BEE1020—BasicMathematicalEconomics

Week6,LectureTuesday11.11.03

Geometricpropertiesofcostfunctions

WELCOME!

tothelectureandgoodluckforyourstudies!

Instructor:

DieterBalkenborg

room

SC49,nexttotheschool’slibrary

Tel.263231

e-mail:D.G.Balkenborg@exeter.ac.uk

homepage:http://www.ex.ac.uk/˜dgbalken/

officehours:Tue,Thu,2-3p.m.

Therulesofthegame:

Scheduleforthefirsttwoterms:

•lectures(twohours):

—Tuesday,11:00a.m-1:00p.m.,SCD

•regularweeklytutorials(classes),onehour:

—Group1:Tuesday,9a.m.,SCE,IannisKrassas

—Group2:Tuesday,10a.m.,SCE,IannisKrassas

—Group3:Wednesday,2p.m.,SCE,GiovanniCaggiano

—Group4:Wednesday,3p.m.,SCE,AxelDreher

Doyouknowyourtutorialgroup?attendance!classexer-

cisesandsolutions

2

•weeklyhomework:distributedintutorial,tobesubmitted

aweeklater,

grading,workedoutsolutions

notsubmittingcountslikenotattendingtheclass(=⇒supple-

mentaryclasses)

submitingroupsuptothreepeople!

•4roundsofsupplementaryclasses(Oct,Nov,Feb,April)

•thirdterm:revisionlecturesbasedonmockexam

forall+

revisionclasses(supplementary)

•2×2-hourexams:January(40marks)+inJune(60marks)

one3-hourresitinSeptemberonallthematerial(notrecom-

mended)

3

•AllmaterialwillbemadeavailableontheWEBsite:

http://www.ex.ac.uk/˜dgbalken/BME/BME.html

Warning:typos

•Thetextbook:L.D.Hoffmann+G.L.Bradley:Calculusfor

business,economicsandthesocialandlifesciences,McGraw

Hill,8th,INTERNATIONALedition(orearlier).

4

Objectiveofthemodule:

Idonotintendtoteachyoustuffyoudon’tneedfor

yoursecondandthirdyear!!!Thisisacourseonmathe-

maticaleconomics.

Yourincentive:

•freechoiceofmodulesinsecondandthirdyear

•notgettingdistractedbythemathematics

Whymath?

1.numbersoccasionallyoccurineconomics

2.AlfredMarshallandtheprinciplesofeconomics

3.Therealworldtoocomplicated.needsimpleandhighlystylized

modelstodevelopintuitions.Usingmathematicalmodelsbyfar

thebestmethod.

5

4.Useofmathematicalmodelsinthesocialsciences:designofle-

galsystems,prosandconsofvotingsystems,moralphilosophy,

disarmament,learning,cognitivepsychology...

5.hardtofoolothers.Youcannothideyourassumptions.

6.homooeconomicusandconstrainedoptimization

Thechallengeforthismodule:

•studentdiversity

•mathematicsisaskillthatneedsregularexercising

6

Overviewonthelecture:

•functionsandtheirdomains,intervals

•theindependentandthedependentvariable

•thegraphofafunction

•linearandquadraticfunctions,polynomialfunctions

•thedifferencequotient

•thetangentandtheslope

•increasinganddecreasingfunctions

•convexandconcavefunctions(upward-anddownwardbowed)

•thefirstandthesecondderivative

memorizetheseconcepts

7

Examplesofcostfunctions

Afunctiondescribeshowonequantitychangesinresponseto

anotherquantity.

totalcostfunctionofafirmsellingnewpaper:

1.quantityofoutput.(independentvariable)

2.totalcostsofproducingtheoutput.(dependentvariable)

descriptionoffunction:

1.byatable,

2.byagraph,

3.algebraicexpression.

8

Example1:Constantmarginalcosts

quantity(in100.000)

01

23

45

67

totalcosts(in1000$)90110130150170190210230

020406080100

120

140

160

180

200

220

TC

12

34

56

7Q

TC(Q)=90+20Q

9

Example2:Increasingmarginalcosts

quantity(in100.000)

01

23

45

67

totalcosts(in1000$)110135170215270335410495

100

200

300

400

500

TC

01

23

45

67

Q

TC(Q)=5Q2+20Q+110

10

Example3:U-shapedmarginalcosts

quantity(in100.000)

012

34

56

7totalcosts(in1000$)5094114122130150194274

20406080100

120

140

160

180

200

220

240

TC

01

23

45

67

Q

TC(Q)=2Q3−18Q2+60Q+50

11

Functions,ConceptandNotation

AfunctionisarulewhichspecifiesforeachobjectinasetA

exactlyoneobjectinathesetB.ThesetAiscalledthedomain

andthesetBtheco-domain.

