Sin

gle

Var

iabl

eCal

culu

san

dApp

licat

ions

Dudle

yCook

e

Trinity

Colleg

eD

ublin

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

Sin

gle

Var

iable

Calc

ulu

sand

Applica

tions

1/

47

EC20

40Top

ic1

-Sin

gle

Var

iabl

eCal

culu

san

dApp

licat

ions

Rea

din

g

1Chap

ters

7.1-

7.3,

9.1-

9.4

and

10of

CW

2Chap

ters

6,7,

and

9of

PR

Pla

n

1Rev

iew

ofsingl

e-va

riab

leca

lculu

s

2O

ptim

izat

ion

3Con

cavi

tyan

dCon

vexi

ty

4A

pplic

atio

ns

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gle

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iable

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Applica

tions

2/

47

Sin

gle

Var

iabl

eCal

culu

s

Afu

nct

ion

isuse

fully

thou

ght

ofas

a“r

ule

”w

hic

hco

nve

rts

anin

put

(den

oted

typic

ally

byx)

into

anou

tput

(den

oted

typic

ally

byy).

Not

atio

nal

ly,we

write

y=

f(x

).In

wor

ds,

“yis

afu

nct

ion

ofx.”

xis

also

calle

dth

ein

dep

enden

tva

riab

leor

the

exog

enou

sva

riab

le.

yis

also

calle

dth

edep

enden

tva

riab

leor

the

endog

enou

sva

riab

le.

Inte

rms

ofec

onom

ics,

itis

impor

tant

tokn

oww

hat

thes

efu

nct

ions

‘look

like’

.

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iable

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Applica

tions

3/

47

Eco

nom

icFun

ctio

ns

Mac

ro-

Nat

ional

inco

me

acco

unting.

The

link

bet

wee

nag

greg

ate

inco

me,

consu

mption

,in

vest

men

t,an

dgo

vern

men

tsp

endin

g.

Y=

C( Y

d) +

I(r

)+G

Mic

ro-U

tilit

yfu

nct

ions.

We

thin

kth

atin

div

idual

sfe

licity

dep

ends

onco

nsu

mption

.W

=u

(c)

Mic

ro-

Pro

duct

ion

funct

ions.

Indiv

idual

firm

sou

tput

dep

ends

onla

bor

.y

=f

(L)

Som

etim

eswe

leav

eth

efu

nct

ions

gener

alan

dot

her

tim

eswe

assu

me

afu

nct

ional

form

.It

alldep

ends

onhow

much

stru

cture

we

wan

tto

put

onth

eunder

lyin

gec

onom

ics.

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tions

4/

47

Slo

pe

ofLin

ear

Fun

ctio

ns

The

sim

ple

stfu

nct

ion

we

consider

isa

linea

rfu

nct

ion.

Alin

eis

are

lation

ship

ofth

efo

rmy

=ax

+b.

Sta

rtat

any

poi

nt

(x0,y

0)

onth

elin

ean

dm

ove

alon

gth

elin

eso

that

the

x-c

oor

din

ate

incr

ease

sby

one

unit.

The

corr

espon

din

gch

ange

inth

ey-c

oor

din

ate

isca

lled

the

slop

eof

the

line.

The

slop

ete

llsus

the

rate

ofch

ange

:how

much

ych

ange

sw

hen

xch

ange

sby

agi

ven

amou

nt.

The

defi

nin

gch

arac

terist

icof

alin

eis

that

this

rate

ofch

ange

isco

nst

ant.

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47

Slo

pe

ofLin

ear

Fun

ctio

ns

Cla

im:

the

slop

eof

y=

ax+

bis

a.

Pro

of:

pic

kx

=x 0

;th

isim

plie

sth

aty 0

=ax

0+

b.

Incr

ease

x 0to

x 0+

∆x.

The

corr

espon

din

gnew

valu

eof

yis

y 1=

a(x 0

+∆

x)+

b.

The

chan

gein

yis

∆y

=y 1

−y 0

=a(

x 0+

∆x)+

b−

(ax 0

+b).

This

sim

plifi

esto

∆y

=a∆

x.

Ther

efor

e,∆

y

∆x

=a

Rat

eof

chan

ge(o

fa

line)

isco

nst

ant.

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47

Slo

pe

ofa

Non

-Lin

ear

Fun

ctio

n

Lin

ear

funct

ions

are

use

fulbec

ause

ther

ear

esim

ple

.H

owev

er,not

man

yec

onom

icphen

omen

aar

elin

ear.

Con

sider

the

follo

win

gnon

-lin

ear

funct

ion

(aquad

ratic)

y=

x2.

