Foundations of Mathematical Economics by Michael Carter | Complete Solutions Manual

Solutions Manual#z



Foundations of Mathematical
#z #z #z



Economics
#z




Michael #zCarter

#z November #z15, #z2002

, c⃝ 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved




Chapter 1: Sets and Spaces

1.1
{ 1, 3, 5, 7 . . . } or { 𝑛 ∈ 𝑁 : 𝑛 is odd }
1.2 Every 𝑥 ∈ 𝐴 also belongs to 𝐵. Every 𝑥 ∈ 𝐵 also belongs to 𝐴. Hence 𝐴, 𝐵 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 𝔑
∙ 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 𝑆 = { 1, 2, 3, 4, 5, 6 }, 𝐸 = { 2, 4, 6 }.
1.5 The player set is 𝑁 = { Jenny, Chris } . Their action spaces are
𝐴𝑖 = { Rock, Scissors, Paper } 𝑖 = Jenny, Chris
1.6 The set of players is 𝑁 = {1, 2 , . . . , 𝑛 .} The strategy space of each player is the set
of feasible outputs
𝐴𝑖 = { 𝑞𝑖 ∈ ℜ+ : 𝑞𝑖 ≤ 𝑄𝑖 }
where 𝑞𝑖 is the output of dam 𝑖.
1.7 The player set is 𝑁 = {1, 2, 3}. There are 23 = 8 coalitions, namely
𝒫(𝑁) = {∅, {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 𝑥 ∈ (𝑆 ∪ 𝑇 )𝑖. That is 𝑥 ∈ / 𝑆 ∪ 𝑇 . This implies 𝑥 ∈/ 𝑆 and 𝑥 ∈ / 𝑇,
or 𝑥 ∈ 𝑆𝑖 and 𝑥 ∈ 𝑇 𝑖. Consequently, 𝑥 ∈ 𝑆𝑖 ∩ 𝑇 𝑖. Conversely, assume 𝑥 ∈ 𝑆𝑖 ∩ 𝑇 𝑖.
This implies that 𝑥 ∈ 𝑆𝑖 and 𝑥 ∈ 𝑇 𝑖. Consequently 𝑥 ∈ / 𝑆 and 𝑥 ∈ / 𝑇 and therefore
𝑥∈/ 𝑆 ∪ 𝑇 . This implies that 𝑥 ∈ (𝑆 ∪ 𝑇 ) 𝑖 . The other identity is proved similarly.
1.9
∪
𝑆=𝑁
𝑖∈𝒞
∩
𝑆=∅
𝑖∈𝒞



1

, c⃝ # z 2001 #zMichael

Solutions #z for #zFoundations # z of # z Mathematical #z Carter
All # z rights
# z Economics # z reserved



𝑥2
1




𝑥1
-1 0 1




-1

Figure #z1.1: #zThe #zrelation #z{ #z(𝑥, #z𝑦) #z: #z𝑥2 #z+ #z𝑦2 #z= #z1 #z}


1.10 The #zsample #zspace #zof #za #zsingle #zcoin #ztoss{ #zis #}z 𝐻, #z𝑇 # z . # z The #zset #zof
#zpossible #zoutcomes #zin # z three #ztosses #zis #zthe #zproduct

{
{𝐻, #z𝑇#z} × #z{𝐻, #z𝑇#z} × #z{𝐻, #z𝑇#z} #z= # z (𝐻, #z𝐻, #z𝐻), #z(𝐻, #z𝐻, #z𝑇#z), #z(𝐻, #z𝑇, #z𝐻),
}
(𝐻, #z𝑇, #z𝑇 #z), #z(𝑇, #z𝐻, #z𝐻), #z(𝑇, #z𝐻, #z𝑇 #z), #z(𝑇, #z𝑇, #z𝐻), #z(𝑇, #z𝑇, #z𝑇 #z)


A # z typical # z outcome # z is # z the # z sequence # z (𝐻, #z𝐻, #z𝑇 #z) # z of # z two # z heads # z followed # z by # z a # z tail.
1.11

