NotesonMacroeconomicTheory - Yale University

1 ,IA52242 August1999 Chapter1 SimpleRepresentativeAgentModelsThischapt erdealswiththemostsimplekindofmacroecono micmodel, ,westudyaneconomyconsistingofarepresen-t ative ,thisisequivalent,undersomecircumstances ,tostudyinganeconomywithmanyidentical ,asinallthemodelswewillstudy,economicage ntsoptimize, ,thetechnologyavailableto rms,andtheendowmentsofresourcesavailable toconsumersand rms,combinedwithoptimizingbehaviorandsom enotionofequilibrium, ,theequilibriumconceptwewilluseiscompeti -tiveequilibrium, ,endowments,andtechnologyThereisoneperio dandNconsumers,whoeachhavepreferencesgiv enbytheutilityfunctionu(c;`);whereciscon sumptionand` ,u( ; )isstrictlyincreasingineachargument,stri ctlyconcave, ,assumethatlimc!

2 0u1(c;`)=1;`>0;andlim`!0u2(c;`)=1;c>0:He re,ui(c;`)isthepartialderivativewithresp ecttoargumentiofu(c;`):Eachconsumerisend owedwithoneunitoftime, ,whichcanberentedto rms,whicheachhaveatechnologyforproducing consumptiongoodsaccordingtoy=zf(k;n);whe reyisoutput,kisthecapitalinput,nisthelab orinput, ,thefunctionf( ; )isstrictlyincreasinginbotharguments,str ictlyquasiconcave,twicedi erentiable, ,produc-tionisconstantreturnstoscale,sot hat y=zf( k; n);( )for >0:Also,assumethatlimk!0f1(k;n)=1;limk!1 f1(k;n)=0;limn!0f2(k;n)=1;andlimn!1f2(k; n)=0 ,wecanatmostdetermineallrelativeprices, ,consumption,leisure, ;andtherentalrateoncapital(again,inunits ofconsumption)isr:Consumer'sProblemEachc onsumertreatswasbeing xed, ,eachsolvesmaxc;`;ksu(c;`) w(1 `)+rks( )0 ks k0N( )0 ` 1( )c 0( )Here,ksisthequantityofcapitalthatthecon sumerrentsto rms,( )isthebudgetconstraint,( )statesthatthequantityofcapitalrentedmus tbepositiveandcannotexceedwhattheconsume risendowedwith,( )isasimilarconditionforleisure,and( ) ,giventhatutilityisincreasinginconsumpti on(moreispre-ferredtoless),wemusthaveks= k0N.

3 And( ) ,andinequi-libriumwewillneverhave`=1;ast hennothingwouldbeproduced, ed, (c;`)+ (w+rk0N w` c);where rst-orderconditions.@L@c=u1 =0@L@`=u2 w=0@L@ =w+rk0N w` c=0 Here,uiisthepartialderivativeofu( ; )withrespecttoargumenti:Theabove rst-orderconditionscanbeusedtosolveoutfo r andctoobtainwu1(w+rk0N w`;`) u2(w+rk0N w`;`)=0;( ) ,`;intermsofw;r;andk0N:Equation( )canberewrittenasu2u1=w; , ,theconsumer'sbudgetconstraintisABD,andh e/shemaximizesutilityatE,wherethebudgetc onstraint,whichhasslope w;istangenttothehighestindi erencecurve,whereanindi erencecurvehasslope u2u1:Firm'sProblemEach rmchoosesinputsoflaborandcapitaltomaximi zepro ts,treat-ingwandrasbeing ,a rmsolvesmaxk;n[zf(k;n) rk wn].

