Solutions Manual Foundations of Mathematical Economics By Michael Carter

Solutions Manual
Foundations of Mathematical Economics

Michael Carter

, ⃝ cFFF2001F MichaelF Carter
SolutionsF forF FoundationsF ofF MathematicalF Economics AllFrightsFreserved




Chapter 1: Sets and Spaces
F F F F




1.1
{F1,F3,F5,F7F. . . F}ForF {F�F ∈ F�F :F �F isF oddF}
1.2 EveryF � ∈ �F alsoF belongsF toF �.F EveryF � ∈
�F alsoF belongsF toF �.F HenceF �,F�F haveFpreciselyF theF sameF elements.
1.3 ExamplesF ofF finiteF setsF are
∙ theF lettersF ofF theF alphabetF {FA,F B,F C,F . . . F ,F ZF}
∙ theF setF ofF consumersF inF anF economy
∙ theF setF ofF goodsF inF anF economy
∙ theF setF ofF playersFinF aF game.F
ExamplesF ofF infiniteF setsF are
∙ theF realF numbersF ℜ
∙ theF naturalF numbersF �
∙ theF setF ofF allF possibleF colors
∙ theF setF ofF possibleF pricesF ofF copperF onF theF worldF market
∙ theF setF ofF possibleF temperaturesF ofF liquidF water.
1.4F �F =F {F1,F2,F3,F4,F5,F6F},F �F =F {F2,F4,F6F}.
1.5 TheF playerF setF isF �F =F {FJenny,FChrisF} . FTheirF actionF spacesF are
��F =F{FRock,FScissors,FPaperF} �F =F Jenny,FChris
1.6 TheF setF ofF playersF isF �F =F 1,{F2 , . .. , F�F . F}TheF strategyF spaceF ofF eachF playerF isF theF setFofF
feasibleF outputs
��F =F {F��F ∈ Fℜ +F :F ��F ≤ F��F}
whereF ��FFisFFtheF outputF ofF damF �.
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1.7 TheF playerF setF isF �F =F {1,F2,F3}. FThereF areF 2 F =F 8F coalitions,F namely
� (�F)F =F {∅ ,F{1},F{2},F{3},F{1,F2},F{1,F3},F{2,F3},F{1,F2,F3}}
10
ThereF areF 2 F coalitionsF inF aF tenF playerF game.
1.8FF AssumeFFthatFF�FF∈ F(�F ∪ F�F)� .FFFThatFFisFF�FF∈/FF�F ∪ F�F.FFFThisFFimpliesFF�FF∈/FF�FFandFF�FF∈/FF�F,ForF�F∈ F
��FandF �F∈ F�F�.F Consequently,F �F∈ F��F∩ F�F�.F Conversely,F assumeF �F∈ F��F∩ F�F�.FThisFFimpliesFFtha
tFF�F ∈ F� �FFandFF�F ∈ F�F� .FFFConsequentlyFF�F∈/FF�FFandFF�F∈/FF�FF andFFtherefore
�∈/F �F ∪ F�F. FThisF impliesFFthatF �F ∈ F(�F ∪ F�F)� . FTheF otherF identityF isF provedF similarly.
1.9
∪
�F =F�
�∈�
∩
�F =F∅
�∈�


1

, ⃝ cFFF2001F MichaelF Carter
SolutionsF forF FoundationsF ofF MathematicalF Economics AllFrightsFreserved


�2
1




�1
-1 0 1




-1
2 2
FigureF 1.1:F TheF relationF {F(�,F�)F :F � F +F � F =F 1F}


1.10F TheF sampleF spaceF ofF aF singleF coinF tossF isF�,F{�F .F The}F setF ofF possibleF outcomesF inFthreeF
tossesF isF theF product
{
{�,F�F} ×F{�,F�F} ×F{�,F�F}F=F (�,F�,F�),F(�,F�,F�F),F(�,F�F,F�),
}
(�,F�F,F�F),F(�,F�,F�),F(�,F�,F�F),F(�,F�,F�),F(�,F�,F�F)


AF typicalF outcomeF isF theF sequenceF (�,F�,F�F)F ofF twoF headsF followedF byF aF tail.
1.11

�F ∩Fℜ+�F =F {0}

whereF0F =F(0,F0 , . . . F,F0)FisFtheFproductionFplanFusingFnoFinputsFandFproducingFnoFoutputs.FTo
F seeF this,F firstF noteF thatF 0F isF aF feasibleF productionF plan.F Therefore,F 0F ∈ F�F.F Also,

