中级微观经济学(尼克尔森).ppt

1 第一讲经济理论的基础 Nicholson Chapter1pp 1 20 2 决策行为的模型化 经济学通常被定义为 在一个稀缺性的环境之下的理性选择行为研究 但是从我们的课本内容来看 往往并不能看到这一点 经济学在其中似乎只能被定义为 在一个变化的环境中对于市场交易和市场行为的研究 近一个世纪以来 经济学的研究方法虽然已经取得了很大的进步 但是结论似乎并没有多大的变化 3 但在另一方面 个人主义方法论在其他社会科学领域的推广应用却给我们带来了很大的收获 而这些推广应用的基石正是经济学的基本逻辑 理性人 在自利本性下的选择 我们在课堂中学习的主要是经济学的分析方法 掌握了这套方法和逻辑 日后我们可以运用它分析任何选择行为 因此 经济学的适用性远远超出了经济学本身的领域 或者说 经济学的研究领域已经不再是传统上认识的范畴了 4 e g thestudyoftheimplicationsofchoiceinasettingofscarcity Politicalwheelinganddealingisseenasaformofcomplexexchange Thecriminaljusticesystemisseenasamethodofpricingillegalactivities Politicians statusseekers altruistsandcriminalsareallseenasrationaldecisionmakerspursuingwell definedgoalsinasettingofscarcity 5 ThepurposeoftheMicrosequenceittoprovideyouwithacoresetoftools ideas intuitions mathematicalmodels andaptitudeforinterpretingtheworldineconomicterms thatwillallowyoutoapproacheconomicproblemswithgreatersophisticationthanyouhavenow Iwilloftenusenontraditionalexamplesasameansofseeingwhetheryoucan see howtoapplytraditionaleconomictoolstonon marketdecisionmaking 6 经济学分析的基本逻辑 经济学总是假定人们在做出某种选择行为 特别是 我们往往总是假定他们在把什么东西力图最大化 最优化 因此 几乎所有的决策选择都可以被模型化为一个最大化的问题 A 在任一决策行为中 你有选择的权利 这些选择就可以对应于最优化问题中的变量 variables 7 B 你当然想做出最优的选择 optimalchoice 这就意味着你试图为你的变量选择一个值 从而使其最大化某种函数 function 目标函数 C 你的选择或许会随着某些情况的变化而变化 你必须依赖于某些情况做出决定 而这是在你的能力之外的事情 你无法改变 比如你只能根据天气来决定穿什么衣服 所以我们必须要特别列明一些情况做为选择的条件 这些条件就是最优化问题的参数 parameter 包络定理 8 D 一旦你建构了一个特定情况下的模型 你就可以估计出个人 理性主体 到底会做出什么样的选择决策 在最优化问题中 这就意味着我们找到了参数函数中变量的最优值 E 一旦特定的环境发生了改变 即参数变化了 那么变量的最优值也会跟着改变 这就是比较静态 comparativestatics 比如需求的改变 不同于需求量的改变 9 F CeterisParibusmeans otherthingsthesame Economicmodelsattempttoexplainsimplerelationshipsfocusontheeffectsofonlyafewforcesatatimeothervariablesareassumedtobeunchangedduringtheperiodofstudy 10 人不能两次跳进同一条河中人不能一次跳进同一条河中如果所有的事情都在变化 就会陷于不可知论的泥潭 世界就无法认识了 经济模型试图从纷繁复杂的经济世界中 抽丝剥茧 找到其所关心的主要变量之间是在存在着内在关系 因此 一次只能研究一种或几种变量的相互关系 且此时其他变量保持不变 11 G 经济命题的 可证伪性 falsification 是命题科学性的关键所在 KarlPopper Therearetwogeneralmethodsusedtoverifyeconomicmodels directapproachestablishesthevalidityofthemodel sassumptionsindirectapproachshowsthatthemodelcorrectlypredictsreal worldevents 12 SuggestedReadings Nagel Ernest AssumptionsinEconomicTheory AmericanEconomicReview May1963 211 219 Caldwell BruceJ ClarifyingPopper JournalofEconomicLiterature March1991 1 33 Harrod RoyF ScopeandMethodinEconomics EconomicJournal48 1938 383 412 13 第二讲征税与补贴的影响 参考书目 pp 270 微观经济学 平狄克 鲁宾费尔德 14 关于税收的两种常见的误解 1 征税1 并不是商品价格上涨1 2 政府提高消费税1 并不是消费者购买商品时多付出1 一 无论是征税还是补贴 都一定是在需求者与供给者之间进行分担 分摊 的 而且 这种分摊效应与征税或者补贴直接面对的对象到底是哪一方无关 二 这种分担的效应取决于供求曲线的形状 尤其是它们两者之间的相对弹性的大小 15 TheImpactofaTaxorSubsidy Theburdenofatax orthebenefitofasubsidy fallspartlyontheconsumerandpartlyontheproducer Wewillconsideraspecifictax 从量税 whichisataxofacertainamountofmoneyperunitsold 16 IncidenceofaSpecificTax Quantity Price Pb PS tax 17 IncidenceofaSpecificTax Fourconditionsthatmustbesatisfiedafterthetaxisinplace 1 QuantitysoldandPdmustbeonthedemandline QD QD Pd 2 QuantitysoldandPSmustbeonthesupplyline QS QS