EconomiX. Disequilibrium, reproduction and money: a Classical approach

Transcription

1 EcoomX h://coomx.fr Docum d Traval Workg Par 04-0 Dsulbrum, rroduco ad moy: a Classcal aroach Carlo B Chrsa Bdard Edh Klmovsky Ao Rbyrol Uvrsé d Pars Ous Narr La Défs (bâm G) 00, Avu d la Réublu 900 NANTERRE CEDEX UMR 735 Tél Fax : 33.(0) Emal : asam.zaroual@u-ars0.fr

2 Dsulbrum, rroduco ad moy: a Classcal aroach Carlo B *, Chrsa Bdard, Edh Klmovsky, Ao Rbyrol Absrac. W cosdr a bscor rroduco modl whch moy s roducd as a ur mas of xchag ssud by a bak a h roducrs' russ. Each caals ams a maxmsg accumulao hs ow scor. Thr las ar basd o avalabl uas ad xcd rcs. Effcv rcs ar drmd by a mark-clarg mchasm. Tmorary dsulbra occur boh hyscal ad moary rms. Th slm of h moary balacs s orad by mas of a rasfr of caal goods. Fal allocaos ad ffcv roducos ar hus drmd. Th dyamcs of h coomy ar hos of a suc of morary dsulbra ad l aar svral ossbls (local or global sably, cycls) ddg o h valus of h aramrs. Kywords. Classcal Rroduco, Moary rcs, Dsulbrum, Growh, Cycl. JEL classfcao: E, E30, E3, O4 * Corrsodg auhor, carlo.b@orag.fr, Uvrsé d Pars Ous Narr La Défs, EcoomX. Uvrsé d Pars Ous Narr La Défs, EcoomX. Uvrsdad Auóoma Mroolaa Azcaozalco (Mxco), EcoomX. Uvrsé d Pars Ous Narr La Défs, EcoomX.

3 Iroduco Th modl sudd hs ar rrss a bscor coomy morary dsulbrum. I s a xso of a rvous modl (B al. 0) srd by Torrs s (8, cha. VI, sc. VI) ad Marx s (885, Book II, cha. XXI) aroachs of rroduco. Is ma ovao s h roduco of moary xchags sad of barr: w cosdr a moary coomy whch moy s h u of accou ad a ur mas of xchag ssud by a bak. Tha ssu s mad a h roducrs russ, ad h roducrs comm hmslvs o rmburs h bak mmdaly afr h xchag. Moy s o a scfc good h al dowms; s o hr a crd or a sor of valu ad s amou s a dogous magud. Tha coco of moy, hough o doma coomc hory, ca b racd back a las o Wcksll (906), s foud os-kysa aroachs ad s also adod by Drèz ad Polmarchaks (000) a gral ulbrum framwork. Ulk hs auhors, howvr, our aalyss roducs a mark mchasm drmg moary rcs boh dsulbrum ad ulbrum. Idvdual dcsos ar ak ddly by ach ag, wh o a ror coordao ad hrfor h aalyss s crd o morary dsulbrum rahr ha ulbrum. Th roduco of moy dos o modfy h vry oo of ulbrum bu allows us o br aalys h workg of a coomy dsulbrum. Th avalably of a mas of xchag allows ach ag o buy goods ddly of hs sals. As a cosuc, moary balacs aar ad mus b sld. Th ags dsulbra ar boh ral (hy do o fulfl hr las) ad moary (osv or gav balacs aar). A suoal rul s roducd ad drms h way o sur h slm of hs moary balacs. Hr w ado a sml rul ad assum ha h slm s mad by mas of a rasfr of owrsh of som caal goods, chos by h caals wh a osv moary balac, a h xs of h ohr caals. Tha rocdur modfs h allocao of rsourcs as had b drmd rvously by h xchag a mark rcs. Th roduco of moy as a ur mas of xchag hus affcs h ral dyamcs of h coomy. I Sco w buld u a modl whch xhbs hs faurs ad sudy s rors. Gv h uas roducd durg rod ad bfor h og of h mark a da (d of Wh ackowldgms o wo rfrs for dald comms o a frs vrso of h ar.

