PDF issue: 2024-09-29

Price Flexibility and Instability in a
Macrodynamic Model with a Debt Effect
ASADA, Toichiro(出版者 / Publisher)
Institute of Comparative Economic Studies, Hosei University / 法政大学比較経済研究所(雑誌名 / Journal or Publication Title)
Journal of International Economic Studies(巻 / Volume)18(開始ページ / Start Page)
41(終了ページ / End Page)
60(発行年 / Year)
2004-03(URL)
https://doi.org/10.15002/00002492

JoumaloflnternationalEconomicStudies（2004),No.】8,41-60

C2004ThelnstituteofComparativcEconomicStudies,HoseiUnivcrsity

Price Ｆ１ｅｘｉｂｉｌｉｔｙａｎｄｌｎｓｔａｂｉｌｉｔｙ●

１ｍａＭａｃｒｏｄｙｎａｍｉｃＭｏｄｅｌ
ｗｉｔｈａＤｅｂｔＥｆｆＥｃｔ

ＴｏｉｃｈｉｒｏＡｓａｄａ

勘C"/〃q/Ebo"omicS,Ｃﾉi哩ｏＵ>'iひ‘だｊ奴Ｔｂね)0,ﾉﾋﾘﾉbα〃

Ａｂｓｔｒａｃｔ

Inthispaper,weinvestigatetheimpactofpricenexibilityonmacroeconomicinstabilityusingas
ouranalyticalfmmeworkamacrodynamicmodclwithaFisherdebteffect・Weintroducetheexpecta‐
tions-augmentedPhillipscurveandtheadaptiveexpectationshypothesisintoamacrodynamicmodeI
withaFisherdebteffect,asdevelopedbyAsada（2001)．Wedemonstrateanalyticallythatbothoftheincreaseofthespeedofpriceadjustmentsandtheincreaseofthespeedofexpectationsadaptation
contributetodestabilizingratherthanstabilizingtheeconomy,andwealsoshowthatatthemtermedi‐
aterangesoftheparametervalues,cyclicaIfluctuationsoccurbymeansoftheHopfbifnrcationtheo‐
remWealsopresentsomenumericaIexampleswhichsupportouranalyticalresults．

1．Introduction

lnthestandardversionofmacroeconomics,itistakenfbrgrantedthatthefnll
employmentequilibriumwillbeattainedautomaticallyifwagesandpricesareflexible，Accordingtothisview,asufTlcienti白emlcrjo〃ofnominalwagesandpricelevels
willcontributetoaniFzclqeaseofouｔｐｕｔａｎｄｅｍploymentwhentheeconomyisin
depressionlfthisviewiscorrect,thecauseofpersistentunemploymentmustbethe
rigiditiesofwagesorprices・Ｔｈｉｓｎｏｔｉｏｎｉｓｎｏｔｕｎｉｑｕｅｔｏｔｈｅｓｏcalled`classical，ｏｒ`neoclassical,school，ｂｕｔｉｓａｌｓｏｓｈａｒｅｄｂｙｔｈｅｔｅｘｔｂｏｏｋｖｅｒｓｉｏｎｏｆｔｈｅ‘Keynesian，
IS-LMmodelanｄｔｈｅ‘newKeynesian，schoolwhichconcentratesonthemicro‐
economicintelpretationofwageorpricerigidities・However,thisconventionalview

apparentlycontradictsthewidespreadexpenencesoftheJapaneseandAsianeconｏ－
ｍｉｅｓｉｎｔｈｅｌ９９０ｓ・Inparticular，ｉｎｔｈｅＪａｐａｎｅｓｅｅｃｏｎｏｍｙｉｎｔｈｅｌａｔｅ1990s，thedownwardpressureofnominalwagesandpricesseemedtocontributetoaggravating

ratherthanmitigatingthedepression,andseemedtodestabilizeratherthanstabilize
theeconomy・ＩｆｗｅｔｒａｃｅｂａｃｋｅｃｏｎｏｍｉｃｈｉｓｔｏｒｙｔｏｔｈｅｐｅｒｉｏｄｏｆｔｈｅＧｒｅａｔＤｅｐｒｅｓ－ｓｉｏｎｉｎｔｈｅｌ９３０ｓ,wecanobserveessentiallythesamephenomenoninmanycountries
includingtheUnitedStates,Britain,GermanyandJapan、Thisphenomenon,whichissometimescalleda`deflationaryspiral，，isgenerallyignoredintheorthodoｘｅｃｏ‐
nomicliterature,whichstressthestabilizingeffectofpriceflexibility,althoughithas
recentlyattractedsomeattentionamongeconomicjoumalists、Inbothl930sandthe

l990s,theimpactoffinancialfactorssucｈａｓｄｅｂｔｏｎｔｈｅｒｅａｌｅｃｏｎｏｍｙｓｅｅｍｓｔｏhave
playedanimportantroleinmacroeconomicperfbrmance・Nevertheless,ｔｈｅｉｍｐａｃｔ
４１

PriceFlexibilityandlnstabilityinaMacmdynamicModelwithaDebtEflcct

ofdebtonmacroeconomicperfbrmanceispracticallyignoredinthestandardversionsofmacroeconomicsI・

However,thereisasomewhathereticaltraditionineconomicthought，whichstressestheimportanceoftheimpactofdebtonmacroeconomicperfbrmance・The

mostfamousexamplesareFisher（1933)，Keynes（1936）andMinsky（1975andl986)．Forexample,Fisher（1933）arguedthatadecreaseofpricesinaperiodof

depressionwillcauseafUrtherdeclineofthedemandfbrgoodsthroughanincrease
oftherealdebtburden．Weshallcallthisnegativeimpactoftheincreaseofthereal
debtburdenonefTectivedemand（ｉｎparticular,oninvestment）ｔｈｅ`Fisherdebt
efTect，．Recently,someauthorshavedevelopedvariousmathematicalmodelswhich
trytocapturetheeconomicinteractionbetweenrealandfinancialfnctors,underthe

inspirationoftheearlierworksbyFisher,Keynes,andMinsky2・Inparticular,Ｋeen(2000）andChiarella,FlaschelandSemmler（2000ａｎｄ2001）studiedthedynamics

ofdebtdeflationinmacroeconomicmodelswithvariablepricesandaFisherdebt
efTect、Inthispaper，wealsoinvestigatetheimpactofpriceflexibilityonmacroeconomicstabilityorinstability，usingasouranalyticalframeworkofamacro‐
dynamicmodelwithaFisherdebtefYCct、Ｉｔｍｕｓｔｂｅｎｏｔｅｄｔｈａｔｏｕｒｍｏｄｅｌｉｓ
ｓｏｍewhatsimilartothoseofChiarella,FlaschelandSemmler(2000and2001),but
thestructureismuchsimpler・Ｉｎｆｎｃｔ，ｏｕｒｍｏｄｅｌｃｏｎｓｉｓｔｓｏｆａｎａｔｍｏｓｔｆＷｅ‐

dimensional（andatleastthree-dimensional）systemofdifferentialequations,while

thoseofChiarella,FlaschelandSemmler（Z000andZOOl）consistofanatleast

seven-dimensional（andatmosttwenty-dimensional）systemofdifferentialequa‐tionsA1thoughourmodelislessgeneralthantheirs,ｉｔｈａｓｍｅｒｉｔｉｎｔｈｅｓｅｎｓｅｔｈａｔ
ｉｔｃａｎｆbcusmoresharplyandclearlyontheessenceoftheproblem・Weintroduce
theexpectations-augmentedPhillipscurveandtheadaptiveexpectationshypothesis

intoamacrodynamicmodelwithaFisherdebtefTectasdevelopedbyAsada(2001）

Thelntroductionofpriceflexibilityintothemodelincreasesthedimensions,sothat
thebasicmodelinthispaper，aspresentedinSection2，consistsofathree‐dimensionalsystemofnonlineardiffbrentialequationsinsteadofatwodimensional

systemasintheoriginalmodelbyAsada(2001)．ＩｎＳｅｃｔｉｏｎ３,weshowanalytically

thatbothanincreaseofthespeedofpriceadjustmentsandanincreaseofthespeed
ofexpectationsadaptationcontributetodestabilizingratherthanstabilizingthe

