August 15, 2023

Examine when and how firm managers should disclose information about the firm’s value in competitive markets, including the timing of such disclosures and whether bad news should be released.

Hi there, there are a couple of requirements for this part as it is a premilineary part for the dissertations.

1. I need at least 20 solid papers which are highly cited and very concrete research papers, please do look at the papers and note this is the Economics dissertations with Economics models and not a business school dissertation. Please type out the instructions in the word document to check out how did you look for the papers and potentially use the Econpaper and other highly cited Economics websites for this.

2. Please provide a overview of how to write this dissertation.

3. Please provide a formatting in the word document into how to set up the menu int he word document.

4. Please do a min-dissertation, when I mean by the mini dissertation I am expecting 5000 words with the key ideas and the key models you wish to see how the dissertation have.

Basically it is like the first draft, here is the descriptions of the first draft:

First draft: Your first draft is about getting words on the page. For example, it may sketch out your first thoughts, arguments and potential structure. You can review these and use them to check: are you focussed on the right topics and questions? Is your structure and line of thought sensible? This is also a good time to set up your format requirements (e.g. page layouts, references).

5. Furter, please find at least 5 dissertations which is theoretical as my dissertation is theoretical not empirical which is in the similar field like this, at the undergraduate, master or PHD level. which gives me a better idea.

Please do not use chatGPT for writing and I am able to spot it out, thank you

If we work well this time, we will work in the next months and even years with other subjects and redrafting the dissertation (which is separate bid), thank you!

Dynamic information disclosure in markets

This topic examines when and how firm managers should disclose information about the firm’s value in competitive markets, including the timing of such disclosures and whether bad news should be released.

Overview article: Milgrom, P. (2008). What the seller won’t tell you: Persuasion and disclosure in markets. Journal of Economic Perspectives, 22(2), 115-131.

Articles:

Kremer, I., Schreiber, A., & Skrzypacz, A. (2022). Disclosing a Random Walk. Available at SSRN 4025592.

Guttman, I., Kremer, I., & Skrzypacz, A. (2014). Not only what but also when: A theory of dynamic

voluntary disclosure. American Economic Review, 104(8), 2400-2420.

Acharya, V. V., DeMarzo, P., & Kremer, I. (2011). Endogenous information flows and the clustering of

announcements. American Economic Review, 101(7), 2955-2979.

Please provide the references at the end of the first draft 5000 words +-/10%

Requirements:

Macroeconomics&InequalityLeeTyrrell-HendryUniversityofEdinburghSeptember2022AbstractThisthesisexploresinequalityanditseffectsonthemacroeconomy.Eachchapterisconcernedwithheterogeneitybetweenhouseholdsorfirmsinsomedimension:thefirstsimplyexploreswhetherastandardmodelcancapturethedegreeofinequalityobservedinthedata;thesecondandthirdtackletheimplicationsofsuchinequalityforsomeaspectofpolicy.InallthreepapersIadopttheapproachnowcommonintheliteratureofviewinginequalityasarisingfromidiosyncraticshocksandincompletemarkets.InthislaysummaryIbrieflydiscusseachchapter.MatchingtheWealthDistributionwithIncomeInequality&Risk.Ishowthatastandardincompletemarketsmodelwithlabourincomeriskalone,whenmodelledaccuratelyusinganewmethod,canbroadlymatchthedistributionofwealthobservedintheUSandUK,whereasaddingcapitalincomeriskdoeslittletoconcentratewealthfurther;bothfindingsareincontrasttothepriorliterature.Moreover,Ishowthatincreasedlabourincomeinequalityandriskcanaccountforaroundathirdoftheriseintheconcentrationofwealthamongthetop1%intheUSsincethe1970s,andabout0.5ppofthedeclineinrealinterestrates.ShouldIStay(inSchool)orShouldIGo(toWork).Howshouldgovernmentscalibratethedesireforredistributionviaprogressivetaxationagainsttheneedtoincentivisethe1

Macroeconomics&InequalityLeeTyrrell-Hendryaccumulationofhumancapitalthroughschooling?Iexplorethisquestioninaheterogeneousagentmodelfeaturingstochastichumancapitalaccumulation,endogeneouslaboursupplyandaneducationchoicemodelledasastoppingtimeproblem,whereagentschooseanoptimalnumberofyearstostudybeforestartingwork.ThelatterfeatureaddressesashortcomingofmuchoftheliteraturethattypicallymodelseducationdecisionsasatimeallocationproblemÐwhichoverstatestheabilityofolderworkerstoinsurethemselvesagainstobsolesenceoftheirhumancapitalandhenceunderstatesthewelfarebenefitsofpublicinsurance,forexamplethroughprogressivetaxationÐorasastylisedproblemwithnoresourceoropportunitycosts,whichunderstatesthebiteoffinancialfrictions,andhencethepotentialwelfaregainsfromsubsidisinghighereducation.Thesocialwelfare-maximisingpolicyfeaturesgeneroussubsidiesforeducationandhighlyprogressivelabourtaxes.Thisresultisrobusttomyriadextensions,includingweakeningtheextentoffinancialfrictionsbyallowingstudentstoborrow.EMForeverBlowingBubbles:GlobalImbalances&theLimitsofFiscalSpace.Whatarethelimitsonhowmuchagovernmentcanborrowwhentherealinterestrateonpublicdebtisbelowthegrowthrateoftheeconomy?Iexplorethisquestioninaframeworkwherehouseholdentrepreneursmakeriskyinvestmentsunderincompletemarkets.Suchrisksinducethemtoengageinprecautionarysaving,whichlowersequilibriuminterestratesandhenceexpandsthefiscalspaceavailabletogovernments.Iexpandonthepriorliteraturebyintroducingemergingmarket(EM)economiesÐwhereentrepreneursfaceevengreaterriskandhenceengageinmoreaggressiveprecautionarysavingÐaswellaslimitstotheprivatesupplyofsafeassets.Bothelementsreduceequilibriumrealinterestratesinthedevelopedworldandfurtherinflatethebubbleinpublicdebt,affordingdevelopedmarket(DM)governmentsevengreaterfiscalspace.Iquantifytheextentoffiscalspaceavailabletogovernments:howmuchdebttheycanissuewithouthavingtorunsurpluses,orhowlargeadeficittheycansustainablyrun.ButIalsoshowthatsuchborrowingcarriesrisks,andgovernmentsmustdesignfiscalrulestoensuretheirdebtisbothstableintheshortrunandsustainableinthelongrun,andthatmoreoverthatsuchpoliciesareingeneralnotPareto-improving.2

MatchingtheWealthDistributionwithIncomeInequality&RiskLeeTyrrell-HendryUniversityofEdinburghSeptember2022(Clickhereforlatestversion)AbstractInthispaperIshowthatincontrasttopriorresearchastandardincompletemarketsmodelcanbroadlymatchthedistributionofwealthobservedinthedata,includingcomingclosetocapturingtheextremeconcentrationatthetop,ifoneaccuratelymodelstheinequalityandriskassociatedwithhouseholdsÕearnings.Moreover,Ishowthataddingcapitalincomeriskdoesnotleadtomateriallygreaterconcentrationatthetop,againincontrasttopriorresearch.Ilastlyshowthatchangesinincomeinequalityandrisksincethe1970sintheUScanexplainperhapsathirdoftheincreaseintheconcentrationofwealthatthetopsincethen.1

MatchingtheWealthDistributionLeeTyrrell-Hendry1IntroductionPriorresearchhasarguedthatthestandardincompletemarketsmodelcannotmatchtheobserveddistributionofwealththroughincomeendowmentshocksalone.InthisshortpaperIshowthatinfactitcandoso,providedoneaccuratelycapturesthedegreeofincomeinequalityandriskobservedinthedata,whichIdousingasimplenewmethodusingsurveypaneldata.WiththisapproachIamabletoverycloselymatchthedistributionofwealthforthebottom95%ofhouseholds,andalsocomeclosetomatchingthedistributionamongthewealthiesthouseholdsaswell,atleastintheUK,wheremorefine-grainedmodellingoftheincomedistributionispossiblewithsurveydata.ThemainproblemwithstandardincomeprocessesÐandwhythisapproachworksincontrastÐisthatanagentÕsprecautionarysavingmotiveisratherweakiftheirincomemean-revertsratherquickly,andinanycasecannotreachextremesÐasisthecaseAR1incomeprocesseswithnormally-distributedshocksÐbecauseofstandardpermanent-incomelogic.However,theactualriskandextremityobservedinlabourincomedataissuchthathighproductivityworkersearnextraordinarysumsforshortperiodsoftime,withconsiderabledownsiderisk,motivatingintenseprecautionarysaving.Priorresearchhasmatchedthewealthdistributionprimarilybyincorporatingatleastoneofthefollowingtwofeatures:1.Riskycapitalincomeinplaceoforontopofriskylabour/endowmentincome.However,Ialsoshowthataddingcapitalincomeriskontopofthelabourincomeprocessdescribedaboveriskdoesnotmateriallyconcentratewealthfurtheroverabaselinemodelwitharealisticlabourincomeprocess.2.Exoticandparticularmodellingstructuresthatstrengthenincentivestosaveforhigh-incomeagentsand/orweakenthemforlow-incomehouseholds.Idescribeavarietyofsuchapproachesbelowinmoredetail,butsufficeittosaythatthesimplifiedapproachofthispaperallowsonetocomeneartomatchingthedesiredfeaturesofthedata2

MatchingtheWealthDistributionLeeTyrrell-Hendrywithoutsomewhatopaquemodellingstructuresthatmaystrainoureconomicintuitionaswellasthelimitsofourcomputationalpower.Lastly,Iuselong-runningUSincomepaneldatasincethe1970stoshowthatchangesintheapparentincomeriskfacedbyhouseholdscanexplainperhapsathirdoftheincreaseintheconcentrationofwealthatthetopsincethen.Huggett(1993)andAiyagari(1994)wereamongthefirsttoquantitativelyexplorehetero-geneousagentmodels.ButthosepapersÐwhichwerenotfocussedonexplainingthewealthdistributionperseÐusedanincomeprocesswithfewstates,andinthelattercaseassumingnormally-distributedshocks,withthevarianceestimatedfromthePSID,andconsequentlycouldnotmatchwelltheobservedfeaturesofthedistributionsofeitherincomeorwealth,particularlytheirheavyParetotailsandhencethehighconcentrationamongthetopfewpercent,asdiscussedexplicitlyinCastanedaetal.(1998)andStachurski&Toda(2019).Macroeconomistshavesincetakenanumberofdifferentapproachestotrytoexplainthisobservedconcentrationofwealth.Whatallsuccessfulapproacheshaveincommonisthattheygivesomehouseholdsanextremedesiretosaveandotherstodissave.Thismaybeachievedthroughmanipulatingtheutilityfromsavingwithheterogeneousand/oridiosyncratically-riskydiscountfactors,likeKrusell&Smith(1998)andCarrolletal.(2017),andoriginallypostulatedbyRamsey(1928)inacasualfinalcommentofhislandmarkpaper.Alternatively,onecanmanipulatethereturnstosaving,likeBenhabibetal.(2011)&(2016),andKhieu&Walde(2018).Thelatterpaperisnotableinthatdespiteitspartialequilibriumsettingandsimpleemployment-unemploymentlabourincomeprocess,heterogeneousreturnsaresufficienttomatchthewealthdistribution,butincorporatingriskycapitalincomerequirescounterfactualÒsuperstarÓstates.Benhabibetal.(2017)andStachurski&Toda(2019)furthermoreshowthatthestandardAiyagari(1994)model(withinfinitely-livedhouseholds)cannotgenerateaPareto-distributedwealthdistributionwithheaviertailsthanthatoftheincomeprocess,whichiswhatweseeinthedata.Idonotarguewiththisresult,butmerelyclaimthatevenwithoutaheavy,Pareto-tailedwealth3

MatchingtheWealthDistributionLeeTyrrell-Hendrydistribution,onecanstillmatchmanyfeaturesofthedatawiththestandardmodel,whichmaybesufficientforcertainkindsofpolicyanalysis.Moreover,Ishowthatreasonablecalibrationsoftheprocessgoverningcapitalincomeriskdonotmateriallyconcentratewealthovermybaselineapproachwithlabourincomeriskalone.Afinalapproachistogivecertainhouseholdsaccesstoasuperior,butsomehowrestrictedsavingtechnology.OneeconomicstoryusuallyofferedÐwhichalsofitsthedataontheoccupationsoftheextremelywealthyÐisentrepreneurship,withoccupationalchoiceandcreditconstraintslimitingaccesstothetechnologyamonglow-productivity,low-wealthhouseholds,e.g.Quadrini(2000)andCagetti&DeNardi(2006).Avariationonthisistoincorporateilliquidassets,asinKaplan&Violante(2014),Kaplanetal.(2018)andLuetticke(2021),whereadjustmentcostsdisincentiviselow-incomeandlow-wealthhouseholdsfromsavinginthehigher-returnilliquidasset.Thealternateapproachtakeninthispaperistodesigntheincomeprocesstogeneratetheappropriateprecautionarysavingbehaviour.Othershavetakenthisapproachtoo,thesimplestofwhichissimplytocalibratetheincomeprocesstomatchthemomentsofthewealthdistribution,asinCastanedaetal.(2003),whouseaproductivityprocesswithanÒawesomestateÓÐreachedby0.04%oftheworkforceÐmorethan1000xthatofthebottomstate,representingthebottom60%ofworkers.Thedownsideisthattheymatchthedistributionofearningsonlyinaverycoarsesense;theirprocessfeaturescounter-factuallyextremeincomeinequalityattheverytopendthatisanorderofmagnitudehigherthanthatobserved.AnalternativethatreliesonamuchmoresophisticatedincomeprocessisduetoLise(2012),whoendogenisesincomeriskandinequalitybyincorporatingon-the-jobsearch.ThisgeneratesprecautionarysavingsbehaviourthroughaÒjobladderÓthatofferssmallincrementalgainsasworkersclimbtheladder,withasmallriskofalargedropinincomeiftheyÒfalloffÓ,i.e.losetheirjobs.ThoseatthetopofthejobladderÐtherichestÐhavemosttoloseandthereforesavethemost.Combinedwithheterogeneousability,thismodelcancome4

MatchingtheWealthDistributionLeeTyrrell-Hendryclosertomatchingthedistributionofwealth.Thedownsidehereisthecomplicatedmodellingstructure(withthestandardsearchmodelbaggage),whichoriginallynecessitatedapartialequilibriumsettingandhencelimitedtheapplicationsofthemodel,particularlyonquestionsofpolicy.ThesimplifiedsearchprocessofKruselletal.(2010),whichintegratesthestandardDMPsearchmodelintotheincompletemarketssettingofAiyagari(1994)achievesthesamegoalofendogenisinglabourincomerisk,butduetothesimplicityofitslabourincomeprocess(employmentvsunemployment),itgeneratesatrivialdispersionofwageincomeamongemployees,andhencealthoughnotexplicitlystatedthedistributionofwealthmatchesthedatapoorly.Otherapproachesmodelthelife-cycleandhumancapitaldecisionsofindividualstoexplaintheincreasesinthelevel,varianceandskewnessofearningsobservedamongagentsovertheirlifetimes,asinHuggettetal.(2006)&(2011).Althoughtheydonotexplicitlytrytomatchtheconcentrationsofincomeandwealthatthetop-endofthedistribution,thisliteratureprovidesamorecompellingeconomicstorytoexplainincomefluctuationsthansimpleexogenousprocesses.DeNardi(2004)highlightedtheimportanceofbequestsandintergenerational(non-)transmissionofhumancapitalinanoverlappinggenerationscontext,essentiallyintroducingsubstantialdownside-riskbyhavinghigh-productivityhouseholdslosepartoftheiraccumulatedhumancapitalfromonegenerationtothenext,motivatinghighbequeststhroughotherwisestandardprecautionarysaving/intertemporalsubstitutionmechanisms,albeittypicallywithnon-homotheticpreferencesforbequests.Withoutthesefactors,purelife-cyclesavingsbehaviourisinsufficienttogeneratetheobservedextremeconcentrationsofwealth,asHuggett(1996)showed.Table1summarisescertainkeycontributionstothisvastliteratureintermsofhowcloselytheywereabletomatchthedistributionofearningsandwealthintheUS(regardlessofwhetherthatwastheirgoal).Thispapercomesclosetoachievingthisaimwithaverysimpleapproachandwithoutmanyofthebellsandwhistlesofthosethatcomecloser.5

MatchingtheWealthDistributionLeeTyrrell-HendryTable1:MatchingthewealthdistributionintheliteratureGiniBottom40%Top5%Top1%USeconomy(2013)Earnings0.672.937.218.8Wealth0.85-0.162.935.5Aiyagari(1994)Earnings0.1032.57.56.8Wealth0.3817.913.13.2Huggett(1996)Earnings0.429.822.613.6Wealth0.740.033.811.1Castanedaetal(1998)Earnings0.3020.610.12.0Wealth0.1332.07.91.7Krusell&Smith(1998)Wealth0.82<8.055.024.0Quadrini(2000)Earnings0.45<36.018.27.9Wealth0.74<16.045.824.9Castanedaetal(2003)Earnings0.633.732.614.9Wealth0.791.448.129.9Cagetti&DeNardi(2006)Wealth0.80<6.060.031.0Benhabibetal(2011)Wealth0.698.052.334.1Lise(2012)Earnings3.632.010.9Wealth-2.140.415.5ThispaperEarnings0.622.336.319.3Wealth0.87-1.256.620.3Source:2013dataonUSearningsandwealthdistributiontakenfromKuhn&Rios-Rull(2016)2BaselinemodelThemodelisthatofAiyagari(1994)withidiosyncraticbutnoaggregaterisk,recastincontinuoustime,`alaAchdouetal.(2022),butwithidiosyncraticincomeshocksmodelledasadiscrete,multi-statePoissonprocess,toallowjumpsandhencetomatchobservedcharacteristicsofincomeshocks.ThereareJidiosyncraticincomestatesandhenceJ(J1)Poissonshocks;ajumpfromeachstatetoeveryotherstate.Estimationofincomeprocess.Sincethemodelisotherwisestandard,IfirstdescribethenovelprocessIusetoestimatetheincomeprocess.6

MatchingtheWealthDistributionLeeTyrrell-HendryOneproblemwiththetypicalincomeprocessesusedinstandardincompletemarketsmodelsisthattheyeitherfeaturenormally-distributedshocks,oronlyafewstates,thatcapturereasonablywellthebodyoftheincomedistribution,butnotthetails,andwhichdonotallowfortheskewnessandleptokurtosisofincomeshocks,asshowninGuvenenetal.(2021).Consequently,theincomeshareofthetop1%isusuallytoolow.Instead,Iexplicitlymodeltheincomeofhighearners,includingthetop1%,andwheredataallows,thetop0.1%.Ishowthisisabletoaccuratelycapturetherighttailoftheincomedistribution.Moreover,Iallowfor(sometimeslarge)jumpsinincome,ratherthanasmoothprocess,whichcancapturetheskewnessandleptokurtosis.Incorporatingboththesefeaturesiscrucialtogeneratingtheadditionalconcentrationofwealthseeninthedata.TomodeltheincomeprocessIusesurveydatatoseparatehouseholdsintodifferentbinsbasedontheirlabourincomeineachyearofthesurvey.Eachbinrepresentsafractileoftheincomedistribution.Ichooseunevenfractiles,withacoarsergradationofproductivityatlowincomelevels,butexplicitlyincorporatingastateforthetop1%ofearnersand(wheredataallows)thetop0.1%.Iestimatethemeanincomeforeachfractile(whichwillstandinfortheirproductivity)andestimatethetransitionprobabilitiesbetweenfractilesdirectlyfromthesurveydata.Ishowtheestimatedannualdiscrete-timetransitionmatrixintheAppendix.ThiscontrastswiththemoretypicalapproachofusingtheSimulatedMethodofMomentstomatchafewmoments(e.g.skewnessandkurtosisofincomechanges)ofartificialdatatothoseobservedinactualdata.MyapproachisroughlyanalogoustotheÒEmpiricalCalibrationÓapproachofCivaleetal.(2017),althoughtheyapplytheirmethodtosyntheticdatageneratedbyadiscretetimeAR(1)processinordertoestimateanincomeprocessusingaSimulatedMethodofMomentsapproach,whereasasIapplymymethodtoactualdata.Theresultofthisestimationisatransitionmatrix(shownintheAppendix,Figure6),governingtheprobabilityoftransitioningfromeachincomefractilefromoneyeartothenext.Iconvertthistransitionmatrixintoitscontinuoustimeanaloguebytakingthematrixlog.Thisresultsinsomenegativeprobabilitiesinoff-diagonals,anissueknownasthe7

MatchingtheWealthDistributionLeeTyrrell-Hendryembeddabilityproblem,whichoccursbecausethetransitionprobabilitiesforcertainfractilesestimatedfromthesampledataare(closeto)zero(Israeletal.2001).IcorrectforthiswiththediagonaladjustmentmethodofInamura(2006):setthenegativevaluestozeroandadjusttheirdiagonalssothateachrowsumstozero.IalsoshowthecontinuoustimetransitionmatrixintheAppendix,Figure7.IusethePSIDfortheUSandtheUnderstandingSocietysurvey(whichIcallbyitsformeracronym,BHPS)fortheUK.SinceIdonotmodelretirement,Iuseonlydataonlabourincome.TheBHPShassince2016accuratelycapturedtheincomesharesofthetop1%and0.1%,albeitwithasmallsamplesizeforthelatter,asshowninTable3.However,thePSIDunder-reportsbothshares,andgiventhesmallsampleitisnotpossibletocapturethetop0.1%atall.Consequently,Irecalibratetheestimatedtop1%incomeleveltomatchthetop1%shareobservedintheSurveyofConsumerFinances(SCF)andadministrativedata.TheSCFaccuratelycapturestheincomesharesoftopearners,butusuallydoesnottrackindividualsovertime,soitisnotpossibletocalculatetransitionprobabilitiesusingthissource.AlthoughIcanthereforematchtheincomeshareofthetop1%usingsupplementalSCFdata,thereisnothingIcandotoadjusttheestimatedtransitionprobabilitiesforthetop1%,thuspotentiallyintroducingerroriftheirtrueincomeriskisverydifferentfromthatobservedinthePSID.Ifforexampleonetakestheviewthatthetop1%ofearnersinthePSIDareactuallythesecond1%(forexample,ifthetop1%aremissedfromthesurveyentirely)thentheobservedtransitionprobabilitiesmayunderstatetheirtrueincomerisk,andhencethestrengthoftheirprecautionarysavingmotive(UKdatasuggeststhetop1%andinparticularthetop0.1%facegreaterincomeriskthanthosejustbelowthem).Tocomparethemodelresultswithwealthconcentrationdata,IusetheWealth&AssetSurveyfortheUK,supplementedwiththeresultsofAlvaredoetal.(2018)tocapturethetopend,andtheSCFfortheUS.AlthoughtheSCFusuallydoesnottrackindividualsovertime,therewasabriefpanelfor2007-09,forwhichKuhn&Rios-Rull(2016)reportanalmost-completetransitionmatrix.8

MatchingtheWealthDistributionLeeTyrrell-HendryIalsoreportresultsforthebaselinemodelusingthistransitionmatrix,althoughthisdataisalsonotideal,forreasonsIdiscussbelow.Recapofstandardincompletemarketsmodel.ThehouseholdÕsproblemistomax-imiseintertemporalutilitysubjecttotheirbudgetconstraint(wealthgrowsbylabourincome,lesstaxesplustransfers,pluscapitalincomelessconsumption),theexogenousprocessforincomeandanexogenousborrowinglimit:V(a0,zj)=max{ct}t2[0,1)E0Z10e⇢tu(ct)dts.t.úat=rat+wzj(1⌧)+Tjctata=Thiscanbere-writtenastheHamilton-Jacobi-Bellmanequation,thecontinuoustimeanalogueoftheBellmanequation:⇢V(a,zj)=maxcu(c)+@aV(a,zj)(ra+wzj(1⌧)+Tjc)+JXk=1jkV(a,zk)(1)wherejjequalsthenegativeofthesumoftheotherjk,suchthateachrowofthetransitionmatrix⇤sumstozero.Thismatrixgovernsthetransitionsbetweenstatesandwillbeestimateddirectlyfrompanelincomedata,asdiscussedbelow.Thebudgetconstraintincludesstate-dependenttransfers,Tj,aflatincometax,⌧,andaborrowingrate(notshown)givenbythelendingrate,r,plusaspread,.ThisHJBequationcanbederivedfromadiscrete-timeBellmanequation,forademonstrationofwhichseeAchdouetal.(2022).ThehouseholdÕsoptimalityconditionsyieldpolicyfunctionsforconsumption,c(a,z),andsaving,úa(a,z),whichgiventhecontinuoustimesettingholdwithequalityeverywhereintheinteriorofthestatespace,withtheboundarycondition@aV(a,zj)u0(ra+wzj(1⌧)+Tj)9

MatchingtheWealthDistributionLeeTyrrell-Hendryholdingattheborrowingconstraint:c(a,zj)=(u0)1[@aV(a,zj)]úa(a,zj)=ra+wzj(1⌧)+Tjc(a,zj)Theevolutionofthejointdistributionofincome(productivity)andwealthisgovernedbytheKolmogorovForwardEquationbelow,andastationarydistribution,ifoneexists,isfoundbysettingthisequaltozero:úg(a,zj)=@a[úa(a,zj)g(a,zj)]JXk=1k6=j[jkg(a,zj)kjg(a,zk)]=0(2)Thefirstterminthisexpressioncapturesthechangeinthedensityfromhouseholdsincreasingordecreasingtheirnetworthbysaving.Thesecondterm(firstpartinsquarebrackets)capturesthedecreaseinthedensityat(a,zj)fromhouseholdswithproductivityzjbeinghitbyanincomeshockatPoissonratejkandmovingtoproductivityzk,andthethirdterm(secondpartinsquarebrackets)capturestheincreaseinthedensityat(a,zj)fromhouseholdswithproductivityzkbeinghitwithanincomeshockatratekjandmovingtoproductivityzj.IclosethemodelinthestandardfashionofAiyagari(1994),withaggregateassetsinequilibriumequaltotheaggregatecapitalstockusedinproduction,andwiththestandardfirmoptimalityconditionsundercompetitivefactormarkets,whereInormalisezaveto1andwhereedenotestheemploymentrate,meaningthemeasureofallexceptthoseinthelowestincomestate:K=JXj=1Zaaag(a,zj)dar=↵zave✓Ke◆↵1w=(1↵)zave✓Ke◆↵Thestationaryequilibriumofthemodelischaracterisedbythevaluefunctionofthehousehold,V(a,z),derivedfromtheHJB,andtheirresultingpolicyfunctions,c(a,zj)andúa(a,zj),whichmaximisetheirlifetimeutilitysubjecttoprices,randw,whichinturnclear10

MatchingtheWealthDistributionLeeTyrrell-Hendryallmarketsgiventhejointdensityfunctiong(a,z),determinedfromtheKFE.Sincethemodelisstandard,IreferthereadertoAchdouetal.(2022)foradiscussionofthenumericalapproachIusetosolveforthemodelequilibrium,thatbeingfinitedifferencemethods.MatchingtheUKwealthdistribution.StartingwiththeUK,forwhichmorerefinedsurveydataareavailable,theestimatedproductivitygridishighlyskewed.Iuse20unevenquantilestotargetincomeshares,includingthetop0.1%andnext0.9%,Theincomestateforthetop0.1%hasproductivitymorethan60xthatofthemean,andthenext0.9%5xthemean.Ihaverebasedthegridsoaverageproductivityis1.Ireservethebottomstateforhouseholdswithzeroornegativeearnings,whichisaround15%ofthetotal.Iinterpretthisstateasnon-employment,althoughthisisnotstrictlythecaseasasmallnumberofhouseholdshavenegativeincomebecauseofself-employmentorbusinesslosses.Thetransitionprobabilitiesarealsoskewed,displayingsignificantdownsideriskforthetop1%and0.1%,whichwillbecrucialforgeneratingthesavingsratesamongwealthyindividualsrequiredtoachieveahighlyconcentratedwealthdistribution.ThedistributionofincomegrowthobservedintheUKdataisalsohighly-non-normal,displayingnegativeskewnessof-0.5andhighleptokurtosisofaround15.IemphasisethatIhaveestimatedtheincomeprocessdirectlyfromthedata,withnoadjustmentstotransitionprobabilitiesorincomelevelsfortheUK;moreoverIhavenotcalibratedtheincomeprocessatalltomatchanyfeaturesofthewealthdistribution.Consequently,thefactthatIamabletomatchcloselythewealthdistributionistrulyaresultofincomeinequalityandincomeriskalone.Asidefromtheincomeprocess,Icalibratethediscountrateto2%perquartertomatchacapital/outputratioofaround3x.Isetthedepreciationrateto1.7%perquartertomatchtheannualinvestment-to-outputratioofaround20%andsetthecapitalshareto40%.Ichooseanunsecuredborrowinglimitofaroundonequarterofannualoutput,inlinewithKaplanetal.(2018)andchooseaspreadoftheborrowingoverthelendingrateof1.65%perquartertomatchtheshareofhouseholdswithnegativewealthofaround9%,slightly11

MatchingtheWealthDistributionLeeTyrrell-HendryhigherthaninthelatestWASdatabutclosetootherestimatesintheliterature(Crawfordetal.2016).IuseaCRRAutilityfunctionwithacoefficientofrelativeriskaversionof1.Isetagovernmenttransferforeachincomequantileequaltothatobservedinthe2017UnderstandingSocietysurveyandsetaflattaxleviedonlabourincomeof20.9%tobalancethegovernmentbudget.Thereisnoothergovernmentspending.Table2:UKmodelcalibrationSymbolParameterValueCalibration⇢Discountrate2%⇠3xcapital/outputratioCoefficientofriskaversion1StandardDepreciationrate1.7%⇠20%investment/outputratio↵Capitalshareofincome40%⇠40%capitalincomeshareBorrowingconstraint4.425%annualoutputBorrowingspread1.65%⇠9%sharewithnegativewealth⌧Incometaxrate20.9%BalancedgovernmentbudgetThemodelmatchestheearnings,incomeandwealthsharesacrossmostofthedistributionalmostexactly,andcomesclosetomatchingthetop1%shareofwealthof20%,asshownintable3.Thetop0.1%holdaround3%ofwealthinthemodel;thereislimiteddataavailablefromsurveys,butAlvaredoetal.2018estimatethisshareataround7.5-8%,whichisalsoconsistentwithapowerlawratioofaround2.5:1betweenthetop10:1%andthetop1:0.1%shares.Themodelappearstounderstatethissharesomewhat.Potentiallythiscouldberesolvedbyincludingastateforthetop0.01%,datapermitting,butthisisspeculation.Itisalsopossiblethatthesmallsamplesizeforthetop0.1%intheBHPSsurveydataledtoerrorintheestimationofthetransitionprobabilities,specificallyunderestimatingtheprobabilityofnegativeshocksandhenceunderestimatingthetruestrengthoftheprecautionarysavingmotive.Nevertheless,theaccuracywithwhichthemodelisabletomatchthebulkoftheincomeandwealthdistributionÐandspecificallythetopwealthsharesÐcomparedtopreviousstudiesdoeshighlighttheimportanceofaccuratelycapturingtheinequalityandriskoftheincomeprocess,specificallythenegativeskewnessandleptokurtosisofincomeshocks.12

MatchingtheWealthDistributionLeeTyrrell-HendryTable3:UKlabourearnings,totalincomeandwealthsharesvsmodelGini<£050%50-90%10%1%0.1%Earnings(model)48.401750.532.510.96Earnings(BHPSdata)48.40.117.250.532.310.85.9Income(model)35.7027.244.328.58.13.9Income(BHPSdata)36.9025.545.529105.4Wealth(model)71.59.24.64055.315.52.7Wealth(WASdata)67.51.66.541.651.919.97.5Source:Bottom90%wealthsharestakenfrom2012WAS;top10%takenfromAlvaredoetal.2018Themodelalsoaccuratelymatchesthesharesoftotalincomeearnedbytheprimeagepopulation,albeitslightlyundershootingforthoseatthetopend,becausethemodelunderstatestheirsharesofwealthandhencecapitalincome.TheLorenzcurvesforearnings(labourincome),totalincomeandwealtharealsoshownbelow,forboththedataandthemodel.Figure1:BaselinemodelLorenzcurvesvsUKdataComparisontootherstandardincomeprocesses.InowcomparetheestimatedPois-sonprocessabovewithtwootherincomeprocessescommonlyusedintheliterature:onewithmean-revertingandonewithpermanent(log-)normally-distributedshocks.Incontinuous13

MatchingtheWealthDistributionLeeTyrrell-HendrytimethesearereferredtorespectivelyasanOrnstein-Uhlenbeck(OU)processandageometricBrownianmotion(GBM).ThestochasticprocessesforproductivityareforanOUprocess(inlogs):dz=(✓ln(z)+122)zdt+zdW,andforaGBM:dz=µzdt+zdW.Icalibratethemean-revertingOUprocesswith✓=0.3tomatchthepersistenceofearningsestimatedintheBHPSdataandavarianceofln(z)of0.5tomostcloselymatchtheLorenzcurveforearnings(thisisslightlylowerthanthemeasuredvarianceoflogearningsofaround0.7),otherwisethecalibrationisthesame,althoughIdropthetaxesandtransfersandofcoursethenotionthatsomeagentsareunemployed;thishasanegligibleimpactonthedistributionalmoments.IdonotrecapitulatethehouseholdÕsHJBequation,ortheKFEgoverningtheevolutionofthejointdistributionofwealthandincome.Sufficeittosaytheyarestandard.IcalibratetheGBMprocesswithµ=0.01,reflectingaveragerealincomegrowthofaround1%/year(alittletoolow),and=0.12.Thevarianceoflogincomegrowsovertimeatrate2t,givingaconditionalvarianceofloglabourincomeamong65year-olds(45inmodelyears)ofaround0.65,whichisbroadlyinlinewithDeaton&Paxson(1994).NotealsothatinthiscaseIuseaperpetualyouthframework,toensureastationarydistributionofincomeandwealthexists;Iassumeagentsdieataconstantrated=1/45,foranexpectedlifetimeof45years.Ifurthermoreassumeagentsarerebornwithnowealthandinitialproductivityofz⇤=0.5;livingagentsareassumedtobuycompetitiveannuitiesthatpaythemdathroughouttheirlifeinexchangeforgivinguptheirassetstotheinsurerupontheirdeath.Inthiscase,asiswell-known(e.g.seeGabaix(2009)),themarginaldistributionofincomehasadouble-Paretodistribution,withthetailsgivenby⇣=⇣µ±pµ2+22d⌘/2.Mycalibrationgives⇣+’1.6,whichisroughlytheobservedParetoright-tailoftheincomedistributionintheUK.Ishowbelowthewealthsharesforvariousearnings,incomeandwealthfractilescomparedtothebaselinemodel,aswellastheLorenzcurvescomparedtotheactualdatafortheUK.Themodelwithmean-revertingshocksperformspoorly.Despitecloselymatchingthe14

MatchingtheWealthDistributionLeeTyrrell-Hendryearningsdistribution,thelowerearningsriskamongtherichmeanstheydolessprecautionarysavingandhencethereismuchlessconcentrationofwealth,sothemodelfallsfarshortofmatchingtheobserveddistributionofwealth.Themodelwithpermanentshocksperformssomewhatbetter.Inparticular,theshareofwealthofthetop1%isclosetothedata,althoughthesenumbersaresensitivetothe(somewhatarbitary)calibration.Moreover,muchoftheconcentrationofwealthinthismodelarisesbecauseahandfulofelderlyhouseholdsaccrueenormouswealthover100+years,whichisobviouslycounterfactual.Amorerealisticageingandbequestprocesscanaddressthisissue,asinDeNardi(2004).AnotheradvantageoftheincomeprocessintheBaselinemodelisthatitcangenerateanaverageannualmarginalpropensitytoconsumeontheorderof20%,morethandoublethatinthemodelwitheithermean-revertingorpermanentnormally-distributedshocks,thoughstillshortofthe50%orsoseeninthedata.Table4:Earnings,incomeandwealthshares:baselinemodelvsotherincomeprocessesGini<£050%50-90%10%1%0.1%EarningsBaseline48.4016.949.333.911.16.1Mean-reverting40.8018.451.829.85.40.7Permanent(analytical)50.8019.636.144.318.87.9Permanent(discretised)48.4020.839.539.78.91IncomeBaseline35.7026.345.228.58.34.2Mean-reverting38.4023.549.5274.50.6Permanent50.9019.63842.411.51.6WealthBaseline71.59.24.640.155.315.52.7Mean-reverting54.510.911.454.6345.30.7Permanent65.409.736.154.220.44.915

MatchingtheWealthDistributionLeeTyrrell-HendryFigure2:Lorenzcurveswithmean-revertingandpermanentshocksvsUKdataComparisontoamodelwithcapital&labourincomerisk.Priorliterature,forexampleBenhabibetal.(2016),havehighlightedtheroleofcapitalincomeriskingeneratingaParetotailinthewealthdistribution.TheyshowanalyticallyinamodelwithoutlabourincomethatcapitalincomeriskalonegeneratesaParetodistributionforwealth.Incombinationwithriskfromuncertainlifetimesinaperpetualyouthmodel,capitalincomeriskgeneratesatwo-tailedÒdoubleParetoÓdistributionforwealth.Althoughtheprecisemodelspecification(nolabourincome)seemspeculiar,theresultsstillapplytomoregeneralmodelswithlabourincomeandlabourincomeriskfortherighttailofthewealthdistribution,sinceinthelimitaswealthgrows,labourincomebecomessmallbycomparison.Iarguehoweverthattheroleofcapitalincomeriskmayhavebeensomewhatoverstatedintheliterature.Forexample,Achdouetal.(2022)showinanAppendixthatamodelwithcapitalincomeriskandasimpletwo-statelabourincomeprocesscangenerateawealthdistributioninlinewiththedata.However,Iwillshowthatinaquantitativemodelwiththerealisticlabourincomeprocessestimatedabove,addingcapitalincomeriskdoesnotresultinamateriallygreaterconcentrationwealthoverthebaselinemodel,andarguablyfitsthebulk16

