J o u r n l o f A c c o u n t i n n M n e m e n t J A M v o l. 4 n o. ( 4 ) Generlize Co-Douls function for three inputs n liner elsticity Cătălin Anelo IOAN Gin IOAN Astrct. he rticle els with prouction function of three fctors with constnt scle return where ech elsticity of two of the fctors is function of first eree. After the exmintion of prmeters conitions ccorin to the xioms of the prouction functions there re compute the min inictors. Also the comintion of fctors is etermine in orer to mximize the totl output uner iven cost. eywors: prouction function Co-Douls Generl spects of the prouction functions In ny economic ctivity otinin result of it implies y efult there is certin numer of resources supposely inivisile neee for the proper functionin of the prouction process. We therefore efine on R the prouction spce for three resources: cpitl - lor n ln or nturl resources s SP() where xsp x() is n orere set of resources. ecuse in prouction process not ny mount of resources re possile we shll restrict the prouction re to suset D p SP clle omin of prouction. In context of the existence of the omin of prouction we put the question of eterminin its output epenin on the level of inputs of D p. It is clle prouction function n ppliction :D p R + ()()R + ()D p. For n efficient n complex mthemticl nlysis of prouction function we impose numer of xioms oth its efinition n its scope. FP. he omin of prouction is convex; FP. (); FP. he prouction function is of clss C on D p tht is it mits prtil erivtives of orer n they re continuous; FP4. he prouction function is monotoniclly incresin in ech vrile; FP5. he prouction function is qusiconcve tht is: (x+(-)y)min((x)(y)) [] xyr p. Dnuius University of Glti Deprtment of Economics ctlin_nelo_ion@univ-nuius.ro Dnuius University of Glti Deprtment of Economics in_ion@univ-nuius.ro 7
J o u r n l o f A c c o u n t i n n M n e m e n t J A M v o l. 4 n o. ( 4 ) From eometric point of view qusiconcve function hvin the property of ein ove the lowest vlue recore t the en of certin sement. he property is equivlent to the convexity of the set - [) R where - [) {xr p (x)}. he min inictors of prouction functions Consier now prouction function: :D p R + ()()R + ()D p. We cll mrinl prouctivity reltive to n input x i : of prouction to the vrition of x i. x i We cll vere prouctivity reltive to n input x i : consumption of unit of fctor x i. x i w x i x i n represents the tren of vrition the vlue of prouction t We cll prtil mrinl sustitution rte of fctors i n j the opposite chne in the mount of fctor j s sustitute for the mount of chne in the fctor i in the cse of constnt level of xi prouction n we hve: RMS(ij). x j We cll elsticity of output with respect to n input x i : x i vrition of prouction to the reltive vrition of the fctor x i. x i x i w xi xi n represents the reltive he Generlize Co-Douls function for three inputs Consierinnow prouction function :D p R + ( )()R + ()D p with constnt return to scle let note. et suppose now tht Consierin the function q such tht: () q From here we fin tht: et F e primitive function of w q q q q q n F of w we hve:. q q q ( )> ( )>. 8
J o u r n l o f A c c o u n t i n n M n e m e n t J A M v o l. 4 n o. ( 4 ) he equtions ecome: or in other wors: Intertin with respect n tht is: ut otin: q F q ln q F ln q we et: q F ln q F q F f ln q F f F f q e q e q F e F f f F f f ' e F ' n from the conition tht F ' F f F f' e f e ecuse ech term is of ifferent vrile n these re inepenent we hve: ut F C f Finlly: implies tht q Ce F C f - constnt (fter n ovious renotin of C). F F we () Ce If now: ( ) + ( ) + we et: ln F F ln therefore: () C F F e 4 he Generlize Co-Douls function for three inputs n liner elsticity Consier now the prouction function: () C e C. ecuse the function is elementry follows tht it is of clss C on the efinition omin. We now hve: 9
J o u r n l o f A c c o u n t i n n M n e m e n t J A M v o l. 4 n o. ( 4 ) Consierin orere Hessin mtrix: () H n the minors: it is known tht if < > the function is qusiconcve. Conversely if the function is qusiconcve then:. In the present cse: 4 4 It is ovious tht. For we shll o restriction of the omin D p such tht. Also reltive to the monotoniclly incresin in ech vrile we hve: n ecuse we must hve lso:. Finlly we hve tht the omin of prouction is: D p {() R }
J o u r n l o f A c c o u n t i n n M n e m e n t J A M v o l. 4 n o. ( 4 ) 5 Min inictors of the Generlize Co-Douls function for three inputs n liner elsticity We cn compute fter section the min inictors for the prouction function efine ove. We hve therefore: he mrinl prouctivity: he vere prouctivity: he prtil mrinl sustitution rte: RMS() he elsticity of output: w w w w RMS() w w RMS() 6 he prolem of eterminin the mximum of prouction in terms of iven totl cost et now the followin prolem: p p mx ( ) p C where C is the totl cost of the prouction which is suppose to e iven constnt. From the rush-uhn-ucker conitions we hve the necessry n sufficient conitions (tkin into ccount tht the restriction is ffine): p p p p p C
J o u r n l o f A c c o u n t i n n M n e m e n t J A M v o l. 4 n o. ( 4 ) From section 5 we et tht the system ecomes: p p p p p C or usin we in from the first two equtions: p p p p p p p p p p p p p p p p p p p p p p p Solvin the lst eqution for n from the first otinin p p we shll fin from: C p p the vlue of * of. Finlly: * * * *. 7 Conclusions he Generlize Co-Douls function for three inputs n liner elsticity is etermine from the conition tht liner elsticity of prouction with cpitl n lor re liner expresse. he prolem of eterminin the fctors of prouction tht mximizes output is reuce to n eqution of thir eree. 8 References. Arrow.J. Enthoven A.C. (96) usi-concve Prormmin Econometric Vol.9 No.4 pp. 779-8. Chin A.C. (984) Funmentl Methos of Mthemticl Economics McGrw-Hill Inc.. Hrrison M. Wlron P. () Mthemtics for Economics n Finnce Routlee 4. Ion C.A. Ion G. () he Extreme of Function Suject to Restrint Conitions Act Oeconomic Dnuius 7 pp. -7 5. Ion C.A. Ion G. () n-microeconomics Ziotto Pulishin Glti 6. Pony P. (999) An Overview of usiconcvity n its Applictions in Economics Office of Economics U.S. Interntionl re Commission 7. Simon C.P. lume.e. () Mthemtics for Economists W.W.Norton&Compny 8. Stncu S. (6) Microeconomics E. Economic uchrest