Solutions Manual Foundations of Mathematical Economics By Michael CarterManual Foundations of Mathematical

Solutions Manual
Foundations of Mathematical Economics

Michael Carter

, ⃝ cXXX2001X MichaelX Carte
SolutionsX forX FoundationsX ofX MathematicalX Economic r AllXrightsXreserve
s d



ChapterX 1:X SetsX andX Spaces

1.1
{X1,X3,X5,X7X. . . X}XorX {X�X ∈ X�X :X �X isX oddX}
1.2 EveryX � ∈ �X alsoX belongsX toX �.X EveryX �∈
�X alsoX belongsX toX �.X HenceX �,X�X haveXpreciselyX theX sameX elements.
1.3 ExamplesX ofX finiteX setsX are
∙ theX lettersX ofX theX alphabetX {XA,X B,X C,X . . . X ,X ZX}
∙ theX setX ofX consumersX inX anX economy
∙ theX setX ofX goodsX inX anX economy
∙ theX setX ofX playersXinX aX gam
e.XExamplesX ofX infiniteX setsX are
∙ theX realX numbersX ℜ
∙ theX naturalX numbersX �
∙ theX setX ofX allX possibleX colors
∙ theX setX ofX possibleX pricesX ofX copperX onX theX worldX market
∙ theX setX ofX possibleX temperaturesX ofX liquidX water.
1.4X �X =X {X1,X2,X3,X4,X5,X6X},X �X =X {X2,X4,X6X}.
1.5 TheX playerX setX isX �X =X {XJenny,XChrisX} . XTheirX actionX spacesX are
��X =X{XRock,XScissors,XPaperX} �X =X Jenny,XChris
1.6 TheX setX ofX playersX isX �X =X{1,X2 , . .. , X�}X . X TheX strategyX spaceX ofX eachX playerX isX theX
setXofX feasibleX outputs
��X =X {X��X ∈ Xℜ +X :X ��X ≤ X��X}
whereX ��XXisXXtheX outputX ofX damX �.
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1.7 TheX playerX setX isX �X =X {1,X2,X3}. XThereX areX 2 X =X 8X coalitions,X namely
� (�X)X =X {∅ ,X{1},X{2},X{3},X{1,X2},X{1,X3},X{2,X3},X{1,X2,X3}}
10
ThereX areX 2 X coalitionsX inX aX tenX playerX game.
1.8XX AssumeXXthatXX�XX∈ X(�X ∪ X�X)� .XXXThatXXisXX�XX∈/XX�X ∪ X�X.XXXThisXXimpliesXX�XX∈/XX�XXandXX
�XX∈/XX�X,XorX�X∈ X��XandX �X∈ X�X�.X Consequently,X �X∈ X��X∩X�X�.X Conversely,X assumeX �X∈ X��X
∩X�X�.XThisXXimpliesXXthatXX�X ∈ X� �XXandXX�X ∈ X�X� .XXXConsequentlyXX�X∈/XX�XXandXX�X∈/XX�XXan
dXXtherefore
�∈/X �X∪ X�X. XThisX impliesXXthatX �X ∈ X(�X ∪ X�X)� . XTheX otherX identityX isX provedX similarly.
1.9
∪
�X =X�
�∈�
∩
�X =X∅
�∈�


1

, ⃝ cXXX2001X MichaelX Carte
SolutionsX forX FoundationsX ofX MathematicalX Economic r AllXrightsXreserve
s d

�2
1




�1
-1 0 1




-1
2 2
FigureX 1.1:X TheX relationX {X(�,X�)X :X � X +X � X =X 1X}


1.10X TheX sampleX spaceX ofX aX singleX coinX tossX isX{�,X�X .}X TheX setX ofX possibleX outcomesX inX
threeX tossesX isX theX product
{
{�,X�X} ×X{�,X�X} ×X{�,X�X}X=X (�,X�,X�),X(�,X�,X�X),X(�,X�X,X�),
}
(�,X�X,X�X),X(�,X�,X�),X(�,X�,X�X),X(�,X�,X�),X(�,X�,X�X)


AX typicalX outcomeX isX theX sequenceX (�,X�,X�X)X ofX twoX headsX followedX byX aX tail.
1.11

