrx of the secod fudametal form h s h = F W The matrx of the shape operator S p s a = k h k g k = 1 W F k 7 The mea curvature H s gve by F k F k F W H= 1 W F 1 W 2 F F F. 18, 8 The Gauss-Kroecker curvature G s G= det h = det F. det g W The sectoal curvature K of the plae secto spaed by / x ad / x s F F F 2 K = 20 W F 2 +F The Rema curvature tesor R satsfes gr x, x x k, x l = F lf k F k F l W Geeralzed Cobb-Douglas Producto Hypersurfaces I ths secto, we gve a oexstece result of mmal geeralzed Cobb-Douglas producto hypersurfaces E +1. Theorem 2. There does ot exst a mmal geeralzed Cobb- Douglas producto hypersurface E +1. Proof. Let M be a geeralzed Cobb-Douglas producto hypersurface E +1 gve by a graph f x 1,...,x = x 1,...,x,Fx 1,...,x, 22 where F s the geeralzed Cobb-Douglas producto fucto Fx 1,...,x =Ax α 1 1 xα. 23 Note that the geeralzed Cobb-Douglas producto fucto 23 s homogeeous of degree h= α.itfollowsfrom 23that F = α x F, F = α α 1 x 2 F,,...,,,...,, F = α α F, =,,=1,...,. x x 24 By assumpto, the producto hypersurface s mmal; that s, H=0.Thus,by18ad10 wehave whch reduces to F 1+ F 2 =, F F F, 25 F + F 2 F F F F =0. 26, Note that, F 2 F F F F = 0 for =.Hece26 becomes F + F 2 F F F F =0. 27 = Substtutg 24to27, we obta α α 1 x 2 F 2 α 2 α =0. 28 =x 2x2 Dfferetatg 28wthrespecttox k,oehas 2α k α k 1 x 3 k whch reduces to α k 1 x 2 k 2α k F 2 α 2 α + 2α k F 2 x k x k =x 2x2 +F 2 α 2 α =x 2x2 Substtutg 28to30, we have α k 1 x 2 k + α α 1 x 2 F 2 α α k +α 2 x 2x2 k F 2 α α k +α 2 x 2x2 k α α k +α 2 x 2 x2 k =0, 29 =0. 30 =0. 31 Moreover, dfferetatg 31wthrespecttox l l =k, we get 2α l α l 1 x 3 l + 2α lα k +2α 2 l x 3 l x2 k F 2 2α l F 2 x l α α k +α 2 x 2 x2 k =0. 32

4 4 Abstract ad Appled Aalyss Combg 32wth31gves α l 1 x 2 l α k +α l xl 2 F 2 + α k 1 x2 k xk 2 + α α 1 x 2 Dfferetatg 33wthrespecttox l aga, oe has =0. 33 α k +α l α l 1 x 3 F 2 + α l 1α 1 +1 l x2 k x 3 =0, 34 l whch yelds ether α l =1l =kor α k +α l F 2 =α l +1x 2 k. 35 Sce F s defed by 23 wthozerocostatsα for = 1,...,,tfollowsthat35smpossble.Hecewehaveα l = 1. Wthout loss of geeralty, we may choose k=1.iths case, α l =1for l =1reduces to Therefore, 33 becomes α 2 = =α =1. 36 α 1 +1 x1 2 F 2 + α2 1 1 x2 l x1 2 =0. 37 The above equato mples that ether α 1 = 1or F 2 =α 1 1x 2 l. 38 It s easy to see that the latter case s mpossble. At ths momet, we obta the fucto F as follows: Fx 1,...,x =Ax 1 1 x 2 x. 39 Cosder 31fork=1,whchtursto 1 x 2 1 Substtutg 39to40, we have F2 xk 2 1 x x 2 =0. 40 A 2 x 2 2 x2 1 x x 2 =x 2 k. 41 I vew of 41, by comparg the degrees of the left ad the rghtoftheequaltywehave=3.hece41becomes A 2 x 2 2 +x2 3 =x2 k, k = 2,3, 42 whch s mpossble sce A =0. Therefore, there s ot a mmal geeralzed Cobb- Douglas producto hypersurface E +1.Thscompletesthe proof of Theorem Geeralzed CES Producto Hypersurfaces I ecoomcs, goods that are completely substtutable wth each other are called perfect substtutes. They may be characterzed as goods havg a costat margal rate of substtuto. Mathematcally, a producto fucto s called a