Testing the Goodwin growth-cycle macroeconomic dynamics in Brazil PDF
Preview Testing the Goodwin growth-cycle macroeconomic dynamics in Brazil
Testing the Goodwin growth-cycle macroeconomic dynamics in Brazil N.J.MouraJr.a,MarceloB.Ribeirob, ∗ aInstitutoBrasileirodeGeografiaeEstat´ıstica–IBGE,RiodeJaneiro,Brazil bInstitutodeF´ısica,UniversidadeFederaldoRiodeJaneiro–UFRJ,RiodeJaneiro,Brazil Abstract 3 This paper discusses the empirical validity of Goodwin’s (1967) macroeconomic model of growth with cycles by 1 0 assumingthattheindividualincomedistributionoftheBraziliansocietyisdescribedbytheGompertz-Paretodistri- 2 bution(GPD).ThisisformedbythecombinationoftheGompertzcurve,representingtheoverwhelmingmajorityof thepopulation( 99%),withtheParetopowerlaw,representingthetinyrichestpart( 1%). InlinewithGoodwin’s n ∼ ∼ a originalmodel,weidentifytheGompertzianpartwiththeworkersandtheParetiancomponentwiththeclassofcap- J italists. SincetheGPD parametersareobtainedforeachyearandtheGoodwinmacroeconomicsisa timeevolving 0 model,weusepreviouslydetermined,andfurtherextendedhere,BrazilianGPDparameters,aswellasunemployment 3 data,tostudythetimeevolutionofthesequantitiesinBrazilfrom1981to2009bymeansoftheGoodwindynamics. Thisis done in the originalGoodwinmodeland an extensionadvancedby Desai et al. (2006). As far as Brazilian ] N data is concerned, our results show partial qualitative and quantitative agreement with both models in the studied G timeperiod,althoughtheoriginaloneprovidesbetterdatafit. Nevertheless,bothmodelsfallshortofagoodempir- icalagreementastheypredictsinglecentercycleswhichwerenotfoundinthedata. We discussthespecificpoints . n wheretheGoodwindynamicsmustbeimprovedinordertoprovideamorerealisticrepresentationofthedynamics i f ofeconomicsystems. - q Keywords: Incomedistribution;Paretopowerlaw;Gompertzcurve;Brazil’sincomedata;Goodwinmodel; [ Growth-cyclemacroeconomics;Fractals 3 v 0 9 0 1. Introduction 1 . IthasbeennotedlongagobyKarlMarxthatcapitalistproductiongrowsoncyclesofboomsandbusts. Duringa 1 boom,profitsincreaseandunemploymentdecreasessincetheworkersareabletogetbetterjobsandhighersalaries 0 3 due to shortage of manpowerto feed the growingproduction. However, this boomis followedby a bust since less 1 unemploymentreducestheprofitmargin,whoserecoveryisachievedbyahigherunemploymentandareductionof : theworkers’bargainingpower. Smallersalariesincreasetheprofitmarginleadingtorenewedinvestmentandthena v i newboomstarts,beingfollowedbyanotherbust,andsoon[1,Chap.25,Sect.1]. X A century later Richard Goodwin[2] proposeda mathematicalmodelwhich attemptsto capture the essence of r Marx’sdynamicsdescribedabove. Inthismodelthebasicdynamicsofacapitalistsociety,asqualitativelydescribed a byMarx,ismodeledbymeansofamodifiedLotka-Volterramodelwherepredatorandpreyarerepresentedbyworkers andcapitalists. GoodwinreplacedtheclassicLotka-Volterradynamicsofnumberofpredatorsandpreysbytwonew variablesu and v, the former giving the workers’ share of total production, which is an indirect way of describing the profit marginof capitalists, and v representingthe employmentrate, which is an indirectway of describing the shareofthosemarginalizedbytheproduction,theunemployedworkers,thatis, theindustrialreservearmyoflabor inMarx’sterminology. Ina boomtheemploymentrate v increasesandthe workers’share u startsto increaseafter a time lag, meaning a decrease in profitmargin. When employmentrate is at its maximumthis correspondsto the lowestprofitmargin,thentheburstphasestartswithadecreaseinv. Atthispointuhadalreadystarteddiminishing. Correspondingauthor ∗ Emailaddresses:[email protected](N.J.MouraJr.),[email protected](MarceloB.Ribeiro) Preprint December11,2013 Theessenceofthemodeliscapturedasaclosedorbitintheu-vphasespace. Clearlythesetwovariablesareoutof phaseintime[3,pp.458-464]. Althoughthe briefdescriptiongivenaboveappearsto indicate thatGoodwinwas able to captureMarx’sobser- vations, the modelhas in