A two-sector model of the business cycle: a preliminary analysis

Gozzi, Giancarlo; Nardini, Franco Working Paper A two-sector model of the business cycle: a preliminary analysis Quaderni - Working Paper DSE, No. 382 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Gozzi, Giancarlo; Nardini, Franco (2000) : A two-sector model of the business cycle: a preliminary analysis, Quaderni - Working Paper DSE, No. 382, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4927 This Version is available at: https://hdl.handle.net/10419/159223 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Atwo-sectormodelofthebusiness cycle:a preliminaryanalysis¤ GiancarloGozzi DipartimentodiScienze Economiche FacoltàdiEconomiay UniversitàdiBologna FrancoNardini DipartimentodiMatematicaperleScienze EconomicheeSocializ FacoltàdiEconomia UniversitàdiBologna February2000 Abstract Inthispaperatwo-sectordynamicmodelofbusiness ‡uctuations ispresented.Itisadisequilibriumdynamicmodelwithtwolawsof evolution(dynamiclaws)builtintoit:pricesofcommoditieschange accordingtothemarketdisequilibriumofsupplyand demand,while quantitieschangeaccordingtothestockdisequilibriumand theshifting ofthedegree ofutilizationofproductive capacityawayfromitstarget value.Investmentby…rmsismodelled byanonlinearaccelerator. Nonlinearityintheinvestmentfunctionmakesthe equilibria of themodelunstableand causesgrowingdisproportionalitiesbetween thetwosectors;business ‡uctuationsaretheoutcomeoftheswitching ¤Theauthorsacknowledgetheirdebt toR.Scazzieriforenlighteningdiscussionsand valuablesuggestions,toS.Zamagni, G.Candela,and F.Zagonariforcommentsand encouragement.Preliminarydraftsofthepaperhavebeen persentedat the2ndEuropean SummerSchool in StructuralChangeand EconomicDynamicheldin SelwinCollege, July1996,at theXX ConvegnoAnnualedel’AMASES heldinUrbino,September1996, at theConvegnoCNRsulleTeorieGenerali inEconomiaheldinTrento,June1997;the authorswishestothankallparticipantstothediscussionsforobservationsandconstructive criticism. yPiazzaScaravilli, 2 I-40126 BOLOGNA ITALY tel+39 051 209 8664 fax+39 051 209 4367 e-mail [email protected] zVialeQuiricoFilopanti, 5 I-40126 BOLOGNA ITALY tel+39 051 209 4364 fax+39 051 209 4367 e-mail [email protected] 1 ofthesystemto a di¤erentregimethatallowstoreduce the existing disproportionality. Thedi¤erentregimesintowhichthe economymaybefound area situationofoverheating and oneofdepression.Afundamentalrolein theswitching ofthe economyisplayed bytwocrucialfeaturesofthe capitalgoodsector: itslimited productive capacityand thetime-lag requiredtoincreaseit. 1Introduction Inthefollowingpageswepresentatwo-sectorsdynamicmodelwithone sectorproducingthe…xedcapitalgoodand theotheroneproducingthe consumptiongood.Themodel isframedincontinuoustime;weshall considerthe casewhenadjustmentofproductive capacitytakestimeand isnot instantaneous. Theaimofourmodel istoprovideamathematicalformulationforthe multisectortheoriesofbusiness ‡uctuationsaccordingtowhichthe causes ofperiodicalcrisesaretobesearchedin periodsofoverproduction, inthe consumptiongoods sectorespecially;theseare, inturn,duetothedi¤erent lengthsoftheproduction processesinthedi¤erentsectorsofthe economic systemand tothedisproportionamongdi¤erentsectorscaused bythese di¤erentlengths1. 1.1Di¤erentmarketsfordi¤erentgoods Since theworking ofeachsinglemarketdependsonthe characteristicsofthe tradedgood2,thetwosectorsturnout tobeasymmetric.Nowadaysthetypicalend-productsofthemanufacturingindustryareratherspecializedand manufacturing and sellingcomeinsubstance underthesame control3;asa consequence “thesellingdepartmentisabletosetasellingprice and make ite¤ectivebyholdingstocks”4.Firmsinthe consumptiongoods sector, therefore,throwininventoriesto…ll gapsbetween demand and supplyand taketheirshortrun decisionstoproduce inorderto approachstockequilibrium5.FollowingHicks6,weassumethat…rmstrytokeepinventoriesnear to a targetvalue. 1EarliercontributionsincludeAftalion(1913),(1927),Fanno(1931),TuganBaranowskj (1913)and Hicks(1974)amongtheothers.To ourknowledgetheonlyattempt toformalize suchclass oftheorieshasbeenthesimple,butinterestingmodelofDelli GattiGallegati (1991). 2Forathorough discussionofthisissuewerefertoHicks(1989),chapters1to 4. 3See Hicks(1989),chapter3;foradiscussionofthestrategicrôleofpricingin highly concentratedindustrieswereferalsotoJudd and Petersen(1986). 