nicholson微观经济理论（答案+ppt+习题库）-220-03-033593-0

LABORSUPPLYINTHISCHAPTERWEEXAMINESOMEASPECTSOFINPUTPRICINGTHATARERELATEDPARTICULARLYTOTHELABORMARKETBECAUSEWEHAVEALREADYDISCUSSEDQUESTIONSABOUTTHEDEMANDFORLABORORANYOTHERINPUTINSOMEDETAIL,WEWILLBECONCERNEDPRIMARILYWITHANALYZINGTHESUPPLYOFLABOR22CHAPTERALLOCATIONOFTIMEINPARTIIWEANALYZEDTHEWAYINWHICHANINDIVIDUALCHOOSESTOALLOCATEAXEDAMOUNTOFINCOMEAMONGAVARIETYOFAVAILABLEGOODSINDIVIDUALSMUSTMAKESIMILARCHOICESINDECIDINGHOWTHEYWILLSPENDTHEIRTIMETHENUMBEROFHOURSINADAYORINAYEARISABSOLUTELYXED,ANDTIMEMUSTBEUSEDASIT“PASSESBY”GIVENTHISXEDAMOUNTOFTIME,ANYINDIVIDUALMUSTDECIDEHOWMANYHOURSTOWORKHOWMANYHOURSTOSPENDCONSUMINGAWIDEVARIETYOFGOODS,RANGINGFROMCARSANDTELEVISIONSETSTOOPERASHOWMANYHOURSTODEVOTETOSELFMAINTENANCEANDHOWMANYHOURSTOSLEEPBYSTUDYINGHOWINDIVIDUALSCHOOSETODIVIDETHEIRTIMEAMONGTHESEACTIVITIES,ECONOMISTSAREABLETOUNDERSTANDTHELABORSUPPLYDECISIONSIMPLETWOGOODMODELFORSIMPLICITYWESTARTBYASSUMINGTHEREAREONLYTWOUSESTOWHICHANINDIVIDUALMAYDEVOTEHISORHERTIMEEITHERENGAGINGINMARKETWORKATAREALWAGERATEOFWPERHOURORNOTWORKINGWESHALLREFERTONONWORKTIMEAS“LEISURE,”BUTTHISWORDISNOTMEANTTOCARRYANYCONNOTATIONOFIDLENESSTIMENOTSPENTINMARKETWORKCANBEDEVOTEDTOWORKINTHEHOME,TOSELFIMPROVEMENT,ORTOCONSUMPTIONITTAKESTIMETOUSEATELEVISIONSETORABOWLINGBALL1ALLOFTHOSEACTIVITIESCONTRIBUTETOANINDIVIDUALSWELLBEING,ANDTIMEWILLBEALLOCATEDTOTHEMINWHATMIGHTBEASSUMEDTOBEAUTILITYMAXIMIZINGWAYMORESPECICALLY,ASSUMETHATANINDIVIDUALSUTILITYDURINGATYPICALDAYDEPENDSONCONSUMPTIONDURINGTHATPERIODCANDONHOURSOFLEISUREENJOYEDHUTILITYH11005UC,H221NOTICETHATINWRITINGTHISUTILITYFUNCTION,WEHAVEUSEDTWO“COMPOSITE”GOODS,CONSUMPTIONANDLEISURETHEREADERSHOULDRECOGNIZETHATUTILITYISINFACTDERIVEDBYDEVOTINGREALINCOMEANDTIMETOTHECONSUMPTIONOFAWIDEVARIETYOFGOODSANDSERVICES2INSEEKINGTOMAXIMIZEUTILITY,THEINDIVIDUALISBOUNDBYTWOCONSTRAINTSTHERSTOFTHESECONCERNSAVAILABLETIMEIFWELETLREPRESENTHOURSOFWORK,THENLH11001HH1100524222THATIS,THEDAYSTIMEMUSTBEALLOCATEDEITHERTOWORKORTONONWORKASECONDCONSTRAINTRECORDSTHEFACTTHATTHEINDIVIDUALCANPURCHASECONSUMPTIONITEMSONLYBYWORKINGLATERINTHISCHAPTERWEWILLALLOWFORTHEAVAILABILITYOFNONLABORINCOMEIFTHEREALHOURLYMARKETWAGERATETHEINDIVIDUALCANEARNISGIVENBYW,THEINCOMECONSTRAINTISGIVENBYCH11005WL223COMBININGTHETWOCONSTRAINTS,WEHAVECH11005W24H11002H224606PARTIVPRODUCTIONANDSUPPLY1PERHAPSTHERSTFORMALTHEORETICALTREATMENTOFTHEALLOCATIONOFTIMEWASGIVENBYGSBECKERIN“ATHEORYOFTHEALLOCATIONOFTIME,”ECONOMICJOURNAL75SEPTEMBER19654935172THISOBSERVATIONLEADSTOTHECONSIDERATIONOFHOWSUCHACTIVITIESAREPRODUCEDINTHEHOMEFORASURVEY,SEERGRONAU,“HOMEPRODUCTIONASURVEY”INOCASHENFELTERANDRLAYARD,EDS,HANDBOOKOFLABORECONOMICS,VOL1AMSTERDAMNORTHHOLLAND,1986,PP273304ORCH11001WHH1100524W225THISCOMBINEDCONSTRAINTHASANIMPORTANTINTERPRETATIONANYINDIVIDUALHASA“FULLINCOME”GIVENBY24WTHATIS,ANINDIVIDUALWHOWORKEDALLTHETIMEWOULDHAVETHISMUCHCOMMANDOVERREALCONSUMPTIONGOODSEACHDAYINDIVIDUALSMAYSPENDTHEIRFULLINCOMEEITHERBYWORKINGFORREALINCOMEANDCONSUMPTIONORBYNOTWORKINGANDTHEREBYENJOYINGLEISUREEQUATION225SHOWSTHATTHEOPPORTUNITYCOSTOFCONSUMINGLEISUREISWPERHOURITISEQUALTOEARNINGSFOREGONEBYNOTWORKINGUTILITYMAXIMIZATIONTHEINDIVIDUALSPROBLEM,THEN,ISTOMAXIMIZEUTILITY,SUBJECTTOTHEFULLINCOMECONSTRAINTSETTINGUPTHELAGRANGIANEXPRESSIONH5112H11005UC,HH11001H926124WH11002CH11002WH,226THERSTORDERCONDITIONSFORAMAXIMUMAREH11005H11002H9261H110050H11005H11002WH9261H110050227DIVIDINGTHETWOLINESINEQUATION227,WEGETH11005WH11005MRSHFORC228HENCEWEHAVEDERIVEDTHEFOLLOWINGPRINCIPLEUTILITYMAXIMIZINGLABORSUPPLYDECISIONTOMAXIMIZEUTILITY,GIVENTHEREALWAGE,W,THEINDIVIDUALSHOULDCHOOSETOWORKTHATNUMBEROFHOURSFORWHICHTHEMARGINALRATEOFSUBSTITUTIONOFLEISUREFORCONSUMPTIONISEQUALTOWOFCOURSE,THERESULTDERIVEDINEQUATION228ISONLYANECESSARYCONDITIONFORAMAXIMUMASINCHAPTER5,THISTANGENCYWILLBEATRUEMAXIMUMPROVIDEDTHEMRSOFLEISUREFORCONSUMPTIONISDIMINISHINGINCOMEANDSUBSTITUTIONEFFECTSOFACHANGEINWACHANGEINTHEREALWAGERATEWCANBEANALYZEDINAMANNERIDENTICALTOTHATUSEDINCHAPTER5WHENWRISES,THE“PRICE”OFLEISUREBECOMESHIGHERTHEINDIVIDUALMUSTGIVEUPMOREINLOSTWAGESFOREACHHOUROFLEISURECONSUMEDTHESUBSTITUTIONEFFECTOFANINCREASEINWONTHEHOURSOFLEISURETHEREFOREWILLBENEGATIVEASLEISUREBECOMESMOREEXPENSIVE,THEREISREASONTOCONSUMELESSOFITHOWEVER,THEINCOMEEFFECTWILLBEPOSITIVEBECAUSELEISUREISANORMALGOOD,THEHIGHEROPTIMIZATIONPRINCIPLEH11128U/H11128HH5007H11128U/H11128CH11128UH5007H11128HH11128H5112H5007H11128HH11128UH5007H11128CH11128H5112H5007H11128CCHAPTER22LABORSUPPLY607INCOMERESULTINGFROMAHIGHERWWILLINCREASETHEDEMANDFORLEISURETHUS,THEINCOMEANDSUBSTITUTIONEFFECTSWORKINOPPOSITEDIRECTIONSITISIMPOSSIBLETOPREDICTONAPRIORIGROUNDSWHETHERANINCREASEINWWILLINCREASEORDECREASETHEDEMANDFORLEISURETIMEBECAUSELEISUREANDWORKAREMUTUALLYEXCLUSIVEWAYSTOSPENDONESTIME,ITISALSOIMPOSSIBLETOPREDICTWHATWILLHAPPENTOTHENUMBEROFHOURSWORKEDTHESUBSTITUTIONEFFECTTENDSTOINCREASEHOURSWORKEDWHENWINCREASES,WHEREASTHEINCOMEEFFECT,BECAUSEITINCREASESTHEDEMANDFORLEISURETIME,TENDSTODECREASETHENUMBEROFHOURSWORKEDWHICHOFTHESETWOEFFECTSISTHESTRONGERISANIMPORTANTEMPIRICALQUESTION3AGRAPHICALANALYSISTHETWOPOSSIBLEREACTIONSTOACHANGEINWAREILLUSTRATEDINFIGURE221INBOTHGRAPHSTHEINITIALWAGEISW0,ANDTHEINITIALOPTIMALCHOICESOFCANDHAREGIVENBYTHEPOINTC0,H0WHENTHEWAGERATEINCREASESTOW1,THEOPTIMALCOMBINATIONMOVESTOPOINTC1,H1THISMOVEMENTCANBECONSIDEREDTHERESULTOFTWOEFFECTSTHESUBSTITUTIONEFFECTCANBEREPRESENTEDBYTHEMOVEMENTOFTHEOPTIMALPOINTFROMC0,H0TOSANDTHEINCOMEEFFECTBYTHEMOVEMENTFROMSTOC1,H1INTHETWOPANELSOFFIGURE221,THESETWOEFFECTSCOMBINETOPRODUCEDIFFERENTRESULTSINPANELATHESUBSTITUTIONEFFECTOFACHANGEINWOUTWEIGHSTHEINCOMEEFFECT,ANDTHEINDIVIDUALDEMANDSLESSLEISUREH1H11021H0ANOTHERWAYOFSAYINGTHISISTHATTHEINDIVIDUALWILLWORKLONGERHOURSWHENWRISESINPANELBOFFIGURE221THESITUATIONISREVERSEDTHEINCOMEEFFECTOFACHANGEINWMORETHANOFFSETSTHESUBSTITUTIONEFFECT,ANDTHEDEMANDFORLEISUREINCREASESH1H11022H0THEINDIVIDUALWORKSSHORTERHOURSWHENWRISESINTHECASESEXAMINEDINCHAPTER5THISWOULDHAVEBEENCONSIDEREDANUNUSUALRESULTWHENTHE“PRICE”OFLEISURERISES,THEINDIVIDUALDEMANDSMOREOFITFORTHECASEOFNORMALCONSUMPTIONGOODS,THEINCOMEANDSUBSTITUTIONEFFECTSWORKINTHESAMEDIRECTIONONLYFOR“INFERIOR”GOODSDOTHEYDIFFERINSIGNINTHECASEOFLEISUREANDLABOR,HOWEVER,THEINCOMEANDSUBSTITUTIONEFFECTSALWAYSWORKINOPPOSITEDIRECTIONSANINCREASEINWMAKESANINDIVIDUALBETTEROFFBECAUSEHEORSHEISASUPPLIEROFLABORINTHECASEOFACONSUMPTIONGOOD,INDIVIDUALSAREMADEWORSEOFFWHENAPRICERISESBECAUSETHEYARECONSUMERSOFTHATGOODWECANSUMMARIZETHISANALYSIS,ASFOLLOWSINCOMEANDSUBSTITUTIONEFFECTSOFACHANGEINTHEREALWAGEWHENTHEREALWAGERATEINCREASES,AUTILITYMAXIMIZINGINDIVIDUALMAYINCREASEORDECREASEHOURSWORKEDTHESUBSTITUTIONEFFECTWILLTENDTOINCREASEHOURSWORKEDASTHEINDIVIDUALSUBSTITUTESEARNINGSFORLEISURE,WHICHISNOWRELATIVELYMORECOSTLYONTHEOTHERHAND,THEINCOMEEFFECTWILLTENDTOREDUCEHOURSWORKEDASTHEINDIVIDUALUSESHISORHERINCREASEDPURCHASINGPOWERTOBUYMORELEISUREHOURSOPTIMIZATIONPRINCIPLE608PARTIVPRODUCTIONANDSUPPLY3IFTHEFAMILYISTAKENTOBETHERELEVANTDECISIONUNIT,EVENMORECOMPLEXQUESTIONSARISEABOUTTHEINCOMEANDSUBSTITUTIONEFFECTSTHATCHANGESINTHEWAGESOFONEFAMILYMEMBER,SAY,THEHUSBAND,WILLHAVEONTHELABORFORCEBEHAVIOROFOTHERFAMILYMEMBERS,SUCHASTHEWIFEWENOWTURNTOEXAMINEAMATHEMATICALDEVELOPMENTOFTHESERESPONSESTHATPROVIDESADDITIONALINSIGHTSINTOTHELABORSUPPLYDECISIONAMATHEMATICALANALYSISOFLABORSUPPLYTODERIVEAMATHEMATICALSTATEMENTOFLABORSUPPLYDECISIONS,ITISHELPFULRSTTOAMENDTHEBUDGETCONSTRAINTSLIGHTLYTOALLOWFORTHEPRESENCEOFNONLABORINCOMETODOSO,WEREWRITEEQUATION223ASCH11005WLH11001N,229WHERENISREALNONLABORINCOMEANDMAYINCLUDESUCHITEMSASDIVIDENDANDINTERESTINCOME,RECEIPTOFGOVERNMENTTRANSFERBENETS,ORSIMPLYGIFTSFROMOTHERPERSONSINDEED,NCOULDSTANDFORLUMPSUMTAXESPAIDBYTHISPERSON,INWHICHCASEITSVALUEWOULDBENEGATIVEMAXIMIZATIONOFUTILITYSUBJECTTOTHISNEWBUDGETCONSTRAINTWOULDYIELDRESULTSVIRTUALLYIDENTICALTOTHOSEALREADYDERIVEDTHATIS,THENECESSARYCONDITIONFORAMAXIMUMDESCRIBEDINEQUATION228WOULDCONTINUETOHOLDASLONGASTHEVALUEOFNISUNAFFECTEDBYTHELABORLEISURECHOICESBEINGMADETHATIS,SOLONGASNISACHAPTER22LABORSUPPLY609INCOMEANDSUBSTITUTIONEFFECTSOFACHANGEINTHEREALWAGERATEWBECAUSETHEINDIVIDUALISASUPPLIEROFLABOR,THEINCOMEANDSUBSTITUTIONEFFECTSOFANINCREASEINTHEREALWAGERATEWWORKINOPPOSITEDIRECTIONSINTHEIREFFECTSONTHEHOURSOFLEISUREDEMANDEDORONHOURSOFWORKINATHESUBSTITUTIONEFFECTMOVEMENTTOPOINTSOUTWEIGHSTHEINCOMEEFFECT,ANDAHIGHERWAGECAUSESHOURSOFLEISURETODECLINETOH1HOURSOFWORKTHEREFOREINCREASEINBTHEINCOMEEFFECTISSTRONGERTHANTHESUBSTITUTIONEFFECT,ANDHINCREASESTOH1INTHISCASEHOURSOFWORKDECLINECONSUMPTIONCONSUMPTIONLEISURELEISUREABC1C1C0C0H1H0H0H1SSU1U1U0U0CH11005W124H11002CH11005W124H11002CH11005W024H11002CH11005W024H11002HHHHFIGURE221“LUMPSUM”RECEIPTORLOSSOFINCOME,4THEONLYEFFECTOFINTRODUCINGNONLABORINCOMEINTOTHEANALYSISISTOSHIFTTHEBUDGETCONSTRAINTSINFIGURE221OUTWARDORINWARDINAPARALLELMANNERWITHOUTAFFECTINGTHETRADEOFFRATEBETWEENEARNINGSANDLEISURETHISDISCUSSIONSUGGESTSTHATWECANWRITETHEINDIVIDUALSLABORSUPPLYFUNCTIONASLW,NTOINDICATETHATTHENUMBEROFHOURSWORKEDWILLDEPENDBOTHONTHEREALWAGERATEANDONTHEAMOUNTOFREALNONLABORINCOMERECEIVEDONTHEASSUMPTIONTHATLEISUREISANORMALGOOD,H11128L/H11128