The generalized marginal rate of substitution

Transcription

1 Jural f Mathematical Ecmics The geeralized margial rate f substituti M Besada, C Vazuez ) Facultade de Ecmicas, UiÕersidade de Vig, Aptd 874, 3600 Vig, Spai Received 31 May 1995; accepted May 1997 Abstract The geeralized margial rate f substituti ccept is defied ad related t the uifrm prperess prperty f a preferece relati 1999 Elsevier Sciece SA All rights reserved JEL classificati: C60 Keywrds: Utility; Prefereces; Margial rate f substituti; Uifrm prperess 1 Itrducti I the mder thery f csumer behaviur, the margial rate f substituti MRS is emplyed t measure the relative margial utility Fr a preferece relati defied a subset X f R, this MRS betwee tw cmmdities i ad j, at a pit xs x,, x 1, is defied as the uatity f cmmdity j that wuld just cmpesate the csumer fr the lss f a margial uit f cmmdity i This is true prvided that the uatity f all ther cmmdities ad the csumer s level f satisfacti are held cstat If the utility fucti u is smth this defiiti f MRS x i j turs ut t be mathematically euivalet t the rati u r u where u is the partial derivative f u with respect t cmmdity k It is i j k ) Crrespdig authr Fax: ; cvazuez@uviges r99r$ - see frt matter 1999 Elsevier Sciece SA All rights reserved PII: S X

2 554 M Besada, C VazuezrJural f Mathematical Ecmics 31 ( 1999) imprtat t te that MRS is defied subject t the cditis that the chages i x ad x d t result i ay chage i utility i j that is, mvemet alg a give idifferece curve is ivlved, ad ly x i ad x j chage, while all ther cmmdities remai cstat The usual defiiti f this ccept fr tw cmmdities seems restrictive give that cmmdity spaces ca be larger Ather way t thik abut this culd be i terms f tw cmmdity budles We defie the geeralized margial rate f substituti GMRS at x as the real umber such that the lsses fr cmmdity budle w1 ca be cmpesated with this amut f ather give cmmdity budle w keepig utility cstat This ti f cmpesati has precedece i the prperess prperty itrduced by Mas-Clell 1986 The GMRS ad the prperty f uifrm prperess are related fr a preferece relati that ca be represeted by a differetiable utility fucti Precisely, if K is a preferece relati defied a pe subset X f a Baach space E, the Õ is a uifrm prper vectr fr K iff GMRS x is buded frm belw by a psitive umber This will be true fr all xgx ad wge, 5w5 s1 It is bserved that the budedess f the margial rate f substituti is ecessary fr the existece f euilibrium i ecmies with ifiitely may cmmdities as is ted by Peleg ad Yaari, 1970, Bewley, 197 with the adeuate assumptis, Mas-Clell, 1986 with the prperess prperty, ad Rustichii ad Yaelis, 1991 amg thers The geeralized margial rate f substituti I this secti we csider the cmmdity space EsR This is later geeralized t Baach lattices Let X;R be a pe ad cvex set, x gx ad let u: X R be a differetiable utility fucti which represets a ctiuus, mte ad strictly cvex preferece relati K defied X The prperties f mticity ad cvexity guaratee the strict psitivity f the partial derivatives f the utility fucti This is essetial t defie the GMRS Csider tw liearly idepedet vectrs w,w g R, where 5w 5s5 w s 4 4 1, ad cmplete w 1,w t a psitive basis Bs w 1,,w fr R Let w:r R be the liear fucti defied by w e sw, where e,,e 4 i i 1 is the stadard basis fr R Thus, w is a C` diffemrphism ad its assciated matrix is the matrix f chage frm B t the stadard basis Let the fucti Fsu`w, thus EF E y i w x s w x with w s w, PPP,w y1 j 1 Ý i i i i js1 Ex j