economy,ａｎｄｗｅａｌｓｏｓｈｏｗｔｈａｔａｔｔｈｅｉｎｔｅｒｍｅｄiaterangeoftheparametervalues，
cyclicalnuctuationsoccurbymeansoftheHopfbifUrcationtheorem、Inotherwords，ｉｎｏｕｒｍｏｄｅｌａｎｉｎｃｒｅａｓｅｏｆｔｈｅｐｒｉｃｅｆｌexibilityisresponsiblefbrmacro‐economicinstability，Thisiscontrarytotheconclusionofthetextbookversionof
macroeconomicswhichignoresthedebteHbct、ＩｎＳｅｃｔｉｏｎ４，ｗｅｐｒｅｓｅｎｔｓｏｍｅ、u‐mericalexampleswhichsupportouranalyticalresults・ＩｎＳｅｃｔｉｏｎ５，weshowthat
themainconclusionofoｕｒｂａｓｉｃｍｏｄｅｌｉｓｒｏｂｕｓｔｅｖｅｎｉｆｗｅｃｏnsideramorecomplicatedmodelconsistingofafbur-dimensionalsystemofnonlineardifferentialequa‐tionsThefInalsectionisdevotedtｏａｎｅｃｏｎｏｍｉｃinteIpretationoftheresultswhich
weobtainedanalyticallyandnumerically．

4２

ToichiroAszndzn

2．Ｆｏｒｍｕｌａｔｉｏｎｏｆｔｈｅｂａｓｉｃｍｏｄｅｌ

Thebasicmodelinthispaperconsistsofthefbllowingsystemofequations3．
●。＝①(g(βZ/,,０－汀．,α))－s/{βZﾉｰﾉ(p,ｄ)ｄ}－{g(β9,ｐ－汀．,。)＋〃}。（１），＝α[の(g(β9,ｐ－兀e,ｄ))＋（１－s/){p〃＋i(p,。)ｄ｝－{S/＋(１－斗)sγ}βg］（２）

汀＝ｅ(ｇ－ｇ*)＋汀．

(３）

７ｔG＝γ(汀－汀．）（４）

wherethemeaningsofthesymbolsareasfbllows.。＝Ｄ/(ＰＸ)＝debt-capitalratio，ＺノーＷＫ＝output-capitalratio，ｗｈｉｃｈｉｓａｌｓｏｃａｌｌｅｄｔｈｅ‘rateofcapitalutilization''９＊＝equilibriumleveloftheoutput-capitalratio,whichcolTespondsto
the`naturalrateofemployment，,Ｄ＝nominalstockoffirｍｓ，privatedebt,Ｐ＝priceleve1,Ｋ＝realcapitalstock,Ｙ＝realoutput(realnationalincome),ｇ＝ｋ７Ｋ＝ratc

ofcapitalaccumulation,Ｐ＝nominalrateofinterestofinterest-bearingsafbassets，
j＝nominalrateofinterestwhichisapplicdtoHrms,privatedebt,汀＝ｊ/P＝rateof

priceinflation,汀c＝expectedrateofpriceinflationThefUnction①(9)istheadjustmentcostfimctionofinvestment,and,α,β,γ,ｅ,似,Ｓハandsrareparameterswhich
willbeexplainedlater・

Next，letusexplainhowtheseequationsarederivedWecanexpressthedy‐

ｎａｍｉｃｌａｗｏｆｍｏｔｉｏｎｏｆｐｒｉｖａｔｅｄｅｂｔａsfbllows．

Ｄ＝｡(9)ｐＫ－Ｓ/(ｍＫ－ｆＤ）（５）whereγiｓｔｈｅｒａｔｅｏｆｐｒｏｆｌｔ（i､e､，γ＝Ｐ/Ｋ，wherePistherealproHt)，ａｎｄ
s/ｉｓｔｈｅｒａｔｅｏｆｍｔｅｒｎａｌｒｅｔｅｎｔｉｏｎｏｆｆirms，ｗｈｉｃｈｉｓａｓｓｕｍｅｄｔｏｂｅｃｏｎｓｔａｎｔ(０＜Ｓ/≦ｌ)4．ThefUnctiond(9)istheadjustmentcostfimctionofinvestment
withtheproperties①'(9)≧１ａｎｄ‘"(9)≧０，asintroducedbyUzawa（1969)，、By

thedefInitionofd,ｗｅｈａｖｅｄ/α＝、/Ｄ－ｊ/ｐ－ｋ/Ｋ＝、/Ｄ－汀－９．

（６）

SubstitutingEq.（５）intoEq.（６）,ｗｅｈａｖｅ

ａ＝①(9)－s/(γ－』｡)－(g＋〃)。（７）
ＡｓｆＯｒｔｈｅｄｙｎａｍｉｃｏｆｔｈｅｇｏｏｄｓmarket，weassumethefbllowingKeynesianquantityadjustmentprocess．

‘＝α(ｃ＋ﾉ２－９)；ｃ＝Ｃ/Ｋ,ﾉz＝Ｅ/Ｋ（８）whereCisrealconsumptionexpenditures,Ｅ＝｡(9)Kisrealinvestmentexpendi‐
turesincludingadjustmentcost,andaisthespeedofadjustmentsinthegoodsmarket,whichisassumedtobeapositiveparameter6・Forconsumptionexpenditures,weadoptthefbllowingKaleckianfbrmulationofthetwo-classeconomy7．
C=ｑ+C７(9)4３PriceFIexibiIityandlnstabilitymaMacrodymamicModeIwithaDebtEfYＥｃｔ

Ｃ【〃＝Ｗ＝Ｙ－Ｐ(10）

Ｃ,＝（１－s,）{(１－s/)Ｐ＋p(Ｖ/，)＋j(、/P)};０＜ｓ,≦1（11）whereC2pisworkers，realconsumption,Criscapitalists，realconsumption，Wisreal
wageincome，Visthenominalvalueofinterest-bearingsafeassets,andsristhe
capitalists，averagepropensitytosave,whichisassumedtobeconstant・Ｅｑ.（10）impliesthatworkersdonotsave,sothatworkers，realconsumptionexpendituresare

identicaltorealwageincomcEq.（11）impliesthatthecapitalistssaveapartoftheirrealincome，whichconsistsofthreeparts，Ｌｅ.，thereceiptofrealdividends((１－s/)Ｐ),realinterestreceiptsoninterest-bearingsafeassets（pWP),andrealinterestreceiptsoncorporatedebt（j､/P)8．
Substitutingequations（９），（10),ａｎｄ（11）intoEq.（８）,weobtainy＝α[①(9)＋(１－sr){p(ＷＰＸ)＋』｡}－{S/＋(１－s/)sr}γ]．（12）Usually,ｊ,therateofinterestof`risky，ａｓsets,ｗｉｌｌｂｅｈｉｇｈｅｒｔｈａｎｐ，andthe
difTerencebetweeniandPwillreHectthedegreeofrisk・Tocapturethisfact,ｌｅｔｕｓ
ａｇｓｍｍｅｔｈａｔ

ｊ＝β＋ど(。)＝ｊ(p,。);ど(ｄ)≧０，ｊｄ＝ど'(ｄ）＞Ofbrd＞０，ｉ`＜Ofbrd＜０．（13）Tosimplifytheanalysis,wefUrtherassumethat

P/Ｙ＝β＝constant,０＜β＜１，（14）

ＷＰＸ＝ツーconstant＞０，(15）

Ｐ＝constant＞０（16）Ｅｑ.（14）meansthattheshareofprofItinnationalincome（Ｐ/Ｙ）isconstant・Asiswellknown,Ｐ/YisanincreasingfUnctionofthemarkupofthepriceonthe
averagewagecost・Ｉｎｆａｃｔ,bydefinitionwehave

P＝Ｚ(zuJV/Ｙ)＝zzp/α;ｚ＞１（17）whereZisthemarkup,ZUisthenominalwagerate,jVislaboremployment,ａｎｄ