MatchingtheWealthDistributionLeeTyrrell-Hendryofthedistributionworse.Householdscannowinvestintwoassets;arisklessbondwithreturnr,andariskyassetÐheldinproportion!,chosenoptimallyÐthereturnofwhichfollowsaBrownianmotion:Rdt+dW.Consequently,totalassetsnowevolvestochastically:da=(ra+(Rr)!ac)dt+!adW.Withsuchpermanentwealthshocks,itisnecessaryagaintoincludesomefeaturetoinduceastationarydistributionofwealth;Iusethesameapproachasabove:aperpetualyouthframeworkwhereagentsdieataconstantrate,d,andarerebornwithnowealth.ThestationarydistributionofwealthagainfollowsaParetodistributionintherighttail:g=ca⇣1,withtheheavinessoftherighttail,⇣,givenby:⇣=±p2+2!22d!22,where=r⇢d+2⇣Rr⌘2and!=Rr2.⇣’1.3roughlymatchestheUSdata,and⇣’1.5theUK.Forcompleteness,IshowtheHJBandKFequations(suppressingnotation):⇢Vj=maxc,!u(c)+@aVj(wzj(1⌧)+Tj+ra+(Rr)!ac)+12@aaVj2!2a2+JXk=1jkVkúgj=@a[úa(a,zj)gj]+12@aa⇥2!2a2gj⇤JXk=1k6=j[jkgjkjgk]=0Themarketclearingconditionsarealsoslightlydifferentforthismodel.HereIinterprettheriskyassetascapitalandtherisk-freeassetasabondinpositivenetsupplyofaround80%ofGDP,i.e.thelevelofgovernmentdebtintheUK.K=JXj=1Zaa!ag(a,zj)daB=JXj=1Zaa(1!)ag(a,zj)daTheresultingincomeandwealthsharesareclosetothedataandwealthisslightlymoreconcentratedthaninthebaselinemodel,althoughthelow/middlepartofthedistributionfitsthedataslightlyworse.17

MatchingtheWealthDistributionLeeTyrrell-HendryTable5:UKearnings,incomeandwealthsharesvslabourandcapitalincomeriskmodelGini<£050%50-90%10%1%0.1%Earnings(model)45.9016.950.732.510.96Earnings(BHPSdata)48.40.117.250.532.310.85.9Income(model)48.5016.850.23310.55.7Income(BHPSdata)36.9025.545.529105.4Wealth(model)84.85.80.426.373.331.36.8Wealth(WASdata)67.51.66.541.651.919.97.5Source:Bottom90%wealthsharestakenfrom2012WAS;top10%takenfromAlvaredoetal.(2018)Figure3:LorenzcurvesofmodelwithlabourandcapitalincomeriskvsUKdataMatchingtheUSwealthdistribution.MovingontotheUS,theestimatedproductivitygridisalsohighlyskewed,althoughnottotheextentofCastanedaetal.(2003):productivityinthetopstateismorethan20xthemean.Duetothelackofgoodpaneldata,thetopincomestateisforthetop1%,notthe0.1%.Theestimatedproductivitygridandsemi-annualtransitionmatrixfortheUSisshownagainintheAppendix,Figure8.Icalibratethesemi-annualdiscountrateto5.8%tomatchtheUScapital/outputratioofaround3x.Isetthedepreciationrateto3%tomatchtheinvestment-to-outputratioofaround18%andsetthecapitalshareto40%.Ichooseanunsecuredborrowinglimitof18

MatchingtheWealthDistributionLeeTyrrell-Hendryaroundonequarterofaverageannualincome,inlinewithKaplanetal.(2018)andchooseaspreadoftheborrowingoverthelendingrateof6%(semi-annualised)tomatchtheshareofhouseholdswithnegativewealthofaround10%.IuseaCRRAutilityfunctionwithacoefficientofrelativeriskaversionof1.Isetagovernmenttransferforeachincomefractileequaltothatobservedinthe2014PSIDdataandsetaflattaxleviedonlabourincomeof6.1%tobalancethegovernmentbudget.Thereisnoothergovernmentspending.Table6:USmodelcalibrationSymbolParameterValueCalibration⇢Discountrate5.8%⇠3xcapital/outputratioCoefficientofriskaversion1StandardDepreciationrate3%⇠18%investment/outputratio↵Capitalshareofincome40%⇠40%capitalincomeshareBorrowingconstraint1.325%annualoutputBorrowingspread6%⇠10%sharewithnegativewealth⌧Incometaxrate6.1%BalancedgovernmentbudgetThemodelmatchesthewealthsharesofthebottom99%ofhouseholdssomewhataccurately,andcangenerateashareofwealthforthetop1%of18%,wellbelowtheirtrueshareofmorethan30%,butstillfargreaterthanpreviousstudieshavemanagedtoreproducewithanormally-distributedincomeprocess.Itisverylikelythattheshortfallamongthetoppercentilesisduetothepoordataontheirtransitionprobabilities.AlthoughwecanmatchtheirincomesharesusingadditionaldatafromtheSCF,wecannotaccuratelycapturetheirincomerisksincethePSIDdoesnotcapturetheseindividualssufficientlywell.ComparingtheUStransitionmatrixtothatfortheUK,thetop1%appeartofaceaslittleashalftheriskoffallingoutofthetopbracket,likelybecausethetop1%inthePSIDdatasetarenotthetrue1%.AlthoughitisalsoconceivablethatforwhateverreasontheUStop1%simplydofacelessincomerisk,andsaveforreasonsotherthanprecautionary,forexamplethosediscussedintheliteraturereview.Toaddsomecredencetothis,belowIalsoincludetheresultsofasecondmodel,wherethetransitionmatrixforearningshasbeeninterpolatedfromthatshowninKuhn&Rios-Rull19

MatchingtheWealthDistributionLeeTyrrell-Hendry(2016),calculatedfrom2007-09SCFdata.Asdiscussedabove,theSCFcapturestheuppertailoftheearningsdistributionmuchbetterthandoesthePSID,soIdonothavetoadjusttheproductivitylevelsforthetop1%.Astheresultsshow,model2isbetterabletomatchthelower⇠95%ofthewealthdistributionthanmodel1,andalsogetsslightlycloserinmatchingtheuppertail.Forcomparability,Iusethesamecalibrationformodel2asmodel1.Unfortunatelythereisnodataonthetransfersreceivedbyeachfractile,soImustdropthetaxesandtransfersfromthismodelandgivethelowestfractile(thebottomquintile)anominalamountofincometoensurethemodelsolves.Thisexplainsthelargefractionwithnegativewealth.Moreover,thisdataisfortheGreatRecessionperiod2007-09,sothetransitionprobabilitiesmaynotberepresentativeofthetypicalidiosyncraticriskfacedbyhouseholdsthroughoutthebusinesscycle.Table7:USlabourearnings,totalincomeandwealthsharesvsmodelGini<£050%50-90%10%1%Earnings(model1)60.5012.141.846.118.9Earnings(model2)61.702.3150.4147.319.3Earnings(SCFdata)67<12.9146.7149.618.8Income(model1)55.6013.945.340.915Income(model2)62.809.342.847.915.3Income(SCFdata)5809.5137.514719.7Wealth(model1)7510.64.134.161.717.9Wealth(model2)87.531.7-1.225.575.720.3Wealth(SCFdata)85100257535.51Bottom40%and40-90thpercentile20

MatchingtheWealthDistributionLeeTyrrell-HendryFigure4:USmodel1LorenzcurvesvsdataAccountingfortheincreaseintopsharesofwealthsincethe1970s.Incomeriskaloneaccountsforsomeoftheadditionalshareofwealthofthetop1%relativetoincomeintheUScomparedtoothercountries,butitcanalsoexplainaroundathirdoftheincreaseintop1%sharesincethe1970s.IshowthisbyperformingthesameexerciseusingPSID/SCFincomequantilesfrom1975andPSIDtransitionprobabilitiesbetween1969and1975.Twodifferencesofnotebetweentheincomeprocessestimatedfrom1970sdatavs2010sdataarethelowershareofnon-employedworkersandthelowershareofincomeearnedbythetop1%intheearlierperiod.Alsonoteworthyisthefactthatthe1%appearedtofacegreaterriskoffallingoutofthetopbracketinthe1970scomparedtotoday.Thiswouldsuggestlessprecautionarysavingandhencealowerwealthsharetoday,howeverthismayjustbeduetoerrorsinestimatingtransitionprobabilitiesduetopoordata.Nevertheless,itdoestentativelysuggestothermotivesforsavingthanprecautionaryareimportantforexplainingtheobservedhighconcentrationofwealthrelativetoincome.OtherthantheincomeprocessIusethesamecalibrationasabove,toisolatetheeffectsofthechangeintheincomeprocess.Theincomeshareofthetop1%wasaround10%in1975(thePSIDunder-reportsthis21

MatchingtheWealthDistributionLeeTyrrell-Hendrybyhalf)andtheirwealthsharearound22%.Todaytheirwealthshareiscloserto35%,anincreaseof15pp.MatchingtheincomeshareasaboveIcangenerateashareofwealthamongthetop1%ofaround10%forthe1970scalibration,comparedto15%forthe2010scalibration.Thusthemodelsuggeststhataroundathird(5pp)oftheincreaseinthetop1%wealthsharecanbeexplainedbychangesintheincomeprocess,includingagreaterincomeshareforthetop1%.Table8:Modellabourearnings,totalincomeandwealthsharesfor1975calibrationGini<£050%50-90%10%1%Earnings(model)47.5018.353.428.38.1Income(model)44.5019.649.4318.3Wealth(model)57.93.712.444.543.110.6Idemonstratethistransitionofinequalitymoreclearlybyintroducingaone-time,unex-pectedpermanentchangeintheincomeprocessin1975.Thetop1%sharerisesfrom10%to15%,withmuchofthetransitionoccurringinthelate70sand1980s,asinthedata,althoughtheincreaseinconcentrationtrailsoffafterthis,incontrastwiththeUSexperience.Thetop10%sharerisesfrom40%tonearly60%;thetransitionisslowerthanthatforthetop1%andmuchlargerthanthatobservedinthedata.Themoreunequalincomeprocessalsoimpliesafallintherealinterestofnearly0.5pp.Thelevelofinterestratesisfarhigherthanobservedinthedata,sincetheprecautionarysavingsmotiveresultsinalargercapitalstockforagiveninterestrate,sotomatchtheobservedcapital-outputratiosonemustraisethediscountrate.Thisissuecanbeaddressedbyincludingariskyasset,asabove,whichgivesthesafeassetaliquiditypremium,loweringtheinterestagain.Nevertheless,thisexercisedoesgiveasenseofthepotentialcontributionofhigherinequalitytowardssecularstagnationandlowerequilibriuminterestrates,ontopof,forexample,thedemographic,productivityandfiscalfactorsidentifiedinEggertssonetal.(2019)andRachel&Summers(2019).Incidentally,thefallininterestratesgoessomewaytooffsettingtheriseininequalitythat22

MatchingtheWealthDistributionLeeTyrrell-Hendrywouldhaveoccurredunderthetransitiontothenewincomeprocessunderpartialequilibrium.Thetop10%sharewouldhaverisentomorethan70%andthatofthe1%toalmost20%,hadinterestratesnotfallen.Figure5:Increaseintopwealthsharesduetoincreasedincomeinequalityandrisk3ConclusionIhavehopefullyshownthatlabourincomeriskalone,properlymeasured,particularlyforhighearners,canhelponematchtheobservedwealthdistributioninastandardincompletemarketsmodeltoahigherdegreeofaccuracythanpastliteraturesuggests.InparticularIusedanewmethodofmodellinglabourincomeriskasamulti-statePoissonprocessbyestimatingcarefullychosenincomefractilesandtransitionprobabilitiesbetweensaidfractilesdirectlyfromsurveydata.Thisapproachinmyopinioncomparesfavourablytootherapproachesthatuseanincomeprocesswithnormally-distributedshocks,albeitperhapsnottomorecomplicatedmodelsthatintroduceothermotivationsforsavingbythewealthy.Ifurthermoreshowedthataddingcapitalincomerisktoastandardmodelwithsuchanincomeprocessdoesnotmateriallyfurtherconcentratewealthoverthebaselinemodel,23

MatchingtheWealthDistributionLeeTyrrell-Hendrycastingsomedoubtoverthetypicalbeliefespousedintheliteraturethatsuchinvestmentriskiscrucialtomatchingthewealthdistribution.Moreover,theobservedchangeinincomeinequalityandriskalonecanalsoexplainasubstantialpartÐperhapsathirdÐoftheincreaseinwealthownedbythetop1%sincethe1970sintheUS,aswellaspartofthedeclineininterestrates.Myconclusionissimple:ifonewantstobuildaheterogeneousagentmodelthatbroadlymatchesthedataonthedistributionofincomeandwealth,onecandosorelativelysimplybytakingtheapproachofthispaperandusingalabourincomeprocessthatcanbeeasilyestimatedfromsurveydata.Onedoesnotnecessarilyneedtoaddinfeatureslikecapitalincomerisk,entrepreneurshipandfinancialfrictions,life-cyclesandbequests,oranyofthemyriadotherapproachestakenintheliterature,whichallhavetheirownmerits,butwhichintroduceadditionalmechanismsandcomplicationsbeyondthestandardmodel..24

MatchingtheWealthDistributionLeeTyrrell-HendryReferencesAchdou,Y.,Han,J.,Lasry,J.-M.,Lions,P.-L.&Moll,B.(2022),ÔIncomeandwealthdistributioninmacroeconomics:Acontinuous-timeapproachÕ,ReviewofEconomicsStudies89,45Ð86.Aiyagari,S.R.(1994),ÔUninsuredidiosyncraticriskandaggregatesavingÕ,QuarterlyJournalofEconomics190(3),659Ð684.Alvaredo,F.,Atkinson,A.B.&Morelli,S.(2018),ÔTopwealthsharesintheukovermorethanacenturyÕ,JournalofPublicEconomics162,26Ð47.Benhabib,J.,Bisin,A.&Luo,M.(2017),ÔEarningsinequalityandotherdeterminantsofwealthinequalityÕ,AmericanEconomicReview107(5),593Ð97.Benhabib,J.,Bisin,A.&Zhu,S.(2011),ÔThedistributionofwealthandfiscalpolicyineconomieswithfinitelylivedagentsÕ,Econometrica79(1),123Ð157.Benhabib,J.,Bisin,A.&Zhu,S.(2016),ÔThedistributionofwealthintheblanchard-yaarimodelÕ,MacroeconomicDynamics20(2),466Ð481.Cagetti,M.&DeNardi,M.(2006),ÔEntrepreneurship,frictionsandwealthÕ,JournalofPoliticalEconomy114(5),835Ð870.Carroll,C.,Slacalek,J.,Tokuoka,K.&White,M.N.(2017),ÔThedistributionofwealthandthemarginalpropensitytoconsumeÕ,QuantitativeEconomics8(3),977Ð1020.Castaneda,A.,DiazGimenez,J.&Rios-Rull,J.V.(1998),ÔExploringtheincomedistributionbusinesscycledynamicsÕ,JournalofMonetaryEconomics42(1),93Ð130.Castaneda,A.,DiazGimenez,J.&Rios-Rull,J.V.(2003),ÔAccountingforearningsandwealthinequalityÕ,JournalofPoliticalEconomy111(4),818Ð857.25

MatchingtheWealthDistributionLeeTyrrell-HendryCivale,S.,Diez-Catalan,L.&Fazilet,F.(2017),Discretizingaprocesswithnon-zeroskewnessandhighkurtosis.Crawford,R.,Innes,D.&OÕDea,C.(2016),ÔHouseholdwealthingreatbritain:Distribution,compositionandchanges2006-12Õ,FiscalStudies37(1),35Ð54.DeNardi,M.(2004),ÔWealthandintergenerationallinksÕ,ReviewofEconomicStudies71,743Ð768.Deaton,A.&Paxson,C.(1994),ÔIntertemporalchoiceandinequalityÕ,JournalofPoliticalEconomy102(3),437Ð467.Eggertsson,G.B.,Mehrotra,N.R.&Robbins,J.A.(2019),ÔAmodelofsecularstagnation:TheoryandquantitativeevaluationÕ,AmericanEconomicJournal:Macroeconomics11(1),1Ð48.Gabaix,X.(2009),ÔPowerlawsineconomicsÕ,AnnualReviewofEconomics1,255Ð294.Guvenen,F.,Karahan,F.,Ozkan,S.&Song,J.(2021),ÔWhatdodataonmillionsofu.s.workersrevealaboutlife-cycleearningsrisk?Õ,Econometrica.Huggett,M.(1993),ÔTherisk-freerateinheterogeneous-agentincomplete-insuranceeconomiesÕ,JournalofEconomicDynamicsandControl17(5-6),953Ð969.Huggett,M.(1996),ÔWealthdistributioninlife-cycleeconomiesÕ,JournalofMonetaryEconomics38(3),469Ð494.Huggett,M.,Ventura,G.&Yaron,A.(2006),ÔHumancapitalandearningsdistributiondynamicsÕ,JournalofMonetaryEconomics53(2),265Ð290.Huggett,M.,Ventura,G.&Yaron,A.(2011),ÔSourcesoflifetimeinequalityÕ,AmericanEconomicReview101(7),2923Ð2954.26

MatchingtheWealthDistributionLeeTyrrell-HendryInamura,Y.(2006),Estimatingcontinuoustimetransitionmatricesfromdiscretelyobserveddata,BankofJapanWorkingPaperSeries7,BankofJapan.Israel,R.B.,Rosenthal,J.S.&Wei,J.Z.(2001),ÔFindinggeneratorsformarkovchainsviaempiricaltransitionmatrices,withapplicationstocreditratingsÕ,MathematicalFinance11(2),245Ð265.Kaplan,G.,Moll,B.&Violante,G.L.(2018),ÔMonetarypolicyaccordingtohankÕ,AmericanEconomicReview108(3),697Ð743.Kaplan,G.&Violante,G.L.(2014),ÔAmodeloftheconsumptionresponsetofiscalstimuluspaymentsÕ,Econometrica82(4),1199Ð1239.Khieu,H.&Walde,K.(2018),Capitalincomeriskandthedynamicsofthewealthdistribution,IZADP11840,IZAInstituteofLaborEconomics.Krusell,P.,Mukoyama,T.&Süahin,A.(2010),ÔLabour-marketmatchingwithprecautionarysavingsandaggregatefluctuationsÕ,TheReviewofEconomicStudies77(4),1477Ð1507.Krusell,P.&Smith,A.A.(1998),ÔIncomeandwealthheterogeneityinthemacroeconomyÕ,JournalofPoliticalEconomy106(5),867Ð896.Kuhn,M.&Rios-Rull,J.V.(2016),2013updateontheusearnings,income,andwealthdistributionalfacts:Aviewfrommacroeconomics,QuarterlyReview1,FederalReserveBankofMinneapolis.Lise,J.(2012),ÔOn-the-jobsearchandprecautionarysavingsÕ,TheReviewofEconomicStudies80(3),1086Ð1113.Luetticke,R.(2021),ÔTransmissionofmonetarypolicywithheterogeneityinhouseholdportfoliosÕ,AmericanEconomicJournal:Macroeconomics13(2),1Ð25.Quadrini,V.(2000),ÔEntrepreneurship,savingandsocialmobilityÕ,ReviewofEconomicDynamics3(1),1Ð40.27

MatchingtheWealthDistributionLeeTyrrell-HendryRachel,L.&Summers,L.H.(2019),Onfallingneutralrealrates,fiscalpolicy,andtheriskofsecularstagnation,Brookingspapersoneconomicactivity.Ramsey,F.P.(1928),ÔAmathematicaltheoryofsavingÕ,TheEconomicJournal38(152),543Ð559.Stachurski,J.&Toda,A.A.(2019),ÔAnimpossibilitytheoremforwealthinheterogeneous-agentmodelswithlimitedheterogeneityÕ,JournalofEconomicTheory182,1Ð24.28

MatchingtheWealthDistributionLeeTyrrell-HendryAppendix:EstimatedtransitionmatricesFigure6:EstimatedUKproductivitystatesanddiscretetimeannualtransitionmatrix29

MatchingtheWealthDistributionLeeTyrrell-HendryFigure7:EstimatedUKproductivityandcontinuoustimequarterlytransitionmatrix30

MatchingtheWealthDistributionLeeTyrrell-HendryFigure8:EstimatedUSproductivityandcontinuoustimesemi-annualtransitionmatrix31

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryUniversityofEdinburghSeptember2022(Clickhereforlatestversion)AbstractIexploreoptimaleducationsubsidiesandprogressivityoflabourtaxesinamodelwithstochastichumancapitalaccumulationandincompletemarkets,endogenouslaboursupplyandaneducationchoicemodelledasastoppingtimeproblem,whereagentschooseanoptimalnumberofyearstostudybeforestartingwork.InapurelyanalyticalBaselinemodelwithtightborrowingconstraintsonstudents,whichleadstoano-tradeequilibriumwithoutsavings,thegovernmentpaysforeducationviatransferstostudentsorÐequivalentlyÐviagrantstouniversities.Thesocialwelfare-maximisingpolicyfeaturesgenerousstudenttransfersandhighlyprogressivelabourtaxes,muchmoresothancurrentlyseenintheUSorEurope.Thisresultisrobusttomyriadextensions,includingaQuantitativemodelwithrelaxedfinancialfrictionswherestudentscanborrowtofinancetheireducation,andwherehencetheequilibriumfeaturesextensiveprecautionarysavingbyworkers.1

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendry1IntroductionThispaperisconcernedwithtwokeymacroeconomicandpublicpolicyquestions:1.Howshouldagovernmentcalibrateredistributivefiscalpolicytomaximisewelfare,weighingthebenefitsofredistribution,socialinsuranceagainstadverseshocksandrevenuegeneration,againstthedeleteriouseffectsofprogressivetaxationonincentivestoworkandtoaccumulatehumancapital?2.Whatmannerandmagnitudeofpolicytofinancehighereducationbestoptimiseswelfare?a)market-settuitionfeeswithtransferstostudents;b)feecaps/subsidiestouniversities(setbyassumptiontoensureequilibriumintheschoolingmarketandavoidrationingofeducation);orc)studentloans,potentiallyalongsidetransfersThesequestionshavebeentackledbefore,butmuchoftheexistingliteraturethatexplicitlyfocusesoneducationandoptimalpolicyprimarilyfeatureseitheratime-allocationproblem,ˆlaBen-Porath(1967),forexampleBenabou(2002),oranotherwisehighly-stylisededucationchoice,forexampleHeathcoteetal.(2017).Theseapproachesalternatelymisstwoimportantfeaturesofeducationalchoice:(1)thatitisoftenaone-timechoice,anditcanbedifficultÐandisindeedrareÐforpeopletore-enterhighereducationlaterinlife;and(2)educationhasrealresourceandtimecoststhatmustbepaidforoutoftheusuallylimitedresourcesavailabletoprospectivestudents.TheformerapproachthenpotentiallyoverstatestheabilityofworkerstoinsurethemselvesagainstadverseshocksÐobsolesenceoftheirhumancapitalÐbyre-enteringeducation,andthusunderstatesthewelfaregainsfrompublicinsuranceviaredistributivetaxation.Onecanviewthisasthehumancapitalcounterpartofthewell-knownbugofsimpleAiyagari-styleincompletemarketsmodels:thathouseholdsengageincounterfactuallyextremepreacutionarysavinginphysicalcapitalthatrendersthewelfareeffectsofadverseshocksÐandthusthewelfarebenefitsofpublicinsuranceÐtrivial;seeforexampleAiyagari(1994),Krusell&Smith(1998),Kruselletal.(2009)andKruselletal.(2010).InessenceItaketheoppositeextremeinthispaper:Itreateducationasastoppingtimeproblem,2

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendrywhereitisimpossibletore-entereducationafterleaving,soÒprecautionaryeducationÓisonlypossibleonce,beforestartingwork.ThesecondapproachÐthatassumeslimitedresourceandopportunitycostsofeducationandhenceassumesawayanyfinancialfrictionsthatmaylimitaccesstoeducationÐlikelyunderstatesthewelfaregainsfromredistributingresourcestostudents.InthefirstpartofthispaperIagaintaketheoppositeextremethatstudentsarecompletelyunabletoborrow.Irelaxthissubsequently,andallowstudentstoborrowuptoaÐpotentiallylargeÐexogenouslimit.Toanswerthesequestionsandaddresstheselimitationsoftheexistingliterature,Idevelopasuiteofincompletemarketsmodelswitheducationalchoicemodelledasanoptimalstoppingproblem.Agentsarebornasstudents,accumulatehumancapitalwhilestudyingandthenoptimallychoosewhentograduate;thereafter,agentsalsochooselaboursupplyandaccumulatehumancapitalwhileworking.IntheBaselinemodel,allagentsareex-antehomogeneousandfaceconstantmortalityrisk;moreover,Imakeassumptionsthatdeliveranequilibriumwithnotradeinbonds,soallagentsconsumetheirincome.Bothstudentsandworkersfaceuninsurable,permanentidiosyncraticshockstotheirhumancapital,althoughwhenstudentsfacenoriskIamabletoderivefullyanalyticalresultsforthevaluefunctions,optimalchoices,equilibriumprices,stationarydistributionsofhumancapitalandaggregatesocialwelfare.ThisBaselinemodelisnotthatrealistic,inparticularitimplieshouseholdsdonotinpracticeinsurethemselvesagainstshocksotherthanbystayinginschoollonger,thuspotentiallyoverstatingthewelfarebenefitsofsubsidisingeducationandofprogressivetaxation.Moreover,itprecludesthemainsourceoffinancingforhighereducationinmostAnglophonecountries:studentloans.However,thesimplemodeldoesclarifysomeofthemechanismsofthepurelyquantitativemodel,whichIsubsequentlyintroduce,andwhichdoesallowstudentstoborrowandfeaturesprecautionarysavinginequilibrium.Ifindthatoptimalpolicyfeatureshighlyprogressivetaxes,farmoresothanintheUSorEuropetoday,aswellasgeneroustransferstostudents.Thesepolicyconclusionsarerobust3

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendrytoabroadrangeofextensionsovertheBaselinemodel.InFigure1,IsummarisetheoptimalpoliciesintheBaselinemodel,severaldiverseextensions,andthefullQuantitativemodel.TheBaselinemodelisrepresentedbythebluedot,andsomeindicativevariationsonitinlightblue;theanalyticalextensionsthatexploreforexampletheeffectsofdecreasingreturnstoscale,ex-anteheterogeneityinability,oradifferentlife-cycleprocesswhereagentsage,arecolouredorange;andthefullQuantitativemodelisinred.InthisBaselinemodelwithnoborrowingbystudentsallowed,andhencenotradeinbondsamongworkerseither,thegovernmentmustfinanceeducationviadirecttransferstostudentsor,equivalently,viagrantstouniversities.Therearelargewelfaregainsfromdoingso:optimalpolicyfeaturesgeneroustransfers,fundedbyhighlyprogressivetaxes.Thegainsfromtransferscomeinpartfromovercomingtheonerousfinancialfrictionfacingstudents.Thegainsfromprogressivetaxationcomebothfromeffectivelyinsuringworkersagainstrepeatedbadshocks,andfromredistributingfromrichtopoor,andthesebenefitstoagreatextentoutweighthecostsofdisincentivisingstudyandworkwhenlaboursupplyisplausiblyelastic.Toshowthis,Ifirstshutdownidiosyncraticriskamongworkers,whichimpliesonlymodestlylessgeneroustransfersandlessprogressivetaxesareoptimal;thenIshutdowngrowthinhumancapitalwhileworking,sothereisnoinequalityamongworkers,whichimpliessubstantiallylower(thoughstillpositive)transfersandregressivelabourtaxesareoptimal;inthatcaseprogressivetaxessimplydisincentiviseeducationandworkwithoutanybenefit.Whenthereturnstoschoolingarerisky,thebenefitsfromtransferstostudentsaresmaller,becausesomestudentsgetlittleoutofschoolingyetarestillincentivisedtoattend,sooptimaltransfersareslightlylower.ThemaindownsideoftheBaselinemodelisthatthefinancialfrictionimposedonstudentsisarguablycounterfactuallystrict:studentsdotypicallyborrowtofinancetheireducation(althougharguablythisisinmanycasesonlypossibleduetogovernmentintervention).Aside-effectofthis(andsomeotherassumptionsdiscussedinthebodyofthetext)isthattheequilibriumoftheBaselinemodelfeaturesnotradeinbonds,sodespiteinprinciplesome4

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryworkerswishingtoborrowtosmoothconsumption,orotherswishingtoinsurethemselvesagainstrisk,interestratesadjustsothatnoneinpracticedoso.IntheQuantitativemodel,Irelaxtheborrowingconstraintonstudents,whichmeansnotonlythatworkersmustbuildwealthtopayofftheirdebts,butthatinequilibriumworkersdoinfactborrowtosmoothconsumptionandsaveinprecautionagainstrisk,whichthusgivesrisetomoreinteresting,diverseandperhapsrealisticbehaviourattheindividuallevel,andadifferentoutcomeattheaggregatelevel.Nevertheless,optimaltransfersintheQuantitativemodelÐthoughsmallerthanintheBaselinemodelÐarestillcomfortablypositive.Andletmeemphasisethateventhoughstudentsareabletoborrowtofinanceseveralyearsofschoolingandconsumption,thesociallyoptimalgovernmentpolicyisstilltoentirelypayforallstudentsÕeducation,plusalittleextra.Optimallabourtaxesalsoremainhighlyprogressive,muchmoresothanthecurrenttaxschedulesintheUS(shownbythedashedlineinthefigurebelow)orEurope.Figure1:SummaryofoptimalpoliciesincoremodelsandextensionsGiventhecoreofthepaperistofindsomeoptimaldegreeofgovernmentintervention,letmeoutlinethejustificationsforgovernmentintervention,andthelimitationsIplaceonthe5

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendrypossibleformsofintervention.Therearetwojustificationsforinterventioninthemodel,relatedtoequityandefficiency:1.Equity.GovernmentsinthemodelareassumedtooperateunderaUtilitariansocialwelfarefunction,soregardlessofanyinefficienciespresent,thegovernmentmaystillwishtointervenetotransferresourcestolesswell-offmembersofsociety,betheypoorerworkersorstudents.InthisregardthepapercontrastswithBenabou(2002),forexample,whoprimarilyfocusesoninterventiontopursueefficientallocations,leavingequitytooneside.2.Efficiency.Therearetwoformsofinefficiencypresentinthemodel:(a)Incompletemarkets.Agentsareunabletofullyinsurethemselvesagainstidiosyn-craticrisks,bothasworkersÐwhereidiosyncraticshockstooneÕshumancapitaldirectlyaffectoneÕswageincomeÐbutalsoasstudents,whereidiosyncraticshockstohumancapitalaffectoneÕsfuturewageincomeandhencetheoptionvalueofgraduating.(b)Borrowingconstraints.IntheBaselinemodel,studentsarecompletelydisallowedfromborrowing;evenwhenborrowingisallowed,thereisstillalimit.Sincestudentsmustpayfortheireducationupfront,thisfrictionisparticularlyconstricting.Financinghighereducationpubliclythroughtransferstostudentsorgrantsforeducationalinstitutionscanthusservetoalleviatetheborrowingconstraint.Idonothoweverallowthegovernmentfreereigntointerveneinmarkets.IrestrictthegovernmentÕsroletothatofwritingtheÒrulesofthegameÓ:designingthetaxandtransferssystemwithconstantpoliciesinordertomaximiseaggregatesteadystatewelfare,measuredinaUtilitariansense.IfurthermoretaketheRamsey(1927)approachinrestrictinggovernmenttaxesandtransferstoaspecificparametricclass:anisoelastictaxscheduleforworkers(alsousedinforexampleHeathcoteetal.2017recently)andhomogeneous,lump-sumtransferstostudents.Ialsoallowfortuitionfeecaps,withsubsidiestouniversitiesthendetermined6

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryendogenouslytobalancesupplyanddemandintheschoolingmarket,butIshowtheseareinfactequivalenttotransfersinthissimplifiedsetting.Idonotallowthegovernmenttoengageindirectlump-sumtransfersbetweenworkers.IdonottaketheMirrlees(1971)approachofdesigninganincentive-compatibletaxsystemforaninformation-constrainedgovernment.AndIdonotexploreeitherthefirst-bestortheconstrainedefficientsolutionofasocialplanner(asinD‡vilaetal.2012andNu–o&Moll2018underincompletemarkets),insteadfocusingontheoptimalpolicyregimeconditionalonindividualhouseholdsoptimisingseparatelyandtransactingfreely.Literaturereview.Thereisarichliteratureincorporatingendogenoushumancapitalaccumulationintomodelswithincompletemarkets.NotablecontributionsincludeHuggettetal.(2006),(2011),whichexplorelife-cyclehumancapitalaccumulationinaBen-Porath(1967)modelwithidiosyncraticabilityinordertodiscusstheextenttowhethertheabilityagentsarebornwith,orthehumancapitaltheyaccumulateduringtheirlifetimes,explainsthedistributionofincomeintheUS,buttheysaylittleaboutpolicyorwelfare.Alonetal.(2020)exploretheeffectofstudentdebtonon-the-jobhumancapitalaccumulationinacombinationBen-PorathandRoymodel,whichtheytaketothedata.Theyfindthatthosewithhigherstudentdebtearnmoreinitially,buttheirreturnstoexperiencearelowerinsubsequentyears.Theyarguethatcreditconstrainedindividualsselectintooccupationswithlessscopeforon-the-jobhumancapitalaccumulation.Theliteratureonoptimaltax&transferpolicyisfartoobroadinscopetocoverhere.InsteadIhighlightsomekeypapersthatfeatureeducationorskillacquisitionprominently.OnelineofresearchexploresparentalinvestmentinchildrenÕseducation,inparttoexplainthepersistenceofearningsacrossgenerations,andalsotodiscussitsimplicationsforoptimaleducationpolicy,usinganoverlappinggenerationsstructurewithexplicitparentalaltruismoverchildrenÕswelfare,startingwithLoury(1981),laterCaucutt&Kumar(2003)andRestuccia&Urrutia(2004),andrecentlyAbbott(2022).Iabstractawayfromthesefeaturesandinsteadconcernmyselfmorewithhighereducationchoiceswhereagentshavemore7

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryautonomyandtypicallyrelylessonparentalresources.Thismayoverlookanimportantchannelforinvestmentinhumancapital;however,atleastRestuccia&Urrutia(2004)andAbbott(2022)findthewelfaregainsfromsubsidiestoparentalinvestmentincollege-agechildrentobeminimal.TheslightlymoredistantlyrelatedworkofKrueger&Ludwig(2013)and(2016)alsofeaturesparentalaltruismandinvestmentineducation,buttakesahighlyquantitativeapproachandisthematicallyclosertomypaper,exploringoptimalprogressivityoftaxationandsubsidiesforeducation,thoughtheirapproachisdistinctfrommine.Theyalsofindhighlyprogressivetaxesandgeneroustuitionsubsidiesareoptimal,butmoderatedsomewhatbythewelfarecostsoftransitioningtosuchapolicy.Analyticalapproachesthatfeatureno-tradeequilibriaorotherwisedispensewithsavingsareacommonfeatureinthisliterature,eveninthepresenceofuninsurablerisks,sobyconstructionmayoverstatethewelfaregainsfromsocialinsurance.Benabou(2002)exploresthewelfareeffectsofeducationfinancepolicyunderincompletemarkets,modellingschoolingchoiceasaconvextimeallocationproblemratherthanastoppingtimeproblem,ˆlaBen-Porath(1967),whichasdiscussedmayunderstatethewelfarecostsofhumancapitallosses.Heathcoteetal.(2017)exploreoptimalprogressivityoftaxation,inamodelthatfeaturesbothinsurableanduninsurableidiosyncraticrisk,endogenouslaboursupply,andastylisededucationalchoiceatthestartofagentsÕlives.Mypaperexploressimilarthematicgroundbutinaverydifferentframework.Inparticular,Itreateducationverydifferently:intheirmodel,skillinvestmenthasnoresourcecostoropportunitycost,onlyautilitycost,whichasdiscussedabovehaslimitations.Aswellascapturingtheseaspectsofeducation,Icanalsodiscussthewelfareeffectsofstudentloansandpoliciesrelatingtoeducationfinance,whichhavenoroleintheirpaper.Furthermore,inthebodyofthepaperIcomparetheirfindingsforoptimalpolicywithmyown,andnotethatsubtlemodellingchoicesintheirpaper,suchasaflatlife-cycleearningsprofile,limittheextentofintergenerationalinequalityandhencethewelfaregainsfromredistributionviaprogressivetaxation,eveniftheydocapturethewelfaregainsfromsocialinsuranceagainstuninsurableprivateriskthatprogressivetaxescan8

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryprovide.Inequalityintheirmodelarisesmoreduetounequalskillacquisition;progressivetaxationdampensthisvariation,butatthecostofloweringoutput.Moreover,boththesepapersonlyexploreno-tradeequilibria,soaremuteontheimplicationsofprecautionarysavingbyworkersorborrowingbystudentsforwelfareandoptimalpolicy.InthefirsthalfofmypaperIalsotaketheno-tradeanalyticalapproach,butinthesecondhalfIallowstudentstoborrow,whichresultsinprecautionarysavingbyworkersinequilibriumtoclearbondmarkets.IshowthatthisdoesnotchangethecorepolicyconclusionsoftheBaselinemodel:thatprogressivetaxationandgeneroustransferstostudentsenhancewelfare.Treatingeducationasastoppingtimeproblemisrelativelyuncommonintheliterature,butnotwithoutprecedent.Card(2001)wasamongthefirsttousethisapproach,albeitforthevastlydifferentpurposeofestimatingthereturnstoschooling.Hogan&Walker(2003),(2007)isperhapsmethodologicallythemostsimilartomypaper:usingapurelyanalyticalapproachtheydiscusstheeffectsofpolicyonschoolingdecisionsinpartialequilibrium,butdonotexplicitlyconsiderdistributionalmattersorthewelfareeffectsofpolicy.Mypaperdevelopstheseaspectsfurther,aswellassupplementingtheanalyticalworkwithaquantitativeapproachtoexploretheroleofstudentdebtandprecautionarysaving.Mellior(2021)maybethemostrecentpapertoexplorethewelfareconsequencesofdifferenteducationfinancingpolicyregimesunderincompletemarkets,inapurelyquantitativepaperthatusesasimilarapproachtothesecondhalfofmypaper.However,theeconomicenvironment,modellingchoicesandthepolicyregimesexploredarequitedifferent,somyanalysisshouldbeseenascomplementinghis,ratherthanextendingit.Forexample,inhispaperagentsstudyforthechancetograduatewithadegreeandajob,whichgivesthemahigher(fixed)wageandlowerchanceofbecomingunemployed;whereasinmypaperagentsaccumulatehumancapitalwhilestudying(andmoreslowlywhileworking).Moreover,hispaperisverycarefultomodelprecisefeaturesofreal-worldstudentloanpolicies,andcomparesthewelfareconsequencesofeach,findingthatwheneducationisaslengthyandcostlyasitiscurrently,government-guaranteedincome-contingentloanscoupledwithtuitionfeesubsidies9