4 rod ad bgg of rod ), ach roducr asks for ad obas a amou of moy ual o h xcd valu of hs suly. A mark mchasm drms h moary rcs ad h hyscal allocaos whch dsulbrum dffr from h xcaos. Th ag s moary rcs also dffr from hs xss ad balacs aar. Th rul rad for balac slms rallocas h goods amog ags. Th fal allocao of mas of roduco s h obad ad h ffcv roducos of rod ar drmd. Sco 3 suds h dyamcs of h rooros roduco ad of moary rcs. Th dyamcs ar hos of a suc of morary dsulbra, o of morary ulbra sc s oly by fluk ha h ags fulfl hr las. Ral ad moary dsulbrum. Producrs las Th hyohss rlad o h ral ar of h modl ar h sam as B al. (0): - Labour dos o aar xlcly, vry workr bg rlacd by h corrsodg wag bask, whch s cororad h mas of roduco. - Thr ar wo scors ad o mhod of roduco r scor. Th chu s rrsd by marx A [ a j ] (, j, ), whr a j s h amou of u j rg o h roduco of o u of good. Marx A s roducv ad dcomosabl ( a 0, a 0). Cosa rurs rval, roduco aks o rod, all caal s crculag ad all goods ar rshabl. - Goods ca b dsosd of frly. - Each caals ams a maxmsg accumulao hs ow scor. 3 - All caalss ar assumd o hav h sam rc xcaos. W may hrfor aggrga hm ad cosdr oly o caals r brach. Th raso of h dcomosably hyohss s ha h accumulao rocss w sudy rss o h dmad by a caals of h u roducd by h ohr. If h wo-good coomy s o-basc, som ag dos o dmad h ohr s good ad ay xchag s xcludd. 3 Maxmsg accumulao s a u usual hyohss h classcal rado, cocd wh h da ha accumulao s h bs way o g fuur rofs. I hs rado, h hyohss ha caalss vs hr ow scor s foud Torrs (8) ad Marx s (885) largd rroduco schms. Tha hyohss xcluds h xsc of a caal mark ad allows us o sola h rol of moy as a ur mas of xchag. 3

5 A a gv da bfor h og of h mark, roducr (, ) has roducd a uay ad hs rc xcaos ar hs sag rc xcaos ar xogously gv). L ad (dx s for xcd ad rfrs o rcs; a b hs lad lvl of roduco (dx s for ooal ad rfrs o uas) for h og rod. Hs xcd budg cosra s wr ( a a ): xcd rocds from h sal of ouu h j j jus comld roduco cao b lss ha h xcd cos of roduco for h forhcomg rod. Accumulao dsrs bg uboudd, hs x a cosra s bdg ad hrfor: a a a a () () Formulas () ad () show ha las oly dd o h rlav rc /, o o moary rcs. Th sam for hr rao / : a a a a (3) whr /. I gral, h caalss las ar o fasbl.. Moy ad mark rcs Moy as h mas of xchag s ssud by a bak a h ags rus. Th amou of moy dmadd by ach ag s ual o h xcd valu of hs urchass of goods, whch slf s ual o h xcd valu of hs suly of goods. Prcs ar masurd rms of moy ha h coomy ackowldgs as h commo u of accou. Each ag comms hmslf o rmburs h moy h rcvd from h bak (hs s why moy dos o aar h x a budg cosra), ad moy s dsroyd afr has accomlshd s crcular flow. As w assum ha h crao of moy, s crculao ad s dsruco ak lac wh a vry shor m rval, h ra of rs s gord. Thr s hr facg of rs urchass by fuur rsourcs (o crd) or facg of fuur urchass by oday s moy (moy s o cosdrd hr as a sor of valu). 4

