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ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015
Money,InterestandPrices
MoneyinGeneralEquilibriumModels,thePriceLevel,InterestRates
andInflation
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 2
Money,thePriceLevelandNominalVariables
• Money,theexistenceofwhichinamoderneconomyisusuallytakenforgranted,performsthreemainfuncJons.
• First,itisaunitofaccount,seconditisauniversallyacceptedmeansofpayment,andthirdly,itisastoreofvalue.
• WhileinmodelswithoutmoneyonecanonlyanalyzethedeterminaJonofrealvariables,suchasthequanJJesofgoodsandservicesproducedandconsumed,andtheirrela9veprices,inmodelswithmoneyonecanalsodeterminenominalvariablessuchasthepricelevel,nominalincome,thelevelofnominalwages,nominalinterestratesandinfla9on.Thesenominalvariablesareexpressedintermsofmoney.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 3
MoneyasaUnitofAccount• Inamonetaryeconomyallpricesaredeterminedandquotedintermsofthe
monetaryunit.Otherwise,economicagentswouldhavetocalculatealltherelaJvepricesofgoodsandservicesinordertoconducttheirtransacJons.
• InaneconomywithNgoodsplusmoney,thereareNmoneyprices.Withoutmoney,economicagentswouldneedtocalculateN(N+1)/2relaJvepricesinordertomaketheirtransacJons.Asthenumberofgoodsandservicesincreases,thenumberofrelaJvepricestobecalculatedgrowsexponenJally.Forexample,ifthereare5goodsandservices,therearefivemoneyprices,and15relaJvepricesofgoodsbetweenthem.With1000goodsandservices,thereare1000moneyprices,and500,500relaJvepricesbetweengoodsandprices.
• MoneythereforehelpstosimplifythecalculaJonofpricesandvalues,andthusfacilitateseconomictransacJonsthroughitsunitofaccountfuncJon.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 4
MoneyasaMeansofPayment• Beingacceptedbyall,moneygreatlyfacilitateseconomictransacJons
anddrasJcallyreducestheircosts.
• Withoutmoney,inordertocompleteatransacJonthesellerofaproductorservicewouldhavetofindabuyerwhowouldbepreparedtoofferinreturnanothergoodorservicethatthesellerwishestoacquire.ThisrequiresthatthereisadoublecoincidenceofwantsinalleconomictransacJons.TransacJonsorthiskindarecalledbarter,whichimplieshugecostsonthepartofeconomicagentsinordertofindsuitablecounter-parJestotheirtransacJons.
• AmoderneconomywouldimmediatelyceasefuncJoningiftherewasnotagenerallyacceptedmediumofexchangeandpayments,becausetransacJoncostswouldbecomeprohibiJve.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 5
MoneyasaStoreofValue• Moneyisastoreofvalue,i.e.ameansofholdingwealth,andis
indeedtheassetthatischaracterizedbygreaterliquidity,asitcanbeuseddirectlyforpaymentsfortheacquisiJonofgoodsandservices.
• Thisisakeyfeatureofmoney,becauseifmoneywerenotastoreofvalue,andlostitsvaluequickly,itwouldnotbegenerallyacceptedasameansofpaymentseither.Thenagain,sincemoneyistheonlystoreofvaluewhichisalsoameansofpayments,bydefiniJonitisthemostliquidstoreofvalue.
• However,asameansofholdingwealth,moneyhastheweaknessthatitdoesnotpayinterest,unlikeotherlessliquidassets.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 6
DefiningtheMoneySupply• Wedefineasmoneythesumofbanknotes,coinsanddepositsin
currentaccountsincommercialbanksheldbyhouseholdsandfirms.
• ThisdefiniJonofmoneysupplyisusuallyknownasM1.Itemphasizesthemoreliquidassetsofhouseholdsandfirms,whichusuallydonotyieldinterest.However,therearebroaderdefiniJonsofthemoneysupply,thatincludelessliquidassetssuchasJmedepositsandotherlessliquiddepositsandsecuriJes(M2,M3etc).
• DepositsofcreditinsJtuJonsandotherinsJtuJonsparJcipaJngintheinterbankmarketandtheforeignexchangemarketarenotconsideredaspartofthemoneysupply.ThesedepositsarenotusedforthetransacJonsofthegeneralpublic.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 7
TheSupplyofMoney• MonetarycondiJonsinmoderneconomiesaredeterminedbycentral
banks.
• Thecentralbankmayaffect,throughavarietyofpolicyinstrumentsatitsdisposal,boththequanJtyofbanknotes(andcoins)incirculaJon,and,indirectly,theamountofdepositsincommercialbanks,whicharealsopartofthemoneysupply.
• AlternaJvely,acentralbankmayfollowaninterestraterule,interveninginthemoneymarketandpeggingnominalinterestrates.Inthisla[ercase,thestockofmoneyintheeconomyisdeterminedbythedemandformoney,andthemoneysupplyadaptstodemandinordertoachievethegoalofthecentralbankregardingthenominalinterestrate.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 8
TheRolesofCentralBanks• CentralbanksarepublicinsJtuJonsthatmanageastate’smoney
supply,interestratesandregulatethecommercialbankingsystem.Inmostcountriesthecentralbankpossessesamonopolyonprin9ngnotes,andmin9ngcoins,whichserveasthestate’slegaltender.InaddiJon,centralbanksusuallyactaslendersoflastresorttothebankingsystemand,inmanycases,thegovernment.CentralbankscanthusdirectlydeterminethecirculaJonofnotes(andcoins)andindirectlytheamountofdepositsincommercialbanks.
• ThedeterminaJonofthemoneysupplybycentralbanksisnotasimpleprocess.ItdependsontherulesunderwhichthecentralbankparJcipatesinmoneyandassetmarketsandregulatesthefinancialsystem,onitsrelaJonswiththegovernment,andonthegoalsenvisagedinitscharter.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 9
TheGoalsandPolicyInstrumentsofCentralBanks
• Themaingoalsofacentralbankarethecontrolofinfla9on,thestabilityofthefinancialsystem,andinsomecases,thesupportofthegeneraleconomicpoliciesofthegovernment.
• InwhatfollowsweshallignoremanyoftheinsJtuJonaldetailsthatrelatetohowacentralbankoperates,andwillmaketwoalternaJvesimpleassumpJons.
• First,weshallassumethatthecentralbankhasfullcontrolofthemoneysupply.ThisisanassumpJonwithalonghistoryinmacroeconomicanalysis,althoughnotparJcularlyrealisJc,ascentralbankshaveimperfectcontroloverthemoneysupply.
• AlternaJvelyweshallassumethatthecentralbankfollowsapolicyofpeggingthenominalinterestrateandcommi^ngtoprovidingunlimitedcredittohouseholds,businessesandcommercialbanksatthisrate.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 10
TheDemandforMoney• Thedemandformoneydependsonthreemainfactors.