𝑌 #z ∩ #zℜ+𝑖 # z = #z{0}

where # z 0 # z = # z (0, #z0,.. . #z, #z0) # z is # z the # z production # z plan # z using # z no # z inputs # z and
# z producing # z no # z outputs. #zTo #z see # z this, # z first # z note # z that # z 0 # z is # z a # z feasible

# z production # z plan. # z Therefore, #z 0 # z ∈ # z 𝑌 #z. # z Also,

0 #z ∈ #z+ ℜ𝑖 # z and # z therefore # z 0 #z ∈+#z 𝑌 # z ∩ #zℜ𝑖 #z.
𝑖
To #zshow #zthat #zthere #zis #zno #zother #zfeasible #zproduction #zℜplan + #zin # z #z, #zwe #zassume
∈ #zℜ
#zthe #zcontrary. #zThat # z is, # z we # z assume # z there # z is # z some # z feasible
+ ##zz production # z plan
#zy # z # z # z
𝑖
# z # z 0 # z . # z This # z implies #zthe #zexistence #z z ∖#z
# of { #z#plan
#za z } #zproducing #za

#zpositive #zoutput #zwith #zno #zinputs. #zThis #ztechnological #zinfeasible, # z so # z that #z𝑦 # z ∈ /
# z 𝑌 #z.


1.12 1. # z Let # z x #z ∈ # z 𝑉 #z(𝑦). # z This # z implies # z that # z (𝑦, #z−x) #z ∈ #z 𝑌 #z. # z Let # z x′ # z ≥ #z x. # z Then # z (𝑦, #z−x′) #z ≤
(𝑦, #z−x) #zand #z free #z disposability #z implies # z that #z (𝑦, #z−x′) #z∈ #z𝑌 #z. #zTherefore #z x′ #z ∈ #z𝑉 #z(𝑦).
2. # z Again # z assume # z x # z ∈ # z 𝑉 #z(𝑦). # z This # z implies # z that # z (𝑦, #z−x) # z ∈
# z 𝑌 #z. # z By # z free # z disposal, #z(𝑦′, #z−x) #z∈ #z𝑌 # z for #zevery #z𝑦′ # z ≤ #z𝑦, #zwhich
# z implies # z that #zx #z∈ #z𝑉 #z(𝑦′). # z 𝑉 #z(𝑦′) #z⊇ #z𝑉 #z(𝑦).


1.13 The #z domain # z of # z “<” # z is # z {1, #z2} #z= #z𝑋 # z and # z the # z range # z is # z {2, #z3} #z⫋ #z𝑌 #z.
1.14 Figure # z 1.1.
1.15 The # z relation #z“is # z strictly #zhigher #zthan” # z is #ztransitive, # z antisymmetric #zand
# z asymmetric. #zIt #zis #znot #zcomplete, #zreflexive #zor #zsymmetric.




2

, c⃝ # z 2001 #zMichael

Solutions #z for #zFoundations # z of # z Mathematical #z Carter
All # z rights
# z Economics # z reserved


1.16 The # z following # z table # z lists # z their # z respective # z properties.
< ≤
√ =
√
reflexive ×
transitive √ √ √
√ √
symmetric ×
√ × ×
asymmetric √ √ √
anti-symmetric
√ √
complete ×
Note # z that # z the # z properties #z of # z symmetry # z and # z anti-symmetry # z are # z not # z mutually # z exclusive.
1.17 Let ~ # z be #zan #zequivalence #zrelation #zof #za∕ #zset ∅ #z𝑋 #z= # z . #zThat #zis, #z∼the #zrelation
# z is #zreflexive, #zsymmetric #zand #ztransitive. #zWe #zfirst ∈ #zshow #zthat #zevery #z𝑥 # z 𝑋
#zbelongs #zto #zsome #zequivalence #zclass. # ~ z Let # z 𝑎 # z be # z any # z element # z in # z 𝑋 # z and

