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SolutionS Manual ’ ’FoundationS oF MatheMa
’ ’
tical econoMicS
’
Michae
l carter
’
, c’’’2001’ Michael’ Carter
Solutions’ for’ Foundations’ of’ Mathematical’ Economics
Chapter’ 1:’ Sets’ and’ Spaces
1.1
{‘1,’3,’5,’7’. . . ’}’or’ {‘ ’ ∈’ ’ :’ ’ is’ odd’}
1.2 Every’ ∈ ’ also’ belongs’ to’ .’ Every’ ∈
’ also’ belongs’ to’ .’ Hence’ ,’ ’ have’precisely ’ the’ same’ elements.
1.3 Examples’ of’ finite’ sets’ are
∙ the’ letters’ of’ the’ alphabet’ {‘A,’ B,’ C,’ . . . ’ ,’ Z’}
∙ the’ set’ of’ consumers’ in’ an’ economy
∙ the’ set’ of’ goods’ in’ an’ economy
∙ the’ set’ of’ players’in’ a’ game.’
Examples’ of’ infinite’ sets’ are
∙ the’ real’ numbers’ ℜ
∙ the’ natural’ numbers’
∙ the’ set’ of’ all’ possible’ colors
∙ the’ set’ of’ possible’ prices’ of’ copper’ on’ the’ world’ market
∙ the’ set’ of’ possible’ temperatures’ of’ liquid’ water.
1.4’ ’ =‘ {‘1,’2,’3,’4,’5,’6’},’ ’ =‘ {‘2,’4,’6’}.
1.5 The’ player’ set’ is’ ’ =‘ {‘J enny,’Chris’} . ’ Their’ action’ spaces’ are
’ =‘ {‘Rock,’Scissors,’Paper’} ’ =‘ Jenny,’Chris
1.6 The’ set’ of’ players’ is’ ’ =‘ 1,{’2 , . . . , ’ ’ . ’}The’ strategy’ space’ of’ each’ player’ is’ the’ set’of’ f
easible’ outputs
’ =‘ {‘ ’ ∈’ℜ+’ :’ ’ ≤’ ’}
where’ ’’is’’the’ output’ of’ dam’ .
3
1.7 The’ player’ set’ is’ ’ =‘ {1,’2,’3 } . ’There’ are’ 2 ’ =‘ 8’ coalitions,’ namely
( ’)’ =‘ {∅,’{1},’{2},’{3},’{1,’2},’{1,’3},’{2,’3},’{1,’2,’3}}
10
There’ are’ 2 ’ coalitions’ in’ a’ ten’ player’ game.
1.8’’ Assume’’that’’ ’’∈’( ’ ∪ ’ ’) .’’’That’’is’’ ’’∈ /’’ ’ ∪ ’
’.’’’This’’implies’’ ’’∈/’’ ’’and’’ ’’∈/’’ ’,’or’ ’∈’ ’and’ ’∈’ ’ .’ Consequently,’ ’∈’ ’∩ ’
’ .’ Conversely,’ assume’ ’∈’ ’∩ ’ ’ .’This’’implies’’that’’ ’ ∈’ ’’and’’ ’ ∈’
’ .’’’Consequently’’ ’∈ /’’ ’’and’’ ’∈/’’ ’’and’’therefore
/’ ’ ∪ ’ ’. ’This’ implies’’that’ ’ ∈’( ’ ∪ ’ ’) . ’The’ other’ identity’ is’ proved’ similarly.
∈
1.9
∪
’ =‘
∈
∩
’ =‘∅
∈
1
, c’’’2001’ Michael’ Carter
Solutions’ for’ Foundations’ of’ Mathematical’ Economics All’rights’reserved
2
1
1
-1 0 1
-1
2 2
Figure’ 1.1:’ The’ relation’ {‘( ,’ )’ :’ ’ +’ ’ =‘ 1’}
1.10’ The’ sample’ space’ of’ a’ single’ coin’ toss’ is’ ,’{ ’ .’ The}’ set’ of’ possible’ outcomes’ int’ hree’ t
osses’ is’ the’ product
{
{ ,’ ’} × ’{ ,’ ’} × ’{ ,’ ’}’=‘ ( ,’ ,’ ),’( ,’ ,’ ’),’( ,’ ’,’ ),
}
( ,’ ’,’ ’),’( ,’ ,’ ),’( ,’ ,’ ’),’( ,’ ,’ ),’( ,’ ,’ ’)
A’ typical’ outcome’ is’ the’ sequence’ ( ,’ ,’ ’)’ of’ two’ heads’ followed’ by’ a’ tail.
1.11
’
’ ∩ ’ℜ
+ =‘ {0}
where’0’ =‘(0,’0 , . . . ’,’0)’is’the’production’plan’using’no’inputs’and’producing’no’outputs.’To’
see’ this,’ first’ note’ that’ 0’ is’ a’ feasible’ production’ plan.’ Therefore,’ 0’ ∈’ ’.’ Also,
0’ ∈’ℜ+’ and’ therefore’ 0’ ∈’ ’ ∩ ’ℜ ’+ .
ℜ +’assume’the’contrary.’That’i
To’show’that’there’is’no’other’feasible’production’plan’in’’’’’ ’,’we
s,’we’assume’there’is’some’feasible’production’plan’y’’’’’’’’ ’’’’’’0∈ ’’.’’’This
+ ∖’implies
ℜ’ ’{‘ } ’the’existen
ce’of’a’plan’producing’a’positive’output’with’no’inputs.’This’technological’infeasible,’ so’ t
hat’ ’∈
/’ ’.