• Thefirstfactoristhepricelevel.Thehigherthelevelofprices,thehigherwillbetheamountofmoneythathouseholdsandfirmswouldwanttoholdfortheircurrentandfuturetransacJons.ThedemandformoneyisusuallyassumedtobeproporJonaltothepricelevel.
• ThesecondfactoristhevolumeoftransacJons.WhenthevolumeoftransacJons,usuallymeasuredbyaggregaterealoutput,increases,householdsandfirmswillneedmoremoneytocarryouttheirincreasedtransacJons.
• Thethirdfactoristhelevelofinterestrates.Banknotespaynointerest.Ontheotherhand,demanddepositsandcurrentaccounts,evenwhentheypayinterest,payaverylowratecomparedtotheyieldsoflessliquidassetssuchasJmedeposits,treasurybillsorbonds.Consequently,thedemandformoneywilldependnegaJvelyonthenominalinterestrate,asthenominalinterestratemeasurestheopportunitycostofholdingmoney.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 11
TheMoneyDemandFuncJon
M d = P ×m(Y ,i)
whereMdisthequanJtyofmoneydemanded,Pisthepricelevel,Yreal
aggregateoutputandincome(GDP)andithenominalinterestrate.misafuncJonincreasinginrealaggregateincomeanddecreasinginthenominalinterestrate.
ThedemandformoneyisproporJonaltothepricelevel,inthesensethatanincreaseinthepricelevelrequiresanincreaseinthequanJtyofmoneybythesameproporJon,inordertocompletethesamenumberoftransacJons.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 12
TheDemandforRealMoneyBalances
M d
P= m(Y ,i)
SincethedemandformoneyisproporJonaltothepricelevel,itcanbeexpressedasademandforrealmoneybalances.Holdingmoneyisusefulforitspurchasingpower.
TherelaJonshipbetweenrealmoneydemandandthenominalinterestrateisnegaJve,becauseholdingmoneybecomesmoreexpensiveasinterestratesrise,sincemoneydoesnotpayinterest.Therefore,householdsandfirmsreducetheamountofmoneyholdingsandincreaseholdingsofsecuriJesandotherinterestyieldingassets.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 13
TheDemandforRealMoneyBalances,theNominalInterestRateandRealIncome
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 14
ShortRunEquilibriumintheMoneyMarket
M s
P= M d
P= m(Y ,i)
TheequilibriumcondiJoninthemoneymarketisforthemoneysupplytobeequaltomoneydemandbyhouseholdsandfirms.
Intheshortrun,withpricesandincomegiven,howthemoneymarketequilibratesdependsonthemodusoperandiofthecentralbank.
Ifthecentralbankfixesthemoneysupply,theequilibraJngmechanismisthenominalinterestrate.Ifthecentralbankfixesthenominalinterestrate,theequilibraJngmechanismisthemoneysupply.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 15
ShortRunEquilibriumintheMoneyMarketwhentheCentralBankFixestheMoneySupply
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 16
AnIncreaseintheMoneySupplyandNominalInterestRates:TheLiquidityEffect
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 17
AnIncreaseinMoneyDemandandNominalInterestRates:TheLiquidityEffect
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 18
ShortRunEquilibriumintheMoneyMarketwhentheCentralBankFixestheNominalInterest
Rate
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 19
LongRunEffectsoftheMoneySupply
P = M s
m(Y ,i)
• Inthelongerterm,thepricelevelalsoadjusts.WecanseethedirecJonofthisadjustmentbyrearrangingtheequilibriumcondiJoninthemoneymarket,andsolvingforthepricelevel.Wethenget,
• Inlongrunequilibrium,aggregaterealincomeandtherealinterestrateareontheirbalancedgrowthpaths.WithconstantinflaJon,nominalinterestratesarealsoconstant.Thus,thefactorsaffecJngthedemandformoneyaregiven,andthemoneysupplydeterminesthepricelevel,withoutaffecJngtheevoluJonofrealvariables.Thispropertyiscalledlong-runneutralityofmoney.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 20
TheNeutralityofMoneyinStaJcGeneralEquilibriumModels
• TheneutralityofmoneyappliestoallstaJcgeneraleconomicequilibriummodelswithflexibleprices.
• Whatdeterminesthelevelofequilibriumrealincome,andotherrealvariablesaretheavailableresources,technology,preferences,thefuncJoningofmarkets,aswellaseconomicinsJtuJonsthatdeterminetotalfactorproducJvityandtheproducJvityofspecificfactors.
• InstaJcgeneralequilibriummodelsrealoutputandincomeandotherrealvariablesdonotdependonthemoneysupply.Moneyismerelya“veil”whichcoverstheeconomy,simplydeterminingnominalvariablessuchasthepricelevel.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 21
TheNeutralityandSuper-neutralityofMoneyinDynamicGeneralEquilibriumModels
• IndynamicgeneralequilibriummodelsweusuallydisJnguishbetweenthe“neutrality”andthe“super-neutrality”ofmoney.
• The“neutrality”ofmoneyreferstotheeffectsofaoneoffchangeinthemoneysupply,andthe“super-neutrality”ofmoneytotheeffectsoftherateofchangeofthemoneysupply.
• Theneutralityofmoneyappliestoalldynamicgeneraleconomicequilibriummodelswithflexibleprices.
• However,asthegrowthrateofmoneysupplyaffectsinflaJonandlong-termnominalinterestrates,itthusaffectsrealmoneydemand.
• The“super-neutrality”ofmoneyholdsinrepresentaJvehouseholdmodels,asmoneydoesnotaffectotherrealvariables,butdoesnotholdinoverlappinggeneraJonsmodels.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 22
LongRunNeutralityofMoneyandMonetaryReforms
• AnalternaJvewaytothinkabouttheneutralityofmoney,istoconsiderwhatwouldbetheimpactofaveryradicalchangeinthemoneysupply.SuchradicalchangestakeplaceinJmesofmonetaryreforms.Anumberofsuchhistoricalexamplesexist,whichsuggestthat,aderamonetaryreform,thepriceleveladjustsimmediatelytothenewmonetarystandard.
• Forexample,inMay1954,therewasaradicalmonetaryreforminGreece.Anewdrachmawascreated,whichamountedto1,000olddrachmas.EssenJallythisamountedtoadirectreducJoninthemoneysupplytoonethousandthoftheoldmoneysupply.AsonewouldexpectonthebasisofourmoneydemandfuncJon,thepricelevelinGreecefellimmediatelytoonethousandthofthepricelevelbeforethereform.Nothingelsechanged,otherthanthelevelofprices.
