Inflation, Endogenous Growth, Transaction Costs, and Varying Discount Rates

Inflation, Endogenous Growth, Transaction Costs, and Varying Discount Rates

著者 MIYAZAKI Kenji

出版者 Institute of Comparative Economic Studies, Hosei University

journal or

publication title

比較経済研究所ワーキングペーパー

volume 153

page range 1‑33

year 2010‑02‑15

URL http://hdl.handle.net/10114/6297

逐次的効用関数。暖昧さ。時間不整合性を考慮した国際マクロ分析シリーズNo.７

□

Transaction

InflationEndogenousGrowth，

Costs,andVaryingDiscountRates

KenjiMiyazaki

Ｌ

InHation)EndogenousGrowth,Transaction Costs，andVaryingDiscountRates＊

KenjiMiyazaki↑

FacultyofEconomics，HoseiUniversity February8，２０１０

Abstract

Thisstudyexplorestherelationbetweeninflationandeconomicgrowth usingthetransactioncostsmodelwithasociallydetermineddiscountrate andalinearproductiontechnology・Evenwhenthelabordecisionisinelas- tic，thisstudydemonstratesthatinflationafIectstheendogenousgrowth Withnonconstanttimepreferences・Inparticular，ifthedegreeofimpa- tienceincreasesintheeconomy-wideaverageratiooftotalassets(thesum ofcapitalandmoney)toconsumption,thenzerorateofmoneysupply couldachievemaximizedendogenousｇｒｏｗｔｈ；therelationshipbetween inflationandeconomicgrowthishump-shaped・Thenumericalexamples confirmsuchahump-shapedrelationship，buttheimpactofinflationon economicgrowthisquantitativelysmalL

KeZﾉT"0,.s：monetarypolicy；impatience；hump-shaped JELclass坂cations：Ｅ５;０４２

＊IwouldliketothankNobusumiSagaraandYOichiGokanfOrtheirhelpMcomments,and HitoshiUtsunomiyafOrexcellentresearchassistance，Ａｌｌｒｅｍａｉｎｉｎｇｅｒｒｏｒｓａｒｅｍｙｏｗｎ・Iam gratefUlfbragrand-in-aidfromtheMinistryofEducationandScience，theGovernmentof Japan．

↑Correspondenceto：KenjiMiyazaki,FEcultyofEconomics,HoseiUniversity,４３４２Aihara，

Machida,Tbkyo,Japan,194-0298;e-mail:miya-kenohosei・ac・jp;tel：＋81-42-783-2591;fax：

＋81-42-783-2611

１

-〆

ｌｌｎｔｒｏｄｕｃｔｉｏｎ

BeginningwithTbbin（1965）andSidrauski（1967)，therelationshipbetween inHationａｎｄｅｃｏｎｏｍｉｃｇｒｏｗｔｈｈａｓｂｅｅｎｏｎｅｏｆｔhecentralissuesinmone- taryeconomictheory・ＷａｎｇａｎｄＹｉｐ（1992）consideredthemoney-in-the- utility-fimction（MIUF）approach,thecash-in-advance（CIA）approach,and thetransactions-costs(TC)approach,anddemonstratedthatwhenlaborde- mandisinelastic，thethreeapproachesleadstosuperneutrality、Whenlabor demandiselastic，inflationgenerallylowersthelevelofcapitalstocksinthe longrun・Ibderivethisconclusion,theyconsideredexogenousgrowthmodels andexaminedtherelationshipsbetweeninHationandthelevelofcapitalstocks ratherthantherateofeconomicgrowth

Severalauthorshaveinvestigatedacorrelationbetweeninflationandthe long-rungrowthrateinanendogenousgrowthframework、ＩｎｔｈｅＣＩＡａｐ－

ｐｒoach,Gomme(1993)introducedatrade-ofIbetweenlaborandconsumption，

andJonesandManuellir(1995)considerednominalrigidities、IntheTCap- proach,DeGregorio(1993)andJhaetaL(2002)usedthetransactiontechnology inwhichthetransaction-costfUnctionincreaseswithconsumptionanddecreas- inginmoneyinDeGregorioandadditionallyincreasinginreａｌｏｕｔｐｕｔｉｎＪｈａ ｅｔａＬＴｈｅｓｔｕｄｉｅｓｃｉｔedaboveingeneralsupportedthenegativeefIectofan increasingmoneysupplyoneconomicgrowth，

Theseimplicationscannot,however,explaintheempiricalevidencereported byGomme(1993)andBullardandKeating(1995)thattheremayexistapos- itivelinkbetweeninflationandthegrowthrateinlow-inflationcountries，Fbr apossibleexplanation,mkuda(1994)andItayaandMino(2003)considered amonetaryversionoftheBenhabibandFarmer(1994)modeLmkuda(1996）

andltayaandMinｏ(2003)adoptedtheCIAapproachandtheTCapproach，

respectively、BothresultsfbundthatwithsuHicientlargelaborexternality，ａ

２

positiverelationshipemerges・However,theirmodelscannotexplainthehump- shapedrelationshipbetweeninHationandeconomicgrowthwithiｎｏｎｅｃｏｕｎｔｒｙ ｗｉｔｈｔｈｅｓａｍｅｅｃｏｎｏｍｉｃenvironment.’

WeexploretherelationshipbetweeninHationandeconomicgrowthusing thetransactioncostsmodelwitｈａｓociallydetermineddiscountrateanda linearproductiontechnology・Evenwhenthelabordecisionisinelastic，this studydemonstratesthatinHationaHectstheendogenousgrowthwithnoncon- stanttimepreferences，Inparticular，ifthedegreeofimpatienceincreasesin theeconomy-wideaverageratioofgeneralassetstoconsumption,thentherela- tionshipbetweeninflationandeconomicgrowthcouldbehump-shaped，which supportsempiricalevidenceslnthecaseoftotalassets(thesumofcapitaland money),zeromonetarypolicyachievesmaximizedeconomicgrowth

Theintuitionisasfbllows・Thehighertheratesofmoneysupplythelower arerealbalancesandthehigherarethetransactioncosts，andthisdiscourages consumptionorincreasestheratioofcapitaltoconsumption、Ａｔｔｈｅｓａｍｅｔｉｍｅ，

highercostofmoneyholdingsdecreasesthemoneysupplyortheratioofmoney toconsumptionWhenaneconomicagentiｓｌｅｓｓｐａｔｉｅｎｔａｓｔｈｅｒａｔｉｏｏｆａｓｓｅｔｓ toconsumptionincreases，theagentbecomeseitherlesspatientifthecapital efIectisstrongerormorepatientifthemoneyeHectisstronger，Ｗｈｅｎｔｈｅｃｏｓｔ ｏfholdingmoneyissufIicientlyhigh，thecapitaleffectisdominant，lowering economicgrowth

Weestablishtheexistenceanduniquenessofabalancedgrowthpath(BGP）

ａｎｄｄｅｍｏｎｓｔｒａｔｅｔｈａｔｓｕｃｈａＢＧＰｉｓｌｏcallystable．、rthermore，weconduct severalnumericalexercisesandcompareourswiththemodelwiththediscount ratesdeterminedinternallybytheindividual、Ournumericalexercisesconfirm

’Oneexplanationfbrthishump-shapedrelationshipwithinthesametechnologyandprefL erenceswasgivenｂｙＤｕｔｔａａｎｄＫａｐｕｒ（1998),ｗｈｏconsideredathree-periodoverlapping modelwithprefCrenceshocks,cash-in-advanceconstrains，irreversibleinvestment，andexoge- noustechnology．

３

thatthehump-shapedrelationshipbetweeninHationandtherateofeconomic growthisestablishedwithasetofplausibleparameters，butshowthattheim- pactofinflatiｏｎｏｎｅｃｏｎｏｍｉｃｇｒｏｗｔｈｉｓｓｍａｌｌｑｕａｎｔitatively、Ourcomparison provesthatevenifthediscountratedependsnotonaggreｇａｔｅｂｕｔｏｎindivid- ualconsumption，thentheresultsofthecomparativeｓｔａｔｉｃｓｏｎｔｈｅＢＧＰａｒｅ observationallyequivalentinbothmodels．

ＴｈｉｓｓｔｕｄｙｉｓｎｏｔｔｈｅｆｉｒｓｔａＬｔｅｍｐｔｔoconsidertherelationshipbetween moneyandgrowthwithvaryingdiscountingrates､Ｋａｍ(2005)consideredthe MIUFmodelwiththediscountratesdependingoneconomy-widetotalassets andfbundthatwhendiscountratesincreaseintotalassets，thenapositive relationshipbetweeninHationandcapitalstocks,aTbbinefIect,emerges・Chen etaL(2008)usedtheMIUFandTCmodelswiththedegreeofimpatienceis internallydeterminedbyindividuals、ChenetaLdemonstratedthatincreasing impatienceinmoney(respconsumption)leadstoalbbineffectintheMIＵＦ