4Hicks(1989),p.24. 5Forthe conceptofstockequilibriumwereferthereadertoHicks(1965),ch.VIII, and Hicks(1974),ch.1. 6See (1974),ch.1. 2 Ontheotherhand weassumethat…rmsinthe capitalgoods sector produce onlyto orderand hence taketheirshortrun decisiontoproduce knowingdemand.Theydonotholdinventories,therefore,and changeprices to o¤setpossiblediscrepanciesbetween demand and their(maximum)productive capacity7. 1.2Investmentand speci…city of…xedcapital Asfarasinvestmentisconcerned,weassumethatpro…tsarenormallyreinvestedinthesectorwheretheyarose,exceptwhen pro…tsinonesectorare solowthat theremunerationofcapitalfallsbelowitsnormal level8.Moreover,whenadisequilibriuminthe consumptiongoodsmarketis signalled byananomalouscapacityutilizationratetheinvestmentiscorrected byan accelerator9,whilethe capitalgoods sectormayrationits supplywhenits productive capacitydoesnotmatchthedemand.Inthis sensethe consumptiongoods sectormaybeformallytreated,atleast toalarge extent,asa sectorofamodelofcompletedisequilibrium10. Itisclearthat the capitalgoods sectorplaysakeyrôlewheneither rationing occursortherateofremunerationofcapital isdi¤erentforthetwo sectors,whereasitsdevelopmentfollowsthedevelopmentofthe consumption goods sectoralongthe equilibriumgrowth path11. Evenifonlyone capitalgoods sectorisconsideredinthemodel, we assumethatallocationsareirreversible;once theallocation hasbeenmade, the capitalgoodsbecomespeci…cand can notbetraded.Thismeans, in particular,thatcapitalgoodscan notbemovedfromonesectortothe other12. 2Assumptionsofthemodel Inourmodel…rmswithineachsectorarealikeand thiswill allowustomake useoftherepresentative…rm’sconcept. Since oneofouraimsisto analysethe emergence ofbottlenecksdueto thelengthofthe capitalaccumulation process,weareledto assumea…xed 7Theseassumptionspresupposethat therateofutilizationofproductive capacityis normallybelowitsmaximumvaluein bothsectors.WerefertoDuménil and Lévy (1991) foradiscussiononthispoint. 8The conceptofnormalremunerationismadepreciseinformulae(9)and (14). 9Theacceleratorinamultisectormodelofbusiness ‡uctuation hasbeenintroduced by Aftalion(1913),Fanno(1931)and Hicks(1974). 10See,forinstance,Duménil and Lévy (1991). 11The evolutionofthesystemisthusfullyendogenous. Ourmodelallowsforthe descriptionofexogenousevolutiontoo, in particularofatechnicalprogress whichchanges the coe¢cientsoftheinputmatrix,wehopetodevelopthisissueinafuturepaper. 12Theassumptionofirreversibleallocationshasalongtraditioninmacroeconomicmodelling;wereferforinstance toSolowetal(1966)and Bliss (1968). 3 coe¢cientrule:anyconcreteunitofcapitalhasa givenoutputcapacity and requiresa givencomplementoflabour, i.e.thereisnosubstitutability betweenlabourand capital. Ontheotherhandlabourisperfectlyelasticandtherealwageisconstant througout theanalysis. Thusweassumethat thetotaloutputYiofthei-thsectorisgiven bya Leontieftypeproductionfunctionwithcomplementaryinputs: Yi=min½biKi;bi liLi ¾;(1) withLidenotingtheinputoflabourinthesector,bitheproductivityof capitaland bi=litheproductivityoflabour,i=1;2. Thestockof…xedcapitalofthei-thsector,Ki,depreciatesataconstant rate±(thesamein bothsectors13); ifIidenotesthegross investmentin thei-thsectorthenthe evolutionofthestockof…xedcapitalengagedin production,_ Kiisgiven by: _ Ki=Ii¡±Ki:(2) Letwbetherealwageand p1theprice ofthe consumptiongood;theincome oftheworkersemployedinthei-thsector,Wi, istherefore: Wi=wp1Li:(3) Weassumenosaving outofwages;thereforethewholeincome(3)will be spentonthe consumptiongoodsmarket. 2.1Theconsumptiongoodsector Inthe consumptiongoodsectorproductioncantakeplace onlyifthe conditionY1p1¡W1>0(4) holds;thismayberewritteninthefollowingform: b1¡wl1>0:(5) Condition(5)meansthat therealwagewiscompatiblewiththeproductive technologyemployedinthe…rstsector14.The evolutionofinventoriesS1in the consumptiongoodsectordependsonthegap betweentotaloutputY1 and demand forconsumptiongoods: _ S1=Y1¡D1:(6) 13The caseofasectordependentdepreciationrate can betreatedwithminorchanges. 14Condition(4)hastoberegardedasasurplusconditionand notasapro…tcondition. 