Solutions Manual Foundations of Mathematical Economics By Michael Carter

Solutions Manual
Foundations of Mathematical Economics

Michael Carter

, ⃝ cyyy2001y Michaely Carter
Solutionsy fory Foundationsy ofy Mathematicaly Economics Allyrightsyreserved




Chaptery 1:y Setsy andy Spaces

1.1
{y1,y3,y5,y7y. . . y}yory {y�y ∈ y�y :y �y isy oddy}
1.2 Everyy � ∈ �y alsoy belongsy toy �.y Everyy � ∈
�y alsoy belongsy toy �.y Hencey �,y�y haveypreciselyy they samey elements.
1.3 Examplesy ofy finitey setsy are
∙ they lettersy ofy they alphabety {yA,y B,y C,y . . . y ,y Zy}
∙ they sety ofy consumersy iny any economy
∙ they sety ofy goodsy iny any economy
∙ they sety ofy playersy iny ay game
.yExamplesy ofy infinitey setsy are
∙ they realy numbersy ℜ
∙ they naturaly numbersy �
∙ they sety ofy ally possibley colors
∙ they sety ofy possibley pricesy ofy coppery ony they worldy market
∙ they sety ofy possibley temperaturesy ofy liquidy water.
1.4y �y =y {y1,y2,y3,y4,y5,y6y},y �y =y {y2,y4,y6y}.
1.5 They playery sety isy �y =y {yJenny,yChrisy} . y Theiry actiony spacesy are
��y =y{yRock,yScissors,yPapery} �y =y Jenny,yChris
1.6 They sety ofy playersy isy �y =y 1,
{ y2 , . .. , y�y }. y They strategyy spacey ofy eachy playery isy they sety
ofy feasibley outputs
��y =y {y��y ∈ yℜ +y :y ��y ≤ y��y}
wherey ��yyisyythey outputy ofy damy �.
3
1.7 They playery sety isy �y =y {1,y2,y3}. yTherey arey 2 y =y 8y coalitions,y namely
� (�y)y =y {∅ ,y{1},y{2},y{3},y{1,y2},y{1,y3},y{2,y3},y{1,y2,y3}}
10
Therey arey 2 y coalitionsy iny ay teny playery game.
1.8yy Assumeyythatyy�yy∈ y(�y ∪ y�y)� .yyyThatyyisyy�yy∈/yy�y ∪ y�y.yyyThisyyimpliesyy�yy∈/yy�yyandyy�yy∈/yy�y,yo
ry�y∈ y��yandy �y∈ y�y�.y Consequently,y �y∈ y��y∩y�y�.y Conversely,y assumey �y∈ y��y∩y�y�.yThisyyim
pliesyythatyy�y ∈ y� �yyandyy�y ∈ y�y� .yyyConsequentlyyy�y∈/yy�yyandyy�y∈/yy�yy andyytherefore
�∈/y �y ∪ y�y. yThisy impliesyythaty �y ∈ y(�y ∪ y�y)� . yThey othery identityy isy provedy similarly.
1.9
∪
�y =y�
�∈�
∩
�y =y∅
�∈�


1

, ⃝ cyyy2001y Michaely Carter
Solutionsy fory Foundationsy ofy Mathematicaly Economics Allyrightsyreserved


�2
1




�1
-1 0 1




-1
2 2
Figurey 1.1:y They relationy {y(�,y�)y :y � y +y � y =y 1y}


1.10y They sampley spacey ofy ay singley coiny tossy isy�,
{y�y .y The
} y sety ofy possibley outcomesy inythr
eey tossesy isy they product
{
{�,y�y} ×y{�,y�y} ×y{�,y�y}y=y (�,y�,y�),y(�,y�,y�y),y(�,y�y,y�),
}
(�,y�y,y�y),y(�,y�,y�),y(�,y�,y�y),y(�,y�,y�),y(�,y�,y�y)


Ay typicaly outcomey isy they sequencey (�,y�,y�y)y ofy twoy headsy followedy byy ay tail.
1.11

�y ∩yℜ+�y =y {0}

wherey0y =y(0,y0 , . . . y,y0)yisytheyproductionyplanyusingynoyinputsyandyproducingynoyoutputs.y
Toy seey this,y firsty notey thaty 0y isy ay feasibley productiony plan.y Therefore,y 0y ∈ y�y.y Also,
0y ∈ yℜ �+y andy thereforey 0y ∈ y�y ∩yℜ �y . +
Toyshowythatythereyisynoyotheryfeasibleyproductionyplanyinyyyyy�ℜy,y+weyassumeytheycontrary.yTh
atyis,yweyassumeythereyisysomeyfeasibleyproductionyplanyyyyyyyyyy∈�yyyyyyy
ℜ y +∖ 0
y{yyy
}.yyThisyimpliesythe
yexistenceyofy ay plany producingy ay positiveyoutputywithy noyinputs.y Thisy technologicalyinfea

