Ch05-《中级微观经济学》范里安-英文版课件

ChapterFiveChoiceChapterFiveChoice1EconomicRationalityTheprincipalbehavioralpostulateisthatadecisionmakerchoosesitsmostpreferredalternativefromthoseavailabletoit.Theavailablechoicesconstitutethechoiceset.Howisthemostpreferredbundleinthechoicesetlocated?EconomicRationalityTheprinci2RationalConstrainedChoicex1x2RationalConstrainedChoicex1x3RationalConstrainedChoicex1x2UtilityRationalConstrainedChoicex14RationalConstrainedChoiceUtilityx2x1RationalConstrainedChoiceUt5RationalConstrainedChoicex1x2UtilityRationalConstrainedChoicex1x6RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti7RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti8RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti9RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti10RationalConstrainedChoiceUtilityx1x2Affordable,butnotthemostpreferredaffordablebundle.RationalConstrainedChoiceUti11RationalConstrainedChoicex1x2UtilityAffordable,butnotthemostpreferredaffordablebundle.Themostpreferred

oftheaffordablebundles.RationalConstrainedChoicex1x12RationalConstrainedChoicex1x2UtilityRationalConstrainedChoicex1x13RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti14RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti15RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti16RationalConstrainedChoicex1x2RationalConstrainedChoicex1x17RationalConstrainedChoicex1x2Affordable

bundlesRationalConstrainedChoicex1x18RationalConstrainedChoicex1x2Affordable

bundlesRationalConstrainedChoicex1x19RationalConstrainedChoicex1x2Affordable

bundlesMorepreferred

bundlesRationalConstrainedChoicex1x20RationalConstrainedChoiceAffordable

bundlesx1x2Morepreferred

bundlesRationalConstrainedChoiceAff21RationalConstrainedChoicex1x2x1*x2*RationalConstrainedChoicex1x22RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isthemost

preferredaffordable

bundle.RationalConstrainedChoicex1x23RationalConstrainedChoiceThemostpreferredaffordablebundleiscalledtheconsumer’sORDINARYDEMANDatthegivenpricesandbudget.Ordinarydemandswillbedenotedby

x1*(p1,p2,m)andx2*(p1,p2,m).RationalConstrainedChoiceThe24RationalConstrainedChoiceWhenx1*>0andx2*>0thedemandedbundleisINTERIOR.Ifbuying(x1*,x2*)costs$mthenthebudgetisexhausted.RationalConstrainedChoiceWhe25RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isinterior.(x1*,x2*)exhauststhe

budget.RationalConstrainedChoicex1x26RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isinterior.

(a)(x1*,x2*)exhauststhe

budget;p1x1*+p2x2*=m.RationalConstrainedChoicex1x27RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isinterior.

(b)Theslopeoftheindiff.

curveat(x1*,x2*)equals

theslopeofthebudget

constraint.RationalConstrainedChoicex1x28RationalConstrainedChoice(x1*,x2*)satisfiestwoconditions:(a)thebudgetisexhausted;

p1x1*+p2x2*=m(b)theslopeofthebudgetconstraint,-p1/p2,andtheslopeoftheindifferencecurvecontaining(x1*,x2*)areequalat(x1*,x2*).RationalConstrainedChoice(x129ComputingOrdinaryDemandsHowcanthisinformationbeusedtolocate(x1*,x2*)forgivenp1,p2andm?ComputingOrdinaryDemandsHow30ComputingOrdinaryDemands-aCobb-DouglasExample.SupposethattheconsumerhasCobb-Douglaspreferences.ComputingOrdinaryDemands-a31ComputingOrdinaryDemands-aCobb-DouglasExample.SupposethattheconsumerhasCobb-Douglaspreferences.

ThenComputingOrdinaryDemands-a32ComputingOrdinaryDemands-aCobb-DouglasExample.SotheMRSisComputingOrdinaryDemands-a33ComputingOrdinaryDemands-aCobb-DouglasExample.SotheMRSis

At(x1*,x2*),MRS=-p1/p2soComputingOrdinaryDemands-a34ComputingOrdinaryDemands-aCobb-DouglasExample.SotheMRSis