(respTC)modeLHowever,bothstudieslayinanexogenousgrowthframework anddidnotgenerateahump-shapedrelationship

Thisstｕｄｙｉｓａｌｓｏｎｏｔｔｈｅｆｉｒｓｔａｔｔｅｍｐｔｔｏｕseanendogenousgrowthmodel withvaryingdiscountrate・PalivosetaL(1997)ａｎｄＭｅｎｇ(2006)investigated therelationshipamonｇｔｈｅＢＧＰａｎｄｔｈｅｆＵｎｃｔｉｏｎａｌｆｂｒｍｓｏｆｔｈefelicityand thediscountrateintheframeworkoflineartechnology、Bothstudiesfbund thatundertheBGPandthediscountratefimctioninvａｒｉａｎｔｔｏｔｈｅＢＧＰ,the discountrateshouldbeconstantorahomogeneousdegreeofzero，ａｎｄthat theelasticityofmarginalfelicitymustbeconstant、Wedemonstratethattheir resultsholdinourmonetaryeconomy・

Ｏｕｒstudiesmakesthreecontributions、First,thismodelgivesasolutionto thepuzzleposedbyGomme(1993)andBullardandKeating(1995),andsheds newlightonmonetarypolicy・Second，weconsiderthemonetarymodelwith

４

～

varyingdiscountrateinanendogenousgrowthframework，whereasPalivoset

aL(1997)andMeng(2006)fbcusedtheirattentiononarealeconomy・Third,we

findthatinsometypesofmodels,ｔｈｅｃｏｍｐａｒａｔｉｖｅｓｔａｔｉｃｓｏｎｔｈｅＢＧＰｈasthe sameresultswhetherconsumptioninthediscountrateisdeterminedexternally andinternally・

Ourpaperisorganizedasfbllows・Section2describesamodeleconomywith theexternallydetermineddiscountrates,andSection3establishestheexistence anduniquenessoftheBGPSection4conductscomparativestaticsanalysis，

andSection5discussesthelocalstability・Section6perfbrmsseveralnumerical exercises，andSection7comparesourmodeltothemodelwithdiscountrates determinedinternallybytheindividual・Section8concludes．

ｚＴｈｅＭｏｄｅｌｗｉｔｈｔｈｅＥｘｔｅｒｎａｌｌｙＤｅｔeｒｍｉｎｅｄＤｉｓ－

ｃｏｕｎｔＲａｔｅｓ

Weconstructanendogenousgrowthmodelaugmentedwithmoneyorrealbal- ances・Therepresentativeagentisinfinitelylivedandendowedwithaninitial capitaMo＞0,aninitialnominalmoneystockMb＞0,ａｎｄanormalizedinitial pricelevelPb＝１．Theproductiontechnologyislinearincapital,whereasthe laborsupplyisinelasticThepopulationoftheeconomystaysconstant・

Therepresentativeagenthasthefbllowinglifetimeutility：

U=ﾉ(~趣低ｗ，

⁽¹⁾

wherectisconsumptionatt，tLrepresentsthefelicity,△trepresentsthecumu- lativediscountrateatperiodt・

ThefblicityfUnctionuiscontinuous,increasing,andconcave・Thecumula行

５

tivediscountratefUnction△ｔｉｓｄｅｔｅｒｍｉｎｅｄｂｙ

(2)

△＝β(Ｃ,Ｘ)，

whereprepresentsthediscountratefimction，Ctistheeconomy-wideaverage levelofconsumptionatt，ａｎｄＸｔｉｓａｎｅｃｏｎｏｍｙ－ｗｉｄｅａｖｅｒａｇｅｌｅｖｅｌｏｆａｓｓｅｔｓ ａｔｔ、Thediscountingratefbreachagentisnotnecessarilyconstant，butitis takenasexogenous・Thesocietydeterminesthetimepreference,ａｎｄｉndividuals acceptsuchasocialnormtomaximizetheirprefbrence、Ｗｅａｓｓｕｍｅ△ｏ＝Ｏａｎｄ Ｏ＜β＜○○ｆｂｒａｌｌＣａｎｄＸＷｅｗｉｌｌｉｍｐｏｓｅｍｏｒｅｒｅｓtrictionsonuandpafter wepresentｔｗｏpropositionsinSection2

Equation(2)indicatesthatnotonlyaggregateconsumption,butalsoassets afIbctthedegreeofimpatience、Weconsidertwotypesofassets：productive andnonproductive・Economy-wideproductiveassets,ｄｅｎｏｔｅｄｂｙＫ,contribute directlytoproductionatamacroeconomiclevel,whereasnon-productiveassets，

ｄｅｎｏｔｅｄｂｙＭ，facilitatetransactions・Inwhatfbllows，werespectivelycallK andMcapitalａｎｄmoney,andthesumofcapitalandmoneyK＋Marereferred toastotalassets・AsthevariableXin(2),wecanconsidertheeconomy-wide generalassets：

Ｘ＝α1Ｋ＋α2Ｍ，

whereα１三Ｏａｎｄａ２二０．Ｗｈｅｎα１＝１ａｎｄα２＝１，thenXrepresentstotal

assets、

Individualmoneyorrealbalances，ｄｅｎｏｔｅｄｂｙｍ，isintroducedintothe modelbyconsideringthecostswithindividualtransactionsTheamountof transactioncostsincreaseswithconsumption，butdecreaseswithrealmoney balances・SuchtransactioncoststechnologyisrepresentedbyT(c,、).rbrthe existenceofabalancedgrowthpath,weassumeT＝s(ｍ/c)c,ｓ三０，s'＜0,s〃＞

６

Ofbrallm/c＞０，limm/c→｡｡s'(ｍ/c)＝０，andlimm/c→Cs'(ｍ/c)＝－．oThis assumptionassuresn＞0,恥くOZc〉０，Thnm〉0,andZLcThzm-噺＝O

Wecalls(ｍ/c)thetransaction-costsfimctionAnexampleis：

s(ｍ/c）＝ｓｏ(ｍ/c)~〃

whereｓｏ＞Oand77＞O Thebudgetconstraintis

[1＋s(ｍ/c)]c＋ｄ＝Ａｈ＋〃－ｍγ2， (3)

whereα＝ｍ＋kisindividualtotalassets,lcisindividualcapital,汀＝Ｐ/P isexpectedratesofinHation，〃islump-sumtransfbrsfromthegovernment，

andAisaconstantparameterrepresentingalineartechnologyofproduction

filnction2

GivenCandX,theeconomicagentchoosesc’ん,andmtomaximize（１）

subjectto(2),(3),andtheboundaryconditionsAo＞０，Ｍｂ＞０，Ｐｂ＝１，ａｎｄ

ｌｉｍ入(ｋ＋、)e~△＝０．sincetheutilityfUnctionisbounded,theoptimization

problemiswell-definedWhenMsthecostatevariableof(3),Pontryagin，s maximumprincipleyields

入＝ｕ'/(1＋ｓ－ｓ'､L/C）

入(Ａ＋汀＋s')＝0 入＝入(β－Ａ）

(4) (5) (6)

Thegovernmentbehavesina(monetary-theoretically)conventionalwaylt ２ＳｉｎｃｅｔｈｅｉｎｄｉｖｉｄｕａｌｌｅｖｅｌｏｆｔｏｔａｌassetsisdenotedbycL，theeconomy-wideaveragelevel oftOtalassetsshouldhavebeenAHowever，thenotationAhasbeenconventionallyusedto representaconstanttechnologyofproductionfUnctionandwefbllowthisconvention、Thus，

astheaverageleveloftotalassets，ｗｅｕｓｅＫ＋〃．

７

printsnominal~moneyataconstantrate／Landrunsabalancedbudgetby transferringseignioragerevenuestotheconsumersinalump-summanner：，ﾉｫｰ

ﾉ４７７１t・

Inequilibrium，ｔｈｅｍoneyandthegoodsmarketsareclear：

ｍ＝い－汀)ｍ A＝Ａｋ－(l＋s形

(7) (8)

