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Consumption and Investment Decision

Material Information

Title:
Consumption and Investment Decision An Analysis of Aggregate and Time-Additive Models
Creator:
Fu, Liang
Place of Publication:
[Gainesville, Fla.]
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (110 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Business Administration
Accounting
Committee Chair:
Demski, Joel S.
Committee Members:
Lin, Haijin
McGill, Gary A.
Sappington, David
Graduation Date:
8/8/2009

Subjects

Subjects / Keywords:
Debt contracts ( jstor )
Expected utility ( jstor )
Financial investments ( jstor )
Investment decisions ( jstor )
Investment risks ( jstor )
Lenders ( jstor )
Library users ( jstor )
Limited liability ( jstor )
Personal wealth ( jstor )
pH ( jstor )
Accounting -- Dissertations, Academic -- UF
consumption, investment
Genre:
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Business Administration thesis, Ph.D.

Notes

Abstract:
In the presence of an investment opportunity, a borrower decides whether to invest. We find that - depending on the borrower's personal wealth and the presence of limited liability - the investment is not always undertaken, which leads to investment distortion. Furthermore, when the borrower has intertemporal consumption preference rather than aggregate consumption preference, the potential investment distortion problem is weakly exacerbated. The change of the borrower's risk preference from constant absolute risk aversion to decreasing absolute risk aversion lowers the occurrence of the potential investment distortion as his personal wealth increases. Furthermore, when the borrower is privately informed about his personal wealth, the Revelation Principle applies and the naive contract which is derived from public information case is optimal. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
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Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2009.
Local:
Adviser: Demski, Joel S.
Statement of Responsibility:
by Liang Fu.

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UFRGP
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Copyright Fu, Liang. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
489206480 ( OCLC )
Classification:
LD1780 2009 ( lcc )

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Iamdeeplyindebtedtomyadvisor,ProfessorJoelS.Demski.ItishisconsistentsupportandguidancethatensuredthesuccessfulcompletionofmyPh.D.program.Hispassioninscholarshiphasgreatlyshapedmyphilosophyonacademiccareer.IwishtothankProfessorHaijinLin,ProfessorGaryMcGill,andProfessorDavidSappingtonfortheirvaluablesuggestionsthatsubstantiallyimprovedthisworkandtheirconstantencouragements.Finally,Iamtrulythankfulformyhusband,JingLiu,forhispatienceandtrust. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 ABSTRACT ........................................ 7 CHAPTER 1INTRODUCTION .................................. 8 2LITERATUREREVIEW .............................. 14 3CARA-BORROWER ................................. 20 3.1Introduction ................................... 20 3.2ConsumptionPreferenceCase1:Aggregate ................. 22 3.2.1TheModel ................................ 22 3.2.2InvestmentBehaviors:Borrow,Self-Invest,No-Invest ........ 22 3.2.2.1Benchmark:rst-bestcontract ............... 23 3.2.2.2Second-bestenvironment ................... 26 3.2.2.3Investmentdistortion ..................... 35 3.3ConsumptionPreferenceCase2:Time-Additive ............... 41 3.3.1NoInvestmentDistortion ........................ 42 3.3.2Second-BestContractandPotentialInvestmentDistortion ..... 44 3.3.3WIInvestmentandPotentialInvestmentDistortion ....... 53 3.4InvestmentDistortionandConsumptionSmoothing ............. 58 4DARA-BORROWER ................................. 60 4.1First-BestContractandNo-LenderInvestment ................ 62 4.2Second-BestLoanAgreement ......................... 67 4.2.1AggregateConsumptionPreference .................. 67 4.2.2Time-AdditiveConsumptionPreference ................ 70 4.3ConsumptionPreferenceandInvestmentDistortion ............. 74 5PRIVATEPERSONALWEALTH ......................... 77 6CONCLUSION .................................... 85 APPENDIX:PROOFS ................................... 87 REFERENCES ....................................... 108 BIOGRAPHICALSKETCH ................................ 110 5

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Table page 3-1Second-BestContractsfortheBorrowerwithAggregateConsumption ...... 34 3-2InvestmentDistortionfortheBorrowerwithAggregateConsumption ...... 40 3-3Second-BestContractsfortheBorrowerwithTime-AdditiveConsumption ... 51 3-4InvestmentDistortionfortheBorrowerwithTime-AdditiveConsumption .... 57 4-1InvestmentDistortionforAggregateConsumption ................. 70 4-2LocalCertaintyEquivalentConsumptionsandLocalRiskPremium ....... 73 4-3InvestmentDistortionforTime-AdditiveConsumption .............. 74 4-4Second-BestContractsInvestmentDistortionfortheDARA-Borrower ...... 76 6

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Inthepresenceofaninvestmentopportunity,aborrowerdecideswhethertoinvest.Wendthat-dependingontheborrower'spersonalwealthandthepresenceoflimitedliability-theinvestmentisnotalwaysundertaken,whichleadstoinvestmentdistortion.Furthermore,whentheborrowerhasintertemporalconsumptionpreferenceratherthanaggregateconsumptionpreference,thepotentialinvestmentdistortionproblemisweaklyexacerbated.Thechangeoftheborrower'sriskpreferencefromconstantabsoluteriskaversiontodecreasingabsoluteriskaversionlowerstheoccurrenceofthepotentialinvestmentdistortionashispersonalwealthincreases.Furthermore,whentheborrowerisprivatelyinformedabouthispersonalwealth,theRevelationPrincipleappliesandthenaivecontractwhichisderivedfrompublicinformationcaseisoptimal. 7

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Thisdissertationexaminesconsumptionandinvestmentbehaviorassociatedwithaninvestmentproject.Twopartiesareinvolved:arisk-averseborrowerandarisk-neutrallender.Thedissertationstartswithaborrowerwhodisplaysconstantab-soluteriskaversion(CARAhereafter)withcertaininitialwealthandeitheraggregateconsumptionpreferenceortime-additiveconsumptionpreferenceasabenchmark.Itfocusesonaninvestmentdecisioninthepresenceoflimitedliabilityandtheresultingpotentialinvestmentdistortionaswellasitsinteractionwiththeborrower'sconsumptionpreference. Bydesign,theinvestmentprojectisalwaystakenwhentheborrower'seortispubliclyobservableandtheassociatedrst-bestcontractalwaysexists.However,whentheborrower'seortchoicebecomesprivate,hedoesnotalwaysundertaketheproject,whichleadstoinvestmentdistortion.Inthisdissertationinvestmentdistortionistheborrower'spassinguptheprotableinvestmentopportunity.Basedonthisconditionanddependingontheborrower'spersonalwealth,investmentdistortionoccursifanyofthefollowingsituationsarise: A. Thelackofaloanagreementbetweentheborrowerandthelenderwhentheborroweriswealth-constrained. B. Thewealth-constrainedborrowerprefersthestatusquotothebestavailableloanagreement. C. Thewealth-abundantborrowerprefersthestatusquoto(a)borrowingfromthelender(iffeasible)and(b)self-nancing. InSituationAandSituationB,thewealth-constrainedborrowerreferstotheborrowerwhodoesnothavesucientpersonalwealthtonancetheinvestment.SituationAimpliesthatthereisnoloanarrangementthatmotivatestheborrowerandletsthelenderrecoupherinvestmentatthesametime.Theculpritforthenon-existenceofsuchanagreementisthejointforcefromtwofactorsinthedesign.Theborroweristheparty 8

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SituationBsuggeststhatfortheborrowertodecidebetweenborrowingfromthelender(giventhatsuchanarrangementisinplace)andstayingwiththestatusquo,heconsidershisnetpayofromtheinvestment.Inparticular,theborrowerreceivestheentireexpectedNPVfromtheinvestmentbutheincursapersonalcostofeort;therefore,ifhisnetpayoisnotsucientlylarge,hemightendupgivinguptheinvestmentanddecidetosettlewiththestatusquo.Theeconomicimplicationbehindthisformofinvestmentdistortionisthattheloanagreement,givenitsexistence,istooriskyfortheborrowertoengageinnancingwiththelender. InSituationC,thewealth-abundantborrowerreferstotheborrowerwhohassucientpersonalendowmenttonancetheinvestmentprojectonhisownwithoutresortingtothelender.Iftheloanagreementfails,theborrowerisleftwiththechoicebetweenno-lenderinvestmentandno-investment.Sincetheborrowerisriskaverse,theinvestment 9

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ThisdissertationanalyzesSituationsAthroughCfortheborrowerwitheitheraggregateconsumptionpreferenceortime-additiveconsumptionpreference.Thediscussionfortheaggregateborrowerprovidesabenchmarkfortheinterplaybetweentheborrower'sinvestmentandconsumptionbehaviorbyderivingconditionsunderwhichinvestmentdistortionarises.Thebenchmarkisimportantinthefollowingtworespects:(a)theroleoftheborrower'slimitedliabilityconstraintsonhisconsumptionismadeexplicitindeterminingthefeasibilityoftheloanarrangementbetweentheborrowerandthelenderwhentheformerhasprivateinformationabouthiseortsupplyand(b)theroleoftheborrower'spersonalwealthisnontrivialinpotentialinvestmentdistortion. Withthebenchmarkfortheborrowerwithaggregateconsumptionpreference,theanalysisrevealsthattheborrower'sconsumptionpreferencehasaneectonhisinvestmentdecision,especiallyontheoccurrenceofthepotentialinvestmentdistortion.Theinvestmentdistortionunderaggregateconsumptionpreferenceimpliesthedistortionundertime-additivepreference,butnotviceversa.Inparticular,iftheborrower'spersonalwealthisstrictlylessthantheinvestmentrequirement,theborrower'spreferenceshavenoinuenceonhisinvestmentbehavior.Intuitively,theborrowerdisplaysCARA,whichwouldimplythathispersonalwealthhasnoeectonhisattitudetowardsriskateachinstantiftheconsumptionproblemweretoextendtomultiperiods.Therefore,conditionsunderwhichinvestmentdistortionarisesareexactlythesamefortheborrowerwithaggregateandtime-additiveconsumptionpreferences.However,iftheborrowerhassucientlylargepersonalwealthtoengageinself-nancing,theconcernforintertemporalconsumptionexacerbatesthepotentialinvestmentdistortionproblem.Theaggravationispronouncedfortheborrower'sno-lenderinvestment.Thereasonisthatwithno-lenderinvestment,themagnitudeoftheborrower'spersonalwealthisimportantinshapingthe 10

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WiththeCARA-borrowerasabenchmark,thesubsequentanalysisintroducesaborrowerwithdecreasingabsoluteriskaversion(DARAhereafter)witheitheraggregateortime-additiveconsumptionpreferencetoinvestigatethechangeoftheborrower'sriskpreferenceonhisconsumptionandinvestmentbehavior.Ingeneral,fortheDARA-borrower,theuniquedistinctionbetweentheCARA-andDARA-borroweristhat,ceterisparibus,thelatterislesslikelytoencounterinvestmentdistortionproblemsassociatedwiththesecond-bestloancontract.TheattributeofthemitigatedpotentialinvestmentdistortionisthattheDARA-borrowerchangeshisattitudetowardsriskashispersonalwealthchanges.Inparticular,astheDARA-borrowerbecomeswealthier,hebecomeslessriskaverse,whichinturnleadshimtoundertaketheinvestmentprojectthatwouldotherwiseberejectedbytheCARA-borrower. Althoughtheborrower'shierarchicaloptimalinvestmentdecisionisinconclusivebetweenborrowingandself-nancing,theloanagreementwiththerisk-neutrallenderprovidesrisksharing,whichisimportanttotherisk-averseborrower.Therefore,weretheborrower'spersonalwealthtobeprivateinformation,howtheloancontractwould 11

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Investmentactivitiesarevitaltoaprosperousandwell-functioningeconomy;however,weoftenseesoundinvestmentopportunitiesbeingpassedup.Thisdissertationdesignsasimpletractabletwo-partymodeltoinvestigatethepotentialattributiontopotentialinvestmentdistortion.Thendingsarespecicallyvaluabletotheentrepreneur-typeofborrower,whomaybewealth-constrainedorrestrictedbyriskpreferencebutwhoseekswaystoengageintheproject.Theborrowerwithaggregateconsumptionpreferencecanbeconsideredtorepresentatraditionalentrepreneur,whosekeyfocusistheaggregatereturnofaninvestmentratherthantheintertemporalgain.Ontheotherhand,theborrowerwithtime-additiveconsumptionpreferenceismorelikelytorepresentsmall 12