PS 18 IncidenceofaSpecificTax Fourconditionsthatmustbesatisfiedafterthetaxisinplace 3 QD QS4 Pd PS tax 19 ImpactofaTaxDependsonElasticitiesofSupplyandDemand Quantity Quantity Price Price BurdenonBuyer BurdenonSeller 20 想一想税收与补贴的份额到底与什么因素相关 是什么使消费者或者生产者能够更多地将税赋转嫁给对方 而把补贴攫取过来 占为己有 21 如果将税赋换成补贴也是一样的 22 Pass throughfraction 转嫁因子 ES ES Ed 表明以高价形式转嫁给消费者的税收份额Forexample whendemandisperfectlyinelastic Ed 0 thepass throughfractionis1 andallthetaxisbornebytheconsumer 当Ed 0 转嫁因子 1 所有税赋由消费者承担 当Ed 转嫁因子 0 所有税赋由生产者承担 TheImpactofaTaxorSubsidy 23 TheEffectsofaTaxorSubsidy Asubsidycanbeanalyzedinmuchthesamewayasatax Itcanbetreatedasanegativetax 补贴等于降价 Theseller spriceexceedsthebuyer sprice 24 25 26 注意 税赋使均衡价格并没有上升T 而只是其3 5 27 第三讲最优值的数学处理方法 Nicholson Chapter2pp 52 55 28 ConstrainedMaximization Supposewewanttochoosex1andx2tomaximizey f x1 x2 subjecttothelinearconstraintc b1x1 b2x2 0WecansetuptheLagrangianL f x1 x2 c b1x1 b2x2 29 经常可以表达为 Maxy f x1 x2 s t c b1x1 b2x2 0 30 ConstrainedMaximization Thefirst orderconditionsaref1 b1 0f2 b2 0c b1x1 b2x2 0Toensurewehaveamaximum wemustusethe second totaldifferentiald2y f11dx12 2f12dx1dx2 f22dx22 31 ConstrainedMaximization Onlythevaluesofx1andx2thatsatisfytheconstraintcanbeconsideredvalidalternativestothecriticalpointThus wemustcalculatethetotaldifferentialoftheconstraint b1dx1 b2dx2 0dx2 b1 b2 dx1Thesearetheallowablerelativechangesinx1andx2 32 ConstrainedMaximization Becausethefirst orderconditionsimplythatf1 f2 b1 b2 wecansubstituteandgetdx2 f1 f2 dx1Sinced2y f11dx12 2f12dx1dx2 f22dx22wecansubstitutefordx2andgetd2y f11dx12 2f12 f1 f2 dx12 f22 f12 f22 dx12 33 ConstrainedMaximization Combiningtermsandrearrangingd2y f11f22 2f12f1f2 f22f12 dx12 f22 Therefore ford2y 0 itmustbetruethatf11f22 2f12f1f2 f22f12 0Thisequationcharacterizesasetoffunctionstermedquasi concavefunctionsanytwopointswithinthesetcanbejoinedbyalinecontainedcompletelyintheset 34 关于quasi concavefunctions的详细讨论可以参看蒋中一的 数理经济学的基本方法 一书P506的相关内容 35 二阶条件的另一表示方法 BorderedHessiandeterminant加边海塞行列式f11f22 2f12f1f2 f22f12 0二阶条件表明MRS递减 无差异曲线凸向原点 原函数为拟凹函数 36 N个未知数的最值问题 37 对于最大化的目标而言 要求d2y必须为负定的 即所有theleadingprincipalminorsofHb 加边主子式 必须遵循 而对于最小值的目标而言 则要求d2y必须为正定的 即所有theleadingprincipalminorsofHb 加边主子式 都为负 蒋中一 P 502清晰 权威 pp 60E2 3 38 请认真阅读蒋书中的P499 504的相关内容 其中之所以与我们教材中关于最大值判定的符号相反是因为那里的切记 39 pp 61e g 40 效用的最大化 一阶条件 41 二阶条件 42 EconomicProfits Totalcostsforthefirmaregivenbytotalcosts C wl vkTotalrevenueforthefirmisgivenbytotalrevenue pq pf k l Economicprofits areequalto totalrevenue totalcost pq wl vk pf k l wl vk 43 EconomicProfits Economicprofitsareafunctionoftheamountofcapitalandlaboremployedwecouldexaminehowafirmwouldchoosekandltomaximizeprofit deriveddemand theoryoflaborandcapitalinputsfornow wewillassumethatthefirmhasalreadychosenitsoutputlevel q0 andwantstominimizeitscosts 44 Cost MinimizingInputChoices Tominimizethecostofproducingagivenlevelofoutput afirmshouldchooseapointontheisoquantatwhichtheRTSisequaltotheratiow vitshouldequatetherateatwhichkcanbetradedforlintheproductiveprocesstotherateatwhichtheycanbetradedinthemarketplace 