At(x1*,x2*),MRS=-p1/p2so(A)ComputingOrdinaryDemands-a35ComputingOrdinaryDemands-aCobb-DouglasExample.(x1*,x2*)alsoexhauststhebudgetso(B)ComputingOrdinaryDemands-a36ComputingOrdinaryDemands-aCobb-DouglasExample.Sonowweknowthat(A)(B)ComputingOrdinaryDemands-a37ComputingOrdinaryDemands-aCobb-DouglasExample.Sonowweknowthat(A)(B)SubstituteComputingOrdinaryDemands-a38ComputingOrdinaryDemands-aCobb-DouglasExample.Sonowweknowthat(A)(B)SubstituteandgetThissimplifiesto….ComputingOrdinaryDemands-a39ComputingOrdinaryDemands-aCobb-DouglasExample.ComputingOrdinaryDemands-a40ComputingOrdinaryDemands-aCobb-DouglasExample.Substitutingforx1*inthengivesComputingOrdinaryDemands-a41ComputingOrdinaryDemands-aCobb-DouglasExample.Sowehavediscoveredthatthemost

preferredaffordablebundleforaconsumer

withCobb-Douglaspreferences

isComputingOrdinaryDemands-a42ComputingOrdinaryDemands-aCobb-DouglasExample.x1x2ComputingOrdinaryDemands-a43RationalConstrainedChoiceWhenx1*>0andx2*>0

and(x1*,x2*)exhauststhebudget,

andindifferencecurveshaveno

‘kinks’,theordinarydemandsareobtainedbysolving:(a)p1x1*+p2x2*=y(b)theslopesofthebudgetconstraint,-p1/p2,andoftheindifferencecurvecontaining(x1*,x2*)areequalat(x1*,x2*).RationalConstrainedChoiceWhe44RationalConstrainedChoiceButwhatifx1*=0?Orifx2*=0?Ifeitherx1*=0orx2*=0thentheordinarydemand(x1*,x2*)isatacornersolutiontotheproblemofmaximizingutilitysubjecttoabudgetconstraint.RationalConstrainedChoiceBut45ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1ExamplesofCornerSolutions-46ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1>p2.ExamplesofCornerSolutions-47ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1>p2.ExamplesofCornerSolutions-48ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1>p2.ExamplesofCornerSolutions-49ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1<p2.ExamplesofCornerSolutions-50ExamplesofCornerSolutions--thePerfectSubstitutesCaseSowhenU(x1,x2)=x1+x2,themost

preferredaffordablebundleis(x1*,x2*)

whereandifp1<p2ifp1>p2.ExamplesofCornerSolutions-51ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1=p2.ExamplesofCornerSolutions-52ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2Allthebundlesinthe

constraintareequallythe

mostpreferredaffordable

whenp1=p2.ExamplesofCornerSolutions-53ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2BetterExamplesofCornerSolutions-54ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2ExamplesofCornerSolutions-55ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2Whichisthemostpreferred

affordablebundle?ExamplesofCornerSolutions-56ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2Themostpreferred

affordablebundleExamplesofCornerSolutions-57ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2Themostpreferred

affordablebundleNoticethatthe“tangencysolution”

isnotthemostpreferredaffordable

bundle.ExamplesofCornerSolutions-58Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions59Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2MRS=0U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions60Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2MRS=-¥MRS=0U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions61Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2MRS=-¥MRS=0MRSisundefinedU(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions62Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions63Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Whichisthemost

preferredaffordablebundle?Examplesof‘Kinky’Solutions64Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1ThemostpreferredaffordablebundleExamplesof‘Kinky’Solutions65Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*Examplesof‘Kinky’Solutions66Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*(a)p1x1*+p2x2*=mExamplesof‘Kinky’Solutions67Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*(a)p1x1*+p2x2*=m

(b)x2*=ax1*Examplesof‘Kinky’Solutions68Examplesof‘Kinky’Solutions--thePerfectComplementsCase(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Examplesof‘Kinky’Solutions69Examplesof‘Kinky’Solutions--thePerfectComplementsCase(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Substitutionfrom(b)forx2*in(a)givesp1x1*+p2ax1*=mExamplesof‘Kinky’Solutions70Examplesof‘Kinky’Solutions--thePerfectComplementsCase(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Substitutionfrom(b)forx2*in(a)givesp1x1*+p2ax1*=m

whichgivesExamplesof‘Kinky’Solutions71Examplesof‘Kinky’Solutions--thePerfectComplementsCase(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Substitutionfrom(b)forx2*in(a)givesp1x1*+p2ax1*=m

whichgivesExamplesof‘Kinky’Solutions72Examplesof‘Kinky’Solutions--thePerfectComplementsCase(a)p1x1*+p2x2*=m;(b)x2*=ax1*.Substitutionfrom(b)forx2*in(a)givesp1x1*+p2ax1*=m

whichgivesAbundleof1commodity1unitand

acommodity2unitscostsp1+ap2;