�X ∩Xℜ+�X =X {0}

whereX0X =X(0,X0 , . . . X,X0)XisXtheXproductionXplanXusingXnoXinputsXandXproducingXnoXoutpu
ts.XToX seeX this,X firstX noteX thatX 0X isX aX feasibleX productionX plan.X Therefore,X 0X ∈ X�X.X
Also,
0X ∈ Xℜ �+X andX thereforeX 0X ∈ X�X ∩Xℜ �X .+
�
ℜ +X,XweXassumeXtheXcontrary
ToXshowXthatXthereXisXnoXotherXfeasibleXproductionXplanXinXXXXX
�
Xℜ X +
.XThatXis,XweXassumeXthereXisXsomeXfeasibleXproductionXplanXy∈XXXXXXXX ∖ X{X } XXXXXX0XX.XXThisXi
mpliesXtheXexistenceXofXaXplanXproducingXaXpositiveXoutputXwithXnoXinputs.XThisXtechn
ologicalXinfeasible,X soX thatX �X∈/X �X.
1.12 1. XXLetXXxX ∈ X�X(�). XXThisXXimpliesXXthatXX(�,X− x)X ∈ X�X. XXLetXXx′X ≥ Xx.XX ThenXX(�,X− x′ )X ≤
(�,X− x)X andX freeX disposabilityX impliesXXthatX (�,X− x′ )X ∈ X�X. XThereforeX x′X∈ X�X(�).
2.XX AgainXX assumeXX xXX ∈ X �X(�).XXXXThisXX impliesXX thatXX (�,X− x)XX ∈ X �X.XXXXByXX freeXX disp
osal,X(� ′ ,X− x)X ∈ X�XX forX everyX � ′X≤ X�,X whichX impliesXXthatX xX ∈ X�X(� ′ ).XX�X(� ′ )X ⊇ X�X(�).
1.13 TheX domainX ofX “<”X isX {1,X2}X=X �X andX theX rangeX isX {2,X3}X⫋X �X.
1.14 FigureX1.1.
1.15 TheX relationX “isX strictlyX higherX than”X isX transitive,X antisymmetricX andX asymmetr
ic.XItX isX notX complete,X reflexiveX orX symmetric.




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, ⃝ cXXX2001X MichaelX Carte
SolutionsX forX FoundationsX ofX MathematicalX Economic r AllXrightsXreserve
s d
1.16 TheX followingX tableX listsX theirX respectiveX properties.
< ≤√XX √=
reflexive ×XX
transitive √ √XX √
symmetric √XX √
×XX
√
asymmetric
anti-symmetric √XX × XX ×
√ √
√X √X
complete ×
NoteX thatX theX propertiesX ofX symmetryX andX anti-symmetryX areX notX mutuallyX exclusive.
1.17 LetXbe∼ XanXequivalenceXrelationXofXaXsetX�X=∕XX.∅ X ThatXis,XtheXrelationXis∼Xreflexive,Xsy
mmetricXandXtransitive.XWeXfirstXshowXthatXeveryX�X�∈ XbelongsXtoXsomeXequivalenceXc
lass.X LetX �X beX anyX elementX inX �X andX letX∼(�)X beX theX classX ofX elementsX equivalentX t
o
�,XthatX is
∼(�)X ≡X{X�X ∈ X�X :X �X ∼ X�X}
Since ∼ isX reflexive,X � ∼ �XandXsoX�∈ X∼ (�).X EveryX � ∈
�X belongsX toX someX equivalenceXclassX andX therefore
∪
�X = ∼(�)
�∈�

Next,X weX showX thatX theX equivalenceX classesX areX eitherX disjointX orX identical,XXthatX
is
∼(�)X ∕=X ∼(�)X ifX andX onlyX ifX f∼(�)X∩X∼ (�) X=X ∅ .
First,X assumeX ∼(�)X∩X∼ (�) X=X ∅ . XThenX �X∈ X∼(�)X butXX�∈
�/ ∼( ). XThereforeX ∼(�)X ∕=X ∼(�).
Conversely,XXassumeXX∼(�)X ∩X∼(�)XX∕=XX∅ XandXXletXX�XX∈ X∼(�)X ∩X∼(�).XXXThenXX�XX∼ X�XXandXXbyXsy
mmetryX �X ∼ X�.XXXAlsoX �X ∼ X�XandXsoX byX transitivityX�X ∼ X�.XXXLetX�X beX anyXelementXi
nXX∼(�)XXsoXXthatXX�XX∼ X�.XXXAgainXXbyXXtransitivityXX�XX∼ X�XXandXXthereforeXX�XX∈ X∼(�).XXXHen
ce
∼(�)X ⊆ X∼(�). XSimilarXXreasoningX impliesXXthatX ∼(�)X ⊆ X∼(�). XThereforeX ∼(�) X=X ∼(�).
WeX concludeX thatX theX equivalenceX classesX partitionX �.
1.18 TheXsetXofXproperXcoalitionsXisX notX aXpartitionXofXtheX setXofXplayers,XsinceX anyX pla
yerXcanX belongX toX moreX thanX oneX coalition.XForX example,X playerX1X belongsX toX theX coa
litions
{1},X {1,X2}XandX soX on.
1.19
�X ≻X�X =⇒ X �X ≿X �X andX �X ∕≿X �
�X ∼ X�X =⇒ X �X ≿X �X andX �X ≿X�
TransitivityX ofX ≿XimpliesX �X≿X� . XWeX needX toX showX thatX �X∕≿X� . XAssumeX otherwise,X th
atXisX assumeX �X ≿X �X ThisX impliesX �X ∼X�X andX byX transitivityX �X ∼X�.X ButX thisX impliesX t
hat
�X ≿X�X whichX contradictsX theX assumptionX thatX �X ≻X� . X ThereforeX weX concludeX thatX �X ∕≿X�
andX thereforeX �X ≻X� . XTheX otherX resultX isX provedX inX similarX fashion.
1.20 asymmetricX AssumeX �X ≻X�.

Therefore
while

3