4 Netpro…tsinthe consumptiongoodsector,¦1,arede…ndedasthedi¤erence betweenthevalueoftotaloutput,Y1p1,and thesumofthewagespaidto thelabour-force and thedepreciationof…xedcapital, W1+±K1p2: ¦1=Y1p1¡(W1+±K1p2);(7) wherep2isthe(unit) price ofthe…xedcapitalgood.Pro…tsaredividedinto twoparts,oneforconsumption,¦c 1,andtheremainingoneforaccumulation, ¦a 1;therefore¦1=¦c 1+¦a 1:(8) Concerningthepro…tsallocatedtoconsumptionweassumethat thispart isproportionaltothevalueofthestockofproductive capitalofthesector; therefore:¦c 1=dK1p2;(9) whered>0isthedividend. Remark1Westress thepoint thataccumulated pro…ts,¦a 1,neednotbe positive.Inthiscase…rmsinthesectorarenotonlyunabletomake (positive)netinvestmentbutalsotoreplace depreciated…xedcapital.Moreover, thesurpluscondition(5)doesnotguaranteesenough output to paythescheduledremuneration(9)ofthe…xedcapital. Whenthishappensthisremunerationinpaidresortingtotheinventories, whenevertheyaresu¢cient15. LetW2bethetotalwagesand ¦c 2thedistributed pro…tsofthe capital goodsector;thedemand forconsumptiongoods,D1, isgiven by: D1=¦c 1+¦c 2+W1+W2 p1:(10) Totalcapitalengagedintheproductionofconsumptiongoods,C1, isthe sumofthevalueof…xedcapitaland thevalueofinventories: C1=K1p2+S1p1:(11) Furthermoreletu=Y1=(b1K1)bethecapacityutilizationrate,s=S1=(b1K1) theinventoriesratio and ·=C1=K1theunitcapitalengagedintheproductionofthe consumptiongoods. 15Theassumptionthat…xed productive capital is speci…ctothesectorpreventscapitaliststomoveitfromasectortotheotherevenwhentheformerloosesitspro…tability.In thisevent thesurpluscondition(5)guaranteesthatgoing on producingdiminisheslosses. Themodelcoversthe caseofcapitaliststhatare entrepreneursinoneofthetwosectors or,moregenerally,capitalistsand entrepreneursineachsectorbelongtothesame class; inthiscasethegross pro…tisdistributedasremunerationand accumulated pro…t. 5 2.1.1Outputand pricedecisionsintheconsumptiongoods sector Productionand price decisionsof…rmsinthe consumptiongoods sector aredeterminedontheonesidebytheneedto avoidthatsuddenjumpsof thedemand maycauseunacceptable changesofpricesofthegoods16 and ontheotherhand to avoidexhaustionaswell asexcessiveaccumulationof stocks,whichareusedasanalternativetoprice changesinorderto o¤set disequilibriainthemarketforconsumptiongoods.Weassumetherefore that,undernormalconditions,…rmshaveamarginofunused productive capacity, i.e.0<¹u<1,with¹urepresentingthenormaldegree ofcapacity utilisation17 and utheactualone. Inthesamemannerweassumethat…rmshaveatargetvalueforthe inventorylevel, ¹s.Whentheactual levelofinventories,s, isbelowits normalvalue,¹s,…rmsrealize that thereisdisequilibriumonthemarketfor consumptiongoodsand reacteitherbyincreasing outputorbyincreasing pricesorboth;aperfectlysymmetricsituationariseswhentheactual level ofinventoriessisaboveitsnormalvalue¹s.Firmschoose,therefore,their behaviouraccordingtothedisequilibriumsignalsthat theyobserve, i.e. marketdisequilibrium(¹s6=s)and disequilibriumin production(¹u6=u). Asfarasthedecisiontoproduce of…rmsinthe consumptiongoods sectorisconcernedweassumethat: _u=F(u;s);(12) whereF2C1isafunction18 suchthatF(¹u;¹s)=0,@F=@u<0,@F=@s<0 and withlim u!0+F(u;s)= +1and lim u!1¡F(u;s)=¡1foranys>0. Equation(12)saysthat…rmsinthe consumptiongoods sectorincreasethe utilizationoftheirproductive capacityifinventoriesand capacityarebelow theirtarget (ornormal)valuesand decreaseitintheopposite case. Concerningpriceswesupposethatp1isadjusted bythe…rmsaccording tothelevelofinventories, i.e.tothedisequilibriumobservedonthemarket; _p1=p1g1(s);(13) whereg12C1isafunctionsuchthatg1(¹s)=0,dg1=ds=g0 1<0and with lim s!+1g1(s)=¡1and lim s!0+sg1(s)= +119. Insuchawayweareabletostudytheproblemofthedynamicbehaviour ofthe economyfarfromthe equilibriumpositionor,alternatively,whenthe normalorlong-termequilibriumofthe economyis(locally)unstable. 16Since theprice isastrategic characteristicofthegood(see subsection1.1forfurther details) too suddenand wide changesmayresultunacceptable. 17ObviuoslyuÀ1,apossiblelowerbound maybeforinstance 0:8. 18Thishypothesesonthebehaviourof…rmsinthe consumptiongoodsectorexclude that themaximumproductive capacityb1K1can be ever reached; inthis sector,therefore, wealwayshaveY1=(b1=l1)L1(see (1)). 19Thisconditionmeansthat…rmsdecidestrongprice increasetopreventexhaustionof resources. 