6 L us assum ha h formao of mark rcs, dod, obys h sml rul ally sad by Callo: Prcs ar fxd by h rooro bw h roduc xosd for sal ad h moy offrd for (755, I.IV.). Tha rul was also usd by Smh ad raars modr hory of sragc mark gams (Shaly ad Shubk, 977). As a cosuc of moary xchags, wo varas of h rul ca b cosdrd: () h whol roduco s brough o h mark; or, () h suly s h dffrc bw ad h uay a rad by roducr as a u for hs x roduco. Th lar s h oly ossbly a barr coomy. I h rmology of sragc mark gams, h frs vara corrsods o h all-for-sal modl, h scod o h offr-for-sal modl. I h frs cas, wash sals occur as h ags ar o boh sds of h mark. I our modl, h dsco has o cdc a ulbrum bu mars morary dsulbrum wh h xcd rcs ad h mark rcs dffr. W cosdr hr h all-for-sal vara whch all roducd uas ar valuad a mark rcs (h alrav modl s brfly sudd Adx ). Th amou of moy dmadd ad s by ag s.th uay of moy brough o mark for urchass s ual o h xcd valu of h dmad d for us by boh ags, as dfd by hr las. Th hyscal uay s brough o ha mark s h roducd uay of ha good. Th Callo rul dfs h mark rc of good a h curr da as h o gv by h xchag of hs uas of moy ad good: d ( a a j j ) s (4) hc h rlav rc rms of good a a a a Th alcao of h Callo rul drms h mark rcs ad h os-mark allocaos of commods. I h modl w cosdr moy s dogous. Th ssud uay M of moy, ual o h oal xss, s basd o xcd moary rcs ( M ) ad, sc M also j j (5) 5

7 rrss oal rcs, h moary rcs sasfy ualy M j j. (If h j j ). Th dogy of moy ls h gral lvl of mark rcs dd o h lvl of xcd moary rcs ad xlas why h ags corrcly fors moary rcs as soo as hy corrcly fors h ulbrum rlav rc. W do T ha rlav rc (dx T s for Torrs). 4 Wh h moary xcd rcs ar ha rgh rooro, ad whavr hr lvl, hy ar slf-fulfllg. Th boh marks ar morary ulbrum: a ha rlav rc ad for h rod w ar cosdrg, h ags las ar m ad boh commods ar rly accumulad. By srg h rlav roduco la gv by (3) o (5) w g h rlav mark rc as a fuco of h rlav xcd rc ad h rlav rs uas. Th Torrs rc s obad by sg h ualy bw ad. Fluks aar, rcs xcaos ar o fulflld. Sc ualy ( ) ( ) 0 j j j holds ay cas, follows j. If h ags rc xcao for good s j grar ha s mark rc, hy ovrvalu ha good ad udrvalu h ohr good j. Formula (4) shows ha d s : h udrvalud commody s xcss dmad ad s calld scarc, whl h ovrvalud commody s xcss suly ad s calld surabuda. Wh h ags rc xcaos dffr from mark rcs, wo ys of dvdual dsulbra aar s of h clarg of boh marks. Th ral dsulbrum srgs from h ag s rol as a buyr: hs msak xcaos lad o hs owg a bask of commods dffr from h o h had schduld. Th moary dsulbrum srgs from hs rol as a sllr: hs xss dffr from hs moary rcs ad a moary balac aars. Th dvdual balac of roducr s osv f good s scarc, gav f o, ad h algbrac sum of h balacs s zro..3 Balacs slm ad ffcv roduco Th ags rac o hs dsulbra, f oly bcaus balacs slms cao b dlayd 4 Torrs (8) mhaszd h hyscal cosras mosd o h rroduco of caal wh h rofs of ach dusry ar rly accumulad h sam scor. 6

8 sc moy s cosdrd as a ur mas of xchag ad o a sor of valu. How do hy rac ad fulfl hr moary commms? A suoal rul s hr rurd, whch rsuls a slm of moary balacs ad os-mark chags h allocao of rsourcs. W ow dscrb a lausbl rul of ha y. Whou loss of graly assum for h mom ha commody s h o whch was udrvalud by h ags (wh good as umrar, ). Th ags osos ar asymmrcal: 5 h roducr of h surabuda good, whos balac a h closur of h mark s gav, cao rmburs h advacs mad by h bak. Th goods h ows mgh b szd by a ludaor. Th suoal rul w adm s h followg: h roducr wh gav moy balac obas h moy h ds o ay back h bak by rasfrrg som of hs ral asss o h ag wh osv moy balac. W assum ha h rasfrrd goods ar valuad a mark rcs ad ha h hyscal comoso of h rasfrrd bask s drmd by roducr, who s a oso o mos : h chooss ordr o maxms hs x ffcv roduco. L us drm h fal allocao of rsourcs wh ha rul s followd. Producr s sals amou o. Sc h roduco cos of vry u of good h x rod s a a, h uay ha hs rrur wll roduc by vsg h whol of hs rcs amous o a a Tha roduco s grar ha h o h had lad bfor h mark. Th xsc of a ga bw xcd ad ffcv uas s a faur commo o all dsulbrum modls. Th rory whch ca b scfcally assgd o h roduco of moy ad h abov balac slm rul s ha h ag wh a osv moary balac succds accumulag hs whol ffcv rofs. Afr h rasfr of h mas of roduco as dcdd by ag, h roducr of h (6) 5 A asymmrc rlaosh bw roducrs s also foud Marx s largd rroduco schms: h caalss of scor I (roduco of mas of roduco) dcd how much hy accumula, hos of scor II (roduco of cosumo goods) ada hr accumulao o h dcso ak by scor I (Marx, 885, Book II, cha. XXI). 7