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryaretheoptimalpolicy.Mypapersimplifiesbutbroadensthepolicyspace,exploringoptimalprogressivityoftaxationandgenerosityofdirecttransferstostudents,alongsidestudentloansandfeesubsidies.2BaselineModel:No-TradeEquilibriumThemodelhasanoverlappinggenerationsstructureincontinuoustime,witheducationalchoicemodelledasastoppingtimeproblem:agentschooseearlyintheirlifetoschoolforacertainnumberofyears,S,andthenstartworking.Aftergraduation,workerschooselaboursupply,n,onlyandcannotre-entereducation.Inprinciple,workersandstudentscanalsochoosehowmuchtoconsumption,c,andsave,úa,however,intheBaselinemodelImakesomeassumptionstodeliverano-tradeequilibriumwithtractableanalyticalresults.First,Iassumeeveryoneisbornwithzerowealth(a0=0).Forstandardintertemporalsubstitutionreasons,thisimpliesstudentswouldliketoborrowÐprovidedthetransfertheyreceivetopayfortheireducationisnottoolargeÐbut,secondly,Iassumetheycannotdoso.Thustheywillcarryzerowealththroughtograduation.Thereafter,inprincipleIallowworkinghouseholdstoborroworsave.However,because,thirdly,thereisnocapitalandbondsareinzeronetsupply,and,fourthly,becausetheirincomeprocessfeaturesstochasticgrowthwithpermanentÐnotmean-revertingÐshocks,theinterestratewilladjusttoclearbondmarketssothattheydonotwishtodoso,implyingtheysimplyconsumetheirpost-taxincome.Thus,despiteworkersÕdesiretoengageinprecautionarysavingtoself-insureagainsttheidiosyncraticrisktheyfacefromtheirstochasticaccumulationofhumancapital,h>0,inpracticetheydonotdoso.HouseholdsInthesimplemodel,householdsareex-antehomogeneousandfacetwosourcesofidiosyncraticrisk:stochastichumancapitalaccumulation,bothwhileastudentandwhileworking,and10

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryuncertainlifetimes.Agentsdierandomly,withconstantmortalityrate,⁄>0,ˆlaBlanchard(1985);inanextensionIexploreafixed,finitelifetime.Humancapital,h,isaccumulatedstochasticallythroughschoolingandworkexperience:dh=Y___]___[1µsh—s≠”sh2dt+‡shdWifschoolingµwhdt+‡whdWifworkingwhereµiØ0,—sÆ1and”iØ0.When—s<1,therearedecreasingreturnstolearning;andwith”i>0inaddition,thereisamaximumlevelofhumancapitalattainable,h=1µs”s211≠—s,attainedintheFaustianlimitifonestaysinschoolforever.When—s=1,asitisforworkers,thehumancapitalprocessisageometricBrownianmotionwithdriftµs≠”s,andonecanderiveanalyticalsolutionsforthevaluefunction,thoughforstudentsthiscaseissomewhatpathologicalandleadstoextremepolicyimplications,soIdonotfocusonit.When‡s=0also,onecanderiveanalyticalexpressionsforthedistributionsofhumancapitalandhencetheequationsdeterminingtheequilibrium,whichImakeuseoflater.Householdincomeconsistsofcapitalincome,ra,andtheirpost-taxandtransferlabourearnings,whichforstudentsissimplytheirtransfernetoftuitionfees,T=÷T≠F,andwhichforworkersisanisoelasticfunctionofpre-taxearnings,·0(wnh)1≠·1,wherepre-taxearningsaretheproductofapiece-ratewage,w,hoursworked,n,andhumancapital,h.ThistaxfunctiondatesbackatleastasfarasFeldstein(1969)andhasbeenusedfrequentlysincethen,includinginHeathcoteetal.(2017)recently.·1œ[≠1,1]isameasureoftheprogressivityoftaxes:when·1=0,taxesareproportionalwithtaxrate1≠·0;when·1>0,taxesareprogressive(andregressivewhen·1<0),withtransferstolowerincomehouseholds,and·0>0determiningtheleveloftaxationrequiredtobalancethebudgetagainstwhateverothergovernmentspendingthereis.ForsimplicityIassumethereisnoothergovernmentspendingbesidestransferstoworkers,studentsoreducationalinstitutions.Laboursupplyisalsoendogenous,allowinghouseholdsasecondmarginofadjustmentinresponsetoprogressivelabourtaxationÐreducingthenumberofhourstheyworkÐin11

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryadditiontospendinglesstimeineducation.Withendogenouslaboursupply,onefacesthedelicatequestionofpreferencespecification.Ifincomeeffectsaretoostrong,forexamplewithseparablepreferenceswithCRRAutilityofconsumptionwithriskaversion“>1,asinMaCurdy(1981),hoursworkedisnegativelyrelatedtoandconvexinhumancapital.Asthisgivesthebroadlycounterfactualresultthathigherincomepeopleworkfewerhours,thisseemslikeanimplausiblecalibration.Ontheotherhand,weakincomeeffectsthatgenerateaplausiblypositiveandconcaverelationshipbetweenhoursworkedandhumancapitalrequireimplausiblylowriskaversion,“<1.Ithusfocussolelyonbalancedgrowthpreferenceswhereincomeandsubstitutioneffectscancel,whichinano-tradeequilibriumwithoutsavingsimpliesconstantlaboursupplyirrespectiveofhumancapital.Inparticular,theutilityfunctionthatIuseisarefinementofthespecificationofKingetal.(1988),andwasalsousedbyTrabandt&Uhlig(2011),andwhichnestsbothextremesofinelasticlaboursupplywithCRRAutilityandtheinfinite-elasticityCobb-Douglasspecification,whileretainingbothconstantrelativeriskaversion,“>0,andconstant,finiteFrischelasticityoflaboursupply,ÏØ0,withoutrestrictingeitherparameter:u(c,n)=Y___]___[c1≠“v(n)≠11≠“wherev(n)=11≠Â(1≠“)n1+1/Ï1+1/Ï2“if“”=1ln(c)≠Ân1+1/Ï1+1/Ïif“=1whereÂØ0isashifterensuringlabourisabad.WhenÂ=0orÏæ0,laboursupplyisinelasticandutilityisstandardCRRAinconsumptiononly.WhenÏæŒÐandÂ=≠11≠“when“>1orÂ=11≠“when“<1ÐtheutilityfunctionreducestoCobb-Douglaspreferences:u(c,n)=c1≠“(1≠Â(1≠“)n)“≠11≠“.TheworkersÕproblem.Workerschooseconsumption,c,hoursworked,n,andsavings,da,tomaximiseutilityovertheirremaininglifetime,subjecttotheirbudgetconstraint,the12

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryexogenousprocessforhumancapitalandanexogenousborrowinglimit,a<0:Vw(aS,hS)=max{ct,nt}tœ[S,Œ)ES5⁄ŒSe≠(ﬂ+⁄)(t≠S)u(ct,nt)dt6s.t.dat=1rat+·0(wntht)1≠·1≠ct2dtdht=µwhtdt+‡whtdWtaØaTheworkersÕproblemcanalsobere-writtenasaHamilton-Jacobi-Bellman(HJB)equation:(ﬂ+⁄)Vw(a,h)=maxc,nu(c,n)+Vwa1ra+·0(wnh)1≠·1≠c2+Vwhµwh+12Vwhh‡2wh2(1)Wecannotinthegeneralcasederiveaclosed-formsolutiontothisHJBequation,soIproceedquantitatively,exceptinsomeimportantspecialcases.Inparticular,intheabsenceofanassetinpositivenetsupply,andwithstudentsunabletoborrow,thereexistsano-tradeequilibriuminwhichallhouseholdsholdsconsumetheirpost-taxincomeandholdnowealth.Forafullerdiscussionofthisno-tradeequilibrium,seetheAppendix.Giventhisno-tradeequilibrium,wecanderiveworkersÕoptimallaboursupplybytheirstaticfirst-ordercondition:un=0,aftersubstitutinginc(h)=·0(wnh)1≠·1fromthebudgetconstraint.Thisyieldsalaboursupplythatisindependentofhumancapitalanddecreasingintheprogressivityoflabourtaxes:n=C(1≠·1)(1+1/Ï)Â“(1+1/Ï)+(1≠·1)Â(1≠“)D11+1/ÏThestudentsÕproblem.Allnewbornsstartasstudentswiththesamelevelofhumancapital,h0=1.Studentsareassumednottobeabletoworkwhilestudying,n=0,sotheironlychoiceishowmuchtosave/consumeandwhentostopeducationandbecomeaworkertomaximiseintertemporalutilitysubjecttotheexogenousprocessforhumancapital13

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryaccumulation:Vs(a0,h0)=maxS,{ct}tœ[0,S)E0C⁄S0e≠(ﬂ+⁄)tu(ct,nt)dt+e≠ﬂSVw(aú(S),hú(S))Ds.t.dat=(rat+T≠ct)dtdht=1µsh—st≠”sht2dt+‡shtdWtnt=0a0=0aØah0=1Theoptimalstoppingtime,Sœ[0,Œ),isassociatedwithathresholdofvaluesofwealthandhumancapital,(aú,hú),outsideofwhichthestudentwillchoosetograduate,andwithinwhichtheywillchoosetostayinschool.DenotebyE(foreducation)theregionof(a,h)-spaceboundedbelowbytheborrowingconstrainta=a(ignoringforamomenttheno-tradeequilibrium)&h=0,andabovebythethresholdcurvehú(aú).For(a,h)œE,studentsprefertostayineducation,sotheirvaluefunctionexceedsthatofworkers;outsidethisregiontheirvaluefunctionsareequal:Vs(a,h)ØVw(a,h)(a,h)œEVs(a,h)=Vw(a,h)(a,h)/œEThestudentsÕproblem,whiletheyremainstudents,with(a,h)œE,canbedescribedbytheHJBequation:(ﬂ+⁄)Vs(a,h)=maxcu(c)+Vsa(ra+T≠c)+Vsh1µsh—s≠”sh2+12Vshh‡2sh2(2)whichholdswithinequalityfor(a,h)/œE.Theseequationsandinequalitiescanbewritten14

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendrymorecompactlyasthefollowingHJBvariationalinequality:(ﬂ+⁄)Vs(a,h)=max;maxcu(c)+Vsa(ra+T≠c)+Vsh1µsh—s≠”sh2+12Vshh‡2sh2,(ﬂ+⁄)Vw(a,h)<(3)Asalludedtoabove,boththeworkersÕandstudentsÕproblemscanbesimplifiedconsider-ablybymakingtheappropriateassumptionstodeliverano-tradeequilibrium.Theorem1.Ano-tradeequilibriumexistsinwhichallagentsconsumetheirincomeandholdzerowealth.Proof.SeeAppendix.Predicatingontheno-tradeequilibrium,wealthcanbedroppedasastatevariablefrombothstudentsÕandworkersÕproblems.Thethresholdatwhichstudentschoosetograduatethusbecomesasinglepoint,húœËh0,h2,whererecallhwasthemaximumattainablehumancapitalifonestaysinschoolforever.Lemma1.IntheBaselinemodelwithano-tradeequilibrium,theworkersÕvaluefunctionis:Vw(h)=Y___]___[1Ÿ(·0(wnh)1≠·1)1≠“v(n)1≠“≠1(ﬂ+⁄)(1≠“)if“”=11ﬂ+⁄Ëln(·0(wnh)1≠·1)≠Ân1+1/Ï1+1/ÏÈ+1≠·1(ﬂ+⁄)2(µw≠‡2w/2)if“=1(4)whereŸ=ﬂ+⁄≠(1≠·1)(1≠“)µw≠12(1≠·1)(1≠“)[(1≠·1)(1≠“)≠1]‡2w,andwherenistheoptimallaboursupply,givenabove.Moreover,thestudentsÕvaluefunctiontakestheform:Vs(h)=u(T)ﬂ+⁄+E(h)whereE(h)satisifiesthefollowingODE:(ﬂ+⁄)E(h)=EÕ(h)1µsh—s≠”sh2+12EÕÕ(h)‡2sh2Proof.PlugtheseintotheworkersÕandstudentsÕHJBequations.15

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryTheparameterŸintheworkersÕvaluefunctionisakindofdiscountratethataccountsforthegrowthandvariabilityofworkersÕfutureincome,andisaninversequadraticfunctionoftaxprogressivity.Note,inthestudentsÕvaluefunction,thefirsttermÐthediscountedutilityfromstudentgrantsnetoffees(T=÷T≠F)Ðrepresentsthevalueofstayingastudentforever,andthesecondtermrepresentstheoptionvalueofgraduating.Asisstandardinstoppingtimemodels,thethresholdhúisdeterminedbythevaluematchingandsmoothpastingconditionsthatthelevelofthestudentsÕandworkersÕvaluefunctionsÐandtheirslopesÐshouldbeequalatthecut-off:Valuematching:Vw(hú)=Vs(hú)≈∆1Ÿ1·0(wnhú)1≠·121≠“v(n)1≠“=u(T)ﬂ+⁄+E(hú)Smoothpasting:Vwh(hú)=Vsh(hú)≈∆1≠·1Ÿ1·0(wnhú)1≠·121≠“v(n)hú≠1=EÕ(hú)(5)Whenthecut-offisbelowthestartinglevelofhumancapital(h0=1),studentswillchoosetostartworkimmediately,sohú=1.Distributionofhumancapital.NowthatwehavesolvedthehouseholdsÕproblem,i.e.solvedforthethresholdlevelofhumancapital,wecansolveforthedistributionofhumancapital,andthedistributionofstudentsandworkers.ThedistributionsofhumancapitalacrossstudentsandworkersisgovernedbytheKolmogorovForwardEquations(KFE):úgs(h)=≠ˆhË1µsh—s≠”sh2gs(h)È+ˆhh512‡2sh2gs(h)6≠⁄gs(h)hœ(0,hú)h0(6)úgw(h)=≠ˆh[µwhgw(h)]+ˆhh512‡2wh2gw(h)6≠⁄gw(h)hœ(0,Œ)hú(7)Thefirsttermsineachequationrepresentthegrowthinhumancapital,thesecondtermsitsvariation.ThefinaltermsrepresenttheÒoutflowsÓ,i.e.thechangeinthemeasureofagentsowingtotheirdeaths.EachgroupalsohasÒinflowsÓofnewagentsatcertainpoints:newbornstudentsattheinitiallevelofhumancapital,h0,andnewworkersatthegraduationthreshold,hú.Inthesteady-state,theseequationswillequal0,givingrisetoastationary16

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendrydistribution.Whenthehumancapitalofstudentsisstochastic,thereistomyknowledgenoclosed-formexpressionforitsstationarydistribution;Idiscussthedeterministiccasewithclosed-formsolutionslater.Supplyside&marketclearingConsumptiongoodsandeducationgoodsaresuppliedbycompetitivefirmsusinglaboursuppliedbyhouseholdsandaconstantreturnstoscaleproductionfunction,fi(Li).Onecanalsoconsidertheslightlymoregeneralspecificationwhereproductionisallowedtofacedecreasingreturns,andwheretheresultingprofitsareassumedtobeeitherpaidtosomeunmodelledfactorofproduction,ordistributedtoforeignownersthatotherwisetakenopartintheeconomy.Thedifferenceisnotqualitativelyimportantfortheequilibriumorpolicyimplications,andcomesattheexpenseoftwoextraequilibriumequations,soIfocusontheCRScaseforsimplicity,butnaturallythiswillmeanthatchangesinpolicyhavenoeffectonrelativeprices:wagesandtuitionfees.IconsidertheDRScaseinanextension.Educationalinstitutions.Educationalinstitutionsproduceschoolingusingtheconstantreturnstoscaletechnology,S=AELE,chargefeesFandreceivegovernmentgrantsGproportionaltotheamountofschoolingsupplied,andchoosetheamountoflabourtoemploytomaximiseprofits.ForsimplicityIassumeworkerswithdifferentquantitiesofhumancapitalareinfinitelysubstitutable,so:ﬁE=maxLE(F+G)AELE≠wLE(8)TheschoolsÕfirst-orderconditionimplieslabourispaiditsmarginalproduct:w=(F+G)AE.Inthefreemarketequilibrium,relativetuitionfees,F,adjustsothat,conditionalonanygrantspaidbythegovernment,totaldemandforschoolingacrossallagentsequalsthat17

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendrysupplied.Oursecondequilibriumcondition:SS©AELE=⁄hú0gs(h)dh©SD(9)Onecanalsoimagineascenariowherethegovernmentscapsorfixestuitionfees,andinsteadsetsgrantsGtoclearthemarketforschooling.Belowwewillseetheseareequivalentanddonotchangetheequilibrium.Thisassumptionthatgrantsclearthemarketwhenfeescannotisconvenient,butoverlooksonepossibleconsequenceoftuitionfeecaps,whichisthatsubsidiesfromgovernmentfallshortofthatneedtoequilibriatemarkets,andhenceeducationisrationed.Iamcurrentlyexploringthisideainseparatework.Firms.Firmsproduceconsumptiongoods,maximisingprofitsinacompetitiveenvironment,operatingapotentiallydifferentCRStechnologytoeducationalinstitutions,butwherethepriceoftheconsumptiongoodisnormalisedto1:ﬁF=maxLFAFLF≠wLF(10)Likewise,firmspaylabourÐwhichisfreetomoveacrosssectorsÐtheirmarginalproduct,implyingthemarginalrevenueproductisequalisedacrosssectors,yieldingproductionefficiencyandalsopinningdowntheequilibriumwageandtuitionfee:w=AFandF=AFAE≠GNote,themodelcanexplaintheriseintuitionfeesintheUSasaformofBaumolÕscostdisease:productivityinthenon-educationsectorrisingrelativetotheeducationsector.Itcannothoweverincorporatedemand-ledrelativepricerises,whereforexamplemoregeneroustuitionfeesubsidies/studenttransfers(orstudentloans)leadtohighertuitionfees;thiswouldrequiredecreasingreturnstoscale,whichIexploreasanextension.EffectivetotallaboursupplyisgivenbythetotalhumancapitalofallworkersÐagain18

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryworkerswithdifferentlevelsofhumancapitalareperfectsubstitutesÐmultipledbytheirworkinghours.Labourmarketequilibriumisthusgivenby:LS©⁄Œ0nhgw(h)dh=LF+LE©LD(11)Bondmarket.Capitalisnotusedinproduction,sohouseholdsÕnetwealthisentirelysavedinbonds,whicharerisk-freeandinzeronetsupply,asinHuggett(1993),withtheinterestrate,r,clearingthemarket:BS©⁄adG(a,h)=0(12)Giventheassumptionsmadeabove,ano-tradeequilibriumprevails,inwhichallagentsholdzerowealth,hencebondmarketclearingholdstrivially.IshowintheAppendixthattheinterestratethatclearsthemarketisr=ﬂ+⁄+“(1≠·1)µw≠12“(1≠·1)(1+“(1≠·1))‡2w(=Ÿ).Government.Thegovernmentreceivestaxrevenuefromlabourtaxes,andredistributesitassubsidiesforstudents,T+F,per-studentgrantsforeducationalinstitutions,G,andpotentiallyalsoastransferstolow-incomeworkersiftaxesareprogressive.Thereisnoothergovernmentspendingorborrowing,sothegovernmentbudgetconstraintisasfollows:Taxrevenue©⁄Œ01wnh≠·0(wnh)1≠·12gw(h)dh=SD(T+F+G)©Studenttransfers(13)wheretheleveloftaxation,·0,balancesthisequation.Socialwelfare.Welfareinthesteadystateofthiseconomyisgivenbytheutilitariansocialwelfarefunction:W©⁄hú0Vs(h)gs(h)dh¸˚˙˝Students+⁄Œ0Vw(h)gw(h)dh¸˚˙˝Workers(14)19

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryStationaryequilibriumandsocialoptimumoftheBaselinemodel.AstationaryequilibriumoftheBaselinemodelisasetofvaluefunctions{Vs(h),Vw(h)},policyfunctions{cs,cw(h),n,hú,S},distributions{gs(h),gw(h)},prices(w,F,r)andgovernmentpolicies(·0,·1,T,G)suchthat:¥Students(2&3)andworkers(1)maximisetheirlifetimeutilitygivenpricesandpolicies;¥Firms(10)andeducationalinstitutions(8)maximiseprofitsgivenpricesandpolicies;¥Thelabourmarket(11),schoolingmarket(9),bondmarket(12)andgoodsmarketallclearandthegovernmentbudgetconstraintholds(13);and¥ThedistributionsofthehumancapitalfoundfromtheKolmogorovforwardequationsofstudents(6)andworkers(7)arebothstationaryovertime.SinceIassumedconstantreturnstoscaleproduction,thewagewandtuitionfeeFarepinneddownbytechnology;therealinterestraterthatclearsthemarketandattainstheno-tradeequilibriumcanbefoundfromtheworkersÕoptimalityconditions;thechoiceofconsumptionishencetrivialÐallagentsconsumetheir(post-tax)income;thechoiceofhours,n,isconstantanddependsonlyontheprogressivityoftaxesandpreferences;andSisisomorphicwithhú.Wethusrequirejusttwoequilibriumequationsintwounknowns,hú(·1,T,G)and·0(·1,T,G),forwhichthevaluematching/smoothpasting/studentHJBequationandthegovernmentbudgetconstraintshallsuffice.Tomyknowledge,theequilibriumhúand·0areuniqueforagivengovernmentpolicy(·1,T,G)(recall·0ispinneddownbythegovernmentbudgetconstraint).Iassumethesepolicyparametersarechosenex-ante.Isearchoverthisparameterspacetofindthesocialoptimum,accordingtothewelfarecriteriaabove(14).Analyticalspecialcase:‡s=0BeforediscussingtheresultsofthefullBaselinemodel,Ifirstconsideraspecialcaseofthemodelabovethatyieldsfruitfulanalyticalresultswherethehumancapitalofstudents20

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryisassumedtobedeterministic,so‡s=0.Withthisassumptionwecanderiveanalyticalexpressionsforthevaluefunctionsofstudentsaswellasworkers,thedistributionsofhumancapitalacrossworkersandstudents,aswellasequilibriumpricesandtaxesandaggregatesocialwelfare.Inparticular,wecannowusethevaluematchingandsmoothpastingconditionstoderiveanimplicitexpressionforthegraduationthreshold,hú.ThefunctionE(h)inthevaluefunctionofthestudentsisstillunknown,butsinceweknowthatatthethresholdthelevelandslopeofthestudentsÕvaluefunctionisthesameasthatoftheworkersÕÐwhichweknowÐwecanderiveanimplicitexpressionforthegraduationthreshold.Notewecouldonlydothiswhen‡s=0becausethesmoothpastingandvaluematchingconditionsdonotprovideanyinformationaboutthesecondderivativeofthevaluefunction,whichappearsinthestudentsÕHJBequationwhen‡s>0.Thiswillserveasourfirstequationdescribingtheequilbrium.Lemma2.Theoptimalgraduationthreshold,hú,isdeterminedbythefollowingimplicitequation:1Ÿ1·0(wnhú)1≠·121≠“v(n)1≠“¸˚˙˝Vw(hú)=1ﬂ+⁄CT1≠“1≠“+1≠·1Ÿhú1·0(wnhú)1≠·121≠“v(n)1µshú—≠”hú2D¸˚˙˝Vs(hú)(15)Proof.UsethevaluematchingandsmoothpastingconditionstoplugtheworkersÕvaluefunction,evaluatedattheoptimalthreshold,intothestudentsÕHJBequation.Forthecasewhen“=1,seetheAppendix.SincethestudentÕsproblemisnowdeterministic,therateatwhichtheyaccumulatehumancapitalisnowgivenbyúh=µsh—≠”h,whichisaBernoullidifferentialequationwithaclosed-formsolution:h(t)=Ëh1≠—0e≠”(1≠—)t+µs”11≠e≠”(1≠—)t2È11≠—.Thususinghú=h(S),wecanfindaclosed-formexpressionfortheoptimalstoppingtime,whichisnowaconstantratherthanadistributionasitisinthestochasticcase,asafunctionofhú:S(hú)=≠1”(1≠—)ln3hú1≠—≠µs”h1≠—0≠µs”4.Goingbacktotheoriginaldefinitionofthevaluefunctionastheexpectationofdiscountedfutureutilityofconsumption,wecanfindasemi-closed-formexpressionforthestudentsÕ21

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryvaluefunction,intermsofthestudentÕscurrenthumancapitalandthethresholdlevel:Lemma3.ThestudentsÕvaluefunctioncanbedescribedintermsoftheircurrenthumancapital,h,andthegraduationthreshold,hú:Vs(h)=u(T)ﬂ+⁄+Ah1≠—≠µs”hú1≠—≠µs”B≠ﬂ+⁄”(1≠—)CVw(hú)≠u(T)ﬂ+⁄D(16)Proof.UsetheintegraldefinitionofVs(h(t))=sSte≠(ﬂ+⁄)(·≠t)u(T)d·+e≠(ﬂ+⁄)(S≠t)Vw(hú);evaluatetheintegralandpluginthevaluesofh(t)andS(hú)foundabove.Asexplainedbefore,thevalueofbeingastudentcomprisestwoterms:thefirstthevalueofbeingastudentforever,thesecondtheoptionvalueofgraduating,previouslydenotedE(h).Thedistributionsofhumancapital.SinceworkersÕhumancapitalevolvesaccordingtoageometricBrownianmotion,itsstationarydistribution,foundfromequation7,hastheformofadouble-Paretodistributionsplitaroundtheinjectionpoint,thecut-offlevelofhumancapital,hú,asdiscussedinGabaixetal.(2016).Lemma4.Thedensityofhumancapitalofworkersis:gw(h)=Y___]___[c1hhú2≠’≠≠1forh<húc1hhú2≠’+≠1forh>húwhere’≠and’+aretherootsof12‡2w’2+1µw≠12‡2w2’≠⁄=0andwherec=’+’≠(’≠≠’+)hú3h1≠—≠µs”sh1≠—0≠µs”s4⁄”s(1≠—s).Thedensityofhumancapitalofstudentsis:gs(h)=⁄µsh—s≠”sh3h1≠—s≠µs”sh1≠—s0≠µs”s4⁄”s(1≠—s)hœ[h0,hú)Proof.ThatthedensityofageometricBrownianmotionwithastabilisingforce(here:death)22

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryfollowsadoubleParetodistributioniswell-knownintheliterature,e.g.Gabaixetal.(2016),andcanbederivedbyguessingthatthedensityhastheformofaParetodistribution,ch≠’≠1,andevaluatingthestationaryKolmogorovforwardequation.Thatwillyieldthequadraticequationabove,verifyingtheguessanddeterminingthetailparametersofthetwoParetodistributions,whicharetherootsofthequadratic.Theconstantcispinneddownbytheconditionthatthesumofthetwodistributions(i.e.thetotalmeasureofallworkersandstudents)equals1:sŒ0gs(h)+gw(h)dh=1.Moreover,thetotalmeasureofworkersissimplythosewholivelongenoughfortheirhumancapitaltoreachthegraduationthreshold,hú,whichisexponentiallydistributedduetothePoissondeathrate.Thusthetotalmeasureofworkersise≠⁄Sandthatofstudents1≠e≠⁄S:⁄Œ0gw(h)dh=e≠⁄S=Qahú1≠—≠µs”sh1≠—0≠µs”sRb⁄”s(1≠—s)Hencewecanfindcbyevaluatingtheintegralontheleftoverthetwopartsofthedensityandsettingitequaltotheexpressionontheright.Thedensityofstudentsoverhumancapitalcanalsobefoundbyachangeofvariableonthedensityofstudentsofaget,whichisexponentiallydistributed,⁄e≠⁄t.ThisdensitycanbecheckedbysubstitutingitintothestudentsÕKolmogorovForwardequation.Thevaluefunctionsandstationarydistributionsofstudentandworkersforthisspecialcaseareplottedbelow:23

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryFigure2:ValuefunctionsandstationarydistributionforworkersandstudentsWiththetotalmeasureofstudentsandthestationarydistributionofworkersÕhumancapitalpinneddown,wecanalsonowevaluatethegovernmentbudgetconstraintexplicitly,yieldinganexpressionfor·0,whichservesasoursecondequilibriumequation:Lemma5.Theparametergoverningtheleveloftaxes,·0,isdeterminedbythefollowingimplicitequation:chú;wnhú511≠’≠≠11≠’+6≠·0(wnhú)1≠·1511≠·1≠’≠≠11≠·1≠’+6<¸˚˙˝Taxrevenue=!1≠SD”(T+F+G)¸˚˙˝Studentstransfers(17)wherec=’+’≠(’≠≠’+)hú3hú1≠—≠µs”sh1≠—0≠µs”s4⁄”s(1≠—s)andSD=1≠3hú1≠—≠µs”sh1≠—0≠µs”s4⁄”s(1≠—s).Proof.Integrateontheleft-handsideofthegovernmentÕsbudgetconstraint(13)overthedensityofworkers,andintegratetheright-handsideoverthedensityofstudents.CalibrationIntheBaselinemodel,Icalibratetheparametersasfollows:24

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryTable1:CalibrationofstructuralparametersinBaselinemodelSymbolParameterBaselineﬂDiscountrate0.01⁄Mortalityrate1/45“Coefficientofrelativeriskaversion2ÏFrischelasticityoflaboursupply0.5ÂLabourdisutilityshifter1µwDriftofhumancapitalwhileworker0.0125µsDriftofhumancapitalwhilestudent0.3—sDecreasingreturnstolearninginschool0.2”sDepreciationofhumancapitalwhilestudent0.18‡wVolatilityofhumancapitalwhileworker0.1‡sVolatilityofhumancapitalwhilestudent0AFFirmproductivity1AEEducationproductivity1Mycalibrationsofthediscountrateandcoefficientofrelativeriskaversionarerelativelystandard.Ichooseamortalityrateof1/45,tomatchtheaverageworkinglife.IsettheFrischlaboursupplyelasticityto0.5,incommonwiththemicroeconometricliterature.IsetthelabourshiftertoÂ=1,whichnormaliseslaboursupplyto1whentaxesareproportional,evenwithpositiveelasticity,anditisalsothevalueittakesintheCobb-Douglaslimitwhen“=2,sinceÂ=≠11≠“=1.Inormaliseproductivityoftherepresentativefirmandeducationalinstitutionsto1forsimplicity.ThisresultsinatuitionfeeequaltothewageofanÒunskilledÓfull-timeworkerÐmeaningonewiththeinitiallevelofhumancapital,h0ÐwhichisatleasttherightorderofmagnitudeintheUS,forexample.TheparametersoftheworkersÕstochasticprocess,µwand‡w,combinedwiththemortalityrate,⁄,pindowntheParetotailsofthepre-taxincomedistribution.Ithereforecalibratetheseparameterstomatchtheright-tailoftheUSincomedistribution,whichhas’+ƒ1.5,andoneadditionalmoment,theestimatedgrowthinthecross-sectionaldispersionoflogincomeoverthelifecycle(Var(ln(ht)≠ln(hú))=‡2w(t≠S)),whichgrowsbyaround1-2%peryear,accordingtoDeaton&Paxson(1994),implying‡w=0.1≠0.14.Thelowerbound‡w=0.1isconsistentwith’+ƒ1.5whenµw=0.0125,soIusethesevaluesinmycalibration.This25

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryimpliesexpectedgrowthinincomeswhileworking,ES[ln(ht)≠ln(hú)]=1µw≠12‡2w2(t≠S),of0.75%peryear.Inthedata,incomestendtogrowfasterearlierintheworkinglife,andplateauaroundmiddleage(whichisnotcompatiblewithastochasticgrowthprocesslikeageometricBrownianmotion)androughlydoubleovertheworkinglife,implyingaconstantgrowthrateofaround1.5%peryear,orµw=0.02when‡w=0.1.However,giventhenatureofthecoupledexponentialageingprocessandstochasticincomegrowthprocess,whichleadstoacounterfactuallylargenumberofveryoldpeoplewithhighincomes,alowergrowthrateseemsthemostreasonabletrade-offtomatchthecross-sectionalmoments.Regardless,Ialsocompareresultswhenµw=0and‡w=0.Iset‡s=0inordertoachieveafullyanalyticalsolution,butexplore‡s=0.1inaquantitativeextensionasarobustnesscheck.IcalibratetheparametersofthestudentsÕdeterministichumancapitalaccumulationprocess,µs,—sand”s,tothereturnstohighereducationestimatedintheliterature.Theseestimatesvarywildlybycountry,cohortandestimationmethodology;andofcourseaveragingoveranentirepopulationasImustdohereobscuresvastdifferencesinreturnsalonglinesofgender,race,fieldofstudyandqualityofinstitution,allwhichImustabstractfromforsimplicitybutwhichneverthelesscouldhaveimplicationsfortheoptimalfinancingofhighereducation.Nevertheless,estimatedreturnsareapproximately8-10%peryearofeducationonaverageforaBachelorÕsdegree(forexampleseeBlundelletal.2000,2001,2005),andperhapsafewpercentatmostforeachyearofaPhD,ifnotnegative(forexampleseevanderSteegetal.2014,Brittonetal.2020).Ithuschooseµs=0.3,”s=0.18and—s=0.2,whichdeliveraround12%growthinhumancapitalperyearinitially,rapidlydecreasingtoaround6%peryearafter3years,oraround9%peryearonaverageforthefirst3yearsofhighereducationÐatypicalBachelorÕsdegreeÐbeforedecreasingtoaround2.5%peryearafter8yearsofstudy.EquilibriumequationsandwelfareProposition1.ThesteadystateequilibriumoftheanalyticalspecialcaseoftheBaselinemodelwhen“”=1isgivenbythefollowingequationsgoverningtheoptimalhumancapital26

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendrygraduationthresholdandthegovernmentbudgetconstraint:hú:1Ÿ!·0(wnhú)1≠·1″1≠“v(n)1≠“¸˚˙˝Vw(hú)=1ﬂ+⁄5T1≠“1≠“+1≠·1Ÿhú!·0(wnhú)1≠·1″1≠“v(n)!µshú—s≠”shú”6¸˚˙˝Vs(hú)(18a)·0:;wnhú5’+’≠(1≠’≠)(1≠’+)6≠·0(wnhú)1≠·15’+’≠(1≠·1≠’≠)(1≠·1≠’+)6<Ahú1≠—≠µs”sh1≠—0≠µs”sB⁄”s(1≠—s)¸˚˙˝Taxrevenue=SU1≠Ahú1≠—≠µs”sh1≠—0≠µs”sB⁄”s(1≠—s)TV(T+F+G)¸˚˙˝Studentstransfers(18b)whererecallthatw=AFandF=AFAE≠Gduetoconstantreturnstoscaleproduction,n=Ë(1≠·1)(1+1/Ï)Â“(1+1/Ï)+(1≠·1)Â(1≠“)È11+1/Ï.Acornersolutionariseswhenhú<h0,inwhichcasetheoptimalgraduationthresholdish0=1,andtheoptimaltaxlevelis·0=(1≠·1≠’≠)(1≠·1≠’+)(1≠’≠)(1≠’+)(wn)1≠·1.Proof.Fortheproofandforthesamesystemofequationswhen“=1,seetheAppendix.Inequilibrium,highertransferstypicallyleadtoahighergraduationthreshold,hú,whichimplieshigheraveragehumancapitalofworkers,butfewerworkers.Inmostoftheregioninwhichanequilibriumexists,highertransfersleadtohigher·0,andsohighertake-homepayforagivenlevelofhumancapital.However,equilibrium·0eventuallydecreasesforveryhighT,astherevenuedemandsofextraschoolingoutweighthehumancapitalgainsandaveragetaxratesthereforemustincrease.AnequilibriummaynotexistforveryhighT,sinceitbecomesoptimaltoremainastudentforever.Toseethis,takethesimplecasewhereworkersretainthehtheygraduatewithuntiltheydie(µw=‡w=0);ignoringtaxesandassumingexogenouslaboursupply(normalisedto1),itisoptimalnevertograduatewheneverTØhú(sincew=1),sothiscannotbeanequilibrium.Duetodecreasingreturnstolearning,húisboundedabovebyh,whichinmycalibrationisaround1.9,soclearlyT?1.9cannotbeanequilibrium;thepresenceofgrowthinhumancapitalwhileworking,risk,taxesandendogenouslaboursupplymeaninfactT27

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendrymustbesomewhatlessthan1.9inorderforanequilibriumtoexist.Figure3:Equilibriumgraduationthreshold,hú,andtaxlevel,·0Corollary1.AggregatesocialwelfareintheanalyticalspecialcaseoftheBaselinemodelisgivenby:Ws=T1≠“≠1(ﬂ+⁄)(1≠“)¸˚˙˝lifetimevalueofbeingastudentSWU1≠Qahú1≠—≠µs”sh1≠—0≠µs”sRb⁄”s(1≠—s)TXV¸˚˙˝measureofstudents+C1Ÿ!·0(wnhú)1≠·1″1≠“v(n)1≠“≠T1≠“(ﬂ+⁄)(1≠“)D¸˚˙˝optionvalueofgraduating⁄ﬂQahú1≠—≠µs”sh1≠—0≠µs”sRb⁄”s(1≠—s)SWU1≠Qahú1≠—≠µs”sh1≠—0≠µs”sRbﬂ”s(1≠—s)TXV¸˚˙˝averageddiscountingWw=SWWWWU1Ÿ!·0(wnhú)1≠·1″1≠“v(n)1≠“¸˚˙˝welfareofgraduate/modalworkerZ¸˚˙˝inequalityamongworkers≠1(ﬂ+⁄)(1≠“)TXXXXVQahú1≠—≠µs”sh1≠—0≠µs”sRb⁄”s(1≠—s)¸˚˙˝measureofworkersW=Ws+WwwhereZ=’+’≠[(1≠·1)(1≠“)≠’≠][(1≠·1)(1≠“)≠’+]Proof.SeetheAppendix.28