AandBmostlysubsetsofthenumberline.domainandco-

domainintegralpartofdefinition

commonnotations:

1.IsaacNewton(1643—1727)andGottfriedWilhelmLeibniz

(1646—1716)—used:

y(x)whereythedependentvariablextheindependentvari-

able.

y(x)=x2+1.

ydependsonx,forx=1y=2,forx=3y=10,y(1)=2

andy(3)=10

12

2.Slightlymoremodern y=f(x)=x2+1.

Theletterfdoesnotrepresentanumber

y=x2+1

|{z}

f(x)

3.Mostmodernandrigourous

f:A−→

Bx7−→

f(x)

f:{x≥1}−→

{y≥0}

x7−→

√ x−1

13

Graphsoffunctions

Thegraphofafunctiony=f(x)isthecurveconsistingofall

points(x,y)=(x,f(x))wherexvariesoverthedomainofthe

function.

Acurveormerelyacollectionofdots?

TheVerticalLineTest:Acurveisthegraphofafunctionif

andonlyifnoverticallineintersectsthecurvemorethanonce.

14

Inversefunctions

interchangethehorizontalandtheverticalaxis.(x,y)thenbe-

comes(y,x),(−2,4)becomes(4,−2).graphismirroredatthe

45◦ -line.

-4-2024

-4-2

24

Invertingagraph.

012345

12

34

5x

squareandsquareroot

U-shapedcurve:squarefunctiony=x2 .C-shapedfailsthe

verticallinetest.Thisissobecauseeverypositivenumbery≥0

hastworootsx=±√

y.Restrictiontonon-negativenumbers:

15

inverseisx=√ y,thesquarerootfunction.Notice:√ y

refers

tothepositiveroot.√ 4

=−2

isincorrect,while( −2)2=4is

correct.

16

costfunctioninExample3invertible,butinverseishardtocom-

pute

20406080100

120

140

160

180

200

220

240

TC

01

23

45

67

Q

Thegraphfrom

Example3.

01234567

Q

50100

150

200

250

TC

Theinvertedgraphfrom

Example3.

17

invertedgraphof

TC(Q)=2Q3−18Q2+48Q+86

isnotgraphoffunction:

050100

150

200

250

TC

12

34

56

7Q

ThegraphofthefunctionTC(Q).

01234567

Q

50100

150

200

250

TC

Theinvertedgraph.

18

Continuousanddifferentiablefunctions

Alldifferentiablefunctionsarecontinuous.

Afunctioniscontinuousifitsgraphcanbedrawninasingle

stroke,withouteverliftingthepen.Thereshouldbeno“jumps”,

atleastoverintervals.

interval:linesegmenty=f(x)=1 x

-4-2024

-4-2

24

x

y=1 xiscontinuous,domainhasholeatx=0.

19

notcontinuousatx=0:isthesignfunction

sign(x)=

+1forx>0

0forx=0

−1forx<0

-1-0.8

-0.6

-0.4

-0.200.20.40.60.81

-4-2

24

x

)( 20

Theintermediatevaluetheorem:Supposey=f(x)isdefined

andcontinuousontheintervala≤x≤b,f(a)<0andf(b)>0.

Thenthereexistsarootbetweenaandb,i.e.,anumbercwith

a<c<bandy=f(c)=0

-2.5-2

-1.5-1

-0.50

0.5

-4-2

24

xa

b

c

(1)

21

Afunctionisdifferentiableifitsgraphhasnokinks.

akink(orcornerpoint)atx=0:

|x|=

xforx>0

0forx=0

−xforx<0=x·sign(x)

theabsolutevaluefunction

-2-101245

-4-2

24

x

.

22

Atakinkthegraphcanhaveseveraltangents,i.e.,severallines

whichtouchthegraphinthispoint.

-2-101245

-4-2

24

x

23

Forafunctiontobedifferentiabletherehastobeauniquetan-

gentateachpointofthegraph.

