Sof Compuing Alrnaivs o Modling and Prdicing Economic Dynamics whn Daling wih Forward-Looing Raional Compiors VASILE GEORGESCU Dparmn of Mahmaical Economics Univrsiy of Craiova 13, A.I. Cuza, 01100 Craiova ROMANIA hp://www.vasil-gorgscu.hom.ro Absrac: - Economics and nginring ar currnly concrnd wih modling and prdicing complx dynamics. Howvr, conomy is par of an anhropic raliy, whras a non-anhropic raliy is acually h fild of nginring. Furhrmor, hs wo filds sm o hav irrconcilabl pismological foundaions. This papr is abou whhr nw flxibl compuaional paradigms, such as Sof Compuing, allow daling wih dynamics inducd by forward-looing human bhaviors, and how diffrn ar hs approachs from h adapiv bacward-looing mchanisms, currnly implmnd in nginring. Whil sandard hypohsis in macroconomics is raional xpcaions, som nw modling framwors provid insids in ral-word mars. Ths volv hrough h inracion bwn compiors who ypically xhibi hrognous blifs and boundd raionally. To modl h complx dynamics mrging from such bhaviors, mor vrsail (nural, gnic, fuzzy, or hybrid) mhods ar rquird. Subsqunly, h problm his papr raiss is o which xn compuaional mhods spanning diffrn filds of raliy ar complian wih h amp o unifying h scinc ha WSEAS has xplicily assumd. Ky-Words: - Sof compuing; Economic dynamics; Raional xpcaions; Prfc vs. boundd raionaliy; Hrognous inracing agns. 1 Epismological considraions An ssnial rquirmn for a prdicion o b rliabl is is assumpions o rmain unalrd during h prdicion horizon. This is ypically h cas whn modling h physical raliy or vn h nonanhropic bio-physical raliy. Th criical cas is whn amping o prdic h forward-looing human bhavior. This implis a rvrbraing raionaliy phnomnon (similar o h rflcion in paralll glasss), which is commonly nown from sragic gams wih compl informaion, whr wo playrs could muually annihila hir acions as long as hy ar abl o raionally anicipa hir innions. Th sam phnomnon has o b considrd whn modling macroconomic dynamics. Th modlr acs as a policy-mar a h macroconomic lvl. Th raional agns ar anicipaiv wih rspc o macroconomic policis rgarding hmslvs, valua hir ffcs on individual businsss and chang accordingly hir bhavior, hus alring h iniial sings: changs in policis induc changs in bhavioral paramrs of h macroconomic modl, which will sar o drif. Thrfor, purly adapiv bacward-looing policis could b infficin. For rliabl prdicions, raional xpcaions nd o b ingrad in h modl, in ordr o capur h forward-looing bhavior of h agns. Th raional xpcaions rvoluion, promod by Lucas criiqu in 1976 ([8]), originad from such ind of pismological considraions. 2 Th raional xpcaions rvoluion Th main moivaion bhind h dvlopmn of raional xpcaions modls was o provid rliabl policy valuaion procdurs. In his sminal papr Economric Policy Evaluaion: A Criiqu (1976), Robr Lucas Jr. argud ha h paramrs of h modls convnionally usd for policy valuaion would shif whn policy changd. Th main rason for his shif is ha xpcaions mchanisms ar adapiv or bacward-looing in convnional modls and hrby unrsponsiv o hos changs in policy ha would b xpcd o chang xpcaions of fuur vns. Hnc, h policy valuaion rsuls using convnional modls would b mislading. An xampl of a bacward-looing macroconomic modl is h following nonlinar dynamic simulanous quaion sysm: f ( y, y 1, x, β ) = ε ~ iid(0, Σ) (1)