Theaggregateconsistencyconditionrequiresc＝Ｃ’ん＝Ｋ，ａｎｄ、＝Ｍ・

Amonetaryequilibriumisasetofpaｔｈ｛Ｃｆ,ＡＷ７Ｗ｢`)に[０，｡｡)thatmaximizes

(1)subjectto(2)ａｎｄ(3)fbrgiveninitialconditions,inwhichthegovernment behaviorcondition，themarketequilibriumconditions，andtheaggregatecon- sistencyconditionholdWecanobtainthefbllowingdynamicsystemofc，ｍ，

andAunderamonetaryequilibrium：

ｓ".ｍ/c２．

Ａ－ｌｏ＋，＋ｓ－ｓ,.ｍ/cｍ，

似十Ａ＋ｓ'，

供ｃｍｌｍ

(9) (１０）

and(8)withtheboundaryconditions,where

SlLm2/C２

＋ 破一Ⅲｃ

一一０

(11）

1＋ｓ－ｓＬｍ/c，

ＮｏｔｅｔｈａｔｔｈｅｒａｔｅｏｆｉｎＨａｔｉｏｎ７Ｔｉｓｎotincludedintheabovedynamicsystem，

butisdeterminedby7r＝阯巾/ｍ＝－Ａ一s'．

８

ＳＡＢａｌａｎｃｅｄＧｒｏｗｔｈＰａｔｈ

Inthissection，weestablishtheexistenceanduniquenessofabalancedgrowth path(BGP）Anon-degeneratebalanced-growth-path(BGP)monetaryequilib- riumisasetofmonetaryequilibriumpaths｛ct,AMuMTt}tE[０，｡｡)suchthatthe

quantityvariablesc,片，ａｎｄｍｇｒｏｗａｔａｃｏｎｓｔａｎｔｒａｔｅ７＞０．ＯｎｔｈｅＢＧＰ，

γ ｜’ ・ｍｌｍｌｌ ｈ－ｋ ｌｌ ・ｃｌｃ

andaconstant7T＝〃－７．NotethattherateofnominalintereｓｔｓｉｓＡ＋7Ｔ＝

Ａ＋似－７＝－s'from(10）Forapositivenominalinterestrates,thegovernment shouldsetthegrowtｈｏｆｍｏｎｅｙｔｏｂｅ以三７－ＡｏｎｔｈｅＢＧＰ，

ＡｓＭｅｎｇ(2006)hasshown,wecaneasilyprovethefbllowmgproposition：

Propositionl：ＩｆａＢＧＰｅｘｉｓｔｓａｎｄｔhediscountratefilnction p(C,Ｘ)isinvarianttotheBGP,then(i)β(0,Ｘ)mustbeconstant orhomogeneousofdegreezeroinCandX,ａｎｄ(ii)theelasticityof marginalfelicitymustbeconstant、

Ｐｒｏｏｆ：TheproofisessentiallythesameasｔｈａｔｏｆＬｅｍｍａ４１ｏｆ Ｍｅｎｇ(2006),andomitted

lnwhatfbllows,ｗｅａｓｓｕｍｅｕ(c）＝ｃ1-｡/(１－び）ｆｂｒＤ＞１，ａｎｄｐ(0,Ｘ）＝

β(Ｘ/C）NotethattheobjectivefUnction(1)iswell-definedinthisutilitycase

becauselimc→｡｡c1-゜/(１－．)＜Ofbr◎＞L

Thediscountingratep(Ｘ/O)dependsontheratioofeconomy-wideassets toconsumption、Ｗｈｅｎβノー０，ｔｈｅｔｉｍepreferencerateisconstant、Ｗｈｅｎ β'＞０，thedegreeofimpatiencedecreaseswiththeeconomy-wideaveragelevel ofconsumption,butitmcreaseswiththeeconomy-wｉｄｅａﾊﾉeragelevelofcapital，

money,ortotalassets,givenallothervariablesfixedlnparticular,β'(Ｘ/C)＞O indicatesthatasthesocietygetswealthierorconsumesless,peoplebecomemore

９

巨夛「ずｇ百一｝○コの）急の四ｍの巨目の二昌己（へ）ＶＰ瓦（へ）ⅣＰ四目□の巨己、（へ）Ⅲロハ』・

二声の目Ｑ〕ＶＰョ『のロの①」自昌＆盆。ご巴ＰｍｍＥ巨官】○口（』＋⑰）⑰ご－助一（廷十ｍ｛）ＶＣさ円 四口己Ｃ自の国の勺・二一の四ｍの巨目目・己の篁二＆蟇の口直Ⅳ○四己の（ぷ）Ⅱ②◎（い）１ｓ 云「声円のへⅡＱ】国桿＋Ｑ図吋母・Ｚ。（の（ず皀口の一○口ｍｐｍ言豈の、○ぐの『自白の己←、の訂（すの四○コ「（す 円昌の。、一ケの曰○口の弩の巨己で一三（○ワの種Ⅳ（澤ｌｑ）座へｑ）（すの口○目旨四一旨←の［の、一門Ｐ（の

】の四｝。「皇の己。、ごくの百毎Ｂ》Ｌ＋一二－三Ⅱ這十（ｑｌ得）」へｑ＋ミＱＶＣざ閂昌一 痘Ⅳｐｌｑ）』へｑ・

ごく耳嵜四ｇ』】（】○口四一四ｍｍ巨閂己己は○口の円の、四Ｈｇ一口、、）ヨ「のｏ②ロで円○ぐの←ケロー←ずの円のの×】切言のＰ 巨自臼巨の□の勺。

日】己昌】の三○円］のｍの二「】］｝旨ｍＳＱの庁Ｈ８ｐｍ巨白官一○口・言のロ、（ｍＣＣ⑦）ケロの①×自己目の旦二の

『二一○○由の○口の巨自〕ご菌○口一○旨○○日の冒四【の四一のｎｏｐｏＢ『・月ケの円昌】。】の一宮①の四目の四ｍ←ずの 曰くの円のの閂昌】○○由の四℃耳Ｐ｝一○ｏｏｐｍ巨日ご←】○ご冒○巨円津四目】の三『○円【すのロロ巨切の（ずの己Ｈｏｇ巨巳】○口

百口○は○口］の』秀．

思

田の←Ｎ」Ⅱ秀へＯＰｐＱＮ口Ⅱコミロ・○ご←すの国の勺》（、））（①））四口』（』ｅ胃の①×で円ののｍの』

旧円。。廉の用言宮の少己己の口＆×。

勺司・己。、三・目醇少の、目〕の、（へ）ＶＣさＨＰ一一へⅡＱ］国］＋Ｑ口いＶＰ 四目」の巨己、（へ）Ⅲロハ』・目ケのご弓の円のの凶の菌四口。ご‐」の、の口の国二旨、国の勺 ざ同色巳迄Ⅳ（澤ｌｑ）』へｑ三二ｑＶ澤・二ケのロ、』ｍ８ｐｍご己》白目 日日の量長宮口＆・口・口］弩・開い（瓦（へ）Ⅳ言己Ｑ］ⅡＣ））吾の三ヶの国の勺

】の巨曰】・自の．固くの巨冒←ずのｏｐｍＱ三】吾Ｑ］ＶＰ←ずの円のの×】の詰四巨已Ｐ巨の国○勺

●三声の口互（へ）ⅣＣｐｐｇ（骨＋の）⑰ご－ｍ一（亡十の｝）ｖＣさ円＆］田母Ｖ○・

ｑ］Ⅱ」’又Ｃ）

二Ⅱ定十』＋功一（鈩）》

望Ⅱ」１つ＋②（鈩）｝一国］》

］①

（］⑭）

（】い）

（】』）

whereｓｏ＞Ｏａｎｄ刀＞o3ThisfunctionalspecificatioｎｉｓｕｓｅｄｉｎＳｅｃｔｉｏｎ６，

wheretheuniquenessofBGPisnumericallyconfirmedinthecasewith/ｕ＜０．

４ComparativeStatics

Inthissection，weinvestigatetheefTectofmonetaｒｙｐｏｌｉｃｙｏｎｔｈｅＢＧＰ、Our modelallowsthegovernmenttocontrolthegrowthrateofmoney・Weconduct acomparativestaticanaｌｙｓｉｓｏｎｔｈｅＢＧＰＷｅｅｘａｍｉｎｅｔｈｅｅｆｆｅｃtofmonetary expansiononeconomicｇｒｏｗｔｈ,theratioofcapitaltoconsumption,ａｎｄｔｈｅｒａtio ofmoneytoconsumption・Ｔｈｅｎ，wedemonstratethatthereexistsanoptimal monetarypolicyinsomecaseWefinallydiscusstherelationshipbetweenthe economicgrowthandtherateofinHation

Weperfbrmthecomparativestaticswithrespectto7,Ｚ,，andZ2inresponse tothemoneygrowthrate仏Ｔｈｅtotaldifferentiationof(12),（13),ａｎｄ（14）

leadstothecomparative-staticresultswithrespectto7，ＺｎａｎｄＺ２ｉｎｒｅｓｐｏｎｓｅ

ｔｏｔｈｅｇｒｏｗｔｈrateofmoney:４

慨Ｈ _p'{αlsl(Z2)/Z,＋α2(1＋s(功))/Z;｝ Ｉ

“'(Z2)/Ｚ１－ａ２ｐ’

α,β,＋⑰(,＋s(Z2))/好

where

Ｑ＝－◎s"(1＋s)/Ｚｆ－ａ１ｐ'(s"＋s'/Z,)－a2P'(1＋s)/Zf．

３１ｎfact，

（,＋s)s〃－s'(似十s'）Ｚ（'＋s)s″－(s′)２

=〃(刀+,)３oz亙り-2+〃s;zﾃﾞﾘｰ2>o

fOrallZ2＞Ｏａｎｄ似三0．

４Thedetailedderivationisavailablebyarequesttotheauthor.