3 M Besada, C VazuezrJural f Mathematical Ecmics 31 ( 1999) y1 Give that EF r E y w x i ) 0, we get the implicit fucti y s g y, y,, y 1 3 verifyig EF y1 j w x Ý w1 x Eg E y js1 Ex j E y y1 1 w x sy sy ' 1 EF E y1 y1 j E y1 w x Ý w x E y Ex js1 j We defie the GMRS betwee the cmmdity budles w1 ad w at x as the real umber y E y re y 1 This is the amut f the cmmdity budle w that wuld be ecessary t cmpesate fr the lss f the cmmdity budle w 1, y1 keepig F cstat ad eual t F w x This utiet, which depeds ly the budles w 1, w ad x, ad t the chse utility fucti, is deted by GMRS x w,w This is, i fact, a geeralizati f the margial rate f 1 substituti betwee tw cmmdities Nte that if w1s ei ad ws e j, this ccept is i accrdace with the defiiti f MRS x i, j Csider sme smth utility fucti u represetig the preferece relati K Nw, we check that GMRS x w,w ca be defied as the utiet f the 1 directial derivatives f u i the directis f w1 ad w, respectively Frm E 1 we btai 1 w1 x PPP w1 x Ex1 Ex Dw u x 1 GMRSw,w x s s 1 1 Dw u x w x PPP w x Ex Ex 1 The ecmic iterpretati f the GMRS ca w be btaied by usig the usual iterpretati f the MRS Dividig the earlier expressi by re x x 1, we btai w1 1 w1 MRS,1 x PPP w1 MRS,1 x GMRSw,w x s 1 1 w w MRS x PPP w MRS x,1,1 The iterpretati f this is as fllws T cmpesate fr the lss f a uit f the secd cmmdity keepig the utility cstat, a trasfer f MRS x,1 uits f the first cmmdity is ecessary Thus, fr the lss f w1 uits f the secd cmmdity, w MRS x 1,1 uits f the first cmmdity are ecessary, ad s I geeral, fr the lss f a cmmdity budle w1 a trasfer f Asw1 1 w MRS x w MRS x 1,1 1,1 uits f the first cmmdity are ecessary I the same way, we btai that a cmmdity budle w culd be cmpesated 1 with the trasfer f Bsw w MRS x w MRS x,1,i uits f the first cmmdity, ie, each uit f the first cmmdity is euivalet t 1rB times the cmmdity budle w Thus, GMRS x w,w sarb is the amut f the 1 cmmdity budle w that is ecessary t cmpesate the lss f the cmmdity budle w 1

4 556 M Besada, C VazuezrJural f Mathematical Ecmics 31 ( 1999) I the geeral case the cmmdity space E is a Baach lattice, give tw liearly idepedet vectrs w,w, csider Ey² w : = ² w : 1 1 =H where H is the tplgical cmplemet f ² w : = ² w : 1 i E Each vectr x g E has a 1 uiue represetati xs x, x, x 3 gr=r=h As befre, csider a differetiable utility fucti u defied a pe set X; E s² w : = ² w : 1 = H The, E ure x 1 x sd u x ad E ure x x sd u x w w By the implicit 1 1 fucti therem there exist pe eighburhds V f x i E ad W f x, x 3 i ² w : = H, respectively, ad a differetiable fucti g:w R such that ad 1 x 1, x, x 3 gv ; u x 1, x, x 3 su x m x 1, x 3 gw ; g x 1, x 3 sx Ex Ex 1 x 1 3 x 'D1 g x, x sy Ex 1 x Ex Thus, fr a ctiuus, mte ad strictly cvex preferece relati defied a pe ad cvex subset X f a Baach lattice E we have the fllwig Defiiti 1 The GMRS betwee tw cmmdity budles w 1,w ge, at a pit x g X relative t the preferece relati K is defied by the utiet GMRS x sd u x rd u x sdu x w rdu x w w,w w w 1, where u is 1 1 ay differetiable utility fucti represetig K Nte that, as was said befre, i the case where EsR, if w1 sei ad w s e, GMRS x cicides with the usual MRS x j w,w i j defiiti 1 Fially, it is ecessary t prve that the defiiti f the GMRS des t deped the differetiable utility fucti u which represets K This is a cseuece f the ext lemma Let u ad Õ be tw differetiable utility fuctis represetig the preferece relati K Lemma 1 If Du( x )/0, DÕ( x )/0, fr all xgx; the there exists a icreasig differetiable fucti w:õ( X) R, such that w X ( t )/ 0, fr all t g Õ( X ), ad usw`õ Prf Let tgõ X Csider xgx such that tsõ x ad defie w t su x The w is well defied ad als is the ly fucti verifyig usw`õ We will prve that this fucti is differetiable, icreasig ad w X t /0, fr all tgõ X Take tgõ X ad xgx such that tsõ x Let w be a uitary vectr such that DÕ x w /0 Let sdõ x w gr ad F:E =R R defied by F l,s sõ xlw ys; where E is a suitable eighburhd f 0 Observe that D F 0,t s /0, ad the, by the implicit fucti therem, there exist a l