• Gradualincreasesinthemoneysupplyinthelongrunhaveeffectssimilartosuchmonetaryreforms.Thetriplingofthemoneysupplyinadecade,inthelongrunhasthesameeffectasamonetaryreforminwhichacurrencyunitisreplacedwiththreeunitsofa“new”currency.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 23
DynamicGeneralEquilibriumModelswithMoney
• TheSamuelson(1958)overlappinggeneraJonsmodel.
• RepresentaJveHouseholdModelswithMoneyintheUJlityFuncJon(PaJnkin1956,Sidrauski1967)
• OverlappingGeneraJonsModelswithMoneyintheUJlityFuncJon(Weil1987).
• Cash-in-AdvanceRepresentaJveHouseholdModels(Clower1967,GrandmontandYounes1972,Lucas1980,1982,Svensson1985).
• Cash-in-AdvanceModelsofOverlappingGeneraJons(Samuelson-Lucas).
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 24
TheSamuelsonOverlappingGeneraJonsModel:MoneyasaStoreofValue
• WeassumethattheeconomyconsistsofsuccessivegeneraJonsofhouseholds,eachofwhichlivesfortwoperiods.
• Everyhouseholdhasexogenousincomey1inthefirstperiodoflifeandy2inthesecondperiodoflife.
• Thisincomeisintheformofanonstorablegood,whichcannotbetransferredfromperiodtoperiod.
• Theonlynon-perishablecommodityismoney,whichcanbeusedasameansofholdingwealth.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 25
TheInter-temporalOpJmizaJonProblemofHouseholds
ΤhehouseholdborninperiodtmaximizestheuJlityfuncJon,
Ut = u(C1t )+ βu(C2t+1) = lnC1t + β lnC2t+1
PtC1t +Mt = PtY1
Pt+1C2t+1 = Mt + Pt+1Y2
undertheconstraints,
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 26
DefiniJonsofVariablesandParameters
C1ishouseholdconsumpJoninthefirstperiodoflifeandC2consumpJoninthesecondperiodoflife.
uisaconcaveuJlityfuncJonandβ=1/(1+ρ)thediscountfactor,whereρisthepurerateofJmepreference.
Mtisthemoneysupply,carriedoverbythehouseholdfromthefirsttoitssecondperiodoflife.Themoneysupplyisequaltothesavingsofhouseholdsintheirfirstperiodoflife.
PtisthemoneypriceoftheconsumpJongoodinperiodtandPt+1themoneypriceoftheconsumpJongoodinperiodt+1.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 27
ConsumpJonofYoungandOldHouseholdsinPeriodt
PtC1t =1
1+ βPtY1 + Pt+1Y2( )
PtC2t = M + PtY2
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 28
GoodsMarketEquilibriumandMoneyDemand
C1t +C2t = Y1 +Y2
MPt
= 11+ β
βY1 −Pt+1PtY2
⎛⎝⎜
⎞⎠⎟
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 29
EquilibriumPriceLevel
MP*
= 11+ β
βY1 −Y2( )
P*> 0⇔ βY1Y2
>1
Thedemandformoney,andhencethepricelevel,willbeposiJveonlyifthediscountedfirstperiodincomeofhouseholdsexceedssecondperiodincome.Itisonlythenthatsavings,andhencemoneydemand,willbeposiJve.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 30
PriceAdjustmentEquaJon
Pt+1 − P*=βY1Y2(Pt − P*)
• Sincethepricelevelisanonpredeterminedvariable,thecondiJonforthestabilityofthedynamicadjustmenttotheequilibriumpricelevelP*isthattherootofthedifferenceequaJonaboveisgreaterthanone.Consequently,thecondiJonfortheexistenceofaposiJveequilibriumpricelevelcoincideswiththecondiJonforthestabilityoftheequilibrium.IfβY1/Y2>1,thenaposiJveequilibriumpricelevelexists,andinaddiJontheequilibriumisasaddlepoint,i.e.dynamicallystable.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 31
ImplicaJonsoftheSamuelsonModel
• TheSamuelsonmodelhasastrikingimplicaJon:Moneyimproveswelfare,becauseitallowshouseholdstoengageininter-temporaltradeandsmoothconsumpJonoverJme.
• Intheabsenceofmoney,consumpJonineachperiodwouldhavetobeequaltocurrentincomeforallgeneraJons.ThisequilibriumisclearlysubopJmalcomparedwiththeequilibriumofamonetaryeconomywhichallowsforconsumpJonsmoothing.
• TheSamuelsonmodelofoverlappinggeneraJonsisoneofthefirstdynamicgeneralequilibriummodelsthatgenerateaposiJvedemandformoneyasastoreofvalue.TheneutralityofmoneyfollowsimmediatelyfromthemoneydemandfuncJon.Moreover,inthismodel,sincethepricelevelisanonpredeterminedvariable,theincreaseinthepricelevelwouldhappenimmediately.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 32
WeaknessesoftheSamuelsonModel
• Thefirstweaknessisthattheequilibriumwehavejustdescribed,whichentailsaposiJvedemandformoney,isnotunique.Thereisasecond,subopJmal,equilibrium,withzeromoneydemand.Thusthedemandformoneyinthismodelisextremelyfragile.
• AsecondweaknessofthismodelisthatthereisnoalternaJvestoreofvalue.Theonlywaytosaveinthismodelisbyholdingmoney.However,ifthereisanalternaJveassetwhichpaysinterest,forexamplebondsorcapital,thenmoneywouldbeostracizedfromthiseconomy,becauseitsonlyroleisasastoreofvalue,andmoneydoesnotpayinterest.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 33
MulJplicityofEquilibriumintheSamuelsonModel
Twolong-runequilibria:
MPt+1
= MPt
Y2βY1 − (1+ β ) M / Pt( )
MP*
= 11+ β
βY1 −Y2( ) andMP**
= 0
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 34
MulJplicityofEquilibriumintheSamuelsonModel
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 35
UniqueEquilibriumiftheOldHavenoIncome
IfY2=0thenthereisauniqueequilibrium,
MP*
= β1+ β
Y1
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 36
MoneyintheUJlityFuncJonofaRepresentaJveHousehold
ThereisarepresentaJvehouseholdmaximizinganinter-temporaluJlityfuncJonoftheform,
Ut = β s−tu(Cs ,Ms
Ps)
s=t
∞∑
undertheconstraint,
Cs +Ms
Ps+ BsPs
= Ys −Ts +Ms−1
Ps+ (1+ is−1)Bs−1
Ps
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 37
LagrangeFuncJonandFirstOrderCondiJon
Et β s−t u(Cs ,Ms
Ps)+ λs
Ms−1
Ps+ (1+ is−1)Bs−1
Ps+Ys −Ts −Cs −
Ms
Ps− BsPs
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟s=t
∞∑
λt =∂u∂Ct
λtPt
= β(1+ it )Etλt+1Pt+1
⎛⎝⎜
⎞⎠⎟
λtPt
= 1Pt
∂u∂Mt
+ βEtλt+1Pt+1
⎛⎝⎜
⎞⎠⎟
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 38
InterpretaJonofFirstOrderCondiJons
• First,isthestaJcfirstordercondiJon,accordingtowhich,themarginaluJlityofconsumpJonshouldinanyperiodisequaltothe“shadowvalue”ofmarginalsavings.EssenJally,thehouseholdshouldbeindifferentatthemarginbetweenconsumpJonandsavings.
