Solutions Manual Foundations of Mathematical Economics ( PDFDrive.com )

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Solutions Manual
Foundations of Mathematical Economics
Michael Carter
November 15, 2002
Solutions for Foundations of Mathematical Economics
c
2001 Michael Carter
All rights reserved
Chapter 1: Sets and Spaces
1.1
{1,3,5,7...}or {𝑛𝑁:𝑛is odd }
1.2 Every 𝑥𝐴also belongs to 𝐵. Every 𝑥𝐵also 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
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
+:𝑞𝑖𝑄𝑖}
where 𝑞𝑖is 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}}
There are 210 coalitions in a ten player game.
1.8 Assume that 𝑥(𝑆𝑇)𝑐. That is 𝑥/𝑆𝑇. This implies 𝑥/𝑆and 𝑥/𝑇,
or 𝑥𝑆𝑐and 𝑥𝑇𝑐. Consequently, 𝑥𝑆𝑐𝑇𝑐. Conversely, assume 𝑥𝑆𝑐𝑇𝑐.
This implies that 𝑥𝑆𝑐and 𝑥𝑇𝑐. Consequently 𝑥/𝑆and 𝑥/𝑇and therefore
𝑥/𝑆𝑇. This implies that 𝑥(𝑆𝑇)𝑐. The other identity is proved similarly.
1.9
𝑆∈𝒞
𝑆=𝑁
𝑆∈𝒞
𝑆=
1
Solutions for Foundations of Mathematical Economics
c
2001 Michael Carter
All rights reserved
0-1 1 𝑥1
-1
1
𝑥2
Figure 1.1: The relation {(𝑥, 𝑦):𝑥2+𝑦2=1}
1.10 The sample space of a single coin toss is {𝐻, 𝑇 }. Thesetofpossibleoutcomesin
three tosses is the product
{𝐻, 𝑇 {𝐻, 𝑇 {𝐻, 𝑇 }=(𝐻, 𝐻, 𝐻),(𝐻, 𝐻, 𝑇 ),(𝐻, 𝑇, 𝐻),
(𝐻, 𝑇, 𝑇 ),(𝑇,𝐻,𝐻),(𝑇,𝐻,𝑇),(𝑇,𝑇,𝐻),(𝑇,𝑇,𝑇)
A typical outcome is the sequence (𝐻, 𝐻, 𝑇 ) of two heads followed by a tail.
1.11
𝑌∩ℜ
𝑛
+={0}
where 0=(0,0,...,0) is the production plan using no inputs and producing no outputs.
To see this, first note that 0is a feasible production plan. Therefore, 0𝑌. Also,
0∈ℜ
𝑛
+and therefore 0𝑌∩ℜ
𝑛
+.
To show that there is no other feasible production plan in 𝑛
+, we assume the contrary.
Thatis,weassumethereissomefeasibleproductionplany∈ℜ
𝑛
+∖{0}. This implies
the existence of a plan producing a positive output with no inputs. This technological
infeasible, so that 𝑦/𝑌.
1.12 1. Let x𝑉(𝑦). This implies that (𝑦, x)𝑌. Let xx. Then (𝑦, x)
(𝑦, x) and free disposability implies that (𝑦, x)𝑌. Therefore x𝑉(𝑦).
2. Again assume x𝑉(𝑦). This implies that (𝑦, x)𝑌. By free disposal,
(𝑦,x)𝑌for every 𝑦𝑦, which implies that x𝑉(𝑦). 𝑉(𝑦)𝑉(𝑦).
1.13 The domain of “<”is{1,2}=𝑋and the range is {2,3}𝑌.
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
Solutions for Foundations of Mathematical Economics
c
2001 Michael Carter
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 𝑋=. That is, the relation is reflexive,
symmetric and transitive. We first show that every 𝑥𝑋belongs to some equivalence
class. Let 𝑎be any element in 𝑋and let (𝑎) be the class of elements equivalent to
𝑎,thatis
(𝑎)≡{𝑥𝑋:𝑥𝑎}
Since is reflexive, 𝑎𝑎and so 𝑎∈∼(𝑎). Every 𝑎𝑋belongs to some equivalence
class and therefore
𝑋=
𝑎𝑋(𝑎)
Next, we show that the equivalence classes are either disjoint or identical, that is
(𝑎)=(𝑏) if and only if f(𝑎)∩∼(𝑏)=.