factseveralshortcomings,the mostsevere onebeing its inabilityto predictquantitatively the above described dynamics (see below). The model was presented simply as an heuristic reasoning capable of givingamathematicaldressingtoMarx’sideas. Itwasbornoutasavisionoftheworldratherthanfromareal-world data-inspiredmodelin a physicalsense. Despite this, or, perhaps,because ofthis, since its formulationGoodwin’s modelhasattractedconsiderabletheoreticalattentioninsomeeconomiccirclesandseveralvariationsoftheoriginal modelwereproposed[see4–18,andreferencestherein]. However,interestinglyenough,almosthalfa centuryafterits proposal,attemptstoactually test thismodelem- piricallyare still extremelylimited. AlthoughGoodwin’sgrowth-cyclemodelis certainlyinfluentialin view of the numberoftheoreticalfollow-uppaperscited above,studiesseekingtoestablish itsempiricalsoundnessare limited onlytoRefs. [10, 19–25]. Thisis a surprisinglyshortlist whenwe considerthe time spansince the model’sinitial proposal. So little interestin empiricallycheckingthe model, especially amongthose who appearto have beense- ducedbyits conceptualaspects, is evenmoresurprisingif we bearinmindthatforthelast 30yearsorso we have been living in an era where large economic databases are easily available digitally, so large-scale checking of this modelagainstempiricaldataceasedlongagotoposeaninsurmountablebarrier. Besides, eventheveryfewstudies whichactuallyattemptedthat,allpointtosevereempiricallimitationsofthemodel,rangingfrompartialqualitative acceptancetototalquantitativerejection.Fromaneconophysicsviewpointitiscuriousthatamodelwithsuchapoor empiricalrecordbecamesoinfluential. Despite this, the model does have some general empirical correspondence to reality on a qualitative level and this justifies furtherempiricalstudies with differentdatabases, data handlingmethodsand/or data type approaches. The basic aim must lie in identifying as clearly as possible where the model performs poorly in order to propose amendmentsandmodifications. Anymodel,especiallythosetheoreticallyseducing,canonlyremainofinterestifit passesthetestofexperience,ifitsurvivesconfrontingitspredictionswithempiricaldata. Ifitdoesnotsurvivethis testthemodelmustbemodified,orabandoned. ThispaperseekstoperformanempiricalstudyoftheGoodwingrowth-cyclemodelusingindividualincomedata ofBrazil. ThestudypresentedherewasdirectlymotivatedbyourpreviousexperienceinmodelingBrazil’sincome distribution, whose resultssuggested a Goodwintype oscillation in the share of the two income classes detectedin the data [26, 27]. Building upon our previousexperiencewith this database, we obtained yearly values of the two mainvariablesof the Goodwinmodel, the laborshare u and the employmentrate v. Nevertheless, differentlyfrom allpreviousapproachesfortestingGoodwin’smodel,herethelaborsharewasobtainedbymodelingtheindividual incomedistributiondatawith theGompertz-Paretodistribution(GPD)andidentifyinguwith theGompertzian,less wealthy,partofthedistribution[27]. Theemploymentratewasalsoestimatedfromthesamedatabase,thatis,from Brazil’sincomedistribution,usingtheconceptofeffectiveunemployment. We show that from 1981 to 2009 u and v do cycle in a form bearing similarities to what the Goodwin model predicts, that is, closed cycles. However, our results show the absence of a single cycling center and also are in completedisagreementwiththeonesforBrazilasreportedbyRef.