ALLOCATIVE EFFICIENCY OF A SPECULATIVE FUTURES MARKET WITH ZERO-PATIENCE ABSTRACT

Transcription

1 ALLOCATIVE EFFICIENCY OF A SPECULATIVE FUTURES MARKET WITH ZERO-PATIENCE Leanne J. USSHER, Queens College of he Ciy Universiy of New York Preliinary Draf. Do no Cie wihou Perission fro he Auhor ABSTRACT This aer eends he resuls of zero-inelligence (ZI) in double aucion arkes for goods o rade in fuures under an illiquid regie of seculaive raders. Desie his ore volaile seing of risk-neural seculaors, shor selling, argin rading, and ulile equilibria, he arificial fuures echange sill converges o he sable heoreical equilibriu. To confor wih he insiuional srucure of a fuures arke, here is no lii order book: he sandard ZI odel of consrained rando lii orders is relaced wih iaien or saisficing arke orders. This odificaion ainains he essence of he ZI rader and builds on he insigh of Gode and Sunder s (993) resuls ha oiizaion is unnecessary o bring abou allocaive efficiency. The addiion of scalers or arke akers creaes a ore liquid regie which increases rading aciviy and reduces he kurosis in rices. Keywords: Zero-Inelligence, Margins, Transacion Ta, Coninuous Double Aucion, Fuures Marke, Agen-based Model, Scalers. JEL Codes: C63, D44, D6 The auhor would like o hank Duncan K. Foley, Rob Aell and Salih Nefci for helful coens and suggesions. All isakes are y own. Address for corresondence: Leanne J. Ussher, Queens College, CUNY, Flushing, NY 367; e-ail: leanne.ussher@qc.cuny.edu.

2 2 INTRODUCTION Price discovery is riarily aribued o new inforaion and he coeiion beween oiizing raders as hey udae heir eecaions and sraegies. Marke icrosrucure suleens his elanaion, recognizing he rading echanis o also lay a crucial role in he equilibraion of suly and deand. A hird disinc ingredien in rice foraion, beyond huan cogniion and he rading echanis, is ha of liquidiy boh a he arke and individual rader level. This aer akes he las wo of hese caegories, wha Sih (982) labeled as insiuional srucure and environen, and considers how efficien he coninuous double aucion (CDA) is wih liquidiy consrained zero-inelligen seculaors. A aricular for of liquidiy is hen added o he arke hrough he addiion of arke akers. The noral rading regie has been defined by Acharya and Schaefer (2006) as one where liquidiy consrains are non-binding, raders are well caialized and fundaenals doinae rice oveens. In conras an illiquidiy regie is one where raders are close o breaching heir caial consrains such as an inernal value-a-risk consrain; a caial adequacy requireen iosed by regulaors; a collaeral consrain iosed by lenders; or a argin requireen iosed by an echange (Ibid). Illiquidiy liis boh arbirage and seculaion as rading becoes cosly and collaeral consrains becoe binding. This ay lead o he forced liquidaion of desired osiions and sudden rice oveens. In he wo good odel below, of cash and fuure conracs, agens are risk neural and seculae on fuure rice changes. Illiquidiy is characerized by he degree o which seculaors are leveraged and he enforceen of real-ie gross seleen (RTGS) which requires seculaors o ark o arke heir osiion afer every rade. Transacion coss of 0. ercen on one-way rades and a 25 ercen argin requireen add o he characerizaion of illiquidiy in he arke. Seculaors will go long or shor in heir derivaive osiion based on heir fied eecaion of fuure rices. Unlike os agen-based odels (ABMs) (see suary of ABMs by Tesfasion 2002) here is no evoluionary rocess in agen behavior, bu here is an evoluion wealh disribuion. In his anner his 2 good odel is closer o he general equilibriu ZI odels of Gode and Sunder (2004) where wealh is elicily odeled, as oosed o heir original arial equilibriu ZI odel of Gode and Sunder (993). Gode and Sunder show how he eleenal forces of wan and scarciy cause Pareo-efficien oucoes even when agens do no aiize and when no evoluionary rocess [in agen behavior] eis (2004, 2). By searaing arke rules fro huan cogniive behavior sudies have shown ha good arke erforance is no deenden on rader learning and raionaliy. This was reviously u forward by Becker (962) analyically wih a Walrasian soluion. Eerienal econoiss who sudied aucion echaniss found he CDA o be a design ha yielded high allocaive efficiency (Sih 962). Couer siulaions have roved very useful in disinguishing he conribuions o rice beween arke srucure and agen behavior. Sih (982) defines hree caegories ha deerine he erforance of a icro syse: he insiuional srucure (he rules ha govern echange), he environen (agens ases, risk rofile and endowens of inforaion and resources), and agen behavior or learning and rading sraegy.

3 3 In he ZI odels of goods or sock rading, he CDA is able o drive rice efficiency wih oucoes ha closely rack he redicions of he Walrasian coeiive equilibriu, wihou he individual oiizaion crieria usually assued in he invisible hand analogy 2. The ZI research rogra ais o aid arke design and rooe our undersanding of he effec arke rules have on efficiency, rice volailiy and liquidiy. Overall, he ZI raders are a ool o isolae and undersand he effec of arke rules on arke oucoes. Undersanding he effecs of arke rules and oher social insiuions is crucial because rules are observable and conrollable, while individual sraegies are inherenly rivae and no direcly conrollable. Theories based on he effec of arke rules are herefore easier o es. The ZI odel rovides a benchark of he srucural effec of arke rules. The radiional sraegic odel in which raders resond fully o changes in arke rules [and rice oucoes] is anoher benchark. The wo bencharks bracke he range in which huan behavior lies (Gode and Sunder 2004,.2). This aer eends he odel of he arke o he rading of fuures 3, where here is shor selling and argin rading, here is no lii book, and he resence of an echange akes real-ie gross seleen ossible. [ Marke disciline is defined by Gode and Sunder as he resricion ha raders are forbidden fro aking buys or sells a a loss because hen hey would no have been able o sele heir accouns] foonoe ha his is no insiued ye]. Adding o his arke srucure he aer focuses on he rading of risk-neural seculaors raher han fundaenal raders. While no relevanly differen fro Gode and Sunder s (993) heerogeneous buyers and sellers, hey do no have he failiar sooh indifference curves of Gode and Sunder s (2004) Edgeworh bo resenaion. By using risk-neural seculaors we hoe o no only generalize he resuls bu also add o he oenial for arke volailiy which will aid in he esing of allocaive efficiency of he CDA. I is ioran o idenify he siilariies beween his seculaive fuures odel and ha of Gode and Sunder s (993, 994, & 2004) so or sock arke, of consrained rando rading. The resuls here reinforce heir sae conclusion ha he CDA is efficien in converging o he Pareo or Walrasian equiliubriu wihou rader oiizaion and, in addiion, even when oenially desabilizing leveraged seculaors eis. The Gode and Sunder ZI consrained odel (993) has raders divided ino buyers and sellers who are given an equal alloen of shares o buy or sell, resecively. Each buyer (seller) has a given resale (cos) value reains fied and creaes he uer bound of heir budge consrain. All raders ainain heir desire o rade unil hey reach heir re-secified alloen of securiies, which is he sae for all raders. Their ariciaion is no guided by oiizaion - iniizaion coss (aiize of rofi) (which are calculaed a he end of he rading eriod). Raher, all raders offer siulaneously a rando bid (ask) ha ranges beween soe noinaed 2 Sandard general equilibriu heory requires srong assuions on agen oiizaion and inforaion o assure he eisence and sabiliy of an equilibriu as in he Arrow-Debreu (954) odel. 3 The ZI odel has been eended by various auhors: Cliff and Bruen (997) who criicized he rice convergence roeries and augened inial inelligence creaing ZI-Plus or ZIP agens. Brewer e al (2002) and Duffy (2006)., bu never for fuures

