Macroeconomic-ps2-solution-2014

ECON6022ProblemSet2SuggestedSolutionsFall2014November4,20141OptimalConsumptionwithFinancialWealthInlecture2,wesawanexamplewherethehouseholdonlylivesfortwoperiods.NowsupposethehouseholdlivesforNperiods,fromPeriod1toN.Householdreceivesanexogenousincomestreamfy1;y2;:::yNg,andaccumulatetheirnancialwealthatineachperiod.Thereturnonrealnancialwealthisr.Lethousehold'sproblembemaxctNt=1t1ln(ct)(1)subjecttoat+1=(1+r)at+ytct(2)ct0(3)aN+10(4)whereatisthenancialwealthatthebeginningofPeriodtanda1=0.1.Interpret(1)(2)(3)and(4).Solution:(1)meansthathouseholdmaximizehisdiscountedlifetimeutilitybyenjoyingconsumptionineachperiod.(2)meansthathouseholdreceivesincomeytandgetagrossreturnof1+roncurrentperiod'snancialwealthineachperiod.Aftercomsumptionofct,householdaccumulatsat+1unitsofnancialwealthtothenextperiod.(3)meansthatconsumptioncannotbenegative,and(4)meansthathouseholdcannotleavebehinddebtattheendofhislife,whichistheNo-Ponzicondition.2.Writedowntheinter-temporalbudgetconstraintandderivetheEulerequation(s).Solution:Theinter-temporalbudgetconstraintfortheconsumerisc1+c21+r+


+cN(1+r)N1=y1+y21+r+


+yN(1+r)N1(1)ThustheLagrangianofhousehold'soptimizationproblemcanbewrittenasL=NXt=1t1ln(ct)+
[y1+y21+r+


+yN(1+r)N1(c1+c21+r+


+cN(1+r)N1)](2)1Foranyt2f1;2;


;Ng,theFirstOrderConditionisFOC(ct):t1
u0(ct)=
1(1+r)t1(3)Thuswehavet
u0(ct+1)=
1(1+r)t(4)Byequ(3)(4)weobtaintheEulerequationu0(ct)=u0(ct+1)

(1+r)(5)or1ct=1ct+1

(1+r)(6)3.Assume=1,r=0.Solvefortheoptimalconsumptionforeachperiod,fctg,t=1;2;:::;N.Solution:With=1andr=0,wearriveattheconclusionthatforallt2f1;2;


;Ngu0(ct)=u0(ct+1)(7)whichimplies,ct=ct+1(8)Combiningequ(8)withthehousehold'sinter-temporalbudgetconstraint,c1+c2+


+cN=y1+y2+


+yN(9)Usingequ(8),weknowthefollowing,N
ct=y1+y2+


+yN(10)wecangettheoptimalconsumptionplanct=1NNXt=1yt(11)wheret=1;2;


;N4.Assume=1,r=0.WhatisthemarginalpropensitytoconsumeinPeriod1?Explaintheintuition.Solution:SinceweknowinPeriodt=1,c1=PNt=1ytNwhichmeansheorsheconsumesafraction(1N)ofpermanentincomeinPeriod1.Ifpermanentincomeincreasesbyoneunit,c1increasesby1N.Sothemarginalpropensitytoconsumeis1N.22PrecautionarySavingsInatwoperiodmodel,supposetheagent'slifetimeutilityfunctionisU(c1;c2)=u(c1)+u(c2),whereu(
)isaconcavefunction.Themarketinterestrateisconstant,r.Theagent'sincomearey1andy2inPeriod1and2,respectively.Theinitialwealthendowmentisw0.1.DerivetheEulerequationinthiscase.Solution:Theagent'sproblemismaxc1U(c1;c2)=u(c1)+u(c2)(12)subjecttoc1+c21+r=w0+y1+y21+r(13)Therst-orderconditionisu0(c1)=(1+r)

u0(c2)(14)whichistheEulerequation.2.Furtherassumethat(r+1)
=1andr=0,solvefortheoptimalconsumption(c1,c2)inPeriod1and2.Solution:If(r+1)
=1,theEulerequationwouldbereducedtou0(c1)=u0(c2)(15)Thenc1=c2=1+r2+r(w0+y1+y21+r):Ifr=0,c1=c2=w0+y1+y22.3.FurtherassumethatincomeinPeriod2isarandomvariable,~y2,whichtakestwovalues,yhandyl,withequalprobability,i.e.,E[~y2]=yh+yl2=y2.Whatistheagent'soptimalconsumptioninPeriod1,iftheutilityfunctiontakesthequadraticform,i.e.u(c)=
c12c2?Isthereanyprecautionarysaving?Whyorwhynot?Solution:Theagent'sproblembecomesmaxc1U(c1;c2)=u(c1)+[0:5u(c2h)+0:5u(c2l)](16)subjecttoc1+c2h1+r=w0+y1+y2h1+r(17)c1+c2l1+r=w0+y1+y2l1+r(18)3Therst-orderconditionisu0(c1)=(1+r)[0:5u0(c2h)+0:5u0(c2l)](19)If(r+1)
=1andr=0,thisconditionwouldbereducedtou0(c1)=0:5u0(c2h)+0:5u0(c2l)(20)Iftheutilityfunctionisu(c)=
c12c2,theoptimalconsumptioninPeriod1wouldsatisfyc1=0:5(ch2)+0:5(cl2);(21)wherech2=w0+y1c1+yh(22)cl2=w0+y1c1+yl:(23)Aftersimplifyingwehave,c1=c2=w0+y1+(yh+yl)=22=w0+y1+y22,whichisthesameasthecasewhenthereisnouncertainty.Therefore,precautionarysavingdoesnotexist.Thereasonisthatthethirdderivativeofthequadraticutilityfunctioniszero.4.(optional)Supposethat~y2isanormallydistributedrandomvariablewithmeany2andvariance2y.Whatistheagent'soptimalconsumptioninPeriod1iftheutilitytakestheCARA:u(c)=1a
eac?Isthereanyprecautionarysaving?Whyorwhynot?(Hint:Recallthatprecautionarysavingequalscertaintyequivalentconsumptionminusactualoptimalconsumption.Tondcertaintyequivalentconsumption,solvetheproblembyassuming~y2equalsy2withcertainty.ArandomvariableXislog-normallydistributed,iflog(X)isnormal.Iflog(X)isanormalrandomvariablewithmeanandvariance2,thenE(X)=exp(+22),whereexp(
)istheexponentialfunction.)Solution:Restatetheproblem,wehavec1=argmaxc1u(c1)+E(u(ec2));(24)subjectto,ec2=w0+y1c1+ey2(25)whereu(c1)+E(u(ec2))=1a
eac1+E(1a
eaec2)(26)=1a
eac11aE(eaec2)(27)Furtherwehave:log(eaec2)=a
ec2=a
(w0+y1c1+ey2)(28)Giveney2N(y2;2y),log(eaec2)N(a
(w0+y1c1+y2);a2
2y).Thenbythefactgivenwehave4thefollowing:E(eaec2)=ea
(w0+y1c1+y2)+a2
2y2(29)Therefore,theproblemtransformstothefollowing:c1=argmaxc11aeac11aea(w0+y1c1+y2)+a22y2(30)FOCgenerates:eac1=ea(w0+y1c1+y2)+a22y2(31)Orac1=a(w0+y1c1+y2)a22y2(32)RearrangetheFOC,wehave:c1=w0+y1+y22a2y4If~y2=y2withcertainty,thentheconsumptionallocationinthecertaintycase,c1=c2=w0+y1+y22(33)Sotheprecautionarysavingwouldbe:c1c1=w0