m/(p1+ap2)suchbundlesareaffordable.Examplesof‘Kinky’Solutions73Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions74ChoosingTaxes:VariousTaxesQuantitytax:onx:(p+t)xValuetax:onpx:(1+t)pxAlsocalledadvaloremtaxLumpsumtax:TIncometax:CanbeproportionalorlumpsumChoosingTaxes:VariousTaxesQ75IncomeTaxvs.QuantityTaxOriginalbudget:p1x1+p2x2=mAfterquantitytax:(p1+t)x1+p2x2=mAtoptimalchoice(x1*,x2*)(p1+t)x1*+p2x2*=m(5.2)Taxrevenue:R*=tx1*Withanincometax,budgetis:p1x1+p2x2=m-tx1*IncomeTaxvs.QuantityTaxOri76Incomevs.QuantityTaxProposition:(x1*,x2*)isaffordableunderincometaxEquivalentto:provethat(x1*,x2*)satisfiesbudgetconstraintunderincometax.Or,budgetconstraintholdsatpoint(x1*,x2*).p1x1*+p2x2*=m-tx1*Whichistrueaccordingto(5.2).Itisnotanoptimalchoicebecausepricesaredifferent.Conclusion:Theoptimalchoicemustbemorepreferredto(x1*,x2*)Incomevs.QuantityTaxProposi77Ch05-《中级微观经济学》范里安-英文版课件78ChapterFiveChoiceChapterFiveChoice79EconomicRationalityTheprincipalbehavioralpostulateisthatadecisionmakerchoosesitsmostpreferredalternativefromthoseavailabletoit.Theavailablechoicesconstitutethechoiceset.Howisthemostpreferredbundleinthechoicesetlocated?EconomicRationalityTheprinci80RationalConstrainedChoicex1x2RationalConstrainedChoicex1x81RationalConstrainedChoicex1x2UtilityRationalConstrainedChoicex182RationalConstrainedChoiceUtilityx2x1RationalConstrainedChoiceUt83RationalConstrainedChoicex1x2UtilityRationalConstrainedChoicex1x84RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti85RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti86RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti87RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti88RationalConstrainedChoiceUtilityx1x2Affordable,butnotthemostpreferredaffordablebundle.RationalConstrainedChoiceUti89RationalConstrainedChoicex1x2UtilityAffordable,butnotthemostpreferredaffordablebundle.Themostpreferred

oftheaffordablebundles.RationalConstrainedChoicex1x90RationalConstrainedChoicex1x2UtilityRationalConstrainedChoicex1x91RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti92RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti93RationalConstrainedChoiceUtilityx1x2RationalConstrainedChoiceUti94RationalConstrainedChoicex1x2RationalConstrainedChoicex1x95RationalConstrainedChoicex1x2Affordable

bundlesRationalConstrainedChoicex1x96RationalConstrainedChoicex1x2Affordable

bundlesRationalConstrainedChoicex1x97RationalConstrainedChoicex1x2Affordable

bundlesMorepreferred

bundlesRationalConstrainedChoicex1x98RationalConstrainedChoiceAffordable

bundlesx1x2Morepreferred

bundlesRationalConstrainedChoiceAff99RationalConstrainedChoicex1x2x1*x2*RationalConstrainedChoicex1x100RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isthemost

preferredaffordable

bundle.RationalConstrainedChoicex1x101RationalConstrainedChoiceThemostpreferredaffordablebundleiscalledtheconsumer’sORDINARYDEMANDatthegivenpricesandbudget.Ordinarydemandswillbedenotedby

x1*(p1,p2,m)andx2*(p1,p2,m).RationalConstrainedChoiceThe102RationalConstrainedChoiceWhenx1*>0andx2*>0thedemandedbundleisINTERIOR.Ifbuying(x1*,x2*)costs$mthenthebudgetisexhausted.RationalConstrainedChoiceWhe103RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isinterior.(x1*,x2*)exhauststhe

budget.RationalConstrainedChoicex1x104RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isinterior.

(a)(x1*,x2*)exhauststhe

budget;p1x1*+p2x2*=m.RationalConstrainedChoicex1x105RationalConstrainedChoicex1x2x1*x2*(x1*,x2*)isinterior.

(b)Theslopeoftheindiff.

curveat(x1*,x2*)equals

theslopeofthebudget

constraint.RationalConstrainedChoicex1x106RationalConstrainedChoice(x1*,x2*)satisfiestwoconditions:(a)thebudgetisexhausted;

p1x1*+p2x2*=m(b)theslopeofthebudgetconstraint,-p1/p2,andtheslopeoftheindifferencecurvecontaining(x1*,x2*)areequalat(x1*,x2*).RationalConstrainedChoice(x1107ComputingOrdinaryDemandsHowcanthisinformationbeusedtolocate(x1*,x2*)forgivenp1,p2andm?ComputingOrdinaryDemandsHow108ComputingOrdinaryDemands-aCobb-DouglasExample.SupposethattheconsumerhasCobb-Douglaspreferences.ComputingOrdinaryDemands-a109ComputingOrdinaryDemands-aCobb-DouglasExample.SupposethattheconsumerhasCobb-Douglaspreferences.