# z let # z # z (𝑎) # z be # z the # z class # z of # z elements # z equivalent # z to

𝑎, #zthat #zis
∼(𝑎) #z≡ #z { #z𝑥 #z∈ #z 𝑋 # z : #z 𝑥 #z ∼ #z 𝑎 #z}
Since ∼ is #zreflexive, #z𝑎∼ 𝑎 #zand #zso #z∈𝑎 # z ∼
(𝑎). # z Every #z𝑎∈ 𝑋 #z belongs #zto #zsome
#zequivalence #zclass #zand #ztherefore
∪
𝑋 #z = ∼(𝑎)
𝑖∈𝑖

Next, # z we # z show # z that # z the equivalence # z classes
# z # z are # z either # z disjoint # z or
identical, # z that # z is
# z

∼(𝑎) #z∕= #z∼(𝑏) # z if # z and # z only # z if # z f∼(𝑎) #z∩ #z∼(𝑏) #z= #z∅.
First, #zassume #z∼(𝑎) #z∩ #z∼(𝑏) #z= #z∅ . #zThen #z 𝑎 #z∈ #z∼(𝑎) #z but #z 𝑎 #z∈
/ # z ∼(𝑏). #zTherefore #z∼(𝑎) #z∕= #z∼(𝑏).
Conversely, #zassume #z∼(𝑎) #z∩ #z∼(𝑏) #z∕= #z∅ #zand #zlet #z𝑥 #z∈ #z∼(𝑎) #z∩ #z∼(𝑏). # z Then #z𝑥 #z∼
#z𝑎 #zand #zby #zsymmetry # z 𝑎 # z ∼ # z 𝑥. # z Also # z 𝑥 # z ∼ # z 𝑏 # z and # z so # z by

# z transitivity # z 𝑎 # z ∼ # z 𝑏. # z Let # z 𝑦 # z be # z any # z element #zin # z ∼(𝑎) # z so # z that # z 𝑦

# z ∼ # z 𝑎. # z Again # z by # z transitivity # z 𝑦 # z ∼ # z 𝑏 # z and # z therefore # z 𝑦 # z ∈ # z ∼(𝑏).

# z Hence

∼(𝑎) #z⊆ #z∼(𝑏). #zSimilar #zreasoning #zimplies #zthat #z∼(𝑏) #z⊆ #z∼(𝑎). #zTherefore
#z∼(𝑎) #z= #z∼(𝑏). #zWe #zconclude #zthat #zthe #zequivalence #zclasses #zpartition #z𝑋.
1.18 The #zset #zof #zproper #zcoalitions #zis #znot #za #zpartition #zof #zthe #zset #zof #zplayers, #zsince
#zany #zplayer #zcan #zbelong #zto #zmore #zthan #zone #zcoalition. #zFor #zexample, #zplayer #z1

#zbelongs #zto #zthe #zcoalitions

{1}, #z {1, #z2} # z and # z so # z on.
1.19

𝑥 #z≻ #z𝑦 # z =⇒ # z 𝑥 #z≿ #z𝑦 # z and #z 𝑦 #z ∕≿ #z𝑥
𝑦 # z ∼ #z 𝑧 # z =⇒ # z 𝑦 # z ≿ #z𝑧 # z and # z 𝑧 # z ≿ #z𝑦
Transitivity #zof #z≿ #zimplies # z 𝑥 #z≿ #z𝑧 . #zWe # z need # z to # z show #zthat #z𝑧 # z ∕≿ #z𝑥 . #zAssume
# z otherwise, #zthat #zis # z assume # z 𝑧 # z ≿ #z 𝑥 # z This # z implies # z 𝑧 # z ∼ # z 𝑥 # z and # z by
# z transitivity # z 𝑦 # z ∼ # z 𝑥. # z But # z this # z implies # z that

𝑦 #z≿ #z𝑥 #z which #zcontradicts #zthe #zassumption #zthat #z𝑥 #z≻ #z𝑦 . #zTherefore #zwe #zconclude #zthat #z𝑧 #z∕≿ #z𝑥
and # z therefore # z 𝑥 #z ≻ #z𝑧 . #zThe # z other # z result # z is # z proved # z in # z similar # z fashion.
1.20 asymmetric # z Assume # z 𝑥 # z ≻ # z 𝑦.


while Therefore
3