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Thestylizedmodelinthisdissertationpartlyresemblestheconventionalagencyproblem(e.g., GrossmanandHart ( 1983 ))inthatthelender-borrowerrelationshipandthecorrespondingoptimalconsumptionsarecounterpartsofthelabormarketprincipal-agentwagecontract.Sincetheborrowerhasthepotentialtonancetheinvestmentfromthelender,theloancontractbetweentheborrowerandthelenderfeaturesmoralhazardinthedebtmarket.Forexample, GaleandHellwig ( 1985 )focusonacontractbetweeninvestorsandentrepreneurs,bothofwhomoperateinacompetitivenancialmarket.Entrepreneursarejustborrowerswhowishtoundertakeriskyventuresbutlackthenecessaryresourcessotheyturntotheinvestorsforexternalnance.Investorsarebanksorothernancialinstitutions.Theyndthattheoptimal,incentive-compatibledebtcontractinamodelofborrowingandlendingwithasymmetricinformationisthestandarddebtcontract.Inthisdissertation,engaginginborrowingisendogenous,especiallyforthewealth-constrainedborrower;thelenderhasrationalexpectationsoftheborrower'sincentivestoshirk(givenhisprivateeortsupply)andmisreport(givenhisprivateinformationaboutwealth);theinteractionbetweentheborrowerandthelenderunavoidablyraisespotentialinvestmentdistortion,inecientrisksharing,andtheborrower'sconcurrentconcernforintertemporalconsumptionwhenhehastime-additiveconsumptionpreference. ThebenchmarkconsumptionandinvestmentbehaviorsfortheCARA-borrowerinthisdissertationarecloselyrelatedto Tirole ( 2006 )'sbasicmodel,wherehemodelsarisk-neutrallenderandarisk-neutralborrowerandinvestigatescreditrationinginthepresenceofawell-functioningnancialmarket.Tirolendsthatthedrivingforceofpotentialcreditrationingwhentheborrowerhasprivateinformationabouthisactionchoiceisthelevelofhiswealthathand.Thisdissertationextends Tirole ( 2006 )byintroducingtherisk-sharingaspectoftheloanagreement.Therisk-averseborrower, 14

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Tirole ( 2006 )byconsideringtheroletheborrower'sconsumptionpreferenceplaysinpotentialinvestmentdistortion.Thendingsofthisdissertationaremorecompellingintwoways.First,itisperhapsmorerealistictoassumearisk-averseborrowerandnon-negativeconsumptionwhentheproblemathandisinvestmentandconsumptionbehavior.Second,sinceconsumptionbehaviorisnotindependentoftheinvestmentdecision,howtheformeraectsthelatterisitselfofparticularinterest. Thisdissertationfocusesontheunder-investmentproblemthatariseswhenaprotableprojectisnotundertaken.Thenanceliteratureexplainsunder-investmentfromacapitalstructureperspective,whichimpliesthatunder-investmentisaresultofinformationasymmetrybetweentheborrowerandthelender(e.g., MyersandMajluf ( 1984 ))andtheprojectisnotundertakenduetocostlyexternalnancing.Thisdissertationisdistinguishedfromthenanceliteraturebyfocusingonthemoralhazardaspectratherthantheadverseselectionaspectoftheinformationproblem.Inparticular,theborrower'smoral-hazard-associatedincentiveproblem,togetherwiththe ( 2006 )alsoassumesthenon-negativityoftheborrower'sincome.Thelimitedliabilitycontractintheliteraturetakesdierentforms. Sappington ( 1983 )showsthattheprincipaldeliberatelyinducesless-than-ecientoutputinastatethatisnotthemostlyproductivewhencompelledtorespecttheagent'sexpostlimitedliability. LewisandSappington ( 2000 )derivetheoptimalcontractwhentheagenthasprivateinformationabouthiswealth. CheandGale ( 2000 )and LewisandSappington ( 2001 )analyzemodelswheretheagentisprivatelyinformedbothofhisabilityandofhiswealth.Amorerecentpaperby Lu ( 2009 )ndsthatrmsexperienceoverinvestmentratherthanunderinvestmentwhenbankruptcyrisk(calculatedasAltman'sZ-score, Altman ( 1968 ))ishigher. 15

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Theaccountingliteraturefocusesontheuseandthe"quality"ofaccountinginformationinmitigatinginvestmentinecienciesdirectly.Forexample, BensandMonahan ( 2004 )and Bushmanetal. ( 2006 )studyhowarm'sdisclosurepolicyandqualitymitigatetheunder-investmentproblem. Biddleetal. ( 2008 )and Lu ( 2009 )exploretheeectsofaccrualqualityandthedisclosureofnon-nancial-statementinformationonover-investment,respectively.Anotherstreamofaccountingliteratureinvestigatestheconnectionbetweenaccountinginformationandinvestmentbehaviorindirectlyfromtheconnectionbetweenaccountingandthecostofcapitalingeneral. Francisetal. ( 2000 )and Aboodyetal. ( 2005 )emphasizetheroleofaccrual(ormoregenerally,earnings)qualityonthecostofcapital. Bertomeuetal. ( 2008 )furtherendogenouslyconnectdisclosurepolicytoitscapitalstructure.However,theroleoftheborrowerassociatedwiththeinvestmentprojectislargelyoverlookedintheaccountingliterature.Asaneconomicagent,theborrowerisimportantintheinvestmentdecisionandthecorrespondingnancingarrangement.Inparticular,hisunobservableeortchoiceandhispersonalwealthendowmentbothsignicantlyaecttheexistenceoftheloanagreement,whichinturn,inuencestheoccurrenceofinvestmentineciency. Investmentdecisionandconsumptionbehaviorarehardlytwoisolatedevents. Breeden ( 1979 )isthepioneerofcontemporaryresearchonconsumptionandnancialmarketinvestmentinwhathasbecomeknownasthe"consumptioncapitalassetpricingmodel(CAPM)".TheintertemporalCAPMisdevelopedby Merton ( 1973 ). Merton ( 1973 )concludesthatwhensecuritieshavestochasticreturns,anindividual'sportfolioholdingsarefoundintermsofhisindirectutilityfunctionforwealthandequilibriumexpectedassetreturnsarecorrespondinglyfoundintermsofaggregatewealthandthereturnsonassetsthatareperfectlycorrelatedwithchangesinthevariousstatevariables. 16

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( 1979 )extends Merton ( 1973 )andintroducestheconsumption-beta2andarguesthatinarationalexpectationsequilibriumassetpricesintheeconomyarefunctionsoftheconsumptionpreferencesofindividualsandtime,whicharenon-stochastic.Theindividual'sproblemistochooseanoptimalrateofconsumptionandanoptimalportfolioofriskyassetstomaximizetheexpectedvalueateachinstant. TheriskyinvestmentprojectisquitesimilartotheriskyportfoliosinBreeden'swork.However,whileBreedenfocusesmoreontheinvestmentofsecuritiesinthenancialmarket,thisdissertationemphasizesthepotentialaspectofinvestmentdistortionwhentheinvestmentdecisionismoreeconomicandinvolvesrealtechnologyandlabor.Moreover,thisdissertationnotonlydiscussestheconnectionbetweenconsumptionbehaviorandinvestmentbutalsoexplorestheimpactofparticularconsumptionpreferencesonthepotentialinvestmentdistortionproblem.Consumptionspending,asdiscussedby DanthineandDonaldson ( 2005 ),denesa"standardofliving,"andmostindividuals,ifnotall,preferastablestandardoflivingfromtimetotime.3Therefore,intertemporalconsumptionpreferenceisnotanunrealisticfocus.Althoughtheborrowerpreferssmoothconsumption,whentheinvestmentdecisionisconsideredatthesametime,consumptionsmoothnessisnotwithoutacatch.Inparticular,intertemporalconsumptionpreferencemakestheborrowermoresusceptibletoinvestmentineciency.Intuitively,toavoidinvestmentdistortiontheaggregateborrowerfacesafewhurdles.First,whetherthepersonalwealthissucientlylargetonancetheprojectwithoutborrowing.Secondisthefeasibilityoftheloanagreement.However,ceterisparibus,fortheborrowerwith Sharpe ( 1964 )and Lintner ( 1965 )CAPMbeta;theformerdenesassetbetastobemeasuredrelativetochangesintheaggregateconsumptionrate,ratherthanrelativetothemarket,whichisthedenitionofthelatter.3Indeed,consumptionhaslongbeenfoundtobesurprisinglysmoothovertime. Hall ( 1988 )ndsthatariseintheinterestrateisnotaccompaniedbyanincreaseinconsumption.Similarly, Deaton ( 1987 )alsondsthatconsumptionissmoothrelativetoincomeandprices. 17

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Inreality,thelendermaynothaveaccesstothepreciseinformationabouthowmuchpersonalwealththeborrowerpossessesmainlyfortworeasons:rst,theborrowercansimplyconcealsuchinformationanditmaybecomealmostimpossibleorprohibitivelyexpensiveforthelendertoconsultathirdpartytorevealsuchinformation;second,eventhoughthelendercanobservetheborrower'sprivatepersonalwealthinformationwithnoadditionalcost,itcanbeproblematicforthelendertodeterminehisexactnetworth.Forexample,theborrower'spersonalwealthcantaketheformsofcash,realassets,oramixofboth.Whilethevalueofcashiseasilydetermined,theremaynotbeamarketvaluefortheassetsortheborrowermayhavebetterinformationofthemarketabilityoftheassets,bothofwhichmakeitdicultforthelendertohavethesameinformationabouttheborrower'spersonalwealthastheborrower. Intheeconomicsliteraturedealingwithinvestmentbehaviorandprivatelyinformedpersonalwealthisnotuncommon. LewisandSappington ( 2001 )investigateanoptimalcontractbetweenaprincipalwhoistheownerofaprojectandanagentwhohastheskillsrequiredtooperatetheproject.Theagentisprivatelyinformedabouthisabilitytooperatetheproject,hiswealth,andhiseortsupplies. LewisandSappington ( 2001 )ndthatthepoweroftheincentiveschemedoesnotalwaysincreaseashiswealthorabilityaloneincreases.Inotherwords,abilityandwealthactasperfectcomplementsindeterminingthepoweroftheincentiveschemeandanagentrequiresthehigherlevel 18

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LewisandSappington ( 2001 ).Theowneroftheinvestmentprojectistheborrower,notthelender.Giventheborrower'sownershipoftheprojectandthefactthathemaybewealth-constrained,amoralhazardproblemintheborrowingsphereisofprimaryconcern;therefore,theborrower'sabilityisnotconsideredasachoicevariable.Thisdissertationismorecloselyrelatedto LewisandSappington ( 2000 ),wheretheyndthattheprincipalwhoisalsotheowneroftheprojectcaninducetheagent,whoisalsotheprojectoperator,totruthfullyrevealhisprivatelyinformedconstrainedpersonalwealthbypromisingahigherprobabilityofoperationand/oragreatershareofrealizedprotthelargerthebondthatapotentialoperatorpostswhenhiseortisessentiallynon-contractible.Inbothstudies,theasymmetricinformationaboutthepersonalwealthleadstoanadverseselectionproblemandthenon-contractiblepersonaleortsupplyrepresentsamoralhazardproblem.Furthermore,theRevelationPrinciple( Myerson ( 1981 ))appliesandatruthfuldirectmechanisminducestruth-telling.Theultimategoalfor LewisandSappington ( 2000 )'sagentandtheborrowerinChapter5istomaximizetheexpectedprotfromtheoperation/project;however, LewisandSappington ( 2000 )ndthattheagenttruthfullyrevealshiswealthbecauseheispromisedahigherprobabilityofoperationand/oragreatershareofrealizedprotthelargerthebondtheagentposts.InChapter5,theborrowerchoosestruthfulrevelationofhiswealthbecausehedoesnotgainbasedonthelimitedreportingstrategyandtheformofhisutilityfunction. 19