11

Ｗｅｈａｖｅａｓｓｕｍｅｄｐ'三OThus,thetermQisnegativewhens"(Z2)＋

s'(Z2)/Z,＞OTheconditions"(Z2)＋s'(Z2)/Z,＞Oisrewrittenas

箸=Ｍ１<-恥"/sＬ

ThisconditionismorelikelytoholdasthedegreeofconcavityofsissuHiciently strongerorastheamountofmoneyisrelativelylessthanthatofcapitalonthe BGPWhens(ｍ/c)＝so(ｍ/c)~刀fbrexample,then-Z2s"(Z2)/s'(Z2)＝り＋Ｌ

ＷｅａｓｓｕｍｅＱ＜Owithinthissection

WhenQ＜Ｏａｎｄｐ'三0,themonetaryexpansionraisestheratioofcapital toconsumption(｡Z,/d似＞Ｏ)andlowerstheratioofmoneytoconsumption

(dIZ2/d似くO）Theeflectofmonetaryexpansiononeconomicgrowthis

壼一,Mz2)z'+・'('+`(z塾)） ^Ｚ?、

^(15）

Notethateconomicgrowthisindependentoftherateofmoneysupplywhen eitherp'＝Ｏｏｒａ，＝α２＝OExceptfbrthesecases，ｔｈｅｓｉｇｎｄｅｐｅｎｄｓｏｎｔｈｅ ｓｉｚｅｏｆａ，ａｎｄα2．Fbrfurtheranalysis,weconsidertwoextremecases：（＝Ｚ１ (α,＝１ａｎｄα2＝O)ａｎｄ（＝Ｚ(α,＝Ｏａｎｄａ２＝１),andwethenreturnto generalcases、

Ｗｉｔｈα，＝１ａｎｄα２＝０，theeflectofmonetaryexpansiononeconomic growthisd7/d似＝－β's'/(Z,Q）Whenthedegreeofimpatienceisanincreas- ingfUnctionoftheratioofcapitaltoconsumption(β'(Ｋ/C)＞O),theeffectof growthratesofmoneyhasanegativeefTectoneconomicgrowthTheintuition isasfbllows，Higherratesofmoneysupplylowerrealbalances,raisethetransac- tioncostsandtherefbrediscourageconsumption・Whenthedegreeofimpatience increasesintheratioofcapitaltoconsumption,decreasingconsumptionmakes

theagentlesspatient,andthuslowerseconomicgrowth(Ａ－ｐ)/ひ．

1２

Ｗｉｔｈα，＝Ｏａｎｄａ２＝１，theefIbctofmonetaryexpansiononeconomic

growthistheopposite(｡γ/d似＝-,0'(1＋s)/(ZfQ)）Whenthedegreeofim-

patienceincreasesintheratioofrealbalancestoconsumption(β'(Ｍ/C)＞Ｏ)，

highermoneyratesofmoneyhaveapositiveeflectoneconomicgrowthIn- tuitively，ｇｉｖｅｎａｃｏｍｍｏｎｇｒｏｗｔｈｒａｔｅ７，thehighercostofholdingmoney decreasestheratioofmoneytoconsumptionZ2from(13).Whenaneconomic agentismoreimpatｉｅｎｔａｓｔｈｅｒａｔｉｏｏｆｍｏｎｅｙｔｏｃｏｎｓｕｍｐtiondecreases，de- creasingZ2makestheagentmorepatient，ａｎｄthusraiseseconomicgrowth u-p)/◎．

Ingeneral(α,＞Ｏａｎｄａ２＞0),boththeeflectsinthecasesofp'(Ｋ/C)＞Ｏ ａｎｄｐ'(Ｍ/C）＞OaremixedThehigherratesofmoneysupplylowerreal balances，raisethetransactioncostsandtherefbrediscourageconsumption,in- creasingZ1、Ａｔｔｈｅｓａｍｅｔｉｍｅｈｉｇｈｅｒｃｏｓｔｏｆｍｏｎｅｙholdingsdecreasestheratio ofmoneytoconsumptionZ2from(13).Whenaneconomicagentislesspatient asX/Cincreases（β'＞Ｏ),increasingZ1anddecreasingZ2makestheagent eitbermorepatientifthecapitalefIectisstrongerorlesspatientifthemoney efIectisstronger、ＴｈｕｓｔｈｅｅｆＩｅｃｔｏｎｅｃｏｎｏｍｉｃｇｒｏｗｔｈｄependsonwhicheflect

isdominant．

RecallthatlimWc→Cs'(ｍ/c)＝－oobyassumption、Thus,asufliciently

highgrowthrateofmoneymakesa1s'(Ｚ)Z,＋α2(1＋s(Ｚ))ｉｎ(15)negative，

anddampenseconomicgrowth7＝(Ａ－ｐ)/aThismaycauseahump-shaped rel帥ionshipbetweenthemoneysupplyandeconomicgrowth・ＷｈｅｎｔｈｅｉｎＨａ－

ｔｉｏｎｒａｔｅｉｓｌｏｗ，、onetaryexpansionhasapositiveeHbctoneconomicgrowth (anbineffect),butsuflicientlyhigherinHation,onthecontrary,decreasesthe economicgrowth(areverseTbbinefIect)．

WefbrmallypresentthefOllowingproposition．

Proposition3:WhenthegovernmentcanseM＝(α1/α2-1)s'(Z2)，

１３

suchmonetarypolicylocallymaximizestheeconomicgrowthrates．

￣

ProofbSeetheAppendix．

Ｗｈｅｎα，三α２，theoptimalrateofmoneyisgreaterthanzeroAsdiscussed inthelastpaｒｔｏｆＳｅｃｔｉｏｎ３，thereexiststheuniqueBGPwhenノｕ二Ｏａｎｄ ｓ(幻)＝so(Z2)-刀.ThisspecificationisusedinSection6

Inparticular,ｗｈｅｎα1＝α２(〈＝α1(Z1＋Ｚ２)),theeconomicgrowthrate 7ismaximｉｚｅｄｂｙ似＝ＯＴｈａｔｉｓ，ifthedegreeofimpatienceincreasesinthe economy-wideaverageratiooftotalassets(thesumofcapitalandmoneyhold- ings)toconsumption,thenazerorateofgrowthofthemoneysupplyachieves maximizedendogenousgrowthNoticethattheinHaｔｉｏｎｒａｔｅｉｓ－７＜Oandthe nominalratesofinterestsisA-7undertheoptimalpolicy・

Weshouldnoticethatthispropositiondoesnotalwaysguaranteethatthe governmentisabletoprintmoneyattherateof(α,/α2-1)s'(ｚ）Whenα1＞

α２，theoptimalgrowthratesofmoneyisnegative,ａｎｄｔｈｅｒａｔｅｃｏｕｌｄｂｅｓｍａller thanA(１－ヶ)/αRemember似三Ａ(１－◎)/ぴisassumedfbrapositivenominal rateofinterests・mrthermore，thepropositionsaysonlythelocalproperties ofoptimalmonetarypolicy、ＩｎＳｅｃｔｉｏｎ６，weconductnumericalexamplesto demonstratetheglobalhump-shapedrelationwithasetofparameters・

Ｆｉnally,weconsidertherelationshipbetweenmoneygrowthandinflation・

Because7T＝似－７，

虫＝,_血＝s,，。('＋s)＋α'β'zf

d似ｄ似ぴs"(1＋s)＋α,p'(s"Z2＋s'Z,)＋α2β'(1＋s）

一旦川筈>Ⅲ

Ｔｈａｔｉｓ，inHationandthemoneysupplyalwayshaveapositivecorrelation Whendγ/d〃＞Ｏ(resp.＜0),monetaryexpansionacceleratestherateofinHa‐

tionmore(resp・less)thanproportionately・Ｗｈｅｎｄ７/d似＝Oうthend7T/d似＝L

AlthoughProposition3showstherelationshipbetweeneconomicgrowtｈａｎｄ

1４

themoneysupplyrate，Section6examinestherelationshipbetweeneconomic growthandinflationratesnumerically．

