IMF-自动化与福利：任务与教育的作用（英）

AutomationandWelfare:TheRoleofBequestsandEducation

PreparedbyManukGhazanchyan,AlexeiGoumilevski,

andAlexMourmouras

WP/24/11

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2024

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©2023InternationalMonetaryFund

WP/24/11

AutomationandWelfare:TheRoleofBequestsandEducation*

PreparedbyManukGhazanchyan,AlexeiGoumilevski,andAlexMourmouras

AuthorizedfordistributionbyAlexMourmouras

January2024

IMFWorkingPapersdescriberesearchinprogressbytheauthor(s)andarepublishedtoelicitcommentsandtoencouragedebate.TheviewsexpressedinIMFWorkingPapersarethoseoftheauthor(s)anddonotnecessarilyrepresenttheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement.

ABSTRACT:Thispaperexaminesthewelfareeffectsofautomationinneoclassicalgrowthmodelswithandwithoutintergenerationaltransfers.Inastandardoverlappinggenerationsmodelwithoutsuchtransfers,improvementsinautomationtechnologiesthatwouldlowerwelfarecanbemitigatedbyshiftsinlaborsupplyrelatedtodemographicsorpandemics.Withperfectintergenerationaltransfersbasedonaltruism,automationcouldraisethewell-beingofallgenerations.Withimperfectaltruism,fiscaltransfers(universalbasicincome)andpublicpoliciestoexpandaccesstoeducationopportunitiescanalleviatemuchofthenegativeeffectofautomation.

JELClassificationNumbers:E13,E62,D64

Keywords:

Automation;Aging;Altruism;FiscalPolicy;Education;OverlappingGenerations

Authors’E-MailAddresses:

mghazanchyan@;

agoumilevski@im

;

amourmouras@

口ManukGhazanchyanisanEconomistintheWesternHemisphereDepartment,AlexeiGoumilevskiisaSeniorScientificComputingEngineerintheInformationandTechnologyDepartment,andAlexMourmourasisDivisionChiefintheAsiaPacificDepartment.

*Presentedatthe28thInternationalConferenceonComputinginEconomicsandFinanceinDallas,Texas,June17-19th,2022.Wethankconferenceparticipantsfortheirusefulcomments.WearealsogratefultoPeterRangazas,MarinaMendesTavares,andJeremyCliftfortheinsightfulcommentsthatshapedourpaper.

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Contents

I.Introduction 1

II.RelevantLiterature 3

III.AutomationandWelfareinOverlappingGenerationModels 5

B.IntroducingaOne-Timetax 9

C.K&SModelwithaBequest 11

D.K&SModelwithBequestandEducation 14

1.PrivateEducation 15

2.PublicEducation 17

IV.ArtificialIntelligence 22

A.K&SModel 22

IV.Conclusions 26

Appendices 27

AppendixA.K&SModelwithBequest 27

AppendixB.ReplicatingKotlikoffandSachs(2012) 31

AppendixC.RobustnessChecks 31

A.References 34

I.Introduction

Automationhasacceleratedinrecentdecades,drivenbyongoingimprovementsincomputingandinformationtechnologiesandassociatedcostreductions.Machinesinawideningrangeofindustriesperformincreasinglycomplextasks,poweredbysophisticated,networkedsoftware.Theaccelerationinautomationanditseconomy-widediffusioninblue-andwhite-collaroccupationsalikeiscreatingnewemploymentcategoriesbutisalsocontributingtowideninginequalityandfuelingdemandforgovernmentpoliciestoreverselong-termincomelossesoflabor.Thislong-standingpromiseandconcernsarevividlyillustratedbythelatestbreakthroughinArtificialIntelligenceinvolvinggenerative,pretrainedtransformers.

Lookingahead,whilethepaceofautomationislikelytocontinue,itseffectsmaybemitigatedbyoffsettingforces.Populationsareagingalmosteverywhere.Intheadvancedeconomies,theworking-agepopulationhasstartedshrinkingforthefirsttimesinceWorldWarII(Spence,2022).Globally,thepopulationofworkingageisexpectedtocontinuetogrowuntilabout2040,buttheratiooftheworkingagepopulationtothetotalisalreadydecliningglobally(Chart).InthecaseofChina,forexample,theworking-agepopulationisexpectedtoshrinkbyafifthoverthenext30years.AsGoodhartandPradhanstress,ourageisoneofdemographicreversalinwhichthe“longglutofinexpensivelaborthathadkeptpricesandwagesdownfordecades,isgivingwaytoaneraofworkershortages,andhencehigherprices”.

