Solutions Manual Foundatio
m m
ns of Mathematical Economics
m m m
MichaelmCarter mNovem
ber m15,m2002
, c⃝ m 2001 m Michaelm Carter
Solutionsm for m Foundatio nsm ofm Math ema ticalm Econo mic Allm rightsm reserve
s d
Chapter 1: Sets and Spaces
m m m m
1.1
{m1,m3,m5,m7 m. . . } mor m{m𝑛m∈m𝑁m : m 𝑛mism oddm}
1.2 Everym 𝑥 ∈ 𝐴malsom belongsm tom 𝐵.m Everym 𝑥 ∈
𝐵 m alsom belongsm tom 𝐴.m Hence m 𝐴,m𝐵 m have m preciselym the m same m ele ments.
1.3 Examplesm ofm finite m setsm are
∙ the mlettersmof m the m alphabetm {mA,m B,m C,m .. . m ,m Zm}
∙ the m setm ofm consumersm inm anm economy
∙ the m setm ofm goodsm inm anm economy
∙ the m setm ofm playersm inm am gam
e.m Examplesm ofm infinite m se tsm are
∙ the m realm numbersm ℜ
∙ the m naturalm numbersm 𝔑
∙ the m setm ofm allm possible m colors
∙ the m setm ofm possible m pricesm ofm copper m onm the m worldm market
∙ the m setm ofm possible m temperaturesm ofm liquidm water.
1.4 𝑆m= m{m1,m2,m3,m4, m5,m6 m } ,m 𝐸m= m { m2, m4,m6 m} .
1.5 The m player m se tm ism 𝑁 m = m {mJe nny,mChrism} . m Their m ac tionm space sm are
𝐴𝑖m= m {mRock, mSc issor s,mPaper m} 𝑖 m = m Jenny,mChris
1.6 The m setm ofm playersm ism 𝑁 m ={m 1,m2 , . . . , }m𝑛m . m The m strategym space m ofm eachm player m ism
the m setm ofm feasible m outputs
𝐴𝑖m= m{m𝑞𝑖m∈mℜ+ m: m𝑞𝑖m≤m𝑄𝑖m}
where m 𝑞𝑖m ism the m outputm ofm damm 𝑖.
1.7 The mplayer msetmism𝑁m= m{1,m2,m3 } .mThere mare m2 3 m= m8 m coalitions, mnamely
𝒫 ( 𝑁) m = m {∅,m{1},m{2},m{3},m{1,m2},m{1,m3},m{2,m3},m{1,m2,m3}}
There m are m 2 10 m coalitionsm inm am tenm player m game.
1.8 Assume m thatm 𝑥m ∈m (𝑆m ∪m 𝑇m)𝑐.m Thatm ism 𝑥m ∈/m 𝑆m ∪m 𝑇m.m Thism impliesm 𝑥m ∈/m 𝑆m and
/m 𝑇m,m or m𝑥m∈m 𝑆𝑐m andm 𝑥m ∈m𝑇m𝑐 .m Conseque ntly,m 𝑥m ∈m𝑆𝑐m ∩m𝑇m𝑐.m Converse ly,m assume
m 𝑥m ∈
𝑐 𝑐
m 𝑥m∈m 𝑆 m∩m𝑇 mm . Thism impliesm thatm 𝑥m ∈m 𝑆𝑐m andm 𝑥 m ∈m 𝑇m𝑐.m Consequentlym 𝑥m ∈/m 𝑆m andm
𝑥m ∈
/m 𝑇m andm therefore
𝑥m∈/m 𝑆m∪m𝑇m. m Thism impliesm thatm 𝑥m ∈m(𝑆m∪ m𝑇m) 𝑐 . m The m other m ide ntitym ism prove dm similar ly.