α＝Ｙ/jVisaveragelaborproductivity,sothatwehavethefbllowingrelationShip．
β＝Ｐ/Ｙ＝（Ｙ－Ｗ)/Ｙ＝１－（Ｗ/Ｙ）＝１－((zu/p)１V/Ｙ}＝１－(l/ご）（18）

Therefbre,βisdeterminedifwesupposethatgisdetermined,fbrexample,ｂｙthe`degreeofmonopoly，inthesenseofKalecki（1971lWeassumethatZisconstantsothatβisalsoconstant、Inthiscase,wecanexpresstherateofprofitas

γ＝Ｐ/Ｋ＝βＹ/Ｋ＝β９．

（19）Inotherwords，therateofprofitisproportionaltotheoutput-capitalratiom

ourmodｅＬＥｑ.（15）ismerelyasimplifyingassumptiontokeepthestructureofthemodel
assimpleaspossibleandavoidunnecessarycomplicationsEq.（16）impliesthatthe
4４

ToichiroAszndzB

nominalrateofinterest（p）isconstantthroughtime・Thismeansthatweareim-

plicitlyassumingthatthemonetaryauthority（centralbank）passivelyaccommo‐

datesthemoneysupplytothedemandfbrmoney，tokeeppconstant・This
hypothesisisinlinewithaPostKeynesian`horizontalist,viewinthesenseofMoore(1988)，、

Toclosethesystem,wemustspecifytheinvestmentfimctionofthecorporate
sector・ＷｅａｓｓｕｍｅｔｈｅｆｂllowinginvestmentfUnctionwithaFisherdebtefYect．

ｇ＝ｇ(γ,ｐ－汀e,。);9,＝０９/0γ＞０，gp-癩＝０９/0(ｐ－汀．)＜０，

９.＝０９/０．＜０（20）ThistypeofinvestmentfimctionhassomemicroeconomicfOundations，Ｆｏｒ
example，ｗｅｃａｎｄｅｒｉｖｅｉｔｆｒｏｍｆｉｒｍｓ，maximizationbehavioroftheexpectednet

cashflowbyusingbothUzawa（1969)，shypothesisofincreasingadjustmentcost(thesocalledPenroseeffect）andKalecki（1937)，shypothesisofincreasingriskof
inVeStment1o，Substitutingequations（13)，（14)，（15)，（16)，（19）ａｎｄ（20）intoequations（７）ａｎｄ（12),weobtainequations（１）ａｎｄ（２）．

Ｎｏｗ,ｌｅｔｕｓｔｕｒｎｔｏｐｒｉｃｅｄｙｎａｍｉｃｓ・Ｗｅａｓｓｕｍｅｔｈｅｆｂｌｌｏｗｉｎｇｔｙｐｅｏｆｅｘｐｅctations-augmentedwagePhillipscurve，

め/ｚＵ＝ど(zノーz/*)＋汀ｃ（21）

whereeisthespeedofthewageadjustment,whichisassumedtobeanonnegative

parameter・Inthisfbrmulation,weadopttherateofcapitalutilization（９）asa
proxyfbrtherateofemployment（e）ortightnessofthelabormarket・Thisproce‐dure,whichisderivedfromFrankeandAsadａ（1994),cansimplifytheanalysisby
savingonestatevariable,therateofemployment・Wecanjustifythissimplificationduetothehighcorrelationbetweenthetwovariablesgandeoverthebusinesscycle(cfFrankeandAsada（1994))．Infact,weshallshowinSection5thattheexplicit

introductionofthevariableeintothemodelonlyaddssomecomplexitytothemodel
withoutaffEctingthemainconclusion

ltfbllowsfromEq.（17）ｔｈａｔ

汀＝ｊ/ｐ＝ＺＭＵ（22）

ifweassumethatthemarkup（Zr）andtheaveragelaborproductivity（α）arecoL
stant・SubstitutingEq．（22）ｉｎｔｏＥｑ．（21)，ｗｅｈａｖｅＥｑ．（３）．Ｅｑ．（４）isa
fbrmalizationoftheadaptiveexpectationshypothesisofthepriceinHation,ａｎｄγisthespeedofadaptation,whichisassumedtobeanonnegativeparameter・

SubstitutingEq.（３）intoequations（１）ａｎｄ（４）,weobtainthefbllowing

systemofthree-dimensionalnonlineardifferentialequations．

（ｉ）。＝。(g(βz/,ｐ－兀喧,‘)）－s/(β９－t(p,。)。｝－(g(βgMo-汀`,〔Z)＋ど(g-g*)＋兀魑}。＝（(d,Z/,汀．;ど）（ii）‘＝α[①(g(β9,ｐ－刀｡,。))÷（１－s『){“＋j(p,ｄ)ｄ｝－(s/＋(１－s/)s,}βz/］＝九(｡,９，汀`，;α）（iii）７t.＝た(zノーｇ*）＝（(g;γ,Ｅ）（Ｓ,）
Ｉｎｔｈｉｓｓｙｓｔｅｍ，ｗｅｃａｎａｄｏｐｔｔｈｅｐａｒａｍｅｔｅｒｓｅａｎｄγａｓｍｅａｓｕｒｅｓｏｆｔｈｅｄｅｇｒｅｅ
４５PriceFIexibilityandlnstabilityinaMacrodynamicModeIwithaDebtEfYbct

ofpriceflexibility、Ｗｅｃａｎａｓｓｕｍｅｔｈａｔｔｈｅｌａｒｇｅｒｅａｎｄγare,themorenexiblearetheprices、Ｉｆｗｅａｓｓｕｍｅｔｈａｔｅ＝γ＝刀Ｇ＝０，ｔｈｉｓｓｙｓｔｅｍｉｓｒｅｄｕｃｅｄｔｏａｔｗｏ‐
dimensionalsystemwithrespecttoｄａｎｄｙ・Thisspecialcasewithfixedpriceswas

studiedbyAsada(2001)．Asada(2001）showedthatdbecomesanincreasingfimc-

tionofZ/and9becomesadecreasingfimctionofdundersomereasonableassump‐
tions・Thisimpliesthatthefbrmalstructureofthistwo-dimensionalsystemis

similartothefamousVolterra‐Lotkasystemofpredator-prey11・Ｉｎｆａｃｔ，ｉｎｔｈｉｓ
ｍｏｄｅldplaystheroleofpredatorandgtheroleofprey,andtheinteractionbetween
thesetwostatevariablescausescyclicalfluctuations（thesocalledMinskycycle）

undersomerangeoftheparametervaluea、Ｗｅｃａｎａｌｓｏｓｈｏｗｔｈａｔｔｈｅｉｎｃｒｅａｓｅｏｆ

atendstodestabilizethesystem（cfAsada（2001))．Inthenextsection,wewill
studyanalyticallyhowtheincreaseofpriceHexibility（increaseofeorγ)aff巳ctsthe
dynamicstabilityorinstabilityofthethree-dimensionalsystem（Ｓ,)．
3．ＡｎａｌｙｓｉｓｏｆｔｈｅｂａｓｉｃｍｏｄｅｌFirst,Ietusconsidertheequilibriumsolutionofthesystem（Ｓ,）whichsatisfies
theconditiond＝。'＝元`＝qTheequilibriumvaluesofd,Z/,ａｎｄ汀caredeterminedbythefbllowingsystemofequations．

（ｉ）（(｡,〃*,刀．）＝０（ii）ん(｡,z/*'7Te）＝０（iii）ｇ＝ｇ＊＞０

Wewillstudythelocalstability/instabilityofanequilibriumpoinｔｂｙａｓｍＪｍｉ"gthattheequilibriumsolutionsucｈｔｈａｔｄ＞Oexists・Itisworthnotingthatthe
equilibrｉｕｍｖａｌｕｅｓｏｆｄ,Zﾉａｎｄ汀earemdepe"｡e"ｒｏｆｔｈｅｐａｒａｍｅｔｅｒｖａｌｕｅｓａ,γａｎｄｅ．TheJacobianmatrixofthissystemａｒｔｈｅｅ９"ｉｌｉ６ｒｊｚＪｍｐｏｊ"ｔｃａｎｂｅｅｘｐｒｅｓｓｅｄａｓ

where

ﾊﾞｰ隅!〃１/h＝｡/i/0.＝(①'(9)-`)g`-9-瀝｡+s'Ｗ+j)。
ﾊﾟ2(e)＝｡/i/,`＝β((①'(9)-.19;-s,)-e`，/１，＝a/i/o汀｡＝－(｡'(9)－.)gﾔﾗﾀﾞ-.,（－）（－）（１）（＋）