５ＬｏｃａｌＤｙｎａｍｉｃｓ

Inthissection，webrieHymentionthelocalstabilityofthebalancedgrowth pathtoexaｍｉｎｅｗｈｅｔｈｅｒｔｈｅｐａｔｈｉｓｓｔａｂｌｅｉｎａｎeconomicsense・

RememberthatZ,＝ん/c,Ｚ２＝ｍ/candsisafilnctionofZ2alone．Ｍひ nipulatingequations(8),(9),ａｎｄ(10)yieldsthefbllowingdiflerentialequation systemofZ1andZ2：

|化Ⅱ'二面燗駒ﾊﾟ;⑥

’

^{Z,/z’ z2/Z２ ●}

１

where

ｓ"(Z2)Z；

γ(Ｚ)＝，＋s(ｚ)－s,(Z2)Z２，

and（＝ａ１Ｚ１＋ａ２Ｚ２,respectively、

LinearizationwithrespecttoZ1，ａｎｄＺ２１ｅａｄｓｔｏ

に汁に二二’

whereJisthe2×２JacobianmatrixofthedynamicsystｅｍａｒｏｕｎｄＺｆａｎｄＺｉｉ Ｗｉｔｈｓｏｍｅａｌｇｅbra,ｗｅｃａｎｓｈｏｗｔｈａｔ

detJ＝－，/{ヮ＋γ(ｚ)｝

ｆｂｒｅａｃｈｃａｓｅＸ＝α1Ｋ＋α2Ｍ，wherenisdefinedintheprevioussectionfbr

1５

eachcase､５since◎＞ＯａｎｄＴ(几）＞０，detJandnhavetheoppositesigns

Similarly,thetracetakes:６

(α,＋α2)β'＋ぴs〃

trJ＝(1＋s)/Zf＋

_ぴ＋７

Ourbefbre-reduceddynamicsystemhastwojumpvariablescandm・There- fbre,Z1andZ2shouldhaveunstablerootsofthecharacteristicfUnctionfbran economicstability・Ｔｈａｔｉｓ，theeconomicstabilityconditioniｓｄｅｔＪ＞Oand trJ＞ＯＢｏｔｈｈｏｌｄｗｈｅｎｎ＜Ｏａｎｄｐ’＞０．Ａsdiscussedintheprevious section,thetermnisnegativeaslongass''(Z2)＋s'(Z2)/Z,＞OInSection6，

ｗｅｃｏｎｆｉｒｍｎ＜Owithaplausiblesetofparameters．

６NumericalExercises

Thissectionconductsnumericalexercisesusingseveralsetsofparameters､７

RememberthatthefblicityfUnctioｎｉｓａｓｓｕｍｅｄｔｏｂｅｕ(c）＝c1-ｸﾞ/(１－ｏ)，

ぴ＞１ｆｂｒａＢＧＰＩｎａｄｄｉｔｉｏｎ，thetransaction-costsfunctionisspecifiedas s(ｍ/c)＝so(ｍ/c)-〃fbrm/c＞0,whereso＞０and〃＞O

nlrthermore,pisassumedtobelinear(β"＝Ｏ)fbrasimpleanalysis・The discountratefUnctionisrepresentedasp(（)＝βo＋p,(<－５０),where５＝α,Ｚ,＋

ａ２Ｚ２ａｎｄ⑰isobtajnedfromthecorrespondingBGPwithabenchmarkrate ofmoneygrowth似oandaconstanttimepreferencepo・clearlyβ'(()＝β,＞O

IbsatisfytheassumptionofPJoposition2,weassumethatpに)＝Ｅ＞Oif Po＋β,に－〈o)＜ＥａｎｄＰ(（)＝A-EifPo＋β,(ﾋﾟｰﾋﾟo)＞Ａ－Ｅ，

WeshouldspecifyaUtheparametersひ,ｐｏ，ｐ,，ノリo，ｓｏ,〃,Ａ,ａｎｄＥ、We ca1ibratetheseparameters,sothatZ,＝２，Ｚ２＝１，７＝0.1and7T＝Ｏ０５ｏｎｔｈｅ

5Thedetailedderivationisavailablebyarequesttotheauthor、

6Thedetailedderivationisavailabｌｅｂｙａｒｅｑｕｅｓｔｔｏｔｈｅａｕｔｈｏｒ、

7ＴｈｅＭａｔｌａｂｃｏｄｅｓａｒｅａｖａｉｌａｂｌｅｂｙａｒequesttotheauthor．

1６

似ｏ

０．１ ０．１ ０．１５ ０．５ ０．６ 0.01 lHblel：ＡＳｅｔｏｆＰａｒａｍｅｔｅｒｓ

ＢＧＰ､Theratioofcapitaltoconsumptionｉｓｔｗｏ１ａｎｄｔｈｅａｍｏｕｎｔｓｏｆｍｏｎｅｙ ａndconsumptionareequivalentoMheＢＧＰ､Theeconomyisgrowingat１０％・

WesetthecoefIicientofrelativeriskaversionひｔｏｂｅｆｉｖｅ、ＷｅａｓｓｕｍｅﾉＣＯ＝0.1．

Withtheendogenousgrowthrateofl０％,thegrowthrateofmoneysupplWLo shouldｂｅ15％・Theremainingparametersso，77,andAaredeterminedfrom (12),(13),ａｎｄ(14)：ｓｏ＝05,り＝1.3,ａｎｄＡ＝０６．RememberthatQ＜０，

discussedinSection4,ifβ'＞Oands''(Z2)＋s'(ｚ)/Z,＞０.Thelastcondition isequivalenttoZ2/Z,二Z2s"/s'＝り＋1,whichissatisfiedinourtransaction filnction(Z2/Z,＝０５＜〃＋１＝２３）

Ｗｈｅｎβ，＝Ｏｏｒｐ＝ｐｏ，theendogenouseconomicgrowthisfivepercent，

independentofgrowthratesofthemonetarysupplyorinflation・Ａｐａｒｔｆｒｏｍ ｐ１＝０，wemvestigatetherelationshipbetweeninflationandeconomicgrowth rates、Ｗｅａｓｓｕｍｅｐ，＝０．１，implyingthatthediscountingfimctionincreases with〈＝Ｘ/Ｃ・WealsosetEtobeOO1sothatO＜β(（)＜Afbrall（＞０.The benchmarkvaluesoftheparametersaresummarizedinlHblel・Withthisset，

wecanfindanon-degeneratingBGP(γ＞０，Ｚ,＞０，Ｚ２＞０)fOrasuHiciently widerageof似．

FigurelconsidersthecasewhereX＝Ｋ(α，＝１ａｎｄα２＝Ｏ)ｏｎｔｈｅ ＢＧＰＷｉｔｈｔｈｉｓｂｅｎｃｈｍａｒｋｓｅｔｏｆparameters，、＜OasdiscussedinSec‐

tion4，ｏｒｄｅｔＪ＞OasdiscussedinSection5・ＡｓｅｘａｍｉｎｅｄｉｎＳｅｃｔｉｏｎ４，

β'(Ｋ/C)＝0.1＞Oindicatesd'1//d似くＯａｎｄｄ汀/d似＞0．Thus,thereexists anegativerelationshipbetweeninHationandeconomicgrowthFigure2de- pictstherelationshipbetweeninflationandeconomicgrowthinthecaｓｅｗｈｅｒｅ Ｘ＝Ｍ(α,＝１ａｎｄα2＝１)．AsdiscussedinSection3,moneta正yexpansion

1７

policyincreasestherateofeconomicgｒｏｗｔｈａｓｗｅｌｌａｓｔｈｅｒａｔｅｏｆｉｎＨａｔｉｏｎ・

Figure3illustratestherelationshipbetweeninflationandeconomicgrowth inthecasewhereX＝Ｋ＋〃（α，＝α２＝１）AsshowninProposition3，

theendogenouseconomicgrowthisl003％maximizedatzerogrowthrateof moneysupply・Reducingtherateofmoneysupplyfroml5％ｔｏ０％increases theendogenousgrowthby0.03％・Sincezeromonetaryexpansionimpliesthat therateofinHationisequaltothenegativerateofgrowth,Figure3showsthat therelationshipishump-shapedattheinflationratesof-10０３％withthegiven setofparameters・

Figure4illustratestherelationshipbetweeninHationandeconomicgrowth inthecasewhereX＝ＡＫ＋Ｍ(α，＝Ｏ６ａｎｄａ２＝１)．Thiscaseimplies thattheagentisaHbctedbythesumofrealbalancesandincome・Aspresented inProposition3，theendogenousgrowthismaximizedwhentherateofmoney

supplyis似＝０４りZ三刀-1＝026．ThemaximizedeconomicgrowthislO０２％

ＴｈｅcorrespondingrateofinHationisl5.98％、Thisnumericalexampleindicates thatacertainpositiveinHationcanachievetheoptimaleconomicgrowth

