The Impac of Traaco Co o Compeve Ecoomy Zhpg Xe Shadog Uvery & Hua Uvery (emal: zpxe@2c.com) Abrac The fucoal relaohp bewee he prce vecor ad coumer Marhalla demad ad/or producer opmal acual ale wll be drecly revealed repecvely by wo ew geomerc mehod geeral equlbrum framewor whou orage. The mpac of raaco co hared by dvdual o hee fucoal relaohp wll be drecly aalyzed. The, proved ha, he demad ad upply fuco of dffere commode wll be o loger mahemacally couou, f x 0 >0 ad/or y 0 j>0, whch maly caued by formao co. Th mae ure ha he raaco co wll ubaally durb he Walraa equlbrum whou orage f o-femal fraco of populao ychroouly hf her deco. Ad he, ele uggeed ha raaco co wll dramacally cu dow ocal welfare. Fally, a reved model wh orage a a heorec way ou for Walraa ecoomy of zero-orage wll be preeed o ugge ha a compeve ecoomy wh zero-orage probably a Nah equlbrum. (JEL: D5, D23, C62, D6, D89) Key word: geeral equlbrum, raaco co, formao co, ocal welfare, orage The radoal Walraa equlbrum ha egleced he aco for orage, whch of coure urealc. Afer Arrow ad Debreu (954) (hereafer, A-D) fr rcly proved he exece of Walraa equlbrum whou orage uder ome relavely rog aumpo, may wor o weae hoe rog aumpo have already doe, e.g., amog oher, Debreu (959) corporaed cogecy he model, Hah (97) creaed a model wh orage ad raaco co, Ma-Colell (974) proved he exece of a compeve equlbrum whou he axom of compleee ad ravy of coumer preferece, Bewley (972) ad Alpra ad Brow (983) expaded he commody pace o fe dmeo Rez pace, Kydlad ad Preco (982) ad Beal ad Ede (993) dcued veore from he perpecve of geeral equlbrum, Quz (984) roduced dvbly of commody o compeve equlbra, ec. Le Kurz (974), he whlom purpoe of my reearch preeed h paper alo wa oly o weae oe of hoe rog aumpo ha raaco co
equal zero he compeve ecoomy, whch fac a mplc preme A-D model. Uforuaely, he reul of my reearch, however, omehow o adly rg ha I am reluca o ay here appear a lgh crac he docre of Walraa equlbrum whou orage f raaco co (epecally formao co) ae o accou. Bu, from he vewpo of cece, f here deed ex ome pobly ha omehg wll defely brea Walraa equlbrum whou orage, wll aurally be very gfca for u o ow wha wll pol valdy o a o loo for a way o amed. Oherwe, he preece ha he Walraa equlbrum docre aboluely problem-free wll obruc u from fdg ou he real ecoomc problem, o ha our ecoomy probably wll really fall o bg rouble. I elf-evde, for he geeral equlbrum framewor, how mpora o weae he aumpo of zero raaco co, becaue oo eay o prove ha raaco co evably prevalg he real world. For ace, o log a we adm ha formao o free he world where formao rcally mperfec ad aymmercal, a lea he co for formao oe or of mpora raaco co. Nowaday, he formao ecoomc ha already ear gfca poo adard mcroecoomc (e.g., ee Sgler (96), Aerlof (970), Damod ad Mrrlee (97), Spece (973, 2002) ad Sglz (975, 979)). The mere py however ha here ll ome gap egregag formao heory from geeral equlbrum heory. Maybe he effor o corporae raaco co (epecally formao co) geeral equlbrum heory ca creae a bae for brdgg he gap. I addo, a ew chool, New Iuoal Ecoomc, ha already gaed a deep gh o may mpora ecoomc ue relaed wh raaco co (ee, e.g., Coae (937), Alcha ad Demez (972), Wllamo (979), Demez (968) ad Cheug (969)). Va embeddg raaco co geeral equlbrum heory, reaoably udoubed ha h model ca gfcaly reveal he exu bewee h chool ad eoclac chool wh a cera exe. Several leraure are led o he effor o coec raaco co o geeral equlbrum. For example, Foley (970) ha dcued he exece of geeral equlbrum wh buyg ad ellg prce ad raaco co; Hah (97) ad Sarre (973) have coruced equece ecoomy model wh budge balacg a each dae; Kurz (974) ha aalyzed raaco co a barer exchage equlbrum model; ec. I eceary o pay more aeo o meo Hah (97) ad Kurz (974). I Hah (97) model wh orage, all mare acve are fully egraed a f hey were uder he corol of a gle frm, o ha he raaco co are codered a a whole ued by a mare acvy ead of eparaely ued by each dvdual. So, he mpac of raaco co o dvdual age deco ca be refleced. I cora o, he pree model wll, by 2
emphazg he mpac o each dvdual deco, embed he raaco co a a deerma of ecoomc flucuao ad ocal welfare. I Kurz (974) model, whch focue oly o a barer exchage ecoomy, he adop a raher cofug cocep, vecor of ale of coumer h, o ha he ha o acle a very complcaed budge correpodece. Tha h cocep cofug becaue doe mea he acual ale of coumer h he mare, bu mea wha he coumer ha o gve up (he acual ale plu raaco co) ead. I cora o, he pree model wll o oe had coder all cae, o he oher had, wll eparae he raaco co from he acual ale a well a combe ale vecor wh purchae vecor by urg he ale o mu purchae. Th paper wll frly prove ha, a compeve ecoomy whou orage where raaco co (epecally formao co) preval, he demad ad upply fuco of dffere commode probably are o loger mahemacally couou. The, oehele, h paper wll ll ugge ha, geeral, he raaco co wll o ubaally durb he equlbrum of a perfecly compeve ecoomy whou orage ule boh a grea umber of dvdual have he ame characerc ad he mare by cocdece a oe of he ocouou po. I oher word, oly f here o-femal fraco of populao ychroouly hfg her deco, he perfecly compeve ecoomy whou orage wll depar from equlbrum. Neverhele, I ugge ha, realy, he rouble caued by formao co wll ehace he probably ha o-femal fraco of populao ychroouly hf her deco. So, Walraa equlbrum whou orage faally eed o be reved. Ad he, I pree a ool o emae approxmaely he lo of welfare caued by raaco co. Fally, I ugge a reved model wh orage 2 a a heorec way ou for Walraa equlbrum whou orage, whch volve aco raferrg commode from ome me perod o oher. Durg my explorg, pred by he hory of offer curve aaly (ee Meade (952) ad Huphrey (995)), I have foruaely come up wh a ew mehod of aalyc geomery o depc he graph of he fucoal relaohp bewee ay coumer Marhalla demad ad he prce vecor uder he geeral equlbrum framewor, va daemblg ha d of Edgeworh Box dagram whch accommodae offer curve. I h way, he raaco co ca be ealy corporaed geeral equlbrum framewor. Meawhle, for produco ecor, I have alo veed a ew geomerc mehod o reveal he fucoal relaohp bewee producer opmal acual ale ad he prce vecor. Thee ew aalyc geomery mehod ca be expeced o become wo Becaue reul from raoal deco, he o-couy h model eeally dffere ha ha caued by dvbly. 2 Th reved model wll mae u a awful ee ha, f he orage codered a a dmeo of raege, Walraa ecoomy of zero-orage wll probably eher fall dequlbrum or ru ou of Nah equlbrum becaue pove orage wll probably mae ome dvdual beer off. Oherwe, here cao be he pheomeo ha people ore ome commode for fuure, ededly or pavely. 3