0F ∈ Fℜ �+
F
andF thereforeF 0F ∈ F�F ∩Fℜ �F . +
ToFshowFthatFthereFisFnoFotherFfeasibleFproductionFplanFinFFFFF�F,Fwe
ℜ +FassumeFtheFcontrary.FThatF
is,FweFassumeFthereFisFsomeFfeasibleFproductionFplanFyFFFFFFFF�FFFFFF∈0FℜFFF.FF+
∖This
F{ F } FimpliesFtheFexist

enceFofFaFplanFproducingFaFpositiveFoutputFwithFnoFinputs.FThisFtechnologicalFinfeasible,F s
oF thatF �F∈/F �F.
1.12 1. FFLetFFxF ∈ F�F(�). FFThisFFimpliesFFthatFF(�,F− x)F ∈ F�F. FFLetFFx′F ≥ Fx.FF ThenFF(�,F− x′ )F ≤
(�,F− x)F andF freeF disposabilityF impliesFFthatF (�,F− x′ )F ∈ F�F. FThereforeF x′F∈ F�F(�).
2.FF AgainFF assumeFF xFF ∈ F �F(�).FFFFThisFF impliesFF thatFF (�,F− x)FF ∈ F �F.FFFFByFF freeFF disposal,F(� ′ ,F−
x)F ∈ F�FF forF everyF � ′F≤ F� ,F whichF impliesFFthatF xF ∈ F�F(� ′ ).FF�F(� ′ )F ⊇ F�F(�).
1.13 TheF domainF ofF “<”F isF {1,F2}F=F �F andF theF rangeF isF {2,F3}F⫋F �F.
1.14 FigureF 1.1.
1.15 TheF relationF “isF strictlyF higherF than”F isF transitive,F antisymmetricF andF asymmetric.FI
tF isF notF complete,F reflexiveF orF symmetric.




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, ⃝ cFFF2001F MichaelF Carter
SolutionsF forF FoundationsF ofF MathematicalF Economics AllFrightsFreserved


1.16 TheF followingF tableF listsF theirF respectiveF properties.
< ≤√FF √=
reflexive ×FF
transitive √ √FF √
symmetric √FF √
×FF
√
asymmetric
anti-symmetric √FF × FF ×
√ √
√F √F
complete ×
NoteF thatF theF propertiesF ofF symmetryF andF anti-symmetryF areF notF mutuallyF exclusive.
1.17 LetFbe ∼ FanFequivalenceFrelationFofFaFsetF�F=F. F∕FThat
∅ Fis,FtheFrelationFisFreflexive,
∼ Fsymme

tricFandFtransitive.FWeFfirstFshowFthatFeveryF�F�Fbelongs∈ FtoFsomeFequivalenceFclass.F LetF
�F beF anyF elementF inF �F andF letF (�)F beF theF class
∼ F ofF elementsF equivalentF to
�,F thatF is
∼(�)F ≡F{F�F ∈ F�F :F �F ∼ F�F}
Since ∼ isF reflexive,F � ∼ �FandFsoF� ∈ F∼ (�).F EveryF � ∈
�F belongsF toF someF equivalenceFclassF andF therefore
∪
�F = ∼(�)
�∈�

Next,F weF showF thatF theF equivalenceF classesF areF eitherF disjointF orF identical,FFthatF is
∼(�)F ∕=F ∼(�)F ifF andF onlyF ifF f∼(�)F∩F∼ (�) F=F ∅ .
First,F assumeF ∼(�)F∩F∼ (�) F=F ∅ . FThenF �F ∈ ∼F (�)F butFF�∈
�/ ∼( ). FThereforeF ∼(�)F ∕=F ∼(�).
Conversely,FFassumeFF∼(�)F ∩F∼ (�)FF∕=FF∅ FandFFletFF�FF∈ F∼(�)F ∩F∼ (�).FFFThenFF�FF∼ F�FFandFFb
yFsymmetryF
�F ∼ F�.FFFAlsoF �F ∼ F�FandFsoF byF transitivityF�F ∼ F�.FFFLetF�F beF anyFelementFinFF∼(�)FFsoFFthatFF�FF
∼ F�.FFFAgainFFbyFFtransitivityFF�FF∼ F�FFandFFthereforeFF�FF∈ F∼(�).FFFHence
∼(�)F ⊆ F∼ (�). FSimilarFFreasoningF impliesFFthatF ∼(�)F ⊆ F∼ (�). FThereforeF ∼(�) F=F ∼(�).
WeF concludeF thatF theF equivalenceF classesF partitionF �.
1.18 TheFsetFofFproperFcoalitionsFisF notF aFpartitionF ofFtheF setFofFplayers,FsinceF anyF playerFc
anF belongF toF moreF thanF oneF coalition.FForF example,F playerF1F belongsF toF theF coalitions
{1},F {1,F2}FandF soF on.
1.19

�F ≻F�F =⇒ F �F ≿F �F andF �F ∕≿F �
�F ∼ F�F =⇒ F �F ≿F �F andF �F ≿F �
TransitivityF ofF ≿FimpliesF �F≿F� . FWeF needF toF showF thatF �F∕≿F� . FAssumeF otherwise,F thatFisF as
sumeF �F ≿F�F ThisF impliesF �F ∼F�F andF byF transitivityF �F ∼F�.F ButF thisF impliesF that
�F ≿F�F whichF contradictsF theF assumptionF thatF �F ≻F� . F ThereforeF weF concludeF thatF �F ∕≿F �
andF thereforeF �F ≻F� . FTheF otherF resultF isF provedF inF similarF fashion.
1.20 asymmetricF AssumeF �F ≻F�.

�F ≻F�F =⇒ F �F ∕≿F�
while

�F ≻F�F =⇒ F �F ≿F �
Therefore
�F ≻F�F =⇒ F �F ∕≻F�

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