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Toimplementtheproject,theborrowermayapproachthelendertoborrowD00.Uponassessingtheprojectproposedbytheborrower,thelendereitherrejectstheproposaloragreestonance,wherethecorrespondingnancingagreementspeciestheewpaymentsfDH;DLgtothelender.Tobeconsistentwith Tirole ( 2006 ),thelenderoperatesinacompetitivenancialmarket.1Afternancingissecured,theborrowerexertsaneortwhichleadstoabinarycashowX2fXH;XLg,thevalueofwhichisrealizedineitheragoodstateorabadstateandXH>XLimpliesthatthecashowfromthegoodstateisstrictlyhigherthanthatfromthebadstate.2TheborrowercaneitherbehavebyexertingthedesiredeortaH>0,whichyieldsaprobabilitypH2(:5;1)ofXH,3ormisbehavebytakingashirkingactionaL=0withanormalizedzeropersonal 20

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Assumption1suggeststhathigheortisalwayspreferredinequilibriumandtheinvestmentprojectisattractivetotheborrower.Sincethelenderoperatesinacompetitivenancialmarket,alltheexpectedNPVfromtheinvestmentaccruestotheborrower;Assumption1restrictstheborrower'sequilibrium"netmonetarygain"fromtheprojecttobestrictlygreaterthanzero. Assumetheborrowerisanexpectedutilitymaximizerwithconstantabsoluteriskaversion(CARAhereafter).Hisutilityfunctiontakestheformofnegativeexponential,whichisincreasingandconcaveinhisgrossconsumption. 21

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where>0istheArrow-Prattrisk-aversionparameterandaistheborrower'spersonalcostofeort.Theborrower'sobjectiveistomaximizehisexpectedutilityfromhisconsumption. Beforeformallydiscussingtheborrower'sinvestmentoptions,theborrower'sconsumptionbehaviorwhennoinvestmentopportunityispresentisbrieydiscussedforcompleteness. Whenthereisnoinvestmentopportunity,theborrowersimplyconsumeshispersonalwealth,andhisoptimalconsumptionischaracterizedasfollows: ( 2005 )discussconsumptionpreferencesformulti-periodcontractsindetailinChapter25. 22

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Theintuitionforthisconsumptionscheduleisstraightforward.Giventhelackofpersonalbankingaccesstheborrower'soptimalchoiceistoconsumeallthewealthathand.Moreover,sincetheborrowerhasaggregateconsumptionpreferencethetotalconsumptionratherthanperiod-by-periodconsumptionisthefocus.TheborrowercanarbitrarilyallocatehispersonalwealthWbetweentwoconsumptioninstantsoftime;therefore,thereisnouniquesolutionforhisperiod-by-periodconsumption. Therisk-neutrallenderbehavescompetitivelyinthenancialmarketinthesensethatthenancialagreement,ifitexists,makeszeroprot.Therefore,inequilibrium,thelender's(IR)constraintisbinding.6Theborrower'sconsumptionisbinary,i.e.,C2fCH;CLg,whichisboundedabovebythesumofhispersonalwealth(W)andthestochasticcash 23

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Finally,CHandCLaretheborrower'smonetaryholdings,whichareusedforconsumptionpurposes;theyareassumedtobenon-negative.Thesenon-negativityconstraintsareessentiallysimilartothelimitedliabilityconstraintsimposedintheagencyproblems(e.g., Sappington ( 1983 )and Innes ( 1990 ));therefore,theyarereferredtoaslimitedliabilityconstraintshereafter.7 3{2 )andthelimitedliabilityconstraints. 3{2 )allsuggestthatthemagnitudeoftheexpectedNPVHiswhatisimportanttotheanalysis;inotherwords,itisthemagnitudeofXLrelativetothatofXHthatmatters,notthemagnitudeofXLitselfthatplaysaroleintheloanagreement.However,intheno-lenderinvestmentanalysislateron,thesignofXLisimportantindeterminingthefeasibilityoftheno-lenderinvestmentincertaincircumstances. 24

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GrossmanandHart ( 1983 )and Holmstrom ( 1979 ).However,italsohasdistinctcharacteristics.Inequality( 3{2 )suggeststhatweretheinvestmenttobetakentheborrower'sexpectedconsumptionwouldbenogreaterthanthesumofhispersonalwealthendowmentandtheexpectedNPVfromtheinvestment.Foragivenproject,theborrower'spersonalwealthWandtheexpectedNPVHarebothxed,whichimpliesthattheborrower'sconsumptioninexpectationhasanupperbound.Thisisquitedierentfromtheconventionalagencyproblemwheretheborrower'sindividualrationalityconstraintdefactoimposesalowerboundontheagent'scompensation.8Aswilllaterbecomemoreclearerinthesecond-bestcontract,thefactthattheborrower'sequilibriumexpectedconsumptionisboundedabovebyaxedamountplaysanimportantroleinthepotentialinvestmentdistortionproblem. Similartowhatisestablishedintheconventionalagencyproblem,therst-bestcontract[ProgramFB]wouldprovidefullinsurancefortheborrowerweretheborrowertosupplyhigheort.Asaconsequence,therisk-averseborrowerdoesnotbearanyriskassociatedwiththecashowfromtheprojectandhisconsumption. 25

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Assumetheinvestmentprojectisviable,theconsumptionscheduleisriskless.ThisisbecauseJensen'sinequalityimpliesthatriskyconsumptionisnotoptimalintherst-bestcontract.Moreover,theborrower'sutilityfunctionisstrictlyincreasingandconcave;therefore,wehaveU(CaH)=U(W+HaH) whichimplies( 3{3 ). Notethattheborrower'soptiontoself-nancetheinvestmentprojectifheiswealth-abundantisnotdiscussedinthissection.Itturnsoutthatwhentheborrower'seortsupplyispubliclyobservableandcontractible,self-nancingisinferiortoborrowing.Intuitively,theloanagreementprovidesafull-insuredrisklessconsumptionfortheborrower,whichwouldbeinfeasibleweretheborrowertochooseno-lenderinvestmentastheprojectresultsinriskycashow.Therefore,therisk-averseborroweralwaysprefersborrowingtoself-nancingwhenhiseortsupplyispubliclyobservable. 26

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Unliketherst-bestcontract,thesecond-bestenvironmentfeaturespotentialinvestmentdistortionandconsumptiondistortion,whicharedenedasfollows: Thelackofamutuallyagreeablesecond-bestcontractintherstsituationimpliesthatnocontractcanbefoundthatmotivatestheborrowertoworkdiligentlyandletsthelenderrecoupherinvestmentatthesametime.Twopossibilitiescancontributetothistypeofinvestmentdistortion:1)theloanagreementisinfeasiblebecausetheredoesnotexistaconsumptionschedulethatsolvestheborrower'smaximizationproblemwithrespecttotheloancontractand2)giventheexistenceofsuchaconsumptionschedulethatmaximizestheborrower'sexpectedutilityinthesecond-bestloanagreement,the 27

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Theonlydierencebetweenthesecond-bestandtherst-bestloanagreementisthemoralhazardconcernbroughtaboutbytheborrower'sunobservableaction,whichrequiresthecorrespondingnancingarrangementtogranttheborrowersucientstakeintheprojecttoinducehimtoworkdiligently.WithaL=0,inequilibrium,theborrowerbehavesifthefollowingincentivecompatibilityconstraint(IChereafter)holds: Theleft-andright-handsidesof( 3{4 )aretheborrower'sexpectedutilityfromhisconsumptionwhenhechoosestoshirkandtodeliverhigheort,respectively;therefore,( 3{4 )ensuresthattheborrowerismotivatedtoworkdiligently.Rearrangingthetermsin( 3{4 )yields Notethattheright-handsideof( 3{5 )isnegativebecausepH>pLandaH>0ensurethesquare-brackettermisstrictlypositive.Ontheleft-handsideof( 3{5 ),since 29

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whichimpliesthat (1pH)(3{6) Thefollowingassumptionisheldthroughoutthischapter. (1pH)alwaysholds. GivenAssumption2,inequality( 3{5 )issimpliedas where=pHexp(aH)pL 3{7 )constraintisensured.11 3{7 )constraintadded,whichis [ProgramSB]maxCH0;CL0pHexp[(CHaH)](1pH)exp[(CLaH)]s.t.pHCH+(1pH)CLW+HCL+1 3{6 ),iswelldenedforthenaturallogarithmfunctionin( 3{7 ). 30

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3{8 )inProposition 1 ensurestheconstraintsetcanbesatised(i.e.,theconstraintsetisfeasible),which,inturn,determinestheexistenceofthesecond-bestcontract[ProgramSB].Closerinvestigationshowsthatcondition( 3{8 )isalsoanecessaryconditionforasolutiontoexistfor[ProgramSB].12Itisevidentthatgiventheexistenceofthesecond-bestsolutiontheborrower'slimitedliabilityconstraintsarebothsatised.Stateddierently,thelimitedliabilityconstraintsaretrivialiftheconstraintsfor[ProgramSB]arefeasiblebecausenegativeconsumptionneverarise.Proposition 1 impliesthataconsumptiondierentialarisesatthesolutiontotheproblem,i.e.,CH>CL,whichisconsistentwiththeconventionalagencyproblemthattheborrowerbearssomerisktobemotivatedtodeliverhigheort. Undercondition( 3{8 ),theborrower'soptimalconsumptionforthesecond-bestcontractshowsthefollowingcharacteristics:CH>CL0 31

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3{8 )holdsatequality.Theborrower'slimitedliabilityconstraintforCHisobviouslyneverbinding.Thus,thefollowingcorollaryisdevelopedtoaddressthelimitedliabilityconstraintsfor[ProgramSB]: 1 concludesthatthesecond-bestcontracteitherhasaninteriorsolutionornosolution;acornersolutionneverarises.13AssumingCHandCLaretheoptimalsolutionsfromtheconventionalagencyproblem,otherthingsbeingequal,constraint( 3{2 )isreplacedbytheusualindividualrationalityconstraintthattheborrower'sexpectedutilityisnolessthanhisreservationutility.Thelimitedliabilityconstraintsintheconventionalagencyproblemworkinthefollowingway:wereCLtobenegative,thelimitedliabilityconstraintforCLwouldbebindingandthecorrespondingCHwouldgoupbytheamountthatkeepstheconsumptiondierentialconstantatwhatitwouldbeifCLwereallowedtobenegative.However,thispracticeisnotapplicablefor[ProgramSB]andthereasonisinconstraint( 3{2 ),whichimposesanupperboundontheborrower's Thenan"interiorsolution"referstoC0andLC=0anda"cornersolution"referstoC=0andLC<0ThisalsoimpliesthatthelimitedliabilityconstraintsforCarenotbindingwhentheinteriorsolutionarises,whereastheyarebindingforthecornersolution. 32

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3{2 )provides Inequality( 3{9 )impliesthatthereisanupperboundforCH.Therefore,foranyCL<0itisinfeasibletoadjustCHupwardstoinduceCL=0.ThisstatesthatthelimitedliabilityconstraintforCLisneverbindingandCL=0when( 3{8 )issatisedatequalityisstillpartoftheinteriorsolutions.Intuitively,Corollary 1 statesthatnoloanarrangementexiststhatprovidesasolutionofCL<0thatcanmotivatetheborrowerandletthelenderrecoupherinvestmentatthesametime.14 3-1 showswhethertheexistencecondition( 3{8 )forthesecond-bestcontractholds.Thethirdcolumnshowsthesecond-bestcontract(labeledas"SB")describedinProposition 1 .15Thelastcolumnshowstheborrower'sconsumptionfrom[ProgramSB]butwithoutthelimitedliabilityconstraints,i.e.,unrestrictedsecond-bestcontract(labeledas"SBU").Assumptions1and2areallsatisedwiththeseparametervalues:H=pHXH+(1pH)XLI=140L=pLXH+(1pL)XLI=20aH=90<1 (1pH)=109:9 3{9 )alonedoesnotleadtothenon-existenceoftheloancontract,itisajointforcewiththelimitedliabilityconstraintonCLthatpotentiallycausesthefailureoftheloanagreement.WereCLtobeallowedtogonegative,afeasibleCHwouldalwaysbefoundtosatisfy( 3{9 ).15Therst-bestcontractcharacterizedinFact2isnotshownbecauseofitsstraightforwardness. 33