whr f dnos a vcor funcion. Th argumns y, y 1, x and β ar vcors of currn and laggd ndognous variabls, xognous variabls and paramrs. ε is a vcor of rrors or shocs, assumd o b inr-mporally indpndn and idnically disribud (iid), wih zro mans and a conmporanous covarianc marix Σ. By conrary, h sin qua non of a forwardlooing modl is h apparanc of forcass of vns basd on informaion availabl bfor h vns a plac, i.., condiional xpcaions of fuur-dad variabls (lads). An xampl of a nonlinar forward-looing modl is as follows: f ( y, y, E( y+ 1 I ), x, β ) = ε ~ iid(0, ) (2) 1 Σ whr E ( y + 1 I ) is h condiional xpcaion basd on all informaion hrough priod and I = { f, y0, K, y 1, x1, K, x, β, Σ} dnos h informaion s a h sar of h priod. Wih y + 1 = E( y+ 1 I ), on can dfin an implici soluion: y ( 1 + 1 ε β = g y, y, x,, ) (3) Whil h bacward-looing modl can b solvd rcursivly, h soluion of h forward-looing nonlinar modl canno b gnrally compud by rcursion. Dpndnc on fuur xpcaions in addiion o pas ralizaions has implicaions in h uniqunss of h soluion and h mhod of is approximaion. Gnrally, h soluion for all priods has o b found simulanously. Th forcas gnrad by his procss will b qual o h xpcaions ha appar in h modl. In his sns, xpcaions ar consisn wih h modl, or quivalnly, xpcaions ar raional. Svral parial soluions hav rid o miiga h criicism on h us of classical conrol hory in conomics, which has bn provd o no b suiabl for daling wih raional xpcaions. A possibl soluion is o formula macroconomic policy as a gam bwn policy-mars and conomic agns. Anohr soluion could b o obsrv and sima h paramrs of h agns bhavior in rspons o policy changs. This dscribs a procss by which h policy-mar uss sochasic conrol mhods o larn abou changs in h bhavior of h agns. Thus h is always on sp bhind h conomic agns. In a dynamic sing, h policy-mar announcs a policy, h agn rsponds and h policy-mar obsrvs h changs in bhavior and us Kalman filr mhods o upda paramrs ovr im and o provid updad man and covarianc simas in ach im priod. Th covarianc marix of paramr simas can hn b usd in dciding on policy lvls for h nx im priod. An xampl of linar quadraic opimizaion for modls wih raional xpcaions and larning could b formulad as follows: find h s of admissibl insrumns U = { u0, u1, K, ut 1} ha minimizs h wlfar loss funcion J T T = E = subjc o h modl x + 1 T β L ( x, u ) + β LT ( x ) (4) 1 0 T 0 = A( θ ) x + B( θ ) u + + C( θ ) z + τ = 1 D τ ( θ ) E x+ τ + ε (5) whr L and L T ar quadraic forms, Dτ ( θ ) is a paramr marix, E is h xpcd sa for x + τ im + τ as sn from im, is h maximum lad in h xpcaions funcions and ε is a whi nois. As a principl, h raional xpcaions hav o b liminad from h modl, in ordr o compu h admissibl s of insrumns. In gnral, solving sochasic conrol problms ha mbd raional xpcaions is a difficul as, whn prsrving h srong hypohsis of prfcly raional agns. This clarly lgiimas h inrs in considring war hypohss and looing a mor vrsail mhods for achiving racabiliy and robusnss. Sof compuing (nural, gnic, fuzzy or hybrid) chnologis may hlp agns o fac limid nowldg and informaion, by involving hm in a larning framwor ha olras boundd raionaliy. 3 Boundd raionaliy and adapiv larning 3.1 Homognous vs. hrognous agns A rducionism has o b implicily assumd in sandard macroconomics for conforming o h Raional Expcaions Hypohsis: h absrac concp of a rprsnaiv agn. This implis ha all agns ar homognous and do no inrac. Undr such assumpion, h dynamics of h aggrga rplica h dynamics of lmns, which ar in quilibrium and xhibi only non sysmaic diffrncs (noiss). Howvr, ral-word mars incorpora agns who ar hrognous in hir dcision bhavior, and ypically do no xhibi prfc raionaliy. This suggss rlaxing h srong hypohsis of raional
xpcaions and adoping a mhodological approach basd on hrognous inracing agns. Hrogniy implis boundd raionaliy and may b moivad by various rasons: incompl informaion, limid capaciy or significan coss of procssing i; changs in chnology and insiuions; poliical vns, rumors, disurbing nws; divrsiy of agn ypology; diffrn agn capabiliis of larning and volving. Th hrogniy of conomic agns and h inracion bwn hm ar capurd by h occurrnc of scaling phnomna and h swd disribuion of svral variabls, such as firms siz, growh ras c. This affcs h concp of macroconomic quilibrium, which dos no rquir any mor ha vry agn is in quilibrium (i.., dos no dpnd on microscopic dails), bu sas ha h sabiliy is rahr an mrgn propry of h aggrga as a whol. A sa of macroconomic quilibrium can b mainaind by a larg numbr of ransiions in opposi dircions. If h sysm is far from quilibrium, slf-organizing phnomna may also occur. On h ohr hand, h imprfc informaion and h sysmaic inracions among agns may produc oupu flucuaions. 