ThenComputingOrdinaryDemands-a110ComputingOrdinaryDemands-aCobb-DouglasExample.SotheMRSisComputingOrdinaryDemands-a111ComputingOrdinaryDemands-aCobb-DouglasExample.SotheMRSis

At(x1*,x2*),MRS=-p1/p2soComputingOrdinaryDemands-a112ComputingOrdinaryDemands-aCobb-DouglasExample.SotheMRSis

At(x1*,x2*),MRS=-p1/p2so(A)ComputingOrdinaryDemands-a113ComputingOrdinaryDemands-aCobb-DouglasExample.(x1*,x2*)alsoexhauststhebudgetso(B)ComputingOrdinaryDemands-a114ComputingOrdinaryDemands-aCobb-DouglasExample.Sonowweknowthat(A)(B)ComputingOrdinaryDemands-a115ComputingOrdinaryDemands-aCobb-DouglasExample.Sonowweknowthat(A)(B)SubstituteComputingOrdinaryDemands-a116ComputingOrdinaryDemands-aCobb-DouglasExample.Sonowweknowthat(A)(B)SubstituteandgetThissimplifiesto….ComputingOrdinaryDemands-a117ComputingOrdinaryDemands-aCobb-DouglasExample.ComputingOrdinaryDemands-a118ComputingOrdinaryDemands-aCobb-DouglasExample.Substitutingforx1*inthengivesComputingOrdinaryDemands-a119ComputingOrdinaryDemands-aCobb-DouglasExample.Sowehavediscoveredthatthemost

preferredaffordablebundleforaconsumer

withCobb-Douglaspreferences

isComputingOrdinaryDemands-a120ComputingOrdinaryDemands-aCobb-DouglasExample.x1x2ComputingOrdinaryDemands-a121RationalConstrainedChoiceWhenx1*>0andx2*>0

and(x1*,x2*)exhauststhebudget,

andindifferencecurveshaveno

‘kinks’,theordinarydemandsareobtainedbysolving:(a)p1x1*+p2x2*=y(b)theslopesofthebudgetconstraint,-p1/p2,andoftheindifferencecurvecontaining(x1*,x2*)areequalat(x1*,x2*).RationalConstrainedChoiceWhe122RationalConstrainedChoiceButwhatifx1*=0?Orifx2*=0?Ifeitherx1*=0orx2*=0thentheordinarydemand(x1*,x2*)isatacornersolutiontotheproblemofmaximizingutilitysubjecttoabudgetconstraint.RationalConstrainedChoiceBut123ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1ExamplesofCornerSolutions-124ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1>p2.ExamplesofCornerSolutions-125ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1>p2.ExamplesofCornerSolutions-126ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1>p2.ExamplesofCornerSolutions-127ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1<p2.ExamplesofCornerSolutions-128ExamplesofCornerSolutions--thePerfectSubstitutesCaseSowhenU(x1,x2)=x1+x2,themost

preferredaffordablebundleis(x1*,x2*)

whereandifp1<p2ifp1>p2.ExamplesofCornerSolutions-129ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2MRS=-1Slope=-p1/p2withp1=p2.ExamplesofCornerSolutions-130ExamplesofCornerSolutions--thePerfectSubstitutesCasex1x2Allthebundlesinthe

constraintareequallythe

mostpreferredaffordable

whenp1=p2.ExamplesofCornerSolutions-131ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2BetterExamplesofCornerSolutions-132ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2ExamplesofCornerSolutions-133ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2Whichisthemostpreferred

affordablebundle?ExamplesofCornerSolutions-134ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2Themostpreferred

affordablebundleExamplesofCornerSolutions-135ExamplesofCornerSolutions--theNon-ConvexPreferencesCasex1x2Themostpreferred

affordablebundleNoticethatthe“tangencysolution”

isnotthemostpreferredaffordable

bundle.ExamplesofCornerSolutions-136Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions137Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2MRS=0U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions138Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2MRS=-¥MRS=0U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions139Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2MRS=-¥MRS=0MRSisundefinedU(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions140Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Examplesof‘Kinky’Solutions141Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1Whichisthemost

preferredaffordablebundle?Examplesof‘Kinky’Solutions142Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1ThemostpreferredaffordablebundleExamplesof‘Kinky’Solutions143Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*Examplesof‘Kinky’Solutions144Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*(a)p1x1*+p2x2*=mExamplesof‘Kinky’Solutions145Examplesof‘Kinky’Solutions--thePerfectComplementsCasex1x2U(x1,x2)=min{ax1,x2}x2=ax1x1*x2*(a)