4 4 floor (ceiling) and heir budge consrain. Traders are seleced and heir bid (ask) for a single uni of he good is subied o he lii order book where hey are ranked, in accordance wih CDA rules, such ha he highes bid (lowes ask) is considered as he curren bid (curren ask). A rade occurs when he new bid equals or eceeds he curren ask (he new ask equals or is less han he curren bid). Following a ransacion he lii order book is cleared and a new round of bids and asks are solicied and he rocess is reeaed. In conras, he odel resened here siulaes liquidiy consrained seculaors who rade fuures conracs and have fied uniforly disribued eecaions on he fuure so rice. The zero-inelligence characerisics are hough o be reserved, bu he sock odel is now alicable o a derivae odel where roises raher han goods are raded for cash. In he siulaions below he fuures arke is for a non-sorable coodiy and he so arke is indeenden of he fuures rading, and is ignored in his rading eriod. All seculaors sar wih an equal endowen of cash o buy or sell fuures conracs 4. The seculaors risk-neuraliy and arked o arke wealh akes he decision buy, sell, or hold 5. A seculaor eecs o ake a rofi by buying low and selling high, relaive o heir so rice eecaion 6. Seculaors will ainain heir desire o buy or sell unil consrained by heir wealh, ransacion coss and argin requireen. As in all fuures arkes, all long lus shor osiions su o zero. Oen oucry, an oral CDA, is used o find he curren bid and ask, which is echnically equivalen o a lii order book of lengh. Seculaor ariciaion in he arke is guided by an iediacy or iaience o ransac using arke orders, raher han he Gode and Sunder rando lii orders ha are ke in he lii order book. I is argued here ha boh sraegies are equivalen, and ha boh oi inelligen oiizaion sraegies. As wih Gode and Sunder s ehasis on arke disciline, seculaors in his odel are consrained fro accuulaing losses by obeying argin calls seled in real-ie. Boh odels oulined above incororae zero inelligence and budge consrained raders in an ae o deerine wheher i is inelligence or arke disciline ha is os ioran for he rocess of rice discovery o he Pareo oiu. Gode and Sunder es 3 CDA arkes: huan consrained, ZI-consrained, and ZI-unconsrained; finding ha he firs wo arkes are alos idenical in heir allocaive efficiency, hereby concluding ha is he consrains wihin a CDA ha was resonsible for he allocaive efficiency of he arke and no inelligence. This aer coares he uli-laeral Walrasian equilibriu (suer inelligence) o he ZI-consrained CDA rice. While he raders here are ZI-C, hey are renaed as zero-aience (ZP) raders o disinguish heir rading wih arke orders raher han lii orders. 7 4 Gode and Sunders (993) only has one good wih no wealh resricions, and Gode and Sunder (2004) have wo goods: green and red chis. This odel has wo goods, cash and fuures conracs, bu he derivaive has a zero-su in aggregae. Transacion coss are a leakage fro he syse. 5 In Gode and Sunder, raders are seleced o be eiher buyers or sellers hence roducing an aggregae deand curve and an aggregae suly curve. Here, seculaors roduce jus an aggregae deand curve ha has boh osiive and negaive sides. This is echnically a oal deand curve, bu i looks siilar o he ecess deand curve of general equilibriu analysis. 6 Jus as he resale or cos valuaions were fied in Gode and Sunder (993), he eecaions of he underlying coodiy valuaion in he fuure so arke is fied, and i is iniially drawn fro a unifor disribuion. 7 Zerio-inelligence/aience (ZIP) raders ay have been an alernaive, bu his anachronis has already been used by Cliff and Bruen (997) for heir zero-inelligence-lus raders.

