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首页 Mirrlees 1974-Notes on Welfare Economics, Inform…

Mirrlees 1974-Notes on Welfare Economics, Information and Uncertainty.pdf

Mirrlees 1974-Notes on Welfare …

zhongwentui
2012-02-21 0人阅读 举报 0 0 暂无简介

简介:本文档为《Mirrlees 1974-Notes on Welfare Economics, Information and Uncertaintypdf》,可适用于经济金融领域

NotesonWelfareEconomics,Information,andUncertaintylINTRODUCTIONThesenotes,althoughnotentirelytentative,arelesssystematicthanIwouldwishTheirpurposeistoshowhow,inaworldwithimperfectandunreliableinformation,theArrowDebreuframeworkforwelfareeconomicsisunsatis­factory,andmatmodelsakintothosenowusedinthetheoryofpublicfinancemaybemoreappropriateshallnordiscussallthereasonsthaturgeonetoextendoravoidthestandardmodelsInparticular,asArrowandothershavepointedout,theproductionofinformationasinmedicalcare,invention,or,presumably,theeducationalsystemrequiresspecialtreatmentbutIshallnotsaymuchaboutitWhatIamgoingtoconsideristheinformationagovernmentmighthaveaboutconsumers,orconsumersaboutgovernmentIshallbethinkingofwelfareeconomicsasamodeofdiscussingalternativegovernmentpolicies,andalsoaspartofthediscussionaboutalternativesystemsofgovernmentThepoliciesagovernmentcanadopt,andthepoliciesitshouldadopt,dependuponinforma­tionaboutconsumers,whattheydoandwhattheyareThus,thefundamentaltheoremofwelfareeconomicsinvitesgovernmentstodistributetohouseholdsquantitiesofresourcesthatareafunctionofwhatthehouseholdsare(notwhattheychoosetodo),inthehopethattherightcompetitiveequilibriwnwillestab­llshitselfIntheArrowDebreutheory,thisdistributionofresourcesiscarriedoutinadvanceofknowingthestateoftheworld,butwithcompleteinforma­tionaboutthecharacteristicsofthehouseholdsAgreatmanyinsuranceandfuturesmarketsarerequirediftheoptimumistobeacompetitiveequilibrium,butIshallnotconcernmyselfdirectlywiththatwellknowndifficultyIshallaskwhatshouldbedoneforreasonstobeexplaincd()ifthedistributiondependsonthestateofnature,and()ifinformationaboutthecharacteristicsofhouseholdsisimperfectIshallalsoincludesomeremarksonpreferencesregardinguncertaintyThjschapterwasoriginallypublishedinMSBalch,DLMcFadden,andSYWu(eds)EssaysonEcmomicBehalliortmderUncertainty(Amsterdam:NorthHolland)ReproducedbylcindpennissionoftheeditorsWelfareEconomicsREDISTRIBUTIONANDRISKTAKINGTheArrowDebreuformulationofwelfareeconomicsacceptseachhouse­hold'sbeliefspossiblyexpressiblebymeansofsubjectiveprobabilitiesinthesamewaythatitacceptsthehousehold'stastesIfamanbelievesstronglybutwronglythattheendoftheworldisathand,hewillbegivenhiswealthnowandallowedtospenditallatonceHewillthenstarve,incircumstanceshebelievedwouldnotoccur,butanArrowDebreuwelfarefunctiondoesnotcareWeshouldliketobeabletodiscusspoliciesforagovernmentthatdoescareaboutsuchoutcomesandis,insomerespects,betterinformedaboutpossiblestatesofnaturethansomeofthehouseholdsforwhichitclaimsresponsibilityAmongthesepolicieswouldbeincomedistributionsthatareafunctionofthestateofnatureAnotherreasonforstudyingsuchdistributionistheimpossibilityofidenti­fyingallstatesofnature'objectively'thisisthephenomenonofmoralhazard,wellrecognizedintheliteratureAfarmercannotperfectlyinsurehiscropagainstadversecircumstances,forthedegreeofadversitycan,inpractice,beassessedperfectlyonlybylookingatthesizeofthecrop,andthatisaffectedbyactionsunderthefarmer'scontrolIfperfectinsurancewerepossible,distributionbygovernmentcould(apartfromthedifficultiesjustmentioned)withoutdis­advantagebeindependentoftbestateofnatureSinceperfectinsuranceisnotpossible,thegovernmentpresumablyoughttorelatedistributiontotheactualoutcomeoftheharvest,asaproxyforthestateofnaturelnthiscase,wedearlyhavewhathascometobecalleda'secondbest'problemthatis,aproblemthatwillnothaveforsolutionthestraightforwardcompetitiveequilibriumfamiliarinsimplerproblemsTheterm'secondbest'maybeabitmisleadingsince,insuchacaseasthatoftherisktakingfarmer­ortheassistantprofessorthefirstbestisanevenmoreunattainabletheor­eticalconstructionthanthe'secondbest'Thefirstcase,of'Allaisoptimality,,seemstohaveaneasyformalsolution,atleastintheextremecasewherehousehold'sprobabilitybeliefsareirrelevant,andpoliciesarediscussedintermsofawelfarefunctionwhichhasasargumentseachhousehold'sutilityineachstateofnature(forexampleexpectedwelfare,withthegovernment'sprobabilities,oryoursormine)Asusual,onewantstohaveshadowpricesforeachcommodityineachstateofnatureProducersshouldmaximizeprofits(interpretingtheshadowpricesconditionaluponthestateofnature)andhouseholdsshouldworkouttheirplansseparatelyforeachstateofnature,inthelightofthesepricesandthegovernment'splansfordistributionofwealth,whichareconditionaluponthestateofnatureFormally,if�isthevectorofh'sconsumptioninstateofnatures,u�=uh(<)ish'sutility,wewanttomaximizeW(u:)=W(ul,ul,,uf,t,)(monotonicallyincreasing),ThisdistinctionbetweenexanteandexpostoptimalityisfairlywellknownSeeTheutiliryfunctionitselfcouldvarywiththestateofnaturesWelfareEconomics,lnformationJandUncertaintywiththecompetitiveequilibriumconditionssatisfiedforeachassignedbudgetb�Then,aggregateexcessdemandsLhX:inthevariousstatesofnaturebeingfeasibletogether,theproductionplan(YhY)(equaltoaggregateexcessdemandsinthevariousstatesofnature)maxjmizes'LPs·Ys,andeachX:maximizesU:subjecttoPs·x!'beingnogreaterthanbfwherePsisthecom­petitivepricevectorforstatesThisisastandardargumentTheproblemassociatedwith'AJlajsoptimality'isthatconsumersmustnottradeinsurance:itmustbeimpossibletotradeaquantityofacommoditycontingentuponadifferentstateIftherewereperfectcontingentmarketsbetweenconsumers,itwouldnotingeneralbedesirabletoallowproducerstotradeinthesesamemarkets:onewouldwanttoseecommoditytaxesimposed,forexampleTheArrowDebreuequilibriumwouldnotbeoptimalWeshallcomeuponthisneedtoprohibitmarketsagainWhenthemoralhazardaspectoftheeconomyisbroughtin,eventhisratherunstablecompetitiveresultisnotoptimalIhavenointerestinggeneralresultsasyet,butthefollowingspeciaJexampleseemstocapturetheessenceofthematterConsideraneconomyofindependentpeasantfarmers,producingcornontheirownfarmswiththeirownlabour,enterprise,andattentionTheprobabilitydensityofcomoutputywhentherebasbeenlabourinputzis{(yz)Thegovernmentrelatesthefarmer'sconsumptjonxtohisoutputthroughafunctionx=c(y),()whichrepresentsredistriburionofoutputbetweenfarmersThefarmers,identicaltooneanother,andeachinterestedonlyinhimself,chooseztomaximizeexpectedutilityju(x,y,z)((y,z)dy,()subjectto()Letustakeutobeaconcaveincreasingfunctionofx,andadecreasingfunctionofyandzIshallalsoassumethatutendstooowhenxtendstozero,sothatfarmerswouldgivefirstprioritytoavoidingzerocon­sumptionThegovernmentaccepts()asitsownwelfarefunction