Recurringglobalpandemicsalsoadverselyaffectlaborsupply,eitherbydepressinggrowthinthelaborforcedirectly(AIDSpandemic),orindirectlybyreducingtheparticipationofolderworkersandothersincontact-intensiveoccupations(pandemicrelatedtoCovid-19).Intheabsenceofmasssouth-north

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migration,robotsmayturnouttobeessentialinmeetingmoreoftheneedsoftheelderlyandreversedeclinesinaggregateoutputandwelfare1.

Thispaperexaminesthecombinedwelfareeffectsofautomationandlowerlaborsupplyusingneoclassicalgrowthmodelswithandwithoutintergenerationaltransfers.Itbeginsbyreplicatingaversionofthewell-knownresultofKotlikoffandSachs(K&S,2012)thataone-timeimprovementinthetechnicalefficiencyofmachinesendsupimmiseratingallfuturegenerations.Thisstrikingresultreliesontwocrucialassumptions.First,machinesareverygoodsubstitutesforunskilledlaborthroughouttheeconomy,sothatimprovedautomationendsupdisplacingworkersandloweringwagesacrosstheboard.Second,therearenooperativeintergenerationaltransfersofanykind,sothattheownersofcapitalendupconsumingtheentirewindfallfromtheimprovementsinautomationduringtheirlifetime(ageneration,roughlythirtyyears).Thepositiveshocktotheefficiencyofmachinesdoesnotraisesaving,depressesinvestmentinphysicalandhumancapital,andsetsinmotionanever-endingcycleofdecliningwelfare.Governmentpolicyisthereforeneededtospreadthiswindfallmoreequitablyacrossfuturegenerations.K&Sconsiderwealthtaxes,inparticularsocializingaportionoftheeconomy’scapitalstockthatallowsthegovernmenttofinanceasustainableincomestream(universalbasicincome)forallfuturegenerations.Resortingtocompulsionisessentialwhengenerationsareselfish,precludinganysortofvoluntaryintergenerationaltransfers.

Infact,privateintergenerationaltransfersaresubstantial,withabouthalfofallhouseholdsplanningtoleaveestates(LaitnerandJuster,2017).Thefirstobjectiveofthepaperistoreassessthewelfareeffectsofautomationinthepresenceofintergenerationaltransfers,bothbequestsandprivatelyandpubliclyfundedschooling.InaversionoftheK&Smodelwithoperativebequests,wefindthatintergenerationaltransfersarepositiveinequilibriumifthestrengthofaltruismexceedsacertainthreshold,mitigatingthenegativeeffectsofautomation.Butwhileitiscomfortabletoknowthatthegainsfromautomationmaybepassedtofuturegenerationswithouttheneedtonationalizecapital,thismodelofperfectaltruismisalsoextreme:manyfamiliesineachgenerationcannotmakeefficienttransferstotheirchildren.

Whatisneededisamorebalancedmodel,onethatfeaturesheterogeneitybothwithinandacrossgenerations,withsomehouseholdsmakingefficientbequestsandothersstuckinacornersolution.WeassesswhetherautomationisimmiseratinginaversionofK@Smodelthatincorporatespurealtruism,andinGlommandRavikumar’smodel(G&R,1992)inwhichparentsmakeinvestmentsintheschoolingoftheirchildren.Westudyhowfiscalandeducationalpoliciescanbestraisewelfare,byalleviatingfinancingconstraintsinthefinancingofhumancapitalinvestmentsandrestoringequalityofopportunity.SimilarresultsobtaininaversionofthemodelusedbyIvanyna,MourmourasandRangazas(2018)inwithtwotypesofhouseholds(thepoor,whoarebequest-constrained)andtherich(whoareunconstrained).

Thepaperthenturnstoananalysisofacombinedshockinvolvingajumpinautomationandasimultaneousreductioninlaborsupplydrivenbydemographics.Asexpected,strengtheningaltruisticbondsraisebequestsandhumancapitalinvestmentsoftheyoung,providinganadditionalstimulustoeconomicgrowth.Inaddition,governmenttransfersoftaxrevenueleviedontherichcanimprovethe

1Businessleadershavealsomadeaconnectionbetweenautomationandagingrecently.OneexamplewastherecentarticleinFortunebyIBMCEO,ArvindKrishnaherehepointstodecliningpopulationstocalmfearsaboutA.I.takingjobs.Hefurtheraddedthat

ultimately“thereisgoingtobejobcreation”withA.I.,asjobswillalsobeaddedinareaswithmorevaluecreation.