mpora fudameal echque mcroecoomc, epecally geeral equlbrum aaly. Baed o he revealed fucoal relaohp boh coumpo ad produco ecor, he demad ad upply fuco wh repec o he prce vecor dffere mare ca be drecly pecfed. The, he cocluded (relave) prce vecor (ee lae ubeco II.B) deermed by he relaed demad ad upply of mare aurally obaed. I h way, he cocep exce demad A-D model ueceary. So, he dcuo of equlbrum exece become exceedgly mple. The mehod I roduce raaco co o my model mply o aalyze mmedaely he mpac of raaco co hared by each dvdual o he fucoal relaohp bewee he prce vecor ad ay age opmal e omal purchae (or acual ale) of dffere commode. I addo, he moey wll be drecly accommodaed he model ad, for he purpoe of coveece ad o deparg from he real ecoomy, pulaed ha ay raaco ca occur oly bewee moey ad oe d of he ormal commody. By he way, worhwhle o meo ha h model wll clearly how ha he aumpo of producer coa reur o cale A-D model problemac. I h paper, eco I wll mahemacally defe he cocep of raaco co. Seco II wll ar from daemblg he Edgeworh Box dagram whch he offer curve of wo rade parer are depced. The, he mple cae where here are oly wo d of commode wh o produco ad o orage, he raaco co wll be embedded he model. Ad he, wll be proved ha, f x 0 >0, he demad ad upply fuco of he mare wll o loger be mahemacally couou. Seco III wll expad he model o a pure exchage ecoomy where he umber of commode more ha wo. Seco IV wll expad he model o a ecoomy wh produco bu ll whou orage. Two ew cocep, dffere prof curve (map) ad feable border, wll be veed, whch wll play wo very mpora role creao of a ew framewor of produco aaly. I h ew aaly framewor, he couy, covexy ad coa reur o cale are ueceary. Moreover, he ew framewor wll ele clearly how ha he aumpo of producer coa reur o cale A-D model problemac. Seco V wll dcu he role of formao co. I wll argue ha oe of he propere of formao co wll o oly reul x 0 >0 ad/or y 0 j>0, bu alo ehace he pobly ha o-femal fraco of populao ychroouly hf her deco. Seco VI wll dcu he exece of Walraa equlbrum whou orage he cae wh raaco co. I wll coclude ha, a perfecly compeve ecoomy, accordg o Kaua Fxed-Po Theorem, uch a equlbrum ca geeral ex ule o-femal fraco of populao ychroouly hf her deco. Seco VII wll creae a approxmae ool o meaure he mpac of raaco co o ocal welfare ad dcu wha hoe 4
age ca do for more beef. I wll coclude ha raaco co wll dramacally cu dow he whole ocal welfare. Ad he, mlar o Hah (97) ad Sarre (973) equece ecoomy o clafy commode, a reved model wh orage wll be preeed o accommodae he aco raferrg commode from prevou me perod o followg perod, whch mea a heorec way ou for Walraa equlbrum whou orage. Baed o h reved model, he codo o avod raaco co ad prce flucuao over me ad o arbrage are dcued. Thee codo mea he compeve ecoomy wh zero-orage a Nah equlbrum. Seco VIII he cocluo. By he way, becaue of oo may fgure ad for he purpoe of eaer uderadg, he yle of h paper ha o be explaaory ead of ha le A-D or Kurz (974). I. The Defo of Traaco Co Th Model Sce R. H. Coae publhed h wor 937, he cocep of raaco co ha gradually become popular he ocey of ecoomc. Due o he grea effor of R. H. Coae, A. A. Alcha, O. E. Wllamo, H. Demez ad S. Cheug, ec., he ew uoal ecoomc, whch he cocep raaco co oe of he mo mpora bedroc, ha already brough u a gh o a alve ew world of heory. I h paper, for geeraly ad rce, however, I have o defe raaco co from he perpecve of mahemac. I h model, coe wh A-D model, he umber of ormal commode wll be a fe umber l ad he leer h wll degae dffere commode; he umber of coumpo u ad produco u wll be fe umber m ad repecvely, ad dffere coumpo u ad dffere produco u wll be degaed by ad j repecvely. For geeraly, however, he moey 3, a a pecal d of commody, hould be explcly ae o accou, 3 I h model, I do dcu why he moey hould be corporaed he framewor, becaue o oe had moey a a medum of raaco preval he real ecoomy ad, o he oher had, moey ca be equvalely deemed a merely a gaugg ool whch wll o ubaally mpac he fal deco of coumpo ad produco hroughou he whole ocey f raaco co egleced. I fac, f le every dvdual edowme of moey be zero ad every dvdual uly fuco be depede of moey, my model wll be equvale o hoe whou drecly coderg moey ( eay o fd ou ha dffere prce heore, uch a A-D model, uually employ ome d of moey called umerare, a he mplcly gaugg rume). The ue why people have o explo moey a a medum her real ecoomc lfe ha draw coderable aeo ecoomc udy. Sdrau (967) cluded moey he repreeave age uly fuco. Clower (967) roduced he cocep cah--advace cora o capure he role of moey a a medum. Broc (974, 990) aumed moey ca lower he raaco co. Kyoa ad Wrgh 5
albe wll ever be really coumed. Le h=0 defe he moey. The, he umber of all commode wll become l+ ad h wll ru from 0 o l. Le p oegave orha R l+ + deoe he prce vecor. I h model, he dcou rae wll be egleced ad moey prce p 0 (f he dcou rae codered, he model hould be reved he way dcued ubeco VII.B). I addo, for he purpoe of expreo, pulaed ha ay rade ca occur oly bewee he moey ad oe d of ormal commody (ome uppor ca be go from Foley (970), Hah (97), Sarre (973) ad Kurz (974b), ec) excep for he cae of pure barer ecoomy dcued eco II. I fac, oe rade of barer erely equvale o wo correpodg moeary rade repecvely bewee he exchaged good ad releva moey, or Hah (97, page 436) word he e of mareg acve a moeary regme wll coa he e uder barer bu o vce vera. The vecor x Eucldea pace R l+ degae ay h coumer omal coumpo ad hh compoe, x h, repree he quay of he hh commody omally coumed by he h coumer. x X, le ha he A-D model. The h coumer real coumpo wll accordgly be degaed by x, x R l+ + (where x 0 0). The upercrp dguhe he omal from he real. The h coumpo u edowme degaed by a vecor, ζ R l+ +, ad hh compoe ζ h. The vecor (x ζ ) called e (omal) purchae of he h coumpo u. If ome of compoe, (x h ζ h ), egave, he ha wll mea he acually ell h d of commody. Aalogouly o coumpo, he vecor y j R l+ degae ay jh producer acual ale ad hh compoe, y hj, repree he quay of he hh commody old by he jh produco u. If y hj egave, he ha wll mea he fac buy h d of commody a pu. The jh producer real oucome of produco wll accordgly be degaed by y j R l+ (where y 0j 0). The upercrp dguhe he acual ale from he real oucome of produco. DEFINITION: For he h coumpo u, durg her acg a compeve ecoomy, he ha o gve up ome reource uch ha he able o realze her e omal purchae, (x ζ ), he wha he gve up called he raaco co hared by her ad degaed by a vecor x c R l+ +. DEFINITION: For he jh produco u, durg her acg a compeve ecoomy, he ha o gve up (993) developed a model of earch heory whch he co emmg from earchg for a uable dvdual o rade wh ubaal for barer ad moey ca gfcaly reolve h problem ( fac h he problem of formao co). More roudly, Walh (998) had made a good dcuo abou he ue of moey. Subeco VII.A h paper wll how how raaco co everely reduce he welfare. Followg h logc, f moey ca help dramacally reduce raaco co, he ure ha moey poee grea rc value, vewed from he perpecve of whole ecoomy. I he ame ee, amog oher, Hah (97) ad Kurz (974b) argue ha becaue of raaco co moey ca effcely ex mare ecoomy. 6