几,(α）＝。/i/０．＝α[｡'(9)g`＋(１－s,)(j`。＋j)]，（＋）（－）（＋）

九2(α)＝a/§/,‘＝αβ[P<Ｗ１－(s'+(1-s')s,)ｌ/23(α)＝｡/i/〃－α9M(),)gf17>ｑ
八2(γ,ど）＝醜/0z/＝７℃≧０

4６

(25）

ToichiroAsundzh

Ｎｏｗ,ｌｅｔｕｓａｓｓｕｍｅａｓｆｂｌｌｏｗｓ．

Assumptionl、

八＜０，（2(0)＞０，（3＞０,九I(α)＜０，and九2(α）＞０．

ASSumption2．

（,/H2(α)－ﾊﾟ2(0)/2,(α）＞０，and/Ｍ３(α)－/Ｍ１(α）＞０．

TheinequalitiesinASSumptionlwillinfactbesatisfiediM'(9),9,,｜gp-jrl,andlgdlaresufficientlylargeaZrhee9"ﾉﾉi6rj"ｍｐｏｊ"r・WecaninterPretAssumPtion2

ineconomictermsasfbllows、ＩｔｉｓｅａｓｙｔｏｓｈｏｗｔｈａｔｔｈｅｆｂｌｌｏｗｉｎｇrelationshiPsare
satisfiedl型(ハル(α)-(2(0)/21(α)}＝αβ[(-({９１－，+(\9;+s')(轌十j)｝Ｗ１９;-1}+s,９１Ｗ)］（26）掴{ハル(α)-Ｍ(α)}＝”'(9)[(g十㎡-Ｗj))gｾﾞﾆﾀﾞＷｊ］（27）TherighthandsideofEq.（26）willbecomepositiveif‘'(9),０，,’9`|,andid

aresufTicientlylarge、ＴｈｅｒｉｇｈｔｈａｎｄｓｉｄｅｏｆＥｑ．（２７）willbecomePositiveif

lgp-jrl，’9`|,andjdaresufTicientlylarge、Thatistosay,thetwoinequalitiesin

Assumption2willinfactbesatisfiedif①'(9),gγ,lgb－，J,'9.1,』｡,ands7aresuffIcientlylargeatrﾉieeqzｲﾉﾉj67mmPoj"L
Thecharacteristicequationofthissystembecomes

j,(入）＝｜lI-ハ｜＝ス3＋α,１２＋α2入＋α3＝０（28）where（ｉ）α,＝－ｊｍＣｅ/,＝－（１－九2(α)，(…←|\f'''１ｻﾙＭ:(二｜＝－た九3(α)＋ハル(α)－（2(ど)ん,(α)，（iii）α3＝－detハーだ{('九3(α)￣(3九](α)}，（iV）αｌａ２－ａ３＝た{/Ｍ１(α)＋/22(α)た3(α)）＋｛－（,－ん(α)）(/i,/22(α)－（2(ど)た,(α)}．

（29）

Ｎｏｗ,wecanprovethefbllowingpropositionunderAssumptionsland2．
Propositionl．

（ｉ）Theequilibriumpointofthesystem（Ｓｌ）islocallyasymptoticallystableiｆ
ａｌｌｏｆｔｈｅｐａｒａｍｅｔｅｒｓａ，γandearepositiveandsuffIcientlysmall．

（ii）Thecquilibriumpointofthesystem（Ｓ,）becomeslocallyunstableifeithcr
4７

PriceFlexibiIityandlnstabilityinaMacrodynamicModeIwithaDebtEffEct

oftheparametersa,γ,orEissufYicientlylarge．
(Proof）（ｉ） Ｗｅｃａｎｅａｓｉｌｙｓｅｅ,ｉｎｖｉｅｗｏｆＡｓｓｕｍｐｔｉｏｎｓｌａｎｄ２,thatalloftheRouth-

Hurwitzconditionsfbrstableroots（asetofinequalities（Ａ２）inthe

mathematicalappendix）aresatisfiedifa,γ,andearesufHcientlycloseto
zero･

(ii）

Ｗｅｈａｖｅｚｚ，＜OfbrasufYicientlylargea、Ｏｎｔｈｅｏｔｈｅｒｈａｎｄ，ｗｅｈａｖｅａ２＜０ｉｆγoreissufTicientlyIarge・Inthesecases,atleastoneofthein‐equalities（Ａ２）inthemathematicalappendixisviolated．□

Thispropositionimpliesthattheincreaseofpriceflexibilitytendsｔｏｄｅｓｍ６ﾉﾉ吃eratherthanstabilizetheeconomyinourmodelwithaFisherdebtefTect,contraryto
theteachingof`classical，ａｎｄ‘neoclassical，macroeconomics・Wewillattemptto
providetheintuitiveeconomicexplanationofthispropositioninSection6
Next,letusinvestigatetheeffectofthechangesoftheparametervalueγonthe

dynamicpropertiesofthesystem,byassumingthataandearerelativｅｌｙｓｍａｌＬＩｔ
ｉｓｅａｓｙｔｏｓｈｏｗｔｈａｔ

ＰＰ}{Ｍ１(α)+九2(α)九3(α))＝－αd'(9)[{(①'(9)-.)9F二F+d)匙
＋αβ(｡'(9)－'}０F二；］（30）（１）（＋）（＋）

TherighthandsideofEq.(30)becomesncgativeifの'(9),|gb-魔|,andlgidlare

sufHcientlylargeandaissufficientlysmalLInsum，（3九l(α)＋九2(α)九3(α）willbccomenegativcifの'(9)，ｌｇｐ－魔'’’9.1,ands,aresufficientlylargeandaissufYYcientlysmalLNow，letuspositthefbllowingaddi-

tionalassumption．

Assumption3．

α＞０，ｅ＞０，α1＝￣（I￣/22(α）＞０，ﾊﾑ(α)￣(2(ど)たl(α)＞０，and（3/Ｈ１(α)＋A2(α)/i3(α)＜qAssumPtion3meansthatthevalueｓｏｆｔｈｅｐａｒａｍｅｔｅｒｓａａｎｄｅａｒｅｒｅｌａtively
smalLUnderAssumptionsl，２，ａｎｄ３，weobtainthefbUowingproposition．
Proposition2、Thereexistsaparametervalueγｏ＞Owhichsatisfiesthefbllowingproperties．

（ｉ）Theequilibriumpointofthesystem（Ｓ,）islocallyasymptoticallystablefbrallγｅ（0,γb),andislocallyunstablefbrallγｅ（70,。｡)，（ii）Ａｔγ＝γo,theHopfbifnrcationoccurs・Inotherwords,thereexistsome
non-constantperiodicsolutionsofthesystem（Ｓ,）atsomeparameterval‐ｕｅｓγ＞OwhicharesufTicientlyclosetoγ0．

（Proof）（ｉ）Letusdeflnethevalueγoasfbllows・ハー｛－（１－A2(α)}(ハル(α)－（2(と)/i'(α)）－E{/Ｍ１(α)＋九2(α)ん3(α)｝
4８

(31）

ToichiroAsaaa

whichispositivefromAssumption3､Ｉｔｉｓｅａｓｙｔｏｓｈｏｗｔｈａｔａｌａ２－ａ３＞Ｏ
ｉｆＯ≦γ＜７０．Ｏｎｔｈｅｏｔｈｅｒｈａｎｄ,ｗｅｈａｖｅｑＩ＞Ｏａｎｄａ３＞OunderAs-

sumptions2and3，Then,wealsohavecJ2＞α3/α,＞０．Therefbre,allof
theRouth-Hurwitzconditionsfbrstableroots(asetofinequalities(Ａ２)inthemathematicalappendix）ａresatisfiedifO≦γ＜７０．Next,letussup‐
posethatγ＞７０．Ｉｎｔｈｉｓｃａｓｅ,ｗｅｈａｖｅａ１ａ２－ａ３＜０，whichimpliesthat
oneoftheRouth-Hurwitzconditionsisviolated(ii）

