高级国际金融学教程习题答案2

FoundationsofInternationalMacroeconomics1Workbook2MauriceObstfeld,KennethRogoff,andGitaGopinathChapter2Solutions1.(a)ThecurrentaccountidentitycanbewrittenasBs+1=(1+r)Bs+TBs.Nowjustplugintheassumedtradebalancerule.(b)Usingtheanswertoparta,foranyξ0,−∞Xs=tµ11+r¶s−tξrBs=−∞Xs=tµ11+r¶s−tξr[1+(1−ξ)r]s−tBt=−ξrBt1−1+(1−ξ)r1+r=−(1+r)Bt.(c)Undertheruleabove,debtgrowswithoutboundifξ1.ButoncethedebtisasbigasY/r,thecountrycanhonoritsforeigncommitmentsonlyifdebtstopsgrowingandconsumptioniszeroforever.Thus,thesuggestedrulemustentailnegativeconsumptionlevelsatsomepoint,whichareinad-missible.Toseedirectlywhy,considertheconstant-outputcase,inwhichTBs=Y−Cs=−ξrBssothatthepaybackruleimpliesCs=Y+ξrBs.NoticethatsinceBs→−∞,Csmustatsomepointbecomenegative.Therulethereforeisconsistentwithintertemporalsolvencyonlyifwecounterfac-tuallyallowfornegativeconsumptionlevels:thepriceofhighconsumptiontodaywouldbeinfeasiblyhightradesurpluseslateron.Ingeneral,supposeoutputgrowsatthegrossrate1+g,sothatYs=(1+g)s−tYt.Unless1+g1ByMauriceObstfeld(UniversityofCalifornia,Berkeley)andKennethRogoff(Prince-tonUniversity).c°MITPress,1996.2c°MITPress,1998.Version1.1,February27,1998.Foronlineupdatesandcorrec-tions,seeffBook.html14isatleastasgreatasthegrossgrowthrateofdebt,whichwasshowntobe1+r(1−ξ)inparta,theexternaldebt-outputratioisunbounded.Thustheminimalpaybackfractionξconsistentwithintertemporalsolvencyandpositiveconsumptionisξ=1−(g/r)(whichispositiveifweassumethatgr).2.(a)TheexpectedutilityEtUtisaweightedaverageoverdifferentlifespans,withweightsequaltothesurvivalprobabilities:EtUt=(1−ϕ)[u(Ct)]+ϕ(1−ϕ)[u(Ct)+βu(Ct+1)]++ϕ2(1−ϕ)hu(Ct)+βu(Ct+1)+β2u(Ct+2)i+....(b)Theresultfollowssimplybyexpandingtheexpressioninpartaandgroupingtermstogether.3.Recallthatwithisoelasticutility,u(C)=C1−1σ1−1σ,σ0,logC,σ=1.UsingtheintertemporalEulerequation,wethusobtain,(1+r)β=1=1Et(µCt+1Ct¶−1/σ).(1)Sinceconsumptionhasaconditionallognormaldistribution,thenaturallogofthegrossconsumptiongrowthrateisconditionallynormallydistributed:logCt+1Ct∼NµEt½logCt+1Ct¾,Vart½logCt+1Ct¾¶.ThusEtµCt+1Ct¶−1σ=Et½exp·−1σlogµCt+1Ct¶¸¾=exp·−1σEt½logCt+1Ct¾+12σ2Vart½logCt+1Ct¾¸.(2)15[Consultfootnote41onp.313ofthebook.Equation(2)followsfromcom-putingthemeanandvarianceoftherandomvariable(−1/σ)log(Ct+1/Ct),whichisnormallydistributedwhenlog(Ct+1/Ct)is.]Combiningeqs.