Solutions Manual Foundations of Mathematical Economics 2024.

Solutions Manual Foundations of
Mathematical Economics


Michael Carter

, ⃝ c j j j2001 j Michael
Solutions j for j Foundations j of j Mathematical j Carter All jrights
j Economics jreserved




Chapter j 1: j Sets j and j Spaces


1.1
{ j1, j3, j5, j7 j... j} jor j { j� j∈ j� j : j� jis j odd j}
1.2 Every j �
∈ � j also j belongs j to j �. j∈Every j � � j also j belongs j to
j �. jHence j �, j� j have jprecisely jthe jsame jelements.
1.3 Examples jof jfinite jsets jare
∙ the j letters j of jthe j alphabet j { jA, j B, j C, j ... j, j Z j}
∙ the j set jof j consumers jin j an j economy
∙ the j set j of j goods jin j an j economy
∙ the jset jof jplayers jin
ja jgame. jExamples jof
jinfinite jsets jare
∙ the jreal jnumbers jℜ
∙ the jnatural jnumbers j�
∙ the jset jof jall jpossible jcolors
∙ the j set j of j possible j prices j of j copper j on j the j world j market
∙ the j set j of j possible j temperatures j of j liquid j water.
1.4 j� j= j{ j1, j2, j3, j4, j5, j6 j}, j� j= j{ j2, j4, j6 j}.
1.5 The j player j set j is j � j = j{ jJenny, jChris j}. jTheir j action j spaces j are
�� j = j{ jRock, jScissors, jPaper j} � j= jJenny, jChris
1.6 The j set j of j players j is{ j � j = j }1, j2,..., j� j . jThe j strategy jspace
j of j each j player j is j the j set jof jfeasible joutputs
�� j= j{ j�� j∈ jℜ + j: j�� j≤ j�� j}
where j�� j jis j jthe j output jof jdam j�.
1.7 The j player jset j is j � j = j{1, j2, j3}. jThere j are j23 j = j8 j coalitions, jnamely
� (�j) j= j {∅ , j{1}, j{2}, j{3}, j{1, j2}, j{1, j3}, j{2, j3}, j{1,
j2, j3}}
There jare j210 j coalitions jin ja jten jplayer jgame.
1.8 j jAssume j jthat j j� j j∈ j(� j∪ j�j)�. j j jThat j jis j j� j j∈/ j j� j∪ j�j. j j
jThis j jimplies j j� j j∈/ j j� j jand j j� j j∈/ j j�j, jor j�j∈ j�� jand j �j∈ j�j�.
j Consequently, j � j∈ j�� j∩ j� j�. j Conversely, j assume j � j∈ j�� j∩ j� j�.
jThis j jimplies j jthat j j� j∈ j�� j jand j j� j∈ j�j�. j j jConsequently j j�j∈/ j
j� j jand j j�j∈/ j j� j jand j jtherefore
�∈/ j� j∪ j� j. jThis jimplies j jthat j� j∈ j(� j∪ j� j)�. jThe jother jidentity jis jproved

, ⃝ c j j j2001 j Michael
Solutions j for j Foundations j of j Mathematical j Carter All jrights
j Economics jreserved
jsimilarly.
1.9
∪
� j= j�
∈�
�∩
� j= j∅
�∈�


1

, ⃝ c j j j2001 j Michael
Solutions j for j Foundations j of j Mathematical j Carter All jrights
j Economics jreserved


�2
1




-1 0 1
�1




-1




Figure j 1.1: jThe j relation j { j(�, j�) j: j�2 j+ j�2 j = j1 j}

1.10 j The j sample j space j of j a j single j {coin j }toss j is j�, j� j . jThe j set j of
j possible j outcomes j in jthree jtosses jis jthe jproduct
{
{�, j�j}×j{�, j�j}× j{�, j�j} j= j (�, j�, j�), j(�, j�, j�j), j(�, j�j, j�),
}
(�, j�j, j�j), j(�, j�, j�), j(�, j�, j�j), j(�, j�, j�), j(�, j�, j�j)


A j typical j outcome j is j the j sequence j (�, j�, j�j) j of j two j heads j followed j by j a j tail.
1.11

� j ∩ jℜ+� j = j{0}

where j0 j= j(0, j0,... j, j0) jis jthe jproduction jplan jusing jno jinputs jand
jproducing jno joutputs. jTo j see j this, j first j note j that j 0 j is j a
j feasible j production j plan. j Therefore, j 0 j ∈ j�j. j Also,
0 j∈ +jℜ � j and j therefore j 0 j∈+ j� j ∩ jℜ � j .
To jshow jthat jthere jis jno jother jfeasible jproduction ℜ + jplan jin j j j j j� j, jwe
jassume jthe jcontrary. jThat jis, jwe jassume jthere ∈ jℜjis ∖ j{
+ j jsome jfeasible
� j}
jproduction jplan jy j j j j j j j j j j j j j j0 j j. j jThis jimplies jthe jexistence
jof ja jplan jproducing ja jpositive joutput jwith jno jinputs. jThis
jtechnological jinfeasible, jso jthat j�j∈/ j� j.
1.12 1. j jLet j jx j∈ j� j(�). j jThis j jimplies j jthat j j(�, j− x) j∈ j� j. j jLet j jx′ j≥ jx. j jThen j j(�,
j− x′ ) j≤
(�, j− x) jand jfree jdisposability jimplies j jthat j(�, j− x′ ) j∈ j� j. jTherefore jx′ j∈ j� j(�).
2. j jAgain j jassume j jx j j∈ j� j(�). j j j jThis j jimplies j jthat j j(�, j− x)
2