20406080100

120

140

160

180

200

220

240

TC

01

23

45

67

Q

Tangentsforthegraphfrom

Example3.

24

Tosummarize,acurveisthegraphofafunctionifitpasses

theverticallinetest.Thefunctioniscontinuousifitsgraph

canbedrawninonestrokeanditisdifferentiableifthegraph

hasnokinks.

25

Fixedcostsandvariablecosts

TC(0)=50inthethirdexample.set-upcostsorfixedcosts

Onedefinesthefixedcostsas

FC=TC(0)

andthevariablecostfunctionas

VC(Q)=TC(Q)−FC

InExample3onehas

TC(Q)=2Q3−18Q2+60Q+50

FC=50

VC(Q)=2Q3−18Q2+60Q

26

050100

150

200

250

FC

12

34

56

7Q

Thefixedcosts.

050100

150

200

250

VC

12

34

56

7Q

Thevariablecosts.

27

Costsarepositiveandincreasing

Obviously,costsarealwayspositivenumbers.

Costfunctionsshouldbeincreasingfunctions(Q1<Q2implies

TC(Q1)<TC(Q2)).oratleastnon-decreasing(Q1<Q2

impliesTC(Q1)≤TC(Q2).

Theorem1SupposeagivenfunctionTC(Q)hasnon-negative

fixedcostsTC(0)andisincreasing.ThenthecostsTC(Q)are

positiveforallQ>0.

28

Linearfunctions

Example1:linearcostfunctionTC(Q)=90+20Q,graphis

a(non-vertical)straightline.

costincreases∆TC

Q0

12

34

56

7TC

90110130150170190210230

∆TC

2020

2020

2020

20

Theseareconstant.Itcosts$2,000moretoprint100,000news-

papersmore.

lessobviouswhentheoutputlevelsinthetablearenotequidis-

tant(notgivenatequaldistances):

Q0

34

711

1217

20TC

90150170230310330390490

∆TC

6020

6080

2010060

29

useratesofchangeorthedifferencequotients

∆TC

∆Q=TC(Q1)−TC(Q0)

Q1−Q0

whereQ0andQ1aredistinctquantities:

Q0

34

711

1217

20∆Q

31

34

15

3TC

90150170230310330430490

∆TC

6020

6080

2010060

∆TC

∆Q

2020

2020

2020

20

Therateofchangeisthesame,whatevertwoquantitiesQ0

andQ1wecompare.

30

Thisrateiscalledtheslopeorgradientoftheline.Economists

speakofconstantmarginalcosts.Inourexamplethemarginal

costsare

∆TC

∆Q=20

µ ×$1,000

100,000

¶ =20(×1p)

Theorem2Alinearfunctionisincreasingifandonlyifits

slopeispositive.

31

fortwodistinctquantitiesQ0andQ1:

1.marginalcosts: m=

∆TC

∆Q=TC(Q1)−TC(Q0)

Q1−Q0

2.GivenafixedquantityQ0andanyotherquantityQ

TC(Q)−TC(Q0)

Q−Q0

=m

orTC(Q)=TC(Q0)+m(Q−Q0).

thepoint-slopeform.

3.Thefixedcostsare

FC=TC(0)=TC(Q0)−mQ0.

ForanyquantityQweobtain

TC(Q)=TC(Q0)−mQ0+mQ=FC+mQ

32

slope-interceptform.Thevariablecostsaresimply

VC(Q)=mQ.

Exercise1Thetotalcostsare$1600forproducing300CDsand

$2000forproducing500CDs.Assumingalinearcostfunction,

determinethemarginalcostsandthefixedcosts.

33

Non-linearcostfunctions

AlsothecostfunctionsinExample2and3areincreasing.Cor-

respondingly,thecostincreases∆TCarealwayspositive,butno

longerconstant.Example2:

Q0

12

34

56

7TC

110135170215270335410495

∆TC

2535

4555

6575

85

Example3:

Q0

12

34

56

7TC

5094

114122130150194274

∆TC

4420

88

2044

80

Costsdifferences∆TCandratesofchange

∆TC

∆Qnolongercon-

stant.