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Second-BestContractsfortheBorrowerwithAggregateConsumption SBU W 3{8 ) N=A N=A 50 29 259 29 200 179 409 179 ConsistentwithProposition 1 ,thesecond-bestcontractisnotalwaysfeasible.ItisevidentfromTable 3-1 thatwhentheborrower'spersonalwealthWis20,thereisnofeasiblesolutionforthesecond-bestcontract.Thisisreectedbythefailuretosatisfycondition( 3{8 )inProposition 1 .16TakingW=20asanexample,wecomparetwosecond-bestcontractscharacterizedinTable 3-1 AtW=20,thecontractSBisobviouslynotfeasible.Thealternativesecond-bestcontractSBU,whichfeaturesthelackoftheborrower'slimitedliabilityconstraints,hassolutionsofCH=229andCL=1.However,thisconsumptionscheduleisnotfeasiblewhenthelimitedliabilityconstraintsareimposed.Weretheborrower'slimitedliabilityconstraintforCLtobebinding,i.e.,CL=0,itwouldfollowfromtheborrower'sICconstraintthatthecorrespondingCHisadjustedupwardsby1tobe230tokeeptheborrowerincentivecompatible.Unfortunately,substitutingCH=230andCL=0intothe( 3{2 )constraintyieldspHCH+(1pH)CL=161W+H=160 whichisaviolationoftheconstraint. 3{8 ),itisnotjustthevariationofWthatmayleadtothenon-existenceofthesecond-bestcontract.Forexpositionalsimplicitytheexample(inTable3-1)focusesontheminimalvariationofoneparameterwhilexingtheotherparameters. 34

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3{2 ))playanimportantroleindeterminingthefeasibilityofcontractSB. Whenthesecond-bestloanSBisfeasibletheoptimalconsumptionfeaturesconsumptiondierentialCH>CLtoensurethattheborrowerismotivated.Furthermore,theconsumptionspreadalwaysstaysat230(i.e.,CHCL=230),reectingtheborrower'sCARAriskpreference.TheconsumptionbehaviorbetweencontractsSBandSBUareidenticalsincetheborrower'sconsumptionarepositiveandthelimitedliabilityconstraintsarethentrivial. 1 ,thefailuretosatisfycondition( 3{8 )contributestotheinfeasibilityoftheloanagreement.Whenthesecond-bestcontractisfeasible,( 3{2 )suggeststhattheexpectedNPVfromtheinvestmentprojectaccruestotheborrower;therefore,theborrower'snetexpectedgainfromtheinvestmentisHaH.ItturnsoutthatnotanylevelofHaHensurestheinvestmentbehavior.Inparticular,thereisaboundaryconditionfortheborrower'snetgainunderwhichheisbetterowithouttheinvestment.Theconclusionsaresummarizedinthefollowingproposition: 35

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ConditionaofProposition 2 addressestheinfeasibilityofthesecond-bestcontract.Thelackofafeasiblesecond-bestcontractisessentiallyaformofcreditrationingforthewealth-constrainedborrower. Tirole ( 2006 )pointsoutthatinsucientpersonalwealthmaycontributetocreditrationingwhentheborrower'seortsupplyisprivatelyobserved.Theargumentisthatsincetheborrowerenjoystheentirebenetofshirkingbutsharestheinvestmentprojectwiththelender,thecontractbetweentheborrowerandthelenderthenrequirestheborrowertocontributemoreendowmenttoensureheismotivated.ConditionaisconsistentwithTirole'scredit-rationingargument.Basically,fortheborrowerwithpersonalwealthW
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3{8 )inProposition 1 ,wehave Forthewealth-xedborrower,astheright-handsideof( 3{10 )increases,condition( 3{8 )ismorelikelytoholdatequality.Therefore,asthesecond-bestcontractbecomesmorerisky,theborrower'sexpectedutilityfromthesecond-bestcontractisdecreasing;heisthenmorelikelytopreferthestatusquototheloanarrangement.Notethattheincreaseintheright-handsideof( 3{10 )ismorelikelytotriggerConditionbtohold,whichmakestheborrowermorevulnerabletoinvestmentdistortion. Whentheborrowerhassucientpersonalwealth,i.e.,WI,theborrower'sdecisionbetweenthesecond-bestcontract,givenitsexistence,andthestatusquoisnotconclusivetodetermineinvestmentdistortionsincetheborrowerhastheoptiontoself-nancetheinvestmentproject.Whenthesecond-bestcontractisinfeasible,thechoiceisonlybetweenno-lenderinvestmentandthestatusquo.LetCENLdenotetheborrower'scertaintyequivalentconsumptionfromno-lenderinvestment,itthenfollowsfromhisconsumptionbehavior17thatCENL=WI+1

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Giventhattheborrower'scertaintyequivalentconsumptionfromthestatusquoisjusthispersonalwealthW,hisdecisiontodeterminewhetherself-nanceisasoundchoiceboilsdowntocomparingtheborrower'snetcertaintyequivalentconsumptionfromno-lenderinvestmentwithW.Inotherwords,wecomparethestochasticcashowfromtheinvestmentCE(X)withtheincurredinvestmentcost,whichisI+aH.Ifthebenetislessthanthecost,i.e., theprojectistooriskyfortheborrowertoengageinself-nancing;therefore,investmentdistortionarises. Whenthesecond-bestcontractisfeasible,investmentdistortionrequiresthefailureofbothno-lenderinvestmentandloanagreementwiththelender.Weconcludethepotentialinvestmentdistortionwithawealth-abundantborrowerasfollows: whereCE(X)=1 ThesecondconditioninPartbofProposition 3 directlyfollowsfromConditionbofProposition 2 ;itistheconditionunderwhichthestatusquoispreferabletotheloanagreement. Whenthesecond-bestcontractisfeasible,thenextlogicalquestioniswhethertheborroweralwaysprefersborrowingtoself-nancing.Thefollowingcorollaryisdeveloped: 38

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NoInvestSecond-BestLoanNo-LenderInvest(3{12) 3 isonlynecessarybutnotsucientfor( 3{12 ).Furthermore,Corollary 2 impliesthatborrowingisnotalwayspreferabletoself-nancingwhentheborroweriswealth-abundant. LetEUSB(EUNL)andRUdenotetheborrower'sexpectedutilityfromthesecond-bestcontract(no-lenderinvestment)andhisreservationutilityfromno-investment.18Thefollowingexampleisdesignedtoillustrateinvestmentdistortion(orthelackofinvestmentdistortion)discussedinProposition 2 andProposition 3 .TheparametervaluesinTable 3-2 followthoseinTable 3-1 unlessotherwisespecied.Withtheparametervalues,thecertaintyequivalentcashowfromtheinvestmentprojectiscalculatedasCE(X)=243.Furthermore,let"ID"standforwhetherthereisinvestmentdistortionandCHandCLbetheoptimalsecond-bestconsumptionderivedinProposition 1 Theupperexhibitrepresentsthewealth-constrainedborrower.CaseAandCaseBbothsuggestinvestmentdistortion;however,theCaseAscenarioreectsConditionaofProposition 2 thatthesecond-bestcontractisinfeasible(W+HpH 2 thatthesecond-bestcontractexists 39

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InvestmentDistortionfortheBorrowerwithAggregateConsumption aH ID A N=A N=A B 226 6 C 176 123 W=180 aH EUNL D N=A N=A E 450 16 F 341 221 2 holdsandtheborrowerachievesahigherexpectedutilitywithborrowingfromthelenderthanconsuminghispersonalholdings(i.e.,EUSB=:28>RU=:82). Thelowergurefocusesonthewealth-abundantborrowerwithW>I.CaseDandCaseErepresentinvestmentdistortionbutbasedondierentgrounds.CaseDreectsPartaofProposition 3 ,wherethesecond-bestcontractisnotfeasible;therefore,investmentdistortionarisesbecauseno-lenderinvestmentisnotpreferabletothestatusquo(CE(X)
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Second-BestLoanNo-LenderInvest(3{15) ForCaseE,( 3{14 )holdsbut( 3{15 )isviolated.Therefore,asreectedbyTable 3-2 ,theborrower'sexpectedutilityfromno-lenderinvestmentishigherthanthatfromthesecond-bestcontractforCaseE.Intuitively,inthepresenceoftherelativelylargepersonalcostofhigheortsupply(aH=107),althoughtheloanagreementprovidesrisk-sharingitalsoenlargestheconsumptiondierentialtomotivatetheborrower,whichmakestheloanagreementveryriskyandinconsequencereducestheborrower'sexpectedutilitydramaticallygivenhisconcaveutilityfunction.Weretheborrowertochooseself-nancing,hisconsumptionfromno-lenderinvestmentwouldbe380and180whenhighandlowcashowsarerealized,respectively.Thisconsumptionscheduleisnotfeasibleforthesecond-bestloanagreementbecausetheborrowerICconstraintwouldbeviolated.Whentheborrowerweighsthebenetbroughtaboutbyrisk-sharingwiththelenderandthecostofrespectingtheimposed(IC)constraint,theborrowermightbeworseosigningacontractwiththelenderthanjustinvestingonhisown. Onthecontrary,CaseFreectsthesituationwherethesecond-bestcontractistheborrower'soptimalinvestmentdecision.Inparticular,condition( 3{15 )issatisedwithCaseF.Here,themoderatepersonaleortcostoftheborrower(aH=60)makesthebenetfromrisk-sharingoutweighthecostfromapplyingtheconsumptiondierentialtokeeptheborrowermotivated. 41

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Theanalysisfortheborrowerwithtime-additiveconsumptionpreferencesharesquiteafewsimilaritieswiththatfortheborrowerwithaggregateconsumptionpreferenceintheprevioussection.Inparticular,regardlessoftheborrower'sconsumptionpreference,hefacessimilarinvestmentoptionsandhisinvestmentbehaviorfeaturesnoinvestmentdistortionorpotentialinvestmentdistortion. ( 1960 )stationarityaxiomimpliesthatintertemporalpreferencesbetweentwoperiodsremainthesame.Therefore,theborrowerwithtime-additiveconsumptionpreferencehasanaggregateutilityfunctionthatcanbeexpressedasthesumofhisperiod-by-periodutility. 42

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Therst-bestcontractfortheborrowerwithintertemporalconsumptionconcernismodeledsimilarlyto[ProgramFB];however,theborrower'sconsumptionconstraintsaredierent.Att=0,theborrowerhasapersonalwealthlevelofW,thisdoesnotnecessarilyindicatesheconsumestheentireWatt=0,hecanchoosetoconsumeW0WandsaveW1=WW00forthefutureperiodt=1.20Thereforeatt=0,theupperboundfortheborrower'sconsumptionishisinitialpersonalwealthnetofhisinvestmentintheprojectandhisstoredwealth,i.e.,CA0W(ID0)W1 43

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[ProgramFBA]maxCA00;CAH0;CAL0exp(CA0)pHexpCAHaH(1pH)expCALaHs.t.CA0+pHCAH+(1pH)CALW+H whereH=pHXH+(1pH)XLI. Theborrowerwithtime-additiveconsumptionpreferencehasaconsumptionschedulethatresemblesthatoftheborrowerwithaggregateconsumptionpreference,andinparticulartheborrower'sintertemporalnetconsumptionaresmooth,whichsuggestsCA0=CAHaH=CALaH where=pHexp(aH)pL 44

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2CAH=W+HpH 2+1 2 45

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where=pHexp(aH)pL CloserinvestigationrevealsthatinthecornerconsumptionsituationonlytheconstraintforCALisbinding.Thedrivingforceofthispossibilityistheborrower'stime-additiveconsumptionpreference.Recallthatfortheborrowerwithaggregateconsumptionpreference,thereisanupperboundonCH,whichisderivedfrom( 3{2 )constraint.Similarly,whentheborrowerhasatime-additiveconsumptionpreference,inequality( 3{16 )alsosuggeststhatCAHisboundedfromabove Moreover,thesecond-bestoptimalconsumptioninProposition 4 impliesthat Therefore,from( 3{19 )and( 3{20 )itispossibleforCAHtobeadjustedupwardswhenthelimitedliabilityconstraintforCALisbindingbecauseofthepresenceofCA0.Intuitively,ifthecontrolproblemgetssevere,theconsumptiondierentialneedstoincreasetoensuretheborrowerdelivershigheort.Thisdoesnotimposeanyadditionalrestrictionaslongastheoptimalconsumptionisstillinterior;however,whenthelimitedliabilityconstraintofCALisbinding,thequestionofwhetherthereexistsafeasibleconsumptionschedulethatsatisestheborrower'sICAconstraintbecomesimportant.As( 3{19 )shows,CA0providessomerangewhichmakesCAHstillbefeasiblewhenthelimitedliabilityconstraintofCALis 46