sible,y soy thaty �y∈/y �y.
1.12 1. yyLetyyxy ∈ y�y(�). yyThisyyimpliesyythatyy(�,y− x)y ∈ y�y. yyLetyyx′y ≥ yx.yy Thenyy(�,y− x′ )y ≤
(�,y− x)y andy freey disposabilityy impliesyythaty (�,y− x′ )y ∈ y�y. yThereforey x′y∈ y�y(�).
2.yy Againyy assumeyy xyy ∈ y �y(�).yyyyThisyy impliesyy thatyy (�,y− x)yy ∈ y �y.yyyyByyy freeyy disposal,y
(� ′ ,y− x)y ∈ y�yy fory everyy � ′y≤ y� ,y whichy impliesyythaty xy ∈ y�y(� ′ ).yy�y(� ′ )y ⊇ y�y(�).
1.13 They domainy ofy “<”y isy {1,y2}y=y �y andy they rangey isy {2,y3}y⫋y �y.
1.14 Figurey 1.1.
1.15 They relationy “isy strictlyy highery than”y isy transitive,y antisymmetricy andy asymmetric
.yIty isy noty complete,y reflexivey ory symmetric.




2

, ⃝ cyyy2001y Michaely Carter
Solutionsy fory Foundationsy ofy Mathematicaly Economics Allyrightsyreserved


1.16 They followingy tabley listsy theiry respectivey properties.
< ≤√yy √=
reflexive ×yy
transitive √ √yy √
symmetric √yy √
×yy
√
asymmetric
anti-symmetric √yy × yy ×
√ √
√y √y
complete ×
Notey thaty they propertiesy ofy symmetryy andy anti-symmetryy arey noty mutuallyy exclusive.
1.17 Letybe ∼ yanyequivalenceyrelationyofyaysety�y=y∕.y y ∅Thatyis,ytheyrelationyisyreflexive,
∼ ysym

metricyandytransitive.yWeyfirstyshowythatyeveryy�y�ybelongs ∈ ytoy somey equivalence yclass.
y Lety �y bey anyy elementy iny �y andy lety (�)y be
∼y they classy ofy elementsy equivalenty to
�,y thaty is
∼(�)y ≡y{y�y ∈ y�y :y �y ∼ y�y}
Since ∼ isy reflexive,y � ∼ �yandysoy� ∈ y∼ (�).y Everyy � ∈
�y belongsy toy somey equivalenceyclassy andy therefore
∪
�y = ∼(�)
�∈�

Next,y wey showy thaty they equivalencey classesy arey eithery disjointy ory identical,yy thaty is
∼(�)y ∕=y ∼(�)y ify andy onlyy ify f∼(�)y∩y∼ (�) y=y ∅ .
First,y assumey ∼(�)y∩y∼ (�) y=y ∅ . yTheny �y ∈ y∼ (�)y butyy�∈
�/ ∼( ). yThereforey ∼(�)y ∕=y ∼(�).
Conversely,yyassumeyy∼(�)y ∩y∼ (�)yy∕=yy∅ yandyyletyy�yy∈ y∼(�)y ∩y∼ (�).yyyThenyy�yy∼ y�yyandyybyysymmet
ryy �y ∼ y�.yyyAlsoy �y ∼ y�yandysoy byy transitivityy�y ∼ y�.yyyLety�y bey anyyelementyinyy∼(�)yysoyy
thatyy�yy∼ y�.yyyAgainyybyyytransitivityyy�yy∼ y�yyandyythereforeyy�yy∈ y∼(�).yyyHence
∼(�)y ⊆ y∼ (�). ySimilaryyreasoningy impliesyythaty ∼(�)y ⊆ y∼ (�). yThereforey ∼(�) y=y ∼(�).
Wey concludey thaty they equivalencey classesy partitiony �.
1.18 They sety ofy properycoalitionsy isy noty ay partitiony ofy they sety ofy players,ysincey anyy playe
rycany belongy toy morey thany oney coalition.yFory example,y playery 1y belongsy toy they coalition
s
{1},y {1,y2}yandy soy on.
1.19

�y ≻y�y =⇒ y �y ≿y �y andy �y ∕≿y �
�y ∼ y�y =⇒ y �y ≿y �y andy �y ≿y �
Transitivityy ofy ≿yimpliesy �y≿y� . yWey needy toy showy thaty �y∕≿y� . yAssumey otherwise,y thatyis
y assumey �y ≿y �y Thisy impliesy �y ∼y�y andy byy transitivityy �y ∼y�.y Buty thisy impliesy that
�y ≿y�y whichy contradictsy they assumptiony thaty �y ≻y� . y Thereforey wey concludey thaty �y ∕≿y �
andy thereforey �y ≻y� . yThey othery resulty isy provedy iny similary fashion.
1.20 asymmetricy Assumey �y ≻y�.

�y ≻y�y =⇒ y �y ∕≿y�
while

�y ≻y�y =⇒ y �y ≿y �
Therefore
�y ≻y�y =⇒ y �y ∕≻y�
3