ome reource uch ha he able o ell all real oucome of her produco, y j, he wha he gve up called he raaco co hared by her ad degaed by a vecor y c j R l+ +. Dffere o Kurz (974), I do egregae ale from purchae, ce a eller wll probably ur o a buyer f he prce ha a eough chage ad vce vera. The, dffere o Kurz (974) (T.2), aumed ha x c f(x ζ ). Bu becaue coumer wll maxmze her uly, he ha o mmze x c. The, x c a fuco wh repec o (x ζ ). Becaue ζ coa, ad x c =x c (x ), where, x c ( ) exogeouly deermed by he mare uo ad he h coumer characerc. Therefore, he h coumer real coumpo, x, wll equal x x c. Smlarly, producer j acual ale, y j, wll equal y j y c j. Of coure, x c or y c j are o rerced by he age edowme. The x =x x c (x )=f c (x ), ad x =f c (x ), or x h=f ch (x h ), where f c ( ) he vere fuco of f c ( ). If x h=ζ h, he x c h=0. I reaoable o ae o accou ha each raaco wll probably requre all d of commode a raaco co. So, each compoe of raaco co wll co of l erm (albe may of hem are zero), becaue pulaed ha ay rade ca occur oly bewee he moey ad he relaed l d of ormal commode. Whou he pulao, here would be l(l+)/2 poeal rade ead of l, whch mea ha each compoe of raaco co wll co of l(l+)/2 erm or a comparavely bgger umber. From aoher perpecve, he gh rade bewee he moey ad he gh commody wll accordgly reul relaed raaco co, x cg, R l+ +. The x c = Σ l g= x cg, ad x c h=x c h(x )= Σ l g= x cg h(x g)= Σ l g= x cg h, h=0,,l. For echque reao, x cg h(x g) wll be pl o wo par. Le x cg h(x g)=x 0g h(x g)+x g h(x g), where x 0g h(x g)= lm xg ζ g x cg h(x g), lm xg ζ g x g h(x g)= lm xg ζ g (x cg h(x g) x 0g h(x g))=0. The, x 0 h= Σ l g= x 0g h, ad x h= Σ l g= x g h. If x g=ζ g, x 0g (x g)=0. Smlarly, y c j a fuco wh repec o y j or y j. There y c j=y c j(y j ), where, y c j( ) alo exogeouly deermed by he mare uo ad he jh produco u characerc. The y j= y j y c j(y j )=g cj (y j ), ad y j =g cj (y j). If y hj=0, he y c hj=0. Aalogouly, here are y c j= Σ l g= y cg j, ad y c hj(y j )= Σ l g= y cg hj(y gj ), h=0,,l. y cg hj(x gj) alo clude wo par. y cg hj(y gj )=y 0g hj(y gj )+y g hj(y gj ), where y 0g hj(y gj )= lm y gj 0 y cg hj(y gj ), 7
lm y g hj(y gj )= lm y gj 0 y gj 0 (y cg hj(y gj ) y 0g hj(y gj ))=0. The y 0 hj= Σ l g= y 0g hj, ad y hj= Σ l g= y g hj. If y gj=0, he y 0g j(y gj )= y 0g j(g cj (0))=0. x 0 ad y 0 j wll play he mo crucal role h model, whch are maly caued by formao co (ee eco V). Sce Kurz (974) eglec h, he ca fd ou mpac o he couy of demad ad upply. II. Traaco Co he Smple Barer Ecoomy I he eoclac ecoomc, ay coumer Marhalla demad defed o be he oluo of he maxmzao model. A her Marhalla demad, he dvdual reache her maxmzed uly, whch called drec uly. A-D or Hurz (974) model ha pad eough aeo o dcu he fucoal relaohp bewee Marhalla demad or producer opmal upply ad he prce uder he geeral equlbrum framewor. Iead, oly oo he oluo of coumer ad producer maxmzao model for graed. Th probably eglec ome problem, e.g., a crcal flaw regardg he aumpo of producer coa reur o cale (ee lae ubeco IV.A). Of coure, drecly revealg h fucoal relaohp decvely gfca. If h fucoal relaohp ca be drecly expoed wh he geeral equlbrum framewor, may ool of paral equlbrum aaly ca be coveely ued for referece. I h ee, wha crcally heurc for me he offer curve 4 aaly he hory of ecoomc. The pree model wll borrow he hough of offer curve a a arg po o explore he fucoal relaohp bewee Marhalla demad ad he prce vecor uder he geeral equlbrum framewor. Th eco wll, for he purpoe of mag a uve ee, ar from he mple pure exchage ecoomy where l=2, =0 ad he prce vecor p, p R 2 +. Here, moey ad orage wll o be codered. I h mple compeve ecoomy, every dvdual a prce-aer who chooe her opmal real coumpo, x *, a her Marhalla demad o a o maxmze her uly uder he gve prce vecor p. Srcly peag, he pree model, ay dvdual preferece eed be couou ad covex. Bu 4 T. M. Huphrey (995) preeed a good revew abou he early hory of offer curve aaly. Offer curve aaly ared from Mll ad Torre cocep of recprocal demad o expuge erm-of-rade deermacy from Rcardo aaly. Afer hem, A. Marhall, E. Y. Edgeworh, A. Lerer, W. Leoef ad J. E. Meade had gfcaly corbued o. Amog all her corbuo, he mo gfca oe for my model Edgeworh combg offer curve wh dfferece map. 8
for he coveece of expreo, ll aumed ha ay dvdual preferece couou ad covex. Th eco wll be dvded o wo par. I par, I beg my erpreao from a barer ecoomy whou raaco co. I par 2, he raaco co wll be embedded he model ad wll be prove ha, he cae where x 0 >0, he demad ad upply fuco of mare wll o loger be couou. A. The Cae whou Traaco Co I h ubeco, aumed ha here o raaco co. I udoubed ha replacg wo coure wh wo coumer a barer ecoomy of wo rader ca drecly place offer curve Edgeworh Box dagram. I oucome he followg. Suppoe here are wo coumer. A how fgure, her orgal po are O ad O 2 ; coumer edowme (ζ,ζ 2 ) ad coumer 2 edowme (ζ 2,ζ 22 ); coumer dfferece map dcaed by hoe old curve ad coumer 2 dfferece map dcaed by hoe dahed curve. Thoe ray pag hrough po ζ dcae he prce rao, whoe lope wll decreae a p creae or p 2 decreae. Gve ay prce rao, he correpodg ray wll be age wh oe of coumer dfferece curve, he po of agecy mu be he coumer opmal choce po uder he gve prce rao. A he prce rao chage, he po of agecy wll mgrae ad he locu wll form a curve, whch coumer offer curve. I he ame way, coumer 2 offer curve ca be depced. Curve OC ad OC 2 are repecvely her offer curve. OC ad OC 2 erec a po E,.e., whe prce rao ha dcaed by he ray ζ, boh coumer opmal choce po are all a po E. The, he equlbrum po emerge. Fgure Offer curve Edgeworh Box dagram Th ory however ca drecly help u o buld our model, albe very heurc. Wha dffere ha here are may coumer our model ead of oly wo. 9