1.12 1. ’’Let’’x’ ∈’ ’( ). ’’This’’implies’’that’’( ,’− x)’ ∈’ ’. ’’Let’’x′’ ≥’x.’’ Then’’( ,’− x′ )’ ≤
( ,’− x)’ and’ free’ disposability’ implies’’that’ ( ,’− x′ )’ ∈’ ’. ’Therefore’ x′’∈’ ’( ).
2.’’ Again’’ assume’’ x’’ ∈’ ’( ).’’’’This’’ implies’’ that’’ ( ,’− x)’’ ∈’
’.’’’’By’’ free’’ disposal,’( ′ ,’− x)’ ∈’ ’’ for’ every’ ′’≤’ ,’ which’ implies’’that’ x’ ∈’
’( ′ ).’’ ’( ′ )’ ⊇’ ’( ).
1.13 The’ domain’ of’ “<”‘ is’ {1,’2}’=‘ ’ and’ the’ range’ is’ {2,’3}’⫋’ ’.
1.14 Figure’1.1.
1.15 The’ relation’ “is’ strictly’ higher’ than”‘ is’ transitive,’ antisymmetric’ and’ asymmetric.’It
’ is’ not’ complete,’ reflexive’ or’ symmetric.
2
, c’’’2001’ Michael’ Carter
Solutions’ for’ Foundations’ of’ Mathematical’ Economics All’rights’reserved
1.16 The’ following’ table’ lists’ their’ respective’ properties.
< ≤’’ =
√ √
reflexive ×’’
√ √’’ √
transitive
√’’ √
symmetric ×’’
√
asymmetric ×’’ ×
anti-symmetric √’’ √ √
√’ √’
complete ×
Note’ that’ the’ properties’ of’ symmetry’ and’ anti-symmetry’ are’ not’ mutually’ exclusive.
1.17 Let’be ∼ ’an’equivalence’relation’of’a’set’ ’=‘. ’∕’ That∅ ’is,’the’relation’is’reflexive,
∼ ’symme
tric’and’transitive.’We’first’show’that’every’ ’ ’belongs∈’to’some’equivalence’class.’ Let’
’ be’ any’ element’ in’ ’ and’ let’ ( )’ be’ the’ class
∼ ’ of’ elements’ equivalent’ to
,’ that’ is
∼( )’ ≡ ’{‘ ’ ∈’ ’ :’ ’ ∼’ ’}
Since ∼ is’ reflexive,’ ∼ ’and’so’ ∈’∼ ( ).’ Every’ ∈
’ belongs’ to’ some’ equivalence’class’ and’ therefore
∪
’= ∼( )
∈
Next,’ we’ show’ that’ the’ equivalence’ classes’ are’ either’ disjoint’ or’ identical,’’that’ is
∼( )’ ∕=‘ ∼( )’ if’ and’ only’ if’ f∼( )’∩ ’∼( ) ’=‘ ∅.
First,’ assume’ ∼( )’∩ ’∼( ) ’=‘ ∅. ’Then’ ’ ∈’∼( )’ but’’ /∈ ∼( ). ’Therefore’ ∼( )’ ∕=‘ ∼( ).
Conversely,’’assume’’∼( )’ ∩ ’∼( )’’∕=‘‘∅’and’’let’’ ’’∈’∼( )’ ∩ ’∼( ).’’’Then’’ ’’∼’
’’and’’b y’symmetry’ ’ ∼’ .’’’Also’ ’ ∼’ ’and’so’ by’ transitivity’ ’ ∼’
.’’’Let’ ’ be’ any’element’in’’∼( )’’so’’that’’ ’’∼’ .’’’Again’’by’’transitivity’’ ’’∼’
’’and’’therefore’’ ’’∈’∼( ).’’’Hence
∼( )’ ⊆’∼( ). ’Similar’’reasoning’ implies’’that’ ∼( )’ ⊆’∼( ). ’Therefore’ ∼( ) ’=‘ ∼( ).
We’ conclude’ that’ the’ equivalence’ classes’ partition’ .
1.18 The’set’of’proper’coalitions’is’ not’ a’partition’of’the’ set’of’ players,’since’ any’ player’ca
n’ belong’ to’ more’ than’ one’ coalition.’For’ example,’ player’ 1’ belongs’ to’ the’ coalitions
{1},’ {1,’2}’and’ so’ on.
1.19
’ ≻ ’ ’ =⇒ ’ ’ ≿’ ’ and’ ’ ∕≿’
’ ∼’ ’ =⇒ ’ ’ ≿’ ’ and’ ’ ≿’
Transitivity’ of’ ≿’implies’ ’≿’ . ’We’ need’ to’ show’ that’ ’∕≿’ . ’Assume’ otherwise,’ that’is’ as
sume’ ’ ≿’ ’ This’ implies’ ’ ∼’ ’ and’ by’ transitivity’ ’ ∼’ .’ But’ this’ implies’ that
’ ≿’ ’ which’ contradicts’ the’ assumption’ that’ ’ ≻’ . ’ Therefore’ we’ conclude’ that’ ’ ∕≿’
and’ therefore’ ’ ≻’ . ’The’ other’ result’ is’ proved’ in’ similar’ fashion.
1.20 asymmetric’ Assume’ ’ ≻’ .
’ ≻’ ’ =⇒ ’ ’ ∕≿’
while
’ ≻’ ’ =⇒ ’ ’ ≿’
Therefore
’ ≻’ ’ =⇒ ’ ’ ∕≻ ’
3
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