Usetangentstoapproximatethegraphnearapoint.rateof

changecanbeapproximatedbyslopeofthetangent

34

tangentat(3,TC(3))=(3,215):

180

200

220

240

260

22.2

2.4

2.6

2.8

33.2

3.4

3.6

3.8

4Q

(2)

goodapproximationofcorrectcostfunctioninExample2forquan-

titiesbetween2(×100.000)and4(×100.000)

35

equationforthetangent:iscalculatedas:

t(Q)=215+50(Q−3)

50(×1p)isthemarginalcosts,theapproximatecostofproducing

a‘small’unitmore.

theexactcostare

(TC(3.000,01)−TC(3))×($1000)

=(215.0005000005−215)×($1000)

=0.0005000005×($1000)=50.0005p

36

Thefirstderivative

Thegradientofafunctiony=f(x)atavaluex0oftheinde-

pendentvariableistheslopeofthetangenttothegraphoff(x)

atthepoint(x0,f(x0)).

Notations:y0 (x0)orf0 (x0)(Newton)or

dydx(x0)or

dydx|x 0or

df dx(x0)(thedifferentialquotient,Leibniz).

tangent:

t(x)=f(x0)+f0 (x0)(x−x0)

37

Thenewfunctionwhichassignstoeachvalueoftheindependent

variablextheslopeofthecorrespondingtangentiscalledthe

(first)derivativeofy=f(x).

20406080100

120

140

160

180

200

220

240

TC

01

23

45

67

Q

Tangentsforthegraphfrom

Example3.

Notations:y0 ,f0 (x)(Newton)or

df dxor

dydx(Leibniz).

The

methodtocalculatederivativesiscalleddifferentiation.

38

Polynomials

Apolynomialofdegreenisafunctionoftheform

f(x)=anxn+an−1xn−1+...+a2x2+a1x1+a0x0

=anxn+an−1xn−1+...+a2x2+a1x+a0

withconstantsan,an−1,···,a0wheretheleadingcoefficientan

isnotzero.anxniscalledtheleadingterm

anda0theconstant

term.Polynomialissumofmonomialsakxk

Specialcases:

constantfunctionsf(x)=a0

linearfunctionsf(x)=a1x+a0

quadraticfunctionsf(x)=a2x2+a1x+a0

cubicfunctionsy(x)=a3x3+a2x2+a1x+a0

39

derivativeofpowerfunctiony=xk:

y0 =

kxk−1

derivativeofapolynomialfunctionf(x)is

f0 (x)=nanxn−1+(n−1)an−1xn−2+...+2a2x2−1

+a1x1−1+0a0x0−1

=nanxn−1+(n−1)an−1xn−2+...+2a2x1+a1+0

derivativeofacubicfunctionisquadratic

thederivativeofaquadraticfunctionislinear

thederivativeofalinearfunctiona1x+a0isconstant

thederivativeofaconstantfunctioniszero.

40

MarginalcostsinExamples2and3

TC(Q)=90+20Q

MC(Q)=dTC

dQ=20

Example2: TC(Q)=5Q2+20Q1+110

MC(Q)=dTC

dQ=2×5Q1+20Q0=10Q+20

41

Example3:

TC(Q)=2Q3−18Q2+60Q+50

MC(Q)=dTC

dQ=3×2Q2−2×18Q+60=6Q2−36Q+60

costincreasesmarginalcosts:Example2:

Q0

12

34

56

7TC

110135170215270335410495

∆TC

2535

4555

6575

85MC

2030

4050

6070

8090

Example3:

Q0

12

34

56

7TC

5094

114122130150194274

∆TC

4420

88

2044

80MC

6030

124

1230

60102

42

Increasingfunctionsandupward-slopedness

CostfunctionTC(Q)inExample2isupward-slopedforpositive

Q:ThemarginalcostsMC(Q)=10Q+20arealwaysbiggerthan

20andhencepositive.

Conjecture1Afunctionisincreasingifandonlyifallits

tangentsareupward-sloped,i.e.,havepositiveslope.

‘almost’true

43

alltangentshavepositiveslope,butfunctionnotincreasing:

-10-8-6-4-20246810

-3-2

-11

23

x

Upward-sloped,butnotincreasing.

Wemustrestrictattentiontointervals

44

tangenttothegraphat(0,1)ishorizontal,butfunctionstrictly

increasing:

-2-10134

-1.5

-1-0.5

0.5

11.5

x

Increasingwithahorizontaltangent.