5 M Besada, C VazuezrJural f Mathematical Ecmics 31 ( 1999) pe eighburhd U f t i R ad a differetiable fucti, l:u R verifyig l t s0 ad Õ xl s yss0 fr i all sgu Mrever, l t sy1r Nw, defiig m:u RPm s su xl s Thus, u xl s smõ x l s, fr all sgu Cseuetly, m cicides with w i a eighburhd f t; ad, because f m is differetiable i Um X t s1r DÕ x w Du x w ) 0, w verifies the therem The, by the lemma, Du x w Dw Õ x `DÕ x w w X 1 1 Õ x DÕ x w1 s s Du x w Dw Õ x `DÕ x w w X Õ x DÕ x w DÕ x w1 s DÕ x w 3 GMRS ad the prperess prperty I this secti we relate GMRS ad the prperess prperty The prperess prperty is uite clse t the idea that the GMRS is buded frm belw I fact, Mas-Clell 1986 itrduced this prperty t impse a priri buds the MRS Fllwig Mas-Clell 1986, a preferece relati K, defied a csumpti set X, is prper at x g X wheever there exists sme Õ) 0 ad sme eighburhd f zer V such that xya ÕzKx with a)0 implies zfav This prperty expresses the ecmic ituiti that the cmmdity budle Õ is desirable, i the sese that the lss f a amut aõ cat be cmpesated with a additial amut a z f ay cmmdity budle z which is sufficietly small Let w,õ be tw liearly idepedet cmmdity budles i E The the ecmic iterpretati says that GMRS x w cmpesates the lss f a uit f the vectr Õ Thus, if Õ is a prper vectr at x, ad VsB 0, is the pe eighburhd assciated, the GMRS x w f V Suppsig w is a uitary vectr, we have GMRS x K We shw this with the help f the fllwig example The idea is frmally stated i Therem 1 Example 1 O XsR the preferece relati represeted by u x, y sxy is, bviusly, prper at each budle xgx Mrever Õs6r,6r is a prper vectr ad GMRS x K6r fr all xgx ad wgr, 5w5 s1 I fact, the biggest ) 0 such that B 0,0, is the pe set crrespdig t Õ is ifgmrs x : xgx, wge, 5w5s14 s6r This ca be easily checked Next, we prve that this result is always true Befre we d s, it shuld be ted that a preferece relati K defied a subset X f a Baach lattice E, is

6 558 M Besada, C VazuezrJural f Mathematical Ecmics 31 ( 1999) prper at x if ad ly if there exist sme psitive vectr Õ)0, ad sme real umber )0, such that xya ÕzKx,a)0 ad zge implies 5 z5ka Ay vectr Õ that satisfies this prperty will be referred t as a prper vectr fr K at x Therem 1 Let E be a Baach lattice Let X be a pe subset f E ad u:x R be a differetiable utility fucti represetig a mte ad cõex preferece-relati K such that Du( x ))0 The ÕgE is a prper vectr fr K at x gx iff there exists )0 such that GMRS x K, fr all wge, 5w5 s1 Prf Suppse that Õ g E is a prper vectr at x with B 0, as the assciated pe set Let wge, 5w5 s1 3 It is kw that GMRS x syd g x, x slim g x h, x 3 Õ,m 1 h 0 y g x, x 3 r yh, where g is the implicit fucti defied by u: X; ² w : = ² Õ: = H ' E R i sme pe eighburhd f x s x 1 w x Õ x 3 ' 1 x, x, x 3 g² w : = ² Õ : =H Mrever, fr h small eugh, it is verified that x h, x 3 belgs t the 3 3 dmai f g ad, cseuetly, g x h, x w x h Õx ;x, the x ;x 1 wyg x, x 3 wg x h, x 3 wx ÕhÕx 3 sx g x h, x 3 yg x, x 3 whõ < 3 Fr h-0, the prperess prperty guaratees that g x h, x yg x, x 3 < < < Gyh s h Thus, fr h-0 i a eighburhd f zer, g x h, x 3 y g x, x 3 r h< G ad cseuetly, < < < GMRS x s D g x, x 3 < G 1 O the ther had, fr h/0, sufficietly small, it is verified that g x h, 3 3 x yg x, x r h -0 This fllws because if h-0 ad g x h, x g x, x, as w,õ g E, the mticity prperty states that x, x, x 3 s g x, x, x, x % g x h, x, x h, x 3, which is impssible Aalgus fr h)0 Thus, GMRS x G0, ad GMRS x K fr all uitary vectr wge Suppse w that there exist )0 ad ÕgE such that GMRS x K, fr all xgx ad fr all wge, 5w5 s1 We prve that Õ is a prper vectr at x Let zge, 5 z5- First we prve x )x a zyõ, fr all a)0 such that x a zyõ gx X 5 5 Csider z s 1r x z, we the kw that GMRS X x s D g x, x 3 Õ, z 1 1 X 3 where x sx z x Õx g² z X : = ² Õ : =H'E ad g is the implicit fucti defied a pe eighburhd f x, x 3 by g x, x z X 3 xõ x 3 ; x that is u g x, x, x, x su x 3 3