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welfareofthepoorandreduceinequalitywithinandacrossgenerationswhenaltruisticlinksbetweengenerationsareweak.

II.RelevantLiterature

Theliteratureonautomationanditseconomicimpactisevolving,withsomeearlierstudiesfromGordon(2012),Cowen(2011),AcemogluandRestrepo(2017,2018),SachsandKotlikoff(2012,2015),Ford,(2015);Freeman,(2015)amongstpessimists,andBrynjolfssonandMcAfee(2014),Autor(2014,2015)amongtheoptimists.Thekeyissueiswhetherautomationreplaceslaborshareandemploymentthroughreplacementofroutinetasksofever-increasingscopeandcomplexityorwhether,onnet,itincreaseslaborparticipationbycreatinghigh-payingjobsinemergingnewoccupations.SomegloomyscenariosforlaborresultingfromartificialintelligenceandsimultaneousautomationbreakthroughsaredescribedinBostrom(2014).Graetzet.al.(2018)examinedtheeconomiccontributionofmodernindustrialrobotsin17countriesfortheperiod1993-2007.

Contrarytothepessimisticview,theseauthorsfoundthattheincreasinguseofrobotsraisedtheannualgrowthofGDPandlaborproductivityby0.37and0.36percentagepoints,respectively.Authorsconcludethatthoserobotsdidnotsignificantlyreducetotalemployment,althoughtheydidreducelow-skilledworkers’employmentshare.Gaaitzenetal.(2020)studiedtheeffectsofadaptationofindustrialrobotsandoccupationalshiftsbytaskcontentinthethirty-sevencountriesfortheperiodfrom2005to2015.Theauthorsfoundthatincreaseduseofrobotsisassociatedwithpositivechangesintheemploymentshareofnon-routineanalyticjobsandnegativechangesintheshareofroutinemanualjobs.Ofcourse,enhancingpolicyincludingR&Dandtheregulatoryplatformsinbothprivateandpublicsectorstosupportdigitaltechnologiesiskeytoimproveproductivity.Whilethe2020-22pandemichelpedtoacceleratethedigitaltransformation,manysectors–includingthepublicsector–arelagging,andhenceconcernsabouttheeffectsofautomationonemploymentwillpersist(Spence,2022).

OnlyafewstudiesexaminedtheeffectofautomationandpopulationagingonthelabormarketasidefromtheclassicalworkbyFreyandOsborne(2017)focusingontheprobabilityofautomationaffectingvariousjobsandoccupations.OneoftheearlieststudiesonautomationandpopulationagingwasbyAcemogluandRestrepo(2017),wheretheauthorsexaminedtherelationshipbetweeneconomicgrowth,populationaging,andautomationatthecountrylevel.

Phiromswad

etal(2022)isamongstthemostrecentstudiestofocusonthoseeffectsbutalsoontheinteractioneffectsofautomationandpopulationagingonthelabormarket.ConsistentwithpreviousliteratureincludingwithGraetzandMichaels(2018)theauthorsfoundstrongevidencethatautomationnegativelyaffectsemploymentgrowthholdingotherfactorsconstant.Theyalsofoundstrongevidencethatthedisaggregatedmeasuresofage-relatedabilitiesaffectemploymentgrowthbutnottheaggregatemeasure.Asexpectedandconsistentwithfindingsthatautomationisstillevolvinginaffectinghighvaluejobs,theauthorsfindthatwithoccupationswithlowscoreonboththeage-appreciatedcognitiveabilityaswellage-depreciatedphysicalability(suchasproductionoccupationsandfoodpreparationandservingrelatedoccupations),thenegativeeffectofautomationonemploymenttendstobestrongest.However,foroccupationswithahighscoreinbothage-appreciatedcognitiveabilityaswellasage-depreciatedphysicalability(suchasprotectiveserviceoccupationsandhealthcarepractitionersandtechnicaloccupations),thenegativeeffectofautomationonemploymenttendstobeweakest.

