Indeterminacy and Sunspots in Macroeconomics · Indeterminacy and Sunspots in Macroeconomics ... , Gerzensee Lectures 2. Reading (c) Roger E. A. Farmer, Gerzensee Lectures •Farmer,

IndeterminacyandSunspotsinMacroeconomics

TuesdaySeptember5th:Lecture4Gerzensee,September2017

©RogerE.A.FarmerWarwickUniversityandNIESR

TopicsforLecture4

• ChoiceUnderUncertainty• Von-NeumannMorgensternPreferences• RiskandIntertemporalSubstitution• ContingentcommoditiesvsArrowsecurities

(c)RogerE.A.Farmer,GerzenseeLectures9/3/17 2

Reading

(c)RogerE.A.Farmer,GerzenseeLectures

• Farmer,TheMacroeconomicsofSelf-FulfillingProphecies[Chapter9]

9/3/17 3

Time,StatesofNatureandGeography

9/3/17 (c)RogerE.A.Farmer,GerzenseeLectures 4

InfiniteGEtheoryutilityisdefinedovertheconsumptionset,𝑋

Householdsmaxautilityfunction,𝑢#(𝑥): 𝑋 ≡ 𝑅*ℓ → 𝑅

Debreupointedthatanelementof𝑥 canbeindexedbydate,stateofnatureandlocation.

Forexample,anappleisnotjustanapple.Itmightbe,forexample,anappleinNewYorkcityonJuly5th 2025ifitrains.

Example


Twodates;1 and2.

Twostatesofnature;𝛼 and𝛽

ℓ = 2 commoditiesateachdate,𝑥 and𝑦

Inthisexampleutilityisdefinedoversixobjects

𝑢# = 𝑢# 𝑥3, 𝑦3, 𝑥5 𝛼 , 𝑥5 𝛽 , 𝑦5 𝛼 , 𝑦5 𝛽

DatesandStates


Period1 Period2 Period3 Period4

𝛼, 𝛼, 𝛼

𝛼, 𝛽, 𝛼𝛼, 𝛼, 𝛽

𝛼, 𝛽, 𝛽

𝛽, 𝛼, 𝛼

𝛽, 𝛼, 𝛽

𝛽, 𝛽, 𝛼

𝛽, 𝛽, 𝛽

𝛼𝛼

𝛼

𝛼

𝛼

𝛼

𝛼

𝛽

𝛽

𝛽

𝛽

𝛽

𝛽

𝛽

PuttingStructureonUtility


Von-NeumannandMorgenstern,InTheoryofGamesandEconomicBehavior,askedhowconsumerswouldchoosegamblesoverascalar;moneyincome

Theiraxiomswerelaterextendedtochoiceovervectors.

9/3/17 7

Lotteries


Alottery,𝐿,isarandomvariabledefinedovertheconsumptionset𝑋.

Forexample,𝐿 mightbeagambleinwhichyoureceive𝑥3 withprobability𝑝 and𝑥5 withprobability1 − 𝑝where𝑥3 and𝑥5 areelementsof𝑋.

9/3/17 8

TheVNMAxioms:


Axiom1: Completeness

Foranytwolotteries𝐿 and𝑀

Either,𝐿 ≻ 𝑀 Or,𝐿 ≻ 𝑀Or,𝐿 ∼ 𝑀

Where≻means“ispreferredto”And∼means“isequivalentto“Also≽means“isatleastasgoodas”

9/3/17 9

TheVNMAxioms:


Axiom2: Transitivity

Foranythreelotteries𝐿,𝑀 and 𝑁

If,𝐿 ≽ 𝑀 and𝑀 ≽ 𝑁then𝐿 ≽ 𝑁

9/3/17 10

TheVNMAxioms:


Axiom3: Continuity


If,𝐿 ≽ 𝑀 ≽ 𝑁 thenthereisarealnumber𝑝 ∈ 0,1suchthat𝑝𝐿 + 1 − 𝑝 𝑁 ∼ 𝑀

9/3/17 11

TheVNMAxioms:


Axiom4: IndependenceofIrrelevantAlternatives


If,𝐿 ≺ 𝑀 thenforany𝑀 andany𝑝 ∈ 0,1

𝑝𝐿 + 1 − 𝑝 𝑁 ≺ 𝑝𝑀 + 1 − 𝑝 𝑁

9/3/17 12

TheVNMAxioms:


Axiom5: ReductionofCompoundLotteries

Forany𝑍,𝑊 andany 𝑝, 𝑞, 𝑟 ∈(0,1]suchthat𝑟𝑞 = 𝑝andanylottery

𝑋 = 𝑞𝑍 + 1 − 𝑞 𝑊

𝑝𝑍 + 1 − 𝑝 𝑊 ∼ 𝑟𝑋 + 1 − 𝑟 𝑊

9/3/17 13

TheExpectedUtilityTheorem


Theorem:(ExpectedUtility)

Ifaxioms1-4hold,thereexistsafunction𝑢 thatassignsarealnumber𝑢 𝑥 toall𝑥 ∈ 𝑋suchthatalottery𝐿 ≡ 𝑝𝑥3 +1 − 𝑝 𝑥5 ispreferredtoalottery𝑀 ≡ 𝑞𝑥3 + 1 − 𝑞 𝑥5 ifandonlyif𝐸 𝑢 𝐿 > 𝐸 𝑢 𝑀 .

Here

𝐸 𝑢 𝐿 ≡ 𝑝𝑢 𝑥3 + 1 − 𝑝 𝑢(𝑥5)

DoTheseAxiomsMakeSense?


Maybe:buttherearemanyobservedviolationsinexperiments.

Importantly:eveniftheaxiomsmakesenseforgamblesoverscalars,theymaynotmakesenseforgamblesovervectors

IfweadoptDebreu’sdefinitionofutility,noviolationofrationalitycaneverbeobserved:Canyouexplainwhy?

ExpectedUtilityinMacroeconomics


TheVNMaxiomswereformulatedforgamblesofascalar:‘money’

Macroeconomistsdefineexpectedutilityoveraninfinitesequence.

Define𝐶K ≡ 𝐶3, 𝐶5, 𝐶L, … , 𝐶K

ThenmacroeconomistsassumethatthereexistsarepresentativepersonwhomaximizestheexpectedvalueofaVNMutilityfunction

𝐸[𝑈] ≡ 𝐸3[𝑢 𝐶K ]



Supposethatuncertaintylivesonatree

Define𝑆K = {𝑆3, 𝑆5, 𝑆L, … 𝑆K}

𝑆K iscalledahistory Itisarandomsequence.

Iftherearetwopossibleeventseveryperiodthen𝑆3and𝑆3haveoneelement,𝑆5 and𝑆5 havetwoelements,𝑆L and𝑆L havefourelements,ansoon.

𝑆L isthesetofthingsthatcanhappenatdate3𝑆L isthesetofpathsbywhichamelementof𝑆L canbearrivedat.Itisabranchofthetree.



Everyrealizationofconsumptionatdate𝑡 isafunctionthatmapsfromthesequence 𝑆K toarealizationof𝐶K

𝑈 ≡ 𝑢 𝐶3, 𝐶5 𝑆5 , 𝐶L 𝑆L , …𝐶K(𝑆K )Here,

𝑢 𝐶3, 𝐶5 𝑆5 , 𝐶L 𝑆L , …𝐶K(𝑆K )

isaVon-NeumanMorgensternutilityfunction

Whenpeoplehaveinfinitehorizons,thisbecomes

𝑈V = 𝐸3{𝑢 𝐶3, 𝐶5 𝑆5 , 𝐶L 𝑆L , … )}



Underexpectedutilitytheory,utilityislinearinprobabilities

𝑈K = 𝐸3 𝑢 𝐶3, 𝐶5 𝑆5 , 𝐶L 𝑆L , …𝐶K(𝑆K }

= W 𝜋#𝑢 𝐶3, 𝐶5 𝑆#5 , 𝐶L 𝑆#L , … , 𝐶K 𝑆#K�

#∈Z



Macroeconomistsalsotypicallyassumethat𝑈 istimeseparable andthatpeoplediscountthefuturewithaconstantdiscountfactor𝛽 ∈ 0,1

