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Solutions Manual Foundations of Mathematical Economics By Michael Carter
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Solutions Manual Foundations of Mathematical Economics By Michael Carter
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Solutions ManualFoundations of Mathematical Economics
Michael Carter
, c⃝ JJJ2001J MichaelJ Carter
SolutionsJ forJ FoundationsJ ofJ MathematicalJ Economics AllJrightsJreserved
ChapterJ 1:J SetsJ andJ Spaces
1.1
{J1,J3,J5,J7J. . . J}JorJ {J𝑛J ∈J𝑁J :J 𝑛J isJ oddJ}
1.2 EveryJ 𝑥 ∈ 𝐴J alsoJ belongsJ toJ 𝐵.J EveryJ 𝑥 ∈
𝐵J alsoJ belongsJ toJ 𝐴.J HenceJ 𝐴,J𝐵J haveJpreciselyJ theJ sameJ elements.
1.3 ExamplesJ ofJ finiteJ setsJ are
∙ theJ lettersJ ofJ theJ alphabetJ {JA,J B,J C,J . . . J ,J ZJ}
∙ theJ setJ ofJ consumersJ inJ anJ economy
∙ theJ setJ ofJ goodsJ inJ anJ economy
∙ theJ setJ ofJ playersJinJ aJ game.J
ExamplesJ ofJ infiniteJ setsJ are
∙ theJ realJ numbersJ ℜ
∙ theJ naturalJ numbersJ 𝔑
∙ theJ setJ ofJ allJ possibleJ colors
∙ theJ setJ ofJ possibleJ pricesJ ofJ copperJ onJ theJ worldJ market
∙ theJ setJ ofJ possibleJ temperaturesJ ofJ liquidJ water.
1.4J 𝑆J =J {J1,J2,J3,J4,J5,J6J},J 𝐸J =J {J2,J4,J6J}.
1.5 TheJ playerJ setJ isJ 𝑁J =J {JJenny,JChrisJ} . JTheirJ actionJ spacesJ are
𝐴𝑖J =J{JRock,JScissors,JPaperJ} 𝑖J =J Jenny,JChris
1.6 TheJ setJ ofJ playersJ isJ 𝑁J =J 1,{J2 , . .. , J𝑛J . }J TheJ strategyJ spaceJ ofJ eachJ playerJ isJ theJ setJofJ
feasibleJ outputs
𝐴𝑖J =J {J𝑞𝑖J ∈Jℜ+J :J 𝑞𝑖J ≤J𝑄𝑖J}
whereJ 𝑞𝑖JJisJJtheJ outputJ ofJ damJ 𝑖.
1.7 TheJ playerJ setJ isJ 𝑁J =J {1,J2,J3}. JThereJ areJ 23J =J 8J coalitions,J namely
𝒫(𝑁J)J =J {∅,J{1},J{2},J{3},J{1,J2},J{1,J3},J{2,J3},J{1,J2,J3}}
ThereJ areJ 210J coalitionsJ inJ aJ tenJ playerJ game.
1.8JJ AssumeJJthatJJ𝑥JJ∈J(𝑆J ∪J𝑇J)𝑐 .JJJThatJJisJJ𝑥JJ∈/JJ𝑆J ∪J𝑇J.JJJThisJJimpliesJJ𝑥JJ∈/JJ𝑆JJandJJ𝑥JJ∈/JJ𝑇J,JorJ𝑥J∈J𝑆𝑐Ja
ndJ 𝑥J∈J𝑇J𝑐.J Consequently,J 𝑥J∈J𝑆𝑐J∩J𝑇J𝑐.J Conversely,J assumeJ 𝑥J∈J𝑆𝑐J∩J𝑇J𝑐.JThisJJimpliesJJthatJJ𝑥J ∈J𝑆 𝑐JJan
dJJ𝑥J ∈J𝑇J𝑐 .JJJConsequentlyJJ𝑥J∈/JJ𝑆JJandJJ𝑥J∈/JJ𝑇JJandJJtherefore
𝑥 ∈/J 𝑆J∪J𝑇J. JThisJ impliesJJthatJ 𝑥J ∈J(𝑆J ∪J𝑇J)𝑐 . JTheJ otherJ identityJ isJ provedJ similarly.