A oo a he offer curve propere are checed, wll be foud ou ha coumer offer curve erely depede of wha are relaed o coumer 2,.e., her ow edowme ad dfferece map wll erely deerme her offer curve. Tha, he Edgeworh Box dagram ca be daembled o wo depede par. Fgure 2 Half of daembled Edgeworh Box dagrm Therefore, le ae h dagram apar, whch he very arg po of my model. Hereafer, we wll maly focu o dvdual a a repreeave ad, for more clear ad more geeral, wll be oherwe arculaed from geeral equlbrum perpecve. So, a alerave approach wll be ued o decrbe he ame apparau he followg. Ad he cocep of offer curve wll be borrowed o dcae he graph of he fucoal relaohp bewee Marhalla demad ad he prce vecor. For ay h dvdual, uder he geeral equlbrum framewor, her dfferece map ca be draw fgure 2. I fgure 2, ζ he h dvdual edowme po ad 0 U her orgal (reervao) uly accordg o ζ, ad m=p ζ =p ζ +p 2 ζ 2. So, very clear ha her budge cora le mu pa hrough her edowme po ζ, ad he lope of he le mu be p /p 2. Le = p /p 2, whch called relave prce ad mea he lope of her budge le (or mu prce rao). Uder h budge cora, he ca mae her uly maxmzed o e U o po E. Whe p creae or p 2 decreae, he budge le wll clocwe ur o he po ζ, ad vce vera. Whe her budge le roae, her be choce po E wll alo vary accordgly. A fgure 3, a p /p 2 vare from 0 o, her po E wll accordgly mgrae ad locu wll form a curve, whch wll be her offer curve more geeral yle eeally mlar o coumer or 2 offer curve fgure. Offer curve here ohg more ha he graph of Marhalla demad wh repec o he (relave) prce, whch reflec he fucoal relaohp bewee he (relave) prce ad he Marhalla demad or opmal e purchae a geomercal dagram. Bu, eceary o oe ha, h dagram, he relave prce wll be pree he form of a parameer equal o mu prce rao ead of he form of a varable dcaed by a ax. Fgure 3 The locu of opmal choce po 0
If =, her opmal choce po wll be a E. If = 2, her opmal choce po wll be a E 2. If = 0,.e. equal he lope of he age of her dfferece curve 0 U a po ζ, her opmal choce po wll be a ζ. Meawhle, eay o be oberved ha, f > 0, he wll ell commody 2 bu buy commody,.e., he wll exchage commody 2 for commody, Fgure 4 The dagram of offer curve ad vce vera. So, a how fgure 4, ca be cocluded, gve ay, whe he ha o chooe he po E (E mu be o her offer curve ad coordae (x,x 2 )) o maxmze her uly, u (x,x 2 ), over e {x x X, px pζ }, he correpodg opmal e purchae vecor, x * ζ, wll be a fuco wh repec o or p,.e., x * ζ = OC (,ζ )=OC (p,ζ )=(OC (,ζ ),OC 2 (,ζ ))=(OC (p,ζ ),OC 2 (p,ζ )), where 5 OC 2 (,ζ )= OC (,ζ ) = (p /p 2 )OC (p, ζ ). I obvou ha, >0, OC (p,ζ )= OC (p, ζ ). For coecy, OC (,ζ ) alo called offer curve fuco of dvdual. Of coure, x *=(x,x 2 )=x (p, pζ ) he h dvdual Marhalla demad fuco wh repec o he prce ad her drec uly fuco wll be defed by v (p, pζ ). B. The Cae wh Traaco Co Now le beg he ecod par o gve up he aumpo of zero raaco co. Th par wll prove ha, he cae where x 0 >0, he demad ad upply fuco of mare wll o loger be mahemacally couou. Relave o zero raaco co world, wha wll chage a world wh ome oegave raaco co? The budge le wll chage, becaue he real erm of coumpo wll be dffere o he omal erm. Of coure, he omal erm, here ll ex he budge cora codo, px = p(x +x c (x )) pζ. So, he real erm, here px pζ px c (x ),.e., he budge le wll become a budge curve, px =pζ px c (x )= pζ px c (f c (x )). x (x ), x <ζ ad x 2>ζ 2 Suppoe ha, x (x )= x + (x ), x >ζ ad x 2<ζ 2 where, x (x ) 0, x + (x ) 0. 5 The reao why I place ζ OC (,ζ ) wll be erpreed ubeco VII.B.
x 2 x 2 px =pζ px =pζ ζ φ px =pζ px c (x ) ζ φ px I =pζ px c (x ) O (a) x O (b) x Fgure 5 The budge curve wo dffere cae I fgure 5a ad 5b, he le, px =pζ, wll be called omal budge le,, ad he curve, px =pζ px c (x ), wll be called (real) budge curve, φ. If x 0 =0, a how fgure 5a, he budge curve wll pa hrough po ζ. If x 0 >0, a how fgure 5b, he budge curve wll o pa hrough po ζ. If x (x ) ad x + (x ) are creag fuco wh repec o x h ζ h, h=,2, he he ew budge cora e wll be covex ( he ee a lle mlar o ha of Kurz (974) (T.)). For coveece, aume ha x (x ) ad x + (x ) are creag fuco wh repec o x h ζ h, h=,2, ad hey are dffereable ad x ( ζ ) x x ( = x ζ ) 2 = x 2 ( ζ ) x = x 2 ( ζ ) x2 =0. Le frly dcu he cae where x 0 =0. A how fgure 6, whe = 0, equal he lope of he age of her dfferece curve.e., he omal budge le ad her dfferece curve 0 U wll be age a po ζ. 0 U a po ζ, Becaue x ( ζ ) x x ( = x 2 ζ ) =0, he, for he budge curve a po ζ here (px (ζ ))= x (pζ px c (ζ )) x = p x (x c (ζ )) x x =0. So, p +p 2 x x 2 (ζ )=0 x x 2 (ζ )= ( 2 = p / p ) = 0. 0 Fgure 6 The cora bewee he opmal choce dffere cae 2
Th mea ha he ew budge curve, φ 0, wll be age wh he dfferece curve wll reul ha her offer curve wll pa hrough he po ζ. Whe 0, a how fgure 6, e.g., whe =, he correpodg budge curve, φ, wll be age wh a hgher dfferece 0 U a po ζ, whch curve U a po E c ; whe = 2, he correpodg budge curve, φ 2, wll be age wh aoher hgher dfferece curve 2 U a po E c 2. Whe vare from 0 o, a how fgure 7, her be choce po E c wll accordgly mgrae ad locu wll form a ew offer curve, OC c, uder h d of Fgure 7 The locu of opmal choce po whe x 0 =0 raaco co. The, her opmal e real purchae x * ζ = OC c (,ζ )= OC c (p,ζ ). Le x * ζ =x *+x c (x *) ζ =OC (,ζ )= (OC c (,ζ )+x c (x *),OC c 2(,ζ )+x c 2(x *)), whch called he omal offer curve fuco (vecor) ad mea her opmal e omal purchae fuco wh repec o. x * her omal Marhalla demad. Obvouly, x * wll be o he correpodg budge curve φ ead of he correpodg omal budge le, bu x * wll be o he omal budge le. I cora o he cae whou raaco co, he omal offer curve fuco wll dffer from he offer curve fuco whou raaco co, OC (,ζ ). Obvouly, boh he po (ζ +OC (p,ζ ), ζ +OC 2(p,ζ )) ad he po (ζ +OC (p,ζ ), ζ +OC 2 (p,ζ )) are o he budge le, a how fgure 7. More gfcaly, however, we ca ele ge he omal offer curve va aoher approach. Le have he h dvdual uly fuco raformed. Le u (x )=u (x x c (x ))= u (x ), where u ( )=u (f c ( )) called her omal uly fuco, a how fgure 8, whch coe Fgure 8 Nomal preferece v real preferece 3