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Asindicatedby( 3{20 ),thelimitedliabilityconstraintfortheborrower'st=1consumptionCAHisneverbinding.Furthermore,Proposition 4 alsosuggeststhatthelimitedliabilityconstraintforCA0isnotbindingaswell.Thereasonissimilartothereason,discussedearlier,whythelimitedliabilityconstraintofCLneverbindsfortheborrowerwithaggregateconsumptionpreference.Again,theupperboundin( 3{19 )prohibitstheexistenceofsuchCAHthatcanprovideamutuallyagreeablesecond-bestcontracttorespecttherequiredconsumptiondierentialwhenbothCALandCA0areforcedtobecornersolutions. FromProposition 4 ,itisevidenttheconsumptiondierentialisunavoidableinthesecond-bestcontract;furthermore,consumptionacrossperiodsisnotsmoothing,whichisconsistentwiththeconsumptiondistortiondenition.Deneutilitysmoothingwhenthefollowingequalityholds:exp(CA0)=pHexpCAHaH(1pH)expCALaH 47

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DeneCEtasthelocalcertaintyequivalentconsumptionfortheborrowerwithtime-additiveconsumptionpreferenceattimet=0;1,respectively,Corollary 3 thenstatesthatexp(CE0)=exp(CA0)exp(CE1aH)=(1pH)expCALaHpHexpCAHaH Equation( 3{21 )indicatesthattheborrower'snett=1localcertaintyequivalentconsumptionequalshist=0localcertaintyequivalentconsumption.Therefore,eventhoughthereisconsumptiondistortionwithrespecttothesecond-bestconsumptionschedule,theborrower'sdesiretohavesmoothconsumptionispreservedinthesensethatsmoothingutilityisnotdistorted. Recallthatinformulatingthetime-additiveborrower'sconsumptionproblem,theborrowerisassumedtoputW10asideasareserveforhist1consumption.Wearguethatoncethesecond-bestcontract[ProgramSBA]isinplace,itisoptimalfortheborrowertoalwaysstoreadditionalconsumptionW1.Considerthefollowingtwopotentialconsumptionschedules: 48

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B. ItisevidentthatConsumptionBehaviorAstoresnoadditionalconsumptionfort=1whileScheduleBdelayssomeconsumptiontillt1.Fortheborrowerwithtime-additiveconsumptionpreference,theconsumptiondierentialatt=1isdeterminedbyhisattitudetowardsrisk,whichisinvariantwithhispersonalwealthholdings;therefore,whetherW1ispositiveornothasnoeectonhisconsumptiondierentialfort=1. ForConsumptionScheduleA,theborrower'speriod-by-periodexpectedutilityis (3{22)pHexp[(XHDH)](1pH)exp[(XLDL)] (3{23) andforConsumptionScheduleB,hisperiod-by-periodexpectedutilityis (3{24)pHexp[(W1+XHDH)](1pH)exp[(W1+XLDL)] (3{25) Itisevidentthat ( 3{22 )( 3{24 ) 49

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3{23 )( 3{25 ) However,giventheconcavityoftheborrower'sutilityfunction,thefollowingrelationshipholds ( 3{22 )( 3{24 )( 3{25 )( 3{23 ) whichisequivalentto ( 3{22 )+( 3{23 )( 3{24 )+( 3{25 ) wheretheequalityholdsatW1=0.Thisindicatesthattheborrower'sexpectedaggregateutilityisweaklyhigherwhenhehasConsumptionScheduleBratherthanA,whichimpliesthatitisoptimalfortheborrowertostoreadditionalconsumptionatt=0. Beforeproceedingtoinvestmentdistortion,thefollowingexampleisprovidedincomparisonwithTable 3-1 .TheparametervaluesfollowthoseinTable 3-1 unlessotherwisespecied.Otherthanthecolumnfortheborrower'spersonalwealthlevels,Table 3-3 consistsofthreeblocks.Therstandthesecondblocksarethesecond-bestcontractcharacterizedinProposition 4 withinteriorandcornerconsumptionindicated,whicharelabeledas"SBI"and"SBC,"respectively.Forminimalvariation,onlyaHischangedtotriggerthecornerconsumption.21Thelastblockisthesecond-bestcontractifthelimitedliabilityconstraintsareabsentgiventhesameborrower'spersonalcostofeortaHasforthecornerconsumption,whichislabeledas"SBUA."Therst-bestcontractforthetime-additiveborrowerfollowsFact4andissmoothingintheborrower'snetperiod-by-periodconsumption;therefore,itisnottabulated. 50

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Second-BestContractsfortheBorrowerwithTime-AdditiveConsumption SBC SBUA CA0 N=A N=A N=A N=A N=A 225 50 19:1 240 9:7 1:2 269:7 0 5:5 265:4 200 94:1 315 84:7 5:9 477:2 0 8:2 475 Whentheborrower'spersonalwealthholdingsis20,thesecond-bestcontractisnotfeasible.22ThisisbecauseforW=20,condition( 3{18 )isviolated. Whenthesecond-bestcontractexists,itisevidenttheconsumptiondierentialispresentandperiod-by-periodconsumptionisnotsmoothing,bothofwhichindicateconsumptiondistortion.ThesecondblockinTable 3-3 providescornersolutionforthesecond-bestcontract.Intuitively,astheborrower'spersonalcostofequilibriumeortgrows,hisdisutilityfromworkingdiligentlyincreases.Theincentivecontractneedstooerawiderconsumptiondierentialtoensuretheborrowerismotivated,whichmakesitmorelikelytosetothelimitedliabilityconstraintforCAL.ThemagnitudeofCA0providessomecushionforsuchconsumptiondierentialadjustment.Inthepresenceoflimitedliability,oncethelimitedliabilityconstraintforCALisbinding,thesecond-bestcontractreallocatesthedistributionofconsumptionacrossperiodstoensurethefeasibilityofsuchaloanagreement. ThereallocationisnotwithoutitsboundarysincethelimitedliabilityconstraintexistsforCA0aswell.ThelastblockofTable 3-3 forW=20reectssuchacase.WereCALtoberestrictedtobenon-negative,wewouldseeCAH=225+j5:3j=230:3 51

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toensuretheconstraint( 3{16 )holds;however,CA0'slimitedliabilityconstraintisviolated,whichimpliesthereisnosolutionfor[ProgramSBA]. Aftercharacterizingthesecond-bestcontract,wenexttacklethepotentialinvestmentdistortionprobleminthesecond-bestenvironment.Similartotheborrowerwithaggregateconsumptionpreference,forthewealth-constrainedborrower(i.e.,W
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where=pHexp(aH)pL InSection3.2.2.3,theconditionunderwhichthestatusquoispreferabletono-lenderinvestmentfortheborrowerwithaggregateconsumptionpreferenceisderivedas whereCE(X)=1 3{26 )isstillasucientbutnotanecessaryconditionforthestatusquotowinoverno-lenderinvestmentwhentheborrower 53

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investmentdistortioncanstillarise.Thisimpliesthatprojectsthatarenotconsideredriskyfortheborrowerwithaggregateconsumptionpreferencemaybeconsideredriskyfortheborrowerwithtime-additiveconsumptionpreference.Inparticular,theborrowerwhodoesnotachievesmoothingutilitiesacrosstimeismorevulnerabletotheriskinessoftheinvestmentproject.Sincefortherisk-averseborrowertheinvestmentresultinginnon-smoothingintertemporalutilitiesisriskierthantheonewhichprovidesotherwise,itisimportantfortheborrowertosmoothhisutilitiesacrosstime.However,withoutaccesstopersonalbanking,itisnotalwaysfeasiblefortheborrowertodoso. Withno-lenderinvestment,theborrower'sintertemporalconsumptionscheduleisCA0WICAHWCA0I+XHCALWCA0I+XL 3{26 )isasucientandnecessaryconditionforProposition 3 inSection3.2.2.3todeterminetheborrower'spreferenceofstatusquotono-lenderinvestment.24Inequilibrium,theborrower'st=1consumptionCAHorCALarebinding.Furthermore,itispossiblethattheborrower'sintertemporalutilitiesaresmoothedwhenC0=WI 54

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ItturnsoutthatCA0isnotbindingwhenthefollowingrelationshipissatised Otherwise,theborrower'st=0consumptionisequaltoWI.Intuitively,otherthingsbeingequal,( 3{28 )ismorelikelytoberealizedifWisrelativelylarge.ForW>>I,CA0isnotbindingandbyreservingsomeofhispersonalwealthfort=1consumption,theborrowerismorelikelytoachievesmoothingperiod-by-periodutilities,whichisstrictlypreferred.Oncetheborrowerhassmoothingutilities,asdiscussedearlier,anyinvestmentdecisionwithtime-additiveconsumptionpreferenceisnotqualitativelydierentfromthatwithaggregateconsumptionpreference;therefore,( 3{26 )isallthatmatterstodeterminewhethertheborrowerprefersthestatusquotono-lenderinvestment.However,itisinconclusivetodeterminetherelationshipbetweeninequalities( 3{28 )and( 3{26 );therefore,condition( 3{26 )isnolongernecessarytoensurepotentialinvestmentdistortion.Theinvestmentdistortionforthewealth-abundantborrowerwithtime-additiveconsumptionpreferenceissummarizedinthefollowingproposition: whereCE(X)=1

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Thenextnumericalexampleshowsinvestmentdistortionfortheborrowerwithtime-additiveconsumptionpreference.TheparametervaluesmainlyfollowthoseinTable 3-2 exceptfortheborrower'spersonalwealthholdings,whicharespeciedintheTable.Thesuperscript"A"isaddedtothenotationtodistinguishtheborrowerwithtime-additiveconsumptionpreferencefromaggregateconsumptionpreference.Theupperexhibitisforthewealth-constrainedborrowerandthelowerexhibitisforthewealth-abundantborrower;hisconsumptionfromthesecond-bestcontractandno-lenderinvestmentarenottabulated. CaseAinvestmentdistortionoccursbecausethesecond-bestcontractisinfeasible,whichmeansthatcondition( 3{18 )failstohold.Notethatcondition( 3{18 )isequivalenttotheonethatdeterminesthenon-existenceofthesecond-bestcontractfortheborrowerwithaggregateconsumptionpreference.CaseBrepresentsinvestmentdistortionduetotheriskinessofthesecond-bestcontract,whichsuggeststhattheborrowerprefersthestatusquototheinvestment.Closecalculationrevealsthattheborrower's"netgain"(i.e.,HaH)is52,whichisnotsucientlylargefortheborrowertoundertaketheinvestmentproject(therequiredboundaryis56).Thisboundaryconditionisalsoequivalenttotheonethatdeterminespassinguponthesecond-bestcontractbasedonitsriskinessfortheborrowerwithaggregateconsumptionpreference.Therefore,theborrower'sinvestmentdecisionisinvariantwithhisconsumptionpreferenceifheiswealth-constrained.Finally,CaseCshowsnoinvestmentdistortionastheborrower'sexpectedutilityfromthesecond-bestcontractishigherthanhisreservationutility. FortheborrowerwithWI,CaseDandCaseEinTable 3-4 bothreectinvestmentdistortion.Thesecond-bestcontractisinfeasibleforCaseDforXL=130; 56

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InvestmentDistortionfortheBorrowerwithTime-AdditiveConsumption W aH 100 B 88 C 50 Case W aH 109 130 E 90 150 F 90 300 3{26 )(i.e.,CE(X)
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Whentheborroweriswealth-abundant,otherthanseekingaloanagreementandstayingput,hecanalsochoosetoself-nancetheinvestmentproject.Giventheborrower'sconcaveandincreasingutilityfunction,theborrowerwithtime-additiveconsumptionpreferencemaximizeshisexpectedutilitywhenhisperiod-by-periodutilitiesaresmoothing.However,theconcernforconsumptionsmoothingmakestheborrower 58