3.2 Using nural nwors for adapiv larning of raional xpcaions Th ponial of sof compuing mhods in gnral, and of nural nwors in paricular, o dal wih nonlinar procss modling and prdicion drivs from wo imporan characrisics: - hir capabiliy o b univrsal approximaors (i.., o sima almos any compuabl funcion on a compac s, providd ha nough xprimnal daa and nough compuing rsourcs ar availabl); - hir olranc for modl misspcificaion, which allows difficul problms, such as h x an spcificaion of funcional form in conomric modl building, o b racabl in a lss sringn and vrsail mannr: h spcificaion of nural nwor archicur (i.., numbr of layrs, numbr of nurons in ach layr, yp of acivaion funcions, and so on). In ordr o b compiiv on mar, agns hav o b abl of forming raional xpcaions. Whn facing modl misspcificaion (i.., h ru nonlinar funcional form of h modl is unnown) hy mus approxima raional xpcaions as a rsul of a larning procss. An auxiliary modl is rquird o accomplish his procss, basd on h assumpion ha i is flxibl nough o rprsn various inds of possibl rlaionships bwn h rlvan variabls. Nural nwors migh b wll suid for his as, du o hir wo characrisics mniond abov. Using h induciv capabiliis of nural nwors, h agns may b abl o larn h formaion of raional xpcaions, wihou h rquirmn of spcifying x an h ru nonlinar funcional form of h modl. To xmplify h way agns larn o form xpcaions, on can considr a cobwb-li modl ([6]), whr h valus of an ndognous variabl y dpnds on a -dimnsional vcor of obsrvabl xognous boundd variabls x Ω x R,, and an unobsrvabl boundd rror ε Ωε R,. Th rducd form of h modl is givn by y = α y + g( x ) + ε (6) whr y dnos h agns xpcaion of h ndognous variabl y in priod, and g (x) is a coninuous funcion for all x Ωx. Th xognous variabls x can b obsrvd bfor h xpcaion y is formd. Th sandard assumpions hold: 2 ε ~ iid, E[ ε ] = 0, E [ ε 2 ] = σ, E[ ε x ] = 0. Givn h rducd form (6) and h prvious assumpions, raional xpcaions ar givn by: y g( x ) + ε g( x ) = E[ y x ] = E = = φ( x ) 1 α (7) 1 α whr φ ( x ) dnos h raional xpcaion funcion and dfins uniquly h raional xpcaions of h ndognous variabl for all x Ω x, as long as α 1. Wihou an x an spcificaion of h funcional form of φ ( x ), agns could us nural nwors as an auxiliary modl for larning xpcaions. A nural nwor wih inpu unis, on hiddn layr consising of m unis and on oupu uni is wll suid for accomplishing his as. Each of h hiddn unis i = 1, K, m rcivs a signal ha is h wighd sum of all inpus x j, ~ j = 1, K,, i.., h i = j = w 1 i, j x j + wi, 0, whr w i, 0 dnos a hrshold valu. In ach hiddn uni, h signal rcivd is ransformd by an acivaion funcion S : R [0,1], S( z) = 1 ( 1+ xp( z) ), such ~ ha h i = S( h i ) is h oupu signal of h hiddn uni i. Finally, h oupu uni rcivs h wighd sum of all hs oupu signals: y = m = h 0, whr q i 1 i +
q 0 dnos a hrshold valu. Th nural nwor dfins a mapping from inpus x j o h oupu y, as follows: y ( w x + w ) + q f ( x, ) m q S 1 1,, 0 0 θ i i j i j j i = = = = (8) q whr x R, θ R, q = 1 + m( + 2), and θ = q, q, w, K, w, q, K, ). ( 0 1 1,0 1, 2 w m, Givn ha nural nwors ar univrsal approximaors, on can assum ha a wll configurd archicur provids, a las horically, nough prdiciv accuracy for agns o larn xpcaions. 