,sothatallthefarmersshouldfinditspoliciesappealingEverythinghasrotakeplaceundertheaggregateproductionconstraintwhich,ontheassumptionofaverylargenumberoffarmers,insensitivetosmaUvariationsinconsumptionandwithstochasticallyindependentproductionpossibilities,canbetakentobejyf(y,z)dyjc(y)f(y,z)dy=()•IfirstcamencrossproblemsofthiskindinconnectionwithpopulationpolicyThatnnalysisisgivenin()WelfareEconomicsThereisnoharminassumingeverythingissuitablydifferentiableandthattherequisiteLagrangemultipliersexistIusethemultiplierrfortheconstraint(),andamultipliersfortheconstraintJUzfdyJufzdy=,()whicharisesfromutilitymaximizationbythefarmersNoticethatwehaveasecondorderconditionfromthatmaximization,A=jU:zfdyJuzf,dyJrudy$()Thefirstorderconditionsforthegovernment'smaximizationarethat,foreachy,uxfrfsuxfzSU:czf=,()fromvariationsofc,and()fromvariationofzNoticethatthislastconditionissimplifiedbyuseof()Itismoreilluminatingtowrite()intheform(rsu:«)fux=s(fzff)()A'firstbest'optimizationwouldofcoursehavemadeuxthesameforallyItistobeexpectedthatif,asintheapplicationIhaveinmindhere,largerzistobeencouraged,cwillbesochosenthat""diminishesasyincreasesToshowthis,itisconvenientfirsttoimposetheconventionthatjyf(y,z)dy=zforallz,andnaturaltosupposethatfrlfisanincreasingfunctionofy,negativeforsmallyandpositiveforlargeyImustalsoassumethatUxz=()()Itfollowsfrom()thatrmustbepositive,sinceuxalwaysis,andtheright­handsideof()sometimesis,positiveWehavetoshowthatsispositiveWriteh(y)=fJf,withtheoptimumzThen(),whichnowreadsrlux=sh(y),tellsusthatxisafunctionofsh(y)andyx=g(sh(y),y),wherethederivativesCoyrWelfareEconomics,Information,andUncertaintyofgareuKt=­ruxxandUxyKl=Uxx()Assumingthatuisconcave,g>withtbisnotation,wecanwrite()intheformr=sArjg(sh(y),y)h(y)fdy()HerewehaveusedthefactthatJyfrdy=,whichfollowsfrombydiffer­entiationSupposethatUxy$()IshallshowthatthisassumptionimpliesthatsispositiveIfwehads$,g(sh(y),y)wouldbeanonincreasingfunctionofy,by(),()and(),and,Ytbeingsuchthath(y)=,jg(sh(y),y)h(y)fdy=j(g(sh(y),y)g(O,yJ)h(y)fdy$ThefirststepisimpliedbyJhfdy=ff,dy=,sinceJfdy=andthesecondisimpliedby()Also,weknowfrom()thatA$Therefore,underassump­tion(),therighthandsideof()isnonpositiveifsisnonpositive:butthatisimpossible,sincer>Ihaveproved,then,thatwhenxandyareweaklycomplementary,inthesensethat()holds,andxandzareindependent,s>an�by(),""isadecreasingfunctionofyInthespecialcaseu)=(whichmighthold,forexample,becausethefarmerdoesnotcareaboutoutputitself,andneedsnolabortofatheritin}wecanfurtherconcludethatxisanincreasingfunctionofyTheassumptionsusedtoobtaintheseresultsareratherstronglysufficient,butitwillbeclearfromtheanalysisthatthe'perverse'case,wherexdecreaseswithy,isnotentirelyimpossibleItcanbeseenfromformula()thatthegovernment'soprimalpolicycisnotgeneraUyHnear,orevenparticularlysimpleToextendtheargument,onewouldexpect,administrativeandpoliticalreasonsapan,torecommendquitecomplicatedallocationrulesformedicalcare,policeprotection,carinsurance,andeducatjonalexpendituresOnecuriousfeatureofruJe()ishighlightedbyOnemightalsothinkthatintheoptimumonewouldb:tvec'(y)<oratleastthatyc{y)wouldchangesignonlyonce,fromnegativetopositive,asyincreasedIhavenotfoundanyniceassumptionsthatIcanproveimplytheseresultsandIsuspectthattheycannoteasily�guaranteedWelfareEconomicsitsresponsetotheapparentlysensibleassumptionthatagriculturaloutputisdistributedlognormalJy:()(Cisaconstant)Equation()impliesthat()Accordingto(),fjftendstooowhenytendstoBurthatis,by(),inconsistentwithanyvalueofsotherthanYets=doesnotgiveanoptimalpolicyThisisobviousbecause(dependingontheutilityfunction)arulefordistributingtheavailableoutputthatleaveseveryonewiththesameUxmayleavenoincentiveforthefarmerswproduceanything!Fromatechnicalpointofview,forthisapparentlywellsetproblem,nooptimumexistsThefactis,inthiscase,thatonedoesbetterthemoreonepenaljzesthosefarmerswhoturnouttohaveverylowoutputThepointcanbemaderigor­ouslybyconsideringwhathappensifallfamerswhoseoutputislessthanasmallnumberreceiveconsumptionf:(anothersmallnumber),whiletheothersreceivewhatrheywouldreceiveinthefirstbestoptimum�llitc•(y)Thiswillbepossibleifitispossibleatthesamerimetoinducefarmerstoadoptthefirstbestoptimumlevelofzcallitz•Forthatwerequire'u(E,y,z•)f:(y,z•)dyu(c•(y),y,z•)f:(y,z•)dyfo",�,(€,y,z•)f(y,z•)dyU:(c•(y),y,z•)f(y,z•)dy=RecollectthatUzisindependentofx,sothatthethirdandfourthtermscanbewritten,together,asJ:ur(y,z•)f(y,z•)dyThuswehavetochoosefandsothatfo'l{u(c•(y),y,z•)u(e,y,z•)}fz(y,z•)dy=fooo{t�(c•(y),y,z•)ft(y,z•)uz(y,t•)f(y,z•)}dyTherightbandsideofthisequationisequaltoWelfareEconomics,Information,andU�tcertaintywherer•=ux(c"'(y),y,z"'),byfirstbestoptimalityandthisexpressionwillnormaJlybenegative,since(c"'y)fzdy=(c"'y)h(y)fdy<byassumption(),andthisisensuredbytheweakassumptionthatUxySUxx•Thus,foranyTJ>,wecanchoose£soastogetz"'chosenYet,givenanynumberM,wecanchooseJsosmallthatfz<Mf(y<),sothatInthisway,wecanapproximateascloselyaswewishtothefirstbestoptimum,byimposingpenalties(presumablyofgreatseverity)onasmallproportionofthepopulationAlthoughthesefarmerssufferseverely,therearesofewofthemthattheirsufferingsareoutweighedbytheencouragementtheirfate,orratherthepro­spectofit,givestofarmerstakingproductiondecisionsItseemsthatmodelsofthiskindcanincertaincasesprovidesomejustificationforextremepunishmentofnegligiblysmallgroupsTheproblemhasbeenpresentedasoneofgovernmentpolicy,butwiththecoincidencebetweenprivateandgovernmentendspostulated,thesolutionmayinsteadbeinterpretedasapredictionofthekindofinsurancesystemthatwouldariseinthesocietyconsideredItisinterestingandimportanttoconsiderfurthersolutionsinwhichthegovernmentadoptscriteriadifferentfromthoseofthefannersexante,onutilitarianoregalitariangroundsbutIdonotconsiderthisfurtherinthepresentnotesTHECHARACTERIZATIONOFHOUSEHOLDSWemaythinkofthestandardproblemofwelfareeconomicsinthefollowingformAhouseholdoftypehhasutilityfunctionu(x,h)intermsofitstradesxwiththerestoftheeconomy(ieexcessdemands)Productionconstrainsy=E,x(h)tolieintheproductionsetYW(u)istobemaximizedThe'funda­mentaltheorem'assertsthattheoptimumisacompetitiveequilibriumifthereisasuitabledistributionofbudgetsdefinedbyafunctionb(h)ThepointIwanttoemphasize,obviousthoughitis,isthattheconsumersarethensupposedtochoosewhattradetheywilldorationallyintermsoftheirownselfinterest,butWelfareEconomicsaresupposedtorevealthenecessaryinformationaboutthemselves,symbolizedbythevariableh,withoutregardtotheirownselfinterestTheusualnotationobscuresthisThefollowingexample,whichhassomecorrespondencewithreality,mayhighlightthedifficultyConsideraneconomywithoneconsumergood,producedwithlabourAhouseholdoftypenprovideslabourofquantitynywhenitworksfortimeyEveryhouseholdhasutilityfunctionu(x,y)intermsofconsumptionxandlabourtimeyThewelfarefunctioniscompletelyseparableintermsofindi­vidualconsumption,sothattheutilityfunctioncanbechosentoensurethatthewelfarefunctionisW=fu(x(n),y(n))f{n)dn,{beingthedensityfunctiongivingthedistributionofabilitynWelfareisquasiconcaveinindividuals'con­sumption,sinceuisconcaveTheoptimumpolicyallocatesconsumptionandtimeasfunctionsofninsuchawaythatJx(n)f(n)dnisproduciblewithlaborinputJny(n)f(n}dnDenotetheoptimumbyx•(n),y•(n)Propositionu•(n)=u(x"'(n),y"'(n))isadecreasingfunctionofnif(andonlyif)labourtimeisastrictlynormalcommodityRecollectthattosaytimeisastrictlynormalcommodityfortheconsumermeans,bydefinition,thatanincreaseinnonlabourincomewould,inamarketeconomy,leadtheconsumertoreducehislaboursupplyItisaveryplausibleassumptionTheproofoftheproposition,whichisroutine,isgivenintheappendixThepointofthepropositionisthatonewouldnaturallyassumethatindividualshavesomecontrolovertheinformationtheyconveytogovernmentabouttheirabilitiesitispresumablyeasiertopretendtolessabilitythanonehasthanitistopretendtomoreInanycase,thereisnoincentivetoprovidetheinformationthatthegovernmentmusthaveifitistobringtheoptimumabout:onthecontrary,thereisanincentiveforanyindividualnottoprovidetheinformationThemodelhassomeunrealisticfeatureswhichservetooverstatethedifficultiesbutitisplausiblethatagovernmentwhichattemptedtorealizetheoptimumofbasicwelfareeconomicswouldfailbecauseofthesedifficultiesInordertocapturethisfeatureofwelfareeconomics,Iproposetoreformulatethebasicproblemintheformoftwolevelmaximization,withhouseholdsmaximizingunderagovernmentimposedconstraint,andthegovernmentchoosingtheconstraintInordertomaximizewelfareLetustheninterpretthevectorxnotonlyastradesbut,moregenerally,asbehaviorThus,ifsomekindsofworkcanbeobserveddirectly,asamountoftimeorenergyspentaswellasthroughproductiveeffects,bothaspectscouldappearinthelistxDenotebykthoseaspectsofindividualsthatarepubliclyknown,independentlyofbehavior(age,sex�placeofbirth,),andbyhthoseaspectswhich,althoughtheyaffectbehaviorthroughtheconsumer'schoices,arenot'visible',atleasttogovern­mentTheindividualwillchoosexsoastomaximizeU(x,h,k),butwillbeIt�neralizesaspecialcasementionedinWelfareEconomics,Information,andUncertaintyconstrained,firstbyhisconsumptionsetX(h,k),andsecondlybythecon­straintsimposedbygovernmentandthemarketsoftheeconomy,A(k)Proofx(h,k)maximizesU(x,h,k)subjecttoxEX(h,k)nA(k)()TheproductionconstraintisthatEx(h,k)EY()(SomeofthecomponentsofEx,correspondingtonontradebehavior,areredundant)Writeu(·)forthefunctionofhandkdefinedbyu(h,k)=U(x(h,k),h,k)ThenthegovernmentseekstomaximizeW(u(·)),subjecttotheconstraints()and(),bychoiceofA{·)Thegovernmentissupposedroknowthenatureofthepopulation,thenumberofpeopleforeachhandkSpecifically,itismostinterestingtosupposehandkcontinuouslydistributedwithdensityfunctionf(h,k)sothattheproductionconstraintshouldbewrittenIx(h,k)f(h,k)dhdkEY(')Thegovernment'smaximandwouJddependbothupontheutilityoutcomeu(·)anduponthedistributionf(·)Forexample,itmighttakethecompletelysepa

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Mirrlees 1974-Notes on Welfare Economics, Information and Uncertainty

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