1.9
∪
𝑆m = m 𝑁
𝑆∈𝒞
∩
𝑆m = m ∅
𝑆∈𝒞
1
, c⃝ m 2001 m Michaelm Carter
Solutionsm for m Foundatio nsm ofm Math ema ticalm Econo mic Allm rightsm reserve
s d
𝑥2
1
𝑥1
-1 0 1
-1
Figure m1.1: m The m relationm {m(𝑥,m𝑦) m: m 𝑥2 m+ m𝑦2 m = m1 m}
1.1 0 The m sample m space m ofm am single m coinm tossm is{m 𝐻,m𝑇}m .m The m se tm ofm possible m outc om
esm inm three m tossesm ism the m product
{
{𝐻,m𝑇m} × m{ 𝐻,m𝑇 m} × m{ 𝐻,m𝑇 m} m= m (𝐻,m𝐻 ,m𝐻),m( 𝐻,m𝐻, m𝑇 m), m(𝐻 ,m𝑇 ,m𝐻),
}
(𝐻,m𝑇,m𝑇m),m(𝑇,m 𝐻,m 𝐻),m(𝑇,m 𝐻,m𝑇 m),m (𝑇,m 𝑇,m 𝐻),m(𝑇,m 𝑇,m𝑇 m)
A m typicalm outcome m ism the m sequence m (𝐻,m𝐻,m𝑇m)m ofm twom headsm followedm bym am tail.
1.1 1
𝑌m ∩mℜ+𝑛m = m{0}
where m 0 m = m (0,m0 , . . . m,m0)m ism the m productionm planm usingm nom inputsm andm producingm nom o
utputs.m Tom see m this,m fir stm note m thatm 0 m ism am fe asible m pr oduc tionm plan.m Therefor e,m
0 m ∈m 𝑌m.m Also,
𝑛m 𝑛m
0 m ∈m ℜ+ andm therefore m 0 m ∈m 𝑌m ∩ + mℜ .
Tom showm thatm there m ism nom other m fe asible m productionm planmℜin+m 𝑛m,m we m assume m the m con
trary.m Thatm is,m we m assume m there m ism some m fe asible m production∈mmℜplan {m m m 𝑛 m m 0 m .m
m m ∖mmy
+
}
Thism impliesm the m existence m ofm am planm producingm am positive m outputm withm nom inputs.m Th
m
ism technologicalm infeasible,m som thatm 𝑦m ∈/m 𝑌m.
1.1 2 1.m Letm xm∈m 𝑉m(𝑦).m This m impliesm thatm (𝑦, m−x) m ∈ m𝑌m .m Letm x′ m ≥m x.m Thenm (𝑦,m−x′) m ≤
(𝑦,m−x)m andm fr ee m disposabilitym implie sm thatm (𝑦,m−x′ )m ∈m 𝑌m. m Ther efore m x ′m ∈m 𝑉 m(𝑦).
2.m Againm assume m xm ∈m 𝑉m(𝑦).m Thism im pliesm thatm (𝑦,m−x) m ∈m 𝑌m.m Bym free m di
sposal,m (𝑦 ′,m−x) m ∈m 𝑌m for m everym 𝑦 ′m ≤ m 𝑦,m w hic hm im plie sm thatm xm ∈m 𝑉m (𝑦′ ).m 𝑉m (
𝑦′)m ⊇m𝑉m(𝑦).
1.1 3 The m domainm ofm “<”m ism {1,m2 }m = m 𝑋m andm the m r ange m ism {2,m3 }m ⫋m 𝑌m .
1.1 4 Figure m 1.1.
1.1 5 The m relationm “ism strictlym higher m than”m ism tr ansitive ,m antisym metr ic m andm asymme t
ric.m Itm ism notm complete,m reflex ive m or m symmetric .
2
, c⃝ m 2001 m Michaelm Carter
Solutionsm for m Foundatio nsm ofm Math ema ticalm Econo mic Allm rightsm reserve
s d
1.1 6 The m followingm table m listsm the ir m respec tive m properties.
< ≤
√ =
√
reflexive ×
transitive √ √ √
symmetric √ √
×
√
asymmetric × ×
anti-symmetric √ √ √
√ √
complete ×
Note m thatm the m proper tiesm ofm symme trym andm anti-sym metr ym are m notm mutuallym exclusive.