3.3 Robus conrol whn daling wih misspcificaion Robus conrol is a promising ool for a policymar who rgards his modl as an approximaion, ha is, an unnown mmbr of a s of unspcifid modls nar his approximaing modl. Imporan sps hav bn mad in h dircion of applying robus conrol o valuaing conomic policis by T. Sargn and L.P. Hanssn ([5]). Insad of assuming ha policy mars now h modl in h form of a ransiion low ha lins h moion of sa variabls o conrols (such as in h ordinary conrol hory), robus conrol hory alrs h mapping from shoc mporal propris o policy ruls. I ss on rul o us for a s of modls ha migh also govrn h daa. Th currnly usd mhods ( H and nropy criria in h frquncy domain, or robus filring) nd som adapaions o incorpora discouning ino h objciv funcionals. 3.3 Capuring and uning nonlinar characrisics by fuzzy conrol Prsumably, h inrs in applying fuzzy conrol o conomic procsss consiss of a las wo advanags: on h on hand, of prscribing conrol acions by linguisic dscripions, and on h ohr hand, of h capabiliy of ransiion from linar o nonlinar mods of conrol, conjugad wih finuning procdurs. W addrssd h lar opporuniy in our prvious papr ([4]), by providing a fuzzy xnsion o h Phillips sabilizaion modl in wo varians: for a closd conomy (using a fuzzy PID conrollr) as wll as for an opn conomy (using a fuzzy sa-fdbac conrollr). Sinc fuzzy conrol can b dscribd as a non-linar mapping, h corrsponding fuzzy conrollr acs as a non-linar conrollr and hnc provids an incrasing flxibiliy. In h firs sag, w focusd on h mulaion of a convnional conrollr (ihr a PID, or a sa-fdbac on) hrough a linar fuzzy conrollr as a sar poin for furhr xploiaions of h full capabiliis of h non-linar fuzzy conrollr. Givn ha a fuzzy conrollr conains a linar conrollr as a spcial cas, i is ru o say ha i prforms a las as wll as h lar. W also brifly suggsd how o ma i non-linar and how o us fin-uning procdurs for achiving h validaion objciv of h conrollr. Th ponial for prforming br dpnds on h dsignr capabiliy o xploi h non-linar opions in h fuzzy conrollr o his advanag. 4 Using sof compuing mhods in a muli-agn modling framwor 4.1 Complx dynamics in financial mars inducd by hrognous xpcaions Th Efficin Mar Hypohsis (EMH) assums idnical invsors who shar raional xpcaions of an ass s fuur pric, and who insananously and raionally discoun all mar informaion ino his pric. Howvr, i is widly nown ha in ralword mars h radrs may xhibi hrognous blifs abou fuur prics of a risy ass ha may considrably dvia from fully raional xpcaions. Financial mars can b viwd as complx voluionary sysms having inrnal dynamics inducd by wo comping rading agns: fundamnaliss, bliving ha prics will mov owards hir fundamnal raional xpcaions valu, as givn by h xpcd discound sum of fuur dividnds; rnd-followrs, bliving ha ass prics ar no complly drmind by fundamnals, bu ha hy may b prdicd by simpl chnical rading ruls, xrapolaion of rnds and ohr parns obsrvd in pas prics. Numrical xprimns and som mpirical vidncs ([3]) hav mphasizd ha hrogniy in blifs may lad o mar insabiliy and complicad dynamics, such as cycls or vn chaoic flucuaions, in financial mars. Ass pric flucuaions ar causd by an ndognous mchanism rlaing h fracion of fundamnaliss and rnd-followrs o h disanc bwn h fundamnal and h acual pric. A larg fracion or wigh of h fundamnaliss nds o sabiliz prics, whras a larg fracion of rnd-followrs nds o dsabiliz prics. Ass pric flucuaions ar causd by h inracion bwn hs sabilizing and dsabilizing compiors. Exprimnal vidncs show ha, undr h hypohsis of hrognous