[25],whoseanalysisemployedHarvie’smethod [22]. Inaddition,weattemptedtoseeifourfindingsbringempiricalsupporttotheDesai-Henry-Mosley-Pemberton (DHMP)extensionoftheoriginalmodel[9]. OurresultsshowthatthisparticularvariationoftheGoodwindynamics hassomeempiricalsoundness,althoughitprovidesasomewhatpoorerdatafitascomparedtotheoriginalmodeland alsoleavesthreeparameterstobedeterminedbyother,stillunknown,meansthantheonesstudiedhere,whereasthe originalmodelleavestwoparametersinasimilarsituation. Weconcludethatthesetwomodelsprovidepartialqual- itativeandquantitativeagreementwithrealdata,atleastasfarasempiricaldatafromBrazilareconcerned,butboth ofthem,andperhapsallvariationsoftheoriginalGoodwingrowth-cycledynamics,requireimportantmodifications andamendmentsbeforetheycanbeconsideredviablerepresentationsoftherealdynamicsofeconomicsystems. Thepaperisorganizedasfollows. Sect.2presentsabriefreviewoftheoriginalGoodwinmodelanditsDHMP extension,focusingmostlyon their dynamicalequations, althoughsome discussion aboutthe underlyingeconomic hypothesesandfoundationsoftheoriginalmodelisalsopresented.InSect.3,afterashortdiscussionaboutmethodol- ogy,wereviewthemainequationsbehindtheGPD.Sect.4analyzestheindividualincomedataofBrazilandpresents theu-vorbitsinthe1981-2009timeperiod. Sect.5providestimevariationsoftheemploymentrateascomparedto workers’shareso thatline fittingsallow ustodeterminesomeofthe unknownparametersofbothmodels. Finally, 2 Sect.6discussestheresultsandpresentsourconclusions. 2. TheGoodwingrowth-cyclemacro-economicdynamics 2.1. Theoriginalgrowth-cyclemodel The model proposed by Goodwin is essentially a Lotka-Volterra predator-prey system of first order ordinary differentialequationswhichcanbewrittenasfollows[2,9,22], u˙ =[ (a+d)+hv]u, (1) − 100 u v˙ = − (a+b) v, (2) " c − # wherethedotdenotesthetimedifferentiationd/dt.Thefiveconstantsa,b,c,d,hcomefromtheeconomichypotheses ofthemodelandaresupposedtoobeythefollowingconditions[3,22], c>0, h>0, ((aa++db))c><0,100. (3) Thesolutionofequations(1)and(2)producesafamilyofclosedorbitswithperiodT,allhavingthepoint(u ,v )as c c theiruniquecenter,accordingtothefollowingequations[22], u =100 (a+b)c, c − vc =(a+d)/h, (4) T =2π √(a+d)[100/c (a+b)]. − (cid:14) Sinceuisthepercentageshareoflabor,orworkers,innationalincomeandvrepresentstheproportionoflaborforce employed,theybothshouldlieinthe[0,100]interval.HerewefollowthenormalizationadoptedinRefs.[26,27]and shallrefertothemaximumshare,orproportion,as100%.Theuppersingularpointv fortheemploymentproportion s isreachedwhenv˙ =0,thenu =100 c(a+b).Similarly,whenu˙ =0wehavev =(a+d)/h.However,if(a+b)is s s − negative,thenu > 100,which,inprinciple,shouldnotbeallowed(foraconceptuallypossible,butsofaruntested, s exception,seeRef.[3],p.461).Similarly,itispossibletohavev >100. s In thismodelu representsthe populationdensityof predatorswhereasv representsthe preypopulationdensity. Thiscanbeseenasfollows. Whenu = 0,u˙ = 0andv˙ > 0. Inotherwords,uremainsequaltozerowhereasvgrows withoutbound,asituationhappeningtothepreypopulationvintheabsenceofpredatorsu. Ontheotherhand,when v = 0, equations(1) and(2) togetherwith conditions(3) showthatv˙ = 0 andu˙ < 0. So, withoutprey(v = 0), the predatorpopulationdecreases(u˙ <0). Themodelisdefinedintermsoffiveparameters.However,oncetheyaregroupedasbelow, a =(a+d), 1 abb21 ===(h11,0000//cc,)−(a+b), (5) theyallowequations(1)and(2)toberewrittenin2theformoftheclassicalLotka-Volterraequations[3], u˙/u= a +b v, (6) 1 1 − v˙/v=a b u, (7) 2 2 − that is, in terms of four parameters which could, in principle, be determined observationally, provided that both variablesandtheirderivativesareobtainedfromrealdata. 3 2.2. TheDesai-Henry-Mosley-Pemberton(DHMP)extension Desai et al. [9] notedthat the originalGoodwinmodelcan producesolutionsoutside the u-v domain[0,100] × [0,100]because,asseenabove,bothu andv cangrowabove100.Thisisthemainreasonwhichledthemtopropose s s amodifiedversionofGoodwin’soriginalmodel,dubbedhereastheDHMPextension. Theyalsorelaxedtwoother economichypothesesassumedintheoriginalmodel. So,intheDHMPextensionallprofitsarenotalwaysinvested andthePhillipscurve,relatingunemploymentandinflationrate,isnon-linear.Thus,thefinalequationsyield, u˙ = (a¯+d¯)+h¯(100 v)δ u, (8) − − h i v˙ = λln(100 u¯) (a¯ +b¯) +λln(u¯ u) v, (9) − − − − nh i o wherea¯,b¯,d¯,h¯,δ,λ,u¯ areconstantsobeyingthefollowingconstraints, δ>0, λ>0, uh(¯a¯<<+(u¯b¯a¯)<+<1d¯0λ)0,l,n u¯ , (10) (cid:18)100u¯−u¯(cid:19)>1(cid:18).100−u¯(cid:19) Ref.