7 M Besada, C VazuezrJural f Mathematical Ecmics 31 ( 1999) As lim g x h, x yg x, x 3 r yh h 0 G, there exists r)0 such that 3 if hg yr,r, the g x h, x yg x, x 3 r yh G Csider hg yr,0, the x ;g x h, x 3 z X x h Õx 3 s g x h, x 3 z X yg x, x 3 z X g x h, x 3 yg x, x 3 X 3 3 g x, x z x ÕhÕxs x yh z ž yh / X 5 5 X yõ Gx yh z yõ )x yh z z yõ sx yh zyõ Csider ad a0 s rr, thus we have prved that fr each z g E with 5 z5-, there exists ad a )0 such that x %x a zyõ 0, fr all ag O the ther had, fr a)a0 such that x a zyõ gx,it is verified that x %x a zyõ, as a cseuece f the cvexity f the preferece relati Nw, let zge Csider z ssup z,0 4 ge The, there exists a 0 )0 such that x %x a z yõ, fr all ag 0,a 0 Thus, we have prved x %x a z yõ, ad, by the mticity, x %x a zyõ By usig agai the cvexity, we btai x %x a zyõ fr all a)0 such that x a zyõ gx That is, if a)0 ad x ya Õa zkx the 5 z5 G Nte that a preferece relati K defied the csumpti set X, is uifrmly prper wheever there exist sme Õ)0 ad sme eighburhd V f zer such that xya ÕzKx with a)0 ad xgx, implies zfav Thus, as a crllary f the therem abve, we btai a ecessary ad sufficiet cditi fr the uifrm prperess prperty Crllary 1 Let E be a Baach lattice Let X be a pe subset f E ad u:x R be a differetiable utility fucti represetig a mte ad cõex preferece relati K such that Du( x ))0, fr all xgx The Õ g E is a uifrm prper vectr iff there exists ) 0 such that GMRS x K, fr all xgx, wge, 5w5 s1 Tw immediate cseueces f this crllary are as fllws: i if Õ is a vectr, such that there exist x g X ad a uitary vectr w g E with GMRS x s 0, the Õ is t a uifrm prper vectr; ii if Õ is a uifrm prper vectr, the biggest )0 such that B 0, is the crrespdig pe set is ifgmrs x : xgx, wge, 5w5s14 Õ,m X Ackwledgemets We are grateful t Carls Herves fr his helpful cmmets ad suggestis This research has bee supprted by DGICYT uder Grat PB C0-01

8 560 M Besada, C VazuezrJural f Mathematical Ecmics 31 ( 1999) Refereces Bewley, TF, 197 Existece f euilibria i ecmies with ifiitely may cmmdities J Ec Thery 4, Mas-Clell, 1986 The price euilibrium existece prblem i tplgical vectr lattices Ecmetrica 54 55, Peleg, B, Yaari, M, 1970 Markets with cutably may cmmdities It Ec Rev 5, Rustichii, A, Yaelis, N, 1991 Edgewrth s cjecture i ecmies with a ctiuum f agets ad cmmdities J Math Ec 0,