NWILLBENEGATIVETHATIS,ANINCREASEINNWILLRAISETHEDEMANDFORLEISUREANDBECAUSETHEREAREONLY24HOURSINTHEDAYREDUCELTOSTUDYWAGEEFFECTSONLABORSUPPLYH11128L/H11128W,WEWILLNDITHELPFULRSTTOCONSIDERTHEDUALPROBLEMTOTHEINDIVIDUALSPRIMARYUTILITYMAXIMIZATIONPROBLEMDUALSTATEMENTOFTHEPROBLEMASWESHOWEDINCHAPTER5,RELATEDTOTHEINDIVIDUALSPRIMARYPROBLEMOFUTILITYMAXIMIZATIONGIVENABUDGETCONSTRAINTISTHEDUALPROBLEMOFMINIMIZINGTHEEXPENDITURESNECESSARYTOATTAINAGIVENUTILITYLEVELINTHEPRESENTCONTEXT,THISPROBLEMCANBEPHRASEDASCHOOSINGVALUESFORCONSUMPTIONCANDLEISURETIMEHH1100524H11002LSOTHATTHEAMOUNTOFADDITIONALSPENDING,EH11005CH11002WL,2210REQUIREDTOATTAINAGIVENUTILITYLEVELSAY,U0H11005UC,HISASSMALLASPOSSIBLEASINCHAPTER5,SOLVINGTHISMINIMIZATIONPROBLEMWILLYIELDEXACTLYTHESAMESOLUTIONASSOLVINGTHEUTILITYMAXIMIZATIONPROBLEMNOWWECANAPPLYTHEENVELOPETHEOREMTOTHEMINIMUMVALUEFORTHESEEXTRAEXPENDITURESCALCULATEDINTHEDUALPROBLEMSPECICALLY,ASMALLCHANGEINTHEREALWAGEWILLCHANGETHEMINIMUMEXPENDITURESREQUIREDBYH11005H11002L2211INTUITIVELY,EACH1INCREASEINWREDUCESTHEREQUIREDVALUEOFEBYL,BECAUSETHATISTHEEXTENTTOWHICHLABOREARNINGSAREINCREASEDBYTHEWAGECHANGETHISRESULTISVERYSIMILARTOSHEPHARDSLEMMAINTHETHEORYOFPRODUCTIONSEECHAPTERS12AND21HERETHERESULTSHOWSTHATALABORSUPPLYFUNCTIONCANBECALCULATEDFROMTHEEXPENDITUREFUNCTIONBYPARTIALDIFFERENTIATIONBECAUSEUTILITYISHELDCONSTANTINTHEDUALEXPENDITUREMINIMIZATIONAPPROACH,THISFUNCTIONSHOULDBEINTERPRETEDASA“COMPENSATED”CONSTANTUTILITYLABORSUPPLYFUNCTION,WHICHWEWILLDENOTEBYLCW,UTODIFFERENTIATEITFROMTHEUNCOMPENSATEDLABORSUPPLYFUNCTIONLW,NINTRODUCEDEARLIERH11128EH5007H11128W610PARTIVPRODUCTIONANDSUPPLY4INMANYSITUATIONS,HOWEVER,NITSELFMAYDEPENDONLABORSUPPLYDECISIONSFOREXAMPLE,THEVALUEOFWELFAREORUNEMPLOYMENTBENETSAPERSONCANRECEIVEDEPENDSONHISORHEREARNINGS,ASDOESTHEAMOUNTOFINCOMETAXESPAIDINSUCHCASESTHESLOPEOFTHEINDIVIDUALSBUDGETCONSTRAINTWILLNOLONGERBEREECTEDBYTHEREALWAGEBUTMUSTINSTEADREECTTHENETRETURNTOADDITIONALWORKAFTERTAKINGINCREASEDTAXESANDREDUCTIONSINTRANSFERPAYMENTSINTOACCOUNTFORSOMEEXAMPLES,SEETHEPROBLEMSATTHEENDOFTHISCHAPTERSLUTSKYEQUATIONOFLABORSUPPLYNOWWECANUSETHESECONCEPTSTODERIVEASLUTSKYTYPEEQUATIONTHATREECTSTHESUBSTITUTIONANDINCOMEEFFECTSTHATRESULTFROMCHANGESINTHEREALWAGEWEBEGINBYRECOGNIZINGTHATTHEEXPENDITURESBEINGMINIMIZEDINTHEDUALPROBLEMOFEQUATION2211PLAYTHEROLEOFNONLABORINCOMEINTHEPRIMALUTILITYMAXIMIZATIONPROBLEMHENCE,BYDENITION,ATTHEOPTIMALPOINTWEHAVELCW,UH11005LW,EW,UH11005LW,N2212PARTIALDIFFERENTIATIONOFBOTHSIDESOFEQUATION2212WITHRESPECTTOWYIELDSH11005H11001H11080,2213ANDUSINGTHEENVELOPERELATIONFROMEQUATION2211FORH11128E/H11128W,WEHAVEH11005H11002LH11005H11002L2214INTRODUCINGASLIGHTLYDIFFERENTNOTATIONFORTHECOMPENSATEDLABORSUPPLYFUNCTION,H11005H20841UH11005U0,2215ANDREARRANGINGTERMSGIVESTHENALSLUTSKYEQUATIONFORLABORSUPPLYH11005H20841UH11005U0H11001L2216INWORDSASWEHAVEPREVIOUSLYSHOWN,THECHANGEINLABORSUPPLIEDINRESPONSETOACHANGEINTHEREALWAGECANBEDISAGGREGATEDINTOTHESUMOFASUBSTITUTIONEFFECTINWHICHUTILITYISHELDCONSTANTANDANINCOMEEFFECTTHATISANALYTICALLYEQUIVALENTTOANAPPROPRIATECHANGEINNONLABORINCOMEBECAUSETHESUBSTITUTIONEFFECTISPOSITIVEAHIGHERWAGEINCREASESTHEAMOUNTOFWORKCHOSENWHENUTILITYISHELDCONSTANTANDTHETERMH11128L/H11128NISNEGATIVE,THISDERIVATIONSHOWSTHATTHESUBSTITUTIONANDINCOMEEFFECTSWORKINOPPOSITEDIRECTIONSTHEMATHEMATICALDEVELOPMENTSUPPORTSTHEEARLIERCONCLUSIONSFROMOURGRAPHICALANALYSISANDSUGGESTATLEASTTHETHEORETICALPOSSIBILITYTHATTHELABORSUPPLYCURVEMIGHTBE“BACKWARDBENDING”THEMATHEMATICALDEVELOPMENTALSOSUGGESTSTHATTHEIMPORTANCEOFNEGATIVEINCOMEEFFECTSMAYBEGREATERTHEGREATERISTHEAMOUNTOFLABORITSELFBEINGSUPPLIEDEXAMPLE221COBBDOUGLASLABORSUPPLYTHECOBBDOUGLASUTILITYFUNCTIONPROVIDESANINSTRUCTIVEEXAMPLEOFTHESEOFFSETTINGSUBSTITUTIONANDINCOMEEFFECTSINLABORSUPPLYDECISIONSSUPPOSETHATHOURLYUTILITYISAFUNCTIONOFCONSUMPTIONANDLEISUREOFTHEFORMUH11005H20857CHH335262217ASBEFORE,THEBUDGETCONSTRAINTISCH11005WLH11001N2218H11128LH5007H11128NH11128LH5007H11128WH11128LH5007H11128WH11128LH5007H11128WH11128LCH5007H11128WH11128LH5007H11128NH11128LH5007H11128WH11128LH5007H11128EH11128LH5007H11128WH11128LCH5007H11128WH11128EH5007H11128WH11128LH5007H11128EH11128LH5007H11128WH11128LCH5007H11128WCHAPTER22LABORSUPPLY611ANDTHETIMECONSTRAINTISHH110051H11002L,2219WHERE,FORSIMPLICITY,WEHAVESETMAXIMUMWORKTIMEEQUALTO1HOURCOMBININGTHESEEQUATIONS,WECANEXPRESSUTILITYASAFUNCTIONOFLABORSUPPLYCHOICEONLYU2H11005CHH11005WLH11001N1H11002LH11005WLH11002WL2H11001NH11002NL2220DIFFERENTIATIONOFU2WITHRESPECTTOLYIELDSTHERSTORDERCONDITIONFORAUTILITYMAXIMUMH11005WH110022WLH11002NH1100502