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59

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TodistinguishtheChapter3borrowerfromtheborrowerdiscussedinthischapter,theborrowerinChapter3islabeledastheCARA-borrower.Chapter3analyzestheCARA-borrower'sconsumptionandinvestmentbehaviorswhilevaryinghisconsumptionpreferences.ThegeneralconclusionisthatpotentialinvestmentdistortionisweaklyaggravatedwhentheCARA-borrowerhastime-additiveconsumptionpreference.Moreover,ifthesecond-bestloanagreementistheoptimalinvestmentdecisionforbothconsumptionpreferences,conditionsunderwhichinvestmentdistortionoccursareidenticalregardlessoftheCARA-borrower'sconsumptionpreference.ThisspecialcaseismainlycausedbythenatureoftheCARA-borrower'sriskpreference.Inparticular,time-additiveconsumptionandinvestmentproblemmaybesimplyconsideredastwoidenticalone-periodconsumptionandinvestmentproblemsfortheCARA-borrower.However,weretheborrower'sriskpreferencetobedierent,suchrepeatedconsumptionandinvestmentdecisionfeaturemaynotbepresent.Inthischaptertheborrowerisassumedtohavedecreasingabsoluteriskaversion(DARAhereafter),whoseutilityfunctiontakestheformofsquare-rootonthenetconsumption.UnlikethenegativeexponentialutilityfunctionwhichistheuniqueutilityformintheCARAclass,thesquare-rootutilityfunctionisonlyoneoftheutilityformsintheDARAclass.Forexpositionalsimplicityandwithalittleabuseofterminology,theborrowerwithsquarerootfunctioninthischapterissometimesreferredtoastheDARA-borrowertoavoidconfusionwiththeCARA-borrower;otherwise,"borrower"inthischapterreferstotheDARA-borrower. TheCARA-borrowerconsumptionandinvestmentanalysesinChapter3showtheimportanceoftheborrower'spersonalwealth.Inparticular,thewealthleveldeterminestheborrower'spotentialinvestmentstrategyandwhethertheloancontractisfeasible.However,oncedierentinvestmentstrategies(i.e.,borrow,no-lenderinvestment, 60

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TheDARA-borrowerfacessimilarinvestmentdecisionsastheCARA-borrower,hehasthechoiceofborrowingfromthelender,engaginginno-lenderinvestment,andconsuminghispersonalwealthandforgoingtheinvestmentproject.Theinvestmentproject,ifengaged,againprovidesabinarycashowX2fXH;XLg.Thelenderisriskneutralandbreaksevenfromherinvolvementintheproject.TheborrowerisexpectedtoworkdiligentlybecauseAssumption1inChapter3appliesforthischapteraswell,whichindicatesthathigheortisalwayspreferredinequilibrium.LetbCandbUdenotetheborrower'saggregatemoneyholdingsacrossperiodsandhisutilityfunction,respectively.Therisk-averseborrower'sutilityisexpressedasbU=bUbCa=q bCa 61

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Thischapterproceedsasfollows.InSection4.1therst-bestcontractandno-lenderinvestmentfortheDARA-borrowerwithaggregateandtime-additiveconsumptionpreferencesarediscussedasbenchmarks.Section4.2characterizesthesecond-bestloanagreementswhentheborrower'seortsupplyisunobservabledependingonhisconsumptionpreferenceandidentiesthesecond-bestloancontractassociatedwithpotentialinvestmentdistortions.Section4.2alsoprovidescomparisonsbetweentheDARA-andtheCARA-borrowerintheirinvestmentbehaviorandtheincidenceofinvestmentdistortion.ThischapterconcludeswithSection4.3,whichprovidesadiscussionoftheeectoftheborrower'sconsumptionpreferenceonhisoptimalinvestmentpolicy.1 62

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bCHaH+(1pH)q bCLaHs.t.pHbCH+(1pH)bCLW+H [ProgramFBDA]maxbCA00;bCAHaH;bCALaHq bCA0+pHq bCAHaH+(1pH)q bCALaHs.t.bCA0+pHbCAH+(1pH)bCALW+H Therst-bestcontractsprovidethefollowingoptimalconsumptionfortheborrower. Intheabsenceofthelender,whentheborrowerhassucientpersonalwealthhehastheoptionofengaginginself-nancing.Theborrowerwhohasaggregateconsumptionpreferencemaximizeshisexpectedutilityfromself-nancing,whichleadstothefollowingprogram: 63

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bCHaH+(1pH)q bCLaHs.t.bCHWI+XHbCLWI+XL However,fromthelimitedliabilityconstraint,itisrequiredthatbCLaH 4{1 )and( 4{2 ),itisrequiredthatpHp 64

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LetCE(X)denotetheborrower'scertaintyequivalentconsumptionwhenW=I,itfollowsthatp 4{3 )inProposition 7 isnotnecessaryfortheborrowertopreferno-lenderinvestmenttonoinvestment.Thereasonisthat( 4{3 )isderivedundertheassumptionthatW=I;therefore,thefailuretosatisfy( 4{3 )doesnotimplytheoccurrenceofinvestmentdistortion. ComparingtheDARA-borrowerwiththeCARA-borrowerintheirno-lenderinvestmentstrategies,itturnsoutthattheconditionwhichleadsbothtypesofborrowertopreferself-nancingisquitesimilar.LetCE(X)betheborrower'scertaintyequivalentconsumptionforW=I,forbothtypesofborrower,thefollowingcorollarycanbedeveloped:

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5 isalsoanecessaryconditionfortheCARA-borrowerbutnotnecessaryfortheDARA-borrower. Next,thepotentialinvestmentdistortioncausedbytheborrower'sunwillingnesstoengageinno-lenderinvestmentisinvestigatedwhentheborrower'sconsumptionpreferencebecomestime-additive. Whentheborrowerhastime-additiveconsumptionpreference,hesolvesthefollowingproblemtojustifyhisno-lenderinvestmentdecision: [ProgramNLDA]maxbCA00;bCAHaH;bCALaHq bCA0+pHq bCAHaH+(1pH)q bCALaHs.t.bCA0WIbCAHWIbCA0+XHbCALWIbCA0+XL 66

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4{4 )with( 4{3 )impliesthattheborrowerrequireshigherCE(X)toengageinno-lenderinvestment(twicetobeexact).Thisisconsistentwiththefactthattheborrowerwithtime-additiveconsumptionpreferencehasintertemporalconsumptionconcern,whichleadstoamoreconstrainedproblem. TheDARA-borrowerinthischapterisassumedtohavedomain-additivepersonalcoststructure.Thiscoststructuremakesitquitediculttoderiveclosed-formsolutionsoftheoptimalconsumptionwithoutimposingfurtherassumptions;however,itisconsistentwiththeCARA-borrowerinChapter3inthecostsphere.Furthermore,thelackofclosed-formexpressionforthesecond-bestloancontractconsumptionhinderstheanalysisofinvestmentdistortionassociatedwiththesecond-bestloanagreement.Therefore,inthissectionnumericalexamplesareappliedtofacilitaterevealingtheinsightsofthesecond-bestloancontractandpotentialinvestmentdistortion. 67

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bCHaH+(1pH)q bCLaHs.t.pHbCH+(1pH)bCLW+H bCH+(1pL)q bCLpHq bCHaH+(1pH)q bCLaH Theborrower's( 4{6 )isbindinginequilibrium,otherwise,therst-bestconsumptionapplywhichleadstodirectlyviolationof( 4{6 )constraintgivenpH>pL.FollowingChapter3,thelenderbreaksevenandallexpectedcashowsfromtheinvestmentprojectaccruetotheborrower;therefore,theborrowerconsumesatmostthesumofhispersonalwealthathandandtheexpectedcashowsfromtheinvestment.Thisimpliesthatthe( 4{5 )constraintisalsobindinginequilibrium.Then,theoptimalconsumptionisderivedfromthefollowingsetofequations:pHbCH+(1pH)bCL=W+HpLq bCH+(1pL)q bCL=pHq bCHaH+(1pH)q bCLaH 68

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Intuitively,personalwealthisofimportanceintheincidenceofinvestmentdistortionassociatedwiththesecond-bestloancontract.TheCARA-borrower'sriskpremiumfortheloanagreementisinvariantwithhispersonalwealthholdings.However,theDARA-borrower'sriskpremiumdecreaseswhenhegetswealthier.Therefore,ceterisparibus,thepotentialinvestmentdistortionislessseverefortheDARA-borrowerashispersonalwealthgrows. LetdRPandRPdenotetheriskpremiumoftheDARA-andtheCARA-borrower'sinthesecond-bestloancontract,respectively.Furthermore,letdEUSB(EUSB),dRU(RU),cID(ID)denotetheDARA(CARA)-borrower'sexpectedutilityfromthesecond-bestloancontract,hisreservationutility,whichisdenedashisutilityfromconsuminghispersonalwealth,andwhetherinvestmentdistortionispresent.TheparametervaluesfollowthoseinChapter3exceptforXH=310.2 4-1 comparestheincidenceofinvestmentdistortionoftheDARA-borrowerwiththoseoftheCARA-borrower.Ingeneral,Table 4-1 indicatesthatinvestmentdistortioncanbemitigatedbychangingtheborrower'sriskpreference.Inparticular,whentheborrower'spersonalwealthisW=50,Case1impliesthatnosolutionexistsfortheDARA-borrower;therefore,investmentdistortionispresent.Furthermore,eventhoughtheloanagreementisfeasiblefortheCARA-borrower,investmentdistortionstilloccursbecauseborrowingistooriskyfortheCARA-borrowertoengagein.Therefore,Case 4-1 69

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InvestmentDistortionforAggregateConsumption W aH EUSB RU ID 50 85 N=A 49:3 135 95 11:5 19:3 11:6 78:8 200 95 14:4 8:7 14:1 78:8 TheDARA-borrowerwithtime-additiveconsumptionpreferencesolvesthefollowingoptimizationproblem: 70

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bCA0+pHq bCAHaH+(1pH)q bCALaHs.t.bCA0+pHbCAH+(1pH)bCALW+HpLq bCAH+(1pL)q bCALpHq bCAHaH+(1pH)q bCALaH dCE1aH=pHq bCAHaH+(1pH)q bCALaHdRP1=EbC1dCE1

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4-2 followthoseinTable 4-1 .Let4dCE(4CE)and4\NCEdenotetheDARA(CARA)-borrower'sdierenceinlocalcertaintyequivalentconsumptionandtheDARA-borrower'sdierenceinnetlocalcertaintyequivalentconsumption,respectively. TheitaliccolumninTable 4-2 indicatesthattheDARA-borrower'st=1localriskpremiumsystematicallydecreaseswiththeincreaseofhispersonalwealthholdings.However,fortheCARA-borrower,hislocalriskpremiumisinvariantwithhispersonalwealthholdings.Specically,forCARA-borrower,anyincreaseofhispersonalwealthisdividedequallyintotwoparts,thersthalfgoestohist=0consumptionC0andthesecondhalfgoestohisexpectedt=1consumption,whichisE(C1).3;4Furthermore,theCARA-borrower'slocalcertaintyconsumptiondierencesareequalandindependentofhispersonalwealthholdings.Thisisreectedbythefactthat However,fortheDARA-borrower,smoothingutilityisnotpresent.Thiscanbeseenfromthefactthat4dCE=dCE1dCE06=aH 3NotethatequallyallocatingwealthacrossperiodsisonlyfeasiblefortheCARA-borrowerwheninteriorsolutionispresent.4Theborrower'sconsumptionatt=0isnotshowninthetable. 72