1.1 7 Letm ~be m anm equivale nce m relationm ofm am se tm 𝑋∕ m=∅m . m Thatm is,m the m relation
∼ m ism reflex i
ve,m symmetric m andm transitive .m We m firstm showm thatm every ∈ m 𝑥m 𝑋mbelongsm tom some m equi
vale nce m class.m Letm 𝑎m be m anym ele me ntm inm~
𝑋m andm letm m (𝑎)m be m the m classm ofm ele men
tsm equivalentm to
𝑎,m thatm is
∼(𝑎)m ≡m {m𝑥m ∈ m 𝑋m : m 𝑥 m ∼ m 𝑎m}
Since ∼ ism reflexive,m 𝑎 ∼ 𝑎mandm som 𝑎 ∈m ∼(𝑎).m Everym 𝑎 ∈
𝑋mbelongsm tom some m equivalence m classm andm therefore
∪
𝑋m = ∼(𝑎)
𝑎∈ 𝑋
Next,m we m showm thatm the m equivalence m classesm are m either m disjointm or m identical,m thatm is
∼(𝑎)m ∕= m ∼(𝑏)m ifm andm onlym ifm f∼(𝑎)m ∩m ∼(𝑏)m = m ∅.
First,m assume m ∼(𝑎)m ∩ m∼(𝑏)m= m ∅ . m The nm 𝑎m ∈m ∼(𝑎)m butm 𝑎m ∈/m ∼(𝑏). m There for e m ∼(𝑎)m ∕= m ∼(𝑏).
Conversely,m assume m ∼(𝑎)m ∩m∼(𝑏)m ∕= m ∅m andm le tm 𝑥m∈m ∼(𝑎)m ∩m∼(𝑏).m The nm 𝑥m ∼m𝑎m andm b
ym symme trym 𝑎m ∼m 𝑥.m Alsom 𝑥m ∼m 𝑏 m andm som bym tr ansitivitym 𝑎m ∼m 𝑏.m Letm 𝑦m be m an
ym ele mentm inm ∼(𝑎)m som thatm 𝑦m ∼m 𝑎.m Againm bym transitivitym 𝑦m ∼m 𝑏m andm therefore
m 𝑦m ∈m ∼(𝑏).m Hence
∼(𝑎)m ⊆m ∼(𝑏).m Similar m re asoningm implie sm thatm ∼(𝑏)m ⊆m ∼(𝑎).m There fore m ∼(𝑎) m= m
∼(𝑏).m We m c onclude m thatm the m e quivale nc e m classe sm par titionm 𝑋.
1.1 8 The m setm ofm proper m coalitionsm ism notm am par titionm ofm the m se tm ofm player s,m since m anym
player m canm belongm tom more m thanm one m coalition.m For m ex ample ,m player m 1 m belongsm tom th
e m coalitions
{1},m {1,m2}m andm som on.
1.1 9
𝑥m≻ m𝑦m =⇒m 𝑥m≿m𝑦m andm 𝑦m∕ ≿m𝑥
𝑦m ∼m𝑧m =⇒m 𝑦m ≿m𝑧m andm 𝑧m ≿m𝑦
Transitivitym ofm ≿m implie sm 𝑥m≿m𝑧 . m We m needm tom showm thatm 𝑧m ∕≿m𝑥 . m Assume m other wis
e,m thatm ism assume m 𝑧m ≿m 𝑥m Thism impliesm 𝑧m ∼m 𝑥m andm bym tr ansitivitym 𝑦m ∼m 𝑥.m Butm th
ism impliesm that
𝑦m≿m𝑥mwhichm contradictsm the m assumptionm thatm 𝑥m≻ m𝑦 . m Therefore m we m conclude m thatm 𝑧m∕ ≿m𝑥
andm therefore m 𝑥m≻ m𝑧 . m The m other m resultm ism prove dm inm similar m fashion.
1.2 0 asymmetricm Assume m 𝑥m ≻ m 𝑦.