• SecondisthedynamicfirstordercondiJon,accordingtowhichthetotalexpectedrealreturnonsavingsshouldbeequaltothepurerateofJmepreferenceofthehousehold.
• ThirdisthedynamicfirstordercondiJonaccordingtowhichthemarginaluJlityofrealmoneybalancesisequaltothedifferenceofthepurerateofJmepreferencefromtheexpectedrealreturnofmoney,takingintoaccountexpectedinflaJonandexpectedcapitalgainsfromachangeinλ.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 39
TheUJlityFuncJonandtheMoneyDemandFuncJon
u = ln γ Ct( )ε + (1− γ ) Mt
Pt
⎛⎝⎜
⎞⎠⎟
ε⎛
⎝⎜
⎞
⎠⎟
1ε
Mt
Pt= γ1− γ
it1+ it
⎛⎝⎜
⎞⎠⎟
− 11−εCt =
γ1− γ
it1+ it
⎛⎝⎜
⎞⎠⎟
− 11−εYt
AssumingthattheperiodicuJlityfuncJonutakestheform,
ImposingthegoodsmarketequilibriumcondiJonCt=Yt,thefirstordercondiJonsimplythat,
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 40
TheCash-in-AdvanceConstraintandMoneyDemand
• Thebasicideaofmodelsinwhichmoneyistheonlymeansofpayment,isthatinordertocompleteanyeconomictransacJon,paymentmustbeinmoney,andinparJcularcash,whichthebuyerholdsinadvanceofthecompleJonofthetransacJon.
• ThisideaisduetoClower(1967),anditsintegraJonintogeneralequilibriummodelsleadstoaclassofmodelsknownascash-in-advancemodels.
• TherestricJonthatthetransacJonmustbepaidwithmoneyheldinadvance,imposesacostofholdingmoney,because,alternaJvely,economicagentscouldholdanotherasset,suchasbonds,whichpaysinterest.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 41
AlternaJveCash-in-AdvanceModels
• ThecashinadvancerestricJoncantakeseveralforms,dependingontheassumpJonsmadeaboutthesequencingoftransacJons.AsimpletradiJonalwayofexpressingthisconstraintistoassumethatspendingcannotexceedthemoneybalancescarriedoverfromtheendofthepreviousperiod(Svensson1985).
• AnalternaJvehypothesisisthateachperiodconsistsoftwodifferentsub-periods.Inthefirstsub-periodagentsvisitafinancialmarket,sayabank,wheretheycanswapinterestbearingassetswithmoney,orborrowcash,andinthesecondsub-periodtheydealinmarketsforgoodsandservices,whichareliabletothecash-in-advanceconstraint(seeHelpman1981,Lucas1980,1982).
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 42
TheCash-in-AdvanceConstraintwhenHouseholdsCanVisitFinancialMarketspriorto
MakingPurchases
At = Mt + Bt
PtCt ≤ Mt
At+1 = Mt + (1+ it )Bt + Pt (Yt −Tt −Ct ) = (1+ it )At − itMt + Pt (Yt −Tt −Ct )
Inthefirstsub-periodagentsvisitafinancialmarket,sayabank,wheretheycanswapinterestbearingassetswithmoney,orborrowcash.Inthesecondsub-periodtheydealinmarketsforgoodsandservices,whichareliabletothecash-in-advanceconstraint.Inthesecondsub-period,householdsalsoreceivetheirexogenousrealincomeYandpaytheirtaxes(netoftransfers)Τ.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 43
TheLagrangianoftheRepresentaJveHouseholdandtheFirstOrderCondiJons
Et β s−t u(Cs )+ν tMt
Pt−Ct
⎛⎝⎜
⎞⎠⎟+ λt (1+ it )
AtPt
+Yt −Tt −Ct − itMt
Pt− At+1Pt
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎞
⎠⎟s=t
∞∑
λt +ν t =∂u∂Ct
= ′u (Ct ) ν t = λtitλtPt
= βEt (1+ it+1)λt+1Pt+1
⎛⎝⎜
⎞⎠⎟
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 44
InterpreJngtheFirstOrderCondiJonsoftheRepresentaJveHouseholdwithaCash-in-
AdvanceConstraint• FirstisthestaJcfirstordercondiJonaccordingtowhichtheopJmal
consumpJonequatesthemarginaluJlityofconsumpJonwiththe“shadowvalue”ofsavingsλ,plustheshadowvalueofmoneyν.TheshadowvalueofmoneyresultsfromtherestricJonforcashinadvanceinordertobuyconsumergoods.
• Secondisthedynamicfirst-ordercondiJon,accordingtowhichthetotalexpectedrealreturnonsavings,includingexpectedinflaJonandexpectedcapitalgains,shouldbeequaltothepurerateofJmepreferenceofthehousehold.
• ThirdisthestaJcfirstordercondiJonaccordingtowhich,theshadowvalueofmoneyshouldbeequaltotheshadowvalueofsavingsJmestheopportunitycostofholdingmoney,whichisnoneotherthanthenominalrate,sincemoneypaysnointerest.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 45
TheEulerEquaJonandtheMoneyDemandFuncJonintheCash-in-AdvanceRepresentaJve
HouseholdModel
′u (Ct )Pt
= β(1+ it )Et′u (Ct+1)Pt+1
⎛⎝⎜
⎞⎠⎟
FromthefirstordercondiJonwecanderivetheEulerequaJonforconsumpJoninamonetaryeconomy.
Mt
Pt= Ct
ThemoneydemandfuncJonisderivedfromthecash-in-advanceconstraint.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 46
TheEulerEquaJonforConsumpJonandtheNominalInterestRatewithLogarithmic
Preferences
1PtCt
= β(1+ it )Et1
Pt+1Ct+1
⎛⎝⎜
⎞⎠⎟
Assuminglogarithmicpreferences,theEulerequaJonforconsumpJoncanbewri[enas,
11+ it
= βEtMt
Mt+1
⎛⎝⎜
⎞⎠⎟
FromtheEulerequaJonaboveandthemoneydemandfuncJonitfollowsthatthenominalinterestrateisdeterminedby
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 47
Cash-in-AdvanceinanOverlappingGeneraJonsModel
WefinallyexaminetheimplicaJonsformoneydemandofacashinadvanceconstraintinavariantoftheSamuelsonoverlappinggeneraJonsmodels.Inthismodel,moneyfuncJonsbothasameansofpaymentsandastoreofvalue.
