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1、Chapter3:TheConsumer’sProblem3.A:BudgetsetT•ConsumptionBundles:x=(x,x,.....x),wherex,i∈Nisthequantityofgood12nini.Hence,XisR.+•PropertiesoftheconsumptionsetX1.Φ≠X2.Xisclosed.3.Xisconvex.4.0∈Xn•BudgetSet:B={x∈R:Px≤w}isthesetofallfeasibleconsumptionbundles
2、forp,w+ntheconsumerwhofacesmarketpricesP=(p,p......p)∈Randhaswealthw.12n•Remark:marketprice(P)•BudgetSetisnonempty(why?(0,0,…0)),closed(?)andbounded(allpricesarestrictlypositive)3.B:Utilityfunctionn•Definition3.1:Areal-valuedfunctionu:R→Riscalledautility
3、function+representingthepreferencerelation≿,ifforallx,,y∈Xu(x)≥u(y)⇔x≿y.•Theorem3.2:Apreferencerelation≿canberepresentedbyautilityfunctiononlyifitisrational.[Figure1.1]1•Axiom3:Continuity.(Mas-colell)Thepreferencerelation≿onXiscontinuousifitispreservedun
4、dernn∞nnlimits.Thatis,foranysequenceofpairs{(x,y)}withx≿yforalln,n=1nnx=limxy=limy,wehavex≿y.n→∞n→∞n(Reny)Forallx∈R,theuppercontourset{y∈X:y≿x}andthelowercontourset+n{y∈X:x≿y}arebothclosedinR.+[Figure1.2]Notallrationalpreferencesarecontinuous!2Example:Le
5、xicographicpreferencerelation.SupposeX=R.Definex≿yifeither+x>yorx=y,andx≥y.Itisrational.(Proof?)111122n⎛1⎞nnnIsitcontinuous?Supposetwosequences:x=⎜1+,1⎟y=()1,2.Thenx≿y.⎝n⎠nnHowever,x=limx=()1,1y=limy=(1,2),wehavey≿x.n→∞n→∞•Theorem3.3:Ifthebinaryrelation≿
6、iscomplete,transitiveandcontinuous,therenexistsacontinuousreal-valuedfunction,u:R→R,whichrepresents≿.+n•Axiom4:Monotonicity.Forallx,,y∈Rifx≥ythenx≿y,whileifx>>ythen+xfy.[Figure1.5]0n0n•Axiom4’:LocalNonsatiation.∀x∈R,andε>0,∃x∈B(x)∩Rsuchthat+ε+0xfx.⇒1.Alo
7、callynonsatiatedconsumerisincome-exhaustive.I.e.,shechoosesconsumptionbundlesontheboundaryofabudgetset.2.LocalNonsatiationrulesouta“thick”indifferencecurve.3.LocalNonsatiationdoesnotruleoutthat“less”ispreferredto“more”.2•Theorem3.3’:Ifthebinaryrelation≿i
8、scomplete,transitive,continuous,andstrictlynmonotonic,thereexistsacontinuousreal-valuedfunction,u:,R→Rwhich+represents≿.Proof:nStep1.Lete≡()1,1,...1.Foreveryx∈R,monotonicityimpliesthatx≿0.Thereisa+usuchthatue>>x.Hence,weha