1.9
∪
𝑆J =J𝑁
𝑆∈𝒞
∩
𝑆J =J∅
𝑆∈𝒞
1
, c⃝ JJJ2001J MichaelJ Carter
SolutionsJ forJ FoundationsJ ofJ MathematicalJ Economics AllJrightsJreserved
𝑥2
1
𝑥1
-1 0 1
-1
FigureJ 1.1:J TheJ relationJ {J(𝑥,J𝑦)J :J 𝑥2J +J 𝑦2J =J 1J}
1.10J TheJ sampleJ spaceJ ofJ aJ singleJ coinJ tossJ isJ𝐻,J{𝑇J .J The}J setJ ofJ possibleJ outcomesJ inJthreeJ
tossesJ isJ theJ product
{
{𝐻,J𝑇J} × J{𝐻,J𝑇J} × J{𝐻,J𝑇J}J=J (𝐻,J𝐻,J𝐻),J(𝐻,J𝐻,J𝑇J),J(𝐻,J𝑇J,J𝐻),
}
(𝐻,J𝑇J,J𝑇J),J(𝑇,J𝐻,J𝐻),J(𝑇,J𝐻,J𝑇J),J(𝑇,J𝑇,J𝐻),J(𝑇,J𝑇,J𝑇J)
AJ typicalJ outcomeJ isJ theJ sequenceJ (𝐻,J𝐻,J𝑇J)J ofJ twoJ headsJ followedJ byJ aJ tail.
1.11
𝑌J ∩Jℜ+𝑛J =J {0}
whereJ0J =J(0,J0, . . . J,J0)JisJtheJproductionJplanJusingJnoJinputsJandJproducingJnoJoutputs.JT
oJ seeJ this,J firstJ noteJ thatJ 0J isJ aJ feasibleJ productionJ plan.J Therefore,J 0J ∈J𝑌J.J Also,
0J ∈Jℜ𝑛J+andJ thereforeJ 0J ∈J𝑌J ∩Jℜ𝑛J . +
ToJshowJthatJthereJisJnoJotherJfeasibleJproductionJplanJinJJJJJ𝑛J,Jwe
ℜ +JassumeJtheJcontrary.JThatJ
is,JweJassumeJthereJisJsomeJfeasibleJproductionJplanJyJJJJJJJJ𝑛JJJJJJ∈0JℜJJ.JJJ+
∖J{J } JimpliesJtheJexist
This
enceJofJaJplanJproducingJaJpositiveJoutputJwithJnoJinputs.JThisJtechnologicalJinfeasible,J s
oJ thatJ 𝑦J∈/J 𝑌J.
1.12 1. JJLetJJxJ ∈J𝑉J(𝑦 ). JJThisJJimpliesJJthatJJ(𝑦,J−x)J ∈J𝑌J. JJLetJJx′J ≥Jx.JJ ThenJJ(𝑦,J−x′ )J ≤
(𝑦,J−x)J andJ freeJ disposabilityJ impliesJJthatJ (𝑦,J−x′ )J ∈J𝑌J. JThereforeJ x′J∈J𝑉J(𝑦 ).
2.JJ AgainJJassumeJJ xJJ ∈J 𝑉J(𝑦 ).JJJJThisJJ impliesJJ thatJJ (𝑦,J−x)JJ ∈J 𝑌J.JJJJByJJ freeJJ disposal,J(𝑦 ′ ,J−x
)J ∈J𝑌JJ forJ everyJ 𝑦 ′J≤J𝑦 ,J whichJ impliesJJthatJ xJ ∈J𝑉J(𝑦 ′ ).JJ𝑉J(𝑦 ′ )J ⊇J𝑉J(𝑦 ).
1.13 TheJ domainJ ofJ “<”J isJ {1,J2}J=J 𝑋J andJ theJ rangeJ isJ {2,J3}J⫋J 𝑌J.
1.14 FigureJ1.1.
1.15 TheJ relationJ “isJ strictlyJ higherJ than”J isJ transitive,J antisymmetricJ andJ asymmetric.JI
tJ isJ notJ complete,J reflexiveJ orJ symmetric.
2
, c⃝ JJJ2001J MichaelJ Carter
SolutionsJ forJ FoundationsJ ofJ MathematicalJ Economics AllJrightsJreserved
1.16 TheJ followingJ tableJ listsJ theirJ respectiveJ properties.
< ≤√JJ √=
reflexive ×JJ
transitive √ √JJ √
√JJ √
symmetric ×JJ
√
asymmetric
anti-symmetric √JJ ×√JJ ×√
√J √ J
complete ×
NoteJ thatJ theJ propertiesJ ofJ symmetryJ andJ anti-symmetryJ areJ notJ mutuallyJ exclusive.