Atγ＝γｏｗｅｈａｖｅａｓｅｔｏｆｃｏｎｄｉｔｉｏｎｓａ，＞０，α３＞０，ａｎｄａ１ｑ２－ａ３＝０，
whichalsoimpliesthata2＝α3/α,＞０．Ｉｎｔｈｉｓｃａｓｅ,thecharacteristic
equation（２８）hasasetofpureimaginaryrootsandanegativerealrootin
viewofTheorem2inthemathematicalappendix・Furthermore，ｗｅｈａｖｅＯ(czla2-a3)/０７＜０，whichimpliesthattherealpartoftheimaginary

rootsbecomesanincreasingfimctionofγａｔγ＝γ0．Thismeansthatallof
theconditionsfbrtheocculTenceoftheHopfbifUrcation（conditions（１）－（３）inTheorem5inthemathematicalappendix）aresatisfIedat
γ＝γ0,sincetheequilibriumsolutionofthesystem（Ｓ,）isindependentof
thevalueofγ・□PrOposition2（ii）impliesthatthecyclicalfluctuationoccursatsomeintermediate
valuesoftheparameterγ、Inotherwords，theendogenousalternationsofbooms

whichentaildebtfinance,anddepressionswhichentaildebtdeflation（debtcycles
orsocalledMinskycycles）occuratsomesetofparametervaluesinourmodeL

4．Anumericalillunstration

Inthissection,wepresentsomenumericalexampleswhichillustratethemain
conclusionsoftheprevioussectionLetusassumethefbllowingparametervalues
andthefimctionalfbrms'３．

ｓ/＝ｓ,＝１，β＝03,ｉ＝ｐ＋0.1.2Mｏ＝008,Z/＊＝0.25,①(9)＝９，

９＝Z/０３－(p-7re)-2.-0.12.（32）Inthiscase,thesystem（ＳＩ）becomesasfbllows．

（ｉ）。＝（Z/03＋兀穆-2.-0.2)(１－．)－０３Z/＋0.08.＋Ｏ１ａ３－Ｅ(z/-0.25)。－汀Ｇｄ（ii）ｚｊ,＝α(z/03＋汀c-2.-0.2-0.3z/）（、）元`＝た(9-0.25）（33）
Fig.１throughFig､SshowthetimetrajectoriesofgandCZcorrespondingtothevariousparametervaluesofa，７，ａｎｄ８，andthecommoninitialcondition

α(0）＝01,Z/(0）＝0.2,ａｎｄ汀⑫(0）＝005.Ｉ４
Ｉｔｓｅｅｍｓｔｈａｔｔｈｅｌｏｎｇｒｕｎｅｑｕｉｌｉｂriumpointisdynamicallystableinａｌｌｅｘａｍ‐
ples・Inotherwords,theconditionsfbrlocalstabilityseemtobesatisfIedintheseexamples・Nevertheless，endogenouscyclicalf1uctuationｓｏｆｇandaoccurinall
cases・Ｔｈａｔｉｓｔｏｓａｙ，theendogenousalternationsofperｉｏｄｓｏｆｂｏｏｍ，ｗｈｉｃｈａｒｅ4９PriceFIexibilityandlnstabilityinaMacrodynamicModeIwitllaDebtElYbct

0.35
0.30

0.25
0.20

0.15

0.10
0.05
tｉｍｅ

0.00
[M〕４(】［

r1

L｣【､0６０m

-0.05
-0.10

Fig.１α＝０．１，７＝0.1,巳＝0.2㈹頭扣奉卯暉、砧帥砧、崎－一一●●●●□■●●●■凸■００００００００００００

tｉｍｅ
ＤＯｆｉ[Ｉ▽
Fig.２α＝０．１，７＝0.1,ｓ＝0.55０ToichiroA塾dzq

0.4

0.3

0.2

Ｖ

0.1

0.0
ｎＪ

】０４０［

ハ

▽

tｉｍｅ

DON/６０【］

－０．１

－０．２

Fig.３α＝０．１，γ＝０．１，巳＝０．８８

0.4

0.3

0.2

0.1

ハ

0.0

tｉｍｅ

【DＯ１４０【］

-0.1

－０．２

Fig.４α＝０．１，γ＝0.18,ｓ＝0.55１PriceFlexibilityandlnstabilityinaMacrodymamicModelwithaDebtEflEct

０５０５０５０５０５０５
４３３２２１１０００ｌｌ

一’一Ｃ０勺●□日０ＤＣ●●●００００００００００００

ハ

ハ

Ｖ

ハ

（

１００

(／

▽ △－－－－▽ｉｆi△‐気ｉ－ｉ壽tｉｍｅ

Fig.５α＝０．２，７＝0.18,Ｅ＝０．５

supportedbydebtfinancing,andperiodsofdebtdeflation（socalleddebtcyclesor
Minskycycles）occurintheseexamples・Weobservethattheturningpointsofd
alwayslagbehindthoseofgintheseexamples・Thismeansthatattheupperturnmg

pointofthebusinesscycle，thedebtburdenstillcontinuestoincrease，ａｎｄａｔｔｈｅ
ｌｏｗｅｒｔｕｒｎｉｎｇｐｏｉｎｔｔｈｅｄｅbtburde、continuestodecreasefbrsometime'５．AcomparisonofFiguresl，２，and3suggeststhattheincreaseofthevalueofparameter
Etendstoshortentheperiodofthecycleandexpanditsamplitude・AcompansonofFigures2and4suggesｔｓｔｈａｔｔｈｅｉｎｃｒｅａｓｅｏｆｔｈｅｖａｌｕｅｏｆγalsohasasimilareffect・Theseobservationsleadustotheconclusionthatroughlyspeaking,increased
priceflexibilitytendstodestabilizethesystem・ThisispreciselythemainpropositionoftheprevioussectionAcomparisonofFigures4and5suggeststhatthe
increaseofthevalueofparameteratendstoshortentheperiodofthecycle、Thesefiguresalsoshowthatanincreaseofthevalueofalessonstheamplitudeofthebusinesscycle，ａｔｌｅａｓｔａｔｓｏｍｅｒａｎｇｅｏｆｔｈｅｃｏｍｂｉｎａtionofparametervalues・In
otherwords,ｔｈｅｉｎｃｒｅａｓｅｏｆｔｈｅｖａｌｕｅｏｆａｈａｓａｓtabilizmgeffbctatleastatsome
raｎｇｅｏｆａ，althoughwealreadyknowthatafUrtherincreaseofawillultimately

destabilizethesystem（cfPropositionl（ii）intheprevioussection)．
5．Ｅｘｔｅｎｓｉｏｎｏｆｔｈｅｍｏｄｅｌ

Inthissection,wefbrmulateanextendedmodelwhichexplicitlyintroducesthe

fburthstatevariable,therateofemployment・LetusreplaceEq.（21）insection2

withthefbllowingmore`natural，fbrmofthewagePhillipscurve．
5２

ToichiroA理dzh

ＺＭＵ＝ど(e-g)＋兀鰹;０＜ｇ＝constant＜１（34）whereｅ＝jWjV8istherateofemployment,jVislaboremployment,ａｎｄハLis
laborsupply、Bydefinition,ｗｅｈａｖｅ

ＩＶ＝｛Ｙ/Ｋ)Ｋ＝（"Ｋ)/α，
WjV

（35）

whereα＝Y7jVisaveragelaborproductivity・SubstitutingEq.（35）intothedefinitionofeanddifYerentiatingwithrespecttotime，weobtainthefbllowinglawof
motionofthevariablee．