[9]givesaclearmeaningtotheparameteru¯asbeing“themaximumshareoflaborthatcapitalistswouldtolerate”, “typically”givenbythelastconstraintequationinthesetofexpressions(10)above.Clearlythisimpliesthatu¯ >50%. OnemustalsonotethatboththeoriginalGoodwinmodelanditsDHMPextensionconsiderthatthelaborshareand profitsare notgiven in terms of money,but in real terms. As we shall see below, this requirementdoesnotpose a problemforourapproachsinceourvariablesarecurrencyindependent[see26]. Asseenabove,theDHMPextensionofGoodwin’sgrowthcyclemodelisdefinedbysevenparameterswhichcan begroupedasbelow, a¯ =(a¯ +d¯), 1 bba¯¯¯21 ===λλh¯,,ln(100−u¯)+(a¯ +b¯), (11) allowingustorewriteequations(8)and(9)as2follows, u˙/u= a¯ +b¯ Vδ, (12) 1 1 − v˙/v= a¯ +b¯ ln , (13) 2 2 − U where u¯ u, (14) U ≡ − andtheunemploymentrategivenby, V 100 v. (15) ≡ − Although the basic motivation for the DHMP extension was to avoid the variables of the model having values above100%,thisdifficultycanbeavoidedifbothuandvaredefinedbyrealdata,inwhichcasethedesiredthreshold willbeachievedbyconstruction.Besides,theDHMPmodelhastheadditionaldisadvantageofrequiringseven,rather thanfive,unknownparameters. 4 2.3. Interpretationoftheconflictingvariables AsseenabovetheGoodwinmodelisessentiallyapredator-preytypeoneandthismeansthatitstwovariables representthe opposing, but interdependent, nature of a predator - prey conflict. This is the reason why this model isalsoknownas“Goodwin’sclassstrugglemodel.” Thenatureofthis“struggle”arisesfromthepossiblewayswe interpretitsvariables. Ononehand,theemploymentratevcanbeidentifiedwiththeworkers’classandtheprofitshareofthe“capital- ists”isthengivenby, U 100 u. (16) ≡ − Inotherwords,U istheshareoftotalnationalincomeobtainedbytheclassthatcontrolsthecapital,theinvestors.In thiscasetheconflictisbetweentheworkersandtheinvestors(capitalists).Thatcanbeseeninthelightofachangeof variablessuchthatwhenu=0,u˙ =0,U˙ =0andv˙ >0,meaningthatwhentheprofitU attainedbyinvestorsremains constant,i.e.,U˙ = 0,theworkers’sharevgrowswithoutboundandrepresentstheprey,whereastheinvestorsU are intheroleofpredators.HereU isassumedtohaveamaximumvalueequalto100%. On the other hand, following Solow [21], employed workers can be identified with the workers’ share u and unemployedworkers with the variable V. In this case the conflict is between employed and unemployed workers. Whenu= 0,u˙ = 0andV˙ < 0. Thisisconsistentwiththeemployedworkforceuintheroleofprey,theunemployed workersV beingidentifiedwiththepredatorsandtheinvestorsaspassivenon-players. However, these interpretationsshould not be taken at their face values as they are dependenton the conditions given by equations (3). Such parameter constraints were, however, not established from an analysis of real-world data,butcamefromentirelyheuristic,andsofarverypoorlytested,reasoning. Inaddition,sinceasseenabovethe variables u and v can be identified in more than one way, this means that such interpretations must be done with careandalwaysinthelightofreal-worlddataanalysisandnotonaspeculativebasis. Asfurtheremphasisofthese difficulties, one may even argue that the constants of the model may not be constants at all, but time dependent variablesthemselves(seebelow). 2.4. OriginsoftheGoodwinmodel Asnotedabove,whendevelopinghismodelGoodwinaimedatputtinginmathematicalformMarx’sconceptual ideasaboutcyclesincapitalism. However,aspointedoutbyKeen[28],Goodwinalsowishedtoshowhowcyclical behaviorcouldarisefromverysimpleeconomichypotheses. Nextweshallpresentasimplederivationofthemodel inordertohighlightthatitresultsfromanextremelysimplifiedrepresentationoftheeconomy. LetK betheamountoffixedcapital(plantandequipment)andY theoutputthataneconomycangenerate. The outputtocapitalratioσclearlyvariesovertimeinacountry,butletusconsideritaconstantasafirstapproximation andwriteitasfollows, σ= K/Y. (17) IfListheamountoflaborforagivenoutput,onecanalsoassumeasfirstapproximationaconstantoutputtolabor ratioa,thatis, a=Y/L. (18) TheamountoflaborcanbewrittenintermsofthepopulationN andtheemploymentratevasfollows, L= Nv. (19) Letwbetheaveragewagevalue.Thenthewagebill,thatis,thetotalamountofwagesinaneconomyisgivenby, W =wL. (20) Atafirstapproximationtheemploymentratecanberelatedtotheriseofwagesasfollows, w˙/w= f (v). (21) 1 Sincethewageshareuisgivenby, u=wL/Y, (22) 5 rememberingthataisconstant,equation(21)becomes, u˙/u= f (v). (23) 1 Thisexpressionreducestoequation(6)if f (v)isassumedtobealinearfunction. 