Thehouseholdborninthebeginningofperiodtlivesfortwoperiods,periodtandperiodt+1.ItreceivesincomeYtandpaystaxesTt,inthefirstperiodoflife,andconsumesinbothperiods.
Itmaximizesaninter-temporaluJlityfuncJonundertheconstraintsthatthepresentvalueofconsumpJonmustbeequaltothepresentvalueofincomenetoftaxes,andthecash-in-advanceconstraintsforeachperiod.
Ut = lnC1t + β lnC2t+1
PtC1t ≤ M1t Pt+1C2t+1 ≤ M 2t+1PtCt +11+ it
Pt+1C2t+1 = Pt (Yt −Tt )
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 48
Cash-in-AdvanceinanOverlappingGeneraJonsModel
AggregateconsumpJonandaggregatemoneybalancesineachperiodaregivenby,
Ct = C1t +C2t Mt = M1t +M 2t
TotalassetsofhouseholdsareequaltoA,andweassumethatyounghouseholdsarebornwithoutassets.Asaresult,allassetsbelongtotheoldhouseholds.ForsimplicityweassumethattaxesTareonlypaidbyyounghouseholds.
Giventhatoldhouseholdsreceivenocurrentincome,theirconsumpJonisequaltotheirassets,which,however,theyneedtoconvertintomoney,inordertopurchaseconsumergoods.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 49
Cash-in-AdvanceinanOverlappingGeneraJonsModel
ConsumpJonofoldhouseholdsisthusgivenby,
C2t =AtPt
Giventhatyounghouseholdsholdnoassets,theyneedtoborrowandconverttheirloanintomoney,inordertofinancetheirconsumpJon.Asaresult,foryounghouseholdsthefollowingconstraintsmusthold,
M1t = PtC1t = −B1t
At+1 = M1t + (1+ it )B1t + Pt Yt −Tt −C1t( ) = Pt Yt −Tt − (1+ it )C1t( )
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 50
Cash-in-AdvanceinanOverlappingGeneraJonsModel
IntroducingassetsinplaceofsecondperiodconsumpJonintheuJlityfuncJon,wefindthatyounghouseholdswillchooseconsumpJonintheirfirstperiodoflifeinordertomaximize,
Ut = lnC1t + β lnPt Yt −Tt − (1+ it )C1t( )
Pt+1
⎛⎝⎜
⎞⎠⎟
C1t =1
1+ βYt −Tt1+ it
FromthefirstordercondiJonsitfollowsthat,
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 51
Cash-in-AdvanceinanOverlappingGeneraJonsModel
AggregateconsumpJoninperiodtisgivenby,
Ct = C1t +C2t =1
1+ βYt −Tt1+ it
+ AtPt
Yt = Ct =1
1+ βYt −Tt1+ it
+ AtPt
Fromequilibriuminthemarketforgoodsandservicesandmoneyitfollowsthat,
Thesecanbesolvedforthepricelevelandthenominalinterestrate.Theneutralityofmoneyholdssincerealincomeisexogenous.
Mt = PtYt
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 52
NominalandRealInterestRates:MoneyintheUJlityFuncJon
Withlogarithmicpreferences,thefirstordercondiJonsforthemaximizaJonoftheuJlityfuncJonoftherepresentaJvehouseholdaregivenby,
λt =γCt
FromthesecondiJons,itfollowsthat,
λtPt
= β(1+ it )Etλt+1Pt+1
⎛⎝⎜
⎞⎠⎟
λtPt
= 1− γMt
+ βEtλt+1Pt+1
⎛⎝⎜
⎞⎠⎟
1PtCt
= β(1+ it )Et1
Pt+1Ct+1
⎛⎝⎜
⎞⎠⎟
1PtCt
= 1− γγ
1Mt
+ βEt1
Pt+1Ct+1
⎛⎝⎜
⎞⎠⎟
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 53
NominalandRealInterestRates:MoneyintheUJlityFuncJon
ThelastfirstordercondiJoncanbesolvedas.
1PtCt
= 1− γγ
β s−tEt1Ms
⎛⎝⎜
⎞⎠⎟s=t
∞∑
GiventhatCt=Ytwhichisexogenous,thisequaJondeterminestheequilibriumpricelevel,asafuncJonofexpectaJonsaboutthefutureevoluJonofthemoneysupply.SubsJtuJnginthemoneydemandequaJonforε=0,andsolvingforthenominalinterestrate,
1+ itit
= γ1− γ
Mt
PtCt
= β s−tEtMt
Ms
⎛⎝⎜
⎞⎠⎟s=t
∞∑
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 54
NominalandRealInterestRates:MoneyintheUJlityFuncJon
ThenominalinterestratesisdeterminedbythecurrentmoneysupplyandexpectaJonsaboutthefuturedevelopmentofthemoneysupply,atarateofdiscountthatdependsonthepurerateofJmepreferenceofthehousehold.Supposetheexpectedgrowthrateofthemoneysupplyisconstantandequaltoμ.
1+ itit
= β s−t 11+ µ
⎛⎝⎜
⎞⎠⎟s=t
∞∑s−t
= 11+ ρ( ) 1+ µ( )
⎛⎝⎜
⎞⎠⎟s=t
∞∑s−t
= (1+ ρ)(1+ µ)(1+ ρ)(1+ µ)−1
Itfollowsthat,
it = (1+ ρ)(1+ µ)−1! ρ + µ
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 55
NominalandRealInterestRates:MoneyintheUJlityFuncJon
Thehigherthegrowthrateofmoneysupplyμ,thehigherwillbethenominalinterestratei,astheexpectedfutureinflaJonratewillbehigher.
ItisworthnoJngthattherealequilibriuminterestrateinthemodelisequaltoρ.Forμ=0,i=ρ.Inthiscase,becausetheexpectedfutureinflaJonrateisequaltozero,thenominalinterestrateequalstheequilibriumrealinterestrate,i.e.thepurerateofJmepreferenceoftherepresentaJvehousehold.