45 Cost MinimizingInputChoices Mathematically weseektominimizetotalcostsgivenq f k l q0SettinguptheLagrangian L wl vk q0 f k l Firstorderconditionsare L l w f l 0 L k v f k 0 L q0 f k l 0 46 Cost MinimizingInputChoices Dividingthefirsttwoconditionsweget Thecost minimizingfirmshouldequatetheRTSforthetwoinputstotheratiooftheirprices 47 Cost MinimizingInputChoices Cross multiplying weget Forcoststobeminimized themarginalproductivityperdollarspentshouldbethesameforallinputs 48 Cost MinimizingInputChoices Notethatthisequation sinverseisalsoofinterest TheLagrangianmultipliershowshowmuchinextracostswouldbeincurredbyincreasingtheoutputconstraintslightly 49 Costsarerepresentedbyparallellineswithaslopeof w v Cost MinimizingInputChoices lperperiod kperperiod C1 C2 C3 50 C1 C2 C3 q0 Theminimumcostofproducingq0isC2 Cost MinimizingInputChoices lperperiod kperperiod 51 第四讲偏好与效用 Nicholson Chapter3pp 65 90 52 AxiomsofRationalChoice CompletenessifAandBareanytwosituations anindividualcanalwaysspecifyexactlyoneofthesepossibilities AispreferredtoBBispreferredtoAAandBareequallyattractive 53 AxiomsofRationalChoice TransitivityifAispreferredtoB andBispreferredtoC thenAispreferredtoCassumesthattheindividual schoicesareinternallyconsistent 54 AxiomsofRationalChoice ContinuityifAispreferredtoB thensituationssuitably closeto AmustalsobepreferredtoBusedtoanalyzeindividuals responsestorelativelysmallchangesinincomeandprices 55 Utility Giventheseassumptions itispossibletoshowthatpeopleareabletorankinorderallpossiblesituationsfromleastdesirabletomostEconomistscallthisrankingutilityifAispreferredtoB thentheutilityassignedtoAexceedstheutilityassignedtoBU A U B 56 Utility UtilityrankingsareordinalinnaturetheyrecordtherelativedesirabilityofcommoditybundlesBecauseutilitymeasuresarenotunique itmakesnosensetoconsiderhowmuchmoreutilityisgainedfromAthanfromBItisalsoimpossibletocompareutilitiesbetweenpeople 57 Utility Utilityisaffectedbytheconsumptionofphysicalcommodities psychologicalattitudes peergrouppressures personalexperiences andthegeneralculturalenvironmentEconomistsgenerallydevoteattentiontoquantifiableoptionswhileholdingconstanttheotherthingsthataffectutilityceterisparibusassumption 58 Utility Assumethatanindividualmustchooseamongconsumptiongoodsx1 x2 xnTheindividual srankingscanbeshownbyautilityfunctionoftheform utility U x1 x2 xn otherthings thisfunctionisuniqueuptoanorder preservingtransformation 59 EconomicGoods Intheutilityfunction thex sareassumedtobe goods moreispreferredtoless Quantityofx Quantityofy x y 60 IndifferenceCurves Anindifferencecurveshowsasetofconsumptionbundlesamongwhichtheindividualisindifferent Quantityofx Quantityofy x1 y1 y2 x2 U1 Combinations x1 y1 and x2 y2 providethesamelevelofutility 61 MarginalRateofSubstitution Thenegativeoftheslopeoftheindifferencecurveatanypointiscalledthemarginalrateofsubstitution MRS Quantityofx Quantityofy x1 y1 y2 x2 U1 62 MarginalRateofSubstitution MRSchangesasxandychangereflectstheindividual