221ORLH11005H110022222THIS,THEN,ISTHISINDIVIDUALSLABORSUPPLYFUNCTIONIFNH110050,THISPERSONWILLWORKH500712H5007OFEACHHOURNOMATTERWHATTHEWAGEISTHATIS,IFNH110050,THESUBSTITUTIONANDINCOMEEFFECTSOFACHANGEINWPRECISELYOFFSETEACHOTHERANDLEAVELUNAFFECTEDEXAMININGINCOMEANDSUBSTITUTIONEFFECTSAMORECOMPLETEANALYSISOFWHYTHISISSOREQUIRESTHATWEEXAMINETHEINCOMEANDSUBSTITUTIONEFFECTSSEPARATELYCALCULATIONOFTHEINCOMEEFFECTINTHESLUTSKYEQUATION2216ISSTRAIGHTFORWARDUSINGTHEOPTIMALCHOICEINEQUATION2222LH11080H11005H20898H11002H20899H20898H11002H20899H11005H11002H110012223IFNH110050,THISINCOMEEFFECTISSIMPLYLH11080H11005H11002,WHERETHENEGATIVESIGNINDICATESTHATTHEINCOMEEFFECTOFANINCREASEINWWILLREDUCEL,BECAUSELEISUREISANORMALGOODCALCULATIONOFTHESUBSTITUTIONEFFECTINTHESLUTSKYEQUATIONISARATHERMESSYPROCESSFIRST,ONEMUSTDERIVEANEXPRESSIONFORINDIRECTUTILITYASAFUNCTIONOFWANDNTHETWOEXOGENOUSELEMENTSINTHEINDIVIDUALSBUDGETCONSTRAINTANDTHENUSETHISTOELIMINATENFROMTHEOPTIMALLABORSUPPLYCHOICEGIVENBYEQUATION2222LUCKILY,YOURAUTHORHASMADETHISCALCULATIONFORYOU5LCW,UH110051H110022224UH5007H20857WH335261H50074WH11128LH5007H11128NNH50074W21H50074W1H50072WNH50072W1H50072H11128LH5007H11128NNH50072W1H50072H11128U2H5007H11128L612PARTIVPRODUCTIONANDSUPPLY5THEINDIRECTUTILITYFUNCTIONISUH11005H11001NH50072H20857WH33526H20857WH33526H50072THISCONSTANTUTILITYLABORSUPPLYFUNCTIONSHOWSTHATIFONLYSUBSTITUTIONEFFECTSAREALLOWED,H11128LC/H11128WISDENITELYPOSITIVEH110052225REPLACINGUWITHITSINDIRECTREPRESENTATIONINTERMSOFWANDNINEQUATION2225SEEFOOTNOTE5NOWYIELDSH11005H110012226HENCE,IFNH110050,H11005,2227ANDTHESLUTSKYEQUATIONH11005H11001LH11005H11002H1100502228SHOWSTHATTHESUBSTITUTIONANDINCOMEEFFECTSAREPRECISELYOFFSETTINGINTHISCOBBDOUGLASCASEQUERYHOWDOTHEMATHEMATICALRESULTSOFTHISPROBLEMCOMPARETOTHEINTUITIVEIDEATHATWITHTHECOBBDOUGLASUTILITYFUNCTIONAPERSONALWAYSSPENDSACONSTANTFRACTIONOFINCOMEONEACHGOODREGARDLESSOFPRICEEXAMPLE222EFFECTSOFNONLABORINCOMEIFNH110140THEINDIVIDUALHASSOMENONLABORINCOME,THEPRECISEOFFSETTINGOFINCOMEANDSUBSTITUTIONEFFECTSSHOWNINEXAMPLE221WOULDNOTOCCURTHEEXPLANATIONFORTHISISTHATTHISPERSONWILLALWAYSCHOOSETOSPENDHALFOFHISORHERNONLABORINCOMEONLEISUREBUTLEISURE“COSTS”WPERHOUR,ANDARISEINWWILLMEANTHATLESSLEISURECANBE“BOUGHT”WITHAXEDNUMBEROFNDOLLARSIF,FOREXAMPLE,NH110052