Solutions Manual Foundations of Mathematical Economics By Michael Carter

Solutions Manual
Foundations of Mathematical Economics

Michael Carter

, ⃝ cooo2001o Michaelo Carter
Solutionso foro Foundationso ofo Mathematicalo Economics Allorightsoreserved




Chapter 1: Sets and Spaces
o o o o




1.1
{o1,o3,o5,o7o. . . o}ooro {o�o ∈ o�o :o �o iso oddo}
1.2 Everyo � ∈ �o alsoo belongso too �.o Everyo � ∈
�o alsoo belongso too �.o Henceo �,o�o haveopreciselyo theo sameo elements.
1.3 Exampleso ofo finiteo setso are
∙ theo letterso ofo theo alphabeto {oA,o B,o C,o . . . o ,o Zo}
∙ theo seto ofo consumerso ino ano economy
∙ theo seto ofo goodso ino ano economy
∙ theo seto ofo playersoino ao game
.oExampleso ofo infiniteo setso are
∙ theo realo numberso ℜ
∙ theo naturalo numberso �
∙ theo seto ofo allo possibleo colors
∙ theo seto ofo possibleo priceso ofo coppero ono theo worldo market
∙ theo seto ofo possibleo temperatureso ofo liquido water.
1.4o �o =o {o1,o2,o3,o4,o5,o6o},o �o =o {o2,o4,o6o}.
1.5 Theo playero seto iso �o =o {oJenny,oChriso} . oTheiro actiono spaceso are
��o =o{oRock,oScissors,oPapero} �o =o Jenny,oChris
1.6 Theo seto ofo playerso iso �o =o {1,o2 , . .. , o�o }. o Theo strategyo spaceo ofo eacho playero iso theo set
oofo feasibleo outputs


��o =o {o��o ∈ oℜ +o :o ��o ≤ o��o}
whereo ��ooisootheo outputo ofo damo �.
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1.7 Theo playero seto iso �o =o {1,o2,o3}. oThereo areo 2 o =o 8o coalitions,o namely
� (�o)o =o {∅ ,o{1},o{2},o{3},o{1,o2},o{1,o3},o{2,o3},o{1,o2,o3}}
10
Thereo areo 2 o coalitionso ino ao teno playero game.
1.8oo Assumeoothatoo�oo∈ o(�o ∪ o�o)� .oooThatooisoo�oo∈/oo�o ∪ o�o.oooThisooimpliesoo�oo∈/oo�ooandoo�oo∈/oo�o,o
oro�o∈ o��oando �o∈ o�o�.o Consequently,o �o∈ o��o∩o�o�.o Conversely,o assumeo �o∈ o��o∩o�o�.oThisooi
mpliesoothatoo�o ∈ o� �ooandoo�o ∈ o�o� .oooConsequentlyoo�o∈/oo�ooandoo�o∈/oo�oo andootherefore
�∈/o �o ∪ o�o. oThiso impliesoothato �o ∈ o(�o∪ o�o)� . oTheo othero identityo iso provedo similarly.
1.9
∪
�o =o�
�∈�
∩
�o =o∅
�∈�


1

, ⃝ cooo2001o Michaelo Carter
Solutionso foro Foundationso ofo Mathematicalo Economics Allorightsoreserved


�2
1




�1
-1 0 1




-1
2 2
Figureo 1.1:o Theo relationo {o(�,o�)o :o � o +o � o =o 1o}


1.10o Theo sampleo spaceo ofo ao singleo coino tosso iso�,
{ o�o .o The
} o seto ofo possibleo outcomeso inothr
eeo tosseso iso theo product
{
{�,o�o} ×o{�,o�o} ×o{�,o�o}o=o (�,o�,o�),o(�,o�,o�o),o(�,o�o,o�),
}
(�,o�o,o�o),o(�,o�,o�),o(�,o�,o�o),o(�,o�,o�),o(�,o�,o�o)


Ao typicalo outcomeo iso theo sequenceo (�,o�,o�o)o ofo twoo headso followedo byo ao tail.
1.11

�o ∩oℜ+�o =o {0}

whereo0o =o(0,o0 , . . . o,o0)oisotheoproductionoplanousingonooinputsoandoproducingonoooutputs.
oToo seeo this,o firsto noteo thato 0o iso ao feasibleo productiono plan.o Therefore,o 0o ∈ o�o.o Also,

0o ∈ oℜ �+o ando thereforeo 0o ∈ o�o ∩oℜ �o . +
Tooshowothatothereoisonoootherofeasibleoproductionoplanoinooooo�ℜ o,+oweoassumeotheocontrary.oT
�
hatois,oweoassumeothereoisosomeofeasibleoproductionoplanoyoooooooo ∈ oℜ o oooooo
+∖ o{o }0oo.ooThisoimpliesot
heoexistenceoofoaoplanoproducingoaopositiveooutputowithonooinputs.oThisotechnologicaloin
feasible,o soo thato �o∈/o �o.
1.12 1. ooLetooxo ∈ o�o(�). ooThisooimpliesoothatoo(�,o− x)o ∈ o�o. ooLetoox′o ≥ ox.oo Thenoo(�,o− x′ )o ≤
(�,o− x)o ando freeo disposabilityo impliesoothato (�,o− x′ )o ∈ o�o. oThereforeo x′o∈ o�o(�).
2.oo Againoo assumeoo xoo ∈ o �o(�).ooooThisoo impliesoo thatoo (�,o− x)oo ∈ o �o.ooooByoo freeoo disposal,
o(� ′ ,o− x)o ∈ o�oo foro everyo � ′o≤ o� , o whicho impliesoothato xo ∈ o�o(� ′ ).oo�o(� ′ )o ⊇ o�o(�).


1.13 Theo domaino ofo “<”o iso {1,o2}o=o �o ando theo rangeo iso {2,o3}o⫋o �o.
1.14 Figureo 1.1.
1.15 Theo relationo “iso strictlyo highero than”o iso transitive,o antisymmetrico ando asymmetri
c.oIto iso noto complete,o reflexiveo oro symmetric.




2

, ⃝ cooo2001o Michaelo Carter
Solutionso foro Foundationso ofo Mathematicalo Economics Allorightsoreserved


1.16 Theo followingo tableo listso theiro respectiveo properties.
< ≤√oo √=
reflexive ×oo
transitive √ √oo √
symmetric √oo √
×oo
√
asymmetric
anti-symmetric √oo × oo ×
√ √
√o √o
complete ×
Noteo thato theo propertieso ofo symmetryo ando anti-symmetryo areo noto mutuallyo exclusive.
1.17 Letobe ∼ oanoequivalenceorelationoofoaoseto�o=o∕o. o∅Thatois,otheorelationoisoreflexive,
∼ osym

metricoandotransitive.oWeofirstoshowothatoeveryo�o�obelongs ∈ otoosomeoequivalenceoclass

.o Leto �o beo anyo elemento ino �o ando leto (�)o be
∼ o theo classo ofo elementso equivalento to
�,othato is
∼(�)o ≡o{o�o ∈ o�o :o �o ∼ o�o}
Since ∼ iso reflexive,o � ∼ �oandosoo� ∈ o∼ (�).o Everyo � ∈
�o belongso too someo equivalenceoclasso ando therefore
∪
�o = ∼(�)
�∈�

Next,o weo showo thato theo equivalenceo classeso areo eithero disjointo oro identical,oothato is
∼(�)o ∕=o ∼(�)o ifo ando onlyo ifo f∼(�)o∩o∼ (�) o=o ∅ .
First,o assumeo ∼(�)o∩o∼ (�) o=o ∅ . oTheno �o∈ o∼ (�)o butoo�∈
�/ ∼( ). oThereforeo ∼(�)o ∕=o ∼(�).
Conversely,ooassumeoo∼(�)o ∩o∼ (�)oo∕=oo∅ oandooletoo�oo∈ o∼(�)o ∩o∼ (�).oooThenoo�oo∼ o�ooandoob
yosymm
etryo �o ∼ o�.oooAlsoo �o ∼ o�oandosoo byo transitivityo�o ∼ o�.oooLeto�o beo anyoelementoinoo∼(�)oo
sooothatoo�oo∼ o�.oooAgainoobyootransitivityoo�oo∼ o�ooandoothereforeoo�oo∈ o∼(�).oooHence
∼(�)o ⊆ o∼ (�). oSimilarooreasoningo impliesoothato ∼(�)o ⊆ o∼ (�). oThereforeo ∼(�) o=o ∼(�).
Weo concludeo thato theo equivalenceo classeso partitiono �.
1.18 Theosetoofoproperocoalitionsoiso notoaopartitionoofotheo setoofo players,osinceoanyo play
erocano belongo too moreo thano oneo coalition.oForo example,o playero1o belongso too theo coaliti
ons
{1},o {1,o2}oando soo on.
1.19

�o ≻o�o =⇒ o �o ≿o �o ando �o ∕≿o �
�o ∼ o�o =⇒ o �o ≿o �o ando �o ≿o �
Transitivityo ofo ≿oimplieso �o≿o� . oWeo needo too showo thato �o∕≿o� . oAssumeo otherwise,o thatoi
so assumeo �o ≿o�o Thiso implieso �o ∼o�o ando byo transitivityo �o ∼o�.o Buto thiso implieso that
�o ≿o �o whicho contradictso theo assumptiono thato �o ≻o� . o Thereforeo weo concludeo thato �o ∕≿o �
ando thereforeo �o ≻o� . oTheo othero resulto iso provedo ino similaro fashion.
1.20 asymmetrico Assumeo �o ≻o�.

�o ≻o�o =⇒ o �o ∕≿o�
while

�o ≻o�o =⇒ o �o ≿o �
Therefore
�o ≻o�o =⇒ o �o ∕≻o�
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