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryStudentsÕwelfarecanbedecomposedintoasumoftwoterms:thelifetimeutilityfrombeingastudentforever,multipliedbythemeasureofstudents;plusthediscountedoptionvalueofgraduating,averagedoverallstudents.WorkersÕwelfarecanbedecomposedintoaproductofthreeterms:thewelfareofagraduate,whichisalsothewelfareofthemodalworker,sincethedistributionofworkersissplitaroundthatpoint;themeasureofworkers;andthetermZ,whichaccountsfortheinequalitybetweenworkers,i.e.therelativemeasuresofworkerseithersideofthemode.Thislasttermconvergesto1whenthedistributionofworkerscollapsestoapoint(’+,’≠æŒ,whenµw=‡w=0)orwhentaxesaretotallyprogressive(·1=1),inwhichcasethereisnoinequalitybetweenworkers.Thecalibrationhere(’+ƒ1.5,’≠ƒ≠3)issuchthat,intheentiretyofthepolicyspaceexplored(i.e.withoutveryregressivetaxes,·1π0,ortotallyprogressivetaxes·1=1)wehavethatZ<1,soinequalityimpliesthewelfareoftheaverageworker,andhencetheaggregate,isbelowthatofthemodalworker.Figure4:WelfarebytaxprogressivityandlevelofstudenttransfersintheBaselinemodelResults:Baselinemodel.Thekeyresultsfromthisbenchmarkcaseare:1.Thelevelofuniversitygrants,G,hasnoeffectontheequilibrium;tuitionfeesfall1:1,29

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendrysogovernmentexpenditureandhenceequilibriumtaxratesareunchanged.2.Iftuitionfees,F,arecappedbythegovernment,butGisappropriatelysettobalancethesupplyanddemandforschoolingÐi.e.suchthatF+G=AEAFÐtheequilibriumisalsounchanged.Thus,financinghighereductionthroughtax-fundeduniversitygrantsorthroughprivately-fundedtuitionaidedbytax-fundedstudenttransfersareequivalent.Ofcourse,studentsrequiresomeleveloftransferstopayforconsumption.3.Socialwelfareismaximisedwithhighlyprogressivelabourtaxes,·1∫0,andrelativelyhighnetstudenttranfers,T.Typically,foragivenprogressivityoftaxes,highertransfersimprovewelfareuptoapoint.Astransfersincrease,theoptimalprogressivityoftaxesalsoincreases,uptotheglobalmaximum(thereddot).4.However,whennetstudenttransfersareverylow,theequilibriumisdegenerate,withnostudents.Inthiscase,welfareisindependentoftransfers.Theoptimaldegreeofprogressivityinthisregionissimilartotheglobalmaximum.Thisregionislargerthemoreprogressivearetaxes.5.Moreover,whenhighstudenttransfersareveryhigh,anequilibriummaynotexist,asitbecomesoptimalnevertograduate.Extremelyprogressivetaxesandextremelyregressivetaxesbothreducethemaximumstudenttransfersthegovernmentcanoffer.Welfareofstudentsvsworkers.Transferstostudentsnaturallyhavenodirecteffectonthewelfareofworkers,soonlyaffectitthroughchangingtheequilibriumhúand·0.Thesehavenoeffectonhoursworked,onlyconsumptionandthemeasureofworkers.30

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryFigure5:WelfareofstudentsvsworkersInsurancevsredistribution.Recalltherearethreedistinctmotivationsforpolicyinter-ventioninthiseconomy:socialinsuranceagainstrisk;redistribution;andfinancingofhighereducation(byredistributingtostudents).Wecandisentanglethesecompetingmotivesbysetting‡w=0,toshutdowntheinsurancemotive,asshowninFigure6.Asonemightexpect,whenthereisnoriskÐandhencenoinequalityamongworkersofagivenageÐthereisnobenefitfromsocialinsuranceandlessbenefitfromredistribution,sooptimaltaxesaredistinctlylessprogressive,thoughnotablystillmoresothanthecurrentUSschedule.31

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryFigure6:WelfareintheBaselinemodelwithnoriskwhileworking,‡w=0Wecanshutdowntheredistributionmotiveentirelybysettingµw=0,soworkersnolongeraccumulatehumancapitalonthejobandthusthereisnoincomeinequalityatallbetweenworkers.TheonlyÒredistributiveÓgoalthatremainsistowardsstudents,i.e.infundingtheirhighereducationandthusovercomingthefinancialfriction(thattheycannotborrow).Inthiscase,optimaltaxesaresomewhatregressive.Notealsothateliminatingwagegrowthwhileworkingsubstantiallyreducesthelevelofstudenttransfersatwhichitbecomesoptimalnevertograduate,thussubstantiallycurtailingtheregioninwhichanequilibriumcanexist.32

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryFigure7:Welfarewithnoinequalityamongworkers,‡w=0&µw=0Clearlythen,whileinthismodelsocialinsuranceagainstriskprovidesacompellingcaseforprogressivetaxation,incommonwithmuchoftheliteratureonthistopic,notleastforexampleHeathcoteetal.(2017),wealsoseethatintergenerationalinequalityÐwhichthatpaperdoesnotfeatureduetoitsassumedflatlife-cycleearningsprofileÐalsomotivatesprogressivetaxation.ThisexplainsmuchofthedisparityintheresultsbetweenmypaperandtheirÕs.Stochasticlearning.WecanexaminenumericallytheslightlymoregeneralversionoftheBaselinemodelwherehumancapitalaccumulatesstochasticallywhilestudying,‡s>0,usingthefinitedifferencemethodofAchdouetal.(2022).WecanseefromFigure8thatqualitativelyandquantitativelytheresultsfromtheanalyticalBaselinemodelgothroughlargelyunchanged:optimaltransfersarehighandoptimaltaxesareprogressive.33

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryFigure8:WelfareintheBaselinemodelwithriskylearningwhilestudying,‡s=0.13AnalyticalExtensionsSupplysidegeneralequilibriumeffectsAconcernonemayhaveisthatexpandingaccesstohighereducationraisescostsandhencetuitionfees,andsoingeneralequilibriumsubsidiesendupcostingagreatdealandimposingamuchgreatertaxburden.TheBaselinemodelneutersthiseffectbyassumingconstantreturnstoscaleproduction,sothatpolicyhasnoeffectonrelativeprices.Irelaxthisassumptionbyinsteadassumingthatproductionofbothconsumptiongoodsandschoolingfacesdecreasingreturnstoscale,fi(Li)=AiL1≠–ii.Onecanascribethisforexampletothepresenceofsomefixedfactorofproduction.Forsimplicity,Iassumethatanyprofitsorpaymentstothisfixedfactorgotooverseasownerswhodonotparticipateintheeconomyinanyotherway.Imoreoverassumeherethatbothsectorsfacethesamedegreeofdecreasingreturns:–E=–F.IntheAppendixIprovidethetwoadditionalequationsthatdeterminetheequilibrium;labourmarketandschoolingmarketclearingconditions.Equilibriumwagesforthemostpartmoveinverselywitheffectivelaboursupply:as34

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendrytheprogressivityoftaxesincreases,allagentsreducetheirlaboursupply,raisingthewage;however,asthegenerosityofstudenttransfersincreases,schoolingincreases,raisingtheaggregateeffectivelaboursupplyandloweringthepiece-ratewage.Tuitionfeesnaturallyincreasewiththenumberofstudents.Figure9:Equilibriumtuitionfees,F,andwages,w,withDRSproductionTheoptimaltaxscheduleremainshighlyprogressive,comparedtothecurrentUSsystem.Asforoptimalstudenttransfers,fromthescaleofthediagrambelow,introducingDRSproductionappearstosubstantiallyreducethemaximumtransfersthatthegovernmentcanofferstudentsbeforeanequilibriumceasestoexist.However,thisispartlyillusory:withDRSproductionboththepiece-ratewageandtheequilibriumtuitionfeearetypicallyfarlessthan1(=AFand=AF/AE),astheywereintheBaselinemodel.Consequently,tointerpretthevaluesofTonthex-axiscorrectly,andcomparethemtotheBaselinecase,oneshouldscalethembyoneofeitherworF;infactthentheoptimalstudenttransfersinthemodelwithdecreasingreturnstoscaleproductionaresomewhatmoregenerousthanintheBaselinemodel.Thus,theintuitionthatsomemayhave,thatsubsidisingeducationinduceshighertuitionfeesandhenceoffsetsanywelfaregainsfromthesubsidies,appearstobemisguided,atleastinthismodel.35

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryFigure10:WelfarewithDRSproductionEx-anteheterogeneityTheBaselinemodelignoresapparentheterogeneity,forexampleinabilityandhenceinthereturnstoeducation,andthusmayoverstatethegainsfromflatsubsidiestoallstudents.IntheAppendixIdiscussinmoredepthaverygeneralanalysisallowingforex-anteheterogeneityinlearningabilityanddisutilityfromlabour.Fornow,Ifocusonheterogeneousability.Inparticular,IenvisionÒlearningabilityÓasaparameter,z,thataugments(z>1)ordiminishes(z<1)thespeedwithwhichanagentaccumulateshumancapital,eitherwhilestudyingorworking.Humancapitalthusaccumulateslikeúht=µszh—st≠”shtwhileastudent,anddht=µwz–htdt+‡whtdWtwhileworking,where–œ[0,1]parameteriseshowstronglyschool-learningabilitytranslatesintoon-the-joblearningability.HereIfocusonaspecialcase,withjusttwotypesofagentswhodifferonlyintheirabilitytolearnwhileinschool(sowith–=0):onewithnormal/highabilityz1=1ÐasintheBaselinemodelÐandonewithlowabilityz0=z,wherezislowenoughthattheseagentswillneverattenduniversity,i.e.S(z)=0orhú(z)=h0.Forsakeofargument,wecanassumez=0,sosuchstudentsareincapableoflearningwhileatuniversity,andwherehencethose36

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryagentswillneverattenduniversity,regardlessofhowgeneroustuitionsubsidiesare.Fortheavoidanceofanyfoundationalmathematicalambiguity,letÕsregardz0=1,sosuchagentslearnon-the-jobatthestandardrate.Thisset-upisconvenientasitallowsustoincorporatethefactthatonlyaroundhalfofstudentsgotouniversity,soletÕsassumehalfofagentsaretypez1andhalfz0.Ofcourse,withz=0,thefractionwhoattenduniversitywillbeeitherhalforzero,dependingonhowgenerousstudenttransfersare,butonecanalsogeneralisetheset-uptoexplaintheincreasingshareofstudentsattendinguniversity.Theeffectofintroducinglow-skilledlearnersistobroadenslightlytherangeofpossibleequilibria.Sincealargefractionofpeopleneverattenduniversity,theyneednosubsidies,soagivenlevelofsubsidiesforthosethatdoattenduniversityrequiresloweraveragetaxes,·0,meaningthevalueofworkingishigher,thusraisingthelevelofstudenttransfersatwhichitbecomesoptimalnevertograduate,andatwhichanequilibriumthereforecannotexist.Inaddition,optimaltaxesaremoreprogressive,althoughingeneraltheeffectofintroducingthisparticularformofex-anteheterogeneityissmall.Figure11:Welfarewithex-anteheterogeneousabilitywhilestudying37

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryAgeingTheBaselinemodelcapturesthetimeopportunitycostofschoolingonlyinaverycrudeway:theperpetualyouthstructuremeansspendinganextrayearinschoolhasauniformtimecost,intheformofahigherdiscountrate,sinceaftertheextrayearonestillhasthesamelifeexpectancyasbefore(1/⁄).Ithusconsideraversionofthemodelwhereagentsliveacertainandfinitenumberofyears,Y,insteadofdyingataconstantrate,sospendinganextrayearinschoolmeansonehasoneyearlesstoliveandhenceearn,limitingthevalueofanextrayearineducationtoaccumulateamarginalamountofadditionalhumancapital.Suchanalterationlikelybetterreflectstheopportunitycostofschooling.Thischangesubstantiallyreducestheaveragewelfaregainstoworkersfromhighertransferstostudents,T,simplybecauseoncethestudentsactuallygraduatetheyhavefeweryearsoflifetoenjoytheirhigherearnings.OptimalstudenttransfersarethereforesubstantiallylowerthanintheBaselinemodel.Optimaltaxesarealsosignificantlylessprogressive,likelybecausethereisamuchsmallerright-tailofworkerswithveryhighincomes.Asiswell-known,theincomedistributionofaperpetualyouthmodelwithpermanentincomeshocksisPareto-distributedbecausethereisatailofworkerswholiveinordinatelylongandsoaccumulateenormoushumancapital;cappingtheirlifespancutsoffthisfatrighttail.Thusthegovernmentraiseslesstaxrevenuefromthesuper-rich,andthisnecessitateshigheraveragetaxes(lower·0)fromthebulkofearners,loweringthewelfaregainsfromprogressivetaxes.Itshouldbepointedouthowever,thattheincomedistributionofcoursedoeshaveathickrighttailinthedata,itsimplyarisesbyothermeansnotcapturedbythemodel(e.g.seeGabaixetal.2016),sothisversionofthemodelmayunderstatethewelfaregainsfromprogressivetaxation.Inanycase,optimaltaxesarestillmoreprogressivethanthecurrentUSsystem.38

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryFigure12:Welfareinmodelwithageing4QuantitativeModel:IntroducingStudentDebtInowmakeonesubtlechangeovertheBaselinemodel:studentscanborrow.Isetaborrowinglimitof5xthetuitionfee.Inadditiontoanyborrowings,studentsstillreceiveagrosstransfer÷T=T+Ffromthegovernment.ThissmallchangemeanstheÒnotradeÓresultisnolongeranequilibrium,andindividualsÕpolicyfunctionsnolongerhaveclosed-formsolutions.Ithusproceedentirelynumerically.IntroducingstudentloansÐandhencenecessitatingthatworkersbuildwealth,bothtopayofftheirdebtsandtoensurebondmarketsclearÐcanchangeindividualbehavioursubstantially.Householdswithlow-to-moderatewealthandincomebehavesimilarlytotheno-tradeequilibrium,butthosewithlowwealthandhighincomesaveconsiderablesumsÐontheorderof30-40%oftheirincomeÐinparttoinsurethemselvesagainstincomerisk,whilethosewithhighwealthandlowincomedissave.ThenewequilibriumalsoevidencesÒprecautionaryworkingÓ,wherelow-wealth,high-incomehouseholdsworklongerhoursÐagainpartlytoinsurethemselvesÐwhilewealthyhouseholdsworkless,owingtostandardwealth39

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryeffects.Figure13:Consumption,labour&savingintheQuantitativemodelvsBaselinenotradeSinceboththeworkersÕandstudentsÕproblemsareotherwisethesameasbefore,Idonotrecapitulatethemhere.ForcompletenessIshowbelowtheKolmogorovForwardequationsthatdescribetheevolutionofthejointdistributionsofhumancapitalandwealthforbothstudentsandworkers.Thestationarydistributionisfoundbysettingthemequaltozero(heresuppressingfunctionnotation):úgs=≠ˆa[úags]≠ˆhË(µsh—s≠”sh)gsÈ≠⁄gs+⁄”(a)”(h≠h0)=0(aú,0]◊[h0,hú)úgw=≠ˆa[úagw]≠ˆh[µwhgw]+12ˆhhË‡2wh2gwÈ≠⁄gw=0[a,Œ)◊(0,Œ)(aú,hú)ComparedtotheBaselinemodelthereisoneadditionaltermineachequation;thefirst,representingthechangeinthemeasuresofstudentsandworkersastheysave.Ihavealsoincludedtheterminthefirstequationthatrepresentsinflowsofnewbornstudentsatthepoint(0,h0);newbornsinheritzerowealthandstartwithhumancapitalh0,whichiscapturedbytheDiracmeasuresath=h0anda=0inthefinalterm.Thesupplysideandmarketclearingconditionsandsocialwelfarefunctionsareagain40

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendrylargelyunchangedfromtheBaselinemodel,soIdonotrecapitulatethemallhere.StationaryequilibriumoftheQuantitativemodel.AstationaryequilibriumoftheQuantitativemodelisasetofvaluefunctions{Vs(a,h),Vw(a,h)},policyfunctions{cs(a,h),cw(a,h),n(a,h),hú(aú)},distributions{gs(a,h),gw(a,h)},prices(w,F,r)andgov-ernmentpolicies(·0,·1,T)suchthatstudents&workersmaximiselifetimeutilityandfirms&educationalinstitutionsmaximiseprofitsgivenpricesandpolicies;allmarketsclearandthegovernmentbudgetconstraintholds;andthedistributionsofthehumancapitalarestationaryovertime.wandFarepinneddownbytechnology,and·0isrestrictedbythegovernmentbudgetconstraint,sothetriple(hú(aú),·0,r)completelydescribesanequilibriumforagivenpolicy(·1,T).Isearchover(·1,T)tofindthepolicythatmaximisessocialwelfare.OnekeydifferenceintheresultingequilibriaoftheQuantitativemodelcomparedtotheBaselineisthattheregionofthepolicyspacewithnostudentsisnowmuchsmaller,evenwhenTissmallbutpositive.TherearenumericalchallengestosolvingthemodelwhenTÆ0(e.g.when÷T=0andsostudentsreceiveno(gross)transfer),sincethosewhoareattheborrowingconstraintandhenceunabletoborrowmorewouldbeunabletoconsume.Economically,itisclearhowtoresolvethis:studentswhohaveborroweduptothelimitandwhocanhencenolongeraffordtopaytuitionfeeswouldÒgraduateÓimmediatelyandstartworking.However,itturnsoutthattheregionofthepolicyspacethat(atleastlocally)maximisessocialwelfarefeaturespositivetransfers,sotheproblemismoot.Results:Quantitativemodel.TheoptimaltaxandtransferscheduleisqualitativelyandquantitativelysimilartotheBaselinemodelanditsmyriadextensions.Optimaltaxesarehighlyprogressive,andonlymodestlylesssothaninprioriterationsofthemodel,withoptimal·1ƒ0.5.Perhapssurprisingly,theoptimaltransfer,foressentiallyanydegreeoftaxprogressivity,isoverwhelmingpositive,albeitsmallerthanintheBaselinemodel.RecallmoreoverthatTisthenettransfer,inexcessofthatneededtocovertuitionfees,evenwithstudentloansavailable.Naturally,thisisfarabovethelevelsofgrantsavailabletomost41

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryprospectivestudentsinmostcountries.Figure14:WelfareinQuantitativemodel5ConclusionInthispaperIhaveexploredthequestionofhowtocalibrateincometaxprogressivityandhighereducationsubsidiestobalancethewelfaregainsfromsocialinsurance,overcomingfinancialfrictionsandredistribution,withthewelfarelossesfromdiscincentivisingworkandstudythatsuchpoliciescanbringabout.Todosowhileovercomingsomeoftheshortcomingsofthepriorliterature,Isetoutanincompletemarketsmodelwithaneducationchoicemodelledasastoppingtimeproblem,withendogenouslaboursupplyandsavingchoices.Ihaveshownthatinsuchamodel,therearesubstantialwelfaregainsfrombothhighlyprogressivetaxationoflabourincomeandhighsubsidiesforhighereducation,comparedtoabaselineofnointervention.OptimalpolicyinthemodelfeaturesmoreprogressivetaxesthancurrentlyobservedintheUSorEurope,aswellaslargesubsidies,evenwhenstudentscanborrowtopayfortheireducation.Ihavefurthershownthatthesewelfareconclusionsarerobusttomyriadextensions42

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryofthemodel,includingdiseconomiesfromscaleineducation,whereexpandingaccesstoeducationmakesitmorecostly;heterogeneityinthereturnstoschooling,i.e.inability;amorerealisticlife-cycleprofile;andlessseverefinacialfrictions,wherestudentcanborrowtosomeextent,whichalsofeaturesamorerealisticequilibriumwithextensiveprecautionarysavingbyworkers.Notably,theformofsubsidyappearsnottobeparticularlyimportantforwelfare:Ihaveshownthatdirecttransferstostudents,whothenpaytuitionfeessetfreelybythemarket,areequivalenttoper-studentgovernmentgrantstouniversities,evenwhengovernmentsexplicitlycaptuitionfees,providedthegrantissufficienttobalancesupplyanddemandforschoolingandavoidrationingofeducation.Financingeducationwithstudentloansalsodoesnotchangetheoptimalityofhighlyprogressivetaxationandsubstantialsubsidiesforeducation.Naturally,furtherresearchoughttoconsideradditionalchannelsbywhichsuchpoliciesmayhaveadverseeffects,forexampleamicro-foundedmodelofon-the-jobhumancapitalaccumulationthatarisesfromoptimisingbehaviour,whereobviouslyincentivestoaccumulatehumancapitalarecrucial,butwhichinthispaperismodelledasanexogenousadhocprocess.Moreover,onemightconsiderincorporatingotherfeaturesoftheeducationdecision,suchasthefieldofstudy,orthequalityofinstitution,orcalibratingthefinancingstructuretomorecloselymatchreal-worldpolicies.Inparticular,aninterestingextention,whichIamexploringinfurtherwork,istoallowforrationingofeducationwhentuitionfeesarecappedbythegovernment,hencebreakingtheequivalencebetweenuniversitygrantsandstudenttransfersinthepresentmodel.Regardless,Ibelievethemodelinthispaperoffersacompellingcaseforrecalibratingtaxandeducationfinancepolicy.43

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryReferencesAbbott,B.(2022),ÔIncompletemarketsandparentalinvestmentsinchildrenÕ,ReviewofEconomicDynamics44,104Ð124.Achdou,Y.,Han,J.,Lasry,J.-M.,Lions,P.-L.&Moll,B.(2022),ÔIncomeandwealthdistributioninmacroeconomics:Acontinuous-timeapproachÕ,ReviewofEconomicsStudies89,45Ð86.Aiyagari,S.R.(1994),ÔUninsuredidiosyncraticriskandaggregatesavingÕ,QuarterlyJournalofEconomics190(3),659Ð684.Alon,T.,Bachus,N.&Wong,A.(2020),Debt,humancapitalallocation,andtheallocationoftalent.Ben-Porath,Y.(1967),ÔTheproductionofhumancapitalandthelifecycleofearningsÕ,JournalofPoliticalEconomy75(4),352.Benabou,R.(2002),ÔTaxandeducationpolicyinaheterogeneousagenteconomy:Whatlevelsofredistributionmaximizegrowthandefficiency?Õ,Econometrica70(2),481.Blanchard,O.(1985),ÔDebt,deficits,andfinitehorizonsÕ,JournalofPoliticalEconomy93(2),223Ð47.Blundell,R.,Dearden,L.,Goodman,A.&Reed,H.(2000),ÔThereturnstohighereducationinbritain:EvidencefromabritishcohortÕ,TheEconomicJournal110(461),F82ÐF99.Blundell,R.,Dearden,L.&Sianesi,B.(2001),Estimatingthereturnstoeducation:Models,methodsandresults,CEEDiscussionPapers0016,CentrefortheEconomicsofEducation,LSE.Blundell,R.,Dearden,L.&Sianesi,B.(2005),ÔEvaluatingtheeffectofeducationonearnings:models,methodsandresultsfromthenationalchilddevelopmentsurveyÕ,JournaloftheRoyalStatisticalSociety:SeriesA168(3),473Ð512.44

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryBritton,J.,Buscha,F.,Dickson,M.,vanderErve,L.,Vignoles,A.,Walker,I.,Waltmann,B.&Zhu,Y.(2020),Theearningsreturnstopostgraduatedegreesintheuk,Governmentsocialresearch,DepartmentforEducationandInstituteforFiscalStudies.Card,D.(2001),ÔEstimatingthereturntoschooling:ProgressonsomepersistenteconometricproblemsÕ,Econometrica69(5),1127.Caucutt,E.M.&Kumar,K.B.(2003),ÔHighereducationsubsidiesandheterogeneity:AdynamicanalysisÕ,JournalofEconomicDynamicsandControl27,1459Ð1502.D‡vila,J.,Hong,J.H.,Krusell,P.&R’os-Rull,J.-V.(2012),ÔConstrainedefficiencyintheneoclassicalgrowthmodelwithuninsurableidiosyncraticshocksÕ,Econometrica80(6),2431Ð2467.Deaton,A.&Paxson,C.(1994),ÔIntertemporalchoiceandinequalityÕ,JournalofPoliticalEconomy102(3),437Ð467.Feldstein,M.(1969),ÔTheeffectsontaxationonrisktakingÕ,JournalofPoliticalEconomy77(5),755Ð764.Gabaix,X.,Lasry,J.,Lions,P.-L.&Moll,B.(2016),ÔThedynamicsofinequalityÕ,Econo-metrica84(6),2071Ð2111.Heathcote,J.,Storesletten,K.&Violante,G.L.(2017),ÔOptimaltaxprogressivity:AnanalyticalframeworkÕ,QuarterlyJournalofEconomics132(4).Hogan,V.&Walker,I.(2003),Educationchoiceunderuncertainty.Hogan,V.&Walker,I.(2007),ÔEducationchoiceunderuncertainty:ImplicationsforpublicpolicyÕ,LaborEconomics14(6),894Ð912.Huggett,M.(1993),ÔTherisk-freerateinheterogeneous-agentincomplete-insuranceeconomiesÕ,JournalofEconomicDynamicsandControl17(5-6),953Ð969.45

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryHuggett,M.,Ventura,G.&Yaron,A.(2006),ÔHumancapitalandearningsdistributiondynamicsÕ,JournalofMonetaryEconomics53(2),265Ð290.Huggett,M.,Ventura,G.&Yaron,A.(2011),ÔSourcesoflifetimeinequalityÕ,AmericanEconomicReview101(7),2923Ð2954.King,R.G.,Plosser,C.I.&Rebelo,S.T.(1988),ÔProduction,growthandbusinesscycles:I.thebasicneoclassicalmodelÕ,JournalofMonetaryEconomics21(2),195Ð232.Krueger,D.&Ludwig,A.(2013),ÔOptimalprogressivelaborincometaxationandeduca-tionsubsidieswheneducationdecisionsandintergenerationaltransfersareendogenousÕ,AmericanEconomicReview103(3),496Ð501.Krueger,D.&Ludwig,A.(2016),ÔOntheoptimalprovisionofsocialinsurance:ProgressivetaxationversuseducationsubsidiesingeneralequilibriumÕ,JournalofMonetaryEconomics77,72Ð98.Krusell,P.,Mukoyama,T.&ahin,A.(2010),ÔLabour-marketmatchingwithprecautionarysavingsandaggregatefluctuationsÕ,TheReviewofEconomicStudies77(4),1477Ð1507.Krusell,P.,Mukoyama,T.,ahin,A.&Smith,A.A.(2009),ÔRevisitingthewelfareeffectsofeliminatingbusinesscyclesÕ,ReviewofEconomicDynamics12(3),393Ð404.Krusell,P.&Smith,A.A.(1998),ÔIncomeandwealthheterogeneityinthemacroeconomyÕ,JournalofPoliticalEconomy106(5),867Ð896.Loury,G.C.(1981),ÔIntergenerationaltransfersandthedistributionofearningsÕ,Econometrica49(4),843Ð867.MaCurdy,T.E.(1981),ÔAnempiricalmodeloflaborsupplyinalife-cyclesettingÕ,JournalofPoliticalEconomy89(6),1059Ð1085.Mellior,G.(2021),Highereducationfunding,welfareandinequalityinequilibrium.46

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryMirrlees,J.A.(1971),ÔAnexplorationinthetheoryofoptimumincometaxationÕ,TheReviewofEconomicStudies38(2),175Ð208.Nu–o,G.&Moll,B.(2018),ÔSocialoptimaineconomieswithheterogeneousagentsÕ,ReviewofEconomicDynamics28,150Ð180.Ramsey,F.P.(1927),ÔAcontributiontothetheoryoftaxationÕ,TheEconomicJournal37(145),47Ð61.Restuccia,D.&Urrutia,C.(2004),ÔIntergenerationalpersistenceofearnings:TheroleofearlyandcollegeeducationÕ,TheAmericanEconomicReview94(5),1354Ð1378.Trabandt,M.&Uhlig,H.(2011),ÔThelaffercurverevisitedÕ,JournalofMonetaryEconomics58(4),305Ð327.vanderSteeg,M.,vanderWiel,K.&Wouterse,B.(2014),Individualreturnstoaphdeducationinthenetherlands:Incomedifferencesbetweenmastersandphds,CPBDiscussionPaper276,CPBNetherlandsBureauofEconomicPolicyAnalysis.47

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryAppendix1:ExtendedproofsProofofTheorem1:Existenceofno-tradeequilibrium.Proof.Toprovetheexistenceofano-tradeequilibrium,ItakethesameapproachasHeathcoteetal.(2017)inshowingthatthepolicyfunctionsdescribedinthebodyofthetextsatisfyalltheconsumersÕoptimalityconditionsandgiverisetoaggregateoutcomesthatsatisfyallmarketclearingconditions.First,recalltheworkersÕHJBequation:(ﬂ+⁄)Vw=maxc,nc1≠“v(n)≠11≠“+Vwa1ra+·0(wnh)1≠·1≠c2+Vwhµwh+12Vwhh‡2wh2Iconjecturethatworkersconsumetheirpost-taxincome,c=·0(wnh)1≠·1,andsupplyconstantlabourofn=Ë(1≠·1)(1+1/Ï)Â“(1+1/Ï)+(1≠·1)Â(1≠“)È11+1/Ï,thelatterderivedinthetextfromtheworkersÕlabouroptimalityconditionassumingtheformerholdsandworkersallholdzerowealth.Theworkersfirst-orderconditionsareasfollows:c≠“v(n)=Vwa≠c1≠“1≠“vÕ(n)=Vwa·0(1≠·1)(wh)1≠·1n≠·1Combiningthesetwoexpressionsandsubstitutingintheassumedpolicyfunctionforcon-sumptionrevealsthatlaboursupplyisindeedgivenbythepriorexpression.Secondly,applyingtheenvelopeconditiontotheworkersÕHJBequationyields:(ﬂ+⁄)Vwa=Vwaaúa+Vwar+Vwhaµwh+12Vwhha‡2wh2wherethefirsttermontheright-handsideiszeroundertheassumedpolicyfunctions,providedwealthisalsozero.RecognisingbyYoungÕsTheoremthatVha=VahandVhha=Vahh,andhencedifferentiatingtheconsumptionfirst-ordercondition(againundertheassumedpolicy48

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryfunction),wefind:Vwah=≠“c(h)≠“≠1cÕ(h)v(n)Vwahh=≠“c(h)≠“≠1cÕÕ(h)v(n)+“(1+“)v(n)c(h)≠“≠2(cÕ(h))2Substitutingtheseintotheenvelopeconditionyields:(ﬂ+⁄≠r)Vwa=≠“c≠“≠1cÕ(h)v(n)µwh+121≠“c≠“≠1cÕÕ(h)v(n)+“(1+“)v(n)c≠“≠2(cÕ(h))22‡2wh2wherec(h)=·0(wnh)1≠·1,cÕ(h)=·0(1≠·1)(wn)1≠·1h≠·1=c(h)·1h≠1andcÕÕ(h)=≠·0(1≠·1)·1(wn)1≠·1h≠·1≠1=≠c(h)(1≠·1)·1h≠2.SubstitutingtheseintotheexpressionforVwa,andthatintotheconsumptionfirst-orderconditionyields:(ﬂ+⁄≠r)c≠“v(n)=≠“c≠“(1≠·1)v(n)µw+12“c≠“(1≠·1)·1v(n)‡2w+12“(1+“)v(n)c≠“(1≠·1)2‡2wMakingtheappropriatecancellationsandrearrangingrevealsthatthisequationholdswhentheinterestrateprevailinginthemarketis:r=ﬂ+⁄+“(1≠·1)µw≠12“(1≠·1)(1+“(1≠·1))‡2wasconveyedinthetext.Tocheckthatthisinterestrateclearsthebondmarketistrivialsincea=0forallagents.Moreover,individualbudgetconstraintsareallrespectedandimplyzerosavingsprovidedallworkersgraduatewithzerowealth,astheassumptionofnostudentdebtguarantees.Theaggregateresourceconstraint/goodsmarketclearingconditionthusclearlyholdstoo.TheworkersÕvaluefunctionsubsequentlyderivedinthebodyofthetextalsosatisfiestheworkersÕHJBequation,againgiventhatundertheassumedpolicyfunctionsavingsareeverywherezero,thusverifyingtheguess.Aggregatelabourdemandandthesupplyofuniversityplacesarebothinfinitelyelastic,giventheassumptionofconstantreturnstoscaleproduction,sowilladjusttoclearthosemarketsprovidedprices(w,F)satisfythesupplysideoptimalityconditions.Thegovernmentbudgetconstraintholdswhen·0takes49

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendrytheequilibriumvaluedescribedinProposition1.Thus,no-traderespectsallhouseholdsÕoptimalityconditionsandclearsallmarketsandishenceanequilibrium.ProofofCorollary1.Proof.Startingfromthegeneralexpressionforaggregatewelfare:Ws©⁄hú0Vs(h)gs(h)dhWw©⁄Œ0Vw(h)gw(h)dhW=Ws+WwPluggingintheexpressionsfortheworkersÕvaluefunction(4),workerwelfarebecomes:Ww=⁄hú0!·0(wnh)1≠·1″1≠“v(n)Ÿ(1≠“)c3hhú4≠’≠≠1dh+⁄Œhú!·0(wnh)1≠·1″1≠“v(n)Ÿ(1≠“)c3hhú4≠’+≠1dh≠AGw=!·0(wn)1≠·1″1≠“v(n)Ÿ(1≠“)chúIhú’≠⁄hú0h(1≠·!)(1≠“)≠’≠≠1dh+hú’+⁄Œhúh(1≠·!)(1≠“)≠’+≠1dhJ≠AGw=!·0(wn)1≠·1″1≠“v(n)Ÿ(1≠“)chúIhú’≠51(1≠·1)(1≠“)≠’≠h(1≠·!)(1≠“)≠’≠6hú0+hú’+51(1≠·1)(1≠“)≠’+h(1≠·!)(1≠“)≠’+6ŒhúJ≠AGw=!·0(wnhú)1≠·1″1≠“v(n)Ÿ(1≠“)chú51(1≠·1)(1≠“)≠’≠≠1(1≠·1)(1≠“)≠’+6≠AGw=C!·0(wnhú)1≠·1″1≠“v(n)Ÿ(1≠“)Z≠ADQahú1≠—≠µs”sh1≠—0≠µs”sRb⁄”s(1≠—s)whereA=1(ﬂ+⁄)(1≠“),Z=’+’≠[(1≠·1)(1≠“)≠’≠][(1≠·1)(1≠“)≠’+],wherethelastlineusesthedefinitionofcandwhereGwrepresentsthetotalmeasureofworkers,Gw=3hú1≠—≠µs”sh1≠—0≠µs”s4⁄”s(1≠—s).Thewelfareofstudentsisslightlymoretaxingtocalculate,aswemustaccountnotonlyforthevalueofbeingastudent,butalsotheoptionvalueofgraduating,andtheeffectivediscountfactorofthatoptionvaluebeforethestudentschosetograduate,averagedacrossthepopulationofstudents.Recallthatfromthedefinitionofthevaluefunctionwhen50

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryhumancapitalaccumulationduringschoolisdeterministic,Vs(h)=sSte≠(ﬂ+⁄)(·≠t)u(cs)d·+e≠(ﬂ+⁄)(S≠t)Vw(hú),theintegralis:⁄Ste≠(ﬂ+⁄)(·≠t)u(cs)d·=C≠1ﬂ+⁄e≠(ﬂ+⁄)(·≠t)T1≠“≠11≠“DSt=T1≠“≠1(ﬂ+⁄)(1≠“)11≠e≠(ﬂ+⁄)(S≠t)2Thefirsttermhererepresentsthevalueofbeingastudentforever.ThesecondtermdependsonS,whichinturndependsonh,andisthediscountedvalueofremainingastudentthatonegivesupwhengraduating.Thuswehave:Ws=⁄hú0;u(T)ﬂ+⁄+e≠(ﬂ+⁄)(S≠t)5Vw(hú)≠u(T)ﬂ+⁄6<gs(h)dh=⁄húh0IT1≠“≠1(ﬂ+⁄)(1≠“)+e≠(ﬂ+⁄)(S≠t)C!·0(wnhú)1≠·1″1≠“v(n)Ÿ(1≠“)≠T1≠“(ﬂ+⁄)(1≠“)DJá≠⁄”s1h1≠—s0≠µs”s2Qah1≠—s≠µs”sh1≠—s0≠µs”sRb⁄”s(1≠—s)≠1h≠—sdhwhereinthesecondlineIhavepluggedinthedensityandvaluefunctionatgraduation,andadjustedthelowerboundofintegrationtoaccountforthefactthatunderdeterministichumancapitalaccumulation,nostudentwillhavelesshumancapitalthantheystartwith.Sincestudentswithdifferentlevelsofhumancapitalareatvaryingdistancesfromthehumancapital(andhencegraduation)threshold,whendiscountingtheiroptionvalueofgraduatingwemustaccountfortheircurrentlevelofhumancapital,andhencedistancefromthethreshold,byreplacinge≠(ﬂ+⁄)(S≠t)=3hú1≠—s≠µs”sh1≠—s≠µs”s4ﬂ+⁄”s(1≠—s)beforeintegratingoverhusingu-substitutionwithu=h1≠—s≠µs”s:51

