Solutions Manual Foundations of Mathematical Economics By Michael Carter

Solutions Manualgr



Foundations of Mathematical
gr gr gr



Economics
gr




Michael grCarter gr

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




Chapter 1: gr g r Sets and Spaces
gr gr




1.1
{gr1, gr3, gr5, gr7 gr. . . gr} gror g r {gr�gr ∈ gr� g r : g r �g r is g r odd gr}
1.2 Every g r � ∈ � g r also g r belongs g r to g r �. g r Every
∈ gr� � g r also g r belongs g r to
g r �. g r Hence g r �, gr� g r havegp
r recisely g r the g r same g r elements.


1.3 Examples g r of g r finite g r sets g r are
∙ the g r letters g r of g r the g r alphabet g r {grA, g r B, g r C, g r . . . g r , g r Z gr}
∙ the g r set g r of g r consumers g r in g r an g r economy
∙ the g r set g r of g r goods g r in g r an g r economy
∙ the g r set g r of g r players grin g r a
gr game.gE
r xamples g r of g r infinite


gr sets g r are
∙ the g r real g r numbers g r ℜ
∙ the g r natural g r numbers g r �
∙ the g r set gr of g r all gr possible gr colors
∙ the g r set g r of g r possible g r prices g r of g r copper g r on g r the g r world g r market
∙ the g r set g r of g r possible g r temperatures g r of g r liquid g r water.
1.4 gr �g r = gr {gr1, gr2, gr3, gr4, gr5, gr6 gr}, g r �g r = gr {gr2, gr4, gr6 gr}.
1.5 The g r player g r set g r is g r � g r = g r {grJenny, grChris gr} . grTheir g r action g r spaces g r are
�� g r = gr{grRock, grScissors, grPaper gr} � g r = gr Jenny, grChris
1.6 The g r set g r of g r players g r is g{r � g r = g r 1,}gr2 , .. ., gr� g r . gr The g r strategy g r space g r of
g r each g r player g r is g r the g r set grof g r feasible g r outputs


�� g r = gr {gr�� g r ∈ grℜ + g r : g r �� g r ≤ gr��gr}
where g r �� grgris grgrthe g r output g r of g r dam g r �.
3
1.7 The g r player g r set g r is g r � g r = g r {1, gr2, gr3}. grThere g r are g r 2 gr = g r 8 g r coalitions, g r namely
� (�gr) g r = g r {∅ , gr{1}, gr{2}, gr{3}, gr{1, gr2}, gr{1, gr3}, gr{2, gr3}, gr{1, gr2, gr3}}
10
There g r are g r 2 gr coalitions g r in g r a g r ten g r player g r game.
1.8 gr gr Assume gr grthat gr gr�gr gr∈ gr(� g r ∪ gr�gr)�. gr gr grThat gr gris gr gr�gr gr∈/ gr gr� g r ∪ gr�gr. gr gr grThis gr grimplies gr gr�
� � �
gr gr∈/ gr gr�gr grand gr gr�gr gr∈/ gr gr�gr, gror gr�gr∈ gr� gr and g r �gr∈ gr�gr . g r Consequently, g r �gr∈ gr� gr∩ gr�
� � � �
gr . g r Conversely, g r assume g r �gr ∈ gr� gr∩ gr�gr . grThis gr grimplies gr grthat gr gr� g r ∈ gr� gr grand gr gr� g r ∈ gr�
�
gr . gr gr grConsequently gr gr�gr∈/ gr gr�gr grand gr gr�gr∈/ gr gr�gr gr and gr grtherefore

�∈/ g r �gr ∪ gr�gr. grThis g r implies grgrthat g r �gr ∈ gr(�gr∪ gr�gr)�. grThe g r other g r identity g r is g r proved g r similarly.
1.9
∪
�gr = gr�
�∈�
∩
�g r = gr∅
�∈�


1

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


�2
1




�1
-1 0 1




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


1.10 g r The g r sample g r space g r of g r a g r single g r coin {g r toss g}r is gr�, gr� g r . gr The g r set g r of
g r possible g r outcomes g r ingtr hree g r tosses g r is g r the g r product

{
{�, gr�gr} ×gr{�, gr�gr} ×gr{�, gr�gr} gr= g r (�, gr�, gr�), gr(�, gr�, gr�gr), gr(�, gr�gr, gr�),
}
(�, gr�gr, gr�gr), gr(�, gr�, gr�), gr(�, gr�, gr�gr), gr(�, gr�, gr�), gr(�, gr�, gr�gr)


A g r typical g r outcome g r is g r the g r sequence g r (�, gr�, gr�gr) g r of g r two g r heads g r followed g r by g r a g r tail.
1.11
�g r
� g r ∩grℜ+ = g r {0}

where gr0 gr = gr(0, gr0 , . . . gr, gr0) gris grthe grproduction grplan grusing grno grinputs grand grproducing grno
groutputs. grTo g r see g r this, g r first g r note g r that g r 0 g r is g r a g r feasible g r production g r plan.

g r Therefore, g r 0 g r ∈ gr�gr. g r Also,

0 g r ∈ grℜ+� g r and g r therefore g r 0 g r ∈ gr�+g r ∩ grℜ �gr .
�
To grshow grthat grthere gris grno grother grfeasible grproduction grplan ℜ +grin gr gr gr gr gr gr, grwe grassume
grthe grcontrary. grThat gris, grwe grassume grthere gris grsome grfeasible ∈ ℜ +∖ { } grplan gry gr gr gr gr
grproduction
�
gr gr gr gr gr gr gr gr gr gr0 gr gr. gr grThis grimplies grthe grexistence grof gra grplan grproducing gra grpositive

groutput grwith grno grinputs. grThis grtechnological grinfeasible, g r so g r that g r �gr∈/ g r �gr.


1.12 1. gr grLet grgrx g r ∈ gr�gr(�). gr grThis grgrimplies grgrthat grgr(�, gr− x) g r ∈ gr�gr. gr grLet grgrx′ gr ≥ grx. gr g r Then grgr(�, gr− x′ ) g r ≤
(�, gr− x) g r and g r free g r disposability g r implies grgrthat g r (�, gr− x′ ) g r ∈ gr�gr. grTherefore g r x′ gr∈ gr�gr(�).
2. gr g r Again gr grassume gr grx gr g r ∈ gr �gr(�). gr gr gr grThis gr gr implies gr gr that gr gr (�, gr− x) gr g r ∈ gr �gr. gr gr
gr grBy gr gr free gr gr disposal, gr(�′ , gr− x) gr ∈ gr�gr g r for g r every g r �′ gr≤ gr�, g r which g r implies grgrthat

′ ′
g r x g r ∈ gr�gr(� ). gr gr�gr(� ) g r ⊇ gr�gr(�).


1.13 The g r domain g r of g r “<” g r is g r {1, gr2}gr= gr � g r and g r the g r range g r is g r {2, gr3}gr⫋ gr �gr.
1.14 Figure gr1.1.
1.15 The g r relation g r “is g r strictly g r higher g r than” g r is g r transitive, g r antisymmetric g r and
g r asymmetric.gI
r t g r is g r not g r complete, g r reflexive g r or g r symmetric.