Ifwe

star

tat

x=

1an

din

crea

sex

by1,

then

ych

ange

sby

3(i.e

.,4−

1).

Ifwe

star

tat

x=

2how

ever

,th

enin

crea

sing

xby

1ch

ange

sy

by5

(i.e

.,9−

4).

Thus

the

sam

ech

ange

inx

lead

sto

diff

eren

tch

ange

sin

y.

We

cannot

ther

efor

edefi

ne

agl

obal

not

ion

ofth

eslop

efo

rnon

-lin

ear

funct

ions.

How

ever

,it

ispos

sible

todefi

ne

anot

ion

ofth

eslop

ew

hic

his

valid

when

the

chan

gein

xis

“sm

all.”

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iable

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Applica

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47

Tan

gent

Lin

e

Con

sider

the

curv

ey

=x

2.

Also

consider

the

line

y=

4x−

4.

This

line

just

touch

esth

ecu

rve

y=

x2

atth

epoi

nt

(x,y

)=

(2,4

).H

ow?

y=

22an

dy

=8−

4.

Such

alin

eis

calle

da

tange

nt

line.

The

tange

nt

line

has

the

prop

erty

that

it“l

ook

sth

esa

me

asth

efu

nct

ion

arou

nd

the

poi

nt

that

itju

stto

uch

es.”

The

tange

nt

line

show

sth

era

teof

chan

geat

apoi

nt

for

smal

lch

ange

s.

The

slop

eof

the

tange

nt

line

isca

lled

the

der

ivat

ive

atth

epoi

nt

x 0.

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iable

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47

Dia

gram

ofth

eTan

gent

Lin

e

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iable

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ulu

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Applica

tions

9/

47

The

Tan

gent

Lin

eas

aLim

itof

Sec

ant

Lin

es

An

alte

rnat

ive

way

ofth

inki

ng

abou

tth

eta

nge

nt

line

isth

atit

isa

limit

ofse

cant

lines

.

Ase

cant

line

isa

line

join

ing

any

two

poi

nts

onth

ecu

rve.

The

tange

nt

line

isth

elin

eto

whic

hth

ese

cant

lines

get

“clo

ser”

asth

epoi

nts

onth

elin

ege

tcl

oser

and

clos

er.

This

issh

own

inth

edia

gram

whic

hfo

llow

sfo

rth

ecu

rve

y=

x2

and

the

poi

nt

(x,y

)=

(2,4

).

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ublin)

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iable

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ulu

sand

Applica

tions

10

/47

Dia

gram

with

Sec

ant

Lin

es

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gle

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iable

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ulu

sand

Applica

tions

11

/47

Der

ivat

ive

The

der

ivat

ive

ofa

funct

ion

f(x

)at

the

poi

nt

x 0is

den

oted

asf′ (

x 0).

The

tota

ldiff

eren

tial

off(x

)at

x 0is

defi

ned

byth

efo

llow

ing.

dy

=f′ (

x 0)d

x

The

tota

ldiff

eren

tial

isa

way

ofunder

stan

din

gth

elo

calra

teof

chan

geof

the

funct

ion

f(x

)ar

ound

the

poi

nt

x 0.

That

is,it

isan

alge

brai

cway

ofden

otin

gth

eslop

eof

afu

nct

ion.

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ublin)

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gle

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iable

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ulu

sand

Applica

tions

12

/47

Eco

nom

icSig

nifica

nce

ofth

eD

eriv

ativ

e:Pro

duct

ion

Cos

ts

Imag

ine

that

yre

pres

ents

the

cost

sof

product

ion

in(in

$)an

dx

the

quan

tity

produce

dby

afirm

.

The

der

ivat

ive

ata

give

nx 0

tells

us

how

cost

sch

ange

inre

spon

seto

ach

ange

inquan

tity

,pr

ovid

edth

ech

ange

issm

all.

For

exam

ple

,if

we

know

that

the

der

ivat

ive

atx

=2

is4,

this

tells

us

that

ifth

equan

tity

produce

dch

ange

sby

asm

allam

ount

dx,th

enth

eim

pac

ton

cost

isgi

ven

appr

oxim

atel

yby

the

tota

ldiff

eren

tial

dy

=4d

x.

Eco

nom

ists

hav

ea

spec

ialfo

rth

eder

ivat

ive

ofth

eco

stfu

nct

ion:

itis

calle

dm

argi

nal

cost

.