ItisworthnoJngthatifμ=-ρ/(1+ρ),i.e.ifthemoneysupplyisreducedatthisrate,thenominalinterestrateisdriventozero.Azeronominalinterestratehasa[racJveproperJesandleadstotheop9malmoneydemandbyhouseholds.
it = (1+ ρ)(1+ µ)−1! ρ + µ
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 56
NominalandRealInterestRates:Cash-in-Advance
IntherepresentaJvehouseholdmodelinwhichmoneydemandresultsfromthecash-in-advanceconstraint,undertheassumpJonoflogarithmicpreferences,thenominalinterestrateisdeterminedby,
it = (1+ ρ)(1+ µ)−1! ρ + µ
11+ it
= βEtMt
Mt+1
⎛⎝⎜
⎞⎠⎟= β 1
1+ µ⎛⎝⎜
⎞⎠⎟= 1(1+ ρ)(1+ µ)
FromthiscondiJon,itfollowsthat,
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 57
TheIndeterminacyofthePriceLevelwhentheCentralBankpegstheNominalInterestRate:
MoneyintheUJlityFuncJon
FromtheequilibriumcondiJonsinthegoodsandmoneymarkets,ifthecentralbankpegsthenominalinterestrateattheleveli0,thentherepresentaJvehouseholdmodelwithmoneyintheuJlityfuncJon,impliesthat,
Mt
Pt= 1− γ
γ1+ i0i0
Yt
Giventhatrealincomeisexogenous,thiscondiJonissaJsfiedforaninfinitenumberofcombinaJonsofΜandP.IfitissaJsfiedforM0andP0,itisalsosaJsfiedforλM0andλP0,foranyλ.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 58
TheIndeterminacyofthePriceLevelwhentheCentralBankpegstheNominalInterestRate:
Cash-in-Advance
FromtheEulerequaJonforconsumpJonandtheequilibriumcondiJoninthegoodsmarket,itfollowsthat,
Pt+1Yt+1 = β(1+ i0 )PtYt
Giventhatrealincomeisexogenous,thiscondiJonissaJsfiedforaninfinitenumberofcombinaJonsofPtκαιPt+1.IfitsaJsfiedforP0andP1,thenitisalsosaJsfiedforλP0andλP1,foranyλ.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 59
InterestRatePeggingandPriceLevelIndeterminacy
ThisindeterminacywasfirstdemonstratedbySargentandWallace(1975)inthecontextofanadhocmacromodelwithraJonalexpectaJons.ThisindeterminacydoesnotonlyariseinrepresentaJvehouseholdmonetarymodels,butinmoneymodelswithraJonalexpectaJons.
Thereasonfortheindeterminacyisthat,underinterestratepegging,thereisnomonetaryanchorwhichcandeterminethepricelevel,asinthecasewherethecentralbankdeterminesthemoneysupply.
Sincethecentralbankiscommi[edtoprovidingunlimitedcreditatanominalinterestratei0,thenthemoneysupplyisdeterminedbythedemandformoney.Neitherthepricelevel,northemoneysupplycanbeidenJfieduniquely.
TheequilibriumcondiJonsforprivateconsumpJonandmoneydemandcanbesaJsfiedwithbothhighpricesandaconsequenthighstockofmoney,andwithlowpricesandaconsequentlowstockofmoney,i.e.virtuallyforanylevelofprices.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 60
DealingwithPriceLevelIndeterminacyunderInterestRateRules:TheWicksellandTaylorRules
Oneofthefirstanswerstothisproblemwasprovidedbythemonetaryeconomistwhofirstrealizeditsexistence,namelyWicksell(1898).Wicksellproposedthat,“Solongaspricesremainunaltered,thebanks’rateofinterestistoremainunaltered.Ifpricesrise,therateofinterestistoberaised;andifpricesfall,therateofinterestistobelowered;andtherateofinterestishenceforthtobemaintainedatitsnewlevelunJlafurthermovementofpricescallsforafurtherchangeinonedirecJonortheother.”(p.189).Thus,Wicksellproposedthatcentralbanksshouldhaveapriceleveltarget,andchangeinterestrateswhenthepriceleveldeviatesfromthistarget.
AlternaJvewaystosolvetheproblemofpricelevelindeterminacywhenthepolicyinstrumentofthecentralbankisthenominalinterestrate,havebeenproposed:InflaJontargeJngrules,nominalincomerules,andmorerecentlytheTaylor(1993),whichisageneralizaJonofWicksell’srule.WeshallexaminetheproperJesofsuchrulesinthelecturesonaggregatefluctuaJonsandmonetarypolicy.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 61
TheFiscalTheoryofthePriceLevel
OnetheoreJcaldevelopmentworthmenJoningistheso-calledfiscaltheoryofthepricelevel(see.Leeper1991,Sims1994andWoodford1994,1995).
Thistheoryarguesthatevenifmonetarypolicyisnotsufficienttodeterminethepricelevel,asunderinterestratepegging,thepricelevelcanbedeterminedatthelevelwhichensuresthatpublicdebt,whichisdefinedinnominalterms,doesnotfollowanexplosivepath.
ApathforthepricelevelthatensuresapathofnominalpublicdebtthatsaJsfiestheinter-temporalbudgetconstraintofthegovernmentissufficientinthosemodelstodeterminethepricelevel.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 62
PriceLevelDeterminacyinOverlappingGeneraJonsModelsandthePigouEffect
ItisalsoworthstressingthattheproblemofpricelevelindeterminacydoesnotariseinoverlappinggeneraJonsmodels.UnliketherepresentaJvehouseholdmodel,whereboththecurrentandthefuturepricelevelarenonpredeterminedvariables,intheoverlappinggeneraJonsmodel,thepricelevelisdeterminedthroughthepredeterminednominalfinancialassetsof“old”households.ThesefuncJonasamonetaryanchorandhelpindeterminingthepricelevel.
IntradiJonalmonetarymodels,thedependenceofconsumpJononthefinancialwealthofhouseholdswascalledthePigoueffect(seePigou1943),ortherealbalanceeffect(seePaJnkin1956).AsSargentandWallace(1975)hadindicatedintheiroriginalanalysisofpricelevelindeterminacy,inthepresenceofaPigouorrealbalanceeffect,theproblemdoesnotariseevenifthecentralbankpegsthenominalinterestrate.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 63
SeigniorageandInflaJon• If the growth of the money supply translates into higher inflation in the longer
term, why don’t governments and central banks keep the rate of growth of the money supply low and stable in order to control and eliminate inflation?
• The answer is that governments often have other policy motives besides the motive of tackling inflation. Perhaps the most important incentive for the issuance of new money by governments is to finance expenditure that they cannot, or do not want to, finance through other methods, such as higher taxes or higher government debt.
• The main cause of all the episodes of high inflation or hyperinflation appears to have been the need of governments to use revenue from money creation (seigniorage) to finance wars and war reparations, revolutions, extraordinary costs related to natural disasters or sudden reductions in their borrowing capacity from financial markets and their capacity to raise revenue from taxes and customs revenues.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 64
HighInflaJonandHyperinflaJon
• WeexploretherelaJonshipbetweenthegrowthrateofmoneysupply,inflaJonandtheneedsofgovernmentstoraiserevenuethroughseigniorage.