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LocalCertaintyEquivalentConsumptionsandLocalRiskPremium DARA CARA E(bC1) E(C1) 152:6 137:3 67:9 119:4 102:6 16:8 60 51 152:7 137:5 67:2 119:9 106:1 16:8 60 isdecreasingwiththegrowthofhispersonalwealthholdings.ThisimpliesthattheDARA-borrower'srequiredlocalcertaintyequivalentconsumptionatt=1,wheretheriskpremiumisnecessarytomotivatehimtosupplyhigheort,isdecreasinginmagnitudewithrespecttohislocalcertaintyequivalentconsumptionatt=0,wherenoriskpremiumispresent.ThismeansthattheDARA-borrowerrequireslessriskpremiumwhenhispersonalwealthholdingsgrow,whichcorrespondstothelocalriskpremiumdRP1column. Insum,whentheborrowerhastime-additiveconsumptionpreference,theDARA-borrowermoveshisconsumptionforwardintimetoreducethelocalriskpremiumassociatedwithunobservableeortsupplyatt=1ashispersonalwealthgrows,whiletheCARA-borrowerhasincentivestoequallyallocatetheincreaseofhispersonalwealthacrosseachconsumptionperiodstosmoothintertemporalconsumptionandretainstheunchangedlocalriskpremium. AlthoughtheDARA-borrower'soptimalconsumptionforthesecond-bestloanagreementisdiculttoderivebecauseofthedomain-additivecostpreference,thefollowingnumericalisdesignedtoshowthedierenceinpotentialinvestmentdistortionfortheDARA-andtheCARA-borrower,respectively. LetaH=80andkeeptheotherparametersthesameasinthepreviousnumericalexamples.Table 4-3 comparestheincidentsofinvestmentdistortionfortheDARA-andCARA-borrowerwhenbothhavetime-additiveconsumptionpreference.DenotedEUASB(EUASB),dRPA(RPA),dRUA(RUA),andcIDA(ID)theDARA-(CARA-)borrower's 73

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InvestmentDistortionforTime-AdditiveConsumption W 100 13:1 21 14:1 39:8 150 17:9 15:6 17:3 39:8 InTable 4-3 ,Case1impliesinvestmentdistortionforbothriskpreferenceswhileCase2displaysadierence.First,theCARA-borrowerstillexperiencesinvestmentdistortioneventhoughhispersonalwealthhasgrown,thisisconsistentwiththeconclusioninChapter3thattheCARA-borrower'sriskpreferencedoesnotvarywithhispersonalwealthholdings;therefore,giventhathehasinvestmentdistortiontobeginwith,hewouldencounterthesameinvestmentdistortionregardlessofthechangeinthemagnitudeofhispersonalwealth.However,asthepersonalwealthincreases,theDARA-borrowernolongerexperiencesinvestmentdistortion;thisisbecausehisattitudetowardsriskchangeswiththeincreaseofhispersonalendowment.AstheDARA-borrowerbecomeswealthier,heislessrisk-averse,whichcanbeinferredfromthedecreasingriskpremium;thisfollowsthatheismorelikelytoundertaketheloanarrangementthatwouldotherwisebetooriskywerehetohavelesspersonalwealth. 74

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LettheDARA-borrower'spersonalwealthbexedatW=140.Table 4-4 compareshisinvestmentdecisionsbetweenborrowingandno-investmentwhenhehaseitheraggregateortime-additiveconsumptionpreference. Giventhesamepersonalwealthlevel,Table 4-4 clearlyshowsthatinvestmentdistortionfortheborrowerwithaggregateconsumptionpreferenceimpliesinvestmentdistortionforthesameborrowerbutwithtime-additiveconsumptionpreference.Whentheborrower'spersonalcostofeortisaH=80,noinvestmentdistortionispresentforeitherconsumptionpreference.However,asaHgrowsto98,investmentdistortionoccursforbothconsumptionpreferences.Moreover,thepotentialinvestmentdistortion 75

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Second-BestContractsInvestmentDistortionfortheDARA-Borrower 12:8 11:8 16:7 12:2 11:8 16:7 11:5 11:8 16:7 Iftheborroweriswealth-abundantandself-nancestheinvestmentproject,similarlogicfollows.Inotherwords,sincetheborrowerwithtime-additiveconsumptionpreferencefacesmorehurdleswithrespecttotheinvestmentbehavior,thepotentialinvestmentdistortionproblemisworsenedastheconsumptionpreferenceemphasizesintertemporalconsumption. 76

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BothChapter3andChapter4discusstheborrower'sconsumptionandinvestmentbehaviorundertheassumptionthattheborrower'spersonalwealthispubliclyobservablebuteortsupplyisunobservable.Thischapterrelaxesthisassumptionandfocusesonthescenariowheretheborrower'spersonalwealthisprivateinformationwhilehiseortsupplyisalsoprivate.Thentheproblembecomesanentanglementofmoralhazardandadverseselection. TheanalysisisfocusedontheCARA-borrowerwithaggregateconsumptionpreference.Thetimelineisrevisedasfollows:attimet=0,amenuofcontractswhichspeciestheCARA-borrower'sconsumptionandthelender'sinvestmentandpayofromtheprojectisoered.TheCARA-borrowersubsequentlyobserveshispersonalwealthW2fWH;WLgwhilethelenderremainsuninformed.However,thelenderrequirestheCARA-borrowerreporthispersonalwealthfW2nfWH;fWLoandpostanup-frontbondintheamountofhisreport.LetW 77

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WeusethefollowingnotationtodenotethechoicevariablesintheCARA-borrower'soptimizingprogram. wherefW2nfWH;fWLo. ( 1985 )and LaontandMartimort ( 2001 )havedetaileddiscussesofRevelationprinciple. 78

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wherefW2nfWH;fWLo. OncetheCARA-borrowerknowshiswealthtype,hemakesreportingandeortsupplychoices.LetEUW;fW;adenotetheCARA-borrower'sexpectedutilitywhenheobserveshispersonalwealthtobeW2fWH;WLg,butchoosestoreportfW2nfWH;fWLoandsupplyaneortofa2faH;aLg.Foreachofhiswealthtypes,hehasthefollowingfeasiblestrategiesandtheassociatedexpectedutilities: Type Report Eort 79

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InChapter3,theCARA-borrower'sconsumptionisupper-boundedbythesumofhisremainedwealthafterputtinghisshareofinvestmentintheprojectandthecashowfromtheinvestmentnetthelender'spayo.Theseconsumptionconstraintsalsoapplyinthischapter: (5{6)CLfWHfWHhID0fWHi+XLDL(fWH) (5{7)CHfWLfWLhID0fWLi+XHDH(fWL) (5{8)CLfWLfWLhID0fWLi+XLDL(fWL) (5{9) TheCARA-borrowerwithprivateinformationthensolvesthefollowingproblemwithrespecttotheloancontract: [ProgramgSB]maxC(fW)0;D0(fW);D(fW)EUWRPjaHs.t.( 5{1 )( 5{2 )-( 5{5 )( 5{6 )-( 5{9 ) 80

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Denethe"naivecontract"astheoptimalloancontracttheCARA-borrowerwouldachieveifhispersonalwealthwerepublic;i.e.,theoptimalsecond-bestloanagreementinChapter3isthenaivecontract.Thefollowingconclusionisdevelopedfor[ProgramgSB]: 3{8 inChapter3).WhentheCARA-borrowerhasprivateinformationabouthispersonalwealth,hisreporting 81

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LettheCARA-borrower'spersonalcostofeortbeaH=30andhisbinarypersonalwealthtakethevalueofeitherWL=140orWH=300.TherestoftheparametervaluesfollowthoseinChapter3.2TheprobabilityoftheCARA-borrowerbeingwealthy,i.e.,hehasapersonalwealthof300isassumedtobe=:7.Theoptimalcontractfor[ProgramgSB]showsasolution:CHfWH=456andCLfWH=403CHfWL=296andCLfWL=243 withanexpectedutilityof:0374.ThisconsumptionschedulecoincideswiththenaivecontractinChapter3whentheCARA-borrower'spublicpersonalwealthis300and140,respectively.Furthermore,notethattheconsumptiondierencesbetweenreportingtruthfully(i.e.,300)andmisreporting(i.e.,140)isthedierencebetweenthereportedwealth(i.e.,300140=160).Therefore,byignoringtheCARA-borrower'sprivatepersonalwealthsourceandoeringhimtheconsumptionschedulethathewouldachieveifhiswealthwerepublic,theCARA-borrowerhasnoincentivetomisreporthiswealth. Chapter3demonstratesinvestmentdistortionariseswhen1)nosolutionexistsfortheloancontractand2)theloanagreementistooriskyandtheCARA-borrowerprefersthestatusquo.SincetheCARA-borrower'sconsumptionbehaviorfollowsthatinChapter3,thecorrespondingpotentialinvestmentdistortionproblemisinvarianttowhetherprivateinformationispresent. 82

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InChapter4,wendthatitisdiculttoderiveclosed-formconsumptionfortheDARA-borrowerbecauseofthedomain-additivecoststructure,thesamedicultyapplieshere.UnliketheCARA-borrowerwhoseriskpreferencestaysthesameregardlessofhispersonalwealth,theDARA-borrowerchangeshisattitudetowardsriskbasedonhiswealth.Inparticular,themorewealthiertheDARA-borroweris,thelessrisk-averseheis,whichimpliesthattheriskpremiumassociatedwiththeloancontractisdecreasingashispersonalwealthincreases.Therefore,thelesspersonalwealththeDARA-borrowerreports,themoreriskyloanagreementhefaces.Thisimpliesthatitisnotinthe 83

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Theborrower'smisreportingstrategyisone-directionalinthischapter.However,over-reportingmaybevaluabletotheborrower.Inparticular,whentheborrowerobserveshispersonalwealthtobelessthanW 84

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Thisdissertationcharacterizesaborrower'sinvestmentandconsumptionbehaviorswithrespecttoaninvestmentprojectandinvestigateswhethertheborrower'sconsumptionpreferenceaectstheinvestmentdecision.Itisconcludedthattheprojectmaybeinecientlyabandoned,whichresultsininvestmentdistortion,andtheinvestmentdistortionproblemisworsenedwhentheborrowerfocusesmoreontheintertemporalconsumptionthanonaggregateconsumption.Theborrower'sinitialwealthendowmentisimportantindeterminingtheformofpotentialinvestmentdistortion.Forthewealth-constrainedborrower,thejointforceofupper-boundedexpectedprotfromtheinvestmentandthelimitedliabilityconstraintsdeterminesthefeasibilityoftheloanarrangementbetweentheborrowerandthelender,which,inturn,isconclusiveaboutthepossibilityofinvestmentdistortion.Theborrower'sconsumptionpreferencehasnoeectontheoccurrenceofinvestmentdistortion.However,thewealth-abundantborrowernotonlyfocusesontheforcenotedabovebutalsoweighsthecertaintyequivalentcashowoftheinvestmentandtheentirepersonalcostofimplementingsuchinvestment,whichistheriskinessoftheinvestmentitself,todeterminewhethertheprojectisundertaken.Theborrower'sintertemporalconsumptionpreferenceisnolongerirrelevantintheinvestmentdecision.Theconditionsunderwhichinvestmentdistortionarisesfortheborrowerwithaggregateconsumptionpreferenceimplythosefortheborrowerwithtime-additiveconsumptionpreferencebutnotviceversa,whichsuggeststhattheinvestmentdistortionproblemisexacerbatedwhenthewealth-abundantborrowerhasintertemporalconsumptionstructure. However,astheborrower'sriskpreferencechanges,theassociatedpotentialinvestmentdistortionproblemdiers.TheDARA-borrowerexperienceslesssevereinvestmentdistortionthantheCARA-borrowerbecausetheincreaseinhispersonalwealthmakeshimlessriskaverse;therefore,otherthingsbeingequal,theloancontractbetween 85

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WhentheCARA-borrowerhasprivateinformationabouthispersonalwealth,theRevelationPrincipleapplieswhichmotivatestheCARA-borrowertotruthfullyreporthisprivateinformation.AnaivecontractwhichignorestheCARA-borrower'sprivatesourceofinformationandtreatsitaspublicinformationisafeasibleandoptimalloanagreement.TheprivateinformationcaseoftheDARA-borrowerremainsanopenissue,thoughthenaivecontractretainsattractivetruthincentives. Inclosing,afewissuesareraised.Itisassumedthattheborrowerhasallthebargainingpowerwhiletheroleofthelenderisquitepassive.Inreal-worldsituations,theborrower,especiallythewealth-constrainedborrower,tendstobethepartythatdoesnothavemarketpowerinthenancialmarket.Therefore,whetherandhowtheresultsofthisdissertationchangeunderalternativebargainingstructuresisstillanopenbutinterestingquestion.Second,thepossibilityofover-reportingwhentheborrower'spersonalwealthisprivatelyobservedwarrantsinvestigation.Over-reportingstrategyinthepresenceofprivateinformationisparticularlyvaluabletothewealth-constrainedborrower,byborrowingtoposttherequiredup-frontbond,theborrowerisabletosecuretheloanagreementandimplementtheinvestmentproject. 86