Weexaminetherobustnessoftheparame1erregardingthemarginaleffectof therelativeassetsonthedegreeofimpatience，β,、Figure5comparesthecase ofp,＝O1withthatofp,＝O8BothcasesconsiderX＝Ｋ＋Ｍ(α,＝１ａｎｄ ａ２＝１),andtheoptimalpoliciesarebothzerogrowthofmoneysupply・The rateofendogenousgrowthisl0.03％ｉｎｔｈｅｃａｓｅｏｆｐ，＝01,whereastherate islO､14％ｉｎｔｈｅｃａｓｅｏｆｐ，＝0.8．ThelargerthemarginalefIbctoftherelative assetonthedegreeofimpatience，thehigherthecorrelationbetweeninHation andeconomicgrowth・However，evenwithmuchhigherp1，optimalmonetary policyimprovestherateofeconomicgrowthonlybylessthan0.2％・

Ｉｎｓｕｍ，wehavepresentedtherelationshipbetweeninHationandeconomic growthwithseveralcasesofvaryingdiscountratesltshouldbenotedthatthｅ

1８

efIectofmonetarypolicyisnotverylargewiththebenchmarksetofparameters、

Figuresland2showthatapproximatelya50％increaseininHation(interms oflOyears)raisesorlowerseconomicgrowthbyaround１％,andFigures3and ４demonstratethatapproximatelya20％increaseininHationhaslessthan0.1％

ｉｍｐａｃｔｏｎｅｃｏｎｏｍｉｃｇｒｏｗｔｈＭｏｎｅｙｉｓｎotsuperneutral，butthemagnitudeis notverylargeinourspecificationofparametersandfilnctionalfbrms．

７ＣｏｍｐａｒｉｓｏｎｔｏｔｈｅＭｏｄｅｌｗｉｔｈｌｎｔｅｒｎallyDe-

terminedDiscountRates

Thissectioncomparesthepreviousmodelwithamodelinwhichconsump- tioninthediscountratesisdeterminedinternallybytheeconomicagent、The modelenvironmentfbllowsthemodelweusedexceptfOrthediscountratebeing determinedby

△＝β(Ｘ/c)，（16）

insteadof(2),whereＸ＝α1Ｋ＋a2MThediscountrateisdeterminedbythe ratioofaggregateassetstoindividualconsumption・Ｗｅｓｈｏｕｌｄｎｏｔｅｔｈａｔｔｈｅ

abovediscountrateisconstantontｈｅＢＧＰ．

ＧｉｖｅｎＸ,theeconomicagentchoosesc’ん,andmtomaximize(1)subject to(3)ａｎｄ(16)Ibsolvetheproblem,weconsidertheHamiltonian：

Ｈ(ＭＭＣ,α,△,入,の,妙)＝e-△{c'-./(１－◎）（17）

+入(AAC－行、＋〃－[1＋s(ｍ/c)]c)－のβ(X/c)＋V)(ｑ－ｍ－Ａ)｝

where入ａｎｄ‘arethecostatevariablesof(3)ａｎｄ(16),respectively・Pontrya戸

1９

gin，smaximumprincipleyields(5),(6),ａｎｄ

c－．－｛1＋s(m/c)－鮒'(７｢L/C)､/c)入十のβ'Ｘ/c2＝Ｏ

●

。＝－Ｔｌ＋の〃．

(18）

Theinitialconditionsareko＞０，Ｍｂ＞Ｏａｎｄ凡＝１，andthetransversality conditionbyMichel(1982)islimt→｡｡Ｈ(t)＝O

AsdiscussedinPalivosetaL(1997),theHamiltonianisindependentoftime ontheoptimalpath，andtheabovetransversalityconditionimpliesthatthe valueoftheHamiltonianiszeroontheoptimalpathHence，

の＝ｕ(c)＋入(肌一万ｍ＋〃－[1＋s(ｍ/c)]c)＋U'(ｑ－ｍ－ｋ）

_β(c/Ｘ）

ThisisinterpretedasthelifetimeutilityontheoptimalpathUsing(18),itis easytoshowdc＝Oorequivalently

1＋(β'Ｘ/c)/{β(１－◎)｝

入＝ｃ－ヶ

1＋s(ｍ/c)－s'(ｍ/c)ｍ/ｃ－(β'〃)(｡Ｘ/c2)．

Ｔｈｅｄｙｎａｍｉｃｓｙｓｔｅｍｏｆｃ，ｍ，andkunderamonetaryequilibriumischar- acterizedby(6),(8),(10),and

1＋(β'（)/{β(１－ぴ)｝

入＝ｃ－ｏ1＋s(Z2)－s'(Z2)Z２－(β'〃)(Z,＋Ｚ２)<(｡/α)， (19）

withtheboundaryconditions,whereＺ,＝ん/c,Ｚ＝ｍ/c,and（＝α,Z,＋a2Z2・

Thedynamicsystemofthemodelwiththeexternallydetermineddiscount ratesischaracterizedby(8),(9),and(10).TheonlydifIbrencefromthemodel withinternallydetermineddiscountratesistheshadowprice，入．Inthecaseof externallydetermineddiscountrates,combining(6)andthetimedifIbrencewith respectto入yields(9).Inthecaseofinternallydetermineddiscountrates,itiｓ

2０

hardtodescribethetimedifTerencewithrespectto(19)ｗｉｔｈ〈＝α,Ｚ,＋a2Z2，

andtherefbreitisdifliculttocharacterizethedynamicsystemwithoutthe shadowprice入．

ＯｎｔｈｅＢＧＰ,however,equation（19)indicatesthaMハーー◎6/ｃ＝－び７．

Thus,thedynamicsystemofequations(6),(8),(10),ａｎｄ(19)arereducedto

(12),(13),and(14)ontheBGPThus,wecanobtajnthefbllowingproposition：

Proposition4：ConsiderthemodelwiththediscountratefUnction determinedby(16）Then,７，Ｚ,,andZ2onthebalanced-growth pathaｒｅｅｘａｃｔｌｙｔｈｅｓａｍｅａｓｔｈｏｓｅｏｆｔｈｅｍｏdelthediscountrates sociallydeterminedbyp(Ｘ/Ｃ)whereＸ＝α1Ｋ＋α2Ｍ

ThecomparativeanalysisconducteｄｍＳｅｃｔｉｏｎ２ｉｓｓｈａｒｅｄｉｎｔｈｅｍｏｄｅｌｗｉth internallydetermineddiscountrates、

Manystudiesdealingwithinternallydetermineddiscountratesassumethat thedegreeofimpatienceincreaseswithconsumption8Thisassumptionisless plausiblefromanempiricalvieｗpoint，ｂｕｔｉｔｉｓｉｍｐｏｓｅｄｆｂｒｄｙｎａｍｉｃｓｔａｂｉｌｉｔｙ Ｏｎｔｈｅｏｔｈｅｒｈａｎｄ，thepreviousmodelassumestheincreasingimpatiencein aratioofaggregategeneralizedassetstoconsumptiontoproducethehump- shapedrelationshipbetweenthemoneysupplyandeconomicgrowthReplacing economy-wideconsumptionbyindividualconsumptionimpliesthatthedegree ofimpatienceisdecreasinginindividualconsumptiongivenaveragegeneralized assets,whichsupportsempiricalevidenceincludingBeckerandMulligan(1997）

However,wehavetwocaveatstothisproposition・First,theequivalentresult holdｓｏｎｌｙｏｎｔｈｅＢＧＰＷｈｅｎｗｅａｔｔｅｍｐｔｔｏｃｏnductdynamicanalysis,wehave todiflbrentiateMn(19)withrespecttotimeEveninlocalstabilityanalysis，

thedynamicsystemistoocomplicated，ａｎｄｓｏｗｅｈａｽﾉｅｔｏｒｅｓｏｒｔｔｏｎｕｍｅｒｉｃａｌ investigation．

８，ａｓ(2003)investigatedthecasethedecreasingimpatiencewithconsumption・Buther interestlayonlyontheexogenousgrowthmodel．

2１

Next，theequivalentresultontheproposｉｔｉｏｎｄｏｅｓｎｏｔｈｏｌｄｅｖｅｎｏｎｔｈｅ ＢＧＰｗｈenanotherindividualvariableapartfromconsumptiondeterminesthe discountrateForexample,consider

△＝β(ｍ/c）

Thedegreeofimpatienceisdeterminedbyrelativeindividualrealbalances・

ThenPontryagin，smaximumprincipleyields(4),(6),and

’