wh her omal preferece erm of omal coumpo. I fgure 8, he dahed curve dcae her real dfferece map, he old curve dcae her omal dfferece map 6, ad here are 0 0 U = U, U = U, 2 2 U = U ad 3 3 U = U. Accordgly, a how fgure 9, uder he ame relave prce, he ca mae her be omal choce a E, ead of E, accordg o her omal dfferece map (whou raaco co, her omal be choce wll be a E accordg o her real dfferece map). I he ame way, her omal offer curve OC ca be depced. Fgure 9 Nomal offer curve wh raaco co v ha whou raaco co Her omal preferece ca be equvalely vewed a ome oher oe real preferece whou raaco co,.e., he ecoomy wh h d of raaco co (x 0 =0) ca be equvalely replaced by a ecoomy whou raaco co. I h cae, herefore, he exece of he equlbrum wll o be durbed. The problem wll are he cae where x 0 >0, whch he demad ad upply fuco wll o loger be mahemacally couou. Le prove he followg. Whe x 0 >0, fgure 8 wll become fgure 0. I fgure 0, he old curve dcae her omal dfferece map ad here ll are 0 0 U = U, U = U, 2 2 U = U ad 3 3 U = U. Bu, fgure 0, dffere from fgure 8, dfferece curve 0 U wll o loger ouch dfferece curve 0 U a po ζ. Of coure, a how fgure, erm of omal preferece, a omal offer curve OC ca alo be draw. However, par of wll be vald. A how fgure, whe =, he budge le wll be age wh 0 U a po A, ad whe Fgure 0 Nomal dfferece map v real dfferece map = 2, he budge le wll be age wh age wh a dfferece curve whoe uly le ha 0 U a po B, hereby, whe 2 <<, he budge le ca oly be be vald, becaue he wll prefer o-radg (he ca eep her reervao uly o arc Aζ B (her uly wll be le ha 0 U. So, here a par of OC, he arc Aζ B, havg o 0 U ). Ad o, he vald par of OC wll become arc 0 U ) o radg a ay po A A ad arc B B. 6 Iuvely, he curvaure of her omal dfferece curve geerally more ha ha of her real dfferece curve, becaue boh x (x ) ad x + (x ) are creag fuco wh repec o x h ζ h (h=,2). However, oe ha h o frmly rgh. 4
Fgure Nomal offer curve he cae x 0 >0 Whe >, dvdual wa o exchage commody 2 for commody ; whe < 2, dvdual wa o exchage commody for commody 2. There are wo pecal cae where ζ or ζ 2 zero. Whe ζ =0, dvdual wll exchage ohg f <. Whe ζ 2 =0, dvdual wll exchage ohg f > 2. Thee wo cae mea wo ymmercal corer oluo. Therefore, (,0) ca be defed a he h dvdual buyg (commody ) erval of relave prce ad (, 2 ) ca be defed a he h dvdual ellg (commody ) erval of relave prce. For he purpoe of explaao he coex below, he upper lm of (, 2 ) for dvdual wll be deoed by ad he lower lm of (,0) for dvdual wll be deoed by b, he whe << (of coure here mu be b for ay dvdual,.e. oe ca oly be a eller or a buyer b or a eural), dvdual wll o ac. The buyg jump-po of prce. called her ellg jump-po of prce ad b called her Gve a prce p, f relave prce <, dvdual wll be called a eller; f >, dvdual wll be b called a buyer; f <<, dvdual wll be called a eural. I a mare, gve ay, amog all b dvdual, here mu be eller, b buyer ad g eural (0 m, 0 b m, 0 g m, ad +b+g=m). Le Ψ defy he e of he eller, Φ he e of he buyer. A vare from 0 o, wll creae ad b wll decreae. I oher word, every dvdual a poeal buyer a well a a poeal eller a he ame me. 5
Whe he mare clear, here mu be Ψ OC (, ζ ) = Φ OC (, ζ ). Bu uforuaely, h equlbrum wll probably o ex, a how he followg. Le ae he demad ad upply of commody a a example. For clarfcao, ome dffere way wll be employed o defy dffere dvdual. Suppoe here are wo poeal le: Oe he poeal eller whle he oher he poeal buyer. Each dvdual mu be mulaeouly o hee wo poeal le ad wll be defed by a bary (, ). ad repecvely ru from o m, ad every dvdual ad are cera bu dffere o oe aoher. Le he leer degae dffere poeal eller alog he eller poeal le ad uppoe whou lo of geeraly ha + m ad he h poeal eller opmal e omal purchae vecor, x * ζ, wll a ew way be deoed by ( OC (, ζ ), OC (, ζ ) 2 ); he leer degae dffere poeal buyer alog he buyer poeal le ad mlarly uppoe ha b b b b + b m ad he h poeal buyer opmal e omal purchae vecor, x * ζ, wll aoher ew way be deoed by b ( OC (, ζ ) b, OC (, ζ ) ). 2 The followg wll prove he demad ad upply fuco wo be couou h cae ha x 0 >0. Gve a relave prce he mare, f >> 2, he here oly real eller; f >> +, he here are real eller; f <, he here are m real eller. Suppoe m >> +, he hoe whoe equece umber are o more ha wll really ell her commody, ad he he mare upply of commody wll be S()= Σ OC (, ζ ). = Becaue whe > dvdual wo really ell bu whe < he wll really ell OC (, ζ ) of commody, ca be cocluded ha Σ OC (, ζ ) o = couou a =, =,,m,.e., a (a) (b) Fgure 2 demad ad upply curve he cae x 0 >0 6
how fgure 2a, S() a ocouou fuco wh repec o. Smlarly, uppoe b b << b b+, hoe whoe equece umber are o more ha b wll really buy b b commody, ad he he mare demad of commody wll be D()= Σ OC (, ζ ), whch alo a = ocouou fuco a hoe po where = (=,,m), a how fgure 2b. b Q.E.D. Wha wll happe he mare h cae? I he mare relave prce ll eady? Whe S()>D(), becaue of he mare prce mecham, p wll ed o fall or wll ed o re, ad vce vera. If a ome * here ju ex S(*)=D(*), a how fgure 3a, * wll be he equlbrum prce ad he mare wll clear. If he cae, however, ha lm S ( )< + lm D ( ) bu + lm S ( )> lm D ( ) (or lm S ( )< + b lm D ( ) bu + b * (a) (b) lm S()> b lm D()), a how b fgure 3b (or 3c), ca be cocluded ha he mare fal relave prce mu be *= (or *= ), or f he cae ha, here b (c) (d) Fgure 3 The deermao of he cocluded relave prce ex boh lm S()< + lm D() + ad lm S()> D(), meawhle lm = b, he he mare relave prce mu be *= = b, a how fgure 3d,.e., he mare prce wll be eady, bu he mare cao clear. If whe =0, here ll rema S()>D(), he *=0, commody free ad o oe wa o exchage commody 2 for commody. I h ee, herefore, * fxed ay cae, or ay *=f (). f ( ) called mare mecham fuco. Becaue he mare wll probably o clear, * ca o be called equlbrum prce aga. Therefore, * wll 7