• Weexamineboththecaseinwhichtherequiredincomefromseignioragecanberaisedonthebalancedgrowthpath,asituaJoninwhichequilibriumturnsouttobecharacterizedbyhighinfla9on,andthesituaJoninwhichtherequiredrevenuefromseigniorageissohigh,thatitcannotberaisedinsteadystateequilibrium,whichcanleadtohyperinfla9on.
• ThegenerallyaccepteddefiniJonofhyperinflaJonisduetoCagan(1956).CagandefinedaaperiodofhyperinflaJonasone“beginninginthemonthinwhichtheriseinpricesexceeds50%andasendinginthemonthbeforethemonthlyriseinpricesdropsbelowthatamountandstaysbelowforatleastayear.”
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 65
EpisodesofHighInflaJonandHyperinflaJon
• ThefirstmodernepisodesofhyperinflaJonoccurredinEuropeintheadermathofWorldWarI,aswellasduringandintheadermathofWorldWarII.
• InthelastfortyyearsveryhighinflaJonandhyperinflaJonreappearedinsomeLaJnAmericancountries,insometransiJoneconomiesaderthecollapseoftheSovietUnionandinsomebelligerentcountriesofAsiaandAfrica.
• Moreover,manycountries,withoutreachingthelevelsofhyperinflaJon,haveexperienceswithhighinflaJonfrom100%to1000%peryearforquitelongperiods.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 66
MonetaryGrowth,InflaJonandSeigniorage:TheDemandforMoneyFuncJon
InordertostudytherelaJonbetweentherateofgrowthofthemoneysupply,inflaJonandrevenuefromseigniorage,weshallstartfromalinearlogarithmicformofthemoneydemandfuncJon.Inequilibrium,thedemandformoneyequalsthesupplyofmoney,soitfollowsthat,
κisaconstant,ethebasisofnaturallogarithms,καιη>0thesemi-elasJcityofmoneydemandwithrespecttothenominalinterestratei.
MP
=κYe−ηi
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 67
MonetaryGrowth,InflaJonandSeigniorage:NominalandRealInterestRates
ThenominalinterestrateisdefinedbytheFisherequaJon,
whereristherealinterestrate,andπeexpectedinflaJon.
RealOutputYisconsideredexogenous,withagrowthrateg+n>0,whiletherateofgrowthofthenominalmoneysupplyMisdenotedbyμ>0.
UndertheseassumpJon,inflaJononthebalancedgrowthpath,saJsfies,
i = r +π e
π = µ − (g + n)
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 68
TheDemandforMoneyandtheRateofGrowthoftheMoneySupply
AssumingraJonalexpectaJons,themoneydemandfuncJoncanthusbewri[enas,
Tofurthersimplifyma[ers,weshallassumethatthegoldenruleappliesonthebalancedgrowthpath,whichimpliesthatr=g+n.UnderthisaddiJonalassumpJon,
MP
=κYe−η(r+µ−(g+n))
MP
=κYe−ηµ
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 69
SeigniorageRevenueandRateofGrowthoftheMoneySupply
Seignioragerevenueisequalto,
AsaproporJonoftotaloutput,seignioragerevenueisdefinedby,
S = M•
P= M
•
MMP
= µ MP
= µκYe−ηµ
s = SY= µ M
PY= µκ e−ηµ
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 70
MaximizingSeigniorageRevenueSeignioragerevenuesasaproporJonoftotaloutput,dependonμ,accordingto,
ThisdependenceisposiJveinμ,uptothepointwhereμ=1/η.Whenμ>1/ηthedependencebecomesnegaJve.Seignioragerevenue,asaproporJonoftotaloutputisthusmaximizedwhenμ=1/η.Maximumseignioragerevenueinsteadystateisthusgivenby,
∂s∂µ
= (1−ηµ)κ e−ηµ
smax =κλe
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 71
TheSeigniorageLafferCurve
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 72
TheRateofGrowthoftheMoneyandMaximumSeigniorageRevenue
• Cagan,usingannualdata,esJmatedthatηliesbetween1/2and1/3.Consequently,heesJmatedthegrowthrateofthemoneysupplythatmaximizesrevenuesfromseigniorage,asapercentageoftotaloutput,andthecorrespondinginflaJon,atbetween200%and300%peryear.
• Assumingthatκ=0.10,themaximumrevenuefromseigniorageasapercentageoftotaloutputisbetween7-11%.ThisisroughlytheesJmateofCagan(1956).
• Fortheperiod1975-1985,SachsandLarrain(1993)esJmatedactualrevenuefromseigniorageatabout5to6.5%forhighinflaJoncountriessuchasItaly,Bolivia,TurkeyandPeru,andmuchlowerforaseriesofothercountries.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 73
HighInflaJonBalancedGrowthPath
• LetusnowconsideragovernmentwhichneedstofundaproporJonofitspublicspendingthroughseigniorage.Wewillassumethatthisfinancingrequirement,asaproporJonoftotaloutputisequaltosE,whichislessthanthemaximumseignioragesmaxthatthegovernmentcanachievebyse^ngthegrowthrateofthemoneysupplyatμ=1/η.
• TherearetwoopJonstoachieverevenueequaltosE.OneiswithagrowthrateofthemoneysupplyμE<1/η,andtheotheriswithagrowthrateofthemoneysupplyμE΄>1/η.
• AssumingthatthegovernmentdislikesinflaJon,itwillchoosethelowestgrowthrateofthemoneysupplythatiscompaJblewiththeobjecJveofraisingrevenuesEfromseigniorage.
• ForaslongasthegovernmentneedstofinanceaproporJonsEofitsoutputthroughseigniorage,theeconomyistrappedinanequilibriumwitharateofgrowthofthemoneysupplyequaltoμEandthecorrespondinghighinflaJon.Forexample,ifthegovernmentwantstoraiseseignioragecorrespondingto6%oftotaloutput,assumingη=1/2,thisimpliesanannualgrowthrateofthemoneysupply(andcorrespondingsteadystateinflaJon)equaltoabout100%.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 74
AHighInflaJonBalancedGrowthPath
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 75
SeigniorageRevenueOutsidetheBalancedGrowthPath:GradualAdjustmentofMoney
DemandorGradualAdjustmentinExpectaJons• Supposenowthatthegovernmentneedstoraiseseigniorage
which,asaproporJonoftotaloutput,ishigherthanthemaximumthatcanberaisedinthesteadystate.WeassumesE>smax.
• Obviouslytherecanbenobalancedgrowthpathinwhichthegovernmentcanraiserevenuesfromseignioragetoexceedssmax.