Aghion-Jones-Jones(2017)studytheimplicationsofartificialintelligenceforeconomicgrowthinlightofreconcilingevolvingautomationwiththeobservedstabilityinthecapitalshareandpercapitaGDPgrowthoverthepastcentury.Theauthorscreatesufficientconditionstogenerateoverallbalanced

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growthwithaconstantcapitalsharethatstayswellbelow100percent,evenwithnearlycompleteautomation.Inotherwords,whileBaumol’scostdiseaseleadstoadeclineintheshareofGDPassociatedwithmanufacturingoragriculture(oncetheyareautomated),thisisbalancedbytheincreasingfractionoftheeconomythatisautomatedovertime.TheauthorsalsostudytheeffectsofintroducingA.I.intheproductiontechnologyfornewideasandthepossibilitythatA.I.couldgeneratesomeformofasingularity,wheretheauthorsneverthelessclaimthattheBaumolthemeherealsoremainsrelevant:evenifmanytasksareautomated,growthmayremainlimitedduetoareasthatremainessentialyetarehardtoimprove.

Pizzinelliandothers(forthcoming)examinetheimpactofArtificialIntelligence(AI)onlabormarketsinbothAdvancedEconomies(AEs)andEmergingMarkets(EMs).TheauthorsproposeanextensiontoastandardmeasureofAIexposure,accountingforAI'spotentialaseitheracomplementorasubstituteforlabor,wherecomplementarityreflectslowerrisksofjobdisplacement.Thentheyanalyzeworker-levelmicrodatafromtwoAEs(USandUK)andfourEMs(Brazil,Colombia,India,andSouthAfrica),revealingsubstantialvariationsinunadjustedAIexposureacrosscountries.TheauthorsfoundthatwhileAIposesrisksoflabordisplacementduetotaskautomation,italsoholdspromiseinitscapacitytoenhanceproductivityandcomplementhumanlabor,especiallyinoccupationsthatrequireahighlevelofcognitiveengagementandadvancedskills.TheauthorsalsofindthatAEsmayexpectamorepolarizedimpactofAIonthelabormarketandarethuspoisedtofacegreaterriskoflaborsubstitutionbutalsogreaterbenefitsforproductivity.

Theextentandformofvoluntaryintergenerationaltransfersisdictatedbythestrengthofintergenerationalaltruismandisanimportantconsiderationinmacroeconomicsthatisrelevantforourpaper.Kotlikoff(2001)providesanexcellentsurveyofkeyworksontheroleofintergenerationalaltruism,includingempiricalfindings—forexample,theresultsofAltonjiandHayashi(1994)whichareconsistentwiththepurealtruismtheory.Acloselyrelatedareaofresearchconcernstheformofhumancapitalinvestments,specificallytherationalebehindeducationorotherbequestsinkind.RazinandRosenthal(1990)showthatfamilytaxationasaresponsetoinformationasymmetrybetweenaparentandachildcouldreducetheneedforgovernmentinterventionandtaxation.HoodandJoyce(2017)provideanexcellentupdatetotheempiricalrelevanceofaltruism.OurpaperismostcloselyrelatedtoMichel,Thibault,andVidal(2004),whostudytheeffectsofaltruismandfiscalpoliciesongrowthinanoverlappinggenerationsmodelinthetraditionofDiamond,andtoGlommandRavikumar(1992)whostudybequestsintheformofhumancapitalinvestmentsinchildren.Westudyprivatelyfundedschoolingforfamilieswithoperativebequestsandpubliclyfundededucation.

Wefindthatgovernmentspendingoneducationpromoteeconomicgrowth.Theseconclusionsaresupportedbyavastvolumeofresearchthatlinkindividuals’educationattainmenttoeconomy-wideprosperity.FabrizioCarmignani(2016)studiedeffectsofgovernmentexpendituresoneducationtoeconomy.AuthorusedtheWorldBank'sWorldDevelopmentIndicatordatabasedataon151countriesfor2000-2010years.Heconcludedthat“increaseineducationexpenditureby1pointofGDPincreasesGDPgrowthby0.9percentagepoints”.Gheraia,Zouheyretal.(2021)investigatedrelationshipbetweenthecostofeducationandeconomicgrowthintheKingdomofSaudiArabiafortheperiod1990-2017.Authorsfoundthatinthelongruntheriseineducationalexpenditureby1%wouldleadtoanincreaseineconomicgrowthby0.89%.SimilarresultswereobtainedbyYahya,Mohdetal.(2012).Authorsanalyzedthelong-runrelationshipandcausalitybetweenthegovernmentexpenditureineducationandeconomicgrowthinMalaysiafortheperiod1970to2010.Theyconcludedthateconomicgrowthispositivelycorrelatedwithfixedcapitalformation,laborforceparticipationandexpenditureineducation.RegardingGrangercausality,theeducationalexpenditure

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istheshort-termGrangercauseofeconomicgrowthandviceversa.MehmetMercanaetal.(2014)performedcointegrationanalysisbetweentherealgrossdomesticproductandtotalexpensestotheeducationforthecaseofTurkeyfortheperiod1970-2012.AuthorsusedAutoregressiveDistributedLagmodelwithboundstesting.Authorsfoundthat1%increaseineducationexpensesincreaseseconomicgrowthby0.3%.