𝑈 = 𝐸3 W𝛽K[3𝑣 𝐶K

V

K]3

Theseareverystrongassumptions

RiskAversionandIntertemporalSubstitution


Riskaversionistheamountyouwouldbewillingtopaytoreceivethemeanofalotteryasopposedtothelotteryitself.Itmeasuresaversiontochangesinconsumptionacrossstates

Whenpreferencesarestateandtimeseparable,riskaversion ismeasuredbythecurvatureof𝑣(𝑥)

𝜌(𝑥) =𝑥𝑣__(𝑥)𝑣_(`)

iscalledtheArrow-Pratt measureofrelativeriskaversion

Riskaversionandintertemporalsubstitution


Intertemporalsubstitutionisthechangeinyourintertemporalconsumptionbundleforagivenchangeintheinterestrate.

Itmeasuresaversiontochangesinconsumptionovertime.

Whenpreferencesarestateandtimeseparable,theintertemporalelasticityofsubstitutionistheinverseoftheArrowPrattmeasureofriskaversion

IntertemporalSubstitution(ITS)


Anoptimizingconsumer(withnouncertainty)wouldchoose

𝑣_ 𝐶K*3𝑣_ 𝐶K

= 𝛽𝑅KWhere𝑅K istherealinterestfactorbetweendates𝑡 and𝑡 + 1.Holdingconstant𝐶KtheITS,𝜂 isdefinedas

𝜂 𝑥 ≡𝑑𝑙𝑜𝑔 𝑥𝑑𝑙𝑜𝑔 𝑅K

≡𝑑𝑙𝑜𝑔 𝑥

𝑑𝑙𝑜𝑔 𝑣_ 𝑥≡𝑣__ 𝑥𝑥𝑣_ 𝑥

RiskAversionandIntertemporalSubstitution(ITS)


Inthewidelyusedparametrizationwhere

𝑣 𝑥 ≡𝑥3[g

1 − 𝜎

Thecoefficientofrelativeriskaversionisaconstant𝜌 𝑥 = 𝜎

andtheITSisitsreciprocal𝜂 𝑥 = 1/𝜎

RationalityasanOrganizingPrinciple


Ifwetakethestrongviewthatacommodityisdate,stateandlocationdependentEVERYGOODISDIFFERENT

Takenliterally,thismeansthatexperimentsarenotpossiblebecausetheycannotberepeated

ButalthoughrationalitydefinedasDebreuChapter7cannotberefuted,rationalitydefinedasVon-NeumanMorgensternpreferencescanbe,andhasbeen,experimentallyrefuted.Thisisthesubjectofbehavioraleconomics

SeparabilityandTimeConsistency


Timeandstateseparabilityaredifferentassumptions

Theyarecommonlyusedbecausetheyimplyaseparateproperty:timeconsistency

Supposethatatdate𝑡 apersonprefers𝐶K*j 𝑎 to𝐶K*j 𝑏 .Timeconsistencysaysthatateveryintermediatedate𝑡 + 𝜏where𝜏 < 𝑠 thechoice𝐶K*j 𝑎 isstillpreferredto𝐶K*j 𝑏

Recursivity andTimeConsistency


Themostgeneralclassofpreferencesthatistimeconsistentcanbewrittenrecursivelyasfollows

𝑈K = 𝑊 𝐶K, 𝐸K 𝑈K*3

If𝑊 isaffine

𝑊 𝑥, 𝑦 = 𝑥 + 𝛽𝑦

WerecoverVNMpreferences

ExpectedUtilityandRecursiveUtility


AdditivelyseparableVNMutilitycanbewrittenrecursivelyas

𝑈K 𝑐K = 𝑣 𝑐K + 𝛽𝐸K 𝑈K*3 𝑐K*3

Thisistheexpectedutilityrepresentationofpreferences

9/3/17 28

Epstein-ZinPreferences


AspecialcaseofrecursiveutilitywasstudiedbyLarryEpsteinandStanZin

𝑈K = 1 − 𝛽 𝑐K3[3q + 𝛽 𝐸K𝑈K*3

3[r3[3q3[r

33[3q

Thisfunctioniswidelyusedinfinancebecauseitisabletoseparateinter-temporalsubstitution(measuredby𝜂)fromriskaversion(measuredby𝜌).