1 TheprofitlevelPisgivenby, P=Y W. (24) − Asafirstapproximationallprofitsareinvested,sotheprofitshareP/Y istheinvestmentΥ. Hence, Υ= P/Y =100 (W/Y)=100 u=U. (25) − − Heretheunitwaschangedto100%duetoourpreviouschoiceofnormalization.Theprofitrateπisgivenby, π= P/K, = Υ=σπ=100 u, (26) ⇒ − whichcanberewritteninfunctionalformasbelow, Υ= f (u). (27) 2 InvestmentisalsotherateofchangeofcapitalK˙/K. So, K˙ Y˙ v˙ Υ= = = +const., (28) K Y v wheretheconstantcomesfromthehypothesisofasteadylaborsupply,e.g., Lchangesexponentially. Summingup wehavethat, v˙/v= f (u), (29) 2 whichreducestoequation(7)if f (u)isassumedlinear. 2 Clearlythemodelresultsfromextremelysimplespecificationsoftheeconomy. But,itissosimplethatitcannot reproducethefrequencypropertiesofoutputgrowthinacertaintimeperiodorthedistributionofrecessionsizesand duration.However,thedynamicstochasticgeneralequilibrium(DSGE)modelsofcyclesadoptedbycurrentneoclas- sicaleconomicscannotdosoeither[29–31],hencewhatisremarkableisthattheveryrestrictedmodelproposedby Goodwinfindsanyempiricalsupportinrealdata[28]. 3. TheGompertz-Paretoincomedistribution Econophysicsis a new research field whose problemsinterest both economistsand physicists. However, when physicists approach a problem traditionally dealt with by economists, they do so under a very different modeling perspective. Although it is uncommon to find methodological issues discussed in physics papers, considering the hybrid nature of econophysicsand the theoretical crisis of the current mainstream economic thought[29–42], it is worthwhile to emphasize the differencesin methodologicalperspectivesbetween physics and economicsregarding modelbuildingand,especially,modelabandoning. Wehavealreadyexpressedsomeofourthoughtsonthistopicin Ref.[26,Sect.3],butafewmorewordsareworthsayingbeforewereviewourapproachtotheincomedistribution problem. Econophysicswasbornandremainsabranchofphysics[43–45], employing,therefore,itscenturiesoldproven epistemologicalmethodology. Itconsidersascientifictheoryasbeingmadebylawsofnature,whicharetheoretical constructs,oftenexpressedinmathematicallanguage,thatcaptureregularities,processes,structuresandinterrelation- ships of reality. Successful physical laws provide good empirical representations, or images, of the real world, of nature,andallowustoreachpredictionsregardingtheoutcomesofprocessesthatdogooninnature. However,by beingimagesofnature,theselawsareobviouslylimitedand,hence,theywillalwaysprovideimperfectrepresenta- tions. Theonlywaywecanascertainhowimperfecttheyareisbypractice,i.e.,bycreatingpragmaticmeasuresof theadequaciesoftheselaws,alwaysempiricallycomparingtheirpredictionswithwhatoccursintherealworld[46]. Inotherwords,goodlawsprovidegoodpredictions,badlawsprovidebadpredictions. Thishasnothingtodowith 6 theextensiveuseofmathematicsbyphysicaltheories.Mathematicsisalanguage,atoolofformallogic,andbyitself has no a priorirelationship with physical, or social, reality. Physicists choose if and which mathematicaltools are requiredtoexpresssomethingobservedinnature.1 Sinceourunderstandingofthetheoriesbehindtheselawschangeswithtime,thesameoccurringwiththemeasures ofadequaciesduetotechnologicaladvances,wemustkeepmeasuringtheadequaciesoftheselawsbyperfectingold measuresaswellascreatingnewones,thatis,constantlyupdatingourtheoriesandmodelsthroughpracticeinorder to find their limits of validity. The theoretical aspects behind these laws, even their metaphysical presuppositions, mustalso be perfectedbysheddingthe inappropriateelementsso thatthe appropriateresidueremains, in