swillingnesstotradeyforx Quantityofx Quantityofy x1 y1 y2 x2 U1 63 IndifferenceCurveMap Eachpointmusthaveanindifferencecurvethroughit Quantityofx Quantityofy U1 U2 U3 64 Transitivity Cananytwoofanindividual sindifferencecurvesintersect Quantityofx Quantityofy U1 U2 A B C TheindividualisindifferentbetweenAandC TheindividualisindifferentbetweenBandC TransitivitysuggeststhattheindividualshouldbeindifferentbetweenAandB ButBispreferredtoAbecauseBcontainsmorexandythanA 65 Convexity Asetofpointsisconvexifanytwopointscanbejoinedbyastraightlinethatiscontainedcompletelywithintheset Quantityofx Quantityofy U1 TheassumptionofadiminishingMRSisequivalenttotheassumptionthatallcombinationsofxandywhicharepreferredtox andy formaconvexset x y 66 Convexity Iftheindifferencecurveisconvex thenthecombination x1 x2 2 y1 y2 2willbepreferredtoeither x1 y1 or x2 y2 Quantityofx Quantityofy U1 x2 y1 y2 x1 Thisimpliesthat well balanced bundlesarepreferredtobundlesthatareheavilyweightedtowardonecommodity x1 x2 2 y1 y2 2 67 UtilityandtheMRS Supposeanindividual spreferencesforhamburgers y andsoftdrinks x canberepresentedby Solvingfory wegety 100 x SolvingforMRS dy dx MRS dy dx 100 x2 68 UtilityandtheMRS MRS dy dx 100 x2Notethatasxrises MRSfallswhenx 5 MRS 4whenx 20 MRS 0 25 69 MarginalUtility Supposethatanindividualhasautilityfunctionoftheformutility U x y ThetotaldifferentialofUis Alonganyindifferencecurve utilityisconstant dU 0 70 DerivingtheMRS Therefore weget MRSistheratioofthemarginalutilityofxtothemarginalutilityofy 71 DiminishingMarginalUtilityandtheMRS Intuitively itseemsthattheassumptionofdecreasingmarginalutilityisrelatedtotheconceptofadiminishingMRSdiminishingMRSrequiresthattheutilityfunctionbequasi concavethisisindependentofhowutilityismeasureddiminishingmarginalutilitydependsonhowutilityismeasuredThus thesetwoconceptsaredifferent 72 ConvexityofIndifferenceCurves Supposethattheutilityfunctionis WecansimplifythealgebrabytakingthelogarithmofthisfunctionU x y ln U x y 0 5lnx 0 5lny 73 ConvexityofIndifferenceCurves Thus 74 ConvexityofIndifferenceCurves IftheutilityfunctionisU x y x xy yThereisnoadvantagetotransformingthisutilityfunction so 75 ConvexityofIndifferenceCurves Supposethattheutilityfunctionis Forthisexample itiseasiertousethetransformationU x y U x y 2 x2 y2 76 ConvexityofIndifferenceCurves Thus 77 HomotheticPreferences IftheMRSdependsonlyontheratiooftheamountsofthetwogoods notonthequantitiesofthegoods theutilityfunctionishomothetic pp 84 277 Perfectsubstitutes MRSisthesameateverypointPerfectcomplements MRS ify x undefinedify x andMRS 0ify x 78 对于Homotheticfunction的理解 Pp 84 277 Homothetic函数的特征就是每一条无差异曲线都有相同的形状 曲线的斜率只与Y X的比率有关 无论其与原点的距离有多远 这样 我们研究消费者的行为 即便是在不同的效用水平上 也不用担心其会有不同的偏好 保证了偏好保持不变 79 Theimportanceofhomotheticfunctionsisthatinsuchasituation oneindifferencecurveismuchlikeanother SlopesofthecurvesdependonlyontheratioY X notonhowfarthecurveisfromtheorigin Indifferencecurvesforhigherutilityaresimplecopiesofthoseforlowerutility Hence