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryWs=⁄húh0Y][u(T)ﬂ+⁄+Ahú1≠—s≠µs”sh1≠—s≠µs”sBﬂ+⁄”s(1≠—s)5Vw(hú)≠u(T)ﬂ+⁄6Z^á≠⁄”s1h1≠—s0≠µs”s2Ah1≠—s≠µs”sh1≠—s0≠µs”sB⁄”s(1≠—s)≠1h≠—sdh=u(T)ﬂ+⁄≠⁄”s3h1≠—s0≠µs”s4≠⁄”s(1≠—s)á⁄húh03h1≠—s≠µs”s4⁄”s(1≠—s)≠1h≠—sdh+5Vw(hú)≠u(T)ﬂ+⁄63hú1≠—s≠µs”s4ﬂ+⁄”s(1≠—s)≠⁄”s3h1≠—s0≠µs”s4≠⁄”s(1≠—s)á⁄húh03h1≠—s≠µs”s4≠ﬂ”s(1≠—s)≠1h≠—sdh=u(T)ﬂ+⁄≠⁄”s3h1≠—s0≠µs”s4≠⁄”s(1≠—s)áC”s⁄3h1≠—s≠µs”s4⁄”s(1≠—s)Dhúh0+5Vw(hú)≠u(T)ﬂ+⁄63hú1≠—s≠µs”s4ﬂ+⁄”s(1≠—s)≠⁄”s3h1≠—s0≠µs”s4≠⁄”s(1≠—s)áC≠”sﬂ3h1≠—s≠µs”s4≠ﬂ”s(1≠—s)Dhúh0=T1≠“≠1(ﬂ+⁄)(1≠“)SU1≠Ahú1≠—≠µs”sh1≠—0≠µs”sB⁄”s(1≠—s)TV+C!·0(wnhú)1≠·1″1≠“v(n)Ÿ(1≠“)≠T1≠“(ﬂ+⁄)(1≠“)DAhú1≠—≠µs”sh1≠—0≠µs”sB⁄”s(1≠—s)⁄ﬂSU1≠Ahú1≠—≠µs”sh1≠—0≠µs”sBﬂ”s(1≠—s)TVwhereagainthefirsttermsrepresentsthevalueofbeingastudentforever,multipliedbythemeasureofstudents;thesecondtermistheoptionvalueofgraduating,discountedbyafactorinverselyproportionaltothelengthoftimetograduationforstudentsatdifferentlevelsofhumancapital,averagedacrossallstudents,i.e.she≠(ﬂ+⁄)(S≠t)gs(h)dh.When“=1,theoptimalgraduationthresholdisdeterminedimplicitlybythejointvaluematchingandsmoothpastingconditions,shownhereagainsubstitutedintothestudentsÕvaluefunctionevaluatedatthethreshold:ln1·0(wnhú)1≠·12≠Ân1+1/Ï1+1/Ï+1≠·1ﬂ+⁄Aµw≠‡2w2B¸˚˙˝(ﬂ+⁄)Vw(hú)=ln(T)+1≠·1ﬂ+⁄1µshú—≠1≠”2¸˚˙˝(ﬂ+⁄)Vs(hú)52

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryAppendix2:ExtensionsSupplysidegeneralequilibriumeffectsThefirstextensionIexploreistointroducedecreasingreturnstoscaleproductionofgoodsandeducation,fi(Li)=Ai(Li)1≠–i,toallowforwhatonemightcallmoreÒinterestingÓsupply-sidegeneralequilibriumeffects.Withthis,relativepricesFandwarenolongerpinneddownbytechnology,soweneedtwoadditionalequilibriumequations,forwhichtheschoolingandlabourmarketclearingconditionswillsuffice:F:AEA(1≠–E)FAEwB1≠–E–E¸˚˙˝SS=1≠Qahú1≠—≠µs”sh1≠—0≠µs”sRb⁄”s(1≠—s)¸˚˙˝SDw:A(1≠–F)AFwB1/–F¸˚˙˝LF+A(1≠–E)FAEwB1/–E¸˚˙˝LE=’+’≠nhú(1≠’≠)(1≠’+)Qahú1≠—≠µs”sh1≠—0≠µs”sRb⁄”s(1≠—s)¸˚˙˝LSEx-anteheterogeneityThesecondextensionincorporatesex-anteheterogeneity,namelyinability,zi,whichherewillenhancethelearningcapabilitiesofstudentsandworkers;andlabourdisutility,representedbytheshifterÂi,whererecallthatworkersÕdisutilityfromlabourisrepresentedbythefunctionvi(n)=11≠Âi(1≠“)n1+1/Ï1+1/Ï2“.Thelatter,ontopofmakingsomeagentsmorewillingtoworkforagivenwage,willalsomeantheywishtofinishstudyingearlierandentertheworkplace,relativetothosemoredistastefulofwork,increasingthesizeoftheworkforce.Iindexboththesevariableswithitoemphasisethattheyvarybetweenindividualsbutareneverthelessconstant.53

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryTheworkersÕandstudentsÕproblemsbecome:(ﬂ+⁄)Vwi(h)=(·0(wnih)1≠·1)1≠“vi(ni)1≠“+Vwhµwz–ih+12Vwhh‡2wh2(ﬂ+⁄)Vsi(h)=T1≠“≠11≠“+Vsh1µszih—s≠”sh2+12Vshh‡2sh2WorkersÕoptimallabourisgivenby:ni=Ë(1≠·1)(1+1/Ï)Âi“(1+1/Ï)+(1≠·1)Âi(1≠“)È11+1/Ïandtheirvaluefunctionby:Vwi(h)=Y___]___[1Ÿi(·0(wnih)1≠·1)1≠“vi(ni)1≠“≠1(ﬂ+⁄)(1≠“)if“”=11ﬂ+⁄5ln(·0(wnih)1≠·1)≠Âin1+1/Ïi1+1/Ï6+1≠·1(ﬂ+⁄)2(z–iµw≠‡2w/2)if“=1whilethatofstudentsis:Vsi(h)=u(T)ﬂ+⁄+Qah1≠—≠ziµs”hú1≠—i≠ziµs”Rb≠ﬂ+⁄”(1≠—)CVwi(húi)≠u(T)ﬂ+⁄DIassumeforsimplicitythatinnatecharacteristicsarediscretelydistributedaccordingtotheprobabilitymassfunction,mi.ThedistributionsofhumancapitalacrossstudentsandworkershavethesameformasintheBaselinemodel:gsi(h)=mi⁄ziµsh—s≠”shQah1≠—s≠ziµs”sh1≠—s0≠ziµs”sRb⁄”s(1≠—s)h0<h<húigwi(h)=Y___]___[mici1hhúi2≠’i≠≠1h<húimici1hhúi2≠’i+≠1h>húiwhereci=’i+’i≠(’i≠≠’i+)húi3hú1≠—i≠ziµs”sh1≠—0≠ziµs”s4⁄”s(1≠—s)andwhere’i+and’i≠arethepositiveandnegativerootsof12‡2w’2i+1z–iµw≠12‡2w2’i≠⁄=0.Theequilibriumequationsnowbecome:54

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryhúi:1Ÿi!·0(wnihúi)1≠·1″1≠“vi(ni)1≠“¸˚˙˝Vwi(húi)=1ﬂ+⁄5T1≠“1≠“+1≠·1Ÿihú!·0(wnihúi)1≠·1″1≠“vi(ni)1µsz–ihú—si≠”shúi26¸˚˙˝Vsi(húi)·0:ÿi;wnihúi5’i+’i≠(1≠’i≠)(1≠’i+)6≠·0(wnihúi)1≠·15’i+’i≠(1≠·1≠’i≠)(1≠·1≠’i+)6<Ahú1≠—i≠ziµs”sh1≠—0≠ziµs”sB⁄”s(1≠—s)mi¸˚˙˝Taxrevenue=ÿiSU1≠Ahú1≠—i≠ziµs”sh1≠—0≠ziµs”sB⁄”s(1≠—s)TV(T+F+G)mi¸˚˙˝StudentstransfersAggregatewelfarehasthesameform,butIdonotrepeatthewholeexpression:W=ÿiC⁄húih0Vsi(h)gsi(h)dhDmi+ÿi5⁄Œ0Vwi(h)gwi(h)dh6miAgeingThefinalextensionintroducesfinitelifetimes,asopposedtoaperpetualyouth,whichgivesthetrade-offofspendinganextrayearinschoolrealbiteandmorecloselyreflectstheopportunitycostofschooling.TreatingdeathasdeterministicatageYallowsustoretainananalyticalsolution.TheworkersÕHJBequationnowbecomes:ﬂVw(h,y)=u(·0(wnh)1≠·1,n)+Vwh(h,y)µwh+12Vwhh(h,y)‡2wh2+Vwy(h,y)AndsowithCRRAutilityasbefore,theirvaluefunctionbecomes:Vw(h,y)=1Ÿ(·0(wnh)1≠·1)1≠“v(n)1≠“11≠eŸ(y≠Y)2≠1ﬂ(1≠“)wherenisthesameasintheBaselinemodelbutwherenowŸ=ﬂ≠(1≠·1)(1≠“)1µw≠‡2w22≠(1≠·1)2(1≠“)2‡2w2.NoticethatiftheageofdeathYistakentobeinfinite,soworkersliveforever,thentheexpressioncollapsestothatoftheperpetualyouthBaselinemodelwitha55

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendrydeathrate⁄=0,asonewouldexpect.TheHJBequationforstudentsbecomes:ﬂVs(h,y)=u(T)+Vsh(h,y)1µsh—s≠”sh2+Vsy(h,y)Rearrangingthisandusingthesmoothpastingconditionstofindthevaluefunctionofthestudentatthecut-off:Vs(hú,yú)=T1≠“≠1ﬂ(1≠“)+(1≠·1)(·0(wnhú)1≠·1)1≠“v(n)ﬂŸhú11≠eŸ(yú≠Y)21µshú—s≠”shú2≠(·0(wnhú)1≠·1)1≠“v(n)ﬂ(1≠“)eŸ(yú≠Y)Equatingthevaluefunctionsofstudentsandworkersatthecut-off(valuematching),wefindanimplicitfunctionforthethresholdvaluesofhumancapitalandage,hú(yú),thatrepresentstheboundaryofwhenstudentsdecidetograduate.Naturally,sinceageingoneyearnowmeansoneyearlessoflifeleft,andhenceadiminishedoptionvalueofgraduating,thefunctionhú(yú)isdecreasinginage.ThesymmetrywiththeBaselinemodelshouldhopefullybeapparent.NoteagainthatasYæŒ,theequationcollapsestothatoftheperpetualyouthBaselinemodelwithadeathrate⁄=0.NotealsothatwithafiniteY,asstudentsapproachtheageofdeath,yæY,thesecondtermdropsout,sothevaluematchingconditionimpliesthattheoptimalgraduationthresholdwillbesuchthattheflowutilityfromworkingequalstheflowutilityofstudying,i.e.u1·0(wnhú)1≠·1,n2=u(T).Inotherwords,near-deadstudentswithnofuturewillchoosetograduatewhentheflowutilitytheycouldgetfromworkingexceedstheflowutilitytheygetasstudents,or(ignoringdisutilityfromlabourmomentarily)whentheirpost-taxwageexceedsthestudenttransfer.Thefunctionhú(yú)describestheoptimalgraduationlevelofhumancapitalforeveryage(Idenoteywithastaronlyforsymmetrywithhú).However,inthecasewherethere56

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryisnoriskwhilestudying,allstudentswillaccumulatehumancapitalatthesamerate,h(y)=Ë1h1≠—s0≠µs”s2e≠”s(1≠—s)y+µs”sÈ11≠—s.Consequently,theywillallreachthethresholdatthesameage,whichIdenoteyúúandcalltheequilibriumgraduationage,andwhichisdeterminedbyequatingthesetwoexpressions,hú(yúú)=h(yúú).Theassociatedlevelofhumancapital,denotedhúú,canbefoundbypluggingyúúintoeitheroftheseequations.Unfortunatelythereisnoclosed-formsolution,buttheequilibriumisshownbelow:Figure15:OptimalhumancapitalgraduationthresholdandaccumulationpathThedensityofallagentsoverageisuniformandsimplyequalto1/Y.Thusthetotalmeasureofstudentsisequaltotheshareofagentswhohavenotyetreachedtheequilibriumgraduationage,yúú,andisthussimplyyúú/Y,withthetotalmeasureofworkersthereforetheremainder,Y≠yúúY.Thedensitiesofworkersandstudentsevolveasfollows:úgs(h,y)=≠ˆhË1µsh—s≠”sh2gs(h,y)È≠ˆy[gs(h,y)]hœ(h0,hú)úgw(h,y)=≠ˆh[µwhgw(h,y)]+ˆhh512‡2wh2gw(h,y)6≠ˆy[gw(h,y)]hœ(0,Œ)57

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-HendryInthestationaryequilibriumthesewillequal0.Theycanthusberearrangedasfollows:ˆy[gs(h,y)]=≠ˆhË1µsh—s≠”sh2gs(h,y)Èhœ(h0,hú)ˆy[gw(h,y)]=≠ˆh[µwhgw(h,y)]+ˆhh512‡2wh2gw(h,y)6hœ(0,Œ)Thefirstequationissimplytheadvectionpartialdifferentialequation,andthesecondistheadvection-diffusionPDE.Thesolutiontothefirstisrelativelysimple.Thedensityofstudentsagedy>yúúisofcoursezero.Thedensityofstudentsagedy<yúúbutwithhumancapitalnotequaltoh(y)isalsozero.Thusthemarginaldensityofstudentsagedy<yúúwithhumancapitalequaltoh(y)isthusequaltothetotaldensityofstudentsagedy<yúú,or1/Y.Inwords,studentsofagivenageallhavethesamelevelofhumancapital,becausetheyaccumulateitatthesamerate,thusthedensityofstudentsatthatagewiththatlevelofhumancapitalissimplythedensityofstudentsatthatage.Achangeofvariablegivesthejointdensityacrosshumancapitalandage:gs(h,y)=Y___]___[1Y1µsh—s≠”shifh=h(y)&y<yúú0otherwiseThesecondPDEalsohasawell-knownsolutionthathumancapitalislog-normallydistributed,withthevarianceincreasinginage:hhúú|y≥LN11µw≠‡2w22(y≠yúú),‡2w(y≠yúú)2.Wecanthususethepropertythatthemomentsofalog-normallydistributedvariablearegivenbyE[hn|y]=(húú)nen(µw≠12‡2w)(y≠yúú)+12n2‡2w(y≠yúú)tofindaggregatevariables.Forexample,tax58

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryrevenueis:Taxrevenue=⁄Yyúú⁄Œ01wnh≠·0(wnh)1≠·12gw(h,y)dhdy=⁄YyúúEËwnh≠·0(wnh)1≠·1|yÈ1Ydy=⁄YyúúÓwnhúúeµw(y≠yúú)≠·0(wnhúú)1≠·1e(1≠·1)(µw≠12·1‡2w)(y≠yúú)Ô1Ydy=wnhúúµwY1eµw(Y≠yúú)≠12≠·0(wnhúú)1≠·1(1≠·1)(µw≠12·1‡2w)Y1e(1≠·1)(µw≠12·1‡2w)(Y≠yúú)≠12Theequilibriumisthusfullydeterminedby3equations:hú(yú):(·0(wnhú)1≠·1)1≠“v(n)Ÿ(1≠“)11≠eŸ(yú≠Y)2¸˚˙˝Vw(hú,yú)=T1≠“ﬂ(1≠“)≠(·0(wnhú)1≠·1)1≠“v(n)ﬂ(1≠“)eŸ(yú≠Y)¸˚˙˝Vs(hú,yú)+(1≠·1)(·0(wnhú)1≠·1)1≠“v(n)ﬂŸhú11≠eŸ(yú≠Y)21µshú—s≠”shú2¸˚˙˝Vs(hú,yú)yúú:hú(yúú)=53h1≠—s0≠µs”s4e≠”s(1≠—s)yúú+µs”s611≠—s¸˚˙˝h(y)·0:⁄Yyúú⁄Œ01wnh≠·0(wnh)1≠·12gw(h,y)dhdy¸˚˙˝Taxrevenue=yúúY(T+F+G)¸˚˙˝StudentstransfersSincesolvingforthisequilibriumisnotastrivialasintheBaselinemodelÐwhichsimplyinvolvessolvingasystemofnon-linearequationsÐletmebrieflyoutlinethe(pseudo-)algorithmIusetosolveit(foragiven·1andT):1.Createagridforyandguessaninitial·02.Findthevaluefunctions,Vw(h,y)andVs(hú,yú),usingtheformulaeabove3.Foreachgridpointiny,findthehúthatsatisfiesthevaluematchingequationusingaroot-findingalgorithm,yieldingavectorfor≠æhú4.Interpolatethediscretemappingy‘æ≠æhú(Iusesplines),yieldingthecontinuousfunction59

ShouldIStay(inSchool)orShouldIGo(toWork)LeeTyrrell-Hendryhú(yú)5.Findtheyúúthatequateshú(yú)andh(y)usingaroot-findingalgorithm,andhencefindhúúfromhú(yúú)orh(yúú)6.Calculatethegovernmentbudgetsurplususingtheequationabove,andupdate·0(Iusebisection)untilitbalancestowithinsometoleranceAggregateworkerwelfareisgivenby:Ww=⁄Yyúú⁄Œ0Vw(h,y)gw(h,y)dhdy=⁄YyúúE[Vw(h,y)|y]1Ydy=⁄YyúúC(·0(wnhúú)1≠·1)1≠“v(n)Ÿ(1≠“)11≠eŸ(y≠Y)2e(ﬂ≠Ÿ)(y≠yúú)≠1ﬂ(1≠“)D1Ydy=(·0(wnhúú)1≠·1)1≠“v(n)Ÿ(1≠“)C1ﬂ≠Ÿ1e(ﬂ≠Ÿ)(Y≠yúú)≠12≠1ﬂe≠Ÿ(Y≠yúú)1eﬂ(Y≠yúú)≠12D1Y≠1ﬂ(1≠“)Y≠yúúYAggregatestudentwelfareis:Ws=⁄yúú0⁄húúh0Vs(h,y)gs(h,y)dhdy=⁄yúú0Vs(h(y),y)gs(y)dy=T1≠“≠11≠“⁄yúú01Ydy+CVw(húú,yúú)≠T1≠“≠11≠“D1Y⁄yúú0e≠ﬂ(yúú≠y)dy=T1≠“≠11≠“yúúY+CVw(húú,yúú)≠T1≠“≠11≠“D1≠e≠ﬂyúúﬂYAndtotalaggregatewelfareacrossallagentsisthesumofthetwo.60

EMForeverBlowingBubbles:GlobalImbalances&theLimitsofFiscalSpaceLeeTyrrell-HendryUniversityofEdinburghSeptember2022(Clickhereforlatestversion)AbstractWhyhastherealinterestrateonpublicdebt,r,recentlyfallenbelowthegrowthrateoftheeconomy,g,andwhatdoesitimplyforthelimitsofgovernmentborrowing?Idiosyncraticinvestmentriskcanmotivateprecautionarysavinginsafeassetslikegovernmentdebt,whichpushesdowninterestrates.ButIarguetwofactorsÐsofarmostlyoverlookedinthisliteratureÐexplainmuchofthedeclineinrvsgoverrecentdecades:(1)theincreasingpresenceofemergingmarket(EM)economies,whoseinvestorsfaceevengreateridiosyncraticriskandhencehavestrongerprecautionarysavingmotives;and(2)limitstotheprivatesupplyofsafeassets.Thesetwoelementsbothafforddevelopedmarket(DM)governmentsgreatercapacitytoborrowanddampenthenegativeside-effectsofdebt-financedgovernmentspendingshocks,liketheresponsetothepandemic.Nonetheless,fiscalspaceisnotunlimited,andgovernmentsshoulddesignfiscalrulesthatensuredebtisbothsustainableinthelong-runandstabilisingintheshort-run.1

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendry1IntroductionWhentherealinterestrateongovernmentdebt,r,isbelowthegrowthrateoftheeconomy,g,thetraditionalarithmeticofgovernmentbudgetconstraintsceasestoapply:thevalueofagovernmentÕsdebtisnolongerequaltothepresentvalueofitsfutureprimarysurpluses.Instead,agovernmentcanrunapermanentdeficitorfinanceaone-offexpenditurewithdebtandrollitoverwithoutsubsequentlyrunningsurpluses.Governmentdebtisabubble,whichwithinlimitscanbeminedforadditionalspending,inthelanguageofBrunnermeieretal.(2021a).Thisisthesituationdevelopedmarket(DM)economiesoftodayfindthemselvesin,asexemplifiedbyrecentestimatesofthenaturalrealinterestrateandtrendgrowthratefromHolstonetal.(2017),showninFigure1.Inthispaper,Itaketheviewthatpartoftherecentdeclineinrvsgisduetoincreasingdemandforsafeassetsfromemergingmarket(EM)economieslikeChina,aswellasdeclinesintheprivatesupplyofsafeassetsfollowingthefinancialcrisis,andIexploretheimplicationsofthesefactorsforthefiscalspaceofDMgovernments.Figure1:r⇤vsgindevelopedmarketeconomiesoverthelast50yearsSource:Estimatesofr⇤andgfromHolstonetal.(2017)areaGDP-weightedaverageoftheirestimatesfortheUS,Canada,UKandEuroarea,usingOECDestimatesofGDPPPP.Thepre-1995Euroareaweightisthesumoftheweightsoftheoriginal11Euroareacountries.2

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryr<ghasledsometoconcludethatgovernmentsofadvancedeconomiesshouldspendmore(Blanchard2019;Furman&Summers2020).Othersarguethatwhileincreasingspendingwithdebtfinancemaybefeasible,itmaynotbewelfare-enhancing,sincetheeconomyisstilldynamicallyefficient,asthemarginalproductofcapitalisstillabovethegrowthrate(Reis2021;Ball&Mankiw2021).Othersareagnosticonwelfare,butsimplytrytoexplainthephenomenon,aswellastherelatedobservationthatgovernmentdebtappearstobevaluedinthemarketsignificantlymorethanthepresentvalueofexpectedfuturesurpluses;thisbubbleisontheorderof3xGDP,accordingtoJiangetal.(2021).Twomainexplanatoryframeworkshaveemergedintheliterature:theoverlappinggenerationsframework;and,morerecently,theidiosyncraticriskframework.Theoverlappinggenerationsframeworkiswell-establishedandgoesbacktoSamuelson(1958)andDiamond(1965),whileTirole(1985)andmorerecentlyFarhi&Tirole(2012)inparticularaddressthepossibilityofassetbubblesinsuchaframework.Insuchmodels,agentsÕneedtosavefortheirretirementiswhatdrivesdowninterestrates.Thisframeworkyieldsanaturalexplanationfortherecentdeclineinrvg:demographicchanges.Todaythepopulationisolder,spendslongerinretirementandperhapsalsoanticipateslessgenerouspublicpensions,necessitatinghighersavingandpushingdowninterestrates.SeeEggertssonetal.(2019)orAuclertetal.(2021)foradiscussionofthisview.ItisthisframeworkthatBlanchard(2019)andothershaveusedtoexploretheimplicationsofr<gforfiscalspace.Thesecond,morerecentstrandoftheliteraturereliesonidiosyncraticriskÐandtheprecautionarysavingitmotivatesÐasthekeymechanismpushingdowninterestrates,startingwithBrunnermeieretal.(2021a)and(2021b),knownhenceforthasBMS.Thesepapersalsotrytoestablishthelimitsoffiscalspace,extendingthefiscaltheoryofthepriceleveltoallowforabubbleingovernmentdebt.Whiletheidiosyncraticriskframeworkcaninprincipleexplainwhyr<g,itdoesnotbyitselfofferacompellingrationaleforwhyrissomuchlessthangtodaythanitwas50yearsago.Onecanarguethatidiosyncraticriskhasincreased;thisishardtoobservedirectly,butwecanpotentiallyseesomeoftheconsequencesofthis3

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrylikehigherincomeandwealthinequality,asdiscussedforexampleinMianetal.(2021c)and(2021b),whofurthermorearguethattheevidencesuggestshigherinequalityexplainsthedeclineininterestratesbetterthandemographics(althoughtheydonotattributethistoidiosyncraticrisk).Wecanalsopointtosometentativecausessuchasthefinancialcrisis,whichobviouslyprecipitatedalargefallinrealinterestrates.MymotivationinthispaperthenistosupplythisrationaleexplainingtheseculardeclineininterestratesÐglobalimbalancesandadearthofsafeassetsÐandhencetostrengthenthecasefortheidiosyncraticriskframework.MoreoverIexploretheconsequencesforthefiscalspaceofgovernments.IthusextendBMSbyincorporatingglobalfinancialmarketsandlimitsonprivatesafeassetsupply,althoughmymodelisrealandnotconcernedwithnominalaspects.Theglobalimbalances-safeassetshypothesisstemsfromtwostylisedfactsthathaveemergedinthelasttwodecades:onerelatingtothedemandforsafeassetsandtheotherrelatingtotheirsupply.Onthedemand-side:thegrowthofChinaandotherEMeconomies.Even20yearsago,Chinaaccountedforlessthan3%ofglobalwealth,todayitaccountsfornearlyafifth(Shorrocksetal.2021).IthasconcomitantlybecomeoneofthelargestholdersofUSTreasuriesandotherDMgovernmentbonds.EMinvestorsÕdemandforsafeassetsmayinpartbebecausetheyfacehigheridiosyncraticriskthaninvestorsindevelopedeconomies,formyriadreasons.Onthesupply-side:thediminishedprivatesupplyofsafeassetssincethefinancialcrisis.IssuanceofAAA-ratedmortgage-backedsecuritiestotalledaround$1trillionayearpriortothefinancialcrisis,onlytobesubsequentlyÒdecisectedÓÑcutinone-tenth.Thesetwofactorshavebeendiscussedintheliteratureonglobalsavingsgluts,suchasAngeletos&Panousi(2011)orCaballeroetal.(2008),(2017)and(2020),butsofartomyknowledgenoonehasdiscussedtheirimplicationsforthefiscalspaceofDMgovernments.InthispaperIthuslinktheliteraturesonidiosyncraticrisk-inducedprecautionarysavingandthatonglobalimbalancesandprovideatractableframeworkwithinwhichtoanalysetheeffectsofbothonthefiscalspaceofgovernments.4

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryAsinBMS,inthispaperinvestorsfaceidiosyncraticriskontheircapitalholdings,whichdrivesprecautionarysavingintorisk-freegovernmentbondsandpullstheinterestratebelowthegrowthrate.Myinnovationistonestthismodelinatwo-countryframework,reminiscentofAngeletos&Panousi(2011).Likethatpaper,Itaketheviewthatthedegreeofidiosyncraticriskfacingentrepreneursinacountryrepresentsthelevelofdevelopmentinthatcountry:lessdevelopedcountriesfacehigherrisk,forexampleowingtoworselegalprotectionsforbusinessownersorlessdevelopedfinancialmarketswithlesscapacityforrisk-sharing.InovellyusethisframeworktoexploretheimplicationsofglobalimbalancesÐdrivenbyidiosyncraticriskÐforfiscalspace.Sinceinvestmentisriskier,theprecautionarysavingmotiveisevenstrongerinthedevelopingworld,sointegrationofglobalcapitalmarketsresultsinemergingmarket(EM)investorslendingtodevelopedmarket(DM)entrepreneursandgovernments,pullingdownDMinterestratesfurther,consistentwiththeobservationthatcapitaltendstoflowfrompoorertorichercountries(Lucas1990,Gourinchas&Jeanne2013).Iintroducetwonotionsoffiscalspaceinthispaper:theÒBubbleBoundÓÑthelevelofpublicdebtbelowwhichthegovernmentfacesnegativeinterestratesandsocanexploititsbubble;andtheÒProfligacyPeakÓÑthemaximumdeficitthatthegovernmentcansustain.IprovethatforDMgovernmentstheBubbleBoundishigherunderintegratedglobalcapitalmarketsthanunderautarky,andIdemonstratethattheProfligacyPeakalsotendstobelarger;moreoverbothare,undercertainweakconditions,increasinginthedegreeofidiosyncraticriskathomeandabroad.Integratedfinancialmarketscanthusmateriallyexpandtheamountoffiscalspaceavailabletogovernmentsinthedevelopedworldcomparedtoautarky,allelseequal:governmentscansustainhigherdebtrelativetoGDPbeforethebubbleburstsandtheyhavetostartrunningprimarysurpluses,andtheycansustainhigherdeficits.However,ifbothDMandEMgovernmentsareattemptingtousethemaximumamountoffiscalspaceavailable,integratedfinancialmarketsdonotnecessarilyofferanadvantageoverautarky.ToborrowandextendthecolourfulmetaphorofBrunnermeieretal.(2021a),financialintegrationallowsgovernments5

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendrytomineanothercountryÕsbubble,butdoesnÕtbyitselfexpandthecombinedsizeofthetwobubbles.Oneimplicationoftheaboveisthatintegrationcanposeriskstodevelopingcountries,iftheyarealreadyclosetotheirBubbleBound.Integratingwithamoredeveloped(lessrisky)country,thatinthenewequilibriumwillbeanetborrower,canleadtocapitalflightandpermanentlyloweroutputintheEMcountry,whilesubstantiallyloweringtheamountoffiscalspaceavailabletoitsgovernment.TheresultthatintegrationcanreducetheoutputofemergingmarketeconomiesandraisethatofdevelopedeconomiesisthereverseoftheresultsinforexampleAngeletos&Panousi(2011),Corneli(2017)&(2021);itisduetosubtlemodellingchoices,butperhapsdemonstratesthefragilityoftheconventionalwisdomandtherisksofintegratedcapitalmarkets.Themodelinthispaperisstylised,butmayofferinsightsintotheperformanceoftheRussianeconomyandthatofotherformerSovietcountriesfollowingthecollapseoftheUSSR.Uponintegratingwiththeglobaleconomy,thesecountriesexperiencedsignificantcapitalflight,asharpfallinoutputandanimmediateneedtoreducegovernmentborrowing,andÐwhenthelatterwasnotachievedÐexperiencedrapidinflationandlaterafinancialcrisis.Thesecondkeyinnovationinthispaperoverpriorresearchonfiscalspaceistoincorporatelimitsontheprivatesupplyofsafeassets.Imodeltheselimitsasacollateral-basedcreditconstraint,`alaKiyotaki&Moore(1997),asareduced-formwaytocaptureconstraintsontheabilityofthefinancialsectortotransformriskybusinessandpersonalloansintorisklesssecurities.Suchconstraintsseemevidentinlightofthecollapseoftheasset-backedsecurities(ABS)marketfollowingthefinancialcrisis,andasshowninFigure1itisfollowingthatcrisisthat(natural)realinterestratesbegantofallsubstantiallybelowtrendgrowth.InthispaperIexploretheconsequencesofsuchconstraintsforgovernmentsÕfiscalspace.IprovethattheBubbleBoundislargerwhentheseconstraintsbind,andshowquantitativelythattheProfligacyPeakisalsohigher.Moreover,creditconstraintssubstantiallylessenthecrowding-outeffectsofpublicdebtforthecountrywhoseentrepreneursareconstrained,since6

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrybyconstructionotherinvestorswhocouldpotentiallybetterusethefundsareconstrainedfromborrowing.NotethatReis(2021)alsofeaturesaborrowingconstraint,butitplaysaslightlydifferentrole,whichIdiscussindepthbelow.Iamnotthefirsttotrytotriangulateconcretelimitsonfiscalspace.Mianetal.(2021a)alsotrytoquantifytheBubbleBoundandProfligacyPeak(underdifferentnames).Theirpaperusesarepresentativeagentframeworkwithbondsintheutilityfunctiontoincentivisesaving.Theyalsoconsidertheimpactofthezerolowerbound,whichhastheeffectofraisingrealratesandreducingfiscalspacewhenitbinds.Kocherlakota(2021)offersabenchmarkcasewheretherearenolimitstofiscalspace.HeconsidersavariationofanAiyagari(1994)modelwithanadditionalextremeidiosyncraticstate,reachedwitharbitrarilylowprobabilitybutwitharbitrarilyhighandconstantmarginalutilityofconsumption.InthisÒall-you-can-eatÓstate,thehouseholdfeelscompelledtoconsumealltheirsavings,forexamplebecausetheysufferanadversehealthshockandmustpaytheirmedicalbills,althoughitisnotgivensuchaninterpretationinthepaper.Thepresenceofsuchastateyieldsextremeprecautionarysavingattheindividuallevel,andawillingnesstoabsorbarbitrarilyhighquantitiesofpublicdebtatnegativeinterestratesattheaggregatelevelÑaninfiniteBubbleBound.Ialsoconsidertheoptimallevelofpublicdebt,addingtoearlierresearchbyAiyagari&McGrattan(1998),Desbonnet&Kankanamge(2016)ormorerecentlyLeGrand&Ragot(2022).Theyhoweverexamineoptimaldebtlevelsinthepresenceofdistortionarytaxesandgovernmentspending,Idothesameinamodelspecificallydesignedtoneutralisethedistortingeffectsoftaxesandspending,andhencetoisolatethewelfareeffectsofpublicdebt.Oneshouldthereforeviewthewelfareresultsinthebodyofthispaperasabenchmarkboundagainstwhichtoevaluatemorequantitativelyrealisticmodels.Ifindtheoptimallevelofpublicdebtispositiveandmuchhigherthanthesepriorauthorsfind,andmuchhigherthanthelevelsobservedinadvancedeconomiestoday;severalmultiplesofoutput.However,debtattheselevelsiswellbeyondtheProfligacyPeakorBubbleBoundnotionsoffiscalspacediscussedabove;thebubbleinpublicdebtisdeflated.7

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryMuchoftherecentresearchinsteadfocusesonwhetherornotdebtissuancecanbePareto-improving,likeBlanchard(2019),Reis(2021),Aguiaretal.(2021)orBrummetal.(2021).Thelatterofwhichalsoevaluatestheeffectsofinternationaltradeandcapitalmarkets,likewisefindingtheyexpandthepotentialforwelfare-enhancingdebtissuance,althoughtheyprimarilyconsiderthecaseoftwosymmetriccountries,ratherthanthatofsystemicglobalimbalances.However,debtissuanceinmymodelisingeneralnotPareto-improving,evenwhenr<g,sinceissuingdebtlikelyraisesr,whichtypicallycrowdsoutcapitalandhencelowersworkersÕwages.TobePareto-improving,thedebtmustbeabletofundincreasesintransferstoworkersthatmorethancompensatefortheirlowerwages;thisonlyoccurswhenr⌧g,muchmoresothanisthecasetoday.Ifurthermoreexplorethedynamicsofthemodel,subjectingtheeconomytoanunexpected,transitory,debt-financed10%ofGDPshocktogovernmentspending,similartotheresponsetotheCovid-19pandemic.Ifindthatfinancialintegrationdampensthenegativeeffectofthespendingshockontheoutputofthedevelopedcountry,relativetoautarky;creditconstraintsfurtherdampenanddelaythenegativeeffectsrelativetotheunconstrainedcase.Butr<gdoesnotgivegovernmentsfreereintospendandborrowmore.Myanalysissug-gestsanumberofcautionsforgovernmentslookingtoexploitlowinterestrates.Governmentsshould:1.Targetasteadystatelevelofdebt,ratherthanthedeficit,inordertoensureasteadystateequilibriumexists2.Respondsufficientlyaggressivelytoexcessivedebt,inordertoensurethesteadystateis(locally)stable3.Followlargespendingshockswithsufficientlyaggressiveausterity,inordertoensuretheeconomyreturnstothe(desired)steadystate,asdynamicsmaybegloballyunstableeveniftheyarelocallystableWhiletheseprinciplesaccord,Ithink,withtheconventionalwisdomand,arguably,withthe8

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryhistoricalrecordinmostdevelopedcoutries(e.g.Bohn1998,Chenetal.2021)andwithcertain(somewhatarbitrary)fiscalrulescurrentlyinplace(butonlydubiouslyenforced),liketheEUÕsMaastrichtcriteria;theyarearguablyinconsistentwiththecurrentprojectedpathoffiscalpolicyintheUS,UKandotherdevelopedeconomies,aswellaswithcertainotherfiscalrulesthathavebeenrecentlyproposedinvariouscountries(e.g.McNicol2017).NotadheringtotheseprinciplesmayresultintheeconomyÒblowingupÓ,withanequilibriumeithernotexisting,orwiththeeconomydivergingawayfromsteadystatefollowingashock.Themodelinthispaperisreal,butinamonetarymodelwithnominaldebtclaims,suchascenariowouldimplyhyperinflation,likethenon-monetarysteadystateofBMS(2021b).Anovelresultinmypaperisthateveniftheeconomyavoidshyperinflationinresponsetolargeshocks,itmaystillfaceaÒdebttrapÓ,whereitapproachesasecondsteadystatefeaturinghigherdebtandloweroutputandremainsthereforanextendedperiod.Itmayprovechallengingtoescapesuchadebttrap,necessitatingsufficientlylargeausteritytoescapeandreturntothenormalsteadystatewithinareasonabletimeframe.Notethatthisappliesevenwhenr⌧g.Insum,thepaperdemonstratesthataccountingforglobalsavingimbalancesandcon-straintsonprivate-sectorsafeassetcreationiscriticalwhenassessingtheamountoffiscalspaceavailabletogovernmentstoday,andoffersatractableframeworkwithinwhichtodothat.TheincreasingimportanceofChinaandtherestofthedevelopingworldinglobalcapitalmarkets,aswellasthelimitedabilityoftheprivatesectortosupplysaidcapitalmarketswithsufficientquantitiesofsafeassets,suggestsgovernmentsofadvancedeconomiesmayhaveconsiderablymorefiscalspaceavailablethaninthepast,thoughstillwithinclearlydefinedlimits.Atthesametime,themodelsuggestscautionforgovernmentsofemergingeconomies:integratingwithglobalisedcapitalmarketscanleadtocapitalflightandsubstantiallyreducetheirabilitytoborrow.9

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendry2ModelofGlobalImbalancesduetoIdiosyncraticRiskThebaselinemodelisasfollows:timeiscontinuous,t2[0,1),andhouseholdscompriseameasureoneeachofinfinitely-livedentrepreneursandworkers.Workersarehomogeneousandcannotsave,soconsumealltheirincomefromlabour,whichtheysupplyinelasticallyinequilibrium.Thefirmsideconsistssolelyofentrepreneurs.Entrepreneurseachoperatetheirownfirmandowncapital,k>0,andhirelabour,l>0,atwage,wt,usingwhichtheyproduceoutput.Theycanalsopurchaseorlendrisk-freebonds,b2R,onwhichtheyearntherealinterestratert.Theirnetwealthisthusa=b+k,andevolvesovertimeasúa=úb+úk.Therearetwocountries,i=H,F,eachwithitsownentrepreneurs,workersandgov-ernments.Iabstractfrominternationaltradeingoodsandcurrencymarketsbyhavingentrepreneursinbothcountriesproducethesamegood,whichtradesatthesameprice,resultingincurrenciesthattradeatparity.TheframeworkborrowsfromAngeletos&Panousi(2011),extendedtoincorporateagovernmentsectorandcreditconstraintsonentrepreneurs.However,unlikethatpaper,Iassumethatentrepreneursthemselvesdonotwork,butinsteadhireworkerswhocannotsave.Imakethesesimplifiyingassumptionsfortworeasons:(1)itallowsforaclosed-formsolutionfortheentrepreneursÕpolicyfunctionsforconsumptionandcapitalinthepresenceofcollateral-basedcreditconstraints;and(2)itallowstheinterestratetobenegative,forreasonsthatwillbecomeclearwhenIanalysetheexistenceofsteadystateequilibriuminsection3,whichisobviouslycentraltomyanalysis.Entrepreneurialprofitsaregivenbyrevenuelesslabouranddepreciationcosts,subjecttoanidiosyncraticshockthatisproportionaltotheentrepreneurÕscapitalstock.Revenuesimplyequalsoutput,producedaccordingtoaconstantreturnstoscaleCobb-DouglasproductionfunctionoftheentrepreneurÕscapitalandlabourdemand,astheconsumptiongoodisthe10