a process very similar to Darwin’s natural selection. And, if there is no appropriate residue left the theoretical construct is abandoned,becomingextinct[49]. Underthisviewpoint,amodelisamorerestrictedtheoreticalconstruct,takingone ortwoelementsabove–regularities,processes,structuresandinterrelationships–,butnotallofthem. Nevertheless, a modelis also subjectto measuresof adequacyand since theyincorporateless elementsthan a theory, it suffersa morerapidprocessofperfectionbyselectionaswellasextinction. Physicistshavebeenfollowingthismethodologicalapproachforcenturiesandasaconsequencetheyhaveamassed alargenumberofphysicaltheoriesthatwereperfectedbygenerationsofphysicists,whokepttheirappropriatekernels butchangedtheir originalelements in variousdegrees, and also to manyother theorieswhich are now superseded. Theoreticalpluralismistacitlyacceptedasanessentialelementforthedevelopmentofphysics. Realsciencestarts fromobservationofnature,eitherphysicalorsocial,andanytheoreticaldiscussionmustkeepreferringbacktoem- pirics,afactorthatlimitsandguidesanytheoreticaldebate,leadingtohealthyrefining,replacingorevenabandoning oftheoriesandmodels[50]. However,itseemsthatthismethodologicalviewpointregardingmodelcheckinghasnotbeenadoptedbyasizable numberof economists. Econophysicistsare often perplexedto witness how ofteneconomistsconfuse their models with reality, showinga behaviorwhich was alreadydescribedas ‘scientific dogmatism’[46]. Thus, theywouldof- ten disregard startling obviousempirical facts rather than change or dismiss their inappropriate theories or models [51,52],showingtoalargeextentanabsolutedevotiontotheoreticaleconomicconstructs,especiallyanempirically unwarrantedobsessionwithequilibrium,inparalleltolittleornoempiricalinterest,oftenkeepingsuchatheoretical worshipevenwhenempiricalevidencethatmightsupportthe theoryis absent. Worse still, evenwhenthereis evi- dencethatdirectlycontradictswhatwouldbepredictedtooccurbyapplyingthetheories[53,pp.2-5]. Somewould say this phenomenonis due to ‘ideological assumptions’, disguised visions of the world under scientific pretenses [48]. Otherscallthisbehavioralmode‘cargoculteconomics’[54]inreferencetothefamousFeynmanspeechabout methodologicallyinadequate, or false science [55, 56]. Nevertheless, the epistemological ideas above, adopted by physicistsalongtimeago,areapparentlybeingslowlyabsorbedintotheeconomicthought[57,58]. Havingstatedourmethodologicalviewpoints,nextweshallreviewthebasichypothesesandequationsbehindthe GPDasadvancedinRefs.[26,27]. 3.1. Definitions Let (x)bethecumulativeincomedistributiongivingtheprobabilitythatanindividualreceivesanincomeless F thanorequaltox. ThenthecomplementarycumulativeincomedistributionF(x)givestheprobabilitythatanindivid- ualreceivesanincomeequaltoorgreaterthanx. Itthenfollowsthat (x)andF(x)arerelatedasfollows, F (x)+F(x)=100, (30) F wherethemaximumprobabilityistakentobe100%.Herexisanormalizedincomeobtainedbydividingthenominal incomevaluesbysome suitablenominalincomeaverage[26]. If bothfunctions (x) and F(x) arecontinuousand F 1Herewetake aviewpoint different fromLawson’s [47]regarding the roleofmathematics ineconomics, aviewpoint basedonthelarger experienceofotherscienceswhichsuccessfullyadoptedmathematicalmodeling,especially, butnotrestricted,tophysics. Theobviousfailures ofmathematicalmodelingineconomicsisaproblemspecifictoacademiceconomicsbecauseitmisinterpretedtheroleoftheoreticalthinkingby meansofacontinuingexcessiveemphasisintheoretical introspectionparalleltoastrongdownplayingoftheempiricalcertification ofmodels. Hudson[48]providesaninterestingaccountofwhyandhowacademiceconomicsreachedthispresentstateofaffairs. Onemustnotethatthe impressiveachievementsofthe20thcenturyintheoreticalphysicswouldneverhadoccurredifphysicistshadignoredempiricstotheextentthat academiceconomistsdo. 