1.17 LetJbe ∼ JanJequivalenceJrelationJofJaJsetJ𝑋J=J. J∕That J ∅ Jis,JtheJrelationJisJreflexive,
∼ Jsymme
tricJandJtransitive.JWeJfirstJshowJthatJeveryJ𝑥J𝑋Jbelongs∈JtoJsomeJequivalenceJclass.J LetJ 𝑎
J beJ anyJ elementJ inJ 𝑋J andJ letJ (𝑎)J beJ theJ class
∼ J ofJ elementsJ equivalentJ to
𝑎,JthatJ is
∼(𝑎)J ≡J{J𝑥J ∈J𝑋J :J 𝑥J ∼J𝑎J}
Since ∼ isJ reflexive,J 𝑎 ∼ 𝑎JandJsoJ𝑎 ∈J∼ (𝑎).J EveryJ 𝑎 ∈
𝑋J belongsJ toJ someJ equivalenceJclassJ andJ therefore
∪
𝑋J = ∼(𝑎)
𝑎∈𝑋
Next,J weJ showJ thatJ theJ equivalenceJ classesJ areJ eitherJ disjointJ orJ identical,JJthatJ is
∼(𝑎)J ∕=J ∼(𝑏)J ifJ andJ onlyJ ifJ f∼(𝑎)J∩J∼(𝑏) J=J ∅.
First,J assumeJ ∼(𝑎)J∩J∼(𝑏) J=J ∅. JThenJ 𝑎J ∈J∼(𝑎)J butJJ𝑎 ∈ ∼(𝑏/ ). JThereforeJ ∼(𝑎)J ∕=J ∼(𝑏).
Conversely,JJassumeJJ∼(𝑎)J ∩J∼(𝑏)JJ∕=JJ∅JandJJletJJ𝑥JJ∈J∼(𝑎)J ∩J∼(𝑏).JJJThenJJ𝑥JJ∼J𝑎JJandJJbyJsymmetryJ 𝑎
J ∼J𝑥.JJJAlsoJ 𝑥J ∼J𝑏JandJsoJ byJ transitivityJ𝑎J ∼J𝑏.JJJLetJ𝑦J beJ anyJelementJinJJ∼(𝑎)JJsoJJthatJJ𝑦JJ∼J𝑎
.JJJAgainJJbyJJtransitivityJJ𝑦JJ∼J𝑏JJandJJthereforeJJ𝑦JJ∈J∼(𝑏).JJJHence
∼(𝑎)J ⊆J∼(𝑏). JSimilarJJreasoningJ impliesJJthatJ ∼(𝑏)J ⊆J∼(𝑎). JThereforeJ ∼(𝑎) J=J ∼(𝑏).
WeJ concludeJ thatJ theJ equivalenceJ classesJ partitionJ 𝑋.
1.18 TheJsetJofJproperJcoalitionsJisJ notJaJpartitionJofJtheJ setJofJplayers,JsinceJanyJplayerJc
anJ belongJ toJ moreJ thanJ oneJ coalition.JForJ example,J playerJ1J belongsJ toJ theJ coalitions
{1},J {1,J2}JandJ soJ on.
1.19
𝑥J ≻J𝑦J =⇒J 𝑥J ≿J𝑦J andJ 𝑦J ∕≿J 𝑥
𝑦J ∼J𝑧J =⇒J 𝑦J ≿J 𝑧J andJ 𝑧J ≿J 𝑦
TransitivityJ ofJ ≿JimpliesJ 𝑥J≿J𝑧 . JWeJ needJ toJ showJ thatJ 𝑧J∕≿J𝑥 . JAssumeJ otherwise,J thatJisJ ass
umeJ 𝑧J ≿J 𝑥J ThisJ impliesJ 𝑧J ∼J𝑥J andJ byJ transitivityJ 𝑦J ∼J𝑥.J ButJ thisJ impliesJ that
𝑦J ≿J𝑥J whichJ contradicts J theJ assumptionJ thatJ 𝑥J ≻J𝑦 . J ThereforeJ weJ concludeJ thatJ 𝑧J ∕≿J 𝑥
andJ thereforeJ 𝑥J ≻J𝑧 . JTheJ otherJ resultJ isJ provedJ inJ similarJ fashion.
1.20 asymmetricJ AssumeJ 𝑥J ≻J𝑦.
𝑥J ≻J𝑦J =⇒J 𝑦J ∕≿J𝑥
while
𝑦J ≻J𝑥J =⇒J 𝑦J ≿J 𝑥
Therefore
𝑥J ≻J𝑦J =⇒J 𝑦J ∕≻J𝑥
3
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