perfect substtute f t s of the form Fx 1,...,x = a x, 43 for some ozero costats a 1,...,a. I the followg, we deal wth the case of mmal geeralzed CES producto hypersurfaces E +1. Theorem 3. A -factor geeralzed CES producto hypersurface E +1 s mmal f ad oly f the geeralzed CES producto fucto s a perfect substtute. Proof. Assume that M s a geeralzed CES producto hypersurface E +1 gve by a graph fx 1,...,x =x 1,...,x,Fx 1,...,x, 44 where F s the geeralzed CES producto fucto Fx 1,...,x =A a ρ xρ γ/ρ, 45 wth A, a >0ad x >0. The geeralzed CES producto fucto 45 s homogeeous of degree γ. By assumpto that the producto hypersurface s mmal, from 18 the mmalty codto s equvalet to F 1+ F 2 =, F F F. 46 We defe aother mmal producto hypersurface as fλx 1,...,λx =λx 1,...,λx,Fλx 1,...,λx, 47 λ R +. Sce the CES producto fucto s homogeeous of degree γ, we coclude that the hypersurface gve by fx 1,...,x =x 1,...,x,λ γ 1 Fx 1,...,x 48 s also mmal. I ths case, the mmalty codto 46 becomes F λ 2 2γ + F 2 =, F F F. 49 Comparg 49wth46, we obtathatetherγ =1or γ =1 ad F =0, 50 F F F =0., 51

5 Abstract ad Appled Aalyss 5 Moreover, t follows from 45 that F =Aγa ρ xρ 2 F =Aγa ρ xρ 1 a ρ xρ γ/ρ 1, 52 ρ 1 a ρ k xρ k +γ 1aρ xρ a ρ xρ γ/ρ 2, k = F =Aγγ ρ a ρ aρ xρ 1 x ρ 1 We dvde t to two cases. a ρ xρ γ/ρ 2, =. Case A γ =1. Substtutg 52 ad54 to51, we get A 3 γ 3 γ ρ, a ρ xρ 3γ/ρ 4 x2ρ 2 =0. If ρ =γ, the the above equato reduces to, whch s equvalet to Hece x2ρ 2 =0, =0. 57 =0. 58 Sce ρ =1,thscasesmpossble. Therefore we coclude that ρ=γ.substtutg53 to 50, we have Aγ γ 1 a ρ xρ 2 a ρ xρ γ/ρ 2 a ρ xρ =0. Notg γ =1,tseasytoseethat59smpossble. 59 Case B γ = 1. We ote that the mmalty codto reduces to F + F 2 F F F F =0. 60 = Substtutg 52 54to60, we have Aρ 1a ρ k xρ k a ρ xρ k = 1/ρ 2 +A 3 ρ 1 a ρ 3/ρ 4 k xρ k a ρ xρ = A 3 1 ρ = k = a ρ x2ρ 2 x ρ 2 x2ρ 2 a ρ xρ 3/ρ 4 =0. a ρ xρ 2 61 Note that ρ=1fulflls 61. Hece, ths case the geeralzed CES productofuctof s a perfect substtute. Ithefollowg,wewllshowthatthecaseρ=1 s mpossble. I fact, f ρ =1,61reducesto a ρ k xρ k a ρ xρ k = +A 2 k =a ρ k xρ k +A 2 = = x2ρ 2 2 2/ρ a ρ xρ 2 a ρ x2ρ 2 x ρ 2 =0. 62 Sce x >0ad a >0for = 1,...,, t follows that the left of equalty 62 s postve. Therefore, t s a cotradcto. Coversely, t s easy to verfy that f the geeralzed CES producto fucto s a perfect substtute the the -factor geeralzed CES producto hypersurface E +1 s mmal. Ths completes the proof of Theorem 3. Remark 4. Remark that Che proved [7] a more geeral result for 2-factor: h-homogeeous producto fucto s a perfect substtute f ad oly f the producto surface s a mmal surface. Ackowledgmets The authors would lke to thak the referees for ther very valuable commets ad suggestos to mprove ths paper.thsworkssupportedbythemathematcaltayua Youth Fud of Cha o , the geeral Facal Grat from the Cha Postdoctoral Scece Foudato o. 2012M510818, ad the Geeral Proect for Scetfc Research of Laog Educatoal Commttee o. W Refereces [1] C. W. Cobb ad P. H. Douglas, A theory of producto, Amerca Ecoomc Revew,vol.18,pp ,1928.

