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Problem Set 2 Solutions
Course: ECON 3010, Fall 2009School: Cornell
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3010
Fall Economics 2010
Professor Daniel Benjamin
Cornell University
ProblemSet2Solutions
1.(Preferences:Moneypumpsandsafetyinmarkets)
[Theideabehindthisquestionistakendirectlyfromarecenttheoreticalresearchpaper:Laibson,
DavidI.,andLeeatYariv(2007),Safetyinmarkets:AnimpossibilitytheoremforDutchbooks,
Caltechmimeo.Ifyouareinterested,feelfreetoreadmoredetails:
http://www.hss.caltech.edu/~lyariv/Papers/DutchBooks.pdf]
(a)Naifistryingtodecidewhattobuyfordessert.ThepossibilitiesareA(applepie),B(brownies),or
C(caramelicecream).Naifhaspreferencesoverdessertandmoney.Someofherpreferences
betweendessertandmoneyarecharacterizedbythefollowingindifferencerelationships:
A~$5
A~(Band$1)
B~(Cand$2)
C~(Aand$3)
Whatdoesitmeanifherpreferencesarecomplete?
Wefirstneedtodefinetheconsumptionsetforthisproblem.Theconsumptionsetforaconsumeris
thesetofallpossibleconsumptionbundles.Notethatpossibledoesnotnecessarilyimply
affordable.Inthisproblem,wehavetwogoods,dessertandmoney.Therearethreepossibilitiesfor
dessert:{A,B,C}.Formoney,anypositiveamountispossible:R+(thisisnotationforthesetofstrictly
positiverealnumbers).Sotheconsumptionsetinthisproblemis{A,B,C}xR+.
Wecannowdefinecompletenessofpreferencesforallocationsinthisconsumptionset.Preferencesare
completeifforanytwobundlesXandYfromtheconsumptionset,oneofthefollowingstatementsis
true:(1)XisatleastasgoodasY,(2)YisatleastasgoodasXor(3)XisatleastasgoodasYandYisat
leastasgoodasX.
(b)AssumethatNaifspreferencesarecomplete,andherpreferencesovermoneyaremonotonic.
WhatcanyouconcludeaboutherpreferencesbetweeneachpossiblepairofA,B,andC?Whataxiom
doherpreferencesviolate?Canyourepresentherpreferenceswithautilityfunction?
WearetoldthatNaifspreferencesarecompleteandthatherpreferenceformoneyismonotonic.In
part(a),westatedwhatitmeansforpreferencestobecomplete.Now,whatdoesitmeanfor
preferencestobemonotonic?Ifsomeonehasamonotonicpreferenceforsomegood,thenshe
considersmoreofthatgoodtobebetter;i.e.tomakeherbetteroff.
Nowwearereadytoanswerthequestion.Wewillarguebycontradiction.Assumethattransitivity
holds.WeretoldthatNaifisindifferentbetweenAand{Band$1}.Bymonotonicityovermoney,Amust
thenpreferAtoB.WealsoknowthatsheisindifferentbetweenBand{Cand$2}.Bytransitivity,she
mustthenpreferAto{Cand$2}.Applyingagainmonotonicityovermoney,NaifmustpreferAtoC.
Finally,NaifisindifferentbetweenCand{Aand$3}.Bytransitivity,NaifmustpreferAto{Aand$3}.But
bymonotonicityovermoney,weknowthatthislaststatementcannotbetrue.Sowereacheda
contradictionandsoNaifspreferencesdonotsatisfytransitivity.
Transitivityisnecessaryforustogofrompreferencestoautilityfunctionrepresentation.Sowecant
representNaifspreferenceswithautilityfunction.
(c)AfellownamedArbitrageurrealizesthathemightbeabletotakeadvantageofNaififhecanget
hishandsononeeachofA,B,andC.Supposehecanproduceeachatacostof$4,sohegoesahead
andpays$12toproduceanA,aB,andaC.FillintheblankstocompleteArbitrageursstrategy:
(i)SellAtoNaifatapriceof$4.99.
(ii)Thenoffertogive__toNaififinexchangeshegiveshimAand$2.99.
(iii)ThenoffertogiveBtoNaififinexchangeshegiveshim__and$1.99.
(iv)ThenoffertogiveAtoNaififinexchangeshegiveshimBand$__.
WouldNaifagreetoeachofthesesteps?WhatisArbitrageursprofit(orloss)fromsteps(i)(iv),
takingintoaccounthiscostsofproduction?Whatifhekeptrepeatingsteps(ii)through(iv)?
Arbitrageur(whowellnowcallArbitforshort)beginsthisprocessbyproducinganA,B,andaCfor$12.
Instep1,ArbitsellshisAtoNaiffor$4.99.NaifacceptsbecausesheisindifferentbetweenanAand$5.
Beforestep2begins,ArbithasaB,aC,and$4.99inrevenue.HenowoffershisCtoNaifforanAand
$2.99.NaifwillacceptthisofferbecausesheisindifferentbetweenaCand{Aand$3}.
Wemoveontostep3.Beforeitbegins,ArbithasanA,aB,and$7.98(4.99+2.99)inrevenue.Henow
offershisBtoNaifinexchangeforaCand$1.99.Naifacceptsthisofferbecausesheisindifferent
betweenaBand{Cand$2}.