5 5 ZP seculaors desire iediacy, rading riarily wih arke orders when here is an eeced gain, or he need o ee real-ie argin requireens. ZP seculaors are saisficing raher han oiizing, and in his sense are void of learning and inelligence. The iaien rading behavior secified here for seculaive raders irrors he rando behavior of fundaenal raders in he Gode and Sunder (993) ZI sock arke odel. Boh are subjec o budge consrains and arke rules bu boh yically rade a rices ha do no oiize heir rofi. The robabiliy of lacing a arke order a a cerain quoed rice in his odel is siilar o having a lii order acceed in he Gode and Sunder odel due o he coon CDA archiecure in boh odels. ZP seculaors are randoly seleced o rade in rounds and will subi a arke order o buy a he curren ask (sell a he curren bid) if heir end-rice eecaion is higher (lower) han his value. Because eecaions are randoly disribued he robabiliy of rading a he curren bid or ask should be siilar o he robabiliy ha God and Sunder s agens rade following heir rando quoes. If he seculaor s end-rice eecaion is wihin he curren bid-ask sread hen hey will no ransac, bu insead beer one of hese quoes, relacing eiher he curren bid or ask wih heir own and hus narrowing he sread, as in Gode and Sunder. The deails of his odel will be elaboraed on in secion 3. The second ajor conribuion of his aer is ha i considers arke liquidiy no jus in ers of rading coss and individual rader liquidiy consrains, bu i builds on he heory ha liquidiy is rovided hrough lii orders and reoved wih arke orders (Schwarz 988) and eends he arke environen wih he rovision of arke akers. In he ne secion we describe he lieraure ha underlies and oivaes he odel. There is a descriion of he CDA on he floor of a fuures echange and a discussion on he definiion of liquidiy and how i deerines he bid-ask sread. Following his, secion 3 elains he odel and derives a ZP seculaor deand curve wih real-ie gross seleen (RTGS). Risk-neural seculaive raders who rade on argin creae he illiquidiy regie and iose arke disciline. ZPs are endowed wih a coon aoun of wealh (cash), he securiy is infiniely divisible, and here is ulile bidding rounds. ZPs are consanly rerading (buying and selling) and quoing bid and ask rices. The algorih used o siulae he CDA is aken fro Chan, e al (998) and odified for a fuures echange wih ransacion coss and argin rading. Seculaor eecaions are ke consan, iicking Gode and Sunder s (993) individual reserve rices reaining consan, and he rading order is randoized, which irrors he rando lii orders in Gode and Sunder. Secion 4 resens siulaions of he ZP arke odel wih and wihou scalers. We find ha he id-oin beween he bid and ask converges o he Walrasian equilibriu rice and he bid-ask sread narrows o he cos of ransacing. The rading aciviy of ZP seculaors increases wih he inroducion of arke akers and rice efficiency becoes ore resilien. In all cases he bid-ask sread narrows o he ransacion a when liquidiy is available. Adding a ee scalers increases he sandard deviaion of rices bu reduces kurosis. The resence of scalers increases he urbulence or rading aciviy along wih an increase in liquidiy (deh of arke. While ro-liquidiy olicies siulae urbulence in rices, hey sill reain close o he coeiive Walrasian equilibriu.

6 6 In general, desie he oenial for individual insabiliy aong he ZP seculaors and heir lack of oiizing behavior, he CDA fuures arke is roven o be a relaively efficien allocaor. This aer suors he view of Gode and Sunder ha individuals ay be iulsive or lack raionaliy and ye arkes can reain orderly and efficien, if he righ arke insiuions are in lace. 2. Oen oucry 2 FUTURES MARKET TRADING In an oen-oucry fuures arke, as described by Silber (984), all bids and offers us be announced ublicly o he i hrough he oucry of buy or sell orders. In aricular, no rearranged rades are eried on fuures echanges. Sric rioriy is ke, where he highes bid rice and he lowes offer ake recedence, and his is known as he inside sread. Lower bidders us kee silen when a higher bid is called ou, and higher offers are silenced when a lower offer is announced, alhough siulaneous offers and siulaneous bids a he sae rice can occur. To increase he robabiliy of eecuion, a rader can raise his bid or lower his offer, and hen oher raders us reain silen. This rule is designed o insure bes eecuion, in he sense ha sales occur a he highes bid rice and urchases occur a he lowes offering, and all bids or offers do no live longer han he oen needed o ake a ransacion. Scalers, also known as locals because of heir echange ebershi, are floor raders who rade on heir own accoun and have low ransacion coss and ore fleible argin requireens han seculaors. Like dealers, in bond or foreign echange arkes, scalers regularly quoe a bid rice a which o buy and an ask rice a which o sell, aking a arke and hereby offering o colee orders quickly, yically a a rice close o he las rice, for hose anious o rade. By insering his sread beween he buy and sell, he scaler hereby receives a rofi for roviding he service of iediacy, which is jus one diension of liquidiy. Scalers ay also rovide deh coensurae wih he quaniy hey are willing o buy or sell. While scalers yically rovide liquidiy, i is ioran o noe ha hey can also consue liquidiy when hey liquidae or offse osiions, by selling a he bid rice or buying a he ask rice. This reducion in liquidiy ay cause eorary insabiliy (Schwarz 988). An ordinary rader (nonscaler) can eiher ender his own ask or bid quoe ha coees wih he scaler, called a lii order, or acce he rice currenly quoed in he arke, called a arke order. When a arke arician acces he arke bid, he is said o hi he bid. When he acces he arke ask, he is said o lif he ask. The following eale highlighing he choices of a nonscaler who wans o buy conracs is aken fro Silber (984,. 940). A coercial hedger can insruc his broker (on he floor) o buy 50 conracs a he arke, in which case he broker lifs he asks of ohers in he i. Alernaively, he coercial hedger can ry o buy ore chealy by insrucing he floor broker o bid for 50 conracs a he revailing bid rice in he i. In he firs case, he arke order uses he iediae eecuion services rovided by he offerers in he i (fro scalers or whoever) consuing liquidiy. In he second case, he bid reresened by he floor broker can be used by ohers o sell ino, hereby roviding liquidiy.

7 7 This sudy firs considers how effecively a financial arke wih asynchronous rading oeraes wihou scalers. Ofen he isach beween buyers and sellers ha yically eiss a any given insan is resolved by soe agens who are willing o lay he role of arke aker and rovide liquidiy. 2.3 Wha is Liquidiy? Liquidiy is defined in any differen ways. A arke is coonly hough of as erfecly liquid if rades can be eecued wih no cos (O Hara 997; Engle and Lange 997). The acadeic lieraure on arke icrosrucure recognizes ha he arrival of rando raders o buy or sell is asynchronous, and arke aciviies are eorally discree. Rarely is here a single rice and research ino liquidiy in a CDA yically focus on he bid-ask sread. A narrower sread eans a ore liquid arke. Modeling he sread is an ereely cole aer, given ha he arkes are coosed of nuerous lii raders (which include dealers and ordinary raders) ebedded in a dynaic, ineracive environen. Such a syse ay bes be odeled wih an agen-based ehodology. The analyical bid-ask sread lieraure (Soll 978; Ho and Soll 98) elains he deand for iediacy fro he asynchronous arrival of rando raders o buy or sell. I is ofen assued ha dealers ariciae in every rade, known as a quoe-driven arke. The behavior of he arke aker or dealer is yically described as a rader who insers a sread beween he buy and sell and hereby receives a rofi for roviding he service of iediacy in wha igh oherwise be a fragened arke. This view of he arke aker, as a rovider of redicable iediacy, was firs foralized by Desez (968) and hen elaboraed on by Garan (976) and any ohers. 8 I is generally acceed ha he bid-ask sread is reresenaive of he risks faced by he dealer as a resul of invenory conrol and asyeric inforaion. When scalers fill arke orders, hey rofi fro iaien raders bu lose o raders ore infored. I is usually concluded ha wih coeiion, he sread is reduced o he dealer s rading coss. This heory has foralized he idea of dealers as he roviders of liquidiy and conrollers of he size of he sread. The size of he reiu charged by iediacy roviders o cover hese eeced coss deerines he size of he sread and hereby he een of illiquidiy in he arke. Invenory conrol coss are assued o be reasonably consan over ie, while risks of asyeric inforaion are no (Engle and Lange 997,. 4). Since his aer ecludes inforaion hen by his arguen suggess ha he bid-ask sread would reain consan. The bid-ask sread easure of liquidiy has gained oulariy (see Fleing 2003), alhough any oher definiions have long been debaed. 9 Schwarz (988) argues ha oo uch ehasis has been ade of arke akers and heir sread. More aenion should be aid o he anner in which ordinary raders suly iediacy o each oher and coee o reduce arke sreads wih he scalers (Cohen e al. 979,. 84). Any rader ha akes arke orders should be seen as reoving liquidiy and hose ha offer lii orders can be characerized as roviding 8 See Soll (985) and Schwarz (988) for furher discussion and references on alernaive views of dealers and scalers. 9 See Bernsein (987), Black (986), and Harris (2003) for alernaive descriions and easures of liquidiy.