heceforh be coformably called cocluded (relave) prce, o maer wheher or o he mare wll clear. Albe he probably of uch a cae ca perhap be codered o doma, o mpoble ha he cae wh o mare clearg wll occur. Noehele, he mare wll o be rouble o log a boh lm OC (, ζ ) ad b lm OC (, ζ ) are o bg eough. I a perfecly compeve ecoomy, + b becaue every dvdual deco aumed o be very mall, he dcrepacy bewee demad ad upply geeral ca be gored. Wha horrble he cae where boh may eller have exacly he ame (or may buyer have exacly he ame ) ad *= (or *= ), becaue he mare wll be bg ure ( wll be furher b b dcued eco VI). The cae abou commody 2 wll be he ame becaue ymmercal bewee commody ad 2. III. Traaco Co a Exchage Ecoomy Now, le expad our model o a relavely more complex ecoomy where l>2, bu ll =0 ad o orage. Smlarly, dvdual are alo prce-aer, who maxmze her real uly uder he gve prce p. Th eco wll prove ha, he cae where x 0 >0, he demad ad upply fuco wh repec o he (relave) prce every mare wll alo be mahemacally ocouou, bu mlarly, he cocluded (relave) prce wll be fxed. A. The Cae whou Traaco Co If here o raaco co, he ecoomy ca be equvalely codered o be ha, a he very begg, every oe exchage her all edowme for moey uder a gve prce, ad he he buy wha he wa uder he ame prce uch ha he ca maxmze her uly. Afer he h dvdual ell her all edowme, her moey wll be M = Σ l h=0 p ζ h h =pζ. The he eed o maxmze her uly u (x ) over he e {x x X,px M }. Suppoe a x =x *, her uly h he op, he her Marhalla demad fuco wll be x *=x (p,m ) ad her drec uly fuco wll be v (p, pζ ). Now he ue, whe l>2, doe here alo ex offer curve le he cae a barer ecoomy? If he awer ye, he he way employg offer curve a equvalece of he radoal way o maxmze uly drecly 8
from he model ha max u (x ) over he e {x x X,px M }? The awer wll be pove f pulaed ha x each exchage ca occur oly bewee oe d of ormal commody ad he moey. Whou lo of geeraly, le ae he rade bewee he moey ad commody a a example. So, p wll emporarly be vewed a varable whle p 2, p 3,, p l coa. Le p defy he vecor (p 2,,p l ), x he vecor (x 2,,x l ) ad ζ he vecor (ζ 2,,ζ l ). Gve ay p, f dvdual ha o chooe her coumpo of commody a x, wha he ca do for her maxmum of uly o decde her coumpo of oher commode accordg o he followg model, max u (x ) x ().. p x =M M p x where M he expedure of moey for x. Suppoe he oluo x *=(x 2 *,,x l *), he her maxmum uly here wll be u (x, x *), ad le v (x,p,p,m )=u (x, x *). If M depede of p ad x, he x * wll be a fuco wh repec o (p,m ) uder Fgure 4 The dfferece map of PIU a gve x, ad he here wll be v (x,p,p,m ) =v (x,p,m )=v (x,m ), ad le call paral drec uly fuco of commody of dvdual or PIU. So, her dfferece map of PIU ca be draw a fgure 4. Now wha looed for wll acually become he oluo of he followg model, max, M x v (x, M ) (2) Fgure 5 The offer curve of commody.. p 0 M +p x M Le l ζ m = ζ 0 Σ h= 2 phζ h +, he M =p 0 ζ +p ζ. By cora wh he cae of he barer ecoomy m meoed above, p 0, M, ζ m, ad v (x,m ) are a he poo exacly mlar o hoe of p 2, x 2, ζ 2 ad u (x,x 2 ) of ubeco II.A repecvely. Therefore, of coure, a mlar offer curve, OC, alo ca be he ame way draw fgure 5. Here, however, he relave prce wll become = p /p 0 = p, ad he edowme po ζ wll become he po ~ ζ. 9
A how fgure 5, gve ay (or p ), here mu be oe correpodg po E o OC a whch he ca maxmze her PIU ad ju equal he drec uly fuco, v (p, pζ ). Therefore, ca be cocluded ha he way employg offer curve equvale o he drec way. Fgure 5 furher how ha, whe = 0, here are x *=ζ ad M *= ζ, meawhle v (x *, M *)= v (ζ, ζ )=u (ζ, x *); whe > 0, he wll buy commody ; whe < 0, he wll ell commody. m Obvouly, x - * ζ wll be a fuco repec o or p,.e., x * ζ =OC (,p,ζ )= OC (p,p,ζ )=OC (p,ζ ), whch called offer curve fuco of commody. Smlarly, whe he rade bewee ay oher ormal commody ad he moey, he correpodg offer curve ca alo be draw he ame way. Le OC h degae he offer curve derved from he exchage bewee he hh commody ad he moey, he he correpodg offer curve fuco wll be OC h (p,ζ ). Hece, a correpodg offer curve fuco vecor ca be defed a OC (p,ζ )=(OC (p,ζ ),,OC h (p,ζ ),,OC l (p,ζ )). m B. The Cae wh Traaco Co Now le ur o he cae wh raaco co. I h cae, he ecoomy ca alo be equvalely codered o be he ame le uppoed he begg of he la ubeco. I oher word, he frly ell all her commode ad he buy wha he wa, bu eed o be added a ew pulao ha f her e omal purchae of ay commody zero he he correpodg raaco co wll be zero. Smlarly, eay o coclude ha f x 0 =0 he exece of he equlbrum wll o be durbed becaue he couy rema udurbed. So, le drecly dcu he cae ha a lea oe of he compoe of x 0 doe o equal zero. I wll be proved ha, he cae where x 0 >0, he demad ad upply fuco wh repec o he (relave) prce every mare wll be mahemacally ocouou. Whou lo of geeraly, le ll focu o he rade bewee commody ad he moey a he repreeave. Le dvde he dcuo o wo ep. I ep, wll be dcued ha, gve ay x ad M, how he raaco co curred by hoe rade repecvely bewee he moey ad oher commode affec PIU reulg from model (). 20
The correpodg raaco co caued by he rade bewee he gh commody (g=2,,l) ad he cg cg l cg cg moey ca be equvalely codered o be M x ) = M ( f ( x )) = Σ = p x ( x ) + x ( x ) = ( g cg g h 2 h h g 0 g l cg cg Σ p x ( f ( x )) + x ( f ( x )) h= 2 h h cg g 0 cg g cg erm of moey. Noe ha M x ) doe clude ( g cg p x ( x ). g So, gve ay x ad M, he model () wll become max u (x ) x (3).. p x + l cg Σ g= 2 M ( f cg ( x g )) =M Suppoe he oluo of he model () x * whch reul PIU ad he oluo of he model (3) x c * whch ur reul PIU c. Accordgly, becaue x fxed a (x, x *), x g, g=2,,l, wll be fxed, ad le x g * defy. Noe ha x g * a fuco wh repec o p, M ad x. Clearly, becaue he budge cora of model (3) gher Fgure 6 Cora bewee wo dffere dfferece map ha ha of model () erm of real coumpo, uder he gve x ad M, here mu be x *> x c *, ad he here mu be PIU >PIU c. Whe x ad M vary, PIU ad PIU c wll chage, he he dfferece map of PIU ad PIU c ca be depced, whch are repecvely how fgure 6, where PIU 0 =PIU c0, PIU =PIU c, PIU 2 =PIU c2, PIU 3 =PIU c3. I ep 2, mlar o ha eco II, le have he h Fgure 7 Cora amog hree dffere dfferece map dvdual PIU c raformed furher. Le v (x,m )=v (x x c (x ), M M c )=v (x,m ), where M he omal quay of M, x c (x )=x c (x l ch l ch )+ Σ x ( x *) ( Σ x ( x *) ca be vewed a h= 2 h h= 2 h par of x 0 ) ad c l c M = x x ) + Σ p x x ). v (x,m ) called her omal paral drec uly c 0 ( h= 2 h h ( fuco of commody wh repec o x ad M, whch deoed by PIU,. A how fgure 7, he dfferece map of PIU dcaed by he old curve, where PIU 0 =PIU 0, PIU =PIU, PIU 2 =PIU 2, 2
