, Solutions Manual
Foundations of Mathematical Economics
Michael Carter
, ⃝ c 2001 Michael Carter
trtrtr tr tr
Solutions for Foundations of Mathematical Economic tr tr tr tr tr All rights reserved tr tr
s
Chapter 1: Sets and Spaces t r t r t r t r
1.1
{1, 3, 5, 7 . . . }or {� ∈ � : � is odd }
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1.2 Every � ∈ � also belongs to �. Every �∈ t r t r t r t r t r t r t r
� also belongs to �. Hence �, � haveprecisely the same elements.
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1.3 Examples of finite sets are tr tr tr tr
∙ the letters of the alphabet {A, B, C, . . . , Z }
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t t r t r t r tr t r tr
∙ the set of consumers in an economy tr tr tr tr tr tr
∙ the set of goods in an economy tr tr tr tr tr tr
∙ the set of players in a game tr tr tr tr tr tr
.Examples of infinite sets are
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∙ the real numbers ℜ tr tr tr
∙ the natural numbers � tr tr tr
∙ the set of all possible colors tr tr tr tr tr
∙ the set of possible prices of copper on the world market
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∙ the set of possible temperatures of liquid water.
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1.4 � = {1, 2, 3, 4, 5, 6 }, � = {2, 4, 6 }.
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t tr tr tr
1.5 The player set is � = {Jenny, Chris }. Their action spaces are
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t tr tr tr t r t r tr
�� = {Rock, Scissors, Paper }
t r tr r
t tr tr tr � = Jenny, Chris
tr tr tr
1.6 The set of players is � ={ 1, 2, .. . , �} . The strategy space of each player is the
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rset of feasible outputs
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�� = {�� ∈ ℜ + : �� ≤ �� }
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t tr r
t tr tr tr r
t tr
where �� is the output of dam �. tr trtr trtr tr tr tr tr
1.7 The player set is � = {1, 2, 3 }. There are 23 = 8 coalitions, namely
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� (�) = {∅ , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
tr t r t r tr tr tr tr tr tr tr tr tr tr tr tr
10
There are 2 tr tr t r coalitions in a ten player game. tr tr tr tr tr
1.8 Assume that � ∈ (� ∪ �)� . That is � ∈/ � ∪ �. This implies � ∈/ � and � ∈/ �, or � ∈
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t tr trtrtr trtr trtr trtr trtr tr r
t tr trtrtr trtr trtr trtr trtr trtr trtr trtr trtr tr tr tr tr tr
�� and � ∈ � �. Consequently, � ∈ �� ∩ � �. Conversely, assume � ∈ �� ∩ � �. This implies
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that � ∈ � � and � ∈ �� . Consequently �∈/ � and �∈/ � and therefore
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�∈/ � ∪ �. This implies that � ∈ (� ∪ �)� . The other identity is proved similarly.
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t tr tr tr trtr tr tr r
t tr r
t tr tr tr tr tr tr tr
1.9
∪
� =� tr tr
�∈�
∩
� =∅ tr tr
�∈�
1
, ⃝ c 2001 Michael Carter
trtrtr tr tr
Solutions for Foundations of Mathematical Economic tr tr tr tr tr All rights reserved tr tr
s
�2
1
�1
-1 0 1
-1
2 2
Figure 1.1: The relation {(�, �) : � + � = 1 }
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t tr tr tr tr tr t r tr tr
1.10 The sample space of a single coin toss is{�, � .}The set of possible outcomes int
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hree tosses is the product tr tr tr tr
{
{�, �} ×{�, �} ×{�, �}= (�, �, �), (�, �, �), (�, �, �),
tr tr r
t tr tr r
t tr tr r
t t r tr tr tr tr tr tr tr tr tr tr
}
(�, �, � ), (�, �, �), (�, �, � ), (�, �, �), (�, �, � ) tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr
A typical outcome is the sequence (�, �, � ) of two heads followed by a tail.
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1.11
� ∩ ℜ+� = {0}
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t r
tr
where 0 = (0, 0 , . . . , 0) is the production plan using no inputs and producing no outputs. To
tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr tr t
see this, first note that 0 is a feasible production plan. Therefore, 0 ∈ � . Also,
r t r t r t r t r t r t r t r t r t r t r t r t r t r tr tr t r
0 ∈ ℜ �+ and therefore 0 ∈ � ∩ℜ � . +
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t r
t r t r tr tr t r tr
tr
To show that there is no other feasible production plan in � ,ℜwe
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+ assume the contrary. Thattr tr tr tr tr tr tr tr trtrtrtrtr tr tr tr tr tr tr
�
is, we assume there is some feasible production plan y
tr tr tr ∈0ℜ . +This
∖ { }implies the exist tr tr tr tr tr tr tr trtrtrtrtrtrtrtr trtrtrtrtrtr trtrtrt trtr
r tr t r tr tr tr
ence of a plan producing a positive output with no inputs. This technological infeasible,
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so that �∈/ � . tr tr tr tr tr
1.12 1. Let x ∈ � (�). This implies that (�, − x) ∈ � . Let x′ ≥ x. Then (�,− x′ ) ≤
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t tr trtr trtr trtr trtr tr tr r
t tr trtr trtr tr r
t trt r trtr tr tr
(�, − x) and free disposability implies that (�, − x′ ) ∈ � . Therefore x′ ∈ � (�).
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t tr tr tr tr r
t tr
2. Again assume x ∈ � (�). This implies that (�, − x) ∈ � . By free disposal, (�
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′ , − x) ∈ � for every �′ ≤ � , which implies that x ∈ � (� ′ ). � (�′ ) ⊇ � (�).
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t trt r tr tr tr r
t tr tr trtr tr tr r
t tr trtr tr tr r
t tr
1.13 The domain of “<” is {1, 2}= � and the range is {2,3}⫋ � . tr tr tr tr tr tr r
t tr t r tr tr tr tr tr r
t tr tr
1.14 Figure 1.1. tr
1.15 The relation “is strictly higher than” is transitive, antisymmetric and asymmetri
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c.It is not complete, reflexive or symmetric.
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2