III.AutomationandWelfareinOverlappingGenerationModels

A.AnAnalyticalTool:TheK&SModel(2012)

Webeginwithasimplemodelfeaturingtwoperiod-livedoverlappinggenerations(OLG).Eachperiodt=1,2,…thepopulationconsistsofyoungandoldhouseholds.Theyoungareendowedwithoneunitofinelasticallysuppliedunskilledlabor.Theyconsumepartofthisincomeandinvesttherestinphysicalcapital(machines,M)andintheirownhumancapital(skilledlabor,S).Machinesandhumancapitalareperfectsubstitutesinsavingportfolios.Inthesecondperiodoflife,householdsrenttheirmachinesandskillsinperfectlycompetitivemarkets,consumingallinterestandprincipal.GrossoutputQisaconstantelasticityofsubstitution(CES)productionfunctionoftheeconomy-widestocksofM,L,andS:

Q=Q(N(uM,L),S)(A.1)

MandLcombineinaCESproductionfunctionwithelasticityεMLtoproduceanintermediateproductN,andNandScombineinaCESproductionfunctionwithelasticityεSNtoproducethefinaloutputQ.Theparameteruisaparametermeasuringthetechnicalefficiencyofmachinery.Ariseinuisapuretechnicaladvance.KotlikoffandSachs(KS,2012)examinewhetherariseinucanreduceeconomicwellbeing,anoutcometheyrefertoas“im-mesmerizingproductivity.”

CompetitivefirmshireM,L,andStothepointwheretheirmarginalproducts(denotedasQifori=M,L,andS)equaltheirmarketwages:Qi=wi.FollowingK&Sthepartialderivativeofwagewithrespecttoproductivityuis,

dln(QL)⁄dln(u)=[εSN−θεML]⁄εML(A.2)

whereθistheshareofskilledlaborintheeconomy,equalto(Qs∗S)⁄Q.Weseethatariseinmachine’sproductivityreducestheunskilledwageifεML>εSN⁄θ.Thus,“im-mesmerizing”productivityismorelikelyif:

.Substitutabilityofmachinesandunskilledlaborishigh(εMLlarge)

.Substitutabilityofintermediategoodsandskilledlaborislow(εSNsmall)

.Theshareofskilledlaborinfinaloutputishigh(θhigh)

BelowwepresenttheoreticalunderpinningsfortheK&Sconclusions.Notethattheincomeoftheyoung,It,iscomprisedofwageswt,partofwhichisinvestedinmachinesandhumancapital.Income

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β

=ct

whenold,It+1,comesfromtheownershipofmachinesandacquiredskills.Thelifetimebudgetconstraintsofgenerationbornatt,whoareyounginperiodtandoldatt+1are:

It=Ltwt=ct+svt(A.3)

It+1=tMt+1+tst+1=Dt+1(A.4)

Herectistheconsumptionwhenyoung,Dt+1isconsumptionwhenold,andsvtissaving,whichisinvestedinmachinesandskills:

svt=Mt+1+st+1(A.5)

Thisallocationofsavingsismadeunderperfectforesighttomaximizeutility,sothatinvestmentinmachinesandskillsarechosentoequalizemarginalproductsofMandStothegrossinterestrateintheeconomy:

Rt=t=t=1+rt(A.6)

HereRtisthegrossrateofreturnandrtistheinterestreturnonmachines.Combiningequations(A.3-4),weget:

ct+D1=It(A.7)

Lifetimeutilityforahouseholdbelongingtogenerationtis:

U=βlog(ct)+(1−β)log(Dt+1)(A.8)

Parameterβisthetimediscountfactor.Itisrelatedtothepatienceoftheagentaboutconsumption-largerβmeanstheagentispredisposedtoconsumemoreduringyouth.Theutilitymaximizingchoicesofconsumptionmustsatisfythefirstordercondition,

1−β

Dt+1

Rt

(A.9)

Turningtotheproductionsideoftheeconomy,wefollowK&SwhopostulateaCobb-DouglasproductionfunctionforfinalgoodsassembledusingskilledlaborSandanintermediateinputN:

Q(N,s)=ANas1−a

(A.10)

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TheintermediatefactorNisalinearcombinationoflaborLandmachinesM,meaningthatthesetwoprimaryfactors,machinesandlaborareperfectsubstitutes.

