Solutions Manual Foundations of Mathematical Economics By Michael Carter

, Solutions Manual vv vv




vv FoundationsofMathematicalEconomics
v v v




Michael Carter
vv

, ⃝c
vv t v v 2001 Michael v v




Solutions for Foundations of Mathematical v v v v v v v v Carter All rights
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Economic
v v reserved
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Chapter 1: Sets and Spaces v v vv vv v v v v




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




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




◻ also belongs to ◻. Hence ◻,◻ haveprecisely the same elements.
v v v v v v v v vv v v v v v v v v v v




1.3 Examples of finite sets are v v v v v v v v




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




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




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




∙ the set of players in a game v v vv vv vv vv vv




.Examples of infinite sets are v v v v v v v v




∙ the real numbers ℜ v v vv vv




∙ the natural numbers v v v v v v




∙ the set of all possible colors v v vv vv vv vv




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




∙ the v v set v v of v v possible temperatures of liquid v v v v v v v v water.
1.4 ◻ = {1,2,3,4,5,6 }, ◻ = {2,4,6 }.
vv vv tr vv v v vv tr




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




◻◻ = {Rock,Scissors,Paper} v v vv v ◻ = Jenny,Chris
vv vv




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




player is the set of feasible outputs
v v v v v v vv vv v v v v




◻◻ = {◻◻ ∈ ℜ+ : ◻◻ ≤ ◻◻}
vv vv vv vv vv vv vv tr v




where ◻◻ is the output of dam ◻. vv vv vv vv v v vv vv




1.7 The player set is ◻ v v v v v v v v v v = {1,2,3}. There are 23 = 8 coalitions, namely
vv vv v v v v v v v v v v v v




(◻) = v v v v {∅ , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
vv vv vv vv vv vv vv v vv vv vv vv vv




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




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




◻◻ and ◻ ∈ ◻ ◻. Consequently, ◻ ∈ ◻◻ ∩ ◻ ◻. Conversely, assume ◻ ∈ ◻◻ ∩ ◻ ◻. This
vv v v vv v v vv v v v v vv v v vv vv vv v v v v v v vv v v vv vv vv vv




implies that ◻ ∈ ◻◻ and ◻ ∈ ◻◻. Consequently ◻∈/ ◻ and ◻∈/ ◻ and therefore
vv vv vv vv vv vv vv vv vv v v v vv vv vv vv vv v v vv




◻∈/ ◻ ∪ ◻. This implies that ◻ ∈ (◻ ∪ ◻)◻. The other identity is proved similarly.
v vv vv vv v v vv
tvv v v vv vv vv vv v v v v vv v v vv vv




1.9
∪
◻ =◻ vv vv




◻∈
∩
◻ =∅ vv vv




◻∈

1

, ⃝c
vv t v v 2001 Michael v v




Solutions for Foundations of Mathematical v v v v v v v v Carter All rights
v v vv vv




Economic
v v reserved vv




◻2
1




◻1
-1 0 1




-1

Figure 1.1: The relation {(◻,◻) : ◻2 + ◻2 = 1} v v vv v v v v vv vv vv vv v v v v v




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




outcomes in t hree tosses is the product
v v v v tr vv v v v v v v v v




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

}
v v v v v v v v




(◻, ◻, ◻), (◻, ◻, ◻), (◻, ◻, ◻), (◻, ◻, ◻), (◻, ◻, ◻ ) vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv




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




1.11

◻ ∩ℜ+◻ = {0} v v v v v vv




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



see this, first note that 0 is a feasible production plan. Therefore, 0
v v v v v v v v v v v v v v v v v v v v v v v v v v




∈ ◻. Also,
v v v v v v v




0 ∈ ℜ◻+ and therefore 0 ∈ ◻ ∩ℜ+◻ .
vv vv v v v v v v v vv vv v v v vv




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


is, we assume there is some feasible production plan y ◻ ∈0ℜ. T∖h{is}implies the exist
vv v v vv v v vv v v v v vv vv
tr

v v
v v v vv vv




+
rtr v v trtrt v v
+ tr r tr tr


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




so that ◻∈/ ◻.
vv v v vv v




1.12 1. Let x ∈ ◻ (◻). This implies that (◻,−x) ∈ ◻ . Let x′ ≥x. Then (◻,−x′) ≤
vv vv vv vv vv v v vv t vv t vv v vv vv vv v v t vv
vv
v v vv v vv




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




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




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




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




1.14 Figure 1.1. vv




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




asymmetri c.It is not complete, reflexive or symmetric.
v v vv vv v v v v v v v v v v




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