PIU 3 =PIU 3. The model (2) wll become x max, M v ( x, M ) (4).. p 0 M + p x M = p 0 ζ +p ζ m By cora wh he cae of he barer ecoomy wh raaco co, p 0, M, ζ m, ad v ( x, M ) are a he poo exacly mlar o hoe of p 2, x 2, ζ 2 ad u (x,x 2 ) of ubeco II.B repecvely. Therefore, mlar o fgure, he mlar offer curve, OC, ca be draw a fgure 8. Whe 2< <, mlarly, he correpodg arc A ~ ζ B of OC wll be vald for he uly of ay po o arc A ~ b ζ B wll be le ha he uly he ca ge oly f he buy bac commody o ζ. Le ubue ad for ad 2 repecvely, he a = b, he wll hf bewee acually buyg ad o buyg commody, whle a =, he wll hf bewee acually ellg ad o ellg commody. repecvely are her buyg ad ellg jump-po of prce of commody. I gfca o compare bewee he cae l=2 ad he cae l>2. I a barer ecoomy, a = (or = ), whle he hf bewee ellg (or buyg) ad o ellg (or o buyg) commody, he mulaeouly wll wholly hf bewee buyg (or ellg) ad o buyg (or o ellg) commody 2. I he cae where l>2, however, a = (or = ), whle he hf bewee ellg (or buyg) ad o b b b ad ellg (or o buyg) commody, he mulaeouly wll oly parly hf bewee buyg (or ellg) ad o buyg (or o ellg) ay d of oher commode. Becaue po A, B ad ~ ζ are dffere for her, ca be cocluded ha h d of hf wll o mae Fgure 8 Nomal offer curve of commody he cae x 0 >0 her drec uly ocouou wh repec o c. Bu, uch d of hf wll mae x x ) (g=2,, l) hf g ( bewee 0 ad x 0 g (g=2,, l), o ha x g *(g=2,,l) evably wll parly chage (h chage o couou wh repec o ) f 0 x g >0 (g=2,,l). 22
I he ame way, oher commode offer curve ca alo be derved from he rade repecvely bewee he relaed commode ad he moey. Le =( 0,,, l )=( p 0 /p 0, p /p 0,, p l /p 0 ). Le OC h (,ζ ) deoe he hh commody offer curve fuco ad le ow focu o he mare of commody h. Smlar o lae ubeco II.B, oao o defy dffere dvdual hould be chaged. I he ame way, le he leer, whch ru from o m, degae dffere poeal eller ad uppoe whou lo of geeraly ha h h h h ad he h poeal eller opmal e omal purchae h + m h of commody h, x h * ζ h, wll be deoed by OC, p, ζ ) (where p h =(p,,p h, p h+,,p l )); he leer h ( h, whch ru from o m, alo degae dffere poeal buyer bu oherwe uppoe whou lo of geeraly ha bh bh bh bh ad he h poeal buyer opmal e omal purchae bh + m b h of commody h, x h * ζ h, wll be deoed by OC, p, ζ ). h ( h Now le prove ha he demad ad upply fuco of commody h wo be couou he cae. Gve a h he mare, f > h > h, he here oly real eller; f h 2 > h > h, he here are h + real eller; f <, he here are m real eller. Suppoe m > h > h, he hoe whoe equece umber h + are o more ha wll really ell her commody h, ad he he mare upply of commody h wll be S h ( h, h h )= Σ OC (, p, ζ ). Becaue whe h > h poeal eller wll o ac bu whe h < h = h h h he wll really ell OC, p, ζ ) h ( h h of commody h, ca be cocluded ha Σ OC (, p, ζ ) h = h h o mahemacally couou a h =. Therefore, S h ( h, h ) a ocouou fuco wh repec o h ad graph exacly le ha fgure 2a. Smlarly, uppoe < h < bh, he hoe poeal buyer whoe equece umber are o more ha b bh b b+ wll really buy commody h, ad he he mare demad of commody h wll be D h ( h, h )= b b h Σ OC (, p, ζ ), whch alo a ocouou fuco a hoe po where h = bh (=,2,,m), = h h ad graph exacly le ha fgure 2b. Q.E.D. So, we have proved ha, he cae where x 0 >0, he demad ad upply fuco wh repec o he (relave) prce every mare wll be mahemacally ocouou. Whe S h ( h, h )>D h ( h, h ), becaue of he mare prce mecham, p h wll ed o fall or h wll ed o 23
re, ad vce vera. If a ome h * here ju ex S h ( h *, h )=D h ( h *, h ), le ha fgure 3a, cocluded relave prce h * wll be he equlbrum prce ad he mare wll clear. If he cae, however, ha lm S h ( h, h )< h h+ lm h h+ D h ( h, h ) bu lm S h ( h, h )> h h lm D h ( h, h ) (or h h lm S h ( h, h )< h bh+ lm D h ( h, h ) bu h bh+ lm S h ( h, h )> h bh lm D h ( h, h )), le h bh ha fgure 3b (or 3c), ca be cocluded ha he mare fal relave prce mu be h *= (or h h *= bh ), or f he cae ha, here ex boh lm S h ( h, h )< h h+ lm h h+ D h ( h, h ) ad lm S h ( h, h )> h h lm D h ( h, h ), meawhle h h = bh, he he mare cocluded relave prce mu be h h * = = bh, le ha fgure 3d,.e., he mare prce wll be eady, bu he mare cao clear. h If whe h =0, here ll rema S h ( h, h )>D h ( h, h ), he h *=0, ad commody h free. So, h * fxed, or ay, h *=f h(), h=,,l. f h( ) called he hh mare mecham fuco. I ca alo coclude ha, o log a boh h lm OC, p, ζ ) ad h h h ( h b h lm OC, p, ζ ) h bh+ h ( h are o bg eough, he mare wll o be rouble. I a perfecly compeve ecoomy, becaue every dvdual codered very mall, he dcrepacy bewee demad ad upply geeral ca be gored. The deal wll be dcued eco VI. IV. Traaco Co a Ecoomy wh Produco Now le furher expad he dcuo o ae produco u wh o orage o accou. May dea h eco were pred by Debreu (959). Smlarly, all producer are prce-aer, who are uppoed o maxmze her prof ad he drbue he prof hey ear bac o he coumer who ow hem. Aalogouly, h eco wll alo prove ha producer opmal deco wll very lely hf ocououly whe he relave prce move acro cera po he cae y 0 j>0, bu mlarly, he cocluded (relave) prce wll alo be fxed. I h eco, a ew mehod wh wo ew cocep, dffere prof curve (map) ad feable border, wll be roduced o he aaly of he fucoal relaohp bewee he (relave) prce ad he opmal real 24
oucome of produco or acual ale, whch eable me o predge ad clarfy gfcaly he decrpo of produco. Smlarly, he cocep of offer curve wll alo be borrowed o dcae he graph of h fucoal relaohp. Va he ew approach, ome aumpo uch a couy ad covexy ca be gve up from he model ad, furhermore, wll eve be ferred ha he aumpo of coa reur o cale problemac. Smlar o A-D model, for ay producer j, here a e Y j of poble produco pla wh whch producer j ca mae her deco of real oucome of produco y j. The he e of whole poble produco chedule Y= Σ Y. j = j Aume ha ay producer poee o orgal edowme. Aume ha ay producer ha o pu a lea oe d of labor f he provde ay pove oupu, whch mea Y I ( Y ) =0 ad Y I Ω =0, where Ω ={x x R l, x 0}. Aume ha, he e Y j cloed ad coag 0. I cora wh A-D model, h model, he aumpo of couy, covexy ad coa reur o cale ca be gve up. A. The Cae whou Traaco Co Now le frly dcu he cae whou raaco co. Here, I wll roduce a bac ew mehod wh wo ew cocep, dffere prof curve (map) ad feable border, o aalyze he deco of a raoal producer. l l l For producer j, her prof fuco π j (p,y j )=π j (,y j )= Σ h= ph yhj = Σ h=( p h / p0 ) yhj = Σ h= h yhj = y j. So, he hould maxmze π j (,y j )= y j over he e Y j. Le fr aalyze e Y j. Le y hj (h=,, l) deoe (y 0j,, y (h )j, y (h+)j,, y lj ). I e Y j, whe y hj fxed, y hj ca vary wh a cloed e, Γ hj (y hj ) R ( eleme wll chage a y hj chage), amog all eleme of whch mmum ad y ) hj he maxmum. y ( hj = y ( hj (y hj ) ad Becaue e Y j cloed, ay y ) hj y ( hj y ) hj = y ) hj (y hj ) are fuco wh repec o y hj. mu be cera ad lmed, oherwe π j (,y j ) ca reach fy. he Becaue he wll maxmze π j (,y j ), aurally, he ha o decde y hj a y ) hj. 25