4 Andthe rst-orderconditionsforanoptimumarethemar ginalproductconditionszf1=r;( )zf2=w;( )wherefidenotesthepartialderivativeoff( ; )withrespecttoargu-menti:Now,giventhatth efunctionf( ; )ishomogeneousofdegreeone,Euler' ,di erentiating( )withrespectto ;andsetting =1;wegetzf(k;n)=zf1k+zf2n:( )Equations( ),( ),and( )thenimplythatmaximizedpro rstisthatwedonotneedtobeconcernedwithhow the rm'spro tsaredistributed(throughsharesownedbycon sumers,forexample).Secondly,supposek andn areoptimalchoicesforthefactorinputs,then wemusthavezf(k;n) rk wn=0( ) andn=n :But,since( )alsoholdsfork= k andn= n forany >0;duetotheconstantreturnstoscaleassumpt ion,theoptimalscaleofoperationofthe erenceforouranalysistosimplyconsiderthec aseM=1(asingle,representative rm),asthenumberof ,c;`;n;k;andpriceswandr; ` ,therearethreemarkets:thelabormarket,the marketforconsumptiongoods, ,given(3), (1 `)=n( )y=Nc( )k0=k( )Thatis, ,thetotalvalueofexcessdemandacrossmarket sisNc y+w[n N(1 `)]+r(k k0).

5 Butfromtheconsumer'sbudgetconstraint,and thefactthatpro tmaximizationimplieszeropro ts,wehaveNc y+w[n N(1 `)]+r(k k0)=0:( )Notethat( )wouldholdevenifpro tswerenotzero, ,ifany2of( ),( ), ( )hold,then( ) ( )issimplyWalras' 'lawstatesthatthevalueofexcessdemandacro ssmarketsisalwayszero,andthisthenimplies that,ifthereareMmarketsandM 1ofthosemarketsareinequilibrium, ,weeliminate( ).Thecompetitiveequilibriumisthenthesolu tionto( ),( ),( ),( ),and( ).Theseare veequationsinthe veunknowns`;n,k;w;andr;andwecansolveforc usingtheconsumer' ,N;isvirtuallyirrelevanttotheequilibrium solution,soforconveniencewecansetN=1, rm,butaswehaveshown,inthiscontextourresu ltswouldnotbeanydi erentifthereweremany ( )toobtainanequationwhichsolvesforequilib rium`:zf2(k0;1 `)u1(zf(k0;1 `);`) u2(zf(k0;1 `);`)=0( )Giventhesolutionfor`;wethensubstitutein thefollowingequationstoobtainsolutionsfo rr;w;n;k,andc:zf1(k0;1 `)=r( )zf2(k0;1 `)=w( )n=1 `k=k0c=zf(k0.)

6 1 `)( )Itisnotimmediatelyapparentthatthecompet itiveequilibriumexistsandisunique, ,generally,isde nedtobesomeallocation(anal-locationbeing aproductionplanandadistributionofgoodsac rosseconomicagents)suchthatthereisnoothe rallocationwhichsomeagentsstrictlyprefer whichdoesnotmakeanyagentsworseo .Here,sincewehaveasingleagent, ctitioussocialplannerwhocandictateinputs toproductionbytherepresenta-tive rm,canforcetheconsumertosupplytheappropr iatequantityoflabor,andthendistributesco nsumptiongoodstotheconsumer,allinawaytha tmakestheconsumeraswello ;`u(c;`)subjecttoc=zf(k0;1 `)( )Giventherestrictionsontheutilityfunctio n,wecansimplysubstituteusingtheconstrain tintheobjectivefunction,anddi erentiatewithrespectto`toobtainthefollow ing (k0;1 `)u1[zf(k0;1 `);`] u2[zf(k0;1 `).]

7 `]=0( )Notethat( )and( )areidentical,andthesolutionwegetforcfro mthesocialplanner'sproblembysubstituting intheconstraintwillyieldthesamesolutiona sfrom( ).Thatis, ,sinceu( ; )isstrictlyconcaveandf( ; )isstrictlyquasiconcave,thereisauniquePa retooptimum, ( )aszf2=u2u1;wheretheleftsideoftheequatio nisthemarginalrateoftransforma-tion, ,ABisequation( ) ,wherethehighestindi ,therepresentativeconsumerfacesbudgetcon straintAFGandmaximizesatpointDwherethesl opeofthebudgetline, w;isequalto u2u1:Inmoregeneralsettings, (FirstWelfareThe-orem). (SecondWelfareTheorem).Thenon-technicala ssumptionsrequiredfor(1)and(2)togothroug hincludetheabsenceofexternalities,comple tenessofmarkets,andab-senceofdistortingt axes( ).