Therefore
while
3
m m
ns of Mathematical Economics
m m m
MichaelmCarter mNovem
ber m15,m2002
, c⃝ m 2001 m Michaelm Carter
Solutionsm for m Foundatio nsm ofm Math ema ticalm Econo mic Allm rightsm reserve
s d
Chapter 1: Sets and Spaces
m m m m
1.1
{m1,m3,m5,m7 m. . . } mor m{m𝑛m∈m𝑁m : m 𝑛mism oddm}
1.2 Everym 𝑥 ∈ 𝐴malsom belongsm tom 𝐵.m Everym 𝑥 ∈
𝐵 m alsom belongsm tom 𝐴.m Hence m 𝐴,m𝐵 m have m preciselym the m same m ele ments.
1.3 Examplesm ofm finite m setsm are
∙ the mlettersmof m the m alphabetm {mA,m B,m C,m .. . m ,m Zm}
∙ the m setm ofm consumersm inm anm economy
∙ the m setm ofm goodsm inm anm economy
∙ the m setm ofm playersm inm am gam
e.m Examplesm ofm infinite m se tsm are
∙ the m realm numbersm ℜ
∙ the m naturalm numbersm 𝔑
∙ the m setm ofm allm possible m colors
∙ the m setm ofm possible m pricesm ofm copper m onm the m worldm market
∙ the m setm ofm possible m temperaturesm ofm liquidm water.
1.4 𝑆m= m{m1,m2,m3,m4, m5,m6 m } ,m 𝐸m= m { m2, m4,m6 m} .
1.5 The m player m se tm ism 𝑁 m = m {mJe nny,mChrism} . m Their m ac tionm space sm are
𝐴𝑖m= m {mRock, mSc issor s,mPaper m} 𝑖 m = m Jenny,mChris
1.6 The m setm ofm playersm ism 𝑁 m ={m 1,m2 , . . . , }m𝑛m . m The m strategym space m ofm eachm player m ism
the m setm ofm feasible m outputs
𝐴𝑖m= m{m𝑞𝑖m∈mℜ+ m: m𝑞𝑖m≤m𝑄𝑖m}
where m 𝑞𝑖m ism the m outputm ofm damm 𝑖.
1.7 The mplayer msetmism𝑁m= m{1,m2,m3 } .mThere mare m2 3 m= m8 m coalitions, mnamely
𝒫 ( 𝑁) m = m {∅,m{1},m{2},m{3},m{1,m2},m{1,m3},m{2,m3},m{1,m2,m3}}
There m are m 2 10 m coalitionsm inm am tenm player m game.
1.8 Assume m thatm 𝑥m ∈m (𝑆m ∪m 𝑇m)𝑐.m Thatm ism 𝑥m ∈/m 𝑆m ∪m 𝑇m.m Thism impliesm 𝑥m ∈/m 𝑆m and
/m 𝑇m,m or m𝑥m∈m 𝑆𝑐m andm 𝑥m ∈m𝑇m𝑐 .m Conseque ntly,m 𝑥m ∈m𝑆𝑐m ∩m𝑇m𝑐.m Converse ly,m assume
m 𝑥m ∈
𝑐 𝑐
m 𝑥m∈m 𝑆 m∩m𝑇 mm . Thism impliesm thatm 𝑥m ∈m 𝑆𝑐m andm 𝑥 m ∈m 𝑇m𝑐.m Consequentlym 𝑥m ∈/m 𝑆m andm
𝑥m ∈
/m 𝑇m andm therefore
𝑥m∈/m 𝑆m∪m𝑇m. m Thism impliesm thatm 𝑥m ∈m(𝑆m∪ m𝑇m) 𝑐 . m The m other m ide ntitym ism prove dm similar ly.