6 2.2Thecapital goods sector Therelationsconnectingtotaloutput,Y2,employment,L2,totalwages,W2, and pro…ts,¦2,ofthe capitalgoods sectorareanalogoustothe corresponding onesinthe consumptiongoods sector: ¦2=Y2p2¡(W2+±K2p2)(14) ¦c 2=dK2p2(15) with¦c 2denotingdistributed pro…tsinthe capitalgoods sectorand ¦a 2= ¦2¡¦c 2pro…ts savedand accumulated. Since itisimpossibletouseinventoriesinordertopaydividends,distributed pro…tsinthe capitalgoods sectorcan notexceedthenetpro…tsof thesector:¦c 2=minfdK2p2;Y2p2¡W2g:(16) Weremarkexplicitlythat theassumptionofanequaldividend perunitof …xedcapitaldin bothsectors should notleadtothe erroneousconclusion that therateofpro…tisthesamein bothsectorsaswell20. Inthiscasetoo weassumethat theproductionofcapitalgoodstake place onlyifthesurpluscondition Y2p2¡W2>0(17) holds(see(4));suchaconditionentailsan upperbound fortherelativeprice p=p2=p1, i.e. p·b2 wl2:(18) We caninterpret thisconditionasan upperbound forthenominalwage ratewp1thatguaranteesitscompatibilitywiththetechnicalproduction conditionsand theprice ofthe capitalgood. 2.2.1Productionand pricedecisionsinthecapitalgoods sector Firmsinthe capitalgoods sectorhavenoreasontoholdinventories,aswe areassumingthat theyproduce to order;thereforeS2=0and thesupplyof capitalgoodsalwayscoincideswithtotaloutputY2.Production decisions areformalizedinthisway: Y2=½D2ifD2·b2k2 b2K2ifD2>b2K2;(19) where:D2=Ia 1+I2(20) 20The caseoftwodi¤erentdividendsinthetwosectorscan betreatedalongthesame lines,withonlyminorchanges. 7 isthedemand forthe capitalgoodandwithI2thatdenotesgross investment inthe capitalgoods sector.Productionofthe capitalgoodsisequalto demand when demand (20)doesnotexceedtheproductive capacityofthe sector.When demand ofthe capitalgoodsexceed productive capacitywe areinasituationofrationing; inthiscaseweassumethatitisthedemand ofcapitalgoodscomingfromthe consumptionsectorthatremain partly unfull…lledandthereforeex-postinvestmentintheconsumptiongoodssector isgiven byI1=b2K2¡I2. Thusthedi¤erentnatureoftheproducedgoodsand theresultingdifferentstructureoftheirmarket21 determinequitedi¤erentcriteriaforthe decisionmaker.Thisistruenotonlyforoutputdecisions,butalsoforprice decisions.In particularthereisnopointfora…rminthe capitalgoods sectortohaveamarginofunused productive capacity,becauseitknows demand and hasnoproblemwithstockequilibrium. Inviewofthisweassumethat…rmsinthe capitalgoods sectorset the price p2accordingtoaWalrasianadjustmentmechanism: _p2=p2g2 µD2¡Y2 Y2 ¶;(21) whereg22C1isafunctionthatsatis…esg2(0)=0,g0 2>0and lim s!+1 g2(s)= +1. Remark2We explicitlynotice that thenormalizationinequation(21)is thesameusedinthede…nitionofuands,since Y2=b2K2wheneverD2> Y2. 2.3Investmentdecisions Investmentdecisionsofthe…rmsplayafundamentalrôleinthedynamic evolutionofthe economy.Two aspectsmustbetakeninto account: 1.the…nancing ofinvestment, i.e.theallocationofcapital(asumof purchasingpower) tothe…rmsbythe capitalists;this…nance isthe sumofthe capitalsetfree bytheproduction process,±Ki(i=1;2), and ofthepro…tsaccumulated; 2.theactualusebythe…rmsofthesumsodetermined. Point1requiresthespeci…cationofamechanismaccordingtowhichcapitalistsallocatetheirpurchasingpowerto…rms.Inourmodelweassume thatcapitalistsdonotknowtherateofpro…tofthedi¤erentsectors;they only knowtherateatwhichinvestmentsareremunerated22.Wefurther 21WereferthereadertoHicks(1989). 22Thisassumptionmustbemodi…edifweareassumingthatsome entrepreneursare alsocapitalists(see remark(1)and therelativenote). 8 there existsatime^ tsuchthat: 8 > > > < > > > : lim t!^ t¡p(t)=0 lim t!^ t¡s(t)= +1 lim t!^ t¡u(t)=^u;(44) where0<^u<¹u.Inthiscasetoo the conditionlim s!+1sg1(s)=¡1yields: lim t!^ t¡s(t)p(t)=0:(45) We call thisalternativebehaviourthedepressionregime. 4Thedepressionregime 4.1 E¤ectsofthedepressionregime Westartbyanalysingthedepressionregime.Inthis sectionweprovethat, aslong asthedepressionlastsand worsens,animbalance betweenthetwo sectorsarisesand theproductionofcapitalgoodsbecomesless and less pro…table.Theproofreliesonshowingthatcondition(34)nolongerholds and thisdoesmeanthat the contractionofinvestmentinthe consumption goods sectorcausesanovercapitalisationofthe capitalgoods sectorand its increasingloss ofpro…tability. Theorem2Iftheinitialconditionslead to a depressionregime(44),(45) thenthere existst=^ t2suchthat35: I2(t)>0t<^ t2(46) andI2¡^ t2¢=0:(47) Proof.Conditions(44)and (45)yield: 8 > > > < > > > : lim t!^ t¡·(t)=1 lim t!^ t¡®(t)=¡d lim t!^ t¡¯(t)=0 MoreoverifI2(t)>0did holdforeveryt>0thencondition(34)would holdand hence bothI1(t)>0and z(t)>0would holdforeveryt>0, 35Thisresultisinaccordance withthewidelyrecognizedevidence thatslumpshit the capitalgoods sectormorerapidlyand deeplythanthe consumptiongoods sector. 