8 8 liquidiy. While arke akers ay be needed in illiquid arkes, hey are no a necessiy for liquidiy (Schwarz 988). Schwarz ehasizes he resiliency diension of liquidiy, raher han coss of ransacing. Resiliency refers o how quickly rices rever o forer levels afer hey change in resonse o large order flow ibalances (Bernsein 987, and Harris 2003, ). For arke akers o sabilize a arke, hey us coi caial or invenory risk, and his ay becoe subsanial. Schwarz warns us ha arke akers injecing liquidiy ino a syse o sabilize rices igh also be jus as quickly wihdrawn a a laer dae if shorages are incurred or if he arke akers seek o rebalance heir orfolios. This issue of sclaers balancing heir invenory osiions is direcly incororaed ino he odel below. 3 A MODEL FUTURES MARKET WITH SPECULATORS, SCALPERS AND HEDGERS 3. Trader Poulaion This fuures arke odel has u o 3 differen raders: seculaors, scalers and hedgers; all conribue o he arke icrosrucure and rice foraion in differen ways. In all cases rading behavior focuses on quaniy and wealh consrains raher han sraegies ha oiize caial gains. We resen a odel of a fuures rading i wih oen-oucry and a coninuous double aucion rading echanis. The odel has wo arkes: a seculaive fuures arke for an underlying non-sorable coodiy (his allows us o ignore he arke) and a residual oney arke. The rice of oney is noralized o, one-way ransacion coss and argin requireens, boh as a ercenage of each rade or osiion, are iosed by he echange on all raders ece locals (scalers). There is no resricion on shor selling, he fuures conrac size is erfecly divisible, and rices are always non-negaive. Near o real ie gross seleen RTGS is aroached as all raders ry o sele wih each oher before each rade akes lace hrough variaion argins. 0 Each ye of rader has heir own quaniy consrains and rules for rading. Zero-aien (ZP) seculaors will rade wih arke orders before considering he laceen of a lii order. All seculaors are risk neural and differ only in heir eecaion of wha he fuures rice should be and heir cuulaive wealh osiions. Eecaions of he ne-eriod 0 Insead of using he close-of-day seleen rice o calculae argin calls, seleen is adjused coninuously hroughou he day and he seleen rice used o calculae argin calls is he average of he bid and ask rice, or id-rice. This eans ha rofis and losses ransfer hands via he echange, beween raders coninuously, reoving he risk of accuulaed losses and rader defaul. The aoun aid is known as he variaion argin. As a silificaion, iniial argin and ainenance argin are considered he sae and he argin requireen is secified as a fied ercenage of he conrac value raher han an absolue dollar value er conrac. By using a argin requireen ha changes wih he ercenage change in rices, we ge closer o he essence of wha he echange considers in seing he argin.

9 9 fuures-conrac rice say consan during he rading eriod. Being risk neural, seculaors yically end u a he corner soluions of heir budge consrain, rading on argin and aiizing heir fuures osiion (long or shor) a every chance hey ge o rade. Scalers are ebers of he echange and oerae on he floor of he echange wihou aying a rading fee. They do no have an oinion on he fundaenal rice and insead ry o buy as low and sell as high as hey can. They wan o aiize he urnover of buys and sells while iniizing heir invenory holding. Scalers refer o lace lii orders (quoes) and o buy a heir bid quoe and sell a heir ask quoe. Scaler aciviy assiss in balancing order flow over he long run, which does rooe rice efficiency, bu i could creae rice insabiliy in he bilaeral CDA, when hey offer liquidiy o sar raders and are hen forced o liquidae heir own invenory holdings wih arke orders. Hedgers lay a liied bu ioran role in seing u he fundaenal deand and suly of conracs in he arke. There are only wo reresenaive hedgers one going long (o cover he eeced urchase of he underlying coodiy in he fuure) and he oher going shor (o cover he eeced sale of he underlying coodiy in he fuure). The quaniy desired is fied above or below a cerain reserve rice regardless of heir wealh osiion. The difference beween he long and shor hedge is he ne hedge, or he ne desired conrac osiion of he hedgers. Hedgers only lace arke orders unil hey fill heir eogenous desired conrac osiion. Once heir fuures osiion is aained, hey so rading. Since he su of all fuures conracs su o zero, he ne hedge will deerine he long-run ne osiion of he reaining rader oulaion. The ne hedge is used here o couner he Cliff and Bruen (997) criicis of he Gode and Sunder (993) ZI odel. They relicaed he ZI odel and found ha he ean rading rice for he ZI-C raders was only close o he o he heoreical equilibriu rice when suly and deand curves were syeric. In general, he rading rice for he ZI-C raders aroached he eeced rice E(P) fro he robabiliy densiy (PDF) funcion given by he inersecion of sellers offer-rice PDF and he buyers bid-rice PDF (Ibid 997,.9). However for a coarison o a Walrasian rice he econoy needs o be closed and wealh needs o be secified. Once he ZI odel becae a wo good odel, as in Gode and Sunder 2004 and his aer, a resul based on eeced rice and robabiliies no-longer holds. To show his, we have siulaed 2 of he 6 arkes below wih a osiive ne hedge of 5000 conracs. Cliff and Bruen (997), by considering only reservaion rices (as resened in he original Gode and Sunder (993) odel), showed ha soe inelligence was required and hey creaed ZI-lus raders which incororaed an adaive behaviour based on he behavior of oher raders. However, his is an unnecessary addiion of inelligence. Raher, iosing arke disciline in a general equilibriu cone is enough o bring abou a convergence o he heoreical Walrasian equilibriu rice. This was shown by Gode and Sunder (2004) and i is suored in he resuls below. Keynes argued ha he ne hedge is yically negaive and jusifies he backwardaion of fuures rices.