9/3/17 29

RepresentativeAgentsandOverlappingGenerations


Timeseparabilityandstateseparabilityarenottheonlystrongassumptions

Asecondstrongassumptionisthattheworldconsistsofafinitenumberoffamilieseachofwhommaximizesutilityoverafinitehorizon

Iwillrefertothatastherepresentativeagentassumption

9/3/17 30

RepresentativeAgentsandEconomicWelfare


Ineconomieswithafinitenumberofinfinitelylivedfamilies,thefirstandsecondwelfaretheoremshold

Ineconomieswithaninfinitenumberoffinitelylivedfamilies,thefirstandsecondwelfaretheoremsbreakdown

Foraproofanddiscussionoftheseresults:seeKehoeandLevine,Econometrica 1985.

9/3/17 31

OverlappingGenerationsandEconomicWelfare


Inoverlappinggenerationseconomiesthewelfaretheoremsbreakdownfortworeasons

Equilibriamaybedynamicallyinefficient.WemetthispossibilityinLecture3

Theremayalsoexistequilibriainwhichnon-fundamentalshocksinfluenceallocations.CassandShellrefertotheseassunspotequilibria.WewillinvestigatethispossibilityinLecture5.

9/3/17 32

WhatisaMarket?


IfwetakeDebreuChapter7asourbenchmark;thereisasinglemarketthatopensatthebeginningoftime

Afinitenumberofinfinitelylivedfamiliestradeaninfinitesetofcontingentcommodities

Thentheworldbeginsandthetradesarerealized

WhatisaMarket?


ThisistheviewthatEdPrescottandRobertLucasaskedustoacceptwhentheyintroducedinfinitehorizon,complete-marketeconomies,witharepresentativeconsumer.

Manypeople(myselfincluded)thinkthatthisisnotaveryusefulwaytothinkaboutmacroeconomics.Inshort:representativeagentmacroeconomicsisnonsense.Readmyblogpost:RealBusinessCycleTheoryandtheHighSchoolOlympics

JohnHicksandTemporaryEquilibriumTheory


InValueandCapital,Hicksformulatedadifferentvision:Timeproceedsinasequenceofweeks

EachweekpeoplecometomarkettotradegoodsTheybringcapitalandfinancialobligationsfromthepast

Theyformbeliefsaboutwhatwillhappeninthefuture

KenArrowandFinancialMarkets


InaReviewofEconomicStudiesarticle,TheRoleofSecuritiesintheOptimalAllocationofRiskBearing,(1964)KenArrowshowedhowtoimplementanequilibriumsequentially.



Considerthefollowingproblemofahousehold

max{`,`v jv }

𝑈 ≡W𝜋#𝑢# 𝑥, 𝑥_ 𝑠#_w

#]3

𝑠. 𝑡. 𝑝 ⋅ 𝑥 +W𝑝y 𝑠#_ ⋅ 𝑥_ 𝑠#_ ≤ 𝑚w

#]3

𝑚 ismoneyincomeand𝑝y 𝑠#_ isavectorofpresentvalueprices



Noticethatwecanwritetheutilityfunctionas

𝑈 𝑥, 𝑥_ 𝑠3_ , 𝑥_ 𝑠5_ , … , 𝑥_ 𝑠w_ ≡W𝜋#𝑢# 𝑥, 𝑥_ 𝑠#_w

#]3

Ifthegoods𝑥_ 𝑠_ aredateandstatedependent,theexistenceofequilibriumisaspecialcaseofLecture2.Inperiod1,peopletradecurrentgoods,theseareelementsof𝑥, andstatecontingentfuturegoods,theseareelementsof𝑥′(𝑠#_) for𝑖 = 1,2…𝑛.