6 6 Abstract ad Appled Aalyss [2]K.J.Arrow,H.B.Cheery,B.S.Mhas,adR.M.Solow, Captal-labor substtuto ad ecoomc effcecy, Revew of Ecoomcs ad Statstcs,vol.43,o.3,pp ,1961. [3] L. Losocz, Producto fuctos havg the CES property, Academae Paedagogcae Nyíregyházess,vol.26,o.1,pp , [4] D. McFadde, Costat elastcty of substtuto producto fuctos, Revew of Ecoomc Studes,vol.30,o.2,pp.73 83, [5] G. E. Vîlcu, A geometrc perspectve o the geeralzed Cobb- Douglas producto fuctos, Appled Mathematcs Letters, vol. 24, o. 5, pp , [6] A. D. VîlcuadG.E.Vîlcu, O some geometrc propertes of the geeralzed CES producto fuctos, Appled Mathematcs ad Computato,vol.218,o.1,pp ,2011. [7] B.-Y. Che, O some geometrc propertes of h-homogeeous producto fuctos mcroecoomcs, Kraguevac Joural of Mathematcs,vol.35,o.3,pp ,2011. [8] B.-Y. Che, O some geometrc propertes of quas-sum producto models, Joural of Mathematcal Aalyss ad Applcatos,vol.392,o.2,pp ,2012. [9] B.-Y. Che, A ote o homogeeous producto models, Kraguevac Joural of Mathematcs,vol.36,o.1,pp.41 43,2012. [10] B.-Y. Che, Geometry of quas-sum producto fuctos wth costat elastcty of substtuto property, JouralofAdvaced Mathematcal Studes,vol.5,o.2,pp.90 97,2012. [11] B.-Y. Che, Classfcato of homothetc fuctos wth costat elastcty of substtuto ad ts geometrc applcatos, Iteratoal Electroc Joural of Geometry, vol.5,o.2,pp , [12] B.-Y. Che, Solutos to homogeeous Moge Ampère equatos of homothetc fuctos ad ther applcatos to producto models ecoomcs, Joural of Mathematcal Aalyss ad Applcatos,vol.411,o.1,pp ,2014. [13] B.-Y. Che ad G. E. Vîlcu, Geometrc classfcatos of homogeeous producto fuctos, Appled Mathematcs ad Computato,vol.225,pp ,2013. [14] A. D. Vlcu ad G. E. Vlcu, O homogeeous producto fuctos wth proportoal margal rate of substtuto, Mathematcal Problems Egeerg, vol.2013,artcleid , 5 pages, [15] B.-Y. Che, Pseudo-Remaa Geometry, δ-ivarats ad Applcatos, World Scetfc, Hackesack, NJ, USA, [16] B.-y. Che, Geometry of Submafolds, MarcelDekker, New York, NY, USA, 1973.

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