Solution Manual for Foundations of Mathematical Economics By Michael Carter A+ verified, ISBN: 978-0262531924

, Solutions Manual d d




d FoundationsofMathematicalEconomics
d d d




Michael Carter
d

, Chapter 1: Sets and Spaces d d d d d




1.1
{1,3,5,7... }or {◻ ∈ ◻ : ◻ is odd} d d d d d d d d d d




1.2 Every ◻∈ ◻ also belongs to ◻. Every ◻∈ d d d d d d d d d




◻ also belongs to ◻. Hence ◻,◻ haveprecisely the same elements.
d d d d d d d d d d




1.3 Examples of finite sets are d d d d




∙ the letters of the alphabet {A, B, C, ... , Z}
d d d d d d d d d d d d




∙ the set of consumers in an economy d d d d d d




∙ the set of goods in an economy d d d d d d




∙ the set of players in a game d d d d d d




.Examples of infinite sets are d d d d




∙ the real numbers ℜ d d d




∙ the natural numbers d d d




∙ the set of all possible colors d d d d d




∙ the set of possible prices of copper on the world market
d d d d d d d d d d




∙ the set of possible temperatures of liquid water.
d d d d d d d




1.4 ◻ = {1,2,3,4,5,6 }, ◻ = {2,4,6 }.
d d tr d d d tr




1.5 The player set is ◻ d d d d d = {Jenny,Chris}. Their action spaces are
d d d d d d




◻◻ = {Rock,Scissors,Paper} d d d ◻ = Jenny,Chris
d d




1.6 The set of players is ◻ ={ d d d d d d t d r d 1,2,...,◻ } . The strategy space of each player is
d d d d d d t r d d d d d d d d




the set of feasible outputs
d d d d d




◻ ◻ = {◻ ◻ ∈ ℜ+ : ◻ ◻ ≤ ◻ ◻ }
d d d d d d d tr d




where ◻ ◻ is the output of dam ◻. d d d d d d d




3
1.7 The player set is ◻ = {1,2,3}. There are 2 = 8 coalitions, namely
d d d d d d d d d d d d d




(◻ ) = {∅ , {1}, {2}, {3}, {1, 2}, {1,3}, {2, 3}, {1, 2, 3}}
d d d d d d d d d d d d d d d




10
There are 2 d d d coalitions in a ten player game. d d d d d




◻
1.8 Assume that ◻ ∈ (◻ ∪ ◻ ) . That is ◻ ∈/ ◻ ∪ ◻ . This implies ◻ ∈/ ◻ and ◻ ∈/ ◻ , or ◻ ∈
t d d d d d d d d d d d d d d d d d d d d d d d d d d d




◻ ◻ and ◻ ∈ ◻ ◻ . Consequently, ◻ ∈ ◻ ◻ ∩ ◻ ◻ . Conversely, assume ◻ ∈ ◻ ◻ ∩ ◻ ◻ . This implies that
d d d d d d d d d d d d d d d d d d d d d d d




◻ ∈ ◻ ◻ and ◻ ∈ ◻ ◻ . Consequently ◻ ∈/ ◻ and ◻ ∈/ ◻ and therefore
d d d d d d d d d d d d d d d d




◻ ∈/ ◻ ∪ ◻ . This implies that ◻ ∈ (◻ ∪ ◻ )◻ . The other identity is proved similarly.
d d d d d d d t d d d d d d d d d d d




1.9
∪
◻= ◻ d d




◻∈
∩
◻ =∅ d d




◻∈


1

, ◻2
1




◻1
-1 0 1




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




1.10 The sample space of a single coin toss is{◻,◻ .} The set of possible outcomes
d d d d d d d d d tr d d d d d




din t hree tosses is the product
tr d d d d d




{
{◻ ,◻ }× { ◻ ,◻ }× { ◻ ,◻ }= (◻ ,◻ ,◻ ),(◻ ,◻ ,◻ ),(◻ ,◻ ,◻ ), tr tr

}
d d d d d




(◻ , ◻ , ◻ ), (◻ , ◻ , ◻ ), (◻ , ◻ , ◻ ), (◻ , ◻ , ◻ ), (◻ , ◻ , ◻ ) d d d d d d d d d d d d d d d




A typical outcome is the sequence (◻,◻,◻) of two heads followed by a tail.
d d d d d d d d d d d d d




1.11

◻ ∩ℜ◻ = {0} d d d d




where0 = (0,0,... ,0) is the production plan using no inputs and producing no outputs. To
d d d d d d d d tr d d d d d d d t



see this, first note that 0 is a feasible production plan. Therefore, 0 ∈ ◻ .
d d d d d d d d d d d d d d d d




Also,
d




0 ∈ ℜ◻ and therefore 0 ∈ ◻ ∩ℜ◻ .
d d d d d d d d d d d




◻
To show that there is no other feasible production plan in
d d d d d d d d d d d ,ℜwe
◻ assume the contrary. That
d d d d


is, we assume there is some feasible production plan y
d d d d d d d d d ∈0ℜ. T∖h{is}implies theexist d d d d


rtr d trtrt d
+
tr r tr tr


ence of a plan producing a positive output with no inputs. This technological infeasible,
d d d d d d d tr d d d tr d




so that ◻ ∈/ ◻ .
d d d d




1.12 1. Let x ∈ ◻ (◻). This implies that (◻,−x) ∈ ◻ . Let x′ ≥x. Then (◻,−x′) ≤
d d d d d d d td td d d d d d td
d
d d d d




(◻,−x) and free disposability implies that (◻,−x′) ∈ ◻ . Therefore x′ ∈ ◻ (◻).
d d d d d d d d d d d d d
d
d d




2. Again assume x ∈ ◻ (◻ ). This implies that (◻ ,−x) ∈ ◻ . By free disposal, (◻
d d d d d d d d d d d d d d d t d d




′,−x) ∈ ◻ for every ◻ ′ ≤ ◻ , which implies that x ∈ ◻ (◻ ′). ◻ (◻ ′) ⊇ ◻ (◻ ).
d d d d d d
d
d d d d d d d d d d d d




1.13 The domain of “<” is {1,2}= ◻ and the range is {2,3}⫋ ◻.d d d d d d d d d d d d d




1.14 Figure 1.1. d




1.15 The relation “is strictly higher than” is transitive, antisymmetric and
d d d d d d d d d




dasymmetri c.It is not complete, reflexive or symmetric. d d d d d d d




2