15 since (34))(33).Inthiscasethelefthand sideof(34)would bepositive foreveryt>0. Ontheotherhand lim t!^ t¡®(t)+b1¯(t)=¡d<0and thustherighthand sideof(34)would become zero atleastinapointta;thiswouldmeanthat I2(ta)=0,again by(34).Whence the contradiction. RemarkItisevident that^ t2isthetimewheninvestmentinthe capital goods sectorisnomorepro…table;thefall oftherateofcapacity utilization hascausedacontractionofinvestmentinthe consumption goods sectorthatresultsinanincreasingexcess oftheproductive capacityoftheothersector,thateventuallycan notremuneratethe …xedcapitalanylonger. 4.2Thesystemduringthedepression Tounderstand the evolutionofthesysteminthedepressionregime,we analysethe changesthat thedynamicalsystem(38)undergoeswhenI2(t)= 0.Since (33)holdsalsoinarightneighbourhood^ t2,from(19),(29),and (27)we can determinethetotaloutputofcapitalgoods: Y2=(®+b1¯u)h(u)K1: >From(16)weobtainthat thedistributed pro…tsofthe capitalgoods sectorinatimeintervalfollowingtheinstant^ t2arereducedto: ¦c 2=Y2¡wpL2 =µ1¡wl2 b2p¶(®+b1¯u)h(u)K1:(48) Since the conditionsunderwhichthe consumptiongoods sectoroperates, remain unchangedthroughout theinstant^ t2,we caneasilydeterminethe demand ofconsumptiongoodsfrom(3),(7),(10),and (48).Bythesame reasoning ofsection2.4itiseasytorealize that the evolutionequationfor theratio ofinventoriesholdsalso after^ t2. Ontheotherhand the evolutionequationfortherelative…xedcapital stockisdi¤erent,since from(48),(14),(15)itiseasytosee that,after instant^ t2,wehave¦a 2=¡±K2;thefall inthepro…tabilityofthe capital goods sectorprevent…rmstoreplace eventhedepreciated …xedcapital. We have,therefore,thefollowingdi¤erentialsystem: 8 > > < > > : _u=F(u;s) _s=· b1p(®+b1¯u)(1¡h(u)) _p=g1(s)p _z=¡(®+b1¯u)h(u)z :(49) 16 The considerationsthatledto(44),(45)and theproofoftheorem2hold alsowith(49).Thediminishingpro…tabilityinthe capitalgoods sectordo notchangethe evolutionofthedepression;thefall intherelativeprice p and theincreaseoftheinventoryratioscontinuealso after^ t2aslong asthe pro…tabilityofthe consumptiongoods sectorpetersout.Thuseventually Ia 1=0;thiscertainlyhappensfort>^ t1(with^ t1>^ t2)duetotheproofof theorem2 and (27). After^ t1investmentinthe consumptiongoods sectorand productionin the capitalgoods sectorcomesto anend togetherwiththedistributionof pro…tsand thepaymentofwagesinthesecond sector. Onthe contrarywe havesupposedthat…rmsinthe consumptiongoods sectorgo on distributing theplanned dividends(9)byresortingtoinventories. Towritedownthenewdynamicalsystemwebegin bynoticingthatif I1=I2=0then_z=0. Since totaloutputofcapitalgoodsiszero,totalwagesand distributed pro…tsofthis sectorareobviouslyzero,W2=¦c 2=0,whiletheanalogous quantitiesinthe consumptiongoods sectorarestill given by(7),(3),and (9)respectively;thedi¤erentialsystemsisthereforethefollowing: 8 > > > < > > > : _u=F(u;s) _s=³1¡wl1 b1 ´u¡d p+±s _p=g1(s)p _z=0 :(50) 4.3 Existenceofaturningpoint >Fromthethirdequationof(50)itiseasytosee that thedeclineofthe relativeprice continuesalsoinatimeintervalfollowing^ t1,whentheproductionofcapitalgoodsdoesnot takeplace,since s(t)>¹sholdsinaright neighbourhoodof^ t1;howeverthesecond equationof(50) togetherwith(45) provesthat theratio ofinventoriesincreasesup to^ t0(with^ t0>^ t1),while afterthat timeitbeginstodecline.Thehigherthelevelhasgoneup,the quickeritsfall will be; itcontinuesatleastuntil s(t)=¹s;at thismoment thefall oftherelativeprice pstopsand anewincreasestarts. The capacityutilizationratealsostopsfallingashortertimeafter^ t0and startsagainincreasingwhiletheinventoriesleveldeclines.All thesefacts allowthesystemto approachoneofthe equilibriumpoints(39). Wefeel itisworth notingthatall these changesare completelyendogenoustothesystemand wedonotneedtosuppose eitherachangeinagents expectationsorintheirbehaviourinorderforthemtotakeplace. 17 5Theoverheatingregime 5.1 E¤ectsoftheoverheating Wemovenowtotheanalysisand descriptionoftheoverheatingregime. Whentherelativeprice isboosted byademand whichexceeds supplyof consumptiongoodsand reducesinventories,the consumptiongoods sector increasesitspro…tsand itsproductive capacityaswell, since expectations areinfavournotonlyofafurtherincreaseofdemand butevenofanincrease exceedingtheoneinthe capitalgoods sector.All thiscan be easilyseen by consideringthat (42)and (43)yield: 8 > > > < > > > : lim t!