9 surabuda good ows ( a ) us of good ad ( a ) of good. I gral, h rooro bw hs uas dos o f h chcal rurms of dusry, so ha som good wll o b rly usd as u. Th x ffcv roduco of good wll amou o a a m(, ) (7) a a L us o ha boh ags rcv from h mark mor of h surabuda good ha hy had lad. 6 Dos h mchasm w cosdr vola h rcl of voluary xchag (s Béassy, 986), whch xrsss ha o ag s oblgd o sll or buy mor ha h had lad a mark rcs? Th uso s rrlva h rs modl for wo rasos. Frs, as h ags las ar basd o xcd rcs whl h dogous mark rcs ar ally ukow, a comarso bw hr xcd ad ffcv osos s rrlva wh hs rcs dffr; ad, scod, h fal allocao rsuls from balac slms whch ar of a suoal aur. Th ag wh a gav moary balac has o da abou h uay of h surabuda good h wll b comlld o rasfr o h ohr ag. H cao kow f h bough oo much or oo ll of a good. Th vry oo of voluary xchag dos o f wh ha framwork..4 Normal ad ahologcal cass I h coomy w sudy, s xcd ha h scarc commody wll b rly usd as u, so ha w hav a a. By mullyg boh sds by ad comarg wh rlao (6), hr coms ualy: a a (8) Ths ualy lls us ha h valu of h surabuda commody usd h roduco of h scarc commody uals h valu of h lar usd h roduco of h formr. Is oworhy faur s ha boh commods lay a symmrc rol ha rlao: had good 6 A smlar homo holds gral sragc mark gams. Shaly ad Shubk (977,. 947) oly ocd ha " s a mar of lg o's somach rahr ha o's urs absorb h flucuaos". 8

10 b h scarc good sad of, w would hav obad h sam formula. For h rmadr of h aalyss, ad whavr h scarc ad h surabuda goods ar, w shall choos a arbrary good, say good, as h umrar for h rlav rc ( / ) ad h rlav uay ( / ). (Tha covo wll b usful h sudy of h dyamcs wh h sam good may bcom alravly scarc or surabuda.) Eualy (8) s h wr as: a a (9) Ths formula shows ha h rlav uay dds oly o h rlav mark rc. Th roduco las ( ) ad ( ), as drmd by formulas () ad (), allow us o calcula h x a xcss dmads as a fuco of. For xaml h x a xcss dmad of good s a ( ) a ( ), ad commody s h x a scarc commody f s xcss dmad s osv. Had h ags kow h mark rcs, h xcss dmad for commody would hav b obad by rlacg by h sam xrsso. Th x os scarc commody s h o xcss dmad a mark rcs: our modl, s ha commody whch s rly accumulad. Usually, h x a ad h x os scarc commods cocd, bu hy may dffr som ahologcal cass. Th ormal cas occurs wh ( T ) ad ( T ) hav h sam sg. I ca b show ha a suffc (hough o cssary) codo for ormaly s wr D 4 a a a a (0) whr D do h drma of h marx A of chcal coffcs. Hc, h ormal cas always occurs f h drma s osv or slghly gav. I h ahologcal cas, formula (9) dos o hold ad h sudy of h dyamcs s mor comlx. A vably roblm may v aar f h roducr of h x a scarc commody cao accumula all hs rofs bcaus h ohr good s shor suly ad, h, a modfcao of h suoal su assumd abov s cssary. For h rmadr of h aalyss, w shall assum ha codo (0) s m. 9

