Utility maximization problem - Wikipedia

Utility maximization is an important concept in consumer theory as it shows how consumers decide to allocate their income. Because consumers are rational, they ... Utilitymaximizationproblem FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Problemofallocationofmoneybyconsumersinordertomostbenefitthemselves Foralesstechnicalintroduction,seeUtility. Thisarticlehasmultipleissues.Pleasehelpimproveitordiscusstheseissuesonthetalkpage.(Learnhowandwhentoremovethesetemplatemessages) Thisarticleincludesalistofgeneralreferences,butitlackssufficientcorrespondinginlinecitations.Pleasehelptoimprovethisarticlebyintroducingmoreprecisecitations.(August2010)(Learnhowandwhentoremovethistemplatemessage) Thisarticleneedsadditionalcitationsforverification.Pleasehelpimprovethisarticlebyaddingcitationstoreliablesources.Unsourcedmaterialmaybechallengedandremoved.Findsources: "Utilitymaximizationproblem" – news ·newspapers ·books ·scholar ·JSTOR(March2011)(Learnhowandwhentoremovethistemplatemessage) (Learnhowandwhentoremovethistemplatemessage) UtilitymaximizationwasfirstdevelopedbyutilitarianphilosophersJeremyBenthamandJohnStuartMill.Inmicroeconomics,theutilitymaximizationproblemistheproblemconsumersface:"HowshouldIspendmymoneyinordertomaximizemyutility?"Itisatypeofoptimaldecisionproblem.Itconsistsofchoosinghowmuchofeachavailablegoodorservicetoconsume,takingintoaccountaconstraintontotalspending(income),thepricesofthegoodsandtheirpreferences. Utilitymaximizationisanimportantconceptinconsumertheoryasitshowshowconsumersdecidetoallocatetheirincome.Becauseconsumersarerational,theyseektoextractthemostbenefitforthemselves.However,duetoboundedrationalityandotherbiases,consumerssometimespickbundlesthatdonotnecessarilymaximizetheirutility.Theutilitymaximizationbundleoftheconsumerisalsonotsetandcanchangeovertimedependingontheirindividualpreferencesofgoods,pricechangesandincreasesordecreasesinincome. Contents 1Basicsetup 1.11)Walras'sLaw 1.1.1Preferencesoftheconsumer 1.1.1.1Complete 1.1.1.2Transitive 1.1.1.3Monotone 1.22)'Bangforbuck' 1.33)Budgetconstraint 1.44)Checkfornegativity 2Atechnicalrepresentation 3Utilitymaximisationofperfectcompliments 4Utilitymaximisationofperfectsubstitutes 5Reactiontochangesinprices 6Reactiontochangesinincome 7Boundedrationality 8Relatedconcepts 9Seealso 10References 11Externallinks Basicsetup[edit] Forutilitymaximizationtherearefourbasicstepsprocesstoderiveconsumerdemandandfindtheutilitymaximizingbundleoftheconsumergivenprices,income,andpreferences. 1)CheckifWalras'slawissatisfied 2)'Bangforbuck' 3)thebudgetconstraint 4)Checkfornegativity 1)Walras'sLaw[edit] Walras'slawstatesthatifaconsumerspreferencesarecomplete,monotoneandtransitivethentheoptimaldemandwilllieonthebudgetline.[1] Preferencesoftheconsumer[edit] Forautilityrepresentationtoexistthepreferencesoftheconsumermustbecompleteandtransitive(necessaryconditions).[2] Complete[edit] Completenessofpreferencesindicatesthatallbundlesintheconsumptionsetcanbecomparedbytheconsumer.Forexample,iftheconsumerhas3bundlesA,BandCthen; A ≽ {\displaystyle\succcurlyeq} B,A ≽ {\displaystyle\succcurlyeq} C,B ≽ {\displaystyle\succcurlyeq} A,B ≽ {\displaystyle\succcurlyeq} C,C ≽ {\displaystyle\succcurlyeq} B,C ≽ {\displaystyle\succcurlyeq} A,A ≽ {\displaystyle\succcurlyeq} A,B ≽ {\displaystyle\succcurlyeq} B,C ≽ {\displaystyle\succcurlyeq} C.Therefore,theconsumerhascompletepreferencesastheycancompareeverybundle. Transitive[edit] Transitivitystatesthatindividualspreferencesareconsistentacrossthebundles. therefore,iftheconsumerweaklyprefersAoverB(A ≽ {\displaystyle\succcurlyeq} B)andB ≽ {\displaystyle\succcurlyeq} CthismeansthatA ≽ {\displaystyle\succcurlyeq} C(AisweaklypreferredtoC) Monotone[edit] Forapreferencerelationtobemonotoneincreasingthequantityofbothgoodsshouldmaketheconsumerstrictlybetteroff(increasetheirutility),andincreasingthequantityofonegoodholdingtheotherquantityconstantshouldnotmaketheconsumerworseoff(sameutility). Thepreference ≽ {\displaystyle\succcurlyeq} ismonotoneifanyonlyif; 1) ( x + ϵ , y ) ≽ ( x , y ) {\displaystyle(x+\epsilon,y)\succcurlyeq(x,y)} 2) ( x , y + ϵ ) ≽ ( x , y ) {\displaystyle(x,y+\epsilon)\succcurlyeq(x,y)} 3) ( x + ϵ , y + ϵ ) ≻ ( x , y ) {\displaystyle(x+\epsilon,y+\epsilon)\succ(x,y)} where ϵ {\displaystyle\epsilon} >0 2)'Bangforbuck'[edit] Bangforbuckisamainconceptinutilitymaximizationandconsistsoftheconsumerwantingtogetthebestvaluefortheirmoney.IfWalras'slawhasbeensatisfied,theoptimalsolutionoftheconsumerliesatthepointwherethebudgetlineandoptimalindifferencecurveintersect,thisiscalledthetangencycondition.[3]Tofindthispoint,differentiatetheutilityfunctionwithrespecttoxandytofindthemarginalutilities,thendividebytherespectivepricesofthegoods. M U x / p x = M U y / p y {\displaystyleMU_{x}/p_{x}=MU_{y}/p_{y}} Thiscanbesolvedtofindtheoptimalamountofgoodxorgoody. 3)Budgetconstraint[edit] Thebasicsetupofthebudgetconstraintoftheconsumeris: p x x + p y y ≤ I {\displaystylep_{x}x+p_{y}y\leqI} DuetoWalras'slawbeingsatisfied: p x x + p y y = I {\displaystylep_{x}x+p_{y}y=I} Thetangencyconditionisthensubstitutedintothistosolvefortheoptimalamountoftheothergood. 4)Checkfornegativity[edit] Figure1:Thisrepresentswheretheutilitymaximizingbundleiswhenthedemandforonegoodisnegative Negativitymustbecheckedforastheutilitymaximizationproblemcangiveananswerwheretheoptimaldemandofagoodisnegative,whichinrealityisnotpossibleasthisisoutsidethedomain.Ifthedemandforonegoodisnegative,theoptimalconsumptionbundlewillbewhere0ofthisgoodisconsumedandallincomeisspentontheothergood(acornersolution).Seefigure1foranexamplewhenthedemandforgoodxisnegative. Atechnicalrepresentation[edit] Supposetheconsumer'sconsumptionset,ortheenumerationofallpossibleconsumptionbundlesthatcouldbeselectediftherewereabudgetconstraint. Theconsumptionset= R + n   . {\displaystyle\mathbb{R}_{+}^{n}\.} (asetofpositiverealnumbers,theconsumercannotpreferencenegativeamountofcommodities). x ∈ R + n   . {\displaystylex\in\mathbb{R}_{+}^{n}\.} Supposealsothatthepricevector(p)ofthencommoditiesispositive, Figure2:Thisshowstheoptimalamountsofgoodsxandythatmaximiseutilitygivenabudgetconstraint. p ∈ R + n   , {\displaystylep\in\mathbb{R}_{+}^{n}\,} andthattheconsumer'sincomeis w {\displaystylew} ;thenthesetofallaffordablepackages,thebudgetsetis, B ( p , I ) = { x ∈ R + n | Σ i = 1 n p i x i ≤ I }   , {\displaystyleB(p,I)=\{x\in\mathbb{R}_{+}^{n}|\mathbb{\Sigma}_{i=1}^{n}p_{i}x_{i}\leqI\}\,} Theconsumerwouldliketobuythebestaffordablepackageofcommodities. Itisassumedthattheconsumerhasanordinalutilityfunction,calledu.Itisareal-valuedfunctionwithdomainbeingthesetofallcommoditybundles,or u : R + n → R +   . {\displaystyleu:\mathbb{R}_{+}^{n}\rightarrow\mathbb{R}_{+}\.} Thentheconsumer'soptimalchoice x ( p , w ) {\displaystylex(p,w)} istheutilitymaximizingbundleofallbundlesinthebudgetsetif x ∈ B ( p , w ) {\displaystylex\inB(p,w)} thentheconsumersoptimaldemandfunctionis: x ( p , I ) = { x ∈ B ( p , I ) | U ( x ) ≥ U ( y ) ∀ y ∈ B ( p , I ) } {\displaystylex(p,I)=\{x\inB(p,I)|U(x)\geqU(y)\forally\inB(p,I)\}} Finding x ( p , I ) {\displaystylex(p,I)} istheutilitymaximizationproblem. Ifuiscontinuousandnocommoditiesarefreeofcharge,then x ( p , I ) {\displaystylex(p,I)} exists,[4]butitisnotnecessarilyunique.Ifthepreferencesoftheconsumerarecomplete,transitiveandstrictlyconvexthenthedemandoftheconsumercontainsauniquemaximiserforallvaluesofthepriceandwealthparameters.Ifthisissatisfiedthen