~ t·(t)= +1 lim t!~ t®(t)=± lim t!~ t¯(t)= +1:(51) Therefore_zsoon becomesnegative. Infull analogywiththe considerationsleadingtotheproofoftheorem 2we canshowthatcondition(36)can notholdforeveryt>0when(51) holds.There exists,therefore,aninstant~ t0suchthateither rationing of capitalgoodsoccursorthe capitalgoods sectorisnolongerabletodistribute theplanned pro…ts. The…rstalternativeoccurswhenthesize ofthe consumptiongoods sectorexceedsthatofthe capitalgoods sectorsomuchthat thelatterhasno longerenough productioncapacityinordertofullysatisfythedemand for itsproducts.Inthiscasethe capitalgoods sectorturnsout tobeundercapitalisedwithrespect tothe consumptiongoods sector. Thesecond alternativeoccurswhentheprice ofconsumptiongoods reaches suchahighlevelthat the capitalgoods sectorisnolongerable to accumulatepro…ts,moreoveritiscompelledtoreduce thedistributed partundertheplannedlevel. Inthiscasethe capitalgoods sectorturnsout tobeovercapitalisedwithrespect tothe consumptiongoods sector. 5.2Thefall ofpro…tsin thecapital goods sector Westartbyexaminingthesecond alternative.Thedi¤erentialsystemis again(49);thereforethe capacityutilizationrateu,theratio ofinventories sand therelativeprice pdonotchangetheirevolutionsafter~ t0,while therelative…xedcapitalstockzcontinuestofall. There exists,therefore, ~ t1>~ t0suchthat (35)doenotholdfort>~ t1and rationing ofcapitalgoods starts.After~ t1ex-postinvestmentI1inthe consumptiongoods sectorequals themaximumproductive capacityofthe capitalgoods sector,since inthis sectornoinvestmentistakingplace after~ t0.ThereforeI1=b2K2and _ K1=b2K2¡±K1whileI2=0and _ K2=¡±K2. 18 Asfarastotaloutput,employment,totalwagesand distributed pro…ts inthe consumptiongoods sectorare concerned,(3),(9)still hold,whilein the capitalgoods sectorwehavenowthefollowingrelations: Y2=b2K2(52) L2=l2K2(53) W2=wl2pK2(54) ¦c 2=µ1¡wl2 b2 ¶b2K2:(55) Themeaning of(52)and (53)isobvious;accordingto(52) the capitalgoods sectorhasreacheditsmamimumcapacityutilizationratewhile(53)yields the employmentrequiredtomaintainsuchalevelofproduction.Totalwages (54)aredetermined using(53)and (3)whiledistributed pro…ts¦c 2aregiven by(16). Inordertodeterminethe changeofinventoriesinthepresentcasewe begin bydeterminingthedemand forconsumptiongoodsusing(9),(21), (10),(54),and (55).Then havinginmind the evolutionofthe capitalstock K1and thesecond equationin(49),weobtainthe evolutionequationfor theinventorieslevel. The evolutionequationfortherelativeprice pis: _p p=_p1 p1¡_p2 p2=g1(s)¡g2 µD2¡Y2 Y2 ¶; whereobviouslyg2>0,whilethe evolutionoftherelative…xedcapitalstock zcan be easilydeducedfromthegrowthrateofthesystem. After~ t1thedynamicalsystemisthereforethefollowing: 8 > > > > < > > > > : _u=F(u;s) _s=³1¡wl1 b1 ´u¡d b1p¡b2 b1pz¡s(b2z¡±) _p=hg1(s)¡g2³(®+b1¯u) b2h(u)1 z¡1´ip _z=¡b2z2 :(56) Therelative…xedcapitalstockzand theratio ofinventoriessaredeclining, atleastforawhile,also after~ t1,whereastherelativeprice pandthe capacity utilizationrateuareincreasing;however,withinacertaintime,eitherthe righthand sideofthe equationforthe evolutionofinventorieschangesits signortherighthand sideofthe equationforthe evolutionoftherelative price changesits sign beforethatofthepreceding one.Inthe…rstcasethe ratio ofinventoriessentersa growth phasewhilepbeginstofall, since the termg1(s)isdecreasingwhereasthetermg2³(®+b1¯u) b2h(u)z¡1¡1´is still increasing.Inthesecond casetherelativeprice pstartsfalling.Inanycase wedenoteby~ t2theinstantwheneitheralternativeoccurs. 19 The…rstalternativeoccurswhenthereaction(21)oftheprice ofthe capitalgoodstothedisequilibriumbetweensupplyand demand isnot too strong; inthiscase,afterinstant~ t2thesystemisdriventowardsoneofthe equilibriumpoints(39).Asinthe caseofdepression,theregimedescribed bythedi¤erentialsystem(56)hastwodi¤erentpossible ends.Eitherthe activitylevelofthe consumptiongoods sectordiminishesuntil theinvestmentIa 1planned by…rmsfallsunderthemaximumproductive capacityof thesecond sectoragain,sothatwe comebacktothedi¤erentialsystem(49); ortherelativeprice falls soquicklythat the capitalgoods sector recoversits pro…tabilityand canstartinvestmentagain.The evolutionofthesystemin thelastcaseisdescribedinthefollowingsection. Therecoveryofpro…tabilityand thebeginning ofanewphaseofinvestmentinthe capitalgoods sectorafter~ t2isthemostlikelyoutcomealso whenthesecond alternativeprevailsand thusweareledtothedi¤erential system(61)below.Theless likelyoutcomeisthat the evolutionofinventoriesundergoesachangein directionsothat thesystemisbroughtbackto thesituation described by(49). 