1.9
∪
𝑆m = m 𝑁
𝑆∈𝒞
∩
𝑆m = m ∅
𝑆∈𝒞
1
, c⃝ m 2001 m Michaelm Carter
Solutionsm for m Foundatio nsm ofm Math ema ticalm Econo mic Allm rightsm reserve
s d
𝑥2
1
𝑥1
-1 0 1
-1
Figure m1.1: m The m relationm {m(𝑥,m𝑦) m: m 𝑥2 m+ m𝑦2 m = m1 m}
1.1 0 The m sample m space m ofm am single m coinm tossm is{m 𝐻,m𝑇}m .m The m se tm ofm possible m outc om
esm inm three m tossesm ism the m product
{
{𝐻,m𝑇m} × m{ 𝐻,m𝑇 m} × m{ 𝐻,m𝑇 m} m= m (𝐻,m𝐻 ,m𝐻),m( 𝐻,m𝐻, m𝑇 m), m(𝐻 ,m𝑇 ,m𝐻),
}
(𝐻,m𝑇,m𝑇m),m(𝑇,m 𝐻,m 𝐻),m(𝑇,m 𝐻,m𝑇 m),m (𝑇,m 𝑇,m 𝐻),m(𝑇,m 𝑇,m𝑇 m)
A m typicalm outcome m ism the m sequence m (𝐻,m𝐻,m𝑇m)m ofm twom headsm followedm bym am tail.
1.1 1
𝑌m ∩mℜ+𝑛m = m{0}
where m 0 m = m (0,m0 , . . . m,m0)m ism the m productionm planm usingm nom inputsm andm producingm nom o
utputs.m Tom see m this,m fir stm note m thatm 0 m ism am fe asible m pr oduc tionm plan.m Therefor e,m
0 m ∈m 𝑌m.m Also,
𝑛m 𝑛m
0 m ∈m ℜ+ andm therefore m 0 m ∈m 𝑌m ∩ + mℜ .
Tom showm thatm there m ism nom other m fe asible m productionm planmℜin+m 𝑛m,m we m assume m the m con
trary.m Thatm is,m we m assume m there m ism some m fe asible m production∈mmℜplan {m m m 𝑛 m m 0 m .m
m m ∖mmy
+
}
Thism impliesm the m existence m ofm am planm producingm am positive m outputm withm nom inputs.m Th
m
ism technologicalm infeasible,m som thatm 𝑦m ∈/m 𝑌m.
1.1 2 1.m Letm xm∈m 𝑉m(𝑦).m This m impliesm thatm (𝑦, m−x) m ∈ m𝑌m .m Letm x′ m ≥m x.m Thenm (𝑦,m−x′) m ≤
(𝑦,m−x)m andm fr ee m disposabilitym implie sm thatm (𝑦,m−x′ )m ∈m 𝑌m. m Ther efore m x ′m ∈m 𝑉 m(𝑦).
2.m Againm assume m xm ∈m 𝑉m(𝑦).m Thism im pliesm thatm (𝑦,m−x) m ∈m 𝑌m.m Bym free m di
sposal,m (𝑦 ′,m−x) m ∈m 𝑌m for m everym 𝑦 ′m ≤ m 𝑦,m w hic hm im plie sm thatm xm ∈m 𝑉m (𝑦′ ).m 𝑉m (
𝑦′)m ⊇m𝑉m(𝑦).
1.1 3 The m domainm ofm “<”m ism {1,m2 }m = m 𝑋m andm the m r ange m ism {2,m3 }m ⫋m 𝑌m .
1.1 4 Figure m 1.1.
1.1 5 The m relationm “ism strictlym higher m than”m ism tr ansitive ,m antisym metr ic m andm asymme t
ric.m Itm ism notm complete,m reflex ive m or m symmetric .
2
, c⃝ m 2001 m Michaelm Carter
Solutionsm for m Foundatio nsm ofm Math ema ticalm Econo mic Allm rightsm reserve
s d
1.1 6 The m followingm table m listsm the ir m respec tive m properties.
< ≤
√ =
√
reflexive ×
transitive √ √ √
symmetric √ √
×
√
asymmetric × ×
anti-symmetric √ √ √
√ √
complete ×
Note m thatm the m proper tiesm ofm symme trym andm anti-sym metr ym are m notm mutuallym exclusive.