7 havecontinuousderivativesforallvaluesofx,wehavethat, d (x)/dx= f(x), dF(x)/dx= f(x), (31) F − and ∞ f(x):dx=100, (32) Z 0 where f(x)istheprobabilitydensityfunctionofindividualincome. Thus, f(x),dxisthefractionofindividualswith incomebetweenxandx+dx. Theequationsaboveleadtothefollowingresults, x (x) (0)= f(w)dw, F(x) F( )= ∞ f(w)dw, (33) F −F Z − ∞ Z 0 x whoseboundaryconditionsare, (0) = F( ) (cid:27) 0, ( F( ) = F(0∞) (cid:27) 100. (34) F ∞ Clearlyboth (x)andF(x)varyfrom0to100. F 3.2. TheGompertz-Paretodistribution(GPD) TheGPDwasproposedinRef.[26]anddiscussedindetailinRef.[27].Itscomplementarycumulativedistribution isformedbythecombinationoftwofunctionswhichcanbeidentifiedwiththetwomainclassesformingmostmodern societies,workersandinvestors(capitalists). Thefirstcomponentdescribesthelowerpartofthedistribution,thatis, thosewhosurvivesolelyontheirwages,theworkers,andisgivenbyaGompertzcurve. Thesecondcomponentof thecomplementarycumulativedistributiondescribesthetailofthedistributionbymeansoftheParetopowerlawand representstheinvestors,thatis,therichcapitalists. Thenwehavethat, e(A Bx) G(x) = e − , (0 x< x), (Gompertz) t F(x)= ≤ (35) P(x) = (x)α ee(A−Bxt) x α, (x x ), (Pareto) t − t ≤ ≤∞ andthecumulativeincomedistributionmaybewrittenasbelow, e(A Bx) (x)=100 e − , (0 x< x), t G − ≤ (x)= (36) F (x)=100 (x)α ee(A−Bxt) x α, (x x ). Herex istheincomevaluethresholdoPfthePareto−regtion,αistheP−aretoindtex≤des≤cr∞ibingtheslopeofthepowerlaw t tail, Bisa thirdparametercharacterizingthe slopeofthe Gompertzcurveand A is anumberwhosevalueis setby boundaryconditions, as follows. SinceG(x) = exp exp(A Bx) , the condition(34) impliesG(0) = 100, then we − havethat, (cid:2) (cid:3) A=ln(ln100)=1.5272. (37) Theterm(x)α ee(A−Bxt) isthenormalizationconstantoftheParetopowerlawandcomesasaconsequenceofcondition t (32),aswellasthecontinuityoffunctions(35)acrossthefrontierbetweentheGompertzandParetoregions,defined tobex= x. t Theequationsaboveallowustofindtheexpressionsfortheprobabilitydensityincomedistribution, g(x) = Be(A Bx) ee(A−Bx), (0 x< x), − t ≤ f(x)= (38) p(x) = α(xt)α ee(A−Bxt) x−(1+α), (xt ≤ x≤∞), 8 aswellastheaverageincomeofthewholepopulationdescribedbytheGPD, x = 1 ∞x f(x)dx= 1 (x)+ αxt ee(A−Bxt) , (39) h i 100Z 100"I t (α 1) # 0 − where, (x) xwg(w)dw= xwBe(A Bw) ee(A−Bw)dw. (40) − I ≡Z Z 0 0 Theparametersα,x andBareallpositiveandtheyfullycharacterizetheGPD.However,duetoconvergencerequire- t ments[26], the expression(39) forthe averageincomeis only validif α > 1. Both α and B can be determinedby lineardatafittingsinceequations(35)canbelinearized.However,x isindependentlyfoundundertheconstraintthat t theboundarycondition(37)issatisfiedtowhateverdegreeofprecisiontheavailabledataallow. TheLorenzcurveoftheGPDhasitsX-axisgivenbythecumulativeincomedistributionfunction (x),whereas F thefirst-momentdistributionfunction (x)definesitsY-axis. Accordingly,theycanbewrittenasfollows[27], 1 F e(A Bx) 100 e − , (0 x< x), x t (x)= f(w)dw= − ≤ (41) F Z0 100 (x)αee(A−Bxt)x α, (x x< ), and − t − t ≤ ∞ (x) I , (0< x< x), t x Thus, (x)varieFs1f(rxo)m=0h1txoi1Z000xawsfw(wel)l.dTwh=eLor1eh0n0zi+cu(rα1v(e−xitsα)αu)seuea(lhAlx−yiBrxet)prxe(s1e−nα)t,edin(axut n≤itxsq<u∞are;,).butthenormal(i4z2a-) 1 F tion(32)impliesthatthesquarewheretheLorenzcurveislocatedhasareaequalto104. TheGinicoefficientunderthecurrentlyadoptednormalizationiswrittenas, Gini=1 2 10 4 ∞ (x) f(x)dx. (43) − 1 − × Z F 0 Consideringnowequations(38)and(42),theGinicoefficienthasthefollowingexpressionintheGPD, Gini=1−2×10−4hBxiZ0xtI(x)e(A−Bx)ee(A−Bx)dx+100ee(A−Bxt) + hxαi(2αx−te12)e((1A−−Bx2t)α). (44) As discussed in [27], we can define the percentage share of the Gompertzian part of an income distribution describedbytheGPDbymeansofequation(42). Thisquantitymaythenbewrittenasfollows, u= (x)=100 α xt ee(A−Bxt). (45) F1 t − (α 1) x − h i Hence,weidentitythepercentageshareofthelowerincomestratadescribedbytheGPDwithGoodwin’slaborshare u. Notethatbydoingso, unolongerrepresentstheindustrialreservearmyoflabor,butinfacttherelativesurplus population since the latter includes not only the unemployed, but