10 0 3.2 Seculaor s Deand Funcion In our odel wih leveraged seculaion, k reresens he lii on how uch larger a seculaor s fuures osiion rice ulilied by he nuber of conracs ( ) can be han a rader s wealh. All siulaions in his aer use k = 4, which eans ha a rader can have u o 4 ies his wealh dedicaed o a long or shor fuures osiion. In oher words he argin requireen is 25%, /k = The collaeral ke in he argin accoun by seculaor i is held as Treasury bills or oney 2 i, reresened here as. Collaeral held us be greaer han he i i argin requireen, / κ for he curren fuures osiion, a all ies (o he een ha rading allows). There will be several ransacion rices hrough ou he day which reresen a rade a eiher a quoed bid or a quoed ask. If here is no enough collaeral in he b argin accoun o ee he argin requireen hen seculaor i will have o liquidae heir osiion wih an offse urchase or sale a heir ne urn o rade. The fuures osiion a rice is aken on by he seculaor as a conrac a ie o sell or buy unis of he underlying coodiy a rice on he so or auriy dae of he fuures conrac. Since he seculaor does no inend on aking delivery on his conrac, he urose of holding his osiion is o fli he osiion and rofi on rice changes. On he basis of a consan rice eecaions i,θ abou he ne ransacion rice +, seculaor i will decide o go eiher long or shor in fuures. Transacion coss are incurred for each one-way rade as a ercenage of he rade value. In he siulaions below his fee 0. ercen and is reresened by v = If he eeced shor-er gain does no coensae he cos of rading over he ne eriod: ( θ - ) v ( - - ) hen he seculaor will hold his curren osiion insead of rading. The rader is yoic and on oening a osiion here is no consideraion of coss incurred for reversing he osiion. Each seculaor i is risk neural and sily aiizes eeced wealh over he eriod o +: a π i, e + = i, θ i i ( ) + 2 There is no ooruniy cos in holding cash since i would receive an ineres on he T-bills, however ineres in his version of our odel is zero.

11 The seculaor s deand curve is derived in Aendi via linear rograing. In suary, seculaor i s deand for fuures in each eriod is a slighly silified version fro Ussher (2004): i i i, θ ( ;,,,, κ,,) i where: = Inra day fuures arke ransacion rice a ie, i - = Previous conrac osiion, ϖ i - = Previous cash osiion in argin accoun following las ransacion, i,θ = Price eecaion θ of he ne fuures rice +, /k = Margin requireen as a ercenage of fuures osiion value, and v = Percenage ransacion a on a one-way rade (aid each way). A fuures deand curve is usually reresened as a sooh downward sloing line fro he o of quadran II o he boo of quadran I in he wo diensional R 2 sace in Figure. This odel roduces a non-linear deand funcion wih inheren corner soluions fro he risk neural seculaors wealh consrains and he regulaory seing of argin liis, ransacion coss and RTGS. Each risk-neural seculaor aiizes he ne eriod s eeced wealh by holding oney as collaeral and buying or selling (going long or shor in) fuures. The decision o buy or sell fuures deends on wheher he seculaor eecs rices o rise or fall, resecively. There is no resricion or disincenive o shor selling (i.e., selling coodiies ha one doesn own). A θ rader will rade only when rice eecaions are far enough away fro acual rices o ay for he one-way ransacion coss. Figure has a zero conrac osiion held over fro las eriod. If a seculaor currenly has a fuures osiion, hen argin calls can lead o forced liquidaion of he osiion when rices ove agains eecaions. The ossibiliy of a backward bending deand funcion, as in Figure 2, is a resul of he collaeral, which underlies deand for, being riced in he sae arke.

12 2 FIGURE A seculaor s deand for fuures as a funcion of, θ wih a as zero osiion -, and rice eecaions of The seculaor will sell (buy) fuures if he eecs he rice o fall (rise) when he sloe of he deand funcion is osiive. The deand funcion has a negaive sloe when urchasing ower is declining fro higher fuures rices or when collaeral is devalued and he seculaor us liquidae ar of his osiion o ainain he argin requireen. A each, he variaion argin is calculaed and ne wealh is adjused. The id-rice is he average of he bid quoe b and ask quoe a : a = ( + b ) / 2 The rofi or loss is calculaed wih rice changes of he id-rice and aid fro he losing agen o he winning agen via he echange clearing house, equivalen o ( ) i Each seculaor esiaes his ne wealh a each given rices ( a, b, ) which deerines heir decision on how any fuures conracs o buy or sell o aiize eeced wealh, while a he sae ie eeing his argin requireen a osiion,, ha is less no ore han ne wealh ulilied by k. The id-rice is used in accouning for ne wealh every eriod, as long as a osiion is held. 3 3 Afer he iniial urchase of a arke order he rader us ay a variaion argin of i i ( )( ). Ioran in his calculaion of variaion argin is ha we kee he disincion beween hose ha rofi by buying a he bid or selling a he ask, versus hose who are considered iaien and sell a he i i ask or buy a he bid. When a conrac is bough and ( ) > 0, if i is bough a he bid wih a lii order hen he variaion argin is osiive ( ) > 0. If however i is bough a he ask wih a arke order hen he variaion argin is negaive ( ) < 0. This resuls in a ransfer of wealh fro he rader who is willing o ay for iediacy o he rader who ges aid for roviding liquidiy and aking he arke. The aiizaion