5.3Insu¢cientoutputof capital goods Whenthe capitalgoods sectorstill makes su¢cientpro…tinorderbothto paytheplanned dividendsand tomake(positive)gross investment,but itsproductive capacitydoesnotsuccedinsatisfyingthedemand,wemust computethesharesintotaloutputofcapitalgoodsofbothsectors,having inmind that thedemand ofthesecond sectoris satis…ed …rst.Fromthe rationingconditionand (52)wehave: I1+I2=b2K2:(57) Theaccumulated pro…tsofthe capitalgoods sectorcan bedetermined using the expressiongivingtheaccumulated pro…tsinthesecond sector: ¦c 2=µ1¡wl2 b2p¶b2K2¡(d+±)K2;(58) whilegross investmentI2carriedoutinthe capitalgoods sectorisgiven by: I2=µ1¡wl2 b2p¡d b2 ¶b2K2:(59) Comparing(57)and (59)we can determinealsothegross investmentI1in the consumptiongoods sector: I1=(wl2p+d)K2:(60) Thedynamicalsystemfort>~ t0isthereforethefollowing36: 36Werecall thatinsection5.1weused~ t0todenotetheinstantfromwhichtheproduction ofcapitalgoodsisnotenoughtosatisfydemand;wehavealsoseeninsection5.2that the dynamicsdescribed by(61)maybereachedalsoinaninstantfollowing~ t0. 20 8 > > < > > : _u=F(u;s) _s=G(u;s;p;z) _p=H(u;p;z) _z=K(u;s;p;z) ;(61) where: G(u;s;p;z)=µ1¡wl1 b1 ¶u+±s¡d b1p¡wl2p+d b1pz(1+b1ps) H(u;p;z)=·g1(s)¡g2 µ(®+b1¯u)h(u) b2z¡wl2p+d b2 ¶¸p K(u;s;p;z)=(b2¡wl2p¡d)z¡(wl2p+d)z2: Thedi¤erentialsystem(61)describesthe evolutionofthesystemaslong as the capitalgoods sectormakespro…ts: p·b2¡d wl2(62) and thedisequilibriumbetween demand and supplypersistsinthemarket forcapitalgoods; z·(®+b1¯u)h(u) wl2+d:(63) The evolution describedbythedi¤erentialsystem(61)ismore complexthan thosewehaveanalysedintheformercases. Our…rstresultisabout the existence ofaturningpointintherisingtrend oftherelativeprice p. Theorem3There exists~ t3>~ t0suchthat_p>0for~ t3<tand_p<0for ~ t3>t. Proof.Assume,onthecontrary,that_p>0foreveryt>~ t0;thenthereexists aninstant~ ta2£~ t0;~ t¤suchthat_z<0fort>~ ta.Whence lim t!~ tz(t)=0. Thereforewehavethat: lim t!~ t (®+b1¯u)h(u) b2z¡wl2p+d b2= +1:(64) Butconditions(13)and (21)onthedynamicsofpricesp1and p2and in particulartheassumptionthatlim s!0+g1(s)= +1togetherwithlim s!+1g2(s)= ¡1leadto a contradiction. 5.4Stationarydisequilibrium The evolutiongiven by(61)hasnotonlytheoutcomedescribedabove37. Infactwearegoingtoshowthat thedi¤erentialsystem(61)mayhavean 37I.e.thereturntowardthe equilibria(39). 21 equilibriumpoint¹´¤=(¹u¤;¹s¤;¹p¤;¹z¤).Becauseofthe conditionH(¹p¤;¹z¤)= 0there existsatrade-o¤betweentherelativeprice ¹p¤and therelative…xed capitalstock¹z¤atsuchequilibriumpoint: ¹z¤=b2¡wl2¹p¤¡d wl2¹p¤+d:(65) Thesameholdstrueforthe conditionG(¹u¤;¹s¤;¹p¤;¹z¤)=0.Bysubstituting thisconditionintoequationK(¹u¤;¹s¤;¹p¤;¹z¤)=0wehave: g1(¹s¤)=g2 µwl2¹p¤+d b2(h(¹u¤)¡1)¶;(66) using againtheabove conditionstogetherwithF(¹u¤;¹s¤)=0weobtainthe equilibriumvaluesforu,s,and p;weget, in particular: ¹s¤=(b1¡wl1)¹p¤¹u¤¡(b2¡wl2¹p¤) (b2¡wl2¹p¤¡d¡±):(67) Itiseasytoprovethatasolutionofthesystem: 8 < :¹s¤=(b1¡wl1)¹p¤¹u¤¡(b2¡wl2¹p¤) (b2¡wl2¹p¤¡d¡±) F(¹u¤;¹s¤)=0(68) existsforevery¹p¤suchthat: ¹p¤¸b2 (b1¡wl1)~u+wl2;(69) if:~u>(d+±)wl2 (b1¡wl1)(b2¡d¡±):(70) Theprecedingconditionis satis…edifthe constant~uthatappearsin(42)is su¢cientlycloseto 1;weanywayassumethat thisisthe case.This solution isalsouniqueifthe equilibriumrelativeprice ¹p¤isnot too large;tobe precisely: ¹p¤<b2¡(d+±) wl2:(71) Inanycasewedenotewith(u(¹p¤);s(¹p¤)) eithertheuniquesolutionofthe system(68)orthesolutionsuchthat the capacityutilizationrate¹u¤islarger and theratio ofinventories¹s¤is smaller.Itisclearthatsolutionswith¹u¤ small and ¹s¤largearenotinterestingforthe caseoftheoverheatingregime. Finallywesubstitute(u(¹p¤);s(¹p¤)) intoequation(66)and determinethe equilibriumrelativeprice ¹p¤. The existence ofatleastonesolution¹p¤satisfying(62)and suchthat the correspondingvector(¹u¤;¹s¤;¹p¤;¹z¤)satis…es(63)obviouslydependson 22 theparticolarformofthefunctiong1and g2.Wethinkthat thereisno pointinspecifyingthese conditions,astheyhavenosigni…canteconomic interpretation.Weprefer, instead,to analysethestabilityofsuchapoint, whenitexists. Todothiswe computethejacobianmatrixJ¤ofthesysteminthe equilibriumpoint(¹u¤;¹s¤;¹p¤;¹z¤),havinginmind