x ( p , I ) {\displaystylex(p,I)} iscalledtheMarshalliandemandfunction.Otherwise, x ( p , I ) {\displaystylex(p,I)} isset-valuedanditiscalledtheMarshalliandemandcorrespondence. Utilitymaximisationofperfectcompliments[edit] U=min{x,y} Figure3:Thisshowstheutilitymaximisationproblemwithaminimumutilityfunction. Foraminimumfunctionwithgoodsthatareperfectcompliments,thesamestepscannotbetakentofindtheutilitymaximisingbundleasitisanondifferentiablefunction.Therefore,intuitionmustbeused.Theconsumerwillmaximisetheirutilityatthekinkpointinthehighestindifferencecurvethatintersectsthebudgetlinewherex=y.[3]Thisisintuition,astheconsumerisrationalthereisnopointtheconsumerconsumingmoreofonegoodandnottheothergoodastheirutilityistakenattheminimumofthetwo(theyhavenogaininutilityfromthisandwouldbewastingtheirincome).Seefigure3. Utilitymaximisationofperfectsubstitutes[edit] U=x+y Forautilityfunctionwithperfectsubstitutes,theutilitymaximisingbundlecanbefoundbydifferentiationorsimplybyinspection.SupposeaconsumerfindslisteningtoAustralianrockbandsAC/DCandTameImpalaperfectsubstitutes.ThismeansthattheyarehappytospendallafternoonlisteningtoonlyAC/DC,oronlyTameImpala,orthree-quartersAC/DCandone-quarterTameImpala,oranycombinationofthetwobandsinanyamount.Therefore,theconsumer'soptimalchoiceisdeterminedentirelybytherelativepricesoflisteningtothetwoartists.IfattendingaTameImpalaconcertischeaperthanattendingtheAC/DCconcert,theconsumerchoosestoattendtheTameImpalaconcert,andviceversa.Ifthetwoconcertpricesarethesame,theconsumeriscompletelyindifferentandmayflipacointodecide.Toseethismathematically,differentiatetheutilityfunctiontofindthattheMRSisconstant-thisisthetechnicalmeaningofperfectsubstitutes.Asaresultofthis,thesolutiontotheconsumer'sconstrainedmaximizationproblemwillnot(generally)beaninteriorsolution,andassuchonemustchecktheutilitylevelintheboundarycases(spendentirebudgetongoodx,spendentirebudgetongoody)toseewhichisthesolution.Thespecialcaseiswhenthe(constant)MRSequalsthepriceratio(forexample,bothgoodshavethesameprice,andsamecoefficientsintheutilityfunction).Inthiscase,anycombinationofthetwogoodsisasolutiontotheconsumerproblem. Reactiontochangesinprices[edit] Foragivenlevelofrealwealth,onlyrelativepricesmattertoconsumers,notabsoluteprices.Ifconsumersreactedtochangesinnominalpricesandnominalwealthevenifrelativepricesandrealwealthremainedunchanged,thiswouldbeaneffectcalledmoneyillusion.Themathematicalfirstorderconditionsforamaximumoftheconsumerproblemguaranteethatthedemandforeachgoodishomogeneousofdegreezerojointlyinnominalpricesandnominalwealth,sothereisnomoneyillusion. Whenthepricesofgoodschange,theoptimalconsumptionofthesegoodswilldependonthesubstitutionandincomeeffects.Thesubstitutioneffectsaysthatifthedemandforbothgoodsishomogeneous,whenthepriceofonegooddecreases(holdingthepriceoftheothergoodconstant)theconsumerwillconsumemoreofthisgoodandlessoftheotherasitbecomesrelativelycheeper.Thesamegoesifthepriceofonegoodincreases,consumerswillbuylessofthatgoodandmoreoftheother.[5] Theincomeeffectoccurswhenthechangeinpricesofgoodscauseachangeinincome.Ifthepriceofonegoodrises,thenincomeisdecreased(morecostlythanbeforetoconsumethesamebundle),thesamegoesifthepriceofagoodfalls,incomeisincreased(cheepertoconsumethesamebundle,theycanthereforeconsumemoreoftheirdesiredcombinationofgoods).[5] Reactiontochangesinincome[edit] Figure5:Thisshowshowtheoptimalbundleofaconsumerchangeswhentheirincomeisincreased. Iftheconsumersincomeisincreasedtheirbudgetlineisshiftedoutwardsandstheynowhavemoreincometospendoneithergoodx,goody,orbothdependingontheirpreferencesforeachgood.ifbothgoodsxandywerenormalgoodsthenconsumptionofbothgoodswouldincreaseandtheoptimalbundlewouldmovefromAtoC(seefigure5).Ifeitherxorywereinferiorgoods,thendemandforthesewoulddecreaseasincomerises(theoptimalbundlewouldbeatpointBorC).