DEFINITION: The e F hj ={y j y j Y j, y hj = y ) hj (y hj )} (h=,,l) he hh upper urface of e Y j. DEFINITION: The e F j = I l F hj h= he jh producer upper urface of e Y j. THEOREM: Ay producer j ca o reach her maxmum of prof f he mae her produco deco oude e F j. The proof mple ad we ca alo refer o Debreu (959). For ace, whe y gj fxed wh F j, f producer j decde her y gj o a g y gj g prof maxmum. y ) gj, ead a ) y gj ) ( y gj < y ) ) gj ), her prof wll be g y gj g y gj ead of y ) ) gj. Becaue g egave, g y gj g y gj < g y gj g y ) gj, hece uch a deco aga Se F j wll be a uper-urface R l, hereby he fucoal form of graph ca be decrbed a F j (y j )=0. So, for producer j, he ue become o maxmze π j (,y j )= y j over he e F j ={y j F j (y j )=0}. Of coure, he po 0 mu be o F j, becaue ay producer ha o pu a lea oe d of labor f he provde ay pove oupu,.e., whe he pu of h d of labor reache op, ull, Y j, all oher compoe of y j mu become 0. Gve ay, he graph of y j =a, where a depede of y j, a uper-plae R l. Whe he uper-plae upward mae a parallel move, a wll creae, ad vce vera. A a vare, we mu be able o fd ou a e, Θ, R +, wh whch, b Θ, a lea oe po of e F j wll be o he uper-plae y j =b. Se Θ called poble prof e. Le b* be he maxmum of b Θ, he b* mu be her maxmzed prof, ad he ereco po bewee e F j ad he uper-plae y j =b*, whch ca be oly oe po or more ha oe po, are her be deco. Θ ca ever be empy, becaue 0 oe ube of e F j ad 0 o he uper-plae, y j =0. Le Π j () degae b* ad y j *() her be deco, whch all are fuco wh repec o. I h way, he producer offer curve ca be obaed. I ay mare of commody h, (h=,,l), defe ha he jh producer opmal ale y hj *()= OC (), ad OC () called he offer curve fuco of commody h of producer j. OC hj hj Now le defe wo ew cocep, dffere prof curve (map) ad feable border, ad he depc h h (, ). DEFINITION: Le π hj = h y hj. The π j (, y j )= y j =π hj h y hj. Whe π j (, y j )=w, dffere prof curve he graph of π hj h y hj =w. Fgure 9 wll how dffere prof map, wh whch, he hgher he hj 26
dffere prof curve poo, he more w wll be. I fgure 9, here are wo map, of whch he dahed map accord wh h = whle he old map accord wh h = 2, 2 <,.e., a h vare from 0 o, he dffere prof map wll clocwe roae., here mu be oe of he correpodg dffere prof curve pag hrough he orgal po O whoe w 0. Le F h j(a)={y j y j F j, y hj =a}. Fgure 9 Idffere prof map Le y hj * or y hj *( h,a) = (y j *( h,a),,y (h )j *( h,a), y (h+)j *( h,a),,y lj *( h,a)) deoe he oluo of model max ( h y hj ) over F h j(a), ad π hj *( h,a)= h y hj *. y hj The, whe y hj =a, here mu be π hj π hj *( h,a); oher word, y hj, π hj π hj *( h, y hj ) feable bu π hj >π hj *( h, y hj ) o feable. DEFINITION: π hj =π hj *( h, y hj ) he boudary of feably of π hj, whoe graph called feable border. Fgure 20 Feable border Feable border wll fuco he way mlar o ha whch budge cora curve doe. Noe ha o eceary for he feable border o pa hrough he orgal po a well a o be couou. A how fgure 20, he area above he feable border ufeable whle he area uder he feable border (cludg he border elf) feable. Now le mae fgure 9 ad 20 ogeher o fgure 2. A how fgure 2, gve ay h, here mu be a lea oe opmal po we ca fd o he feable border whoe correpodg π j (,y j ) relavely he mo. For ace, whe h = 0, he be po a he orgal po O, whoe w 0; whe h =, he be po abca a, whoe w w ; whe h = 2, he be po abca b, whoe w w 2 ; whe h = 3, he be po abca c, whoe w w 3. I h way, herefore, he hh Fgure 2 The opmal choce po uder dffere relave prce 27
offer curve ca be depced a how fgure 22. There are ome pecal cae eceary o be dcued. I he cae where, a how fgure 23, whe y hj 0, π hj *( h, y hj ) depede of y hj, he feable border a horzoal le ad, whe y hj >0, π hj *( h, y hj )=, he, whe h <0, he opmal po abca mu be 0, whoe w Fgure 22 The hh offer curve whe y 0 j=0 π hj *( h,y hj ). I he cae where, a how fgure 24, he prof reur o cale wh repec o he pu of y hj o coa, ead, he fr perod of pu cale creag ad afer ha become decreag. The, whe h creae from 0 o 0, we ca draw offer curve from po A, bu whe h Fgure 23 The cae ha < 0, for ay po he feable area, here π j (,y j ) 0, hece he ha o π hj *( h, y hj ) depede of y hj mae her deco a orgal po o eep her π j (,y j ) o egave. I oher word, he relave prce 0 her produco oppg po, whch mea ha her deco of pu of y hj wll be o more ha a. I he cae where, a how fgure 25, whe h 0, we ca depc offer curve from po B, bu whe h > 0, for ay po feable area, here π j (,y j ) 0, hece he ha o decde her choce a orgal po O. Fgure 24 The cae where prof reur o cale wh repec o he pu of y hj o coa, ead, he fr perod of pu cale creag ad afer ha become decreag. Fgure 25 The cae mlar o ha of fgure 24 bu ha y hj oupu. Fgure 26 The cae combg ha fgure 24 ad fgure 25 I he cae where, combed ha fgure 24 ad 25 ogeher, a how fgure 26, whe A < h < B, for ay po feable area, here π j (,y j ) 0, he her be choce alo mu be a po O. I he laer hree cae, whch are realc, whe he relave prce h vare acro 0, (or eher A or B 28
fgure 26), he wll hf her deco of y hj bewee zero ad a or b. I fac, hee cae are pervave he realy. Therefore, hee cae her deco wll o be couou a 0 (or eher A or B ). However, he cae o whch we mu pay more aeo ha he reur o cale eep coa hroughou whole covex e Y j,.e., e Y j a covex coe, le ha A-D model. I h cae, π hj *( h,y hj ) mu be y hj, where =π hj *( h, ) f π hj *( h, ) 0 (f π hj *( h, )<0, he mu decde her be choce a po O). A how fgure 27, f h >, her be choce of y hj wll be ; f h <, her be choce wll be + ; f h =, her be choce ca be a ay po ad here mu be π j (,y j ) 0. So, h way ca be cocluded ha he aumpo of coa reur o cale problemac for A-D model, becaue here a coradco bewee maxmzao of prof ad he aumpo of perfecly compeve ecoomy. Fgure 27 The cae wh coa reur o cale Fgure 28 The cae wh creag reur o cale Aalogouly, he cae where he reur o cale eep creag hroughou whole o-covex e Y j alo problemac. I h cae, he ufeable area covex. A how fgure 28, a ad b repecvely are he lope of wo correpodg aympoe. Whe h < a, apparely, her be choce of y hj hould be + ; whe h > b, her be choce of y hj hould be ; whe b < h < a, her be choce of y hj ca be eher + or. Therefore, we ca coclude ha all of hee cae ca o yeld a couou offer curve fuc wh repec o. The, whch cae ca yeld a couou offer curve fuco? I eay o fd ou ha oly he cae where boh he reur o cale eep decreag hroughou whole covex e Y j ad feable border wll pa hrough oegave par of he ordae ax (.e. π hj *( h, 0) 0). B. The Cae wh Traaco Co Now, le ar he dcuo of he mpac of raaco co. Afer raaco co ae o accou, he prof fuco mu be ured o 29