Test bank for Foundations of Mathematical Economics By Michael Carter Verified Chapters Complete Newest Version

Foundations Of Mathematical Economics (Mit
Press) First Edition By Michael Carter




TEST BANK

, ⃝ c 2001 Ṁichael Carter
Solutions for Foundations of Ṁatheṁatical Econoṁics All rights reserved

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 saṁe eleṁents.
1.3 Exaṁples of finite sets are
∙ the letters of the alphabet { A, B, C, . . . , Z }
∙ the set of consuṁers in an econoṁy
∙ the set of goods in an econoṁy
∙ the set of players in a gaṁe.
Exaṁples of infinite sets are
∙ the real nuṁbers ℜ
∙ the natural nuṁbers �
∙ the set of all possible colors
∙ the set of possible prices of copper on the world ṁarket
∙ the set of possible teṁperatures of liquid water.
1.4 � = { 1, 2, 3, 4, 5, 6 }, � = { 2, 4, 6 }.
1.5 The player set is � = { Jenny, 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 feasible outputs
�� = { �� ∈ ℜ + : �� ≤ �� }
where �� is the output of daṁ �.
1.7 The player set is � = {1, 2, 3}. There are 23 = 8 coalitions, naṁely
� (� ) = {∅ , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
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There are 2 coalitions in a ten player gaṁe.
1.8 Assuṁe that � ∈ (� ∪ � )� . That is � ∈/ � ∪ � . This iṁplies � ∈/ � and � ∈/ � ,
or � ∈ �� and � ∈ � �. Consequently, � ∈ �� ∩ � �. Conversely, assuṁe � ∈ �� ∩ � �. This
iṁplies that � ∈ � � and � ∈ � � . Consequently � ∈/ � and � ∈/ � and therefore
�∈/ � ∪ � . This iṁplies that � ∈ (� ∪ � )� . The other identity is proved siṁilarly.
1.9
∪
�=�
�∈�
∩
�=∅
�∈�


1

, ⃝ c 2001 Ṁichael Carter
Solutions for Foundations of Ṁatheṁatical Econoṁics All rights reserved


�2
1




�1
-1 0 1




-1
2 2
Figure 1.1: The relation { (�, �) : � + � = 1 }


1.10 The saṁple space of a single coin toss is �,{� . The
} set of possible outcoṁes in
three tosses is the product
{
{�, � }×{�, � }×{�, � } = (�, �, �), (�, �, � ), (�, � , �),
}
(�, � , � ), (�, �, �), (�, �, � ), (�, �, �), (�, �, � )


A typical outcoṁe 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 � . +
To show that there is no other feasible production plan in ℜ �+ , we assuṁe the contrary.
That is, we assuṁe there is soṁe feasible production plan y ∈ ℜ �+∖ { }0 . This iṁplies
the existence of a plan producing a positive output with no inputs. This technological
infeasible, so that � ∈/ � .
1.12 1. Let x ∈ � (�). This iṁplies that (�, − x) ∈ � . Let x′ ≥ x. Then (�, − x′ ) ≤
(�, − x) and free disposability iṁplies that (�, − x′ ) ∈ � . Therefore x′ ∈ � (�).
2. Again assuṁe x ∈ � (�). This iṁplies that (�, − x) ∈ � . By free disposal,
(� ′ , − x) ∈ � for every �′ ≤ �, which iṁplies that x ∈ � (�′ ). � (�′ ) ⊇ � (�).
1.13 The doṁain 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, antisyṁṁetric and asyṁṁetric.
It is not coṁplete, reflexive or syṁṁetric.




2

, ⃝ c 2001 Ṁichael Carter
Solutions for Foundations of Ṁatheṁatical Econoṁics All rights reserved


1.16 The following table lists their respective properties.
< ≤√ √=
reflexive ×
transitive √ √ √
syṁṁetric √ √
×
√
asyṁṁetric
anti-syṁṁetric √ ×
√ ×
√
√ √
coṁplete ×
Note that the properties of syṁṁetry and anti-syṁṁetry are not ṁutually exclusive.
1.17 Let ∼be an equivalence relation of a set �∕ =∅ . That is, the relation∼ is reflexive,
syṁṁetric and transitive. We first show that every �∈ � belongs to soṁe equivalence
class. Let � be any eleṁent in � and let (�)
∼ be the class of eleṁents equivalent to
�, that is
∼(�) ≡ { � ∈ � : � ∼ � }
Since ∼ is reflexive, � ∼ � and so � ∈ ∼ (�). Every � ∈ � belongs to soṁe equivalence
class and therefore
∪
�= ∼(�)
�∈�

Next, we show that the equivalence classes are either disjoint or identical, that is
∼(�) ∕= ∼(�) if and only if f∼(�) ∩ ∼(�) = ∅ .
First, assuṁe ∼(�) ∩ ∼(�) = ∅ . Then � ∈ ∼(�) but �∈
�/ ∼( ). Therefore ∼(�) ∕= ∼(�).
Conversely, assuṁe ∼(�) ∩ ∼(�) ∕= ∅ and let � ∈ ∼(�) ∩ ∼(�). Then � ∼ � and by
syṁṁetry � ∼ �. Also � ∼ � and so by transitivity � ∼ �. Let � be any eleṁent in
∼(�) so that � ∼ �. Again by transitivity � ∼ � and therefore � ∈ ∼(�). Hence
∼(�) ⊆ ∼(�). Siṁilar reasoning iṁplies 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
can belong to ṁore than one coalition. For exaṁple, player 1 belongs to the coalitions
{1}, {1, 2} and so on.
1.19
� ≻ � =⇒ � ≿ � and � ∕≿ �
� ∼ � =⇒ � ≿ � and � ≿ �
Transitivity of ≿ iṁplies � ≿ � . We need to show that � ∕≿ � . Assuṁe otherwise, thatis
assuṁe � ≿ � This iṁplies � ∼ � and by transitivity � ∼ �. But this iṁplies that
� ≿ � which contradicts the assuṁption that � ≻ � . Therefore we conclude that � ∕≿ �
and therefore � ≻ � . The other result is proved in siṁilar fashion.
1.20 asyṁṁetric Assuṁe � ≻ �.

� ≻ � =⇒ � ∕≿ �
while

� ≻ � =⇒ � ≿ �
Therefore
� ≻ � =⇒ � ∕≻ �

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