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrynumeraire.AnentrepreneurÕsflowprofitsarethus:d⇡t=k↵tl1↵twtltktdt+ktdWtwhereWtisaBrownianmotion,representingtheidiosyncraticrisktotheentrepreneurÕsprofitsthataveragesoutacrossagents.Onecaninterpretthismoststraighforwardlyasadepreciationshocktotheircapitalstock.>0isthestandarddeviationofthisshock.Thereisnoaggregaterisk,butIdoexploreMITshocksunderaperfectforesightequilibrium,soIallowprices,andhencetheentrepreneursÕvaluefunctionsandoptimalpolicies,tovaryovertime.↵2(0,1)representsthecapitalintensityofproductionand>0the(average)rateofdepreciation.Theentrepreneurconsumesandsavesoutofprofitsandinterestfromtheirrisk-freesavings,sotheirbudgetconstraintis:dat=d⇡t+(rtbtct)dtPlugginginprofitsandsubstitutingoutbondholdingsgives:dat=k↵tl1↵twtlt(rt+)kt+rtactdt+ktdWtWecanimmediatelysolveforoptimallabourdemandandreplaceinthebudgetconstrainttosimplify:lt=✓1↵wt◆1↵ktEntrepreneursmayalsofacecreditconstraints,suchthattheycannotborrowmorethanafraction✓2[0,1]oftheircapital:b✓k,orequivalentlythattheircapitalholdingsmustbenomorethanafraction=11✓1oftheirnetworth:ka.Intheabsenceofthecollateralconstraint,entrepreneursalsofacealessstrictno-Ponziconditiontoruleoutinfiniteaccumulationofdebt.ThepreciseformthistakesisnotimportantÐasimple(non-positive)lowerboundonnetwealthwillsuffice:aaÐsincewithPonzischemesruledout,itwill11

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryalwaysbeoptimaltokeepwealthpositive,giventheabsenceofnon-financialincome.EntrepreneursÕproblem.TheentrepreneursÕproblemistomaximisetheirlifetimeutility,discountedatrate⇢>0,bychoosingconsumption,capitalandlabourdemand,giventheirinitialwealthattimet0,a0>0,andsubjecttotheir(stochastic)budgetconstraintandcollateralconstraintorno-Ponzicondition:V(t0,a0)=max{ct,kt}t2[t0,1)Et0Z1t0e⇢(tt0)u(ct)dts.t.dat=((Rtrt)kt+rtatct)dt+ktdWtktatwhereRt=↵⇣1↵wt⌘1↵↵istheexpectedreturnoncapital.IassumeentrepreneurshaveCRRApreferences,u(c)=c11,with1thecoefficientofrelativeriskaversion.NoteIhavealreadysubstitutedouttheentrepreneurÕslabourdemandproblem,sotheresultingproblemissimplyanalogoustoaMertonmodel:chooseafractionofwealthtoinvestintheriskyasset,afractionintherisk-freeasset,andaflowconsumptionrate.Thisproblemcanbemoreeasilysolvedrecursively,andtothisendcanbeexpressedasaHamilton-Jacobi-Bellmanequation:⇢V(t,a)=maxc,kau(c)+@aV(t,a)((Rtrt)k+rtac)+12@aaV(t,a)2k2+úV(t,a)(1)[email protected]Õproblemcanbesolvedwithasinglestatevariable,individualwealtha,sincecapitalandbondwealthareperfectlyfungible(theproblemwouldstillonlyrequireonestatevariableevenifcapitalforexamplewerenotperfectlyfungible,providedadjustmentcostswereconvex).However,sinceIallowforMITshocksanddeterministictransitiondynamics,thevaluefunctionandtheentrepreneursÕpolicyfunctionsdependonpricesandhencemaychangeovertime.SinceentrepreneurshaveCRRApreferences,theirvaluefunctionwillhavethatsameformÐisoelasticinnetwealthÐand12

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrytheirpolicyfunctionswillbelinearinnetwealth.ThepresenceofaborrowingconstraintofthisformpresentsnoissueforfindingananalyticalsolutiontotheentrepreneursÕproblem.Whentheconstraintdoesnotbindthesolutionissimplythatoftheunconstrainedproblem,andwhenitdoesbind,thesolutioncanbefoundsimplybysubstitutingthebindingconstraintk=adirectlyintotheHJBequation.Inessence,theentrepreneursÕoptimalinvestmentdecisionismyopic;theyÒseeÓtheconstraintonlywhenitbinds,asshowninVila&Zariphopoulou(1997).Lemma1.TheentrepreneursÕvaluefunctionisasfollows:V(t,a)=8>><>>:mt1a1if6=1Et+1⇢ln(a)if=1whereEtisdeterminedbythefollowingODE:úEt=⇢Et+1ln(⇢)1⇢✓(Rtrt)!t+rt122!2t◆TheentrepreneursÕpolicyfunctionsare:c(t,a)=8>><>>:mtaif6=1⇢aif=1k(t,a)=min⇢Rtrt2,a⌘!tab(t,a)=(1!t)awheretheirmarginalpropensitytoconsume,mt,isdeterminedbythefollowingODE:úmtmt=mtrt+rt⇢+1(Rtrt)!t+2(1)2!2tTheirEulerequationis:dctct=✓rt⇢+Rtrt!t+2(1)2!2t◆dt+!tdW13

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryTheentrepreneursÕoptimalchoicesofconsumption,andhencetheiroptimalpathsofwealthgiveninitialwealtha0,alsosatisfythetransversalitycondition:limt!1Et0⇥e⇢(tt0)u0(ct)at⇤=0.Proof.SeeAppendix.RepresentativeworkerÕsproblem.Entrepreneursdonotworkinthismodel,topreservetheanalyticalsolution.InsteadIassumethatentrepreneurshirehand-to-mouthworkerswhoareunabletosaveandwhoalsopaytaxes/receivetransfers,⌧t,thatareproportionaltotheirincomeandwhichwillbeonepossiblesourceofgovernmentdeficits/surpluses.Asaresult,theworkersÕproblemissimpleanddoesnotaffectthedynamicsoftheentrepreneursÕwealth/capitalaccumulationandconsumption.Iassumeworkerssupplyoneunitoflabourinelastically.Sinceworkershavenowealththisisequivalenttoassumingtheyhavepreferencesconsistentwithbalancedgrowth.InthiscasetheworkersÕconsumption-labouroptimalityconditionwouldbe:ctl’t=wt(1⌧t),where’istheinverseFrischelasticityoflaboursupply,whichgiventheyconsumetheirincomeresultsinconstantlaboursupplylt=1.3AggregateEquilibriumSincewehavearepresentativeworkerwhosuppliesaunitlabourinelastically,aggregatelaboursupplyisalsoLt=1.CombiningthiswithentrepreneursÕaggregatelabourdemandÐandhenceimposingequilibriuminthelabourmarketÐyieldsanexpressionforthewagerateintermsoftheaggregatecapitalstock:wt=(1↵)K↵t.Giventhatallentrepreneursareex-anteidentical,andgiventhattheirpolicyfunctionsarealllinearintheirnetwealth,aggregationissimple;theaggregateeconomysimplybehaveslikeascaled-upversionofanindividual,withidiosyncraticriskwashingoutintheaggregate.Thecorrespondingaggregatevariablesaredenotedwithcapitalletters:Ct=Z10c(t,a)g(t,a)daKt=Z10k(t,a)g(t,a)daBt=Z10b(t,a)g(t,a)da14

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryThereisadistributionofwealth,g(t,a),whichonecancharacterise,butithasnoeffectontheaggregateeconomy.Inaddition,creditconstraintswilleitherbindforallentrepreneursinacountryornone.Therearetwocountriesinthemodel,denotedHandFforHomeandForeign,representingthedevelopedworldandthedevelopingworld,andfromhereonIsubscriptquantitiesandpricesinagivencountrywiththeirrespectiveindexorwithiingeneral.Themodelisclosedthroughbondmarketequilibrium:anAutarkicequilibrium,whereeachcountryÕsbondmarketclearsseparately,withdomesticbondholdings,Bit,equallingdomesticpublicdebtoutstanding,Dit;oranIntegratedequilibrium,wherebondmarketsclearglobally.Definition1.AnAutarkicequilibriumintheeconomywithhand-to-mouthworkersisasequenceforaggregatevariables{Kit,Bit,Cit}1t=0fori=H,F,aggregatedfrompolicyfunctions{ki(t,a),bi(t,a),ci(t,a)}1t=0,whichmaximiseagentsÕlifetimeutilitygivenprices{rit,Rit,wit}1t=0,andwhereallmarketsclear,includingbondmarketsineachcountryBit=Dit.NotethatsinceworkersÕconsumptionhasnoeffectonaggregatedynamics,Iignoreitindiscussionoftheequilibrium;aggregateconsumptionreferstothatofentrepreneursalone.Definition2.AnIntegratedequilibriumisasequenceforaggregatevariables{Kit,Bit,Cit}1t=0fori=H,F,aggregatedfrompolicyfunctions{ki(t,a),bi(t,a),ci(t,a)}1t=0,whichmaximiseagentsÕlifetimeutilitygivenprices{rt,Rit,wit}1t=0,andwhereallmarketsclear,includingglobalbondmarketsBHt+BFt=DHt+DFt.Proposition1.Theaggregatedynamicsofthetwo-countrymodelwithhand-to-mouthworkersaregivenbythefollowingsystemofnon-linearordinarydifferentialequationsandstatic15

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryequations:úKit+úBit=RitKit+ritBitCit(2)úCit/Cit=rit⇢+Ritrit!it+2i(1)2!2it(3)Rit=↵K↵1it(4)!it=min⇢Ritrit2i,i(5)wherei=H,Fforallequations.Governmentsinbothcountriessatisfythefollowingrealbudgetconstraint:úDit=ritDit(TitGit)(6)whereTitandGitaredetermineddependingonthespecificationofthemodel.UnderAutarky,thelocalrisk-freeinterestratesadjusttoclearlocalbondmarkets:Bit=Dit(7)UnderIntegratedfinancialmarkets,thereisasinglerisk-freeraterHt=rFt=rtthatclearstheglobalmarket:BHt+BFt=DHt+DFt(8)Proof.Equation(2)comesfromaggregatingtheentrepreneursÕbudgetconstraint.Equation(3)istheaggregateEulerequation,derivedbyaggregatingtheindividualÕsEulerequationgivenabove.(4)isanexpressionfortheexpectedreturnoncapital,derivedfromthatexpressionabove(intermsofwt),withtheworkersÕlaboursupplyconditionsubstitutedin,thusitalreadyimposeslabourmarketequilibrium.Equation(5)istheaggregatedoptimalportfolioshareincapital.Theorem1.UnderbothAutarkyandIntegration,anequilibriumpricevectorexistsandis16

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryunique.Proof.UnderbothAutarkyandIntegration,theexpectedreturnoncapitalandthewageratearegivenbyRi(Ki)=↵K↵1iandwi(Ki)=(1↵)K↵i.Itremainstodeterminetheequilibriumrisk-freerealinterestrate,r.UnderAutarky,combiningequations4and5with7,bondmarketequilibriumisgivenbyBi=⇣2i↵K↵1iri1⌘Ki=Di,orrearranging,theinterestrateisrAi(Ki,Di)=↵K↵1iKiKi+Di2i.UnderIntegration,combiningthosesameequationswith8,bondmarketequilibriumbecomes:B(KH,KF,r)=max⇢2H↵K↵1Hr1,✓HKH+max⇢2F↵K↵1Fr1,✓FKF=DClearlyonlythetotalstockofpublicdebtmatters,sothestatevariablesKH,KF,DpindowntheequilibriuminterestrateviathemappingrI(KH,KF,D).Thismappinghasnoclosedform,butisuniquebytheimplicitfunctiontheorem,since@@r[B(KH,KF,r)]6=0anywhereinthepositiveorthantof(KH,KF).Notethatwhencreditconstraintsbind,sayintheHomecountry,theequilibriuminterestratebecomesrI,CC(KH,KF,D)=↵K↵1FKFKF+D+✓HKH2F,withtheindicesreversedifconstraintsbindintheForeigncountry.Ihaveyettospecifygovernmenttaxesandspending,anditisactuallynotnecessarytodosotocharacterisethesteadystategiventheassumptionsIhavemade;onlytheprimarysurplus/deficitandthelevelofpublicdebtmatterfortheequilibriumÑakindofreverseRicardianequivalence.Taxes/transferstoworkersareproportionaltolabourincome,andsogiventhepreferencesofworkersoutlinedabovehavenodistortingeffectssinceincomeandsubstitutioneffectscancel.Moreover,governmentspendingmerelycrowdsoutworkersÕconsumption,sincetheypayallthetaxes,butbecauseworkerscannotsavedoesnotbyitselfdistorttheequilibrium.Thisresultispeculiartothisspecificationofthemodel,whereforsimplicityIhaveassumedconsumptionandcapitalgoodsaretotallyfungible.Ifthere17

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrywereadjustmentcostsoncapital,thenthiswouldnotholdandtherewouldbeadditionaldistortionsfromgovernmentspending,asisthecaseinBMS(2021b).ABubbleinGovernmentDebt.IntegratingthegovernmentÕsflowbudgetconstraintgivesitsintertemporalequivalent:Dt=Z1teRstr⌧d⌧(TsGs)ds|{z}PVofprimarysurpluses+limT!1eRTtr⌧d⌧DT|{z}BubbletermThisholdsintheabsenceofaggregaterisk,soprimarysurplusesarediscountedattherisk-freerate,andintheabsenceofseignoragerevenues,i.e.whentheinterestrateonmoneyisthesameasthatongovernmentdebt.Thelatterlikelyapproximatelyholdstruetodayinmostdevelopedcountries,asmostcentralbanksnowpayinterestonreservesÐwhichtodaycomprisethebulkofhigh-poweredmoneyÐnotfarbelowthatondebt.Thereisnothingtoruleouttheexistenceofabubbleinthismodel;thelasttermneednotequalzeroowingtosometransversalityorno-Ponzicondition,andinsteadystatewillindeedbepositivewhenevertheinterestrateisbelowthegrowthrateoftheeconomy,r<g,asdiscussedinBMS(2021b).InthispaperIabstractfromlongrungrowth(g=0),asanumberofsourceshighlightthatfactorsotherthandeclininggrowtharemostimportantforexplainingthedeclineinglobalrealinterestrates,forexampleRachel&Smith(2015),Rachel&Summers(2019a),(2019b),Mianetal.(2021c)orHolstonetal.(2017),(2020),whoupdateLaubach&Williams(2003)inestimatingthecausesofthedeclineinthenaturalrateofinterest.Thisisnotcompletelywithoutlossofgenerality,sincewhen6=1rfluctuates:1withg,butsinceformuchofthispaperIwillbeconsideringcalibrationswith=1orcloseto1,thelossisnotmaterial.Therelevantcriterionforabubbleisthusr<0,whichasshownbelowholdswheneverdomesticandorforeigninvestmentissufficientlyrisky.Inthesteadystate,theequationbecomes:18

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryDt=srlimT!1er(Tt)sr|{z}PVofprimarysurpluses+limT!1er(Tt)DT|{z}Bubbleterm=sr+limT!1er(Tt)⇣DTsr⌘|{z}Must=0inSSwheres=TGnowdenotestheprimarysurplus.Inthelongrun,forasteadystatetoexistthesetermsmustexactlyoffseteachotherbyhavingD=sr,orúD=0inthegovernmentÕsflowbudgetconstraint.Thus,abubbleinpublicdebt(r<0)doesnotmeangovernmentscanissuehowevermuchtheywant.Inthelongrun,therealvalueofpublicdebtmuststabilise,butstablepublicdebtisconsistentwithpersistentdeficits,equaltorD,ifrisnegative.Moreover,asCochrane(2021)highlights,thereisnodiscontinuityatr=g;themaximumsustainableprimarydeficitsmoothlyincreasesby”astheinterestratemovesbelow0.IfagovernmentweretomaintainapersistentdeficitlargerthanrtDt,governmentdebtwouldaccumulate,andbybondmarketequilibriumsowouldentrepreneursÕbondholdings,creatingapersistentcrowding-outofcapital,untiltherealvalueofcapitalandhenceoutputreachedzero.A(positive)steadystateequilibriumwouldnotexist.Thisisbecausethemodelhereisentirelyreal.IfIweretointroducepricesandmakegovernmentdebtanominalclaim,theproblemmayreconcileitselfthroughhyperinflation,analogoustothenon-monetaryequilibriuminBMS(2021a,2021b).SteadyStateInthesteadystate,equations2,3and6areequaltozeroinbothcountries.Basedonthediscussionabove,IproceedtoanalysethesteadystatehavingalreadyimposedúDt=0,implicitlyassumingtheprimarysurplus,s,adjuststoaccommodatechangesinrtokeepDconstantandignoringout-of-steady-statedynamics.Icanalsodropequation2,sinceitisnotnecessarytodeterminethesteadysteady,andIcanthussimplifytheotherfourequationsintothreeequationsinKH,KFandr,andultimatelyintooneequationinrasafunctionofD,asbelow,alsoassumingpreferencesarethesameinbothcountries.Proposition2.ThesteadystateunderbothAutarkyandIntegrationisdescribedbythe19

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryfollowingequationsforthecapitalstockandbondmarketequilibriumintermsoftheinterestrate(s):8i=H,FKi(r)=✓↵r++imax{µ(r),µci(r)}◆11↵(9)whereµ(r)⌘q21+(⇢r)andµc(r)⌘⇢rii(1)2ii.BondmarketclearingunderAutarky:8i=H,FBi(ri)=Di=)✓2i↵Ki(ri)↵1ri1◆Ki(r)=DiBondmarketclearingunderIntegratedfinancialmarkets:BH(r)+BF(r)=DH+DF=)max⇢Hµ(r)1,✓HKH(r)+max⇢Fµ(r)1,✓FKF(r)=DH+DF(10)Proof.Followsimmediatelyfromthesteadystateofthedynamicequilibriumequations.Notethatµ(r)andµc(r)arealsotheSharperatios,µ=Rri,whencreditconstraintsdonÕtanddobind,respectively.NotealsothatunderAutarkycreditconstraintscannotbind,providedpublicdebtisnon-negative.Doesasteadystateexistandisitunique?Inthelongrun,thegovernmenthasonechoicetomake:howmuchdebttoissue,orhowbigadeficittorun;notboth.Withthechoiceofone,thesteadystatebudgetconstraintpinsdowntheother,D=sr.However,althoughanyequilibriumcanbeattainedbyeitherchoice,theyarenotequivalent.Targetingthelong-runlevelofdebtismuchmorelikelytopindownauniquesteadystate;oneisguaranteedtoexistunderverymildconditions,anditisverylikelytobeunique,asIshowbelow.Intargetingthedeficitasteadystatewillalmostcertainlynotbeunique,andmaynotexistatall.ThusIassumehenceforththatgovernmentstargetthelong-runlevelofdebt.20

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryIntheabsenceofcreditconstraints,theexistenceofasteadystateÐwhetherunderAutarkyorIntegrationÐisrelativelyeasytoprove,andamildsufficientconditionguaranteeingexistenceisthatinvestmentinatleastonecountryissufficientlysafe:Theorem2.UnderbothAutarkyandIntegratedfinancialmarkets,intheabsenceofcreditconstraintsasteadystateforagivenlevelofglobalpublicdebtexistswhen9i:⇢+>2i2(1).Proof.SeeAppendix.Ifthisconditionisnotsatisfied,asteadystatedoesnotexistforarbitrarylevelsofpublicdebt,butdoesexistÐinfacttwoexistÐforsufficientlyhighlevels.Incharacterisingthedynamicsofthemodellateron,itwillbecomeapparentthattheeconomywilltendtowardsthesehigh-debtsteadystates.Uniquenessofthesteadystateisconsiderablyhardertoestablish,evenwithoutcreditconstraints.Asufficientconditionisthatbondholdingsaremonotonicallyincreasingintheinterestrate,whichislikelytobethecasewhentheaboveconditionholds,thoughhardtoprove.AnothersufficientbutonerousconditionunderIntegratedmarketsisasfollows:Corollary1.UnderIntegratedfinancialmarkets,intheabsenseofcreditconstraintsasteadystateforagivenlevelofglobalpublicdebtisuniquewhenonecountryisanetborrower,i.e.9i:r⇤<⇢2i2(1+).Proof.SeeAppendix.Inthepresenceofcreditconstraints,provingtheexistenceofasteadystateismorechallenging;asthebelowcalibrationssuggest,asconstraintsbindmoretightlythesteadystatemaynolongerbeunique(e.g.whenH=1.25inthecalibrationbelow),ormaynotexistatall(e.g.whenH<1.2,notshown).Figure2belowshowssomeexamplesofthepossibleexistenceanduniquenessorotherwiseofsteadystateinthemodel.Thefigureshowsaggregatebondholdingsineachcountry21

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendry(HomeÐthedevelopedmarketÐinblue,andForeignÐtheemergingmarketÐinred),andglobally(black).Thehorizontaldashedlinerepresentsglobalpublicdebt;hereitiszero,butadjustitupordownfreelyinyourhead.FormorereasonablecalibrationsÐinparticularwheninvestmentisnottoorisky,asspecifiedinTheorem2Ðaggregatebondholdingsinacountryare(monotonically)increasingintherisk-freeinterestrate.Whenthisconditionisviolated,bondholdingsareU-shaped.TheintersectionbetweeneachcolouredlineandthedashedlineistheAutarkicsteadystateineachcountry;theintersectionbetweenthethickanddashedblacklinestheIntegratedsteadystate.WhentheydonotintersectÐasmayhappenwhentheconditioninTheorem2isviolatedÐthereisnosteadystateequilibriumatthechosenlevelofpublicdebt.ThecalibrationinFigure2issuchthatparametersareequalinbothcountries,with↵=0.36,=2,⇢==0.06and=1.5,exceptF=0.4andH=0.1,unlessotherwisestatedinthefigure.Existenceofsteadystatethusrequires(ignoringcreditconstraintsforonemoment)thatforatleastonecountry,i<p⇢+’0.346.Fexceedsthisthresholdinallthreepanels;Hisbelowitinthefirsttwo(andcreditconstraintsarenotsostrict)sothatatleastonesteadystateexists,butexceedsitinthelast,sonosteadystateexists,exceptforhighlevelsofpublicdebt.22

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryFigure2:Exampleswhenasteadystateexistsandisunique,isnotunique,ordoesnotexistNotehoweverthattheuniquenessofthesteadystatehereispartlyanartifactofimposingúD=0beforespecifyingthedynamicsofdebtandhenceignoringitsout-of-steady-statedynamics.Inspecifyingthosedynamics,onemusttakeastandontheÒtargetÓsteadystateÐsincethisdeterminestaxrevenuesandhencethedeficit/surplusÐwhichwillpindownaúD=0locusthroughthephasespace;theintersectionofthislocuswiththeotherlociwillthendeterminethesteadystate.WhenIcometodiscussdynamicslateron,itwillbecomeapparentthatevenwhentheconditioninTheorem2holds,thesteadystatemaynotbeunique.Letusbrieflyconsiderthealternativeoftargetingthelong-rundeficit/surplus,withthelevelofdebtdeterminedendogenouslyasD=sr.Inthiscase,steadystateisnotguaranteedtoeitherexistorbeunique,evenwhenglobalbondholdingsareÒnicelyÓbehaved,i.e.monotonicallyincreasingintheinterestrate.AsFigure3belowsuggests,targetingasurplusisconsistentwithtwosteadystates,onewithapositivelevelofdebtandanotherwithnegativepublicdebt;asmalldeficitisalsoconsistentwithtwosteadystates,bothwithpositivedebt;butforalargedeficitnosteadystateexists.Thesesteadystatescorrespond23

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrytothesameonesthatexistwhenchoosingthecorrespondinglevelofdebt,D=sr,butmayhavedifferentstabilityproperties,whichsuggestscautionforgovernmentsthattargetthedeficit.Badshockscouldleadtheeconomytoconvergeonanothersteadystatewithloweroutputand/orwelfare,andamiscalculationoftheamountoffiscalspaceavailable,orapermanentchangeinthedegreeofrisk,couldleadtonosteadystateexistingatall,withtheeconomyspiralingtowardscollapseorhyperinflationuntiladifferentfiscalruleischosen.NoteFigure3appliesinAutarky,orunderIntegrationwhenbothcountriesfollowthesamedebt-ordeficit-targetingstrategy.Whenonecountrytargetsthedeficitbutanotherthedebt,theequilibriumwillstilllooklikethepanelontheright,butwiththehyperbolashiftedupbythedebttargetofthesecondcountry.Figure3:SteadystatemaynotexistorbeuniquewhentargetingthedeficitCrowdingOutorCrowdingIn?InastandardrepresentativeagentRamseymodelwithnon-distortionarytaxation,Ricardianequivalenceholds,soissuanceofdebtdoesnotbyitselfaffectequilibriumandsodoesnotcrowdoutcapital.Herehowever,publicdebtmayeithercrowdoutorcrowdincapital.ConsiderthelocusofthecapitalstockasafunctionoftheinterestrategiveninProposition2.Thepointonthiscurveatwhichtheeconomy24

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryrestsisgivenbytheinterestratethatensuresbondmarketequilibrium.Ifbondholdingsaremonotonicallyincreasingintheinterestrate,thenissuingmorepublicdebtresultsinahigherinterestrate.Whetherornotthiscrowdsoutcapitaldependsonwhethertheequilibriumliesinthedownward-slopingorupward-slopingregionofthecurve.Theinflectionpointisreachedwhen÷ri=⇢121+2i;abovethisinterestrate,issuingmoredebtcrowdsincapital.Thereisthusalevelofpublicdebtabovewhichissuingmoredebtcrowdsincapital,althoughforsensiblecalibrationsthislevelissohighÐseveralmultiplesofGDPÐthatwecanreasonablyignoreit.Autarky.UnderAutarky,andwithnogovernmentdebt,thereisnonetborrowing,soentrepreneursmustallocatealltheirwealthtocapital:!i=Riri2i=1.Combiningwithequation9,thisyieldsanequilibriuminterestrateofrA,NDi=⇢(1+)22i,whichnoteputstheeconomynecessarilytotheleftoftheinflectionpoint,÷ri,soissuingdebtfromthispointcertainlycrowdsoutcapital.Inthepresenceofgovernmentdebt,theequilibriuminterestratebecomesrA,D=⇢(1+)2(1 )22,where =DK+Ddenotestheshareofgovernmentdebtintotalwealth,whichclearlytendstotheno-debtinterestratewhen =0andto⇢inthelimitwheregovernmentdebtcomprisesallwealth( !1).NotethenthattheeconomyreachesthecrowdingoutÐcrowdingininflectionpointwhen÷r=rA,D,whichoccurswhen =1+,soforexamplewhen=1,governmentdebtmustaccountforhalfoftotalwealth,orontheorderofthreetimesoutputifthecapitalstockisalsothreetimesoutput,asinthedata.Wecanseethisgraphically.Wecanrewriteequation9intermsofthecapital/outputratio,KY(r)=↵r++µ(r).Wecanalsoderiveasecondequationforthecapitalstock,conditionalonbondmarketequilibriumforagivenlevelofpublicdebt:KY(r)=↵r++(1 )2.TheequilibriumcapitalstockrelativetooutputunderAutarkyisgivenbytheintersectionofthesetwocurves:KY⇤=↵⇢++2((1 )12(1 )2(1+)),whichnaturallydependsonthelevelofdebtthrough .SuchequilibriaareshowninFigure4:theleftpanelshowsbondmarketequilibriumforvariouslevelsofpublicdebtandtheassociatedlevelsofcapital;therightpanel25

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrytranslatesthesebondmarketequilibriaintolociofcapital,withtheintersectiondenotingtheequilibrium.Asdiscussed,issuingdebtfirstcrowdsout,thencrowdsincapitalbeyondacertainpoint.Figure4:Autarkyequilibrium:debtcrowdsoutcapital,exceptathighlevelsNotethatunderAutarkyandwithoutpublicdebt,theequilibriumcapitalstock,KY⇤=↵⇢++12(1)2,ishigherunderincompletemarkets(>0)thanthatundercompletemarkets(=0)when>1,astheequilibriumliesinthedownward-slopingpartofthecapitallocus.ThishaspreviouslybeennotedbyAngeletos(2007),excepttherethepresenceoflabourincomeforentrepreneursalteredtheconditionunderwhichthisholds,soinfactforreasonablecalibrationsthecapitalstockwasinfactlowerunderincompletemarkets.Thecontraryresultinthispaperissomewhatfrustratingfortheempiricalrealismofthemodel,becauseasAngeletos&Panousi(2011)pointout,itmeansriskiercountriesÐi.e.lessdevelopedcountriesÐwillhaveahighercapitalstockrelativetooutputthanmoredevelopedcountriesunderAutarky.Inthedatatheoppositeistypicallytrue;richercountriesaremorecapitalintensivethanpoorerones.ThemodelreplicatesthisstylisedfactunderIntegration,butnotunderAutarky.Thoughwedonothaveparticularlygooddataforcountriesthatare26

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrycompletelydivorcedfromglobalcapitalmarkets,likeNorthKorea,SomaliaortheformerSovietUnion,itseemsunlikelythatsuchcountrieswouldbreaktheinternationalmouldandpossesshigherrelativecapitalstocks.Thatsaid,thepresenceofrelativelyhighpublicdebtinthesecountriesoffersapotentialrouteoutofthedilemma,andinanycaseitisquestionablethatamodelbasedonoptimisingentrepreneursoperatingunderfreemarketsshouldapplytosuchcountriesinthefirstplace.Integration.Thereisunfortunatelynoclosed-formsolutionfortheequilibriuminterestrateunderIntegration.However,wecanstillcharacterisetheequilibrium:firstly,providedgovernmentdebtisnottoogreatortoosubstantiallynegative,theinterestrateunderAutarkyintheriskier,less-developedcountry,rAF,mustbelowerthanthatinthedevelopedcountry,rAH,forthereasonsestablishedabove.Secondly,ifglobalbondholdingsarewell-behaved,i.e.monotonicallyincreasingintheinterestrate,thentheequilibriuminterestrateunderIntegrationmustliebetweenthetwointerestratesthatwouldprevailunderAutarky,rAF<rI<rAH.Consequently,financialIntegrationmustlowertheequilibriumcapitalstockinthedevelopingcountryandraiseitinthedevelopedcountry,againprovidedbothcountriesareinthedownward-slopingpartofthecapitallocus.Again,theobverseofthisresultwasnotedinAngeletos&Panousi(2011),wheretheircalibrationplacesbotheconomiesontheupward-slopingpartofthecapitallocus.Thisemphasisesthefragilityofcertainpropertiesofequilibriuminthismodel,butIdonotfocusontheseanddonotrelyontheminthesubsequentdiscussiononfiscalspace.Thepresenceoftwocountriesnowmeansthatitispossibleforonecountrytobeanetborrower.Forexample,withoutgovernmentdebt,ifForeignriskisgreaterthandomesticriskthenHomeentrepreneurswillbenetborrowers,since!H>1requiresthattheequilibriuminterestunderfinancialIntegrationbestrictlylessthanthatundertheAutarkyrI,ND<rA,NDH=⇢(1+)22H.Equivalently,thisrequiresthattheForeignAutarky,No-DebtequilibriuminterestratebebelowthatofHome.GiventhattheequilibriuminterestrateunderIntegrationliesbetweentheAutarkyrates,forHomeentrepreneurstobenetborrowers27

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryrequiressimplythatH<F.InFigure5belowIshowsuchanequilibrium:thedashedlinesrepresentaggregatebondholdings,thethicklinesthecapitalstock;thebluelinesrefertothedevelopedmarket,redtotheemerging;thethickblacklineagainrepresentsaggregateglobalbondholdings.ForsimplicityIassumethereisnopublicdebtineithercountry,sotheequilibriuminterestratesoccurwhenbondholdingscrossthezerohorizontal.Figure5:IntegratedequilibriumwithnopublicdebtwhenH<FNowthatacountrycanbeanetborrower,itispossibleforcreditconstraintstobind.FromthesteadystateEulerequation,onecanimmediatelyseethatwhencreditconstraintsbindthecapitalstockisstrictlylowerthanitwouldbeintheirabsence.Fromthesteadystatebondmarketclearingcondition,onecanseethatintheabsenceofnetglobalgovernmentdebt(DH+DF=0)andwithH6=F,bondholdingscanbepositiveinatmostonecountry;entrepreneursinonecountryarenetdebtors,thoseintheothernetcreditors.Withpositivenetpublicdebt,itispossibleforentrepreneursinbothcountriestobenetlenders,butclearlynotbothnetborrowers.Consequentlycreditconstraintswillbindinatmostonecountry.28

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryTestingtheModelagainsttheDataThekeyempiricalfactthismodelisdesignedtoexplainispersistentnegativerealinterestrates,andinparticularthedeclineintherealinterestrateoverthelast20years.ThisitnaturallydoesifoneconsidersthegrowthofChinaandotheremergingmarketsoverthatperiodtobeatransitionfromanequilibriuminwhichthedevelopedworldwasessentiallyinÕautarkyÕÐoreffectivelysogiventhedeminimiswealthheldbyagentsinthoseemergingeconomiesÐtooneinwhichthedevelopedandemergingmarketsareintegrated(howeverimperfectly)andmorebalancedinsize.Themodelnaturallymakesotherempiricalpredictions,whichwecantestagainstthedata,forwhichIusethePennWorldTables,version10.0(Feenstraetal.2015),andtheWorldInequalityDatabase(Chanceletal.2021).ThemostobviousempiricalpredictionistheglobalimbalancesofthetitleÐthatinequilibriumcapitalflowsfromdevelopingcountriestodevelopedcountries.Thisempiricalfactiswell-establishedandindeedwastheprimarymotivationforAngeletos&Panousi(2011).TheimagebelowistakendirectlyfromGourinchas&Jeanne(2013)andshowscapitalflowsagainstproductivitygrowthover1980-2000for68non-OECDcountries.Afewotherquasi-anecdotaldatapointsalsolendcredencetothetheory,forexamplethatChinaisthelargestforeignholderofUSTreasuries.However,itshouldbenotedthatarelatedpredictionofthemodelisthatrichcountriesshouldhavetradeandnetforeignassetdeficits,andpoorcountriessurpluses;whilethisholdsforthekeymotivatingexamplesliketheUS,UKandChina,itdoesnotholdacrossallcountries,withtheGermaniccountriesandoil-richstatesbeingnotableexceptions.Moreover,withlong-rungrowth,onabalancedgrowthpathnetforeignassetswouldremainconstantasashareofoutput,sothenetlendercountry(thedevelopingcountry)wouldhavegrowingnetforeignassets,i.e.wouldruncurrentaccountsurpluses,whilethenetborrowingcountry(developedcountry)wouldruncurrentaccountdeficits;thisagainistruefortheUS,UKandChina,butisatoddswiththecross-countrycorrelationofcurrentaccountbalances.Iabstractfromgrowthinthemodel,socurrentaccountbalancesarezeroeverywhereinsteady29

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrystate.Figure6:GlobalimbalancesbetweenrichandpoorcountriesareconsistentwiththemodelSource:Gourinchas&Jeanne(2013)Thesecondmajorempiricalpredictionofthemodelisthatriskier,lessdevelopedcountrieswillhavelowercapital-outputratiosthandosafer,developedcountriesinanequilibriumwithgloballyintegratedcapitalmarkets.Thistooweobserveinthedata(Caselli2005),andthiscannotbeexplainedforexamplebystandardgrowththeoriesthatexplainincomedifferencesbetweencountriesonabalancedgrowthpathsolelythroughdifferencesintotalfactorproductivity.Apositivecorrelationbetweenincomesandcapitalintensityisalsoapredictionofmodelsfeaturinginvestment-specifictechnologicalgrowth,butsuchmodelsalsopredictlowerreturnsoncapitalinlessdevelopedcountries,whilemyidiosyncraticrisk/globalimbalancesmodelpredictstheopposite;thedataispatchierhere,andthecorrelationisweak,butdoessomewhatsupporttheimbalanceshypothesis,althoughseeGourinchas&Jeanne(2013)andCaselli&Feyrer(2007)forargumentsagainstthisview.Iseetherisk/globalimbalanceshypothesisandtheOLG/demographicshypothesisasverymuchcomplementaryexplanationsfortherecentfallinrealinterestrates.Nevertheless,there30

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryaresomerevealingpatternsinthedatathatcanhelpusdiscriminatebetweenthetwo.TheOLG/demographicshypothesisdoesnottomyunderstandingimplyanythinginparticularaboutthedegreeofoverallwealthinequalityeitherwithinorbetweencountries.TheglobalimbalancesmodelhoweverpredictsaParetoright-tailforthewealthdistribution,whichunderintegratedbondmarketsshouldbeidenticalinbothcountrieswhencapitalisunconstrained;however,whenrichcountriesareborrowing-constrained,themodelpredictstheyshouldhavelowerwealthinequalitythanpoorercountries.Thisislargelyborneoutbythedata,althoughthecorrelationisonlyweaklysignificant,andthereareafewnotableexceptionssuchastheUSandseveralMiddleEastcountries.Figure7:RichercountrieshavehighercapitalintensityandlowerwealthinequalitySource:GDPperworkeristheoutput-siderealGDPatcurrentPPPsfromthePWT10.0,dividedbyemployment.Thecapital/outputratioisthecapitalstockatcurrentPPPs,dividedbythesamemeasureofoutput,alsofromthePWT10.0.Thetop1%wealthshareisfromtheWorldInequalityDatabase.31