relations(65),(66),(67) and thesecond of(68)linkingthe coordinatesofthispoint. Itiseasytosee38 that therootsofthe correspondingcharacteristicpolynomialP¤(¸)obviouslydepend onthe explicitexpressionofthederivatives appearinginthethirdrowofmatrixJ¤; ifweassume,forinstance,thatat the equilibriumpoint thefunctionsg1and g2arebothstationarythenthe thirdrowisnull and amongtherootsofthe characteristicpolynomialoneis obviouslyzero,anotherisnegativeand two ofthemhavenegativerealpart. Hence thestabilityofthe equilibriumpointdependscruciallyonthesecond derivative. Itistherefore clearthat the equilibriumpoint,wheneveritexists,may belocallyattractive39.Inthiscasethesystem mayremaininanoverheating regimeand both pricesp1and p2riseat thesamerateofgrowth,whilethe capacityutilizationrateremainsabovethedesiredtargetand inventories remain belowtheirnormal level. Lastbutnotleast thedisequilibriuminthe capitalgoodsmarketnever petersout.We call this situationoverheatingstationarydisequilibrium. 6Conclusions Themain purposeofthepaperhasbeento analysethestabilityofequilibriumofacapitalisteconomybyconsideringthedi¤erentregimesthat characterize itsworking:normalequilibrium,overheating and depression. Theproblemhasbeen dealtwith,at theanalytical level, intermsofa fourdimensionaldeterministicdynamicalsystemframedincontinuoustime withthedegree ofcapacityutilizationofthe consumptiongoods sector,the stockofinventoriesofthe consumptiongoods sector,therelativeprice of commoditiesand thestructureofthe…xedcapitalstocksas statevariables. Thedynamicalsystemhasacontinuumofequilibriumpoints(¹u;¹s;¹p;¹z) where¹u;¹scharacterize thenormalequilibriuminrealtermsand ¹p= ª(¹z). >Fromtheanalyticalpointofviewthe existence ofacontinuumofequilibria dependsonthefact thatquantitiesdonotreactdirectlyon prices. >Fromthe economicpointofviewthenormalequilibriumofthemodel exhibitstheusualpropertiesofsuchacon…gurationofthe economy:productive capacityofthesectorsisutilizedat thenormal level, stocksofin38Wesketchaproofinthemathematicalappendix. 39Inthiscasetoo wearedealingwiththelocalstabilityanalysisofanequilibriumin proportionsinthesensediscussed byBoggio(1993). 23 ventoriesareat theirnormal leveland pricesofcommoditiesare constant, i.e.thereisnoin‡ation. The existence ofa onedimensionalmanifoldofequilibriaisperhaps unusualbutfarfrombeensurprising asthe e¤ectsofdisproportionamong sectorscan bebalanced byasuitableprice structure,atleastwithingiven bounds.Theseboundscan notbeovercomewithoutunderminingpro…tand growthratesofsomesectors,togetherwiththe equilibriumofthesystem. Thestabilityanalysis showsthat thenormalequilibriumis(locally)asymptoticallyunstable;the economycanevolve,therefore,accordingtotwo di¤erentregimes.The…rstregimeiswhatwe call overheating; itsmain featuresareastockofinventoriesthatdiminishcontinuouslyintime,an increasingdegree ofcapacityutilizationand arelativeprice continuously increasing.Thesecond regimeiswhatwe call depression; itscharacteristics arearisinglevelofinventories,fallingpricesofconsumptiongoodsand a decreasingdegree ofcapacityutilization. Letusconsiderthe caseofdepression.Themodelshowsthatasthe economystaysinsucharegimeforasu¢cientlylongtimethenanincreasingdisproportion betweensectorsarisesthatmakesinvestmentinthe capitalgoods sectorless and less pro…table.Inotherwordswehavethat thereductionofinvestmentsinthe consumptiongoods sectordetermines anoverdimensionedcapitalgoods sector;the excess ofproductive capacity that, infact,characterisesthe capitalgoods sectorwill eventuallybringto ahalt theaccumulation process. Thingsaretheotherwayaround inthe caseoftheoverheatingregime. Theincreasingrelativeprice causesaredistributionofpro…tsinfavourof the consumptiongoods sector;thiseventuallycausesproblemstothe capital goods sectoreitherontheproductionsideoronthepro…tside.The consequence ofthisiseithertherationing ofthe capitalgoods sectororthearrest ofaccumulation duetoalackofpro…ts. 7Mathematicalappendix 7.1Proofoftheorem1 Simple calculationsyieldthefollowingjacobianmatrixJofthelinearized system,evaluatedatoneofthe equilibriumpoints(39): J= 0 B B B @ @F(¹´) @u@F(¹´) @s0 0 ¡¹· ¹pb1 ¡¹®+b1¹ ¯¹u¢¹ h0000 0 ¹p¹g0 10 0 @H(¹´) @u@H(¹´) @s@H(¹´) @p@H(¹´) @z 1 C C C A:(72) Todeterminethe eigenvaluesofthejacobianmatrixJitis su¢cient towrite the explicitexpressionofthelastpartialderivativeofthefunctionHinthe 24