xpcaions among radrs h mrging dynamics of ass pric changs dramaically, wih bifurcaion rous o srang aracors, spcially if swiching o mor succssful sragis bcoms mor rapid. 4.2 Nural nwor basd agns daling wih conomric forcas modls Nural nwor basd conomric forcas modls ar frqunly mbddd in a muli-agn modling framwor. Thy assum conncionis rlaionships bwn h inpu signals and h arg valus. Boh fd-forward and rcurrn nural nwors can b usd for muli-agn modling, dpnding upon h dynamic bhavior of h mar. Thy allow capuring nonlinar characrisics and ar abl of fiing a wid rang of forcas srucurs. Agns can build a pric forcas a im, E p + 1, j, using a nwor raining wih svral inpus including svral laggd prics, and rad prics avragd ovr all agns from arlir priods. Agns can b randomly machd and rad occurs whn agn pairs hav diffrn xpcd fuur prics. Thy hn spli h diffrnc and rad a h pric in bwn hir wo xpcd valus. In h modl in [1], h agns us fd-forward nural nwors o prdic upcoming soc pric movmns. Basically, h auhors inroduc wo inds of mar paricipans: smar and naiv agns. Smar agns prdic soc pric movmns on h basis of 3-layr fd-forward nural nwor wih four inpu signals. Inpus o h 3-layr nwor ar h mos rcn mar prics π 1, π 2 and h formr ransacion prics P ij, 1, P ij, 2. In conras, h forcas modls of naiv agns ar simplifid: Naiv agns only rly on h mos rcn mar pric π 1 in ordr o forcas h fuur dvlopmn of h soc pric. Th undrlying nural nwor archicur consiss only of a singl inpu nuron conaining h mos rcn mar pric and on oupu nuron compuing h pric forcas. A muli-agn modl basd on rcurrn nural nwors is prsnd in [10]. Th modl includs hr yps of agns, valu radrs, momnum radrs, and nois radrs. Th agns plac hr funds ihr in a risy soc paying a sochasic dividnd or in a rislss bond. Valu invsors bliv ha h acual soc pric rflcs h discound sram of all fuur dividnds. Momnum radrs and nois radrs ar chnicians who only considr hisorical pric parns. Whil momnum and nois radrs ar modld as rulbasd agns who rfr o chnical rading ruls, valu radrs form hir xpcaions on h basis of Elman's rcurrn nural nwors. Ths rcurrn nural nwors incorpora so-calld conx unis, which fd h informaion of prvious acivaion valus bac ino h nwor. Mor prcisly, valu invsors us rcurrn nural nwors o prdic h dividnd growh of h risy ass. Afrwards, h mar pric of h risy ass is simad on h basis of h dividnd growh by using Gordon's consan dividnd growh modl. Anohr nural nwor basd approach can b found in [7]. In his arificial soc mar, h agns invs hir funds ihr in a rislss bond or in a risy ass paying a sochasic dividnd. As a spcialy, h agns hav diffrn dcision maing horizons: som agns ar long-rm invsors whil ohrs rly on shor-rm planning horizons. According o hir planning horizon, h agns choos from a broad spcrum of forcas modls ha ar fid o hisorical daa. Forcas modls and agns ar hrfor sparad. Each forcas modl is composd of a simpl fd-forward nwor incorporaing on hiddn nuron and a limid numbr of inpu signals. Th inpu signals consis of chnical and fundamnal indicaors. Th nural nwors ar volvd by using a gnic algorihm. As i can b sn from hs xampls, h dcision maing schms of conomric agns do no incorpora smanic spcificaions of h undrlying cogniiv procsss in rms of.g. prcpion, inrnal procssing and acion. In ohr words, h conomric forcas modls of h agns mrly assum conncionis rlaionships bwn h inpu signals and h arg valus. 4.3 Cogniiv sysm basd modls, larning, and voluion Cogniiv sysm basd agn modls ar an approach of capuring h smanic spcificaions of h agns' dcision schms in a srucural framwor. Mor prcisly, h dcision maing of an agn is modld by a basic cogniiv sysm. Th cogniiv sysm incorporas hr propris: prcpion, inrnal procssing and acion. Ths propris consiu ncssary condiions for a cogniiv sysm and may includ various ohr faurs. As a srucural rprsnaion of such a cogniiv sysm, on may rfr o im-dlay rcurrn nural nwors. Th cogniiv procss gnras no only xpcaions of h mar pric, bu also concr rading dcisions. Sinc a cogniiv agn has a spcific objciv funcion (.g. uiliy maximizaion), h rsuling acions ar always goal-orind. As i can b sn from his oulin, h buying and slling dcisions of a