人(Ａ＋汀十s')＋のβ'/c＝０

insteadof(5).Withabitofalgebra,9ｗｅcanshowthattheBGPcharacterizes

(12),(14),and

{1-器三;|(二鵠}ＦＡ+鰹十｡'+号{'筈}=器娑）

insteadof(13).Thus,itismuchmorediHiculttofindconditionsfbrtheexistence ^b

ofBGP．

ｓＣｏｎｃｌｕｄｉｎｇＲｅｍａｒｋｓ

lbexploretherelationshipbetweeninHationandeconomicgrowth，wehave usedthetransaction-costsmodelwithasociallydetermineddiscountrateand alinearproductiontechnology・ＷｅｈａｖｅｄｅｆｉｎｅｄａＢＧＰａｎｄｐｒｏｖｅｄｔｈａｔthere uniquelyexistsanon-degenerateBGPwhenweassumeincreasingimpatience intheratioofgeneralassetstoconsumptionandseveralotherconditions，ａｎｄ ｔｈａｔｓｕｃｈａＢＧＰｉｓｌｏｃａｌｌｙｓｔａｂｌｅＷｅｈａﾊﾞﾉefbundthatinflationalfectsthe endogenousgrowthinthecaseofnon-constanttimepreferences・specifically,iｆ

9Thedetailedderivationisavailablebyarequesttotheauthor．

2２

thedegreeofimpatienceincreasesintheeconomy-wideaveragetotalassetsto consumptionratio,thenazerorateofgrowthofthemoneysupplyachievesthe maximizedendogenousgrowth

Wehaveconductedseveralnumericalexercisestoconfirmthehump-shaped relationshipbetweentherateofinflatｉｏｎａｎｄｔｈｅｒａｔｅｏｆｅｃｏｎｏｍｉｃｇｒｏｗｔｈｗith asetofplausibleparameterslnparticular，wehavedisplayedsuchahump- shapedrelationship，butwehavediscoveredthattheimpactofinHationon economicgrowthisquantitativelysmall、Finally,ｗｅｈａｖｅｃｏｍｐａｒｅｄｏｕｒｍｏｄｅｌｓ ｔｏｔｈｅｍｏdelswiththediscountratesdeterminedinternallybytheindividual，

andwehaveprovedthatifthediscountratedependsontheindividualcon- sumption,theresultsofthecomparativestaticsontheBGPareobservationally equivalentinbothmodels・

Wepointouttwoextensions・First，whereaswehaveusedthetransaction－

cost（TC）modelinthisstudy,wewouldapplyourtheorytothemoney-in‐

the-utility（MIUF)modeLFeenstra(1986)demonstratedafilnctionalequiv‐

alencebetweenTCandMIUFmodelswithaninelasticlaborsupplyanda constanttimeprefbrenceinanexogenousgrowthframework、Ｗｅｎｅｅｄｔｏｅｘ－

ａｍｉｎｅｗｈｅｔｈｅｒｓｕｃｈｆUnctionalequivalenceestablishesinourframeworkWhen aequivalentｆＵｎｃｔｉｏｎａｌｆｂｒｍｉｎｔｈｅＭＩＵＦｉｓｎｏｔｐｒｏｐer，aquantitativeequiv- alence(ＷａｎｇａｎｄＹｉｐ,1992)isworthinvestigatingWhenwefindaproper fimctionalfbrm，inwhichanon-degeneratingmonetaryBGPexists，thenit wouldbeinterestingtoexaminewhetherinHationandeconomicgrowthstill haveahump-shapedrelationships、

Second,althoughwehaveconcludedthattheimpactofinflationoneconomic groｗｔｈｉｓｓｍａｌｌ,ourchoosingofparametersandfUnctionalfbrmsdoesnotcap- turetherealeconomyverywelLInparticular,weshouldestimatethemarginal efIectoftherelativeassetsonthedegreeofimpatience1p1，inamoreaccurate

2３

way･Ibestimatethemarginaleffectofassetsontimepreferences，manyem- piricalstudiesincludingBeckerandMulligan(1997)usemicroeconomicdata，

andconsidersanindividuallydetermineddiscountrate・Forthecalibrationof ourmodel，internationalmacroeconomicpaneｌｄａｔａｗｏｕｌｄｂｅｍｏｒｅsuitable・

EmpiricalanalysisusingsuchdataisanotherfUturetask．

Ａｐｐｅｎｄｉｘ

ＰｒｏｏｆｏｆＰｒｏｐｏｓｉｔｉｏｎ２：

Ｆｒｏｍ(12),７＝(Ａ－ｐ(（))/ｏ＞(Ａ－Ｐ)/◎＞0.Wecombine(12),(13),and(14）

toreducetwoequations：

(１－ぴ)Ａ－叩一ぴs'(幼)，

似十s'(几）

β(α,Z,＋a2Z2）

１＋s(幼）

(20）

(21）

Ｚ１

ItsufIicesfbrtheprooftoexaminewhetherthereexistsapairofZ，＞Ｏａｎｄ Ｚ２＞Osatisfying(20)ａｎｄ(21)．

SetanarbitrarWL三(１－び)A/Ctobefixed､Then,bythepropertyofs'，

thereexistsaZ2＞Osuchthats'(Z2)＋ルＡ(１－び)/ひ＝ONotethaM2can takeinfinitywhen〃＝(１－ヶ)A/a

Becauses'(Z2)＋仰ぐｓ'(Z2)＋ﾉﾙ＝Ａ(１－ぴ)/◎＜ＯｆｂｒａｌｌＺ２ｅ(0,几)，

equation(21)isrewrittenas：

１＋s(ｚ）

z,＝（(z2)＝－ 似十s'(Z，)， (22）

withlimz2→o（(恥)＝Ｏand

1＋s(ｚ）ひ{1＋s(ｚ)｝

linL（(ｚ)＝－

Ｚ２→Ｚ２ ＞0．

(似十s'(幼)）（ひ－１)Ａ

2４

Thederivativeof（'(Z2)is：

s"(1＋s)－s'(似十s'）

（'(Z2)＝ (/川十s')２

Thatis,（'(Ｚ)＞０or（(幼)increasesmonotonicallyifs"(1＋s)－s'ぃ＋s')＞Ｏ

ｆＯｒａｌｌＺ２・

Tbexamineequation(20),weconsiderthecasewithnon-constanM･Sub- stituting(22)into(20)leadsto

β(α,（(Z2)＋a2Z2)＝(１－ぴ)Ａ－叩一◎s'(Z2)，

(23）

Then,theright-handsideof(23)decreasesinZ2e(0,易)with：

豊里｡(１－。)卜。ルCs'(z2）→ｏｑ

ｌｉｎＬ(１－ぴ)Ａ－ｏルワs'(Z2）‐十ｏ

ｚ2→Ｚ２

Thus,aslongasO＜β＜Ａ,thereexistsatleastoneZ2satisfying(23)．

Whenpisconstant,oranincreasingfUnctiononlyofZ2(α,＝O),thenwe canobtainauniqueZ2e(0,Ｚ２).Ｈｏｍ(22),wecanobtainauniqueZ,＞0．

Next,considerthecasewherea,＞Ｏａｎｄｐ'(（)三0.Ｗｈｅｎs"(1＋s)－s'(似十 s'）＞0,ｔｈｅｎ（(Z2）increasesmonotonically・Therefbre,theleft-handsideof (23)increasesinZ2e(0,Ｚ)with：

墓聖｡IC(α'〈(幼)＋a2Z2）→IC(O)〉０，

linUo(α1（(Z2)＋a2Z2）→β(αlひ{1＋s(Z2)}/{(グー1)A}＋a2Z2)＞ｐ(O）

Ｚ２→Ｚ２

Thus,thereexistsauniqueZ2satisfying(23),givingauniqueZ,＞０from(22）

2５

ProofofProposition3：

Asdiscussedbefbre，thefirstderivativeof7withrespeｃｔｔｏ似ｉｓｚｅｒｏｗｈｅｎ ａ２(1＋s(Z2))＋α1s'(Z2)Z,＝0．SincM＋s'(ｚ)＋(1＋s(Z2))/Z,＝Ｏfrom (13)ａｎｄ(14),ｄ７/d似＝Ｏat〃＝(α1/α2-1)s'(Z2).SincedZ1/叩＝(ひs'/Z1-

a2/o')/nanddz/“＝(α,β'＋o(1＋s)/Z?)/Q,thesecondderivativeof7with

respecttMaM＝(α,/α2-1)s'(Z2)is：

尚一脈 -Ｍ+仏､+ｗＭ１孟孟 β’。[α2(1＋s(Z2))＋α1s'(Z2)Zl]＿{α2(1＋s(Ｚ))＋αｒ _{ｚｆｎｄ１似}

孟[.,ざ筈十(叩，+･川筈ｌ

而ＭＷ+`(,+霞)/z:)]－余叩州Zn等

〆

Sincen＜Ｏ,ｐ'＞OjanMZ2/d/L＜0,ｔｈｅＭ２７/d似２isnegative;γislocally

maximizedintheneighborhoodof〃＝０．

Refbrences

[1IBecker,０ｓ,andOBMulligan(1997)"Theendogenousdetermination oftimeprefbrence.”QuqrterIyJ0urnqlQ/Economics,ＶＯＬ122,ｐｐ､729-758.