[6] Boundedrationality[edit] forfurtherinformationsee:Boundedrationality Inpractice,aconsumermaynotalwayspickanoptimalbundle.Forexample,itmayrequiretoomuchthoughtortoomuchtime.Boundedrationalityisatheorythatexplainsthisbehaviour.Examplesofalternativestoutilitymaximisationduetoboundedrationalityare;satisficing,eliminationbyaspectsandthementalaccountingheuristic. Thesatisficingheuristiciswhenaconsumerdefinesanaspirationlevelandlooksuntiltheyfindanoptionthatsatisfiesthis,theywilldeemthisoptiongoodenoughandstoplooking.[7] Eliminationbyaspectsisdefiningalevelforeachaspectofaproducttheywantandeliminatingallotheroptionsthatdon'tmeetthisrequiremente.g.priceunder$100,colouretc.untilthereisonlyoneproductleftwhichisassumedtobetheproducttheconsumerwillchoose.[8] Thementalaccountingheuristic:Inthisstrategyitisseenthatpeopleoftenassignsubjectivevaluestotheirmoneydependingontheirpreferencesfordifferentthings.Apersonwilldevelopmentalaccountsfordifferentexpenses,allocatetheirbudgetwithinthese,thentrytomaximisetheirutilitywithineachaccount.[9] Relatedconcepts[edit] TherelationshipbetweentheutilityfunctionandMarshalliandemandintheutilitymaximisationproblemmirrorstherelationshipbetweentheexpenditurefunctionandHicksiandemandintheexpenditureminimisationproblem.Inexpenditureminimisationtheutilitylevelisgivenandwellasthepricesofgoods,theroleoftheconsumeristofindaminimumlevelofexpenditurerequiredtoreachthisutilitylevel. Theutilitariansocialchoiceruleisarulethatsaysthatsocietyshouldchoosethealternativethatmaximizesthesumofutilities.Whileutility-maximizationisdonebyindividuals,utility-summaximizationisdonebysociety. Seealso[edit] Choicemodelling Expenditureminimisationproblem Optimaldecision Substitutioneffect Utilityfunction Lawofdemand Marginalutility References[edit] ^Levin,Jonothan(2004).Consumertheory.Stanforduniversity.pp. 4–6. ^Salcedo,Bruno(2017).Utilityrepresentations.Cornelluniversity.pp. 18–19. ^abBoard,Simon(2009).Utilitymaximizationproblem.Departmentofeconomics,UCLA.pp. 10–17. ^Choice,preferenceandUtility.Princetonuniversitypress.n.d.p. 14. ^abUtilityMaximizationandDemand.UniversityofMinnesotalibrary.2011.pp. chapter7.2. ^RiceUniversity(n.d.)."Howchangesinincomeandpricesaffectconsumptionchoices".Pressbooks.Retrieved22April2021.{{citeweb}}:CS1maint:url-status(link) ^Wheeler,Gregory(2018).boundedrationality.StanfordEncyclopediaofPhilosophy. ^"Elimination-By-AspectsModel".MonashUniversity.2018.Retrieved20April2021.{{citeweb}}:CS1maint:url-status(link) ^"Whydowethinklessaboutsomepurchasesthanothers?".Thedecisionlab.2021.Retrieved20April2021.{{citeweb}}:CS1maint:url-status(link) Externallinks[edit] AnatomyofCobb-DouglasTypeUtilityFunctionsin3D Rulesformaximisingutilitybylumenlearning Anexampleofutilitymaximisation UtilitymaximisationdefinitionbyEconomicshelp ApplicationofautilityfunctionbyInvestopedia DefinitionofsubstitutegoodsbyInvestopedia Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Utility_maximization_problem&oldid=1098390878" Categories:OptimaldecisionsUtilityMathematicaloptimizationBusinessandeconomicsportalHiddencategories:CS1maint:url-statusArticleswithshortdescriptionShortdescriptionwithemptyWikidatadescriptionArticleslackingin-textcitationsfromAugust2010Allarticleslackingin-textcitationsArticlesneedingadditionalreferencesfromMarch2011AllarticlesneedingadditionalreferencesArticleswithmultiplemaintenanceissues Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages 한국어ລາວ日本語РусскийУкраїнськаTiếngViệt中文 Editlinks