wecanstudythebehaviorofanindividualwhohashomotheticpreferencesbylookingonlyatoneindifferencecurveoratafewnearbycurveswithoutfearingthatourresultswouldchangedramaticallyatverydifferentlevelsofutility 80 homogeneity pp 116 Moregenerally afunctionf X1 X2 Xn issaidtobehomogeneousofdegreekiff tX1 tX2 tXn tkf X1 X2 Xn foranyt 0 Themostcommoncasesofhomogeneousfunctionsarek 0andk 1 Iffishomogeneousofdegreezero doublingallofitsargumentsleavesfunchangedinvalue Iffishomogeneousofdegree1 doublingallitsargumentswilldoublethevalueoff Weshallencounterfunctionshomogeneousofdegree1inPartIV 81 HomogeneousFunction Afunction f X1 X2 Xn ishomogeneousofdegreekiff mX1 mX2 mXn mkf X1 X2 Xn 82 Stigler G TheDevelopmentofUtilityTheory JournalofPoliticalEconomy59 pts 1 2 August October1950 307 327 373 396 83 第五讲效用与生产函数的几种常见形式 Nicholson pp 80 83 Chap 11 12 84 几个常用的希腊字母 delta epsilon sigma rho 85 pp 81 86 ExamplesofUtilityFunctions Cobb DouglasUtilityutility U x y x y where and arepositiveconstantsTherelativesizesof and indicatetherelativeimportanceofthegoods 87 ExamplesofUtilityFunctions PerfectSubstitutesutility U x y x y Quantityofx Quantityofy Theindifferencecurveswillbelinear TheMRSwillbeconstantalongtheindifferencecurve 88 ExamplesofUtilityFunctions PerfectComplementsutility U x y min x y Quantityofx Quantityofy TheindifferencecurveswillbeL shaped Onlybychoosingmoreofthetwogoodstogethercanutilitybeincreased 89 ExamplesofUtilityFunctions CESUtility Constantelasticityofsubstitution Theelasticityofsubstitution isequalto1 1 Perfectsubstitutes Fixedproportions 0所谓的替代弹性 为无差异曲线的曲率 即过曲线上两点切线的正向夹角与这两点之间弧长的比值 90 ExamplesofUtilityFunctions CESUtility Constantelasticityofsubstitution utility U x y x y when 0andutility U x y lnx lnywhen 0Perfectsubstitutes 1Cobb Douglas 1 0Perfectcomplements 0 91 pp 281 92 TheLinearProductionFunction Supposethattheproductionfunctionisq f k l ak blThisproductionfunctionexhibitsconstantreturnstoscalef tk tl atk btl t ak bl tf k l AllisoquantsarestraightlinesRTSisconstant 93 TheLinearProductionFunction lperperiod kperperiod Capitalandlaborareperfectsubstitutes 94 FixedProportions Supposethattheproductionfunctionisq min ak bl a b 0Capitalandlabormustalwaysbeusedinafixedratiothefirmwillalwaysoperatealongaraywherek lisconstantBecausek lisconstant 0 95 FixedProportions lperperiod kperperiod Nosubstitutionbetweenlaborandcapitalispossible 0 96 Cobb DouglasProductionFunction Supposethattheproductionfunctionisq f k l AkalbA a b 0Thisproductionfunctioncanexhibitanyreturnstoscalef tk tl A tk a tl b Ata bkalb ta bf k l ifa b 1 constantreturnstoscaleifa b 1 increasingreturnstoscaleifa b 1 decreasingreturnstoscale 97 Cobb DouglasProductionFunction 98 Cobb DouglasProductionFunction TheCobb Douglasproductionfunctionislinearinlogarithmslnq lnA alnk blnlaistheelasticityofoutputwithrespecttokbistheelasticityofoutputwithrespecttol 99 CESProductionFunction Supposethattheproductionfunctionisq f k l k l 1 0 0 1 increasingreturnstoscale 1 decreasingreturnstoscaleForthisproductionfunction 