45

Theorem3Acontinuouslydifferentiablefunctionisincreas-

ingon

anintervalifandonlyifitsfirstderivativeisnon-

negativeintheintervalandnotconstantlyzeroonanysubin-

terval.

Theorem4Acontinuouslydifferentiablefunctionisdecreas-

ingon

anintervalifandonlyifitsfirstderivativeisnon-

positiveintheintervalandnotconstantlyzeroonanysubin-

terval.

46

0.9

0.951

1.051.1

-0.4

-0.2

00.2

0.4

0.6

0.8

11.2

1.4

x

Non-decreasing,butnotincreasing.

47

Theorem5Afunctionisinvertibleifandonlyifitisin-

creaingordecreasing.

Summary:Thefirstderivativemeasureshowsteeplyafunc-

tionincreases.Increasingfunctionshavepositivederivatives,de-

creasingfunctionshavenegativederivatives.

48

Strictconvexityandconcavity

InExample2costincreasesarethemselvesincreasing:

Q0

12

34

56

7TC

110135170215270335410495

∆TC

2535

4555

6575

85

∆2 TC

1010

1010

1010

theincreaseoftheincrease(∆2 TC=

∆(∆TC))isalwaysposi-

tive.

Costsareaccelerating,

increasingmarginalcosts

thecostsofproducingoneunitmoreishigherwhenmoreis

produced.

strictlyconvexfunction

–Thegraphisupward-bowed.

–Thetangentsgetsteeperfrom

lefttoright,i.e.,theirslopes

49

areincreasing.

marginalcostsMC(Q)=10Q+20areincreasing

100

200

300

400

500

TC

01

23

45

67

Q

ThetotalcostsinExample2

020406080100

MC

12

34

56

7Q

Increasingmarginalcosts.

50

upward-bowed(likeacup^):strictlyconvex

downward-bowedgraph(likeacap_):strictlyconcave

“strictly”:properlycurved

alinearfunctionisbothconvexandconcave,butnotstrictly

convexorstrictlyconcave.

theword“cave”appearsinconcave:

con-

cave

51

Example3doesnotexhibitincreasingmarginalcosts:Thecost

increases∆TCarefirstdecreasingandthenincreasing.

Q0

12

34

56

7TC

5094

114122130150194274

∆TC

4420

88

2044

80

∆2 TC

-24

-12

012

2436

52

20406080100

120

140

160

180

200

220

240

TC

01

23

45

67

Q

Example3

20406080100

MC

01

23

45

67

Q

U-shapedmarginalcosts.

Thetotalcostsfunctionisstrictlyconcavefor0≤Q≤3and

strictlyconvexfor3≤Q.

53

calculuscanhelptodecidewhetherafunctionisconvexorcon-

cave usethesecondderivative(thederivativeofthederivative)ofa

function.

Newton:y00 (x),f00 (x)Leibnizusedd2 ydx2ord2 f dx2.

InExample2

d2 TC

dQ2=dMC

dQ

=d(10Q+20)

dq

=10>0.

InExample3

d2 TC

dQ2=dMC

dQ

=d¡ 6Q2

−36Q+60¢

dq

=12Q−36=12(Q−3)

whichisnegativeforQ<3andpositiveforQ>3.

54

Theorem6Thefollowingstatementsareequivalentforatwice

continuouslydifferentiablefunctiononaninterval:

a)Thefunctionisstrictlyconvexontheinterval.

b)Itsfirstderivativeisincreasingontheinterval

c)Itssecondderivativeisnonnegativeontheintervaland

neverconstantlyzeroonanysubinterval.

Theorem7Thefollowingstatementsareequivalentforatwice

continuouslydifferentiablefunctiononaninterval:

a)Thefunctionisstrictlyconcaveontheinterval.

b)Itsfirstderivativeisdecreasingontheinterval

c)Itssecondderivativeisnonpositiveon

theintervaland

neverconstantlyzeroonanysubinterval.

55

Summary:Afunctionisconvex(upward-bowed)ifitstan-

gentsgetsteeperfrom

lefttoright.Thelattermeansthatitsfirst

derivativeisincreasingandhencepositivelysloped.Thusconvex

functioncorrespondstoincreasingfirstderivativeandthelatterto

positivesecondderivative.Correspondingly,concave(downward-

bowed)functionshavedecreasingfirstderivativesandnegativesec-

ondderivatives.

56