＠/ｅ＝（ｊＷｊｖ)－(jVWvB）＝（9,/g)＋(」て/Ｋ)－(｡/α)＿(jvB/Ａｐ＝(ｚＷ)＋g(βz/,p-"｡,ｄ)－(｡/α)－(jＭＶＰ（36）Ontheotherhand,wehavethefbllowingexpressionfrｏｍＥｑ.（17）ｉｆｗｅａｓ－

ｓｕｍｅａｃｏｎｓｔａｎｔｍａｒｋｕｐ

汀＝，/ｐ＝（zMU)－(｡/α）（37）

Fromequations（34）ａｎｄ（37）weobtainthefbllowing`pricePhillipscurve,、
汀＝ど(e-g)－(｡/α)＋兀曾

（38）

Inthissection,weadoptthesimplifyingassumptionofconstantgrowthratesof
laborsuPPlyandlaborProductivity,fbllowingthetraditionofthebld，growththeory・Thisimpliesthat16／VWVR=",=constant≧0,(39)。/α＝〃2＝constant≧０．（40）Inthiscase,ｔｈｅsystem（Ｓ,）inSection2mustbemodifiedtothefbllowing

fbur-dimensionalsystemofnonlineardiffbrentialequations．

（ｉ）ｊ＝の(g(β9,ｐ－汀`,ｄ))－s/{βg-j(p,。)α｝－{９１(βz/,ｐ－汀｡,。)＋ど(e-g)－"2＋汀`}。＝Ｅ(d,９，汀､,ｅ;ど）（ii），＝α[①(g(βz/,p-7Tc,。))＋(１－sγ)(pし＋j(p,。)α｝－{s/＋(１－s/)s,)βz/］＝Ｆｈ(｡,９，刀｡;α）（iii）７t｡＝γ{ど(e一彦)－"2｝＝Ｆｈ(e;γ,ｅ）（iv）６＝｛Ｅ(｡,Z/,汀．;α)/g＋g(βZ/Mo-刀.,。)－(〃,＋"2))ｅ＝ＦＷｚ,z/,汀`,ｅ;α）（S,）

Theequilibriumpointofthissyste、（。*,ｇ*’7Ｔｅ*,ｅ*）isdeterminedbythe

fbllowingsystemofequations．

（ｉ）。(〃,＋〃2)－s/{βg-j(p,。)。)－(〃,＋〃2＋汀｡)ｄ＝Ｏ（ii）｡(〃,＋〃2)＋（１－s,）{Pし＋j(P,。)ｄ}－{S/＋（１－s/)Sγ}βZ/＝０（iii）ｇ(βz/,,０－汀｡,α）＝〃,＋〃２（iv）ｅ＝ｇ＋("2/ど）（41）
TheJacobianmatrixofthissystemαＺｒ/lee9zｲﾉﾉi6rjwmpoj"ｔｍａｙｂｅｗｒｉｔｔｅｎａｓ
５３

PriceF1exibilityandlnstabilityinaMacrodynamicModelwithaDebtEfTect

ⅧEiilmlliMii■１１⑭

whereFh＝Ｍ(9)-.)，`-("!+"2)イ+s'(#,d+j)，Ｆｈ=β{Ｗ)-.}(餡-乢
凡＝－(#い-.)，ﾔｺﾞｰ`ハ(α)＝α[`'(9)g`+(1-s,)Ｍ+j)’:iご悶鰍±剛`鯰芒蝿ｺﾞﾋ

6．Ｄｉｓｃｕｓｓｉｏｎｏｆｔｈｅｒｅｓｕｌｔｓ

ｌｎＣｈａｐｔｅｒｌ９ｏｆＴ１/ｉｅＧｅ"eグロノmheory，Keyneswroteasfbllows．

“ThedepressinginHuenceonentrepreneursoftheirgreaterburdenofdebtmay

partlyoffSetanycheerfUlreactionsfromthereductionofwages・Indeedifthefall
ofwagesandpriceｓｇｏｅｓｆａｒ，theembarrassmentofthoseentrepreneurswhoare

heavilyindebtedmaysoonreachthepointofinsolvency，－withseverelyadverse

efYどctoninvestment.，，（Keynes（1936）ｐ264）“Itwouldbemuchbetterthatwages
5４

ToichiroA且a｡、shouldberigidlyfixedanddeemedincapableofmaterialchanges,thanthatdepressionsshouldbeaccompaniedbyagradualdownwardtendencyofmoneywages.，，(Keynes（1936)ｐ265）“Thereis,therefbre,nogroundfbrthebeliefthataHexible

wagepolicyiscapableofmaintainingastateofcontinuousfUllemployment;…The

economicsystemcannotbemadeselfLadjustingalongtheselines.，，（Keynes（1936）
ｐ267）

ItisclearfromthesequotationsthatKeynesdidnotconsiderthecauseofthe
persistentunemploymentandtheeconomicfluctuationtoberigiditiesofwagesor
prices、ThisassertionbyKeyneshimselfiscontrarytotheteachingsofclassical，
neoclassical,ａｎｄｏｆｔｈｅｓｏｃａｌｌｅｄｎｅｗＫｅｙｎｅsianmacroeconomics、Inthispaperwe
havefbrmulatedamacrodynamicmodelwhichprovidessometheoreticalfbunda‐
tionstothissomewhathereticalassertionbyKeynes，Themaindestabilizingmecha‐
nismofpriceflexibilityinourmodｅｌｉｓｂａｓｅｄｏｎｔｈｅ`FisherdebtefTect，,whichmay
berepresentedschematicallyasfbllowsI8．

（ｇ↓)－汀↓－．＝Ｄ/(ｐＫ)↑－９↓＝（Z/↓）（ＦＤE）
Furthermore,wehavethefbllowingotherdestabilizingefTectthroughchanges
oftheexpectedrealrateofinterest，viachangesofthepriceexpectations，whichis
calledthe`Mundelleffbct'１，．

（Z/↓)＝汀↓－汀`↓＝(,０－汀`)↑＝ｇ↓＝（ｇ↓）（ＭＥ）
Thesetwoeffbctshavepositivefeedbackmechanismswhichareundoubtedly
destabilizingInotherwords，intheseprocesses，adecreaseofZノinducesafUrther
decreaseofgzo・Needlesstosay，therealsoexistcounteractingnegativefeedbackmechanisms
whichhavestabilizingeffbctsFirst,ｔheincreaseofrealcorporatedebtduetothe
fallofthepricelevelduringtheperiodofdepressionimpliesanmcreaseofthereal

wealthofcreditors、Thus,itdoesnotonlyhaveanegativeimpactonthecorporate
investmentexpenditure,butwillalsoinduceanincreaseoftheconsumptionexpendi‐
tureofcreditors・Ｔｈｉｓ`wealthefTect，isexplicitlyincomoratedinourmodeLHowever,itisdoubtfillwhetherthelatterefYectisstrongenoughtothatitoutweighthe
fbrmer、Ｉｎｆａｃｔ,inourmodelthestabilizing`wealthefTect,isrelativelyweakcom‐
paredtothedestabilizing`debtefTect，、Ｔｈｅｓｅｃondstabilizingmechanism,whichhasbeenneglectedmthispapeｒｕｐｔｏｎｏｗ，ｉｓｔｈｅｓｏｃａｌｌｅｄ`KeynesefTect,whichworks
throughthechangeofthenominalrateofinterest・Ifweassumethestandard

KeynesianrealdemandfimctionfbrmoneyLD＝ｐ(,o)Ｙ（④'(p）＜０),theequilibriumconditionfbrthemoneymarketcanbeexpressedas

jＷＰ＝･(p)Ｙ（43）ｗｈｅｒｅ〃isthenominalmoneysupply、Solvingthisequationwithrespectｔｏｐ，wehavethefbllowingstandardtypeｏｆｔｈｅ`LMequation，．

ｐ＝ｐ(叩);ｐ'(机)＝１/①'(p)＜０，ｍ＝１Ｗ(ＰＹ）（44）
Ｉｎｏｕｒｍｏｄｅｌ,itwasassumedthatthemonetaryauthorityadoptsthepolicyof
keepingthenominalrateofinterest（p）constant,whichimplicitlymeansthatthe
monetaryauthorityadoptsamonetarypolicyofkeepingmconstant・Ｉｔｉｓｏｂｖｉｏｕｓ