13 3 Seculaor is shor fuures - = -60 Seculaor is long fuures - = +60 FIGURE 2 A seculaor s deand curve wih eiher a shor or long saring osiion: - = 5000, θ = 50, and k = 2 for each grah 3.3 The Bidding and Trading Process Wihin he CDA, seculaors and scalers (if included) are seleced randoly for a sequence of bilaeral rading wih non-relaceen in each round, so ha each rader has an equal chance of rading and rades every round. The hedgers are laced las in his sequence, which reresens one round. The inraday eriod of fuures rading has several rounds of quoing or ransacing, a he bid or ask rice. Quaniies raded and heir ransacion rices are regisered a each ie. Cenral o he rading rocess is he aucion ha siulaes he oen-oucry on he floor of an echange, leading o ransacions and hus ransacion rices. I is a groing echanis where boh bid and ask rices adjus and where ou-of-equilibriu rades ake lace when an agen agrees o sell conracs (hi he bid) o anoher agen who is bidding for he, or when anoher agen decides o buy conracs (lif he ask) fro he agen who is asking for he. This rocess of quoing and rading is reeaed any ies, giving each arke arician he chance o quoe and rade several ies and fill his orders. No new inforaion is brough ino his rocess; eecaions reain consan. The coeiive bidding algorih resened here for he ZP seculaors is drawn fro several sources. The anner in which seculaors coee and how heir rice eecaions inerac wih he bid-ask sread during he bidding rocess coes fro Chan e al. (998) and Yang (2002). An ioran odificaion o heir odel, aar fro keeing eecaions consan, is he resence of risk-neural seculaors wih collaeral consrains and ransacion coss. Hedgers ac siilarly o seculaors bu only lace arke orders o fill eecaions and do have consraining argin requireens. Hence, hedgers do no coee in he bid-ask sread. Anoher algorih, derived fro Silber (984), is resened for our scalers, ehasizing invenory conrol and noncoeiive behavior. θ of eeced wealh by he seculaor only akes ino accoun he eeced change in he rade rice ( ) wihou aniciaing wheher he ransacion is by arke order or lii order.

14 4 This asynchronous bilaeral bidding rocess allows wo or hree raders o ariciae a any one ie: offering, or beering, lii order quoaions or carrying ou arke order rades. Agens ake urns in enering ino he iner-dealer arke o quoe rice and quaniy, o ransac, or o ei. A round is coleed when all agens have ariciaed once, wih he hedgers coing las. This is reeaed for a differen rando sequence of scalers and seculaors for ore han 50 rounds. The reeiion or rading rounds reresens coeiion wihin he rice echanis and hels he convergence o equilibriu of arke deand and suly. This bidding and bilaeral rading rocess is deailed ne. 3.4 Aucion Algorih for a ZP Seculaor i,θ ZP seculaor i s reserve rice is his eeced rice,, lus he one-way ransacion a v. Half of he bid-ask sread is ofen hough of as a easure of he cos of eecuing a arke order (he difference beween he id-oin rice and he ayen rice). We shall reresen his rice difference by he lowercase leer s. The size of his half sread is acually endogenous o he bilaeral rading rocess. A ies when here is no bid or ask, a seculaor will announce his own noncoeiive lii order and increase he half sread on he basis of eecaions ( S v) i,q. In his case, S is a ercenage of he ransacion fee. If S is greaer han 00%, hen he new lii order will guaranee ha a new hi or bid occurs wih a deand differen fro zero. We resen he rading algorih wih hree raders: iner-dealers k and j (which could be a seculaor or scaler) and a new ZP seculaor, enran i. In his resenaion, agen i reresens a seculaor who deands iediacy and will always refer o rade wih a arke order raher han a lii order if ossible. ZP seculaor i eners he arke and winesses he curren bid:ask a b ( : ) sread and akes a rade choice under he following four scenarios. Scenario. (Figure 3a) The ask, offers, a ie. j a, and bid, k b,, currenly eis wih non-zero i, θ j, a. If > seculaor i will os a arke order and buy a his ask rice lif he ask quoe. i, θ k, b 2. If <, seculaor i will os a arke order and sell a his bid rice hi he bid quoe. k, b i, θ j, a k 3. If and (, b j, < a + )/ 2, seculaor i will os a sell lii order a a rice of (+ S v) i,q and hus quoe his own ask, relacing agen j k, b i, θ j, a k 4. If and (, b j, a + )/ 2, seculaor i will os a buy lii order a a rice of (+ S v) i,q, and hus quoe his own bid, relacing agen k

15 5 i,q Ouside Marke New enran Seculaive rader i deciding wha o do i will os arke order o buy - lif ask quoe i will os own lii order Bid ( i,q - s) i will os own lii order Ask ( i,q + s) i will os arke order o sell - hi bid quoe j ask j,a idoin k bid k,b Inside Quoes a : j, a k, b = 2 s Inside Marke Traders k and raders j, yically arke akers, wih bes buy and bes sell quoed. FIGURE 3a Scenario, in which boh coeiive quoes bid and ask eis in he arkelace rior o new enran Scenario 2. (Figure 3b) Only he bes ask, k long is zero as ( k ) 0. j a,, eiss; ha is, a k b, deand o go. If, i, θ > j, a seculaor i will os a arke order, buy a his ask rice. 2. If i, θ j, a b, seculaor i will os a buy lii order i, a a rice of (- S v) i i > i,q, bu only if ecess deand a his rice is ( ) 0 Scenario 3. (Figure 3b) Only he bes bid, j shor is zero as ( ) 0 j k b, eiss; ha is, a j a, deand o go. If i, θ < k, b, seculaor i will os a arke order and sell a his bid rice; 2. If i, θ k, b i a, seculaor i will os a sell lii order, a a rice of (+ S v) i i < i,q, bu only if ecess deand a his rice is ( ) 0