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendry4TheLimitsofFiscalSpaceItisconvenienttoexpressequation10,whichcharacterisesbondmarketequilibriumsolelyasafunctionoftheinterestrate,intermsofHomeDebt/GDP:DHYH=max⇢Hµ(r)1,✓H↵r++Hmax{µ(r),µcH(r)}+max⇢Fµ(r)1,✓F↵r++Fmax{µ(r),µcF(r)}DFYFá✓r++Hmax{µ(r),µcH(r)}r++Fmax{µ(r),µcF(r)}◆↵1↵(11)Wecanusethisequationtodeterminetheequilibriumsteadystateinterestrate,ascertainthedeficit/surplusthatagovernmentmustruntomaintainthatsteadystate,andhenceestablishthelimitsoffiscalspaceavailabletogovernments.Thisallowsustotraceoutalocusofdeficitsforanygivenlevelofpublicdebt,whichissimilarinspirittowhatBrunnermeieretal.(2021a)calltheÒDebtLafferCurveÓ,anamewhichIborrow,althoughwhichintheirpaperreferstoarelationshipbetweendeficitsandissuance.Figure8showstheDebtLafferCurveforacountryunderAutarkyforthreelevelsofidiosyncraticrisk.Withlowrisk,precautionarysavingisrelativelyweak,andtheequilibriuminterestrateispositiveexceptatverylowlevelsofpublicdebt,hencethereisnobubble.Withhigherrisk,precautionarysavingisstronger,equilibriuminterestratesarelowerandthereisalargebubble,affordingthegovernmentconsiderablefiscalspace.Oneusefulnotionoffiscalspaceisthelevelofdebt(relativetoGDP)atwhichtheinterestrateiszero,andhencebelowwhichthereisabubble,allowingthegovernmenttorunaprimarydeficitindefinitely.IcallthistheÒBubbleBoundÓ:Definition3.TheBubbleBound,denotedDi,isthemaximumsustainablelevelofpublicdebtbelowwhichagovernmentcanrunaperpetualdeficit,i.e.wherer=0.Anotherusefulnotionoffiscalspaceisthemaximumprimarydeficit(relativetoGDP)thatagovernmentcansustain.IcallthistheÒProfligacyPeakÓ:32

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryDefinition4.TheProfligacyPeak,denotedsi,isthemaximumsustainableprimarydeficit:si=maxDHrDHs.t.BH(r)+BF(r)=DH+DFNotethatattheProfligacyPeak,theelasticityoftheinterestratetopublicdebtequalsnegative1,@DH[r(DH)DH]=0()drdDHDHr=1.InFigure8IalsohighlighttheBubbleBoundandProfligacyPeakforthedifferentlevelsofrisk.Figure8:TheDebtLafferCurveunderAutarkyWithoutCreditConstraintsWecanexaminethecasewithoutcreditconstraintsmorecloselyandachievesomeusefulanalyticalbenchmarkresults.Thesingleequilibriumequation11becomes:DHYH=✓Hµ(r)1◆↵r++Hµ(r)+✓Fµ(r)1◆↵r++Fµ(r)DFYF✓r++Hµ(r)r++Fµ(r)◆↵1↵Lemma2.UnderAutarky,theBubbleBoundofdebt/GDP,belowwhichthegovernmentcan33

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrysustainpositiveprimarydeficits,isgivenby:DiYiA=✓iµ01◆↵+iµ0whereµ0⌘µ(0)=q2⇢1+.Lemma3.UnderFinancialIntegrationandwithoutcreditconstraints,theBubbleBoundofdebt/GDPisgivenby:DHYHI,UC=✓Hµ01◆↵+Hµ0|{z}DH/YHA+24✓Fµ01◆↵+Fµ0|{z}DF/YFADFYF35✓+Hµ0+Fµ0◆↵1↵|{z}YF/YHProof.TheseLemmasfollowimmediatelyfromequation11aftersettingr=0andrearranging.OneimmediateconsequenceofthisisthattheBubbleBoundishigherunderIntegrationthanunderAutarkywhenForeignriskissufficientlyhighandwhenForeignpublicdebtissufficientlylow:Theorem3.TheBubbleBoundishigherunderIntegrationthanunderAutarkywhenForeignpublicdebtisbelowitsownAutarkicBubbleBound.WhenForeigngovernmentshavenodebt,thisholdswhenoverseasinvestorsarenetlenders,i.e.when2F2⇢(1+).Proof.ThefirstpartfollowsimmediatelyfromLemma3.WhenDFYF=0,DHYHI,UCDiYiAifFµ01>0,or2F2⇢(1+).However,evenifthisresultholdsandtheBubbleBoundishigherunderIntegration,themaximumsustainabledeficitmaystillbelower,asshowninFigure9below.ForthemaximumdeficittobehigherunderIntegrationthanunderAutarky,Foreignriskmustbehigherstillthantheconditionabove.34

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryFigure9:IntegrationraisestheBubbleBoundwhenForeigninvestmentisriskyenoughAclosecorollaryofthisresultisthattheBubbleBoundisincreasinginbothforeignanddomesticrisk.Corollary2.TheBubbleBoundistypicallyincreasinginForeignrisk,F.AsufficientconditionunderwhichthisistrueisthatForeignentrepreneursarenetborrowers,i.e.when2F2⇢(1+).AnecessaryconditionisthatF⇣(1↵)µ0+µ0⌘1↵,whichholdsforallreasonableparametervalues,evenifForeigninvestorsarenetlenders.Proof.ThederivativeofthebubbleconditionwithrespecttothedegreeofForeignriskis:@@F DHYHI,UC!=↵µ0(+Hµ0)↵1↵(+Fµ0)1↵1+✓Fµ01◆↵↵1(+Hµ0)↵1↵(+Fµ0)1↵11µ0>0()µ0+✓Fµ01◆1↵1(+Fµ0)1µ0>0whichispositivewheneitheroftheconditionshold.Corollary3.UnderAutarky,theBubbleBoundisalwaysincreasingindomesticrisk,H;35

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryunderIntegration,asufficientconditionforthisiswhendomesticinvestorsarenetborrowers,i.e.when2H2⇢(1+),andForeignpublicdebtisbelowitsAutarkicBubbleBound.Proof.Forthefirstpart:@@H DHYHA!=µ0↵(+Hµ0)1+✓Hµ01◆↵(+Hµ0)2µ0>0()µ0+✓Hµ01◆(+Hµ0)1µ0>0()µ0+µ0>0whichalwaysholds.Forthesecondpart:@@H DHYHI,UC!=µ0↵(+Hµ0)1+✓Hµ01◆↵(+Hµ0)2µ0+24✓Fµ01◆↵+Fµ0|{z}DF/YFADFYF↵µ01↵✓+Hµ0+Fµ0◆2↵11↵>0whichiscertainlypositivewhenHµ01,or2H2⇢(1+),andDFYF⇣Fµ01⌘↵+Fµ0,althoughthisiscertainlynotanecessaryconditionandtheresultwilllikelyholdevenwhendomesticinvestorsarenotnetborrowers.36

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryFigure10:TheBubbleBoundriseswithHomeandForeignriskTheProfligacyPeakisthesecondkeynotionoffiscalspace.AsIhaveshownabovequantitatively,whetherornotthemaximumsustainabledeficitrisesorfallsasaresultofintegrationismorenuanced,andtheconceptescapesformaltheoremsasfarasIcantell.However,itisclearquantitativelythattheProfligacyPeakishigherunderIntegrationwhenForeignriskissufficientlyhigh,asshowninFigure11below,andmoreoverisincreasinginbothHomeandForeignrisk.37

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryFigure11:TheProfligacyPeakishigherunderIntegrationwhenEMriskishighWithCreditConstraintsAsdiscussedabove,giventhatentrepreneursinagivencountryareex-anteidentical,andbecauseofthelinearityoftheirpolicyfunctions,allentrepreneursbehaveasscaled-upordownversionsofoneanother.Consequently,entrepreneursinagivencountrywillbeeitherallborrowersoralllenders,dependingontherisktheyfaceandtheriskfacedbyentrepreneursintheothercountry.Assuch,creditconstraintswillbindonentrepreneursinatmostonecountryatatime,andiftheydobindtheywillbindforallentrepreneursinthecountry.Theconditionforcreditconstraintstobebindingis↵K↵1ir2ii,orr<⇢(1+)2(ii)2,whichnotecertainlyputstheequilibriuminthedownward-slopingpartofthecapitallocus.Withcreditconstraints,theexpressionfortheBubbleBoundisasfollows:Lemma4.UnderFinancialIntegrationandwithcreditconstraints,thedebt/GDPBubbleBoundisgivenby:DHYHI,CC=↵maxnHµ01,✓Ho+Hmax{µ0,µcH0}+24↵maxnFµ01,✓Fo+Fmax{µ0,µcF0}DFYF35✓+Hmax{µ0,µc0}+Fmax{µ0,µcF0}◆↵1↵38

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryProof.Thisagainfollowsimmediatelyfromequation11aftersettingr=0andrearranging.TheBubbleBoundisnownolongergenerallyincreasinginHomeorForeignrisk,butasthefollowingtheoremshows,whencreditconstraintsbindtheBubbleBoundishigherthanintheirabsence.Theorem4.TheBubbleBoundinthepresenceofcreditconstraintsisatleastaslargeasitiswithoutcreditconstraints,providedForeignpublicdebtisbelowitsAutarkicBubbleBound.Proof.TakingwithoutlossofgeneralitythecasewhereconstraintsbindatHome,soHomeentrepreneursarenetborrowers(H<µ0)andForeignnetlenders(F>µ0).ComparingtheBubbleBoundsunderIntegrationwithandwithoutcreditconstraints:DHYHI,CCDHYHI,UC()max⇢Hµ01,✓H↵+Hmax{µ0,µcH0}+✓Fµ01◆↵+Fµ0DFYF✓+Hmax{µ0,µc0}+Fµ0◆↵1↵✓Hµ01◆↵+Hµ0+✓Fµ01◆↵+Fµ0DFYF✓+Hµ0+Fµ0◆↵1↵Thesecondtermonthelefthandsideonoftheinequalityisclearlyatleastasgreatasthatontherightduetothepresenceofthemaxoperatorinthenumerator(theµsarecertainlypositive,beingtheSharperatios,assomelightalgebrademonstrates).Thefirsttermonthelefthandsideisalsoatleastasgreatasthatontheright,toseethisnotethefollowing(andrecallthatHµ01<0byassumption):max⇢Hµ01,✓H↵+Hmax{µ0,µcH0}✓Hµ01◆↵+Hµ0maxnHµ01,✓HoHµ01|{z}1+Hmax{µ0,µcH0}+Hµ0|{z}1AsimilarlogicprovesthecasewhenconstraintsbindforForeignentrepreneurs.39

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryFigure12:CreditconstraintsraisetheBubbleBoundwhentheybindforeithercountryWithcreditconstraintsitisagaindifficulttoderiveexplicittheoremsregardingthesizeofthemaximumsustainabledeficitÐtheProfligacyPeakÐbutthequantitativeevidencebelowsuggestsitishigherwhenconstraintsbind,asintheÒwingsÓofFigure13below.Whenonecountryisverysafeandtheotherveryrisky,creditconstraintsbind,pushtheequilibriuminterestratedownandallowthegovernmenttorunaperpetualdeficitontheorderof1%ofGDP,whenabalancedbudgetwouldhavebeenrequiredintheabsenceofconstraints.40

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryFigure13:TheProfligacyPeakishigherwhencreditconstraintsbindTheroleofcreditconstraints.ItisimportanttonotethatthemechanismatplayherebywhichcreditconstraintsaffecttheequilibriumisfundamentallydifferentfromthatinKiyotaki&Moore(1997).Inthatpaper,anegativeproductivityshocklowersthepriceofland/capital,whichinturntightenstheborrowingconstraintandlowersproductivecapacityfurther,amplifyingthecontraction.Here,creditconstraintsonDMentrepreneurslimitthecompetitionforsafeassets;forcingsavingstoberedirectedtogovernmentbondsratherthantoprivateborrowers,loweringtheinterestrateandexpandingthebubble.TheroleofcreditconstraintsherealsodiffersslightlyfromReis(2021).Inhispaperentrepreneurshavedifferentqualities,withhigherqualityentrepreneurshavingbothhighermarginalproductandalsolower(no)idiosyncraticrisk.Withoutcreditconstraints,thehighqualityentrepreneurswillborrowfromlowqualityonesandwillultimatelydominateproduction,effectivelycompletingmarkets.Thustheinterestrateonbondswillequalthemarginalproductofcapital,whichexceedsthegrowthrate,eliminatingthebubbleingovernmentdebtandthusfiscalspaceÐasmeasuredaboveÐdisappears.Creditconstraintsthereforeservetostoptheconcentrationofproductionamongthehighqualityentrepreneurs.41

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryWhatmypaperdistinctivelyshowsisthatevenifborrowingcannoteffectivelycompletemarkets,creditconstraintsstilllowertheequilibriumrisk-freerealinterestrateÐandexpandthebubbleÐsimplybyrestrictingthesupplyofsafeassets.Creditconstraintsalsolimitthecrowdingouteffectsofpublicdebtissuanceinthecountrywhentheybind,andmayexacerbatethemelsewhere.Figure14belowshows,foranexamplecalibrationwhenHomeentrepreneursarecreditconstrained,theequilibriuminterestratesandcapitalstocksunderfinancialintegrationbeforeandafteranissuanceofpublicdebtontheorderof80%ofGDP.Theequilibriummovesfromclearlycapital-constrainedtojustatthekinkwhereconstraintsarenolongerbinding.Thisentailsalossofoutputontheorderof4%fortheHomecountrybut8%fortheForeigncountry.Withoutcreditconstraints,thelosseswouldbecloserto6%domesticallyand4%overseas.Figure14:IssuanceofpublicdebtwhenentrepreneursarecreditconstrainedThecalibrationrequiredtogeneratetheabovefigureisnotnecessarilyrealistic,withamaximumloan-to-value(LTV)ratioforentrepreneursofonlyaround✓H=0.2.TothisIhavetworesponses:1.Firstlythemodelishighlystylised,andamorerealisticmodelfeaturing,forexample,42

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrystochasticidiosyncraticentrepreneurialproductivity,similartotheworkofQuadrini(2000)andCagetti&DeNardi(2006),wouldfeaturebindingcreditconstraintswithrealisticLTVratiosforsufficientlyhighlyskilledentrepreneurs.Thisnotwithstandingthecritiquethatsuchcollateralconstraintsmaynotbequantitativeimportantfromamacroeconomicperspective,e.g.Kocherlakota(2000)andCordoba&Ripoll(2004),orthatalternativecreditconstraintsonprofitabilityorcashflowseemtobemoreprevalentandbindingmorefrequentlyinthedata,asintheworkofDrechsel(2021).2.Secondly,theappropriatemomentconditionwithwhichtocalibrate✓isarguablynotthemaximumloan-to-valueatwhichcommercialbanksarewillingtoofferloanstosmallbusinessessecuredonphysicalcapital.InsteadtheconstraintreflectslimitsonbanksÕabilitytotransformriskyloansintorisk-freesecurities;entrepreneurialdebtiscompletelyrisk-freeinthemodelandthusareaprivatesourceofsupplyofsafeassets.SuchlimitsarisefrominformationalasymmetriesandmoralhazardandareexpoundeduponinBMS(2021b).WhenconstraintsbindatHomeinthemodel,thetotalprivatesupplyofsafeassetsis✓HKHYH;thetotalsupplyofsafeassetsbyUSfinancialinstitutionshoversaround60Ð75%ofGDP(Gourinchas&Jeanne2012),whichwithKHYH’3Ð4implies✓H’0.2.NotethatissuanceofpublicdebtbytheForeigncountrylowerstheBubbleBoundandProfligacyPeakfortheHomecountry.ConsideranexamplewherebothcountriesareidenticalandHomehasaBubbleBoundof2xGDPwhenthereisnoForeignpublicdebt.AsForeignpublicdebtrises,however,theBubbleBoundfalls.Ifbothcountriestriedtoexploitthebubblefully,conditionalontheactionsoftheother,theirBubbleBoundwouldbeperhaps1xGDP,thesameasinAutarky.Thuswhenbothcountriesaretryingtoexploitthebubble,IntegrationdoesnotnecessarilyconferanadvantageoverAutarky;itallowsonecountrytomineanothercountryÕsbubble,whichisadvantageousiftheirbubbleisbiggerthanyoursandtheydonotorcannotexploitit,butdoesnotbyitselfincreasethecombinedsizeofthetwobubbles.However,thisisonlytruewhencreditconstraintsdonÕtbindinequilibrium;ifthey43

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrydo,theIntegratedbubbleislargerthanthetwoAutarkybubblescombined.OneimmediateimplicationofTheorem4isthatthetotalamountofdebtthatcanbesustainedwithzerointerestratesunderIntegration(aglobalBubbleBound,ifyoulike)ishigherwhencreditconstraintsarepresentandbindthanwhentheywouldbindbutarenotpresent,andhencehigherthanthatwhichcanbesustainedbythetwocountriescombinedunderAutarky.TheexampleinFigure15showssuchascenario.Figure15:Foreigndebtissuancelimitstheabilitytomineforeignbubbles5WelfareandtheOptimalQuantityofPublicDebtInowturntowelfareandtheoptimalquantityofgovernmentdebt.Iconsideronlysteadystatewelfare,withconstantdebtpolicies,ratherthanallowingforarbitrarytime-varyingdebtpolicies,foreaseofcomparisonwiththeabovesectionsandwiththepriorliterature.Optimaldebt/GDPwillingeneralbemuchhigherthantheBubbleBoundorProfligacyPeak,becausethesehingeonexploitingthebubbleinpublicdebt,butthewelfarecalculationstradeoffthetaxationofworkersÐwhichhaslimitedcostshereduetotaxesbeingnon-distortionaryÐagainsttheincomeandself-insurancebenefitsofpublicdebtforentrepreneurs,whichare44

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryconsiderable.OnechallengeofassessingthewelfareattheaggregatelevelinthismodelisthatthestochasticprocessthatgovernsentrepreneursÕwealthisageometricBrownianmotionÐarandomwalkinlogsÐwhichdoesnothaveastationarydistribution.Thedistributionofwealthamongentrepreneursisthuslognormalandincreasinginvarianceovertime.Thisisnotaproblemforanalysingmacroeconomicaggregates,sincethelinearityofentrepreneursÕpolicyfunctionsimpliesthedistributionofwealthdoesnotmatterforaggregates,butitdoesmatterforwelfare.Thetypicaltrickinthesecircumstancesistoaddastabilisingforcetolimitthisdivergence,forexamplePoissondeathshocks,whichIusehere,withnewbornsinheritingsomelowerlevelofwealth,e.g.theaverage.Thisgivesrisetoastationarydouble-Paretodistributioninwealth,asdiscussedinforexampleGabaix(2009)andGabaixetal.(2016).Idonotmicrofoundthisinheritanceprocess,butonecouldexplicitlymodelabequestmotiveandinheritancetaxestogeneratesuchasprocess,asinBenhabibetal.(2016).Notethattheparameter⇢aboveshouldnowbeinterpretedasacomposite⇢=÷⇢+p,where÷⇢istheactualsubjectivetimediscountrate,andpisthePoissondeathrateofentrepreneurs,whichIcalibrateto1/50,foranaverage50-yearlifespan.Inaddition,Iset=1toensureboththatasteadystateexistsandthattheaveragewelfareofentrepreneursisfinite.Consequently,totalsocialwelfareinsteadystateisgivenbythefollowingProposition:Proposition3.SteadystatewelfareincountryiisdescribedbyaUtilitarianwelfarefunction,withweight$onthewelfareoftherepresentativeworkerand1$onthewelfareofentrepreneurs:Wi=$Vwi+(1$)Z10Vei(a)gi(a)da=$1⇢ln((1↵)Ki(r)↵rDi)11+’+(1$)mi(r)1(Ki(r)+Bi(r))1Z(r)wheregi(a)isthedensityofthestationarywealthdistribution,Z(r)=⇣i+⇣i(1⇣i)(1⇣i+)and45

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrywhere⇣i±=12±q122+2p(i!i)2aretheParetotailparameters,andwhereIhavesuppressednotationdenotingthedependenceofronDi.Proof.SeeAppendix.Iusethismeasureofwelfaretofindtheoptimalquantityofpublicdebt.Iassumethattheentiredeficitisusedtopayproportionaltransferstoworkers,⌧=rD(1↵)K↵(ortaxesifthelevelofdebtnecessitatessurpluses).Ifinsteadweweretoassumedeficitspaidforgovernmentspending,whichisnotvaluedbyagents,therewouldbelesswelfarebenefittoissuingdebt.However,giventhatmuchofthepersistentdeficitshighlightedbyfiscalauthoritiesofadvancednationsoverthecomingdecadeswillariseduetotransfers,likepensions,Ibelievethisisareasonablebenchmark.Therearethusseveralcountervailingdirectandindirecteffectsofdebtissuanceonwelfare,correspondingtohowDentersintotheabovedirectly,andalsohowitaffectsrandhenceK,B,mandZ.Issuingmoredebthasthefollowingdirecteffect:¥Raisesworkerconsumptionthroughtransfersifthedeficitgrows,@D[r(D)D]=r0(D)Dr>0,i.e.D/YisbelowtheProfligacyPeak;orlowersconsumptionbyraisingraxesifnotProvidedbondholdingsarewell-behaved,issuingmoredebtraisesinterestrates,whichhasthefollowingindirecteffects:¥Crowdsoutcapital,reducingincomeforentrepreneursandwagesforworkers,unlessdebtissufficientlyhighthattheequilibriumisintheupward-slopingpartofthecapitallocus¥Providesmoreliquidity/risk-freeincometoentrepreneursiftheyarenetsavers,orreducestheirborrowingotherwise,bothofwhichreducethevolatilityoftheirnetworthandhenceconsumption46

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryIftheElasticityofIntertemporalSubstitutionisnotequaltoone,1/6=1,thenissuingmoredebtalsohasthefollowingeffects:¥Raisesthemarginalpropensitytoconsumeofentrepreneurs,mi,if1<1,andviceversa¥Changesthedistributionofwealthamongentrepreneursbyfatteningtherighttail,⇣0+(r)<0,andthinningthelefttail,⇣0(r)>0,soZ0(r)>0if1/<1,andviceversaBreakingRicardianequivalence.BeforequantifyingtheoptimallevelofdebtIwanttofirstaddressacuriosityofthismodelcreatedbytheseparationofworkersandentrepreneurs:Ricardianequivalencedoesnotevenwithoutuninsurablerisk.Toilluminatethematter,considerthespecialcaseofautarkyundercompletemarkets(=0),withlogutilityforbothworkersandentrepreneurs.ThecapitalstockinsteadystatebecomesK=⇣↵⇢+⌘11↵.ConsumptionofworkersbecomesCw=w(1⌧)=(1↵)K↵(1⌧)wherethetaxrateisdeterminedinequilibriumbyD=⌧wr!⌧=⇢1↵DY,takingDYasexogenous,chosenbythegovernment;yieldingCw=(1↵⇢DY)K↵.Itreatworkerlaboursupplyasex-anteinelastichere,foreaseofcomparison;thisiswithoutlossofgeneralitysincedisutilityfromlaboursupplysimplyactsasashifteronwelfareanddoesnotinteractwithdebtissuance.Becausemarketsarecomplete,thereturnsonbothcapitalandbondsmustbeequal,R=r=⇢,soentrepreneursÕconsumptionbecomesCe=RK+rD=↵K↵K+⇢D=⇢(K+D).Anotherwaytoseethisisthatm=⇢when=0.Welfareinthiscompletemarketsspecialcaseforthistwo-agentmodelisgivenby:WCM,TA=$1⇢ln✓✓1↵⇢DY◆K↵◆+(1$)1⇢ln✓⇢✓K+DYK↵◆◆OnecantreatthisasthegovernmentÕsobjectivefunctionandtakefirstorderconditionsto47

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryfindoptimaldebtissuanceinsteadystate,andpluginareasonablecalibrationtogive:DY⇤=(1$)1↵⇢$↵⇢+=0.5á0.640.06+0.5á3’3.8However,inthemorestandardrepresentativeagentRamseymodelwhereworkersandentrepreneursareoneandthesame,andwhereproportionallabourtaxesarenon-distortionarynowbecauselaboursupplyisinelastic(lump-sumtaxeswithelasticlaboursupplyofcoursegivesasimilarresult),welfareisinsteadgivenby:WCM,RA=1⇢ln((1↵)K↵+⇢K)whereinthetwotermsrelatingtopublicdebtcancel,sowelfareisindependentofpublicdebtissuance,givingthestandardRicardianequivalenceresult.ItisthusworthwhilenotingthatRicardianequivalencedoesnotholdinthistwo-agentmodel,evenwithnodistortionsandcompletemarkets,simplybecauseoftheseparationofworkersandentrepreneurs.Intuitively,Ricardianequivalenceholdsinthestandardmodelbecauseagentsanticipatethattheywillhavetorepaythedebtthroughfuturetaxes,exactlyoffsettingtheincomereceivedfromholdingbondsinpresentvalueterms.Whentheagentsreceivingtheinterestincomearedifferentfromtheagentsbeingtaxed,theequivalencefallsapart.Whatisthewelfare-maximisinglevelofdebt?ArangeofcalibrationssuggestthatunderbothAutarkyandIntegration,welfareismaximisedatdebtlevelsontheorderofseveralmultiplesofGDP(4-6x)ÐfarabovecurrentlevelsandfarabovethelevelsassociatedwiththeBubbleBoundortheProfligacyPeaknotionsoffiscalspacediscussedabove.Thisisalsosubstantiallyhigherthanoptimaldebtlevelsfoundinpreviousstudiesfeaturingincompletemarkets,suchasAiyagari&McGrattan(1998)orDesbonnet&Kankanamge(2016),partly48

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrybecausethesestudiesfeaturehighlydistortionarylabourandcapitaltaxationandgovernmentspending,whileIpurposefullyneutralisetheseeffects.Figure16belowshowswelfareasafunctionofthedebt/GDPratioinmymodel,forgivenlevelsofrisk,andmoreovershowsthatthewelfare-maximisingdebtlevelisincreasinginbothHomeandForeignrisk.Animportantexceptionoccurswhencreditconstraintsbind.Thepresenceofbindingcreditconstraintsincreaseswelfare,sincetheleverageofentrepreneursisconstrained,andhencesoisthevolatilityoftheirwealthandthusconsumption.AsthepanelontherightofFigure16suggests,whenForeigninvestmentissufficientlyrisky,creditconstraintsbindandraiseHomewelfare.Publicdebtissuanceinthiscasemaybeimmediatelydetrimentaltowelfare,sozerodebt/GDPmaybea(local)maximumforwelfare.Ofcoursemypaperabstractsfromwhetherornotthegovernmenthastheabilityorwilltofulfillsuchlargedebtobligationsbyrunninglargeenoughsurpluses,ormarketparticipantsÕperceptionsthereof.Nevertheless,themodeldoeshighlightthatpublicdebtitselfisusuallynotharmfultowelfare,indeeditisbeneficial,exceptatenormouslyhighdebtlevels.Inotherwords,thecrowding-outeffectsofpublicdebtareveryweak,eventhoughRicardianequivalencedoesnothold.Figure16:Thewelfare-maximisinglevelofpublicdebtmaybeseveraltimesGDP49

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryIsdebtissuancePareto-improving?Muchoftheliteraturesofarhas,inthetraditionofoverlappinggenerationsmodels,focusedonwhetherintergenerationaltransfers(viapublicdebt)arePareto-improving,likeBlanchard(2019)orAguiaretal.(2021).Thisisrelatedtodynamicinefficiency,whichoccurswhenr<ginsimplemodelswithoutrisk,likeDiamond(1965).However,asReis(2021)hasshown,inthepresenceofrisk,r<gdoesnotsufficetogeneratedynamicinefficiency,whichinsteadrequiresthatthemarginalproductofcapitalbebelowthegrowthrate,R<g,whichisamuchmoreonerouscondition.Inmymodelhowever,neitheroftheseconditionsarerelevantfordeterminingwhetherdebtisPareto-improving,becauseoftheseparationofentrepreneursandworkers.SinceworkersÕwelfaredependsonlyontheirconsumption(theirlaboursupplybeinginelastic),andsincebecausetheycannotsave,theirconsumptionequalstheirpost-tax/transferwageincome;issuingmoredebt,bycrowdingoutcapitalandloweringthewage,necessarilymakesworkersworseoff,unlessrissufficientlynegativethatthegovernmentcanrunapersistentdeficitandusethatdeficittofundtransferstoworkersthatarelargeenoughtooffsetthedeclineinthewage.Asalludedtoabove,debtmustbebelowthelevelassociatedwiththeProfligacyPeakiftransfersaretoincreasewhenissuingmoredebt.Itmustbefurtherbelowstillandwithrsufficientlynegativeiftheincreaseinthesetransfersistobelargeenoughtooffsetthedeclineinthewagefromthecrowdingoutofcapital.Figure17showstheParetofrontierstracedoutbydifferentequilibriawithdifferentlevelsofgovernmentdebtfortwoseparatecalibrationsofthemodel:onerealisticcalibrationinblue,withH=0.3andF=0.5;andoneinredwithH=0.5andF=0.5.Onlytheredlinebendssufficientlythatthegovernmentcanincreaseworkerwelfarebyissuingmoredebt;theseequilibriafeaturerinthenegativedouble-digits,thoughwithR>0still.Therealisticcalibration,whichwithdebt/GDParoundthecurrentlevelimpliesarealinterestrateclosetocurrentmarketrates,featuresaParetofrontierthatisalwaysdecreasinginworkerwelfare:issuinggovernmentdebtisnotPareto-improving.50

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryFigure17:DebtissuanceisnotPareto-improvingunlessr⌧0Onecouldalternativelyconsideranotheruseofgovernmentdeficits:subsidiesforcapitalaccumulation.Providedrislowenoughandhencethedeficitlargeenough,thesubsidiesforcapitalmaybesubstantialenoughtooffsetthedisincentivetoholdingcapitalfromahigherrthatgreaterdebtissuancebringsabout,potentiallyraisingwagesforworkers.Thiswouldhowevercomeattheexpenseofgreaterrisk-takingbyentrepreneurs,dampeningtheaggregatewelfaregains.Regardless,mymodelgivesaverydifferentflavourtothediscussionsofthewelfarebenefitsofdebtissuance.Muchofthediscussionsofarhasfocusedonthepotentialtransferstotheoldorthepoor,ortheutilitygainsfromtheinsurancebenefitsfromexpandingpublicservices.Butinmymodelmostofthewelfaregainsactuallycomefromprovidingaliquid,safeassetthatentrepreneurscanusetopartiallyinsurethemselvesagainsttheriskstheyaloneface.Strategicinteractions.Thepresenceoftwogovernmentssimultaneouslytryingtoissuetheoptimalquantityofpublicdebtgivesrisetosomelimitedstrategicinteraction.Againconsideringonlythesteadystate,supposethegovernmentofcountryimaximisesthewelfareofcountryi,givenabove,subjecttobondmarketequilibriumandconditionalonthepublic51

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrydebtofcountryj.Theoptimalityconditiongivesabest-responsefunctionofsteady-statedebtasafunctionofthedebtoftheothercountry,D⇤i(Dj),whichasFigure18belowsuggests,appearstobelinear,althoughIhaveasyetfoundnoexpressionforit.TheNashequilibriumis(D⇤H(D⇤F),D⇤F(D⇤H)),andisshownbelowforH=0.3andF=0.5,andfeaturesdebtbetween3-5xGDP,clearlyfargreaterthanlevelsobservedtoday,butalsofarbeyondthepointwherer<g.Figure18:Nashequilibriumdebtwhenbothcountriesmaximisewelfareis>3xGDP6DynamicsFollowingaGovernmentSpendingShockInowturntoanalysingthedynamicsofthemodelfollowinganunexpected,one-timepersistentbuttransitorydebt-fundedincreaseingovernmentspendingabovethesteadystatelevel,G,suchasthatmanygovernmentshaveundertakeninresponsetotheCovid-19pandemic.Governmentspendingfollows:Gt=G+et(G0G),whereG0>G.Sincethestockofpublicdebtistheonlygovernmentchoicethataffectstheequilibrium,thegovernmentcouldneutralisetheeffectsofgovernmentspendingshocksbypayingforthembyraisingtaxesandbalancingthebudget.Butthen,inthismodelgovernmentspendinghasnovalue,soadhering52

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryrigidlytothelogicofthemodelthenofcoursethegovernmentshouldnotincreasespendinginthefirstplace.Butletusconsidertheeffectsofdebt-fundedspendingshocksnonetheless,ifonlybecauseweobservetheminpractice,regardlessofwhethertheyareoptimalordesirableresponsestobusinesscyclesorothershocks.ThusfarIhavenotmadeexplicitthedynamicprocessgoverningtaxation,asitisnotrelevantforthesteadystate.However,ifgovernmentspendingissimplymean-revertingandtaxratesareconstant,thengovernmentdebtdynamicsmaybeunstable.Tostabilisethesystem,IaddacomponenttotaxrevenuesthatrespondstothelevelofdebtinexcessofthetargetlevelÐbutnotdirectlytogovernmentspendingÐtoensuretaxrevenuesriseinresponsetoexcessivedebt:Tt=rD+G+(DtD).TheequationsgoverningthedynamicsofthemodelarelaidoutinfullinProposition1,butwecansimplifythemconsiderably.AssuminglogutilitysimplifiesmattersbyanorderofmagnitudesincetheMPCsimplifiesto⇢andishenceexogenous;aggregate(entrepreneurial)consumptionthereforeonlydependsontotalwealth,At=Kt+Bt.InAutarky,sincebondholdingsmustequalgovernmentdebt,Bt=Dt,whichisalsoastatevariable,consumptionispredetermined.Logutilityalsoservestoguaranteeexistenceofasteadystateequilibrium,asdiscussedabove.TheAutarkycaseunderlogutilityisparticularlyilluminating,becausethedynamicscanbeexpressedbyasystemofjusttwoODEs,andhencecanbevisualisedbyaphasediagram,withtwoinitialconditions.InAppendix2,Iexplorethecasewith6=1,whereinthedynamicsaremuchthesameasthelogutilitycase,butIalsoallowfortaxrevenuestobeamoregeneralfunctionofthestateofthedomesticeconomy,whichcangiverisetomoreexoticdynamicsandpotentiallyevenrealindeterminacyofequilibrium.Proposition4.TheaggregatedynamicsofthemodelwithlogutilityunderAutarkyaregiven53

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrybythefollowingsystemofordinarydifferentialequationsin(Kt,Dt):úKt=↵K↵tKt⇢(Kt+Dt)+T(Dt)GtúDt=rA(Kt,Dt)Dt(T(Dt)Gt)whererA(Kt,Dt)=↵K↵1tKtKt+Dt2andT(Dt)isgivenabove.Proof.SeeAppendix.>0isgenerallyneededtoensurelocalstabilityofthesteadystate;turningitfroma(nunstable)saddle-pathtoasink.Notehoweverthat>0isnotsufficienttoensureglobalstabilityoftheeconomy.mustbesufficientlylarge,theshocktogovernmentspending,G0G,sufficientlysmallandtransitory,andtheinitialconditionsufficientlyclosetothesteadystate.TheroleofisreminiscentoftheparametergoverningtheresponsivenessofsurplusestoexcessivedebtinLorenzoni&Werning(2019),althoughthereitpinsdownauniqueequilibrium,whereashereitstabilisesthedynamicsaroundthesteadystate,andperhapsrulesoutmultiplesteadystates,whiletheequilibriumitselfisunique(i.e.thereisauniquemappingfromstatestopricesr(KH,KF,D)).Inpractice,=0.1seemssufficienttoensurestabilityforagovernmentspendingshockevenontheorderofmagnitudeof25%ofGDP.Figure16showsthephasediagramforthiseconomy;iftaxesaresufficientlyreactivetoexcessivedebt,i.e.islargeenough,thenthevectorfieldislocallystableinalargeenoughneighbourhoodaroundthesteadystatethattheeconomyreturnstothesteadystateast!1.Ifnot,theequilibriumisunstableandtheeconomydiverges,whichinamonetarymodelwouldentailhyperinflationandtherealvalueofthenominaldebtclaimgoingtozero.54

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryFigure19:PhasediagramunderAutarkyandtrajectoryafterspendingshockProposition5.TheaggregatedynamicsofthemodelwithlogutilityunderfinancialIntegra-tionaredeterinedbythefollowingsystemofODEsin(KHt,KFt,Dt):úAit=↵K↵1itr(KHt,KFt,Dt)Kit+(r(KHt,KFt,Dt)⇢)Aiti={H,F}úDt=(r(KHt,KFt,Dt)r)Dt(DtD)+GHtGH+GFtGFwhereAit=minn↵K↵1itr(KHt,KFt,Dt)2i,io1Kitforeachi={H,F}andwherer(KHt,KFt,Dt)isdeterminedimplicitlybythebondmarketequilibriumequationinTheorem1:Dt=max⇢2H↵K↵1Hr(KHt,KFt,Dt)1,✓HKHt+max⇢2F↵K↵1Fr(KHt,KFt,Dt)1,✓FKFtwithrsimplybeingtheinterestrateinsteadystate.Proof.SeeAppendix.Iomitthephasediagramhere,sinceitiseasiertosimplyimaginethreeplanesintersecting55

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryoneanother,withthevectorfieldsallpointingtowardstheintersection(thesteadystate)whenH=F=issufficientlylarge.UnderIntegration,theresponseofoutputtothegovernmentspendingshockisdampenedwhenthespenderisfinanciallyintegratedwithariskiercountry.Theresponseisfurtherdampenedwhentheprivatesupplyofsafeassetsismateriallyconstrained,i.e.whenthecreditconstraintsbind.Figure20:IntegrationandcreditconstraintsdampenthenegativeeffectsofspendingshocksStability.Itisfruitfultoexamineformallythestabilityofthesteadystateinresponsetoshocks.Asalludedtoabove,theresponsivenessoftaxestoexcessivedebtiscriticaltoensuringlocallystabilityofthesteadystate.WecanexaminelocallystabilitythroughtheJacobianofthedynamicsystem(s)above.SinceeverythinginthissectionandthenextrelatestoboththeAutarkicandIntegratedeconomies,IsimplifythediscussionbyfocusingonthecaseofAutarky.Localdynamicsaroundthesteadystatearegivenbythefollowingsystemoflineardifferentialequations:Proposition6.ThedynamicsoftheeconomyaroundthesteadystateunderAutarkyare56