11 3 Dyamcs 3. Ral dyamcs of dsulbra W hav aalyzd h bhavour of h modl wh a rod, gv h jus comld roducos ad rc xcaos. Th modl drms h mark rcs ad, afr h balac slms, a w allocao of us bw h ags. A hyohss o h formao of xcaos allows us o mak h lk bw hs mark rcs ad h xcd rcs for h x rod, ad hrfor o df h dyamcs of h modl. Isrg (3) (5) gvs h rlav rc as a fuco of ad. To df h dyamcs w assum ha h xcd mark rc a som da s h ffcv rc a h rvous da:. (Ths sac xcaos hyohss s oly rad for s smlcy.) Sc w ar sudyg dyamcs, h m dx s ow wr dow xlcly, so ha hs rlao s wr as:,,. () Wh h rooro s ha of h Prro-Frobus row-gvcor ( *,) of marx A ad h rc xcaos ar h corrsodg colum-gvcor, all goods ar accumulad, h rc ad uay xcaos ar m ad h coomy follows a rgular growh ah a maxmum ra (vo Numa growh ra). Ohrws, h rooro vars from a rod o h x, hr ar balac slms ad som goods ar xcludd from accumulao. As goods whch ar scarc durg som rod ar surabuda durg ohr rods, rlao (9), whch holds ddly of h aur of h scarc good, lays a ssal rol h sudy of h dyamcs of h rlav uay. Th voluo of h ffcv rlav uay bw coscuv rods ca b sudd by cosdrg h lad rlav uay. O h o had, formula (3) xrsss h rlav uay lad a da as a fuco of h rvous roducd uay ad h xcd rlav rc, ual o h rvous mark rc. Tha mark rc also rsuls from h rvous uay by mas of ualy a a (accordg o rlao (9) ald a da ). O h whol, wh xlc m dcs, h rlao bw h lad uas a da ad hos 0

12 roducd a da s wr: a a a a f ( ) a a a a O h ohr had, formula (5) gvs h rlav rc (ad hrfor h rlav uay, as a cosuc of rlao (9)) as a fuco of h xcd rc ad h lad roduco. Wh sac xcaos, o obas: () a a a a g( ) (3) Th combao of formulas () ad (3) dfs h dyamcs of. Th xlc formula a a a a a a a a a a a a a a 3 3 s far from bg aracv, bu h coomc homa a sak bcom clarr wh o ks md h rad dcomoso: - h lad rlav uay crass wh h acual rlav uay a h rvous rod (formula ()); - h acual rlav uay crass wh h lad rlav uay f D 0, ad dcrass f D 0 (formula (3)). Th sg of D adms a coomc rrao: s osv f a / a s grar ha a / roduc, ad gav h ohr cas. (4) a,.. f ach dusry maks a rlav grar us of s ow Clarly ough, h uu saoary o * of h dyamcs (4) corrsods o h rooro of h gvcor vcor ( *,) as * s rsrvd by fucos f ad g. Morovr, sc fuco g s boudd by a / a ad a / a, h dyamcs dfd by (4) ar vr xlosv. Bu svral voluos ar ossbl: s show Adx ha h dyamcs dd o h sg of D: If D s osv, h sysm covrgs owards a vo Numa growh ah. If D s gav, hr ha covrgc s local (ad may b global), or hr xss a lm cycl of ordr wo, ddg o h rao bw h scod ad h frs (or doma) gvalu of marx A. Thaks o rlao (9), h dyamcs of h rlav rc ar bascally h sam as hos of h rlav uas. I arcular, cas of covrgc, h rlav rc ds owards h rgh

13 gvcor ( *,) of marx A,.. owards h rc of roduco as dfd by h Classcals. 3. Nomal dyamcs L us cosdr h ormal cas wh covrgc owards ulbrum. Accordg o rlao (9), h dyamcs of h rlav uay ad h rlav rc ar dcal. W look hr a absolu maguds, vz. h roducd uas of ach good ad h omal rcs. Rahr uxcdly, hr xss a asymmry bw h dyamcs of uas ad hos of rcs. Th rul adod for uas maks rfrc uao (7) o fuco m ad s o amabl o a sml aalycal sudy. By coras, h voluo of omal rcs s much mor rgular. By subsug h ags las () ad () uals (4), h sac xcaos hyohss () lads o h duco formulas:, a a a a ( a a ), (5) (whr, a a a a a a,, ad, (6), ). Th maguds o h lf-had sds of hs uals, rrs h flao facors bw das ad. I urs ou ha chags moary rcs ar xlad by ral facors (rvous rlav rc ad rvous rlav uay). By usg (9) o lma, h sam rlaoshs df h voluo of omal rcs by h duco formulas: a a a a,,, a, a, a, a,,,, a, a, a, a, A doublg of omal rcs a da 0 lads o h sam rooroal chag a ay da. I h cas of covrgc, h omal rcs a ulbrum ar also doubld. Tha s why, f o rs o fd drcly h log-rm saoary rcs from uaos (7), urs ou ha hs wo uaos rduc o o, whos soluo s h rlav rc * as gv by h Prro-Frobus gvcor of marx A. (7)