GerardDebreuandContingentCommodityEquilibrium


UndertheinterpretationofDebreuChapter7thereisasinglemarketthatoccursinperiod1.Peoplechoose 𝑥, 𝑥_ 𝑠3_ , … 𝑥_ 𝑠w_ tosolve

𝑚𝑎𝑥𝑈 𝑥, 𝑥_ 𝑠3_ , 𝑥_ 𝑠5_ , … , 𝑥_ 𝑠w_

𝑠. 𝑡. 𝑝 ⋅ 𝑥 +W𝑝y 𝑠#_ ⋅ 𝑥_ 𝑠#_ ≤ 𝑚w

#]3Theequilibriumiscalledacontingentcommodityequilibrium

KenArrowandSequentialMarketEquilibrium


Arrowasked:whathappensifpeopletradesequentially?Onceinperiod1 andonceinperiod2.Hedefinedasecurity[nowcalledanArrowsecurity]tobeapromisetopay$1 instate𝑠# ifandonlyifstate𝑖 occurs.

Ifthereare𝑛 Arrowsecurities,wesaythatmarketsarecomplete.



𝑚𝑎𝑥W𝜋#𝑢# 𝑥, 𝑥_ 𝑠#_w

#]3

𝑠. 𝑡. 𝑝 ⋅ 𝑥 +W𝑄#𝑎# ≤ 𝑚w

#]3

𝑝 𝑠#_ ⋅ 𝑥_ 𝑠#_ ≤ 𝑎#,𝑖 = 1,2…𝑛

Peoplesolvethefollowingsequentialproblem

Here,𝑎# isapromisetopay𝑎#dollarsinperiod2.Itcanbepositiveornegative.

If𝑎# isnegativewesaythattheconsumerhasshortedsecurity𝑖

The𝑝 𝑠#_ arespotpricesinperiod2 state𝑖

EquivalenceofContingentCommoditiesandArrowSecurities


Theconsumerfacesasequenceofconstraints

1 𝑝 ⋅ 𝑥 +W𝑄#𝑎# ≤ 𝑚w

#]32 𝑝 𝑠#_ ⋅ 𝑥_ 𝑠#_ ≤ 𝑎#,𝑖 ∈ 1,2… , 𝑛 ,

Bysubstitutingfor 𝑎# in (1) from 2 weget

𝑝 ⋅ 𝑥 +W𝑄#𝑝 𝑠#_ ⋅ 𝑥_ 𝑠#_ ≤ 𝑚w

#]3

𝑄#isascalarand𝑝 𝑠#_ isavector

EquivalenceofContingentCommoditiesandArrowSecurities


Noticethatundercompletemarketsthehouseholdfacesasinglebudgetconstraintandwhen

𝑝y 𝑠#_ ≡ 𝑄#𝑝 𝑠#_

Thebudgetconstraintsarethesameforthetwoproblems.



InArrow’sformulationofmarkets(followingHicks)thereisasequenceoftrades.Ateachdate,peopletradegoodsandsecurities.

Theorem:Ifpreferencesaretimeconsistent,andifmarketsarecomplete,thesetofequilibriaundersequentialmarketsisthesameasthesetofequilibriaundercontingentcommoditytrades.

Summary

• Generalequilibriumtheoryisapowerfultool• Theideaofanequilibriumisverygeneral• Itsummarizesanimportantsocialidea• Peoplemaketheirowndecisions• Theyareconstrainedbysociety• Societyisitselfthesumofthosedecisions


Documents

Indeterminacy and Sunspots in Macroeconomics · Indeterminacy and Sunspots in Macroeconomics ... , Gerzensee Lectures 2. Reading (c) Roger E. A. Farmer, Gerzensee Lectures •Farmer,