[2IBenhabib,Ｊ､,ａｎｄＲＦｎｒｍｅｒ（1994)“Indeterminacyandincreasingre- turns.，，JournalけＥｃｏｎｏｍｉｃTheory,ＶＯＬ63,ｐｐ､19-41.

[31Bullard,』.,andJKeating(1995)``Thelong-runrelationshipbetweenin‐

Hationandoutputinpostwareconomies.，，JOur7Dq1QfMOM川Economics，

ＶＯＬ５１，pp477-496．

[4]Chen,Ｂ－Ｌ,Ｍ､Hsu,ａｎｄＯ－ＨＬｉｎ(2008)"InHationandgrowth:impa tienceandaqualitativeequivalent.，，JOurM⑰Money,伽djt,ａｎｄＢａｎＡ－

ｍ９,ＶＯＬ40,ｐｐ、1310-1323.

2６

[5ＩＤas,Ⅲ（2003)“Optimalgrowthwithdecreasingmarginalimpatience，，

ＪｏｕｒｎｑｌＱ/Econo77zjcDZ/7w7zicsCMOon伽【,ＶＯＬ27,ｐＰ､1881-1898.

１６１DeGregorio,Ｊ(1993)"InHation,taxation,andlong-rungrowthVoumuノ ｑ/MoMqryEconomjcs,Vol､31,ｐｐ､271-298.

[7]Dutta,Ｊ､,andKapur,Ｓ・（1998)"LiquidityprefCrenceandfinancialinter- mediation.”比ujeuﾉｑ/EconomjcStMes,Ｖ01.65,ｐｐ､551-572

[8IRenstra,Ｒ､Ｃ・（1986)"FnnctionalEquivalencebetweenLiquidityCosts andtheUtilityofMonｅｙ１ＶＯＬ６５,ｐｐ、271-291.

[9]Fllkuda,Ｓ､(1996)"Moneyandeconomicgrowthwithincreasing-returns-to- scale.，，ThelnstituteofEconomicResearch，HitotsubashiUniversity・Ｄｉｓ－

cussionPaperno320．

[10]Gomme,Ｐ.（1993)"Moneyandeconomicgrowthrevised：measuringthe costsofinHationinanendogenousgrowthmodeL，，JournalけMonetary Economics,ＶＯＬ３２，pp51-77．

[11]Itaya,ＪａｎｄＫＭｉｎｏ(2003)"Inflation,transactioncostsandindetermi‐

nacyinmonetaryeconomieswithendogenousgrowth.，，ＥＣＯ〃o77Dicq,ＶＯＬ７０，

pp451-470．

[12]Jha，Ｓ、Ｋ，PingWang，ａｎｄＣｈｏｎｇＫＹｉｐ（2002）“Dynamicsina

transactions-basedmonetarygrowthmodel.”JOumqJQ/EconomicＤｙｎｑｍ－

ｉｃｓａｎｄＯｏｎｔＭ,Ｖ０１．２６，ｐｐ､616-635.

[13]Jones,Ｌ､EandManuellir,Ｒ､Ｅ(1995)``GrowthandtheefIectsofinfla- tion.，，ＪＯｕｒｎａｌｑ/EconomicＤ"ｎｑｍｊｃｓｑＭＣｍｔＭ,Ｖ01.19,ppl405-1428．

[l41Kam,Ｅ(2005)"AnoteontimepreferenceandthelbbinefIect.，，Economics LetteTs,ＶＯＬ89,ｐｐ、127-132

2７

[l5IMeng,Ｑ（2006)"Impatienceandequilibriumindeterminacy.，'ＪOur〃αｌけ ＥｃｏｎｏｍｉｃＤＺ/nqmicsqMCio7ztM,ＶＯＬ30,ｐｐ、2671-2692

[161Michel,Ｐ.(1982)"Onthetransversalityconditionininfinitehorizonopti- malproblem.”Econometrjca,Ｖ０１．５０，pp975-985．

[17]Palivos,Ｔ､,PWang,andJ､Zhang(1997)``Ontheexistenceofthebalanced

growthequilibrium.”InternationalEconomicReujeuﾉ,Ｖ01.38,ｐｐ､205-224.

[18]Sidrauski,Ｍ(1967)"InHationandeconomicgrowth,Ｔ/ＤｅＪ０勉ｒｎｑＭＰｏ－

ノiticalEcono77W,Ｖ０１．７５，ｐｐ、796-810.

[19]Ibbin,』（1965)“Moneyandeconomicgrowth，，EconoMmc(B,ＶＯＬ３３，

pp671-684．

[201Wang,Ｐ.,ａｎｄＯＫＹｉｐ（1992)“Altemativeapproachestomoneyand growth.，，ＪＤｕｒＭｑ/Money,Cmlit〔MBankin9,ＶＯＬ24,ｐｐ､553-562.

2８

１０．８

1０６

10.4

２ ０ ８

１ ０ ９

１ （率）⑫①》⑤』二］季Ｃ」四⑫。。このｍＣつこ］

9.6

9.4

9.2 －４０ -20 ０２０４０

Inflationrates（%）

6０ 8０

Figurel：ＴｈｅｒｅlationontheBGPbetweeninHationandeconomicgrowthin thecaseofa1＝１ａｎｄα２＝０

Notes:TherateofeconomicgrowthislO％whenthemoneysupplyrateis15％．

2９

OG

･％9Is1e9UMIddnsKeuomall4YｴalPYL％O1s1W9ALoｴS01uIonooaJoa肥ユａｌｌＬ:剛OＡＩ

０

Ｉ＝ＺＵｐｕＵ０＝ＩＤＪＯｅｇＵＯｅＩＩ４ ^● ｍｌｌ１仏ojd3olIuouooapIruuoI肥Uuluee川ａｑａＤＨｅＩＩリuouo1wIeエａｌｌＬ：ＺｅｴｎＨＨ

（%）99]B」uO1ﾕﾛlＪｕｌ ＯＬＯ９０９肺OＣＯＺＯＩ ００１－ＯＺ－OG－

Ｚ､６ ｒ６ 9６

、コユ。ｍの．。この、『。重【。『、庁の⑤（率） ０１ ⑤、０１ 、） ＱＵ ｎＵ アリ

ｒＯＩ ９．０１ ８０１

10.05

1０

9．９５

９ ５・８ ５ ８ ７ ９ ９ ９ ９

（率）ｍ①］ロ」二］エ。』釦⑫ゴロ色①画ＣｐＥ］

9.7

9．６５

9.6 1０２０３０

lnflationrates（%）

4０５０６０７０

－３０-20－１００

Figure3：ＴｈｅｒｅlationontheBGPbetweeninflationａｎｄｅｃｏｎｏｍｉｃｇｒｏｗｔｈｉｎ ｔｈｅｃａｓｅｏｆａ，＝１ａｎｄα２＝１

jVotes:TherateofeconomicgrowthislO％whenthemoneysupplyrateis15％．

3１

配

･％ＱＴｓ１ｅ１ＥエバIddnsKeuomaII4TI311AA％ＯＩｓＩ叩Jykoｴ目olmouooeJoa肥エａｌＩＬ:５３$oAＩ

Ｏ

Ｉ＝ｚＤＰＩｒｕＧ.Ｏ＝IDJoes1goell4 uPIIMhoj【S0IulouooaPlruuoI1jgUu1ueojYL1eqdDH3111uouoIWIe1ellェ：ｆｅｍＨ３Ｈ

●

（%）９９４B」ｕＯ１１Ｂｌ｣uｌ ＯＬＯ９０９０ｹＯＧ‐OＺＯＩ 0０１－ＯＺ－OE-

9９．６ と６ gＬ６

、。二○ｍの。ＣＥｍｍ『◎壬汁。『四片の、（率） ＲＵ Ｒ） 負ｕ虞ＵｎＵ 点Ｕ ｎＵ ｎ。Ｒｕ かゴ

0１ 9０．０１ 101

1０１５

１ ５

０ ０

１ ０

１ （率）い①］⑩』二］エロ』四⑭ゴロ臣②凶。ごこ］

1０ -15 -10 －５

Inflationrates（%）

0 ５

Figure5：ＴｈｅｃｏｍｐａｒｉｓｏｎｏｎｔｈｅＢＧＰｂｅｔｗｅｅｎｔｈecaseswithp，＝Ｏ１ａｎｄ ｐｏ＝０８

Notes:TherateofeconomicgrowthislO％whenthemoneysupplyrateis１５％．

3３