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1 andCorollary 1 where=pHexp(aH)pL RearrangingtermsoftheconstraintsprovidesCHCL+1 (A-3)CHW+H(1pH)CL IfW+HpH (A-5) GiventhelimitedliabilityconstraintthatCL0,if( A-5 )holds,theconstraintsetof( A-3 )and( A-4 )isneversatised;whichbycontradiction,suggeststhatthereexistsnofeasiblesolutionifW+HpH Therefore,[ProgramSB]hasfeasiblesolutionif (A-6) Assumecondition( A-6 )holds,thefollowinganalysisshowstheexistenceofthesolutionanditscharacterizations. Notethattheobjectivefunctionin[ProgramSB]isconcaveandcontinuous,givencondition( A-6 ),theconstraintsetiscompact,thenbasedontheWeierstrassTheorem,thereexistsasolutionfor[ProgramSB]. 87

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A-1 )isnotbinding,thelenderisgettingrentfromherrepayment,whichcontradictsacompetitivenancialmarketassumption.If( A-2 )isnotbinding,therst-bestcontractconsumptionscheduleCH=CLapplies,whichleadstoaviolationof( A-2 )foranyaH>0.Thenthesolutionfor[ProgramSB]issolvedfromthefollowingsetofequationspHCH+(1pH)CL=W+HCL+1 (A-7)CL=W+HpH (A-8) Withsolution( A-7 )and( A-8 ),therst-orderconditionswithrespecttoCHandCLarebothzero,whichimpliesthat( A-7 )and( A-8 )areinteriorsolutions.QED. 2 1 andCorollary 1 Partb.LetCESBandCERUdenotetheborrower'scertaintyequivalentconsumptionfromthesecond-bestcontractandfromconsuminghispersonalwealthathand,respectively.GivenProposition 1 'soptimalsecond-bestcontract,CESBandCERUarecharacterizedasCESB=W+HpH (A-9)CERU=W where=pHpL 88

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IfHaH
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(A-11) whereCE(X)=1 GivenCERU=W 2 ,wecompareno-lenderinvestmentwithno-investment.Notethatthepersonalcostofeortisconsideredfortheformer.Then,ifCE(X)CESBaH>CENLaH 4 andCorollary 3

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where=pHexp(aH)pL RearrangingtermsoftheconstraintsprovidesCAHCAL+1 (A-14)CAHW+HCA0(1pH)CAL IfW+HpH (A-16) GiventhelimitedliabilityconstraintthatCA0;CAL0,if( A-16 )holds,theconstraintset( A-14 )and( A-15 )isneversatised,whichbycontradiction,suggeststhatthereexistsnofeasiblesolutionifW+HpH Therefore,[ProgramSBA]hasfeasiblesolutionif (A-17) Assumecondition( A-17 )holds,thefollowinganalysisshowstheexistenceofthesolutionanditscharacterizations. Notethattheobjectivefunctionin[ProgramSBA]isconcaveandcontinuous,givencondition( A-17 ),theconstraintsetiscompact,thenbasedontheWeierstrassTheorem,thereexistsasolutionfor[ProgramSBA]. Notethat>1;itthenfollowsfromtheborrower'sICAconstraintthatCAHCAL+1 91

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Inequilibrium,bothconstraintsin[ProgramSBA]arebinding.If( A-12 )isnotbinding,thelenderisgettingrentfromherrepayment,whichcontradictsacompetitivenancialmarketassumption.If( A-13 )isnotbinding,therst-bestcontractconsumptionscheduleCA0=CAHaH=CALaLapplies,whichleadstoaviolationof( A-13 )foranyaH>0.Therefore,wehaveCAL+1 SubstitutingCAHin( A-18 )into( A-19 )andsolvingforCA0 Given( A-18 )and( A-20 ),theborrower'speriodt=0expectedutilityis Substituting,theborrower'sproblembecomes [ProgramSBA0]maxCAL0expW+HpH

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(A-23)CALL0CAL=0 (A-24) where( A-24 )isthecomplementaryslacknesscondition.Solving( A-23 )and( A-24 )yieldstheoptimalsolutionofCAL: IfW+HpH 2;L0CAL=0 (A-25) Otherwise (A-26) where=pHpL Therefore,( A-25 )and( A-26 )aretheinteriorandcornersolutionsforCAL,respectively.SubstitutingCALin( A-25 )into( A-21 )and( A-22 ),( )=( ) Moreover,substitutingCALin( A-25 )into( A-19 )and( A-20 ),optimalconsumptionfor[ProgramSBA]isCA0=W+HpH 2CAH=W+HpH 2+1 2 NotethataH+1 A-26 )into 93

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A-19 )and( A-20 ),theoptimalcornerconsumptionfor[ProgramSBA]isCA0=W+HpH Notethatfortheaboveconsumptionschedule,given( A-17 ),theborrower'slimitedliabilityconstraintsareallsatised.Moreover,therst-orderconditionwithrespecttoCA0for[ProgramSBA]is0;whichimpliesthattheoptimalCA0isaninteriorsolution.QED. 4 a. b. HaH
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Whenthesecond-bestloancontractprovidescornerconsumption,giventheconcavityoftheborrower'sutilityfunction,hisexpectedutilityisstrictlylowerthanthatunderinteriorconsumption.Therefore,ceterisparibus,ifHaH1and<1,thenpH (A-32) However,given( A-32 ),inequalities( A-30 )and( A-31 )cannotcoexist.ThisimpliesthatgivenHaHpH Forthesecond-bestinteriorconsumption,investmentdistortionisnotpresentwhenexp(CA0)exp(W thenbasedonCorollary 3 and( A-29 ),exp(CA0)pHexpCAHaH(1pH)expCALaH2exp(W whichindicatesaabsenceofinvestmentdistortion.QED. 95

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5 4 Thefollowingderivestheconditionsunderwhichtheborrowerprefersno-investmenttono-lenderinvestment.Theborrower'sno-lenderinvestmentprogramwhenhehastime-additiveconsumptionpreferenceis [ProgramNLA]maxCA00;CAH0;CAL0exp(CA0)pHexpCAHaH(1pH)expCALaHs.t.CA0WICAHWCA0I+XHCALWCA0I+XL (A-33)LCAL=(1pH)expCALaH3+6=0 (A-34)Rearrangingtermsfor( A-33 )and( A-34 ),givennon-negativemultipliers,2;3>0 Thentheborrower'sconsumptionconstraintsatt1arebothbinding,i.e.,CAH=WCA0I+XH 96

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SubstitutingthebindingCAHandCALintotheobjectivefunction,theprogrambecomesmaxCA00exp(CA0)+expCA0exp[(WIaH)]EU(X)s.t.CA0WI whereEU(X)=(1pH)exp(XL)pHexp(XH).TakingthederivativeandsolvingforCA0, 2 (A-38) whereCE(X)=1 Given( A-37 ),( A-38 )onlyappliesif Otherwise,for itfollowsthatCA0=WI A-39 )holds,given( A-35 ),( A-36 ),and( A-38 )theborrower'soptimalconsumptionisderivedasCA0=WIaH+CE(X) 2CAH=WI+aHCE(X) 2+XHCAL=WI+aHCE(X) 2+XL

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2RUA=2expW 2
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A-40 )hastohold,i.e.,CE(X)+IaH>W.Summarizingcase2impliesthatforXL0,investmentdistortionarisesif 2WI+aH,nosolutionexistsfortheborrower'sno-lenderinvestmentbehavior.WhenCE(X)WI+aH,investmentdistortionanalysisfollowstheanalysisofCase1above. Second,ifXL
PAGE 100

NotethatatW=I,theborrowerconsumesthestochasticcashowsfromhisinvestment;therefore,letCE(X)denotetheborrower'scertaintyequivalentconsumptionatW=I,thenitfollowsthatp ForanyWI,giventheborrower'sdecreasingabsoluteriskaversionpreference,itisalwaysthecasethatif( A-41 )holds,thefollowingrelationshipistrue:EUDNL(WI)p

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A-42 )isthesucientconditionfortheborrowertoengageinno-lenderinvestmentovernoinvestmentintheabsenceofthelenderforanyWI.QED. 5 :DirectlyfollowstheProofofProposition 3 inChapter3andtheProofofProposition 7 above.QED. 8 : whichcanbesimpliedasp whereCE(X)followsthedenitionintheProofofProposition 7 anddenotestheborrower'scertaintyequivalentconsumptionatt=1.Therefore,atW=I,investmentdistortionarisesif 101

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A-43 )issucienttoavoidinvestmentdistortionintheabsenceofthelender.QED. 6 : [ProgramSBD]maxbCHaH;bCLaHpHq bCHaH+(1pH)q bCLaHs.t.pHbCH+(1pH)bCL=W+HpLq bCH+(1pL)q bCL=pHq bCHaH+(1pH)q bCLaH bCHaH+(1pH)s TakingthetotalderivativewithrespecttoaHyieldsdobj daH=1 224pH1 bCHaH+(1pH)1 2pH0@1 bCHaH1 daH<0 102

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A-44 )withrespecttoWimpliesthatdobj dW>0 QED. 9 AndtheCARA-borrower'sexpectedutilitiesfromeachreportingandeortsupplystrategyare: 103

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(A-47)CLfWHfWHhID0fWHi+XLDL(fWH) (A-48)CHfWLfWLhID0fWLi+XHDH(fWL) (A-49)CLfWLfWLhID0fWLi+XLDL(fWL) (A-50) Notethatfrominequalities( A-47 )and( A-48 ),thefollowingrelationshipholds: 104

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A-45 ),( A-51 )becomes Similarly,given( A-49 )and( A-50 ),togetherwith( A-46 ),wehave Therefore,theCARA-borrower'sconsumptionconstraints( A-47 )to( A-50 )andthelender'sindividualrationalityconstraints( A-45 )and( A-46 )becometheCARA-borrower'sexpectedconsumptionconstraints( A-52 )and( A-53 ). IgnoringtheCARA-borrower'slimitedliabilityconstraints,let160betheLagrangianmultipliersfortheCARA-borrower'sfourincentivecompatibilityconstraintsandtheexpectedconsumptionconstraints( A-52 )and( A-53 ).TakingthederivativesoftheLagrangianfunctionwithrespecttotheCARA-borrower'sconsumptionwhenhighwealthisreportedCfWH,wehave (1pH)=5 (1pH) However,thisiscontrarytotheassumptionthatpH>:5;therefore,5>0.Giventhat5>0,if2=0,wehaveexphCHfWHiexp(aH)(+1+3)=5exphCLfWHiexp(aH)(+1+3)=5

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where=pHexp(aH)pL Similarly,takingthederivativesoftheLagrangianfunctionwithrespecttotheCARA-borrower'sconsumptionwhenlowwealthisreportedCfWL,wederivethat4;6>0,whichimpliesthatthefourthincentivecompatibilityconstraintand( A-53 )arebinding:EUWL;fWL;aH=EUWL;fWL;aLpHCHfWL+(1pH)CLfWL=fWL+H 106

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GiventhatfWH>fWLW Notethattheoptimalsolutionsfor[ProgramgSB]areexactlythesameastheoptimalconsumptionfromthenaivecontractwhentheCARA-borrower'spublicobservablepersonalwealthareWHandWL,respectively.Therefore,thenaivecontractisoptimalfor[ProgramgSB].QED. 107

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LiangFuearnedherBachelorofArtsinInternationaleconomicsfromSouthwesternUniversityofFinanceandEconomicsin1998.SheearnedherMasterofArtsineconomicsfromKentStateUniversityin2002.ShejoinedthePh.D.programofFisherSchoolofAccountingattheUniversityofFloridaandshereceivedherdegreeofDoctorofPhilosophyinAccountingin2009. 110