Solution Manual for Foundations of Mathematical Economics First Edition by Michael Carter, All Chapters.

Solutions rManual
rFoundations rof


rMathematical


rEconomics




Michael

r Carter r

, Chapter r 1: r Sets r and r Spaces


1.1
{r1, r3, r5, r7 r... r} ror r{r𝑛 r∈ r𝑁 r : r 𝑛 r is rodd r}
1.2 Every r 𝑥∈ 𝐴 r also r belongs r to r 𝐵. r Every
∈ r 𝑥 𝐵 r also r belongs r to r 𝐴. rHence
r 𝐴, r𝐵 r haverprecisely rthe rsame relements.
1.3 Examples rof rfinite rsets rare
∙ the r letters r of r the r alphabet r {rA, r B, r C, r ... r , r Z r}
∙ the r set r of r consumers rin r an reconomy
∙ the r set r of r goods rin r an r economy
∙ the rset rof rplayers rin ra
rgame.rExamples rof rinfinite rsets
rare
∙ the rreal rnumbers rℜ
∙ the rnatural rnumbers r𝔑
∙ the rset rof rall rpossible rcolors
∙ the r set r of r possible r prices r of rcopper r on r the r world rmarket
∙ the r set r of r possible r temperatures r of r liquid r water.
1.4 r𝑆 r = r{r1, r2, r3, r4, r5, r6 r}, r 𝐸 r = r{r2, r4, r6 r}.
1.5 The r player r set r is r 𝑁 r = r{rJenny, rChris r}. rTheir r action r spaces r are
𝐴𝑖 r = r{rRock, rScissors, rPaper r} 𝑖 r = rJenny, rChris
1.6 The r set rof rplayers ris r 𝑁{ r = r 1, r2},..., r𝑛 r . rThe r strategy rspace r of reach
r player r is r the r set rof rfeasible routputs
𝐴𝑖 r = r{r𝑞𝑖 r ∈rℜ+ r: r 𝑞𝑖 r ≤r𝑄𝑖 r}
where r 𝑞𝑖 r ris r rthe r output rof rdam r 𝑖.
1.7 The r player r set r is r 𝑁 r = r{1, r2, r3}. rThere r are r23 r = r8 r coalitions, rnamely
𝒫(𝑁 r) r= r {∅, r{1}, r{2}, r{3}, r{1, r2}, r{1, r3}, r{2, r3}, r{1, r2, r3}}
There rare r210 r coalitions rin ra rten rplayer rgame.
1.8 r rAssume r rthat r r𝑥 r r∈ r(𝑆 r ∪r𝑇 r)𝑐. r r rThat r ris r r𝑥 r r∈/ r r𝑆 r ∪r𝑇 r. r r rThis r rimplies
r r𝑥 r r∈/ r r𝑆 r rand r r𝑥 r r∈/ r r𝑇 r, ror r𝑥 r∈ r𝑆𝑐 rand r 𝑥 r∈ r𝑇 r𝑐. r Consequently, r 𝑥 r∈ r𝑆𝑐 r∩ r𝑇
r𝑐. r Conversely, r assume r 𝑥 r∈ r𝑆𝑐 r∩ r𝑇 r𝑐. rThis r rimplies r rthat r r𝑥 r ∈ r𝑆 𝑐 r rand r r𝑥 r ∈ r𝑇 r𝑐. r
r rConsequently r r𝑥 r/ ∈ r r𝑆 r rand r r𝑥 r/
∈ r r𝑇 r rand r rtherefore
𝑥/∈ r𝑆 r∪r𝑇 r. rThis rimplies r rthat r 𝑥 r ∈r(𝑆 r∪r𝑇 r)𝑐. rThe rother ridentity ris rproved rsimilarly.
1.9
∪
𝑆 r= r𝑁
𝑆∩
∈𝒞

, 𝑆 r = r∅
𝑆∈𝒞

1

, ⃝
c r r r2001 r Michael
Solutions r for r Foundations r of r Mathematical r Carter All rrights
r Economics rreserved



𝑥2
1




-1 0 1
𝑥1




-1




Figure r 1.1: rThe r relation r {r(𝑥, r𝑦) r: r 𝑥2 r+ r 𝑦2 r = r1 r}

{ r is r}𝐻, r𝑇 r . rThe r set r of r possible
1.10 r The r sample r space r of r a r single r coin r toss
r outcomes r inrthree rtosses ris rthe rproduct
{
{𝐻, r𝑇 r}× r{𝐻, r𝑇 r}× r{𝐻, r𝑇 r}r= r (𝐻, r𝐻, r𝐻), r(𝐻, r𝐻, r𝑇 r), r(𝐻, r𝑇 r, r𝐻),
}
(𝐻, r𝑇 r, r𝑇 r), r(𝑇, r𝐻, r𝐻), r(𝑇, r𝐻, r𝑇 r), r(𝑇, r𝑇, r𝐻), r(𝑇, r𝑇, r𝑇 r)


A r typical r outcome r is r the r sequence r (𝐻, r𝐻, r𝑇 r) r of r two r heads r followed r by r a r tail.
1.11

𝑌 r ∩rℜ+𝑛 r = r{0}
where r0 r= r(0, r0 ,... r, r0) ris rthe rproduction rplan rusing rno rinputs rand rproducing rno
routputs. rTo r see r this, r first r note r that r 0 r is r a r feasible r production r plan.
r Therefore, r 0 r ∈ r𝑌 r. r Also,
0 r∈ rℜ+𝑛 r and r therefore r 0 r ∈ r𝑌 r+∩rℜ𝑛 r .
To rshow rthat rthere ris rno rother rfeasible rproduction rplan rin ℜ +r r r r r𝑛 r, rwe rassume rthe
rcontrary. rThat ris, rwe rassume rthere ris rsome rfeasible rproduction ∈ rℜ +r ∖rplan
r{
ry r r r r r r
r}
r r r r r r r r0 r r. r rThis rimplies rthe rexistence rof ra rplan rproducing ra rpositive
𝑛

routput rwith rno rinputs. rThis rtechnological rinfeasible, rso rthat r 𝑦 r/ ∈ r𝑌 r.
1.12 1. r rLet r rx r∈r𝑉 r(𝑦). r rThis r rimplies r rthat r r(𝑦, r−x) r∈r𝑌 r. r rLet r rx′ r≥rx. r rThen r r(𝑦, r−x′ ) r≤
(𝑦, r−x) rand r free rdisposability rimplies r rthat r(𝑦, r−x′ ) r∈r𝑌 r. rTherefore rx′ r∈r𝑉 r(𝑦).
2. r rAgain r rassume r rx r r∈ r 𝑉 r(𝑦). r r r rThis r rimplies r rthat r r(𝑦, r−x) r r∈ r 𝑌 r. r
r r rBy r rfree r rdisposal, r(𝑦′ , r−x) r∈r𝑌 r r for revery r 𝑦 ′ r≤r𝑦 , rwhich rimplies r rthat
rx r∈r𝑉 r(𝑦′ ). r r𝑉 r(𝑦′ ) r⊇r𝑉 r(𝑦).
1.13 The rdomain rof r“<” ris r{1, r2}r= r𝑋 r and rthe rrange ris r{2, r3}r⫋ r 𝑌 r.
2