14 Cosdr ow h ffcs of a xogous o rooroal shock o xcd moary rcs, wh h coomy s assumd o b a ulbrum a h al da, all goods bg rly accumulad. As h rlav xcd rc s chagd, h ags dmads ar modfd: som good bcoms surabuda ad s o oally accumulad; moary mbalacs also aar, whch rur moary slms. Effcv roducos, as wll as hr rooro, ar modfd. Th coomy s hus submd o ral ad moary dsulbra. I arcular, h goods xcludd from accumulao do o corbu o coomc growh. Ths homa las all alog h rasoal dyamcs ul a w ulbrum s rachd h log ru. Th coomy h rcovrs s al vo Numa ra of growh (o log-ru ra ffc), bu h ral ffcs of h shor-rm shock rms of lvls of roduco ar rma: h w growh ah s lowr ha h rvous. Thr s a log-ru lvl ffc, wh o cach u. Cocluso W hav cosdrd a bscor coomy whch h caalss am a maxmsg accumulao hr ow scor. Th ags dcsos ar basd o xcaos ad ar cofrod wh h cosras rsulg from h avalably of us ad wh h ga bw xcaos ad achvms. Our modl aks moy o accou, moy bg cosdrd as a ur mas of xchag ssud by a bak a h ags russ (dogous moy). I s of mark clarg, h dscracs bw xcd ad ffcv mark rcs mafs hmslvs by boh ral (ags g budls dffr from hos hy had lad) ad moary (moary mbalacs aar) dsulbra. Th coomy hus movs alog a suc of morary dsulbra. Th sudy of h ral dyamcs shows ha, h ormal cas, h ah followd by h rlav roduco covrgs owards h ulbrum rooro or owards a cycl of ordr wo. Th bhavour of ha moary coomy dffrs from ha of a barr coomy obyg ohrws h sam ruls (B al. 0): v f ulbrum s h sam boh modls, rlav rcs dffr morary dsulbra as wll as h amous of goods ad hr allocaos bw ags. Th sam for h dyamcs, ad hrfor moy mars. Why dos moy, wh cosdrd as a ur mas of xchag, mar? Bcaus moary mbalacs mus b sld a ach rod. Byod h arbrarss of h cular rul w hav adod, h basc da s ha h rsc of a mas of xchag rurs h dfo of 3

15 suoal ruls rlav o s ssu ad o balac slms. Th h allocao of us bw ags ad h dyamcs ar affcd by h xsc of moary mbalacs. Tha aalyss of h rol of a ur mas of xchag corass for sac wh Kys s oso comarg, h hrs, a ral-xchag coomy o a moary coomy : accordg o Kys, moy, as log as s rgardd as a cov mas of ffcg xchags, s ural s ffc bcaus s us dos o affc h ral hgs (Kys, 933,. 408). Th dfo of suoal ruls rlav o h ssuac of moy ad o balac slms comls ad modfs h workg of a modl of a mark coomy. Ours s a am o formalz a bscor coomy ad may srv as a bass for a macrocoomc sudy srd by a Classcal aroach. I calls for varas cocrg lgal ruls ad for xsos: for sac facg moary balacs by rasfrrg scurs sad of ral caal, or alravly roducg crd ad rs ras. Adx : Dyamcs of h rlav uay Th sudy cocrs h dyamcs of h rooro / bw roducd goods. Formula () gvs h ooal rao as a fuco of h rvous ffcv rao ( f ( ) ) ad formula (3) gvs h ffcv rao as a fuco of h ooal rao ( g( )). Th dyamcs of h ffcv rao ar dfd by h comos fuco h g f gv by formula (4). Calculag h drvav shows ha f s crasg; as g s crasg f D 0 ad dcrasg f D 0, h sam for fuco h. L do h doma gvalu of marx A of chcal coffcs ad h ohr gvalu. W hav D, (accordg o h Prro-Frobus horm), ad h sg of D s ha of. By cosruco, * s h saoary soluo of () ad of (3): s h uu osv fxd o of fucos f, g ad h. ( *,) bg h doma row-gvcor of A, w hav uals a* a * ad a* a. Prooso If D 0, h saoary ulbrum s globally sabl. 4