2

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


1.16 The g r following g r table g r lists g r their g r respective g r properties.
< ≤√ gr g r=√
reflexive ×gr g r
transitive √ √ gr g r √
symmetric √ gr g r √
×gr g r
√
asymmetric
anti-symmetric √ gr g r × √
gr g r ×
√
√ g r √ g r
complete ×
Note g r that g r the g r properties g r of g r symmetry g r and g r anti-symmetry g r are g r not g r mutually g r exclusive.
1.17 Let gr∼be gran grequivalence grrelation grof gra grset ∕ gr∅�gr= gr. g r That gris, grthe∼ grrelation gris
grreflexive, grsymmetric grand grtransitive. grWe grfirst grshow ∈ grthat grevery gr�gr�grbelongs grto
grsome grequivalence grclass. g r Let g r �g r be g r any
∼ g r element g r in g r � g r and g r let g r (�) g r be g r the
g r class g r of g r elements g r equivalent g r to

�, grthat gr is
∼(�) g r ≡ gr{gr�g r ∈ gr� g r : g r �g r ∼ gr�gr}
Since ∼ is g r reflexive, g r �∼ �grand grso gr∈�∼ (�). g r Every g r∈�� g r belongs g r to g r some
g r equivalencegrclass g r and g r therefore
∪
�g r = ∼(�)
�∈�

Next, gr we g r show g r that g r the equivalence g r classes
g r g r are g r either g r disjoint g r or
g r identical, gr gr that g r is

∼(�) g r ∕= g r ∼(�) g r if g r and g r only g r if g r f∼(�) gr∩gr∼ (�) gr= g r ∅ .
First, g r assume g r ∼(�) gr∩gr∼ (�) gr= gr ∅ . grThen g r �gr∈ gr∼ (�) g r but grgr�∈
� grTherefore g r ∼(�) gr ∕= g r ∼(�).
/ ). ∼(
Conversely, gr grassume gr gr∼(�) g r ∩gr∼ (�) gr gr∕= gr gr∅ grand gr grlet gr gr�gr gr∈ gr∼(�) g r ∩gr∼ (�). gr gr grThen gr gr�gr gr∼
gr�gr grand gr grb
ygrsymmetry g r � g r ∼ gr�. gr gr grAlso g r � g r ∼ gr�grand grso g r by g r transitivity gr� g r ∼
gr�. gr gr grLet gr � g r be g r any grelement gr in gr gr∼(�) gr grso gr grthat gr gr� gr gr∼ gr�. gr gr gr Again gr grby gr

grtransitivity gr gr�gr gr∼ gr� gr grand gr grtherefore gr gr�gr gr∈ gr∼(�). gr gr gr Hence

∼(�) g r ⊆ gr∼ (�). grSimilar grgrreasoning g r implies grgrthat g r ∼(�) g r ⊆ gr∼ (�). grTherefore g r ∼(�) gr= g r ∼(�).
We g r conclude g r that g r the g r equivalence gr classes g r partition g r �.
1.18 The grset grof grproper grcoalitions gris gr not gra grpartition grof grthe gr set grof grplayers,
grsince gr any gr playergc r an g r belong gr to g r more gr than g r one g r coalition. grFor g r example, g r player

gr 1 g r belongs g r to g r the g r coalitions

{1}, g r {1, gr2} grand g r so g r on.
1.19

�g r ≻ gr� g r =⇒ gr �g r ≿ gr � g r and g r � g r ∕≿ gr �
� g r ∼ gr� g r =⇒ gr � g r ≿ gr � g r and g r � g r ≿ gr �
Transitivity gr of gr ≿ grimplies g r �gr≿ gr�. grWe g r need g r to g r show gr that g r �gr∕≿ gr�. grAssume
g r otherwise, g r thatgir s g r assume g r � g r ≿ gr � g r This g r implies g r � g r ∼ gr� g r and g r by g r transitivity
g r � g r ∼ gr�. g r But g r this g r implies g r that

� g r ≿ gr� g r which g r contradicts g r the g r assumption g r that g r �g r ≻ gr�. g r Therefore g r we g r conclude g r that g r � g r ∕≿ gr �
and g r therefore g r �gr ≻gr�. grThe g r other g r result g r is g r proved g r in g r similar g r fashion.
1.20 asymmetric g r Assume gr �g r ≻ gr�.

Therefore
while

3