16 6 i,q i,q i will os arke order o buy - lif ask quoe j ask a i will os lii order Ask (+ Sv) i,q j ask a i will os lii order Bid (- Sv) i,q k buy b bu k = 0 bu j =0 k buy b Inside Quoes i will os arke order o sell - hi bid quoe Inside Quoes Scenario 2 Scenario 3 FIGURE 3b Scenario 2 in which an ask bu no bid eiss rior o new enran and Scenario 3, in which a bid bu no ask eiss rior o new enran Scenario 4. If no bid or ask effecively eiss; ha is a he ask quoe j b, and a he bid quoe, k, ( k ) 0 ( ) 0 j k j a,,. The new enran seculaor will os boh a buy and a sell lii order a (- S v) i,q and/or (+ S v) i,q resecively, as long as his bid is quoed for a buy of greaer-han-zero conracs, and he ask is o sell greaer-han-zero conracs. If his is no he case hen he curren bid-ask reains, even hough boh raders have zero deand, and enran i eis o join he queue o rade again laer. In his odel, under Scenario 2 (Scenario 3) he seculaor endering he bes bid (ask) igh have had rices ove agains hi; for eale, if he is long (shor) and rices fell (rose). They ay reain offering a bid (ask) rice o buy (sell), bu a a quaniy of zero. Now he wans o offse his osiion and sell (buy) so ha ecess deand is less (greaer) han zero. k k, b Scenario 2: ( ] k ) 0 [ where j j, a Scenario 3: ( ] ) 0 j [ where k is a funcion of j is a funcion of k b, j a, Effecively under Scenario 2 (Scenario 3) agen k (agen j) falls silen and will evenually be i, θ j, a i, θ k, b relaced by a new enran, as long as he new enran has < (has > ) and as i i, θ i i i, θ long as [ ( + Sϖ ) ] > 0 (as long as [ ( ) i Sϖ ] < 0 ) oherwise agen k (agen j) will reain. Only when agen k (agen j) is relaced and eis he arke will he be given he chance o saisfy argin requireens by liquidaing heir osiion wih a arke order, in urn, in he rando rading round.

17 7 This odel considerably changes he Chan e al (Ibid) rules, which ehasize he anner in which rice foraion feeds back ino he arke by agens udaing heir eecaions, o one where rice foraion feeds back ino he arke via quaniy consrains, argin requireens, and invenory conrol. The odel allows for leveraged rading and shor selling and akes he ehod of seleen a cenral variable of he odel. 3.5 Aucion Algorih for a Scaler Scalers will ry o charge as high a rice as ossible when selling and as low a rice as ossible when buying, while sill coeing wih oher raders o ake a sale or urchase. Only he highes bid and lowes ask are heard in he rading i. All oher noncoeiive quoes us reain silen. Since seculaors us coee on rice; only seculaors willing o narrow he inside-arke bid-ask sread can quoe such lii orders. Scalers balance arke order flow by using he iner-dealer arke o offse heir own invenory ecesses. Taking a loss in order o liquidae an unbalanced invenory osiion ay force oher iner-dealer scalers o also liquidae, and his dries u liquidiy in he arke unil rices are odified o equae deand and suly. The scaler algorih we use is a silified version of one saed in Sid (985). The objecive is o buy low a he bid and sell high a he ask, aiizing a rofi equal o he urnaround of invenory ulilied by he sread. A he sae ie, he scaler will iniize invenory risk wih a very sile invenory conrol echanis. There is a aiu ne invenory ceiling K for each scaler. Neing ou he long and shor rades by a single agen consolidaes he invenory. Scaler invenory is ke below a osiion lii K: n K K for scaler n In acual arkes K aybe as sall as one conrac and could be differen for differen scalers. In his odel all scalers have K=0. When a scaler eners he rading floor fro he rando sequence, if his invenory is less han his aiu lii K, he always has he righ o relace any agen in he iner-dealer arke by sily aching he agen s quoed bid and ask. This is in conras o seculaors who us offer a beer rice o relace he agens in he iner-dealer arke. If, however, he scaler s invenory is on his lii, hen he scaler will lace a arke order o offload all invenory, if ossible. The scaler algorih is one of sile invenory conrol: New enran scaler n: n. If K < < K, relace curren arke akers and quoe boh bid and ask a he curren b quoaions, a and,. k j 2. If shor and K hi he arke bid for a aiu n 3. If long and n K lif he arke ask for a aiu n n, and os no quoes, and os no quoes

18 8 The dealer invenory conrol odel oulined here, where a scaler will choose o ake a arke order raher han change his lii order rices, is in conras o he ore coonly acceed invenory conrol odels such as Garan (976) and Aihud and Mendelson (980). These auhors resen dealers as changing heir bid and ask quoaions o induce an ibalance of incoing orders, in order o reduce invenory. Hasbrouck (2003) quesions his laer odel and clais ha as a general rule, os eirical analyses of invenory conrol refue his ehod of changing he quoe for invenory conrol. He argues ha a dealer who would ursue his rice adjusen echanis would be signaling o he world a large his desire o buy or sell. This would u hi a a coeiive disadvanage (Ibid 2003,. 78). This silified echanis does no ouch on inforaion signaling, ye i does avoid his secific criicis. 3.6 Aucion Algorih for a Reresenaive Hedger Hedgers are only concerned abou filling heir eeced sales or urchases a he so dae via arke orders in fuures. They always coe las in each round of he rando sequence of seculaors and scalers. Hedger Scenario:. The fuure urchaser of he coodiy a so, agen q, will lif he ask, q aiu ask quoe quaniy, in each round unil arke buy order is filled = 2. The fuures seller of he coodiy a so, agen r, will hi he bid, r aiu bid quoe quaniy in each round unil arke sell order is filled = q* k b, r* j a,, for, for he Since seculaors and scalers do no usually offer large size lii order conrac los, i ay ake several rounds for each hedger o finalize heir urchases or sales. The hedgers conribue so called fundaenals o he seculaive arke. 3.7 The Trading Sequence In he siulaions which follow, rading begins wih a rando ordering of 20 seculaive agens all wih $0,000 cash and, when included, 0 scalers wih zero cash. The wo reresenaive hedgers coe las in his sequence, which, once coleed, is called a rading round. Seculaors have equal endowens and he heerogeneous eecaions are aken fro a unifor disribuion q ~U(00,30). Seculaors coe ogeher, along wih hedgers and scalers, for a lengh of 500 bilaeral rades and rices. The quaniy raded is he lesser of he arke and lii order deands ha are crossed in he CDA. Two randoly seleced raders (seculaor or scaler) begin wih iniial rando arke quoes, b a se 0 oins aar, e.g 0 = 20 : 0 = 30. A new enran, randoly seleced fro he reaining raders (bu no a hedger), eners he floor o eiher acce or beer he rices quoed.