16 Proof If *, fuco h( x) x, whch oly vashs a x *, s osv for x * ad gav for x *. As fuco h s crasg, * mls h( ) h( *) * ad h suc covrgs moooously owards *. Th sam f h al oso 0 s such ha 0 *. Wh D 0, as w ow assum, fuco f s crasg bu h raco of h mark as dscrbd by g s dcrasg, so ha h slf s dcrasg. Prooso For D 0 ad ( ) 0 * such ha:, hr xss a ghbourhood V, - ay rajcory sarg from V covrgs owards h ulbrum * (local sably); - ay rajcory sarg ousd V covrgs owards a cycl of ordr wo. 5 of Proof By usg h scfc rors of *, a sml calculao shows ha g ( *) D. From f ( x) x / g( x) o obas f( *). As h g f, w hav h( *) f ( g( *)) g( *) f ( *) g ( *). Th local sably codo h'( *) h rducs o h'( *), whch s m wh ( ) 0. I ha cas, sc h s dcrasg, h dyamcs osclla aroud *. L us cosdr h fuco k h h, whch s such ha k( ). Fuco k s crasg ad boudd ad may adm fxd os ohr ha *. Ths fxd os go ars: f s a fxd o of k grar ha *, h( ) s aohr fxd o smallr ha *. L 3, c. b h fxd os of k grar ha * 0. Th fxd os smallr ha * ar 3 wh h( ). Th os ar h zros of fuco k( ), * ad gav o *,,.. As k( *) h( *), h fuco k( ) s osv o, mor grally osv o, ad gav o

17 Fuco k sds V, closr o * ha o slf ad h sg of k( ) shows ha s, hc h covrgc rsul o V. For 0, boh ss, ad, ar rsrvd by k ad h sg of fuco k( ) shows ha h succssv rasforms of a o d owards blogg o ay of hs ss d owards h lm cycl. Thrfor h succssv rasforms by h of a o,. Prooso 3 If D < 0 ad ( ), ay rajcory whch dos o sar from * ds owards a cycl of ordr wo. (Th argums ar smlar o hos of h rvous roof.) Proosos ad 3 ar llusrad by h dagrams blow. * * 3 3 Adx : Th offr-for-sal modl Th offr for sal modl s h moary vrso of h ral modl xamd B al. (0). Th ags las ar dcal h offr-for-sal modl ad h all-for-sal modl. Formulas () ad () cocrg ooal uas, formulas (6) ad (7) dfg h roducd uas as fucos of ffcv rcs as wll as rlao (9) h ormal cas sll hold. Th ma dffrc ls h suly ad dmad fucos: d a j j, hc s a ad 6

18 d a s a d a s a hrfor h rlav rc s a a a a By srg () ad () (8) o obas a a a a a a Th dyamcs of h rlav rc follow from hyohss () o h formao of xcaos, from whch rlao (9) allows us o dduc h dyamcs of h rlav acvy lvl: w hav h( ) or, uvally, h( ), wh a h( ) a a a I coms as o surrs ha * s h uu fxd o of h. Bu fuco h s hr boudd or cssarly moooous: calculao shows ha h sg of s drvav s ha of xrsso a a ( D a a ) a a. Thrfor, h s dcrasg f D 0 bu s o cssarly crasg f D 0, so ha h dyamcs may b vry volvd. For a local sudy a ghbourhood of h ulbrum, o calculas h drvav of fuco h a *: h'( *) h h * ( *) * *(l ) *( ) h ( ) a* a a * a ( *) * D h ( ) Thrfor, h local dyamcs ar usabl f 0, oscllag ad sabl f 0, mooo ad sabl f. a a (8) 7

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