First, assume (𝑎)∩∼(𝑏)=. Then 𝑎∈∼(𝑎) but 𝑎/∈∼(𝑏). Therefore (𝑎)=(𝑏).
Conversely, assume (𝑎)∩∼(𝑏)=and let 𝑥∈∼(𝑎)∩∼(𝑏). Then 𝑥𝑎and by
symmetry 𝑎𝑥. Also 𝑥𝑏and so by transitivity 𝑎𝑏. Let 𝑦be any element
in (𝑎)sothat𝑦𝑎. Again by transitivity 𝑦𝑏and therefore 𝑦∈∼(𝑏). Hence
(𝑎)⊆∼(𝑏). Similar reasoning implies that (𝑏)⊆∼(𝑎). Therefore (𝑎)=(𝑏).
We conclude that the equivalence classes partition 𝑋.
1.18 The set of proper coalitions is not a partition of the set of players, since any player
can belong to more than one coalition. For example, player 1 belongs to the coalitions
{1},{1,2}and so on.
1.19
𝑥𝑦=𝑥𝑦and 𝑦𝑥
𝑦𝑧=𝑦𝑧and 𝑧𝑦
Transitivity of implies 𝑥𝑧. We need to show that 𝑧𝑥. Assume otherwise, that
is assume 𝑧𝑥This implies 𝑧𝑥and by transitivity 𝑦𝑥. But this implies that
𝑦𝑥which contradicts the assumption that 𝑥𝑦. Therefore we conclude that 𝑧𝑥
and therefore 𝑥𝑧. The other result is proved in similar fashion.
1.20 asymmetric Assume 𝑥𝑦.
𝑥𝑦=𝑦𝑥
while
𝑦𝑥=𝑦𝑥
Therefore
𝑥𝑦=𝑦∕≻ 𝑥
3
Solutions for Foundations of Mathematical Economics
c
2001 Michael Carter
All rights reserved
transitive Assume 𝑥𝑦and 𝑦𝑧.
𝑥𝑦=𝑥𝑦and 𝑦𝑥
𝑦𝑧=𝑦𝑧and 𝑧𝑦
Since is transitive, we conclude that 𝑥𝑧.
It remains to show that 𝑧𝑥. Assume otherwise, that is assume 𝑧𝑥. We
know that 𝑥𝑦and transitivity implies that 𝑧𝑦, contrary to the assumption
that 𝑦𝑧. We conclude that 𝑧𝑥and
𝑥𝑧and 𝑧𝑥=𝑥𝑧
This shows that is transitive.
1.21 reflexive Since is reflexive, 𝑥𝑥which implies 𝑥𝑥.
transitive Assume 𝑥𝑦and 𝑦𝑧. Now
𝑥𝑦⇐⇒ 𝑥𝑦and 𝑦𝑥
𝑦𝑧⇐⇒ 𝑦𝑧and 𝑧𝑦
Transitivity of implies
𝑥𝑦and 𝑦𝑧=𝑥𝑧
𝑧𝑦and 𝑦𝑥=𝑧𝑥
Combining
𝑥𝑧and 𝑧𝑥=𝑥𝑧
symmetric
𝑥𝑦⇐⇒ 𝑥𝑦and 𝑦𝑥
⇐⇒ 𝑦𝑥and 𝑥𝑦
⇐⇒ 𝑦𝑥
1.22 reflexive Every integer is a multiple of itself, that is 𝑚=1𝑚.
transitive Assume 𝑚=𝑘𝑛 and 𝑛=𝑙𝑝 where 𝑘, 𝑙 𝑁. Then 𝑚=𝑘𝑙𝑝 so that 𝑚is a
multiple of 𝑝.
not symmetric If 𝑚=𝑘𝑛,𝑘𝑁,then𝑛=1
𝑘𝑚and 𝑘/𝑁. For example, 4 is a
multiple of 2 but 2 is not a multiple of 4.
1.23
[𝑎, 𝑏]={𝑎, 𝑦, 𝑏, 𝑧 }
(𝑎, 𝑏)={𝑦}
1.24
(𝑦)={𝑏, 𝑦, 𝑧 }
(𝑦)={𝑏, 𝑧 }
(𝑦)={𝑎, 𝑥, 𝑦 }
(𝑦)={𝑎, 𝑥 }
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