cogniiv agn ar no dducd from a spcific conomric forcas modl which only prsums a conncionis or, possibly, an ad-hoc funcional rlaionship bwn xrnal influncs and h mar dvlopmn. Rahr, h dcisions of h agn ar formulad in rms of h undrlying cogniiv sysm. This is ruly a mor ralisic approach of modling h agns' bhavior. Hrognous bhavior may also origina from h larning and voluion of h agns. Muli-agn modls ha rfr his sourc of hrogniy ypically incorpora agns who ar ndowd wih s of dynamic rading ruls. Th rul ss ar volvd by gnic algorihms. In such a modl, agns ar vry homognous a h sar in hir abiliis and sragy srucurs. During h mar procss, h agns larn and dvlop mor sophisicad rading sragis. In ohr words, diffrncs in bhavior and sragy volv ndognously as h mar runs, and hus agn hrogniy bcoms a changing faur of h mar. Exampls including gnic algorihms as a sourc of hrogniy ar givn in [2] and [7]. Bsids gnic algorihms, on may also rfr o gradin basd larning chniqus in ordr o gnra hrognous dcision bhavior. Firs, h agns may simply diffr in hir undrlying larning chniqus. Diffrn ways of larning should lad o hrognous agns. An inrsing qusion is how h agns volv hir rading sragis, adap o changing mar condiions and larn how o improv hir bhavior. Alrady mniond, larning is also an imporan sourc of hrognous dcision bhavior. Gnrally spaing, 'larning' or 'adapiv bhavior' mans ha h agns ar abl o modify pars of hir dcision maing schms in a spcific mannr. By his, h agns adap hir bhavior o changing mar condiions. Th larning and adapiv bhavior has diffrn purposs. For xampl, rul-basd agns may ihr dvlop complly nw rading sragis or chang a fw paramrs of alrady xising ons. Anohr possibiliy is ha rul-basd agns swich bwn diffrn rading sragis as h mar volvs. This corrsponds o conagious dcision maing or so-calld hrding bhavior. Morovr, forcasing agns larn by fiing h fr paramrs of hir forcas modls o hisorical daa. During h larning, h agns gnra a srucural hypohsis of h undrlying mar dynamics, i.. hy ry o idnify invarian srucurs ou of varying im sris. Th gnrad srucurs allow h agns o prdic h fuur dvlopmn of h mar pric. 5 Conclusion In his papr, w mainly inndd o confron h ara of Economic Dynamics and Conrol wih is roos and som conncd filds, and o anchor h formal rsuls in pismological considraions, as a sin qua non condiion for dply undrsanding h undrlying mchanisms and h prsn and fuur major rnds. Subsqunly, h papr also ampd o valua h ponial of som flxibl ransdisciplinary paradigms for h always acual Arisol s propnsiy o unifying h scinc. Whn aing a loo a hs modrn ools, w ar rahr opimisic. Howvr, whn xploring h pismological foundaions of h wo qualiaivly diffrn filds (h anhropic on and h nonanhropic on), w find ssnial rasons o rmain mpraly pssimisic. Rfrncs: [1] Blrai A., Margaria S. Trna P., Nural Nwors for conomic and financial modlling, Inrnaional Thomson Prss, London, 1996. [2] Bihahn J., Nissn V. (Eds.), Evoluionary Algorihms in Managmn Applicaions, Springr Vrlag, Hidlbrg, 1995, pp. 290-304. [3] Broc W., Homms C., A raional rou o randomnss, Economrica, No.65, 1997, pp.1059-1095. [4] Gorgscu V., Capuring and Tuning Nonlinar Characrisics of Economic Sabilizaion Sysms by Fuzzy Conrol Tchniqus, Compuaional Economics, Kluwr Acadmic Publishrs, Vol.19, No.3, 2002, pp. 247-271. [5] Hanssn L., Sargn T., Misspcificaion in Rcursiv Macroconomic Thory, Princon Univrsiy Prss (forhcoming). [6] Hinmann M., Adapiv larning of raional xpcaions using nural nwors, Journal of Economic Dynamics and Conrol, No.24, 2000, pp.1007-1026. [7] LBaron B., Evoluion and Tim Horizons in an Agn-Basd Soc Mar, Macroconomic Dynamics, Vol. 5, 2001, pp. 225-254. [8] Lucas R.E. Jr., Economric Policy Evaluaion: A Criiqu, Carngi-Rochsr Confrnc Sris, No.1, 1976, pp.19-46. [9] Soy N., Lucas R.E. Jr., Rcursiv Mhods in Economic Dynamics, Harvard Univ. Prss, 1989 [10] Yang J., Hrognous Blifs, Inllign Agns, and Allocaiv Efficincy in an Arificial Soc Mar, Compuing in Economics and Financ, No.612, 1999.