Solutions Manual Foundations of Mathematical Economics By Michael Carter

, Solutions Manual
Foundations of Mathematical Economics

Michael Carter

, ⃝ c 2001 Michael Carter
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Solutions for Foundations of Mathematical Economic tr tr tr tr tr All rights reserved tr tr


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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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∙ 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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1.5 The player set is � = {Jenny, Chris }. Their action spaces are
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�� = {Rock, Scissors, Paper }
t r tr r
t tr tr tr � = Jenny, Chris
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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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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}}
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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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�� 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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1.9
∪
� =� tr tr



�∈�
∩
� =∅ tr tr



�∈�


1

, ⃝ c 2001 Michael Carter
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Solutions for Foundations of Mathematical Economic tr tr tr tr tr All rights reserved tr tr


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�2
1




�1
-1 0 1




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



{
{�, �} ×{�, �} ×{�, �}= (�, �, �), (�, �, �), (�, �, �),
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}
(�, �, � ), (�, �, �), (�, �, � ), (�, �, �), (�, �, � ) 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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where 0 = (0, 0 , . . . , 0) is the production plan using no inputs and producing no outputs. To
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see this, first note that 0 is a feasible production plan. Therefore, 0 ∈ � . Also,
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0 ∈ ℜ �+ and therefore 0 ∈ � ∩ℜ � . +
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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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(�, − x) and free disposability implies that (�, − x′ ) ∈ � . Therefore x′ ∈ � (�).
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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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1.13 The domain of “<” is {1, 2}= � and the range is {2,3}⫋ � . tr tr tr tr tr tr r
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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