麻省理工行为金融学讲义14127_lec4_noise

14.127Lecture4XavierGabaixFebruary26,20041BoundedRationalityThreereasonstostudy:•Hopethatitwillgenerateaunifiedframeworkforbehavioraleconomics•Somephenomenashouldbecaptured:difficult­easydifference.ItwouldbegoodtohaveametricforthatArtificialintelligence•Warning—alotofeffortspendonboundedrationalitysinceSimonandfewresults.Threedirections:Analyticalmodels•—Don’tgetallthefinenuancesofthepsychology,butthosemodelsaretractable.Processmodels,e.g.artificialintelligence•—Rubinsteindirection.SupposeweplayNash,givenyourreactionfunc­tion,mystrategyoptimizesonbothoutcomeandcomputingcost.Rubinsteinprovessomeexistencetheorems.Butitisverydifficulttoapplyhisapproach.•Psychologicalmodels—Thosemodelsaredescriptivelyrich,buttheyareunsystematic,andoftenhardtouse.Human­computercomparisonHumanmind1015operationspersecond••Computer1012operationspersecondMoore’slaw:every1.5yearscomputerpowerdoubles•Thus,every15yearscomputerpowergoesup103•Ifwebelievethis,thenin45yearscomputerscanbe106morepowerfulthanhumans•Ofcourse,we’llneedtounderstandhowhumanthink•1.1AnalyticalmodelsBoundedRationalityasnoise.Consumerseesanoisysignalq˜=q+σε•ofquantity/qualityq.BoundedRationalityasimperfectmonitoringofthestateoftheworld.•Peopledon’tthinkaboutthevariablesallthetime.Theylookupvariablekattimest1,...,tnBoundedRationalityasadjustmentcost.Callbyθtheparametersofthe•world.—NowIamdoinga0andκ=costofdecision/change—Ichangemydecisionfroma0toa∗=argmaxu(a,θt)iffu(a∗,θt)−u(a0,θt)κ1.1.1ModelofBoundedRationalityasnoiseRandomutilitymodel—Luce(psychologist)andMcFadden(econometri­•cianwhoprovidedeconometrictoolsforthemodels)—ngoods,i=1,...,n.—Imaginetheconsumerchoosesmaxqi+σiεii—What’sthedemandfunction?Definition.TheGumbeldistributionGis•F(x)=P(εx)=e−e−xandhavedensityf(x)=F′(x)=e−e−x−x.���IfεhastheGumbeldistributionthenEε=γ0,whereγ≃0.59isthe•Eulerconstant.•Proposition1.SupposeεiareiidGumbel.ThenPmaxεi+qi≤lnn+q∗+x=e−e−xi=1,..,nwithq∗definedaseq∗=1eqi.ThismeansthatnMn=maxεi+qi=dlnn+q∗+ηi=1,..,nandηisaGumbel.��ProofofProposition1.CallI=Pmaxi=1,..,nεi+qi≤y.•Then•I=P((i)εi+qi≤y)=Πin=1P(εi+qi)≤y∀Thus,•lnI=�P(εi+qi≤y)andlnP(εi+qi≤y)=lnP(εi≤y−qi)=−e−(y−qi).�Thus•eqilnI=�(y−qi)=−e−y−e−Using•eq∗1�eqi=nwehavelnI=−e−yneq∗=[y−lnn−q∗]−e−whichprovesthatIisaGumbel.QED����DemandwithnoiseDemandforgoodn+1equals•Dn+1(q1,...,qn+1)=Pmaxεn+1+qn+1i=1,..,nεi+qiwhereqiistotalquality,includingthedisutilityofprice.•Proposition2.eqn+1Dn+1(q1,...,qn+1)=�n+1.i=1eqiIngeneral,eqjDj=Pεj+qjmaxεi+qi=�n+1i=ji=1eqi��ProofofProposition2.