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iable

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ulu

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Applica

tions

13

/47

Rem

arks

Sin

ceth

era

teof

chan

geal

ong

acu

rve

isch

angi

ng

const

antly,

the

der

ivat

ive

has

tobe

com

pute

dse

par

atel

yat

each

pos

sible

valu

eof

x.

The

der

ivat

ive

isth

us

alo

calphen

omen

on:

itte

llsus

som

ethin

gab

out

the

rate

ofch

ange

inth

enei

ghbor

hood

ofa

poi

nt,

but

itgi

ves

no

info

rmat

ion

abou

tth

era

teof

chan

gegl

obal

ly.

For

exam

ple

,th

ein

form

atio

nth

atth

eder

ivat

ive

ofy

=x

2(i.e

.,dy

=2x

dx)

atx

=2

is4

tells

us

that

the

rate

ofch

ange

is4

when

xis

“clo

se”

to2.

Itdoes

not

give

any

info

rmat

ion

abou

tth

era

teof

chan

geat

x=

10,

and

soon

.

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ublin)

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gle

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iable

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ulu

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Applica

tions

14

/47

Der

ivat

ive

asa

Fun

ctio

n

Not

ice

that

we

can

also

thin

kof

the

der

ivat

ive

itse

lfas

afu

nct

ion

ofx.

Giv

ena

funct

ion

y=

f(x

),th

eder

ivat

ive

funct

ion

sim

ply

asso

ciat

esto

ever

yx

the

slop

eof

the

tange

nt

line

atx.

Typ

ical

ly,w

hen

we

talk

abou

tth

eder

ivat

ive,

we

mea

nth

eder

ivat

ive

asa

funct

ion.

So

when

we

wan

tto

talk

abou

tth

eva

lue

ofth

eder

ivat

ive

ata

poi

nt

x,we

shal

lm

ention

itby

sayi

ng

“the

der

ivat

ive

atx

is..

.”N

otat

ional

ly,gi

ven

afu

nct

ion

y=

f(x

),we

shal

lden

ote

the

der

ivat

ive

funct

ion

as,

f′ (

x)

ordy

dx

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iable

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Applica

tions

15

/47

Rul

esfo

rCom

puting

Der

ivat

ives

Ther

ear

ea

num

ber

ofru

les

for

com

puting

der

ivat

ives

ofsp

ecifi

cfu

nct

ions:

1f(x

)=

a⇒

f′ (

x)

=0.

[con

stan

t]

2f(x

)=

mx

+b⇒

f′ (

x)

=m

.[lin

ear

funct

ion]

3f(x

)=

xn⇒

f′ (

x)

=nx

n−1

.[p

ower

funct

ion]

4g(x

)=

cf(x

)⇒

g′ (

x)

=cf

′ (x).

5h(x

)=

f(x

)+g(x

)⇒

h′ (

x)

=f′ (

x)+

g′ (

x).

6h(x

)=

f(x

)g(x

)⇒

h′ (

x)

=f′ (

x)g

(x)+

f(x

)g′ (

x).

[pro

duct

rule

]

7h(x

)=

f(x

)g(x

)⇒

h′ (

x)

=f′ (

x)g

(x)−

f(x

)g′ (

x)

[g(x

)]2

,pr

ovid

edg(x

)�=

0.

[quot

ient

rule

]

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Applica

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/47

ASim

ple

Exa

mpl

e

Con

sider

the

follo

win

g:y

=f(x

)=

αx−2

.W

eca

nal

sow

rite

this

asy

=f(x

)=

α x2.

Inth

efirs

tca

sewe

can

use

the

pow

erru

le.

Inth

ese

cond

case

,we

use

the

quot

ient

rule

.

Cas

e1:

y′ =

0·x

−2+

(−2)x

−2−1

·α=

−2α

/x

3<

0.

Cas

e2:

y′ =

0·x2

−α(2

x)

(x2)2

=−2

α/x

3<

0.

Inei

ther

case

we

get

the

sam

ere

sult.

As

xrise

s,y

lower

s.W

eal

sokn

ow,w

hen

x=

0,y

=0

and

when

x>

0,y

>0

and

when

x<

0,y

<0.

Inte

rms

ofw

hic

hru

leyo

uuse

,it

isbes

tto

use

the

sim

ple

ston

e(c

ase

1in

this

exam

ple

).

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iable

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Applica

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17

/47

Sim

ple

Eco

nom

icExa

mpl

es

Con

sider

the

product

ion

funct

ion,y

=f

(L).

Ass

um

eth

efo

llow

ing

funct

ional

form

:y

=12

L2−

L3.

By

(3),

(4)

and

(5),

dy dL

=24

L−

3L2

An

alte

rnat

ive

form