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrygivenbythefollowingsystemoflineardifferentialequationsin(Kt,Dt):0B@úKtúDt1CA=264↵2K↵1⇢⇢+rKDr+DK2(K+D)23750B@KtKDtD1CAProof.CalculatetheJacobianmatrixofthesystemofODEsinProposition4.Noter=↵K↵1K2K+DandrK=↵(↵1)K↵22K+D+K2(K+D)2.TheeigenvaluesoftheJacobianmatrixindicatewhetherornotthesteadystateislocallystable.Forsufficientlylargethedynamicsarestable(botheigenvaluesnegative);ifr⌧0,=0maybesufficientforlocalstability.Ifistoolow,oneeigenvaluemaybepositiveandthesystembecomesasaddlepath,whichisunstableheresincebothKtandDtarestatevariables.Notethatthetraceofthematrixisverylikelytobenegativeifr<0,andthedeterminantisalsolikelytobenegative,unlessissufficientlylarge.Negativetraceandpositivedeterminantentailsthatbotheigenvaluesarenegative,andhencestability,butanegativetraceandnegativedeterminantentailsonepositiveandonenegativeeigenvalue,andhenceinstability.Multiplesteadystatesanddebttraps.Notethatevenwhenthesteadystateisuniquewhenthetaxandspendingfunctionsareundefined,whenthesefunctionsaredefinedintermsofendogenousvariableslikeDt,thentheseadditionaldynamicscanintroduceasecondsteadystate,usuallyfeaturinglowercapitalandhigherdebt(aÒdebttrapÓ)andwhichistypicallyunstable.However,undercertainconditionstheexistenceofmultiplesteadystatescangiverisetointerestingdynamicsinthefaceoflargeshocks.Withlocalstability,smallshockswillresultintheeconomytendingbacktothemainsteadystateast!1;however,largeshockscanpushtheeconomyoutofthestablezone(ingreenbelow)andintotheunstablezone(inred).Thismayresultintheeconomyrapidlydivergingawayfromsteadystate(thehyperinflationscenario);butifr⌧gandtaxesdonotrespond,orrespondonlymodestly,toexcessdebt,thedynamicsaroundthesecondsteadystatemaybesufficientlyslowthat57

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendrytheeconomymaygetÒstuckÓthereforanextremelylongtime.Inthisscenariotheeconomyfacesadebttrap,whichfeatureshigherdebtandloweroutputthanthestandardsteadystate.Itishardtogetoutofthedebttrapnaturally;itisaknife-edgecaseastowhethertheeconomyreturnstothenormalsteadystateorexplodes,andinanycasethismaytakedecadesorevencenturies.Itmaybenecessarytoengageinausterityinordertoreturntothenormalsteadystatewithinareasonabletimeframe.Figure21:LargeshocksmaypushtheeconomyintoaDebtTrap7ConclusionInevaluatingthefiscalspaceofgovernmentstoday,wemustconfronttwoquestions:whyhastheinterestrateonpublicdebtfallensomuch,inspiteofrisingdebtandhighdeficits;andwhyhasthisfalloccuredprimarilyinthelasttwodecades?TheanswerIbelieveliesinnascentglobalimbalances:greaterdemandforsafeassets,precipitatedbytheriseofChinaandotheremergingmarketeconomies,andthelimitedifnotdecliningprivatesupplyofsafeassets,asexemplifiedbythecollapseoftheasset-backedsecuritiesmarketduringthefinancialcrisis.InthispaperIhaveprovidedahighlytractableenvironmentinwhichtomodelthese58

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrytwofeaturesanddrawnsimplebutpolicy-relevantanalyticalandquantitativeconclusionsaboutthelimitstothefiscalspaceavailabletogovernments.Thekeytake-awayforpolicymakersisthatprecautionarysavingbyEMinvestorsandthelimitedprivatesupplyofsafeassetscanbepowerfulforcesgeneratingsignificantfiscalspace,butthisfiscalspaceislimited,andgreatcareisneededtodesignfiscalpoliciesthatarebothsustainableinthelong-runandstabilisingintheshort-run.Governmentsshouldtargetthelong-runlevelofdebt,notthedeficit.Thisdebtlevelcanbelargeandstillsustainableandconsistentwithdeficits,providedr<g,butgovernmentsmustbewillingtoadapttheirlong-runspendingandtaxationplanstoaccommodatechangingconditions;inotherwords,adjustdeficitstohitthedebttarget.Governmentscanborrowsubstantiallyinresponsetoeconomiccrises,butmustbesufficientlyaggressiveinreducingtheirborrowingsubsequently.Theycanreturntorunning(primary)deficitsinthelongrun,butintheaftermathofaspendingshocktheymustrunsmallerdeficits,perhapsevensurpluses,toensurestabilityandavoidinflationaryspiralsordebttraps,eveniftheprevailinginterestrateisbelowthegrowthrate.Acomplementaryifnotcompetinghypothesistotheglobalimbalancesthesisforwhyr<gtodayisthedemographicthesis,whichstatesthatcomparedto50yearsago,theworldpopulationtodayisonaverageolderandthussavesmore.Whatthendothesehypothesesimplyforthefiscalspaceofgovernmentsinthefuture?Goodhart&Pradhan(2020)assertthatcontinuedageingofthepopulationimpliesadeclineinglobaldemandforsafeassetsasolderworkersstarttorundowntheirsavingsinretirement,whileAuclertetal.(2021)taketheopposingview.Whateverthecase,inmyviewtheglobalimbalanceshypothesisimpliesifanythingmorefiscalspaceinfuture.Emergingmarketeconomiesarelikelytocontinuetogrowinimportance,becomingalargershareofworldoutputandholdingalargershareofworldwealth.However,despitethistheriskinessofinvestmentintheseeconomiesmaynotdiminishgreatlyinthenearfuture,inpartbecausesomuchoftheriskinvolvedstemsfromunderlyingstructuralandpoliticaldrivers,likerisksofpoliticalupheavalor59

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrypolitically-motivatedconfiscationorredistributionofassetsthatdonÕtnecessarilydiminishwitheconomicgrowth.Nevertheless,thelinkbetweeneconomicdevelopmentandtheriskinessofinvestmentisnotclear,anditmaybethecasethatfinancialdevelopmentoccuringalongsideeconomicdevelopmentallowsforgreaterrisk-sharingamonginvestors,thusreducingtheextentorbiteofidiosyncraticrisk.Ontheotherhand,thesafe-assetstatusofUSorotherdeveloped-countrygovernmentbondsisnotguaranteed,anditisnotablethatcapitalflowsfromemergingmarketstothedevelopedworldhavetoanextentrecededsincethefinancialcrisis,andinthelastyearorsoitappearsEMholdingsofDMgovernmentbondshavealsodeclinedsomewhat.Eitherway,idiosyncraticriskstoinvestorsandtheresultingglobalimbalances,aswellaslimitsontheabilityoftheprivatesectortocreatesubstitutesafeassets,arelikelytoremainimportantdeterminantsofthefiscalspaceavailabletogovernments,andthuswarrantfurtherresearch.60

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryReferencesAguiar,M.,Amador,M.&Arellano,C.(2021),Microrisksandparetoimprovingpolicies.Aiyagari,S.R.(1994),ÔUninsuredidiosyncraticriskandaggregatesavingÕ,QuarterlyJournalofEconomics190(3),659Ð684.Aiyagari,S.R.&McGrattan,E.R.(1998),ÔTheoptimumquantityofdebtÕ,JournalofMonetaryEconomics42(3),447Ð469.Angeletos,G.-M.(2007),ÔUninsuredidiosyncraticinvestmentriskandaggregatesavingÕ,ReviewofEconomicDynamics10(1),1Ð30.Angeletos,G.-M.&Panousi,V.(2011),ÔFinancialintegration,entrepreneurialriskandglobaldynamicsÕ,JournalofEconomicTheory146(3),863Ð896.Auclert,A.,Malmberg,H.,Martenet,F.&Rognlie,M.(2021),Demographics,wealth,andglobalimbalancesinthetwenty-firstcentury.Ball,L.&Mankiw,N.G.(2021),Marketpowerinneoclassicalgrowthmodels.Benhabib,J.,Bisin,A.&Zhu,S.(2016),ÔThedistributionofwealthintheblanchard-yaarimodelÕ,MacroeconomicDynamics20(2),466Ð481.Blanchard,O.(2019),ÔPublicdebtandlowinterestratesÕ,AmericanEconomicReview109(4),1197Ð1229.Bohn,H.(1998),ÔThebehaviorofu.s.publicdebtanddeficitsÕ,TheQuarterlyJournalofEconomics113(3),949Ð963.Brumm,J.,Feng,X.,Kotlikoff,L.J.&Kubler,F.(2021),Wheninterestratesgolow,shouldpublicdebtgohigh?,NBERWorkingPapers28951.Brunnermeier,M.,Merkel,S.&Sannikov,Y.(2021a),Debtasasafeasset:Miningthebubble.61

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryBrunnermeier,M.,Merkel,S.&Sannikov,Y.(2021b),Thefiscaltheoryofthepricelevelwithabubble.Caballero,R.J.,Farhi,E.&Gourinchas,P.-O.(2008),ÔAnequilibriummodelofglobalimbalancesandlowinterestratesÕ,AmericanEconomicReview98(1),358Ð93.Cabellero,R.J.,Farhi,E.&Gourinchas,P.-O.(2017),ÔThesafeassetsshortageconundrumÕ,JournalofEconomicPerspectives31(3),29Ð46.Cabellero,R.J.,Farhi,E.&Gourinchas,P.-O.(2020),Globalimbalancesandpolicywarsatthezerolowerbound.Cagetti,M.&DeNardi,M.(2006),ÔEntrepreneurship,frictionsandwealthÕ,JournalofPoliticalEconomy114(5),835Ð870.Caselli,F.(2005),AccountingforCross-CountryIncomeDifferences,inP.Aghion&S.Durlauf,eds,ÔHandbookofEconomicGrowthÕ,Vol.1ofHandbookofEconomicGrowth,Elsevier,chapter9,pp.679Ð741.Caselli,F.&Feyrer,J.(2007),ÔTheMarginalProductofCapitalÕ,TheQuarterlyJournalofEconomics122(2),535Ð568.Chancel,L.,Piketty,T.,Saez,E.,Zucman,G.&al.(2021),Worldinequalityreport2022,Technicalreport.Chen,X.,Leeper,E.M.&Leith,C.(2021),Strategicinteractionsinu.s.monetaryandfiscalpolicies.Clarida,R.,Gali,J.&Gertler,M.(1999),ÔThescienceofmonetarypolicy:AnewkeynesianperspectiveÕ,JournalofEconomicLiterature37(4),1661Ð1707.Cochrane,J.H.(2021),TheFiscalTheoryofthePriceLevel.62

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryCordoba,J.&Ripoll,M.(2004),ÔCreditcyclesreduxÕ,InternationalEconomicReview45(4),1011Ð1046.Corneli,F.(2017),Mediumandlongtermimplicationsoffinancialintegrationwithoutfinancialdevelopment,Bankofitalyworkingpapers.Corneli,F.(2021),ÔFinancialintegrationwithoutfinancialdevelopmentÕ,AtlanticEconomicJournal49(2),201Ð220.Desbonnet,A.&Kankanamge,S.(2016),Publicdebtandaggregaterisk.Diamond,P.A.(1965),ÔNationaldebtinaneoclassicalgrowthmodelÕ,TheAmericanEconomicReview55(5),1126Ð1150.Drechsel,T.(2021),Earnings-basedborrowingconstraintsandmacroeconomicfluctuations.Eggertsson,G.B.,Mehrotra,N.R.&Robbins,J.A.(2019),ÔAmodelofsecularstagnation:TheoryandquantitativeevaluationÕ,AmericanEconomicJournal:Macroeconomics11(1),1Ð48.Farhi,E.&Tirole,J.(2012),ÔBubblyliquidityÕ,ReviewofEconomicStudies79(2),678Ð706.Feenstra,R.C.,Inklaar,R.&Timmer,M.P.(2015),ÔThenextgenerationofthepennworldtableÕ,AmericanEconomicReview105(10),3150Ð3182.Furman,J.&Summers,L.H.(2020),Areconsiderationoffiscalpolicyintheeraoflowinterestrates.Gabaix,X.(2009),ÔPowerlawsineconomicsÕ,AnnualReviewofEconomics1,255Ð294.Gabaix,X.,Lasry,J.,Lions,P.-L.&Moll,B.(2016),ÔThedynamicsofinequalityÕ,Econo-metrica84(6),2071Ð2111.Goodhart,C.&Pradhan,M.(2020),TheGreatDemographicReversal:AgeingSocities,WaningInequality,andanInflationRevival,PalgraveMacmillan.63

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryGourinchas,P.-O.&Jeanne,O.(2012),Globalsafeassets,BISWorkingPapers399,BankforInternationalSettlements.Gourinchas,P.-O.&Jeanne,O.(2013),ÔCapitalflowstodevelopingcountries:TheallocationpuzzleÕ,ReviewofEconomicStudies80(4(285)),1484Ð1515.Holston,K.,Laubach,T.&Williams,J.C.(2017),ÔMeasuringthenaturalrateofinterest:InternationaltrendsanddeterminantsÕ,JournalofInternationalEconomics108(1),39Ð75.Holston,K.,Laubach,T.&Williams,J.C.(2020),Adaptingthelaubachandwilliamsandholston,laubach,andwilliamsmodelstothecovid-19pandemic.Jiang,Z.,Lustig,H.,vanNieuwerburgh,S.&Xiaolan,M.Z.(2021),Theu.s.debtvaluationpuzzle.Kiyotaki,N.&Moore,J.(1997),ÔCreditcyclesÕ,JournalofPoliticalEconomy105(2),211Ð248.Kocherlakota,N.R.(2000),ÔCreatingbusinesscyclesthroughcreditconstraintsÕ,QuarterlyReview,FederalReserveBankofMinneapolis24,2Ð10.Kocherlakota,N.R.(2021),Publicdebtbubblesinheterogeneousagentmodelswithtailrisk.Laubach,T.&Williams,J.C.(2003),ÔMeasuringthenaturalrateofinterestÕ,ReviewofEconomicsandStatistics85(4),1063Ð1070.LeGrand,F.&Ragot,X.(2022),Shouldweincreaseordecreasepublicdebt?optimalfiscalpolicywithheterogeneousagents.Lorenzoni,G.&Werning,I.(2019),ÔSlowmovingdebtcrisesÕ,AmericanEconomicReview109(9),3229Ð3263.Lucas,R.E.(1990),ÔWhydoesnÕtcapitalflowfromrichtopoorcountries?Õ,AmericanEconomicReview80(2),92Ð96.64

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryMcNicol,I.(GeneralSecretaryoftheLabourParty).(2017),LabourÕsfiscalcredibilityrule,Technicalreport.Mian,A.,Straub,L.&Sufi,A.(2021a),Agoldilockstheoryoffiscalpolicy.Mian,A.,Straub,L.&Sufi,A.(2021b),Thesavingglutoftherich.Mian,A.,Straub,L.&Sufi,A.(2021c),Whatexplainsthedeclineinr⇤?risingincomeinequalityversusdemographicshifts.Quadrini,V.(2000),ÔEntrepreneurship,savingandsocialmobilityÕ,ReviewofEconomicDynamics3(1),1Ð40.Rachel,L.&Smith,T.D.(2015),Seculardriversoftheglobalrealinterestrate,BankofEnglandWorkingPaper571.Rachel,L.&Summers,L.H.(2019a),Onfallingneutralrealrates,fiscalpolicy,andtheriskofsecularstagnation,Brookingspapersoneconomicactivity.Rachel,L.&Summers,L.H.(2019b),Onsecularstagnationintheindustrializedworld,Brookingspapersoneconomicactivity.Reis,R.(2021),Theconstraintonpublicdebtwhenr<gbutg<m.Samuelson,P.A.(1958),ÔAnexactconsumption-loanmodelofinterestwithorwithoutthesocialcontrivanceofmoneyÕ,JournalofPoliticalEconomy66(6),467Ð482.Sargent,T.J.&Wallace,N.(1975),ÔRationalexpectations,theoptimalmonetaryinstrument,andtheoptimalmoneysupplyruleÕ,JournalofPoliticalEconomy83(2),241Ð254.Shorrocks,A.,Davies,J.&Lluberas,R.(2021),Creditsuisseglobalwealthdatabook2021,Technicalreport.Tirole,J.(1985),ÔAssetbubblesandoverlappinggenerationsÕ,Econometrica53(6),1499Ð1528.65

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryVila,J.-L.&Zariphopoulou,T.(1997),ÔOptimalconsumptionandportfoliochoicewithborrowingconstraintsÕ,JournalofEconomicTheory77(2),402Ð431.66

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryAppendix1:ExtendedproofsProofofLemma1:Proof.TheentrepreneurÕsvaluefunctioncanbeexpressedrecursivelybytakingthederivativeofthesequenceformwithrespecttotime,orsimilarvariationalargument:V(t,a)=max{cs,ks}s2[t,1)EtZ1te⇢(st)u(cs)ds|at=a⇢V(t,a)=maxc,kau(c)+limdt!01dtEt[dV(t,a)]ThisyieldsHamilton-Jacobi-Bellmanequation,whereintheconditionalexpectationcanbeexpandedusingItoÕsLemma:⇢V(t,a)=maxc,kau(c)+@aV(t,a)((Rtrt)k+rtac)+12@aaV(t,a)2k2+úV(t,a)Fortheremainderoftheproofitisconvenienttoseparatethetwomaximisationproblems:⇢V(t,a)=maxc{u(c)@aV(t,a)c}+maxka⇢@aV(t,a)(Rtrt)k+12@aaV(t,a)2k2+@aV(t,a)rta+úV(t,a)WithCRRAutility,u(c)=c11,thefirst-orderconditionsaregivenbythefollowing.Inparticular,theoptimalinvestmentdecisionismyopic;theconstraintonlyplaysarolewhenitbinds,asshownbyVila&Zariphopoulou(1997)intheirProposition4.2:c=@aV(t,a)1k=8>><>>:@aV(t,a)(Rtrt)@aaV(t,a)2ifunconstrainedaifconstrained67

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryWhen6=1,IguessthattheentrepreneurÕsvaluefunctionhastheform:V(t,a)=mt1a1,whichimplieslinearpolicyfunctions:c(t,a)=mtak(t,a)=!ta=8>><>>:Rtrt2aifRtrt2<aifRtrt2b(t,a)=(1!t)aOnecanverifythisguessbypluggingitintotheHJBequationandverifyingthattheequationstillholdswithequality,whichitdoeswhenthemarginalpropensitytoconsume,mt,isdeterminedbytheODEgiveninthelemma;thisODEcanbederivedbydividingbothsidesoftheHJBequationby1mta1andrearranging:úmtmt=mtrt+rt⇢+1(Rtrt)!t+2(1)2!2tWhen=1,IguessthevaluefunctionhastheformV(t,a)=Et+1⇢ln(a),whichbythesameprocessyieldstheODE:úEt=⇢Et+1ln(⇢)1⇢✓(Rtrt)!t+rt122!2t◆where!tisthesameasabove.TheentrepreneurÕsEulerequationcanbefoundbydifferenti-atingtheirpolicyfunctionwithrespecttotime,dctct=úmtmt+datat,andpluggingintheaboveODEformtandthebudgetconstraint:dctct=✓rt⇢+Rtrt!t+2(1)2!2t◆dt+min⇢Rtrt2,dWWithlogarithmicutility,theentrepreneursÕchoicesandpathsforwealthtriviallysatisfythetransversalitycondition:limt!1Et0he⇢(tt0)1⇢atati=0.ForCRRAutilitymoregenerally,theconditionstillholdsprovidedtheeconomyisatortendstothesteadystateinthelong-run,andprovidedthemarginalpropensitytoconsumeispositive,m>0(formorediscussionofwhichseebelow),andprovided⇢12(1)2!2,forwhenthesehold:68

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrylimt!1Et0⇥e⇢(tt0)ma1t⇤=limt!1exp(1)s12(1)2!2⇢ (tt0)=0,wheresisthedriftofsavings,andwhichequalszerointhelimitbyvirtueoftheeconomytendingtowardssteadystate.ProofofTheorem2:Proof.Toguaranteeasteadystateequilibriumexists,weneedthattotalbondholdingsarebelowthetotalstockofpublicdebtforsomesufficientlylowinterestrate,andaboveforsomesufficientlyhighinterestrate,i.e.BH(r)+BF(r)<DH+DFandBH(r+)+BF(r+)>DH+DF.Recallthatbondholdingsinagivencountrycompriseaproductofaweightonbondholdings(relativetocapital)andthelevelofthecapitalstock:Bi(r)=1!i(r)!i(r)áKi(r).Itiseasytoseethatasr!⇢frombelow,bondholdingsforeithercountrybecomearbitrarilylarge.Thebondweight,1!i(r)!i(r)=iµ(r)1,isclearlypositiveandinfacttendstopositiveinfinitysincelimr”⇢µ(r)!0(creditconstraintswillnotbindheresowecanignorethemaxoperator),andthecapitalstockwillapproachitscompletemarketslevelinthislimit:limr”⇢K(r)=⇣↵⇢+⌘11↵.Thusbondholdings8ilimr”⇢Bi(r)!1.Itremainstoshowthatforinterestratessufficientlylowthatbondholdingsforatleastonecountrytendtonegativeinfinity;ifthisholdsinonecountry,totalbondholdingswillalsotendtonegativeinfinity,sincebondholdingsintheothercountrywillbefinite.Consequently,wewillhaveshown(sinceallthesefunctionsarecontinuous),thatbondmarketequilibriumwillholdforsomer⇤foranyfiniteamountoftotalpublicdebt.Intheabsenceofcreditconstraints,thereexistsanr<0<⇢suchthatwhenapproachedfromabovethecapitalstocktendstoinfinity;thisristheinterestratesuchthattheexpressioninthedenominatorofthecapitalstockequalszero,i.e.r++iq21+(⇢r)=0,orr⌘2i1+r⇣+2i1+⌘2+2⇢1+2i2.Thus8ilimr#rKi(r)!1.Forinterestratesbelowthislevel,thecapitalstockisundefined.Moreover,sincetheweightonbondsismonotonicallyincreasinginr(@@rh1!i(r)!i(r)i=iµ(r)2µ0(r)>0),thereisalwaysanör<⇢sufficientlylowsuchthattheweightonbondsisnegative.Thisoccurswheniµ(ör)1=0,i.e.69

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryör⌘⇢(1+)22i.Hence,inordertoguaranteeasteadystateequilibriumexists,allweneedisthatör>r.Ifso,theremustexistanr⇤suchthattotalbondholdingsequalanygivenfiniteleveloftotalpublicdebt.Ifnot,theweightonbondswillbecomenegativeataninterestratewherethecapitalstockisundefined,orratherbondholdingswillalwaysremainpositiveandwilltendtopositiveinfinityasr#r,soasteadystateequilibriummaynotexistforarbitrarylevelsofpublicdebt,thoughitwillexistforsufficientlyhighlevelsofpublicdebt.Theconditionör>rimplies⇢(1+)22i>2i1+r⇣+2i1+⌘2+2⇢1+2i2.Thissimplifiesthroughsometediousalgebrato⇢+>2i2(1),whichistheconditionstated.Notethatif1,theconditionalwaysholds,soasteadystateequilibriumalwaysexists.Therighthandsideisalsoclearlyincreasinginand,sohigherlevelsofriskaversionorriskmakethisconditionhardertosatisfy.Notealsothatentrepreneurialconsumptionmustalsoobviouslybepositiveinorderfortheequilibriumtoexist.When1thisisalwayssatisfied,andwhen>1itissatisfiedwhenForeignriskisnottoogreat,since(ignoringcreditconstraintsforsimplicity)themarginalpropensitytoconsumeinthesteadystateisgivenby:m=r+⇢r+121(R(r)r)22>0m=r+21+(⇢r)>0=)r⇤8>><>>:<2⇢1if<1()r⇤<⇢<2⇢1>2⇢1if>1i.e.theconditionm>0requirestheequilibriuminterestrate,r⇤,tobebelowsomevaluethatisalwaysgreaterthan⇢,butinequilibriumther⇤<⇢,asistypicalformodelswithincompletemarkets.When=1,theMPCisequalto⇢,sotheconditionistriviallysatisfied.Inorderfortheconditiontoholdwhen>1,theequilibriuminterestratemustbeabovesomenegativevalue,r⇤>2⇢1.Usingfromabovethatr⇤>ör,asufficientconditiontoensurem>0when>1isthus⇢>2i2(1).70

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryThepotentialforasteadystatetonotexistconstrastswithAngeletos&Panousi(2011),wherethepresenceoflabourincomeforentrepreneursensuresthatbondholdingstendtonegativeinfinityasinterestratesapproachzerofromabove,ensuringanequilibriuminterestratealwaysliesbetween0and⇢.Thisisbecauseintheirset-upbondholdingsdependnegativelyonhumancapital,whichsinceentrepreneuriallabourincomeisrisklessisdiscountedattherisk-freerateandhenceisproportionaltotheinverseoftheinterestrateinthesteadystate.Obviouslyensuringtheinterestrateisalwayspositivedefeatsthepurposeofmyanalysishere,sothisisonereasonIdispensewithentrepreneuriallabourincome,asoutlinedabove.ProofofCorollary1:Proof.LetusconsiderthecasewhereHomeentrepreneursarenetborrowersandForeignentrepreneursarenetlenders,i.e.Hµ(r)1<0<Fµ(r)1,whichrequiresthatH<F.Globalbondholdingsaregivenby:BH(r)+BF(r)=✓Hµ(r)1◆✓↵r++Hµ(r)◆11↵+✓Fµ(r)1◆✓↵r++Fµ(r)◆11↵=DH+DFWewillproceedbyprovingthatinthisscenarioglobalbondholdingsarestrictlyincreasingintheinterestrate,r.Differentiatingwithrespecttor:@@r[BH(r)+BF(r)]=Hµ(r)2µ0(r)✓↵r++Hµ(r)◆11↵+1↵1✓Hµ(r)1◆✓r++Hµ(r)↵◆1↵111+Hµ0(r)↵Fµ(r)2µ0(r)✓↵r++Fµ(r)◆11↵+1↵1✓Fµ(r)1◆✓r++Fµ(r)↵◆1↵111+Fµ0(r)↵Onecaneasilynotethatthefirstandthirdtermsarepositive,sinceµ0(r)=12q21+(⇢71

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryr)1/2<0.Moreover,thesecondtermisalsopositive,sincebyassumptionHµ(r)1<0,whichimpliesr<⇢2H2(1+).whichinturnimpliesthatr<⇢2H21+,or1+Hµ0(r)>0(andofcourse↵<1).Finally,theforthtermisalsopositive,sincebyassumptionFµ(r)1>0,andr<⇢2H21+impliesthatr>⇢2F21+sinceF>H,whichinturnimplies1+Fµ0(r)<0.AsimilarargumentholdsforthecasewhereForeignentrepreneursarenetborrowersandHomenetlenders.Clearlyhoweveritisnotnecessaryforthisconditiontoholdforbondholdingstobemonotonic,norisitnecessaryforbondholdingstobemonotonicforequilibriumtobeunique,butthemorerestrictiveconditionsunderwhichtheequilibriumisuniquearenotparticularlyrevealing.IntheAutarkycase,therearetomyknowledgenosimpleandrevealingconditionsthatsufficetoguaranteeuniqueness.Inthiscase,theequilibriumconditionsimplifiesto:B(r)=✓µ(r)1◆✓↵r++µ(r)◆11↵=Dwhichyieldsauniquer⇤whenitsderivativeispositive:@@r[B(r)]=µ(r)2µ0(r)✓↵r++µ(r)◆11↵+1↵1✓µ(r)1◆✓r++µ(r)↵◆1↵111+µ0(r)↵>0Assumingpublicdebtispositive,itmustbethecasethatµ(r)1>0,sotoguaranteemonotonicitywithoutdissectingeachtermmorethoroughlyrequires1+µ0(r)<0,orr⇤>⇢221+,orthattheequilibriumliesintheupward-slopingpartofthecapitallocus,whichingeneralIhaveshownabovedoesnothold,exceptforveryhighlevelsofpublicdebt.Nevertheless,bondholdingsareverylikelytobemonoticallyincreasingintheinterestrateif1,andwhenriskisnottoohighif>1.72

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryProofofProposition3:Proof.First,wecanfindanexpressionforthestationarydistributionofwealthoveren-trepreneursfromtheKolmogorovForwardequation(KFE),usingtheassumptiondiscussedinthetextthatentrepreneursdieatratepandarerebornwiththeaveragelevelofwealth,A:úg(t,a)=@a[s(a)g(a)]+122@aa⇥!(a)2g(a)⇤pg(a)=0where,s(a)=(Rr)!a+racdenotesaveragesaving(i.e.ignoringshockstowealth,whichiscapturedbythesecondterm).Sincebothconsumption,c=ma,andcapitalholdings,k=minnRr2,oa(denoting!⌘minnRr2,o),areproportionaltowealth,wehavethats(a)=sa.ThefirstthingtonoteisthatthefirsttermintheKFEdropsout,becauseinthesteadystates(likeaggregatesavings)equalszero,s=(Rr)!+rm=0,toseethisnotethatinsteadystate,theODEdeterminingtheMPCsimplifiesto:m=rr⇢+1(Rr)!2(1)2!2Thus:s=r⇢+1(Rr)!+2(1)2!2=0whichisexactlytheconditionsatisfiedbythesteadystateEulerequation,úC=0.Itiswell-established(e.g.seeGabaix(2009))thatadouble-ParetodistributionsatisfiestheKFEequation.Toseethis,plugintheguessg(a)=a⇣1:0=122@aa⇥!2a1⇣⇤pa⇣10=⇣2⇣2p(!)273

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryThispolynomialhastwosolutions,⇣±=12±q122+2p(!)2,whichrepresentthetwotailparametersforeachbranchofadouble-Paretodistributionaroundthere-entrypoint,A.Wecannowfindtheconstantbyusingtheconditionthatthedistributionmustintegrateto1,theunitmassofentrepreneurs:1=ZA0⇣aA⌘⇣1da+Z1A⇣aA⌘⇣+1da1=A✓1⇣+1⇣◆sothat,=⇣+⇣(⇣⇣+)ATheremainderoftheproofisasfollows:Wi=$Vwi+(1)Z10Vei(a)gi(a)da=$1⇢ln(Cw)L1+’1+’+(1$)Z10mi1a1gi(a)da=$1⇢ln((1⌧i)(1↵)K↵i)11+’+(1$)mi1Z10a1✓aAi◆⇣1da=$1⇢ln✓✓1↵rDiYi◆K↵i◆11+’+(1$)mi1(Ki+Bi)1⇣i+⇣i(1⇣i)(1⇣i+)whichistheequationgiveninthetext,whereIhaveusedinevaluatingtheintegralthatnewentrepreneursarebornwithwealthequaltotheaverageamongallentrepreneurs,i.e.Ai.ProofofProposition4:Proof.Takeequation2andsubstituteinequations4andentrepreneursÕaggregatedpolicyfunctionforconsumption,Cit=⇢(Kit+Bit),whilealsousingbondmarketclearing,7:úKit+úDit=↵K↵itKit+ritDit⇢(Kit+Dit)74

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryFurthermore,subtractequation6frombothsides,yielding:úKit=↵K↵itKit⇢(Kit+Dit)+T(Dit)Gitwhichistheexpressiongiven.OnecanarriveatthesameequationbymanipulatingtheEulerequation(equation3)inasimilarfashion.Theseconddifferentialequationisobviouslyimmediatefromequation6,substitutingintheequilibriuminterestraterAi(Ki,Di)=↵K↵1iKiKi+Di2i.ProofofProposition5:Proof.Againstartingwith2,thistimedefiningAit⌘Kit+Bit,andagainsubstitutinginequations4andCit=⇢(Kit+Bit)andthistimealsoequation5,onearrivesatthefirstODEofProposition5:úAit=↵K↵1itrtmin⇢↵K↵1itrt2i,iAit+(rt⇢)Aiti={H,F}Addingtogetherthegovernmentdebtflowequations6forbothcountries,andusingthedefinitionsofT(Dit)ÐwhereforsakeofsimplicityIamassumingH=F=ÐonearrivesatthesecondODE:úDt=(rtr)Dt(DtD)+GHtGH+GFtGFThetwostaticequationscanbederivedimmediatelyfromequations5and8bysubstitutinginequation4andusingthatBit=(1!it)Ait,yielding:Kit=min⇢↵K↵1itrt2i,iAiti={H,F}Dt=✓1min⇢↵K↵1Htrt2H,H◆AHt+✓1min⇢↵K↵1Ftrt2F,F◆AFt75

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-HendryAppendix2:ExtensionsDynamicswhen6=1andTit=T(Kit,Dit):NowIexplorethedynamicsoftheequilibrium,allowingfornon-logCRRAutilityandfortaxrevenuestobeamoregeneralfunctionofthestateofthedomesticeconomy.AsIwillshow,thiscanresultinthesteadystatebeingunconditionallystable,withallnegativeeigenvalues,implyingindeterminacyofequilibrium;manyfunctionsC(t)convergetothesteadystateinthelongrunandhenceareconsistentwiththetransversalitycondition.ThedynamicsofequilibriumunderAutarkynowbecome:úKt=↵K↵tKtCt+T(Dt,Kt)GtúDt=rA(Kt,Dt)Dt(T(Dt,Kt)Gt)úCt/Ct=rA(Kt,Dt)⇢+121+↵K↵1trA(Kt,Dt)22whereagainrA(Kt,Dt)=↵K↵1tKtKt+Dt2andT(Dt,Kt)isacontinuouslydifferentiablefunctionwithTD0.TheJacobianofthissystemevaluatedatthesteadystateisasfollows:0BBBB@úKtúDtúCt1CCCCA=266664↵2K↵1+TKTD1rKDTKrDD+rTD0hrK+1+(Rr)2(RKrK)iChrD1+(Rr)2rDiC03777750BBBB@KtKDtDCtC1CCCCAConsiderforsimplicitycalibrationswherethesteadystateisunique.Sincewehavetwostatevariables,KandD,andonejumpvariable,C,thesystemisconditionallystablewhentwoofitseigenvaluesarenegativeandoneispositive;thetransversalityconditionthensufficestoruleoutexplosivesolutionsandpindownauniqueequilibriumpathforconsumption,capitalanddebt.Iffewerthantwoeigenvaluesarenegative,thesystemislocallyunstableandperturbationsfromthesteadystateÐforexampleduetoagovernmentspendingshockÐ76

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendrycanputtheeconomyonanunstablepath,asdiscussedinthemainpartofthetext.Thereishoweveranewpossibilityofindeterminacyofequilibrium,whereconsumptionisnownotpinneddownuniquelyasafunctionofcapitalanddebt;ifalleigenvaluesarenegativethenthesystemisunconditionallystable,meaningwithinsomerangeanychoiceofconsumptionisconsistentwiththetransversalityconditionandwillasymptotetothesteadystate.Itshouldbenotedthatthisindeterminacyisdistinctfromthepurelynominalindeterminacyencounteredfrequentlyinmonetaryeconomieswithflexiblepricesoperatingunderaninterestraterule(Sargent&Wallace1975);heretheindeterminacyisreal,notnominal.ButthecauseofthisrealindeterminacyisalsodistinctfromthatwhicharisesinNewKeynesianmodelswithnominalrigiditiesandpassiveinterestratepolicy(Claridaetal.1999).SinceindeterminacyherearisesduetothespecificationofthegovernmentÕsfiscalrule,onemighttermthisfiscalindeterminacy.Therearetomyknowledgenosimpleconditionsunderwhichindeterminacyorinstabilityarises;itismorefruitfulsimplytocalculatetheeigenvaluesnumericallytoascertaintheirsigns,especiallyintheequilibriumwithintegratedglobalcapitalmarkets.However,boththeindeterminacycaseandtheinstabilitycasearisewhenthedeterminantoftheJacobianmatrixaboveisnegative,whichimpliesthateitheralleigenvaluesarenegativeoronlyoneis,sincethedeterminantofamatrixistheproductofitseigenvalues|J|=123.Looselyspeaking,thedeterminantisnegativewhendebtgrowthrespondsonlyweaklytohigherdebtbutrelativelystronglytohighercapital,whileatthesametimeconsumptiongrowthrespondsstronglytohigherdebtbutweaklytohighercapital:|J|=h@K⇣úD⌘@D⇣úC⌘@D⇣úD⌘@K⇣úC⌘iContrarily,whenthedeterminantispositive,wearelikelytoencounterthecaseofconditionalstability,withonepositiveeigenvalueandtwonegative.Itispossiblebutunlikelythatalleigenvaluesarepositive,sincethetraceoftheJacobianÐwhichisthesumoftheeigenvalues,tr(J)=1+2+3Ðislikelytobenegative(implyingatleastoneeigenvalueisnegative),77

GlobalImbalances&TheLimitsofFiscalSpaceLeeTyrrell-Hendryandwillbeprovidedtaxrevenuesrespondsufficientlystronglytoexcessivedebtandarenottoo-rapidlyincreasingincapital:tr(J)=@K⇣úK⌘+@D⇣úD⌘78

Collepals.com Plagiarism Free Papers

Are you looking for custom essay writing service or even dissertation writing services? Just request for our write my paper service, and we'll match you with the best essay writer in your subject! With an exceptional team of professional academic experts in a wide range of subjects, we can guarantee you an unrivaled quality of custom-written papers.

Get ZERO PLAGIARISM, HUMAN WRITTEN ESSAYS

Why Hire Collepals.com writers to do your paper?

Quality- We are experienced and have access to ample research materials.

We write plagiarism Free Content

Confidential- We never share or sell your personal information to third parties.

Support-Chat with us today! We are always waiting to answer all your questions.

Examine when and how firm managers should disclose information about the firm’s value in competitive markets, including the timing of such disclosures and whether bad news should be released.

Related Posts

John Doe has decided to clone himself. He is sterile

Prepare a 2- to 3-page paper (not counting the cover and reference pages) discussing how the Family and Medical Leave Act, the Pregnancy Discrimination Act, the Americans with Disabilities Act, safety laws, or other state laws might apply to the following scenario.

Capital Budget Methodologies