• However,foraJme,andastheeconomyadjuststowardsthebalancedgrowthpath,thegovernmentmaybeabletoraiseseignioragerevenuesgreaterthansmax.Thiscouldhappenifforexamplethereisgradualadjustmentinthedemandformoney,orgradualadjustmentininflaJonaryexpectaJons.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 76
SeigniorageRevenueOutsidetheBalancedGrowthPath:GradualAdjustmentofMoney
Demand• Supposethatthedemandformoneydoesnotadjustimmediatelytoitssteady
stateleveladerachangeinthenominalinterestrate,butonlyadjustsgradually.Thus,whenthenominalinterestrateincreases,moneydemandistemporarilyhigherthaninthesteadystate.
• Inthiscase,duringtheadjustment,themonetarybaseuponwhichtheinflaJonarytaxisimposedishigherthanthesteadystatemonetarybase.Consequently,duringtheadjustment,asμincreases,seignioragerevenueswillexceedsmaxbecauserealmoneybalancesarehigherthanonthebalancedgrowthpath.
• Asthedemandformoneydecreasesgradually,thegovernmentshouldconstantlyincreasetherateofmonetaryexpansionandtheconsequentinflaJon,tobeabletohavetherequiredhighrevenuesfromseigniorage.Thiscanleadtoanexplosivepathfortherateofgrowthofthemoneysupply,andaconsequenthyperinflaJon.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 77
GradualAdjustmentinMoneyDemand
Assumethat,intheshortrun,realmoneydemandadjustsgraduallytowardsitssteadystatevalueaccordingto,
d lnm(t)dt
= m•(t)
m(t)=ψ lnm*− lnm(t)( )
where,ψisthespeedofadjustment,and,
m(t) = M (t)P(t)Y (t)
m*= MPY
=κ e−ηµ
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 78
MonetaryGrowthandtheGradualAdjustmentinMoneyDemand
SubsJtuJngforthedeterminantsofsteadystatemoneydemandm*,theadjustmentofmoneydemandtakestheform,
m•(t)
m(t)=ψ lnκ −ηµ(t)− lnm(t)( )
InordedtoachieveaconstanttargetsEforseignioragerevenue,therateofgrowthofthemoneysupplymustbeequaltotherateofreducJonofthedemandforrealmoneybalances
sE = µ(t)m(t) µ•(t)
µ(t)= − m
•(t)
m(t)and
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 79
DynamicAdjustmentoftheRateofGrowthoftheMoneySupply
inordertomaintainrevenuesfromseigniorageconstantasapercentageoftotalincomeatthelevelsE,thegrowthrateofthemoneysupplymustkeepincreasingconJnuously,atthesamerateasthedeclineofrealmoneydemandrelaJvetooutput.FromthepreviousequaJonsthisimpliesthat,
µ•(t)
µ(t)= −ψ lnκ − ln sE + lnµ(t)−ηµ(t)( )
sE = µκ e−ηµ ≤ smax
AnecessaryandsufficientcondiJonforstabilizingtherateofgrowthofthemoneysupplyisthat,
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 80
TransiJontoHyperinflaJon
IfsE>smaxthentherateofgrowthofthemoneysupplyincreasesconJnuously.Aderapointitstartsincreasingatanincreasingrate,withtheresultanexplosivepathfortherateofgrowthofthemoneysupplyandinflaJon.TheresultishyperinflaJon.
µ•(t)
µ(t)= −ψ lnκ − ln sE + lnµ(t)− λµ(t)( )
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 81
HighInflaJonandHyperinflaJonwithParJalAdjustmentinMoneyDemand
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 82
RequiredSeigniorageRevenue,MoneySupplyGrowthandInflaJon• Inthecasewherethefinancingneedsofthegovernmentfromseigniorage
arelessthanorequaltothemaximumpossibleonthebalancedgrowthpath,thentherateofgrowthofthemoneysupplystabilizesataratethatmayindeedentailsignificantinflaJon,butinflaJonisstableanddoesnotevolveintohyperinflaJon.
• However,ifthefinancingneedsofgovernmentexceedthemaximumthatissustainableonthebalancedgrowthpath,then,asthegovernmenttriestoraisethenecessaryrevenuefromseigniorage,therateofgrowthofthemoneysupplygraduallyaccelerates,inordertokeepupwiththedecliningmonetarybase,andtheeconomyfallsintoastateofhyperinflaJon.ThereasonisthatinflaJongraduallyreducesthedemandformoneyrelaJvetototaloutput,andthegovernmentneedsaneverincreasinggrowthrateofthemoneysupplyinordertobeabletocollecttheneededseignioragerevenue.
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 83
ConclusionsontheDeterminantsofHighInflaJon
• Ourbasicanalysisexplainswhy,inmanycases,inflaJonmaybedriventoveryhighlevels.Thisisduetotheinabilityofagovernmenttofinanceitsspendingfromotherrevenuesources,suchastaxaJonorborrowingfromthemarkets,anditsneedtouseseigniorage,i.erevenuefrommoneycreaJon.
• TheanalysisalsoexplainswhyeventhoughinflaJonmayreachveryhighlevels,itisnotnecessarythatitwillevolveintoanexplosivehyperinflaJon.Forthistohappen,thefinancingneedsofthegovernmentmustbesohighthattheyexceedthemaximumlevelthatcanbefinancedthroughseigniorageonthebalancedgrowthpath.
• Finally,theanalysisemphasizesthecentralroleoffiscalproblemsasthemainrootcausesofbothhighinflaJonandhyperinflaJon.AsignificantprecondiJonfortacklinghighinflaJonorhyperinflaJonistopursuereformsthataddresstheunderlyingfiscalproblems(Sargent1982).
ProfGeorgeAlogoskoufis,DynamicMacroeconomicTheory,2015 84
FinalConclusions• InthislecturewehaveanalyzedtheroleandfuncJonsofmoney.Moneyperformsthree
funcJons.First,itisaunitofaccount,second,itisagenerallyacceptedmeansofpayment,and,thirdly,itisastoreofwealth.
• WefirstreviewedthebasicfuncJonsofmoneyandthefactorsthatdeterminethedemandforandsupplyofmoney.Weanalyzedtheconceptofshortrunequilibriuminthemoneymarket,assumingthatthecentralbankfollowsapolicyofeithertargeJngthemoneysupplyorpeggingnominalinterestrates,andalsodefinedthenoJonofthelong-termneutralityofmoney.
• Wethenfocusedonanumberofdynamicgeneraleconomicequilibriummodelswithmoney,inordertoanalyzethedeterminaJonofthepricelevelandnominalinterestratesandalsoanalyzedthelongrelaJonshipbetweenthemoneysupply,thepricelevelandinflaJon.
• FinallyweexaminedthefiscalincenJveforincreasingthemoneysupplyanditseffectsoninflaJon.ThemostimportantmoJveforsustainedlargeincreasesinthemoneysupplybygovernmentshasbeentheincenJvetouseseigniorageinordertofinancegovernmentexpenditurethatcouldnotbefinancedbyothermethods,suchasaddiJonaltaxesorgovernmentbonds.