ｔｈａｔｔｈｅ`Keyneseffect,ｄｏｅｓｎｏｔｗｏｒｋｉｎｔｈｉｓｃａｓｅ・However,ｔｈｅｒｅｉｓａｒｏｏｍｆｂｒｔｈｅ5５

PriceF1exibilityandlnstabilityinaMacrodynamicModelwithaDebtEfYEct

Keyneseffectifthemonetaryauthorityadoptsanalternativepolicy,fbrexample,tokeepthegrowthrateofnominalmoneysupplyQMf)constantratherthanthe

nominalrateofinterest・Ｉｎｔｈｉｓｃａｓｅ,weobtainthefbllowinglawofmotionfbrthevariable醜bydifYerentiatingthedefinitionalequation、＝Ｍ/(ＰＹ）＝１Ｗ(PZK）

withrespecttotime，

巾/、＝（〃/Ｍ)－(j/p)－(`/z/)－(１t/Ｋ）＝〃－汀－(，/Z/)－９(βZ/,ｐ－汀｡,zZ）（45）

where/U＝ｊＷＭ＝constant・Ifwereplacetheassumptionp＝constantwithcqua‐tions（44）ａｎｄ（45),thesystem（Ｓ,）istransfbrmedtoafbur-dimensionalsystem，

andthesystemS2becomesafive-dimensionalsystem・First,letusconsiderthelong
runequilibriumso1utionSubstituting巾＝，＝Oandg＝〃,＋〃2mtoEq.（45),we
obtainthefbllowingexpression．

〃一万一(〃!＋〃2)＝０（46）Eq.（46）characterizesthelongrunequilibriumpositionofbothofthesystemwith

afixednominalrateofinterestａｎｄｗｉｔｈａｆｉｘｅｄｇｒｏｗｔｈｒａｔｅｏｆｔｈｅｎｏminalmoney
supply・However,thecausalityofdeterminationisdifferentbetweentwosystems，Inthesystemwithafixednominalrateofinterest，thisequationbecomesthefbr‐

mulawhichdeterminestheequilibriumgrowthrateofnominalmoneysupplycorre‐
spondingtotheequilibriumrateofpriceinflation,whichispre-determined,whilein
thesystemwithafixedgrowthrateofnominalmoneysupply,theequationbecomes
thefbrmulawhiｃｈｄetelminestheequilibriumrateofpriceinflationcorresponding
totheexogenouslygivengrowthrateofmoneysupply・Whatcanwesayaboutthe
out-ofLsteady-statedynamicsofthesystemwithaconstantgrowthrateofmoney
Supplyandvariablenominalrateofinterest？Ｈｏｗｄｏｅｓｔｈｅｃｈａｎｇｅｏｆｔｈｅａｓｓｕｍptionfromconstantnominalrateofinteresttoconstantgrowthrateofnominal
moneysupplyaffectthedynamicstabilityofthesystem？
WithoutcommittingourselvestoafOrmalanalysis，wecanobservethatthis

modificationofthemodelwillincreasethestabilityofthesystem，atleastpoten-

tially、Infact,thefO11owingstabilizingnegativefeedbackeffect,whichiscalledthe

KeyneseffEct，，willworkifthesensitivityofmoneydemandwithrespecttothechangeoftherateofinterestkD'(p)｜issosmallthatthedecreaseof汀caninduce
adecreaseoftheexpectedrealrateofinterestP-7Tethroughthesharpdecreaseof
thenommalrateofinterestlo,ｉｎｓｐｉｔｅｏｆｔｈｅｆａｃｔｔｈａｔ汀ealsofalls．

（Z/↓)→汀↓＝?〃↑→,o↓＝(lo-7T｡)↓－９↑－（ｇ↑）（KE）
But,itisquiteunlikelythattheKeyneseffbctwilloutweightheMundellefTect
whentheeconomyisindepression・Ingeneral，ｔｈｅＦｚｏｍｉ"αノｒａｔｅofinterestcannot

mllbelowzero・Ｉｎｆａｃｔ，thesensitivityofpwithrespecttoｔｈｅｃｈａｎｇｅｏｆ?〃will

approachtozerobecausekD'(p)lbecomesalmostinfiniteifpisalreadycloseto

zero・Ｉｎｔｈｉｓｓｉｔｕａｔｉｏｎｏｆｔｈｅｓｏｃａｌｌｅｄ‘liquiditytrap,，theKeyneseffectdoesnot
work・Ｆｏｒexample,intheJapaneseeconomyinthelatel990s,theexpectedrealrate
ofinterestwasstillrelativelyhighbecauseofthefearofpricedenation,althoughthe
JTomj"ロノｒａｔｅｏｆｉnterestwasnearlyzero・Furthermore，thecorporatedebtburden
washighenoughAsaresult，corporateinvestmentwashelddowntoquitealow
leveLThisrecentJapaneseexperienceisconsistentwiththetheoreticalconjecture5６

ToichiroAmd2

derivedfromourmodeL

Ourtheoreticalinvestigationofthedynamicinteractionbetweenrealandfinancial

factorssuggeststhatthepolicyrecommendationtoincreasetheflexibilityofwages
andpricesduringaperiodofdepression,withoutanactivecountercyclicaldemand
managementpolicMislikelytoleadtodisastrousresults．
Acknowledgment

Thisisarevisedversionofapaperwhichwaswrittenwhiletheauthorwasa
visitingscholarattheUniversityofBielefeldinGermany・Anearlierversionofthis
paperwaspublishedasDiscussionPaperNo､４８３，DepartmentofEconomics,Uni‐versityofBielefeld(February2002),andwaspresentedatthe6hlnternationalCon‐

ferenceonMacroeconomicAnalysisandlntemationalFinancewhichwasｈｅｌｄat

theUniversityofCreteinGreece（２５May,2002)．Theauthorisgratefilltothe
UniversityofBielefeldfOrthegenerousprovisionofitsfacilitiesThisresearchwas

financiallysupportedbytheJapanMinistryofEducation，Culture，Sports,Science
andTechnology（Grant-in-AidfbrScientifHcResearch（B）15330037）andChuo
UniversityJointResearchGrantO382．

MathematicalAppendix
lnthisappendix,wepresentsomemathematicaltheoremswhichareusefUlfbr
thederivationofthepropositionsinthetext．

Theoreml.（Routh-Hurwitzconditionsfbrathree-dimensionalsystem）
Ａｌｌｏｆｔｈｅｒｅａｌｐａｒｔｓｏｆｔｈｅｒｏｏｔｓｏｆｔhecharacteristicequationj,(几）＝ス3＋α1入2＋α2入＋α3＝Ｏ(Ａ１）

ａｒｅｎｅｇａｔｉｖｅｉｆａｎｄｏｎｌｙｉｆａｓｅｔｏｆinequalities

aj＞0（ノー１，２，３),αlZZ2￣α３＞O(Ａ２）issatisfIed．
Theorem2・Thecharacteristicequation（Ａ１）hasapairofpureimaginaryroots±んj(ノーＶ－１，/Z≠０）ヴα"doJMyヴasetofconditions
a2＞０，CZla2-a3＝０（Ａ３）issatisfiedlnthiscase,wehavetheexplicitsolutionスーーα,,±Z伝．
Theorem3.（Routh-Hurwitzconditionsfbrafbur-dimensionalsystem）
Ａｌｌｏｆｔｈｅｒｅａｌｐａｒｔｓｏｆｔｈｅｒｏｏｔｓｏｆｔhecharacteristicequation

j2(入）＝ス'＋６１１３＋６２入2＋６３ス＋６４＝Ｏ(Ａ４）

ａｒｅｎｅｇａｔｉｖｅｉｆａｎｄｏｎｌｙｉｆａｓｅｔｏｆinequalitiesbj〉０(ノー1,2,3,4,),bIM3-biD1-b;＞O(Ａ５）

issatisfied．

5７

PriceF1exibilityandlnstabilityinaMacrodynamicModelwithaDebtEIYbct
Theorem4･Thecharacteristicequation（A4）hasapairofpureimaginaryrootsandtwo

rootswithnon-zerorealpartsヴα"ｄｏ"(yヴeitherofthefbllowingsetofconditions(Ａ）ｏｒ（B）issatisfIed．