1 1 1 linearproductionfunction fixedproportionsproductionfunction 0 Cobb Douglasproductionfunction 100 ElasticityofSubstitution Theelasticityofsubstitution measurestheproportionatechangeink lrelativetotheproportionatechangeintheRTSalonganisoquant 即k l相对变化率对于RTS相对变化率的反应程度 Thevalueof willalwaysbepositivebecausek landRTSmoveinthesamedirection 101 替代弹性与MRS的区别 Theelasticityofsubstitution 说明一种要素投入对于另一种要素投入的替代能力如何 通常的MRS只是知道递减 但是并没有确切地反映出两种要素的替代能力到底是多少 102 ElasticityofSubstitution lperperiod kperperiod q q0 istheratiooftheseproportionalchanges measuresthecurvatureoftheisoquant pp 279F 11 3 103 技术替代率是切线的斜率 而技术替代弹性则指的是曲线的曲率 弯曲程度 即指过曲线上两点切线的正向夹角与这两点之间弧长的比值 104 ElasticityofSubstitution If ishigh theRTSwillnotchangemuchrelativetok ltheisoquantwillberelativelyflatIf islow theRTSwillchangebyasubstantialamountask lchangestheisoquantwillbesharplycurvedItispossiblefor tochangealonganisoquantorasthescaleofproductionchanges 105 ElasticityofSubstitution Generalizingtheelasticityofsubstitutiontothemany inputcaseraisesseveralcomplicationsifwedefinetheelasticityofsubstitutionbetweentwoinputstobetheproportionatechangeintheratioofthetwoinputstotheproportionatechangeinRTS weneedtoholdoutputandthelevelsofotherinputsconstant 106 通常CES函数使用一个分配权重 0 1 表示投入的相对重要性 当规模报酬不变且 0时 这个函数收敛为一个Cobb Douglas函数 与K L的收入分配相关 107 Stigler G J TheDivisionofLaborIsLimitedbytheExtentoftheMarket JournalofPoliticalEconomy59 June1951 185 193 Douglas P H AreThereLawsofProduction AmericanEconomicReview38 March1948 1 41 TheCobb DouglasProductionFunctionOnceAgain ItsHistory ItsTesting andSomeEmpiricalValues JournalofPoliticalEconomy84 October1976 903 916 Machlup F OntheMeaningofMarginalProduct ReprintedinAmericanEconomicAssociation ReadingsintheTheoryofIncomeDistribution Philadelphia Blakiston 1951 pp 158 174 108 第六讲需求函数 Nicholson Chap 5 109 Ordinarydemandfunction 马歇尔需求函数 已知效用函数 在收入约束条件下 求效用最大化 110 效用的最大化 一阶条件 111 马歇尔需求函数 112 113 In a theindividual sutility maximizingchoicesofXandYareshownforthreedifferentpricesofX P X P X andP X In b thisrelationshipbetweenPXandXisusedtoconstructthedemandcurveforX ThedemandcurveisdrawnontheassumptionthatPY I andpreferencesremainconstantasPXvaries 114 两个重要的性质 一 任何商品的需求都是价格和收入的单值函数 single valuedfunction 二 需求函数是关于价格与收入的零次齐次函数 homogeneousofdegreezero 注意比较前面的homothetic 第一个性质源自效用函数的严格拟凹性 因此对应于一组特定的价格与收入 只能有一组能够获得最大效用的商品 第二个性质表明 收入与价格同方向同比例变化不会影响消费者的行为 115 Compensateddemandfunction 希克斯需求函数 已知效用函数 在特定的效用约束条件下 求成本最小化 116 同样根据一阶条件可以得到 这个函数同样也是关于价格的零次齐次函数 117 118 ThecurvehXshowshowthequantityofXdemandedchangeswhenPXchanges holdingPYandutilityconstant Thatis theindividual sincomeis compensated soastokeeputilityconstant Hence hXreflectsonlysubstitutioneffectsofchangingprices 119 两种效用函数的比较 120 Thecompensated hX anduncompensated dX demandcurvesintersectatP xbecauseX isdemandedundereachconcept ForpricesabovePX theindividual sincomeisincreasedwiththecompensateddemandcurve somoreXisdemandedthanwiththeuncompensatedcurve ForpricesbelowP x incomeisreducedforthecompensatedcurve solessXisdemande