1.1 7 Letm ~be m anm equivale nce m relationm ofm am se tm 𝑋∕ m=∅m . m Thatm is,m the m relation
∼ m ism reflex i
ve,m symmetric m andm transitive .m We m firstm showm thatm every ∈ m 𝑥m 𝑋mbelongsm tom some m equi
vale nce m class.m Letm 𝑎m be m anym ele me ntm inm~
𝑋m andm letm m (𝑎)m be m the m classm ofm ele men
tsm equivalentm to
𝑎,m thatm is
∼(𝑎)m ≡m {m𝑥m ∈ m 𝑋m : m 𝑥 m ∼ m 𝑎m}
Since ∼ ism reflexive,m 𝑎 ∼ 𝑎mandm som 𝑎 ∈m ∼(𝑎).m Everym 𝑎 ∈
𝑋mbelongsm tom some m equivalence m classm andm therefore
∪
𝑋m = ∼(𝑎)
𝑎∈ 𝑋
Next,m we m showm thatm the m equivalence m classesm are m either m disjointm or m identical,m thatm is
∼(𝑎)m ∕= m ∼(𝑏)m ifm andm onlym ifm f∼(𝑎)m ∩m ∼(𝑏)m = m ∅.
First,m assume m ∼(𝑎)m ∩ m∼(𝑏)m= m ∅ . m The nm 𝑎m ∈m ∼(𝑎)m butm 𝑎m ∈/m ∼(𝑏). m There for e m ∼(𝑎)m ∕= m ∼(𝑏).
Conversely,m assume m ∼(𝑎)m ∩m∼(𝑏)m ∕= m ∅m andm le tm 𝑥m∈m ∼(𝑎)m ∩m∼(𝑏).m The nm 𝑥m ∼m𝑎m andm b
ym symme trym 𝑎m ∼m 𝑥.m Alsom 𝑥m ∼m 𝑏 m andm som bym tr ansitivitym 𝑎m ∼m 𝑏.m Letm 𝑦m be m an
ym ele mentm inm ∼(𝑎)m som thatm 𝑦m ∼m 𝑎.m Againm bym transitivitym 𝑦m ∼m 𝑏m andm therefore
m 𝑦m ∈m ∼(𝑏).m Hence
∼(𝑎)m ⊆m ∼(𝑏).m Similar m re asoningm implie sm thatm ∼(𝑏)m ⊆m ∼(𝑎).m There fore m ∼(𝑎) m= m
∼(𝑏).m We m c onclude m thatm the m e quivale nc e m classe sm par titionm 𝑋.
1.1 8 The m setm ofm proper m coalitionsm ism notm am par titionm ofm the m se tm ofm player s,m since m anym
player m canm belongm tom more m thanm one m coalition.m For m ex ample ,m player m 1 m belongsm tom th
e m coalitions
{1},m {1,m2}m andm som on.
1.1 9
𝑥m≻ m𝑦m =⇒m 𝑥m≿m𝑦m andm 𝑦m∕ ≿m𝑥
𝑦m ∼m𝑧m =⇒m 𝑦m ≿m𝑧m andm 𝑧m ≿m𝑦
Transitivitym ofm ≿m implie sm 𝑥m≿m𝑧 . m We m needm tom showm thatm 𝑧m ∕≿m𝑥 . m Assume m other wis
e,m thatm ism assume m 𝑧m ≿m 𝑥m Thism impliesm 𝑧m ∼m 𝑥m andm bym tr ansitivitym 𝑦m ∼m 𝑥.m Butm th
ism impliesm that
𝑦m≿m𝑥mwhichm contradictsm the m assumptionm thatm 𝑥m≻ m𝑦 . m Therefore m we m conclude m thatm 𝑧m∕ ≿m𝑥
andm therefore m 𝑥m≻ m𝑧 . m The m other m resultm ism prove dm inm similar m fashion.
1.2 0 asymmetricm Assume m 𝑥m ≻ m 𝑦.
Therefore
while
3