（Ａ）６，＝０，ｂ３＝0,ａｎｄ６４＜０．

（B）６，６３＞0,64≠0,andbIb2b3-bib1-6i＝０
Theorem5.（Hopfbifnrcationtheoremfbran〃dimensionalsystem）
Letji＝／(〃;Ｅ),ｒＥＲ",ｅｅＲｂｅａｓｙｓｔｅｍｏｆａｎルdimensionaldifferential
equationswithaparameteraSupposethatthefbllowingproperties（１）-（３）aresatisfHed．

（１）Thissystemhasasmoothcurveofequilibria／(工*(e);ど)＝０．

（２）Thecharacteristicequationlｽ１－町(r*(eo);ＥＣ)|＝Ohasapairofpure
imaginaryrootsス(どo),ス(CO）ａｎｄnootherrootswithzerorealparts,where〃(工*(EC);ＥＣ）istheJacobianmatrixoftheabovesystemat（r*(Ｅ０)，ＥＣ）withtheparametervalueEo．

（３）α{Reｽ(ど)/dど}'－０≠０，whereRe1(E）istherealpartofス(E)．

Then,thereexistsacontinuousfimctiｏｎｅ(γ）ｗｉｔｈど(0)＝ＥＣ,andfbrallsufTi‐

cientlysmallvaluesofγ≠Othereexistsacontinuousfnmilyofnon-constantperi‐
odicsolutionsZ(/,γ）fbrtheabovedynamicalsystem，whichcollapsestotheequilibriumpoint工率(ＥＣ）ａｓγ－－０Theperiodofthecycleisclosｅｔｏ２汀/1ｍｽ(eo)，wherelmス(eo）istheimaginarypartofス(EC)．

Theoremsl’３，ａｎｄ５ｉｎｔｈｉｓａｐｐｅｎｄｉｘａｒｅｗｅｌｌｋｎｏｗｎａｍｏｎｇresearchersof

dynamiceconomics(cfGandolfb（1996）andLorenz（1994))．Ontheotherhand，Theorems2and4（andinparticular,Theorem4）mayberelativelylittleknown

amongeconomists,althoughtheirresultsarequiteusefillfbrourpurposes・Forthe

proofofTheorem2,seeAsada（1995）ａｎｄＡｓａｄａａｎｄＳｅｍｍｌｅｒ（1995)．Asadaand
Yoshida（2003）containstheproofofTheorem4
Notes
１

２

３４
５６
７８

5８

Forahistoryoffinancialcrises,seeKindleberger（1989）ａｎｄＷｏｌｆｍｎ（1994)．
See,fbrexample,Asada（l999and2001),Chiarella,FlaschelandSemmler(2000and2001)，Chiarella，Flaschel，ＧｒｏｈａｎｄＳｅｍｍｌｅｒ（2000）Ｃｈａｐ２，DelliGattiand
Gallegati（1994)，F1ascheLFrankeandSemmler（1997）Chap､１２，FrankeandSemmler（1989),Ｋｅｅｎ（2000),andPalley（1996)．Nasica（2000）containsanexcel‐
lentsurveyofrecentdevelopmeｎｔｓｉｎｔｈｉｓａｒｅａｏｆｒｅｓｅａｒｃｈ
Ａｄｏｔｏｖｅrasymboldenotesthederivativewiｔｈｒｅｓｐｅｃｔｔｏｔｉｍｅ
Ｗｅａｓｓｕｍｅｔｈａｔｔｈｅｒｅａｒｅｎｏｉｓｓｕｅｓｏｆｎｅｗｓｈａｒｅｓ，andfbrsimplicityweneglectthe
repaymentofdebL
SeealsoAsada（l999and2001）ａｎｄＡｓａｄａａｎｄＳｅｍｍｌｅｒ（1995)．
Inthisfbrmulation，weneglectgovernmentexpendituresandintemationaltradefbr
simplicity・Inotherwords,weabstracttheproblemsofpublicdebtandfbreigndebtto
concentrateonthatofdomesticprivatedebt、
SecKalecki（1971)．
Ifthecapitalistsaredebtorsandthecorporatesectorasawholeisacreditor,ｗｅｍｕｓｔ
ｈａｖｅＤ＜０，ｂｕｔｗｅｃａｎｅｘｐｅｃｔｔｈａｔｉｎａ‘normarsituation，ｗｅｈａｖｅＤ＞０，meaningToichiroAsada

that，thecorporatesectorasawholebecomesadebtorandthecapitalistsbecome
creditors
９

Lavoie（1995）assertsthattherealrateofinterestshouldbetreatedasanexogenous

variablefbrPostKeyesiananalysi＆However,wedonotfbllowhisapproach,because
itisinfactmoredifTIcultfbrthemonetaryauthoritytocontroltherealratherthan
nominalrateofinterestinaneconomywithvariabｌｅｐrices・Incidentally,inthefinal
section,wewillbrieHydiscussthe`verticalist'ｃａｓｅｉｎｗｈｉｃｈｔｈｅｇｒｏｗｔｈｒａｔｅｏｆｎｏｍｉnal

1０
１１
１１

２３

1４

moneysupplyratherthanthenominalrateofinterestisassumedtobeconstant・Forthederivation,seeAsada（1999ａｎｄ2001)．ＳｅｅａｌｓｏＡｓａｄａａｎｄＳｅｍｍｌｅｒ（1995)．

FortheVolterra-Lotkasystem,seeGandolfb（1996）Chap､２４andLorenz（1993）

Ｃｈａｐ２・

AIIvalucsof乃areevaluatedatthecquilibriumpoｉｎＬ
ＷｅｄｏｎｏｔｃｌａｉｍｔｈａｔｏｕｒｓｉｍｐｌｅｎｕmericalexamplesarequantitativelyrealistiQThe
purposeofthissectionisonlytoillustrateandmakevisiblethemainqualitativecon‐

clusionswhichwerederivedanalyticallyintheprevioussection，WeadoptedEuler，salgorithmandthetimemtervaljt＝０.lfbrthecomputersimulation・However，wefbundthatwecanobtainessentiaｌｌｙｔｈｅｓａｍｅｒｅｓｕｌｔｓｅｖｅｎｉｆｗｅ
ａｄｏｐｔthetimeintervaljt＝００１．

1５

Intheseexamples,insomeperiodsofthebusinesscycleDbecomesnegative、Inotherwords，thecorporatesectorbecomesacreditorratherthandebtormtheseperiods・However，thisfeatureisnotessentialfbrthequalitativepatternofmovementofthe

1６

Therecentlydeveloped`newgrowththeory，fbcusesontheendogenousdeterminationof〃Iａｎｄ〃2,butthisisnotthethemeofthispaper・Incidentally,Ｅｑ.(40）impliesthat
weareassuminga`Harrodianneutral,exogenoustechnicalprogress・
Ｄｕｅｔｏｌａｃｋｏｆｓｐａｃｅ,theproofisomitted・ＷｅｃａｎｍａｋｅｕｓｅｏｆＴｈｅｏｒｅｍｓ３,４，and5
inthemathematicalappendixtoprovethesepropositions、

mainvariables・

1７

1８
１１
２０

FortheFisherdebtefYCct,seeChiarella,FlaschelandＳｅｍｍｌｅｒ（2000andZOO1)．

FortheMundelleffect,seeFlaschel,ＦｒａｎｋｅａｎｄＳｅｍｍｌｅｒ（1997〕Ｃｈａｐ７・

Insomerecenteconomicliterature,theimportanceofpositivefとedbackmechanismsor

selfLreinfbrcingmechanismsinthedynamiccontextiswellrecognized・See,fbrexam-ple,Agliardi（1998)．Itisworthnoting,incidentally,thatthesepositivefeedback
mechanismsworkinperiodsofinHationarｙｂｏｏｍａｓｗｅｌｌａｓｉｎｐｅｒｉｏｄｓｏｆｄｅｐｒｅｓsion．
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6０