8 TheFirstWelfareTheoremisquitepowerful,an dthegeneralideagoesbackasfarasAdamSmith' ,ifwecansuccessfullyexplainparticularphe nomena( )usingacompetitiveequilibriummodelinwhic htheFirstWelfareTheoremholds, , ,itcanbemucheasiertoobtaincom-petitiveeq uilibriaby rstsolvingthesocialplanner'sproblemtoobt aincompetitiveequilibriumquantities,andt hensolvingforprices, ,intheaboveexample,acompetitiveequilibri umcouldbeobtainedby rstsolvingforcand`fromthesocialplanner's problem,andthen ndingwandrfromtheappropriatemar-ginalcon ditions,( )and( ).Usingthisapproachdoesnotmakemuchdi erencehere,butincomputingnumericalsoluti onsindynamicmodelsitcanmakeahugedi ,weuseu(c;`)=c1 11 +`;where >0measuresthedegreeofcurvatureintheutili tyfunctionwithrespecttoconsumption(thisi sa\constantrelativeriskaversion"utilityf unction).

9 Notethatlim !1c1 11 =lim !1dd [e(1 )logc 1]dd (1 )=logc;usingL'Hospital' ,usef(k;n)=k n1 ;where0< <1:Thatis, 'sproblemhereisthenmax`([zk 0(1 `)1 ]1 11 +`);andthesolutiontothisproblemis`=1 [(1 )(zk 0)1 ]1 +(1 ) ( )Asinthegeneralcaseabove, ;from( ),wegetc=[(1 )1 (zk 0)]1 +(1 ) ;( )andfrom( ),wehavew=[(1 )1 (zk 0)] +(1 ) ( )From( )and( )clearlycandwareincreasinginzandk0:Thatis, ,fromequation( )thee (andthereforeonemployment) ectgoesdependsonwhether <1or >1:With <1;anincreaseinzorink0willresultinadecreaseinleisure,andanincreaseinemployment,butthee ectsarejusttheoppositeif >1:Ifwewanttotreatthisasasimplemodelofthe businesscycle,where uctuationsaredrivenbytechnologyshocks(ch angesinz); ,aggregateoutput,aggregateconsumption, , ,thattherealwagewillbeprocyclical(itgoes upwhenoutputgoesup), ,andshows,inasomewhatmoregeneralsensetha ntheaboveexample, ,weconsiderasimpli edtechnology,y=zn; , 'sprob-lemforthiseconomyisthenmax`u[z(1 `);`];andthe rst-orderconditionforamaximumis zu1[z(1 `);`]+u2[z(1 `).]

10 `]=0:( )Here,incontrasttotheexample,wecannotsol veexplicitlyfor`;butnotethattheequilibri umrealwageisw=@ `;applytheimplicitfunctiontheoremandtota llydi erentiate( )toget[ u1 z(1 `)u11+u21(1 `)]dz+(z2u11 2zu12+u22)d`=0:Wethenhaved`dz=u1+z(1 `)u11 u21(1 `)z2u11 2zu12+u22:( )Now,concavityoftheutilityfunctionimplie sthatthedenominatorin( )isnegative, ,itiseasytoconstructexampleswhered`dz>0; andwhered`dz<0:Theambiguityherearisesfromopposingincomeandsubstitutione ,ABdenotestheresourceconstraintfacedbythesocialplanner,c=z1(1 `);andBDistheresourceconstraintwithahigherlevelofproductivity,z2>z1:Asshown,thesocialoptimum(alsothecompe titiveequilibrium)isatEinitially,andatFa ftertheincreaseinproductivity,withnochan gein`buthigherc:E ectively,therepre-sentativeconsumerfaces ahigherrealwage,andhis/herresponsecanbed ecomposedintoasubstitutione ect(EtoG)andanincomee ect(GtoF).