PERHOURANDWIS10PERHOUR,EQUATION2222SHOWSTHATTHISINDIVIDUALWILLWORKLH11005H11002H110052229THISPERSONSPENDS1OFHISORHERNONLABORINCOMEONLEISUREEACHHOURATAWAGEOF10,THIS1WILLBUYH5007110H5007OFANHOUROFLEISUREIF,ONTHEOTHERHAND,WH110055,THE1WOULDBUYH5007120H5007OFANHOURANDLH11005H11002H110052230WITHNONLABORINCOME,THEREFORE,THEINCOMEANDSUBSTITUTIONEFFECTSARENOTOFFSETTINGTHESUBSTITUTIONEFFECTDOMINATES,ANDAFALLINWAGESREDUCESHOURSOFWORK3H5007102H5007101H500724H5007102H5007201H500721H50074W1H50074WH11128LH5007H11128NH11128LCH5007H11128WH11128LH5007H11128W1H50074WH11128LCH5007H11128WNH50074W21H50074WH11128LCH5007H11128WUH50072W3/2H11128LCH5007H11128WCHAPTER22LABORSUPPLY613CHANGESINNONLABORINCOMEANDTHEEFFECTSOFSUBSIDIESCLEARLY,THEEFFECTOFANINCREASEINNINTHISPROBLEMISTOREDUCEHOURSOFWORKWITHNH110054ANDWH1100510,FOREXAMPLE,H5007130H5007OFANHOUROFWORKISSUPPLIEDFORNH1100510,HOURSOFWORKWOULDFALLTOZEROIFNISINTERPRETEDASANINCOMESUBSIDYFROMTHEGOVERNMENT,THISMODELTHENPROVIDESSOMESUPPORTFORTHEPURPORTEDNEGATIVELABORSUPPLYEFFECTSOFINCOMEMAINTENANCEPROGRAMSQUERYSUPPOSETHEGOVERNMENTSETSNAT4BUTREDUCESNBYHALFOFEACHDOLLAREARNEDTHATIS,NH110054H11002WOULDTHISSCHEMEINCREASEORDECREASELABORSUPPLYRELATIVETOAATGRANTOF4REGARDLESSOFLABORSUPPLYMARKETSUPPLYCURVEFORLABORWECANCONSTRUCTAMARKETSUPPLYOFLABORCURVEFROMINDIVIDUALLABORSUPPLYDECISIONSATEACHPOSSIBLEWAGERATEWEWOULDADDTOGETHERTHEQUANTITYOFLABOROFFEREDBYEACHINDIVIDUALTOARRIVEATAMARKETTOTALONEPARTICULARLYINTERESTINGASPECTOFTHISPROCEDUREISTHATASTHEWAGERATERISES,MOREINDIVIDUALSMAYBEINDUCEDTOENTERTHELABORFORCEFIGURE222ILLUSTRATESTHISPOSSIBILITYFORTHESIMPLECASEOFTWOINDIVIDUALSFORAREALWAGEBELOWW1NEITHERINDIVIDUALCHOOSESTOWORKWLH50072614PARTIVPRODUCTIONANDSUPPLYCONSTRUCTIONOFTHEMARKETSUPPLYCURVEFORLABORASTHEREALWAGERISES,THEREARETWOREASONSWHYTHESUPPLYOFLABORMAYINCREASEFIRST,HIGHERREALWAGESMAYCAUSEEACHINDIVIDUALINTHEMARKETTOWORKMOREHOURSSECOND,HIGHERWAGESMAYINDUCEMOREINDIVIDUALSFOREXAMPLE,INDIVIDUAL2TOENTERTHELABORMARKETREALWAGEREALWAGEREALWAGEHOURSHOURSTOTALLABORSUPPLYAINDIVIDUAL1BINDIVIDUAL2CTHEMARKETW1W2W3S1S2SFIGURE222CONSEQUENTLY,THEMARKETSUPPLYCURVEOFLABORFIGURE223CSHOWSTHATNOLABORISSUPPLIEDATREALWAGESBELOWW1AWAGEINEXCESSOFW1CAUSESINDIVIDUAL1TOENTERTHELABORMARKETHOWEVER,ASLONGASWAGESFALLSHORTOFW2,INDIVID