�n+1Observethatj=1Dj=1.•Note•Dn+1(q1,...,qn+1)=Pmaxεn+1i=1,..,nεi+qi′whereq′=qi−qn+1.iThus,•Dn+1(q1,...,qn+1)=Ee−e−(εn+1−lnn−q∗)������Calla=lnn−q∗.Then•−Dn+1(q1,...,qn+1)=Ee−e−(εn+1+a)(x+a)e−e−x−xdx=e−e−(x+a)f(x)dx=e−e−(x+a)−e−x−xdx=e−e−x(e−a+1)−xdx=e−e−CallH=1+e−aandre­writetheaboveequationas•Dn+1(q1,...,qn+1)e−e−x−lnH=−xdxe−e−x−lnH(x−lnH)lnHdx=−e−�Notethat�b ��b•e−e−y−ydy=e−e−yaaThus• Dn+1(q1,...,qn+1)=e−lnHe−e−x−lnH�+∞dx1 =1=1= =1−∞H 1+e−a1+elnn+q∗1+neq∗1=eqn+1eqn+1= =ieqn+1+eqn+1�in=1eqi′�n+11+�in=1eq′ i=1eqiQEDDemandwithnoisecont.Thisiscalled“discretechoicetheory”.•—ItisexactforGumbel.—ItisasymptoticallytrueforalmostallunboundeddistributionsyoucanthinkofflikeGaussian,lognormal,etc.��Dividingtotalqualityintoqualityandpricecomponents•D1=Pq1−p1+σε1maxii=2,...,nq−pi+σεiwhereεiareiidGumbel,σ0.Then•�q1−p1eσiD1=Pq1−p1+ε1maxqi−p+εi�=�nqi−piσi=2,...,nσσi=1eThisisveryoftenusedinIO.•Optimalpricing.Anapplication—example•Supposewehavenfirms,n≫1.Firmihascostcianddoes•max(pi−ci)Di(p1,...,pn)=πii�����•Denotetheprofitbyπiandnotethatqj−pji�lnπi=ln(pi−ci)+qi−p+ln�eσσandσ∂11qi−pi∂pilnπi=−c−σ+−e−qj−pjpiieσ11=piin−c−σ+O�1So11•pii−c−σ≃0andunitprofitspi−ci=σThusdecisionnoiseisgoodforfirms’profits.SeeGabaix­Laibson“Com­•petitionandConsumerConfusion”Evidence:cardealerssellcarsforhigherpricestowomenandminorities•thantowhitemen.Reason:differenceinexpertise.Thereislotsofotherevidenceofhowfirmstakeadvantageofconsumers.SeepaperbySusanWoodwardonmortgagerefinancingmarkets:unsophisticatedpeoplearechargedmuchmorethansophisticatedpeople.��Whataboutnon­Gumbelnoise?Definition.AdistributionisinthedomainofattractionoftheGumbelif•andonlyifthereexistsconstantsAn,BnsuchthatforanyxlimPmaxεi≤An+Bnx=e−e−x.i=1,...,nn→∞whenεiareiiddrawsfromthegivendistribution.Fact1.Thefollowingdistributionsareinthedomainofattractionofa•Gamble:Gaussian,exponential,Gumbel,lognormal,Weibull.Fact2.Boundeddistributionsarenotinthisdomain.•Fact3Powerlawdistributions(P(εx)∼x−ζforsomeζ0)are•notinthisdomain.��Lemma1.FordistributionsinthedomainofattractionoftheGumbel¯•F(x)=P(εx)takeF=1−F(x)=P(ε≥x),andf=.F′ThenAn,BnaregivenbyF¯(An)=1n1Bn=nf(An)Lemma2•limPmaxεi+qi≤An+Bny+q∗=e−e−yi=1,...nn→∞nwitheq∗/Bn1�eqi/Bnn=n��•Proposition.D1=Pq1−p1+σε1maxii=2,...,nq−pi+σεi¯Forn→∞,limD1/D1=1whereq1−p1/Bnσ¯D1=eσqi−pi/Bnσ≃D1.�nσi=1e���•Example.Exponentialdistributionf(x)=e−(x+1)forx−1andequals0forx≤−1.then,forx−1F¯(x)=P(εx)=x∞e−(x+1)dy=(x+1)∞=e−(x+1)=f(x).−e−xThusF¯(An)=1,nandAn=−1+lnnandBn=nf(1An)=1��=F¯=1√2πe−s22•Example2.Gaussian.f(x)(x)x∞ds.Forlargex,,e−x22√2.πxRe