N=L+u1M(A.11)

Hereuisthetechnicalefficiencyofmachinery.Highervaluesofu1indicateimprovementsinautomation.Themarginalproductsofthefactorsare,

QL=Aa(N⁄s)a−1

QM=Aau1(N⁄s)a−1

Qs=A(1−a)(N⁄s)a

Assuming“perfectforesight”,QM=Qs,weget,

aus=(1−a)N

and

QM=Qs=Aau1a−1

wt=Aa()a−1

(A.12)

(A.13)

(A.14)

Solvingsavingsequation(A.5),noarbitrageassumption(A.6),firstordercondition(A.9),andusingequationforintermediateproduction(A.11),wecanderive:

ct=βLtwt

(A.15)

Dt+1=(1−β)RtLtwt

svt=(1−β)Ltwt

Mt+1=asvt−(1−a)Lt⁄u1

st+1=(1−a)svt+(1−a)Lt⁄u1

MachinesMtandskillsstareproportionaltosavingsandincrease/decreasewithgrowthofproductivityfactoru,respectively.

Weareinterestedintheeffectsontheeconomyofdifferentcombinationsofshockstoproductivityandlabor,especiallyinthepossibilityofimmiseratingequilibriaovertimeinwhichinequalitywithinandacrossgenerationsrises.

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PropositionA1:Improvementsinautomationbenefittheownersofcapital(theinitialoldgeneration)andlowerthewelfareofallsubsequentgenerationsofworkers.Apositiveshockintheefficiencyofmachinesleadstohigherconsumptionandincomeoftheold.Itsimultaneouslylowersthewageincomesofunskilledworkersandlowerstheirlifetimeconsumptionandutility.Intergenerationalinequalityrises.

BelowweassumethatCobb-Douglasproductionfunctionhasweightsa=0.74onlaborand0.26onmachinesandβ=0.5onskills.ThissetupproducestheresultsinFigure1.InFigure1weassumethatproductivityunexpectedlyrisesby35%inperiodt=3whilelaborsupplyremainsthesame.Inthethirdperiod,MandNareunchangedbecauseoftheproductivityshock,astheyweredeterminedbythesavingdecisionsoftheyounginthefirstperiod.Thecurrentwageofunskilledworkers(i.e.,theearningsoftheyoungofgeneration3)declinesfrom2.82to2.61astheresultoftheriseinu1.ThereturnsonMandSbothrises,andtheoldgeneration(thatownsbothMandS)experiencesaboominincomewhileyounggenerationexperiencesabust.Consumptionoftheoldriseswhileconsumptionandsavingoftheyoungdeclines.ThispushesdownthefuturecapitalstocksofMandS.Byperiod5theeconomyreachesanewequilibriumcharacterizedbylowerwages,lowerskills,lowerM,andhighertotaloutputthaninthebaseline.Theratioofearningsofskilledworkersrelativetounskilledworkers,ws/wl,ispermanentlyraisedfrom1to1.35.

FollowingK&S,wetraceshowtheimplicationsofimprovedautomationforlifetimeutilityacrossgenerations.Theutilityofgeneration3isslightlybelowthatofgeneration1,aswageshavedeclinedwhilethereturnsonsavinghaveincreased.Yettheutilityofgenerations4andlaterishigherthanthebaselineutility.Youngunskilledworkerslosefromtheriseinmachineproductivity.Theproductionofgeneration3soarsasthisgenerationbenefitsfromhighreturnstobothMandS.Theanticipatedcaseproducesslightlydifferentbutqualitativelysimilarresults.

Forpiece-wiseconstantproductivityu1thesevariablesarepiece-wiseconstant.Consumptionreaches

stationarylevelwhenct=Dt+1,orwhen(1−β)Rt=β.Thissetsconditionfortotalfactorproductivity,A=1−a.Itinverselydependsonproductivityfactoru1.

Figure1.TimedependenceofvariablesforSachsandKotlikoffsettings.Atp