also those unable to work. Such identification allows the description of the Goodwin variables in terms of measurable quantities connected to different income classeswhoseempiricalvaluescanbeobtained,forinstance,fromtheLorenzcurves. Thisconnectioncanbemade clearerbytheinversionofequation(45), 1 ux ee(A−Bxt) =1 t . (46) α −(100 u) (x) 9 − I t Due to the high non-linearityof this expressionone can only use it to determineα if the values of u, B and x are t knowntoaveryhighdegreeofaccuracy. Theequation(46)linkstheParetoindexαtoparameterswhicharesolelydeterminedintheGompertziansegment of the distribution: the cutoff value x, the Gompertzian percentage share u and its distribution slope B. In other t words,equation(46)linkstheincomedistributionofthelowerandupperclassesformingasociety,showingclearly their dynamical inter-dependency. If we consider that temporal changes in the income distribution do take place, we can no longer consider these quantities as parameters. Some of them, or perhapsall of them, oughtto be time dependentvariables(seebelow). The GPD requires α > 0. In addition, an average income is only possible if α > 1. Considering these two conditionsinequation(46)weconcludethat, ux ee(A−Bxt) 0< t <1 and u<100. (47) (100 u) (x) − I t Remembering equation (16) the last condition is equivalent to U > 0, which means that an income distribution describedbytheGPDisonlypossibleinasystemwhereinvestorshaveanonzeroshareofthetotalincome. 3.3. Exponentialapproximation AsshowninRefs.[26,27],theupperpartoftheGompertzcurvecanbeapproximatedbyanexponentialandthis allowsustotakethissubdivisionoftheGompertzcurveasrepresentingthemiddleclasspresentinmostsocieties. In otherwords,inthisapproachoftheincomedistributioncharacterizationofasocietyweassumethatthemiddleclass isjusttheupperechelonofthewagelaborclass. Thus,forBx> A,e Bx <1andx< x wehavethat, − t G(x) 99+e Bx, − ≈ (x) 1 e−Bx, (48) gG(x) ≈B−e−Bx, ≈ which are already normalized to obey the boundary conditions (34). If the lower stratum of a society is formed essentially by a verylarge middle class, one can in principle write all equationsshown in Sect. 3.2 in terms of the approximations(48),althoughinsuchacasewecanexpectacertaindegreeofdistortioninthedistributionsinceall modernsocietiesseemtohaveacertainpercentageofverypoorpeople,howeversmallthispercentagemaybe. 4. CyclesintheincomeandemploymentdataofBrazil Publicly available individual income distribution data of the Brazilian population have allowed Moura Jr. and Ribeiro[26]todeterminetheGPDparametersfrom1978to2005afteracarefulhandlingofthedata.ChamiFigueira, MouraJr.andRibeiro[27]extendedthisanalysistoincludeincomedatafor2006and2007,aswellasshowinghow the GPD producesresults compatible with those obtaineddirectly from the raw data, that is, withoutassuming the GPD, with errormarginsup to 7%. In this workwe furtherextendthese two previousanalyzesto includedata for 2008and2009,butdisregardingtheresultsfor1978and1979duetotheirunreliability[27]. Table1presentsthethreeGPD parametersB, x andαfollowedbytheunemploymentrate[V],Ginicoefficient t and the percentage share of the Gompertziancomponentof the distribution. B and α were obtained by linear data fittingwhereas x wasdeterminedsuchthatalinearfitwouldproducetheboundarycondition(37)withdiscrepancy t ofabout2%.LorenzcurvesweregeneratedfromtherawdistributionforeachyearallowingthecalculationoftheGini coefficientwithoutassumingtheGPD,denotedhereas[Gini]inordertodistinguishitfromtheoneobtainedassuming the GPD in equation (44). Once x was found it became possible determine [u] directly from the raw data, that is, t withoutusingequation(45). Similarly,[V]denotestheunemploymentdatawithoutanydistributionassumptions,[v] isobtainedusingequation(15)and x istheunemploymentincomethresholdusedtocalculate[V](seebelow). The d timederivativesaregivenbytheexpressions, d d [u˙]= [u], [v˙]= (100 [V]). (49) dt dt − 10
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