Headingintostep4,ArbithasanA,aC,and$9.97(4.99+2.99+1.99)inrevenue.HenowoffersNaifan
AinexchangeforaBand$0.99.NaifacceptsbecausesheisindifferentbetweenAand{Band$1}.Soby
theendofstep4,ArbithadaB,aC,and$10.96inrevenue.
Histotalprofitfromsteps1through4is$1.04(10.9612).NotehoweverthatArbitfinishesthese4
stepswithaBandaC.Hecouldthereforegobacktostep2andkeeponrepeatingsteps2through4
forever.Hisrevenuewillincreasewithoutbound,allowingprofittoapproachinfinity.
(d)Someeconomistshaveconcludedthatitisreasonabletoassumepeopledonthavepreferences
likeNaifsbecauseitispossibletotakeadvantageofconsumerslikeher(viaamoneypumplikein
part(c)).Therearetwoversionsofthisargument.Oneversionisthatifsomebodydidhaveher
preferences,thatpersonwouldquicklygobankruptandsowouldnthaveanyeffectontheeconomy.
Thesecondversionisthatifsomebodydidhaveherpreferences,thenwewouldseealotofmoney
pumpsgoingontherealworld;sincewedontseethem,thennobodyhasthosepreferences.Choose
oneofthesearguments,andexplainitinalittlemoredetail.
Letsbeginbyexpandingonthefirstargument.IfapersonlikeNaifexists,thenshewouldgobankrupt
bythecyclesmentionedinpart(c).Withoutanymoney,Naifwouldnotbeabletomakeexchangesin
theeconomyand,asaresult,wouldhavenoinfluenceonit.Inotherwords,consumerslikeNaifwould
neverparticipateinmarketsforlongenoughforthemtohaveanynoticeableeffectonthem.
ThesecondargumentbeginsbysayingthatifpeoplelikeNaifexisted,thenmoneypumpswouldbe
everywhere.Sincewedonotseemoneypumpsintherealworld,peoplelikeNaifdontexist.This
argumentrestsonthe(reasonable?)suppositionthatwheneverthereare(infinite!)profitstobemade,
someonewillsurelytakeadvantageoftheopportunity.
(e)Sofar,wehavebeenassumingthatArbitrageurisamonopolysupplierofgoodsA,B,andCtoNaif
thatis,thereisnooneelsewhoiscompetingwithArbitrageurtoprovidethosegoods.Now
supposeinsteadthereisacompetitivesupplyofgoodsA,B,andCtoNaifi.e.,thereareaninfinite
numberofsupplierscompetingwitheachothertoselltheirgoodstoNaif.Assumethateachsupplier
canproduceA,B,andCatacostof$4(thesameasArbitrageur).ExplainwhyArbitrageurcannow
chargeNaifnomorethan$4forA.ExplainwhyArbitrageurcannotchargeNaifanythinganymoreto
maketradesbetweenA,B,C.
SupposethatArbitdecidestochargeNaifmorethan$4foranA.Ifhedoesthis,thenanotherproducer
cancomeinandundercutArbitbyasmallamountandstillmakeaprofit.Theseundercutscontinueto
happenuntilpriceispusheddowntothemarginalcostof$4.
Usingthesameargumentfromabove,wecanconcludethatArbitwillnolongerbeabletochargeNaif
anythingtomaketradesbetweenA,B,andC.IfheofferedNaifanytradeoftheformXinexchangefor
{Yand$m},whereX=A,BorC,Y=A,BorCandm>0,anotherarbitrageurwouldenterthemarketand
undercuthim.Thisisbecauseitcoststhesameprice($4)toproduceanyofA,BorC(thelogicislaidout
morecarefullyinpart(f)ifthisisnotclear).
(f)IfArbitrageurfollowedthroughwiththesequenceoftradefrompart(c)(butchargingonlyas
muchasthemarketwouldallow),howmuchprofitwouldhemake?Explainwhymarketseliminate
moneypumpsandtherebyactuallyallowindividualslikeNaiftopersistinhavingoddpreferences.
Inacompetitivemarket,ArbitcanchargeNaifonly$4foranA(Step1).InStep2,hecanofferNaifaC
foramaximumofA.Toseethat,assumeArbitoffersNaifaCfor{Aandx}wherex>0chosenarbitrarily.
ThensinceitcostsasmuchtoproduceAandC,anotherarbitrageurwouldenterthemarketand
undercutArbitwiththefollowingoffertoNaif:Cinexchangefor{Aandy}where0<y<x.Asisevident,
theonlystateofrestforthisproblem(orequilibrium)isforArbittoofferNaifCinexchangeforA.By
thesamelogic,inStep3ArbitwillofferNaifBinexchangeforC.InStep4,hewillofferherAin
exchangeforB.
Soaftergoingthroughsteps1to4laidoutin(c),wefindthatArbithasrevenueof$4(fromsellinghisA
instep1)andcostsof$12(thetotalcostofproducingA,B,andC).Hisprofitistherefore$8.Repeating
steps2through4doesntchangeanythingbecausewehavealreadyarguedthatthiscantleadtomore
revenue.
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