(1)and(2)aboveandtakingnaturallogsoftheresult,wearriveatEt½logCt+1Ct¾=12σVart½logCt+1Ct¾orlogCt+1−logCt=12σVart{εt+1}+εt+1,whereεt+1≡logCt+1−Et{logCt+1}.Sinceεt+1isanormalrandomvari-ablethatisuncorrelatedwithpastinformation(becauseitisapurefore-casterror),itisalsostatisticallyindependentofthatinformationontheassumptionthatthepastinformationitselfisgeneratedbyajointlynormal(i.e.,Gaussian)stochasticprocess.Inthatcasetheconditionalvarianceintheprecedingequationactuallyisatime-invariantconstant,sothenaturallogofconsumptionfollowsarandomwalkwithaconstantdriftequalto12σVar{εt+1}.4.(a)Usingeq.(32)inChapter2,wecanwriteCt+1−Ct=r(Bt+1−Bt)+r1+r∞Xs=t+1µ11+r¶s−(t+1)Et+1Ys−∞Xs=tµ11+r¶s−tEtYs.ThecurrentaccountidentitygivesBt+1−Bt=Yt+rBt−Ct=Yt−r1+r∞Xs=tµ11+r¶s−tEtYs,whichcanbesubstitutedintothepreviousequationforconsumptiontogivetheresultthatthechangeinconsumptionequalsthepresentvalueofchangesinexpectedfutureoutputlevels.16(b)IftheprocessforoutputfollowsYt+1−Yt=ρ(Yt−Yt−1)+²t+1,then(Et+1−Et)Yt+1=²t+1,(Et+1−Et)Yt+2=(1+ρ)²t+1,(Et+1−Et)Yt+3=(1+ρ+ρ2)²t+1,andsoon.Therefore,forst,(Et+1−Et)Ys=1−ρs−t1−ρ²t+1.(c)Substitutingthelastexpressionintotheequationforthechangeincon-sumptionderivedinparta,wegetthefollowingCt+1−Ct=r1+r²t+1Ã11−ρ!+Ã11−ρ!11+r+...−ρ1−ρ−ρ2(1+r)(1−ρ)−...#=r1+r²t+1(1+r)(1−ρ)r−ρ1−ρÃ1+r1+r−ρ!#=1+r1+r−ρ²t+1.(3)Asaresult,providedthat0ρ1,thedesiretosmoothconsumptionmakesconsumptioninnovationsmorevariablethanoutputinnovations.(d)Thecurrentaccountidentityfordatet+1isCAt+1=Bt+2−Bt+1=Yt+1+rBt+1−Ct+1.BecauseYt+1−EtYt+1=²t+1and,byeq.(3)frompartcabove,Ct+1−EtCt+1=Ct+1−Ct=1+r1+r−ρ²t+1,theprecedingcurrentaccountidentitygivesacurrentaccountinnovationof²t+1−1+r1+r−ρ²t+1=−ρ1+r−ρ²t+10.17Thus,apositiveoutputinnovationleadstoacurrentaccountdeÞcit,asclaimedattheendofsection2.3.3inthebook.5.WorkbackwardfromtheequationCAt+1−∆Zt+1−(1+r)CAt=²t+1,where²t+1isuncorrelatedwithdatetorearlierinformation.Takingexpec-tationswithrespecttodatetinformationyieldsEtCAt+1−Et∆Zt+1−(1+r)CAt=0.ThepreviousequationcanberearrangedtoexpressCAtasCAt=11+rEtCAt+1−11+rEt∆Zt+1.Throughforwardrecursivesubstitution(andusingthelawofiteratedcondi-tionalexpectations)weobtainCAt=−∞Xs=t+1µ11+r¶s−tEt∆Zs[becauseasj→∞,³11+r´jEtCAt+j→0].ThisisCampbell’s(1987)“savingforarainyday”equation,eq.(43)inChapter2.Theequationcanalterna-tivelybederivedusingthelagandleadoperatormethodologydescribedinsupplementCtoChapter2.StartagainwithCAt+1−∆Zt+1−(1+r)CAt=²t+1andtakeexpectationswithrespecttodatetinformationtogetEtCAt+1−Et∆Zt+1−(1+r)CAt=0.UsingtheleadoperatorwewritethisasL−1CAt−L−1∆Zt−(1+r)CAt=0,18or,dividingby1+randrearranging,asµ1−11+rL−1¶CAt=−11+rL−1∆Zt.Inversionofthelagpol