19 9 If a bid or ask is acceed, a rade is done and a ransacion rice occurs for he arke order by he new enran. If, insead, he enran relaces a bid or ask or boh, hen a new se of b a quoaions : (bid:ask) is creaed, wih no ransacion rice. A sequence of quoes, and ransacion rices, is generaed as each agen eners he arke during he rading round, wih only ransacion rices and volues regisered. Reeaing he round, drawing a new rando sequence of seculaors and scalers each ie, creaes an iner-day rading session. This rading sequence is suarized here:. Seculaors are iniialized wih iniial wealh and rando rice eecaions. Two b a randoly seleced seculaors or scalers begin wih iniial quoes of 0 : 0 and heir resecive buy and sell quaniies (which ay be zero), given heir eecaions. 2. The rando sequence of seculaors and scalers o ener he arke wih nonrelaceen is deerined, wih hedgers coing las. 3. Wih one or wo agens quoing a bid-ask sread, he new enran can eiher subi a new bid or ask, acce he eising bid or ask, or hold (ass). 4. A ransacion occurs when he eising bid or ask orders are acceed and he ransacion rice is recorded accordingly. The ransacion is he iniu of he quaniies roosed for echange by each bilaeral rader. 5. A each oin, id-oin rices are used o calculae seculaor budge consrains in real ie. On he basis of he as ransacion rice, each agen s wealh is udaed, aking accoun of all argin calls (rofis and losses). 6. Ses 3 hrough 5 are reeaed for n ies, n = nuber of raders (one round). 7. Ses 2 hrough 6 are reeaed for N ies, N = nuber of rounds. 8. Final arke rice is recorded as he 500h ransacion rice for his rading session.

20 20 4 SIMULATIONS CDA TRADING Si arkes are siulaed: he firs 3 are wihou scalers (arkes, 2 and 3). All arkes have 20 ZP seculaive agens saring wih $0,000 cash, and wo reresenaive hedgers. In 4 arkes he e ane ne hedge is se o zero, and in he oher 2 arkes i is se o 5000 which eans ha he hedgers will be 5000 fuures long by he ie hey so rading. Trading begins wih a rando ordering of seculaors (and scalers if included) in each rading round, he hedgers ener wih arke orders a he end of his round. Each round has an unsecified nuber of quoes and coninues unil bilaeral rades and ransacion rices are roduced a which ie rading ends. In all arkes is se o 500, and here are anywhere fro 000 o 2000 rading rounds. The quaniy raded is he lesser of he arke and lii orders ha are crossed (hi or lifed) in he CDA. The 20 ZP seculaors have eecaions aken fro a unifor disribuion q ~U(00,30) in arkes and 4, and a binoial disribuion in arkes 2, 3, 5 and 6, q ~{U(00,0),U(20,30)}. The frequency disribuions of hese wo ZP seculaor eecaion ses are resened in Figure 4. By coaring wo differen ses of eecaion disersion we hoe o see wha iac his ay have on he bid-ask sread. frequency Marke & 4: Mean q frequency Mean FIGURE 4 Two differen disribuions of he ZP seculaor eecaions. q The aggregae deand curves and ransacion and equilibriu rice series for a single run, and he average of several runs and heir sandard deviaion for several differen ordering sequences, of he sae raders and heir sae eecaions, ino he CDA are resened in figures 5 and 6. Markes, 2 and 3 are in he resecive coluns of Table, and have no scalers. Where as Markes 4, 5 and 6 are in he resecive coluns of Table 2, and have 0 scalers included in he rading wih an invenory caaciy of K=±0. The las wo coluns of each figure has a binoial disribuion of eecaions and he very las colun also has a ne hedge of 5000 fuures conracs. The order of ZP enry ino he arke akes quie a big difference o he rice ah. The Walrasian equiliubriu, is defined as he dark (blue) flaes line in he ables in row 3. Where he Walrasian equilibriu does ove abou, i is alos always due o he resence of ulile equilibriu, and he juing fro one equilibriu o he oher.

21 2 Coninuing rice volailiy in hese arkes is due liquidiy consrains and i is significanly differen fro volailiy ha coes eogenous inforaion shocks or adaive eecaions. There are considerable rice and quaniy feedbacks. While for an individual high-risk seculaive rader, arking o arke is a cauionary ac and reduces counerary risk, i can also resul in a volaile arke rice which ay be greaer he higher he seleen frequency is (Farer e al. 2004). If raders are on heir budge consrains, hen hey will liquidae soe of heir osiion when rices ove agains he in order o say wihin heir argin requireens. This creaes backward-bending deand funcions, as inroduced in Secion 3.2 and can lead o sikes or large shifs in he equilibriu rice series. Firs we consider he visual rice convergence of he CDA rades o a heoreical Walrasian equilibriu rice for 6 differen arkes. The unifor disribuions of reservaion rices for he ZP seculaors led o a unique equilibriu which was sable over he eriod in he arke wihou scalers (arke ) and had a endency o drif over he arke wih scalers (arke 3). In he case where reservaion rices were binoial he heoreical Walrasian rice was quie ofen a ulile equilibriu. Doinaed by 2 sable equilibriu oins eiher side of he unsable equilibriu when he ne hedge was zero, and a ofen a single sable and unsable fied oin when he ne hedge was equal o Desie he ulile equilibrius, in boh cases he CDA converged o he sable oin. I ook longer o converge in he case of he 5000 ne hedge. When scalers were added o hese 3 arkes, he CDA roduced a rice series ha had a larger sandard deviaion, bu a saller aoun of kurosis. The heoreical rices were ore sable wih he resence of scalers in he binoial case (arke 2 versus arke 4). In he arke wih a ne hedge of 5000 he rices converged a a uch quicker rae o he uch higher rice. While his rice was no yically susained, ofen oving back down closer o seculaor eecaions, i was a consan graviaion oin for he CDA rices. This shows ha he Cliff and Bruen criique (997) ha rices in he ZI odel will aroach eeced rices raher han he Walrasian equilibriu is inaccurae. I is rue ha eecaions and heir disribuion do iac rices, bu graviaion is o he Walrasian equilibriu even when here is zero inelligence. The een ha bilaeral rices converge o he Walsrasian equilbriu or areo oiu can also be a easure of he efficien allocaion of our arke. Drawing fro Sih (962) 4 we ake our ransacions rices j : j =, and forulae a rice convergence easure α = 00 s / P where P is he ean of he heoreical equilibriu rice given by he inersecion of he aggregae suly and deand curves a ie : 00σ α where σ = = ( ) j j Pj P = j = j We calculae α for each individual run, resened in able 3. The closer α is o zero he close he CDA rice series is o he rice ha equaes suly and deand for each oin on average over 2 4 Sih (962) creaed a rice convergence easure, α = 00 s0 / P 0 where P 0 is he saic heoreical equilibriu 2 ( ) σ 0 = j j P rice given by he inersecion of he aggregae suly and deand curves, and 0. Since he odel here has a dynaic conce of wealh he heoreical Walrasian equilibriu is also evolving over ie. =