Solution Manual for Foundations of Mathematical Economics By Michael Carter A+ verified, ISBN: 978-0262531924

Solutions Manual
Foundations of Mathematical Economics

Michael Carter

, ⃝ cwww2001w Michaelw Carte
Solutionsw forw Foundationsw ofw Mathematicalw Economi r Allwrightswreserve
cs d



Chapterw 1:w Setsw andw Spaces

1.1
{w1,w3,w5,w7w. . . w}worw {w�w ∈ w�w :w �w isw oddw}
1.2 Everyw � ∈ �w alsow belongsw tow �.w Everyw �∈
�w alsow belongsw tow �.w Hencew �,w�w havewpreciselyw thew samew elements.
1.3 Examplesw ofw finitew setsw are
∙ thew lettersw ofw thew alphabetw {wA,w B,w C,w . . . w ,w Zw}
∙ thew setw ofw consumersw inw anw economy
∙ thew setw ofw goodsw inw anw economy
∙ thew setw ofw playerswinw aw gam
e.wExamplesw ofw infinitew setsw are
∙ thew realw numbersw ℜ
∙ thew naturalw numbersw �
∙ thew setw ofw allw possiblew colors
∙ thew setw ofw possiblew pricesw ofw copperw onw thew worldw market
∙ thew setw ofw possiblew temperaturesw ofw liquidw water.
1.4w �w =w {w1,w2,w3,w4,w5,w6w},w �w =w {w2,w4,w6w}.
1.5 Thew playerw setw isw �w =w {wJenny,wChrisw} . w Theirw actionw spacesw are
��w =w {wRock,wScissors,wPaperw} �w =w Jenny,wChris
1.6 Thew setw ofw playersw isw �w ={w 1,w2 , . .. , w�}w . w Thew strategyw spacew ofw eachw playerw isw the
w setwofw feasiblew outputs


��w =w {w��w ∈ wℜ +w :w ��w ≤ w��w}
wherew ��wwiswwthew outputw ofw damw �.
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1.7 Thew playerw setw isw �w =w {1,w2,w3}. wTherew arew 2 w =w 8w coalitions,w namely
� (�w)w =w {∅ ,w{1},w{2},w{3},w{1,w2},w{1,w3},w{2,w3},w{1,w2,w3}}
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Therew arew 2 w coalitionsw inw aw tenw playerw game.
1.8ww Assumewwthatww�ww∈ w(�w ∪ w�w)� .wwwThatwwisww�ww∈/ww�w ∪ w�w.wwwThiswwimpliesww�ww∈/ww�wwandw
� � � � �
w�ww∈/ww�w,w orw�w∈ w� w andw �w∈ w�w .w Consequently,w �w∈ w� w∩ w�w .w Conversely,w assumew �w∈ w�
� � �
w∩w�w .wThiswwimplieswwthatww�w ∈ w� wwandww�w ∈ w�w .wwwConsequentlyww�w∈/ww�wwandww�w∈/ww�ww a
ndwwtherefore
�∈/w �w ∪ w�w. wThisw implieswwthatw �w ∈ w(�w ∪ w�w)� . wThew otherw identityw isw provedw similarly.
1.9
∪
�w =w�
�∈�
∩
�w =w∅
�∈�


1

, ⃝ cwww2001w Michaelw Carte
Solutionsw forw Foundationsw ofw Mathematicalw Economi r Allwrightswreserve
cs d

�2
1




�1
-1 0 1




-1
2 2
Figurew 1.1:w Thew relationw {w(�,w�)w :w � w +w� w =w 1w}


1.10w Thew samplew spacew ofw aw singlew coinw tossw is{w�,w�w}.wThew setw ofw possiblew outcomesw i
nwthreew tossesw isw thew product
{
{�,w�w} ×w{�,w�w} ×w{�,w�w}w=w (�,w�,w�),w(�,w�,w�w),w(�,w�w,w�),
}
(�,w�w,w�w),w(�,w�,w�),w(�,w�,w�w),w(�,w�,w�),w(�,w�,w�w)


Aw typicalw outcomew isw thew sequencew (�,w�,w�w)w ofw twow headsw followedw byw aw tail.
1.11

�w ∩wℜ+�w =w {0}

wherew0w =w(0,w0 , . . . w,w0)wiswthewproductionwplanwusingwnowinputswandwproducingwnowoutp
uts.wTow seew this,w firstw notew thatw 0w isw aw feasiblew productionw plan.w Therefore,w 0w ∈ w�w
.w Also,
0w ∈ wℜ �+w andw thereforew 0w ∈ w�w ∩wℜ �w+
.
�
ℜ + w,wwewassumewthewcontrar
Towshowwthatwtherewiswnowotherwfeasiblewproductionwplanwinwwwww
�
y.wThatwis,wwewassumewtherewiswsomewfeasiblewproductionwplan∈wywℜwwwwwwww
w +∖ w{w } wwwwww0ww.wwThis
wimplieswthewexistencewofwawplanwproducingwawpositivewoutputwwithwnowinputs.wThiswtec

hnologicalwinfeasible,w sow thatw �w∈/w �w.
1.12 1. wwLetwwxw ∈ w�w(�). wwThiswwimplieswwthatww(�,w− x)w ∈ w�w. wwLetwwx′w ≥ wx.ww Thenww(�,w− x′ )w ≤
(�,w− x)w andw freew disposabilityw implieswwthatw (�,w− x′ )w ∈ w�w. wThereforew x′w∈ w�w(�).
2.ww Againww assumeww xww ∈ w �w(�).wwwwThisww impliesww thatww (�,w− x)ww ∈ w �w.wwwwByww freeww di
sposal,w(� ′ ,w− x)w ∈ w�ww forw everyw � ′w≤ w�,w whichw implieswwthatw xw ∈ w�w(� ′ ).ww�w(� ′ )w ⊇ w�w
(�).
1.13 Thew domainw ofw “<”w isw {1,w2}w=w �w andw thew rangew isw {2,w3}w⫋w �w.
1.14 Figurew 1.1.
1.15 Thew relationw “isw strictlyw higherw than”w isw transitive,w antisymmetricw andw asymmet
ric.wItw isw notw complete,w reflexivew orw symmetric.



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, ⃝ cwww2001w Michaelw Carte
Solutionsw forw Foundationsw ofw Mathematicalw Economi r Allwrightswreserve
cs d
1.16 Thew followingw tablew listsw theirw respectivew properties.
< ≤√ww √=
reflexive ×ww
transitive √ √ww √
symmetric √ww √
×ww
√
asymmetric
anti-symmetric √ww × ww ×
√ √
√w √w
complete ×
Notew thatw thew propertiesw ofw symmetryw andw anti-symmetryw arew notw mutuallyw exclusive.
1.17 Letwbe∼ wanwequivalencewrelationwofwawsetw�w∕= w w∅. w Thatwis,wthewrelationw∼
iswreflexive,ws
ymmetricwandwtransitive.wWewfirstwshowwthatweveryw�w∈�wbelongswtowsomewequivalence
wclass.w Letw �w bew anyw elementw inw �w andw let
∼ w (�)w bew thew classw ofw elementsw equivalen
tw to
�,wthatw is
∼(�)w ≡w{w�w ∈ w�w :w �w ∼ w�w}
Since ∼ isw reflexive,w �∼ �wandwsow�∈ w∼ (�).w Everyw � ∈
�w belongsw tow somew equivalencewclassw andw therefore
∪
�w = ∼(�)
�∈�

Next,w wew showw thatw thew equivalencew classesw arew eitherw disjointw orw identical,wwthat
w is

∼(�)w ∕=w ∼(�)w ifw andw onlyw ifw f∼(�)w∩w∼(�) w=w ∅ .
First,w assumew ∼(�)w∩w∼(�) w=w ∅ . wThenw �w∈ w∼(�)w butww�∈
�/ ∼( ). wThereforew ∼(�)w ∕=w ∼(�).
Conversely,wwassumeww∼(�)w ∩w∼(�)ww∕=ww∅ wandwwletww�ww∈ w∼(�)w ∩w∼(�).wwwThenww�ww∼ w�wwandwwb yw
symmetryw �w ∼ w�.wwwAlsow �w ∼ w�wandwsow byw transitivityw�w ∼ w�.wwwLetw�w bew anywelemen
twinww∼(�)wwsowwthatww�ww∼ w�.wwwAgainwwbywwtransitivityww�ww∼ w�wwandwwthereforeww�ww∈ w∼(�).www
Hence
∼(�)w ⊆ w∼(�). wSimilarwwreasoningw implieswwthatw ∼(�)w ⊆ w∼(�). wThereforew ∼(�) w=w ∼(�).
Wew concludew thatw thew equivalencew classesw partitionw �.
1.18 Thewsetwofwproperwcoalitionswisw notw awpartitionw ofwthew setwofw players,wsincew anyw pl
ayerwcanw belongw tow morew thanw onew coalition.wForw example,w playerw 1w belongsw tow thew c
oalitions
{1},w {1,w2}wandw sow on.
1.19
�w ≻w�w =⇒ w �w ≿w �w andw �w ∕≿w �
�w ∼ w�w =⇒ w �w ≿w �w andw �w ≿w �
Transitivityw ofw ≿wimpliesw �w≿w� . wWew needw tow showw thatw �w∕≿w� . wAssumew otherwise,w th
atwisw assumew �w ≿w�w Thisw impliesw �w ∼w�w andw byw transitivityw �w ∼w�.w Butw thisw impliesw t
hat
�w ≿w�w whichw contradictsw thew assumptionw thatw �w ≻w� . w Thereforew wew concludew thatw �w ∕≿w �
andw thereforew �w ≻w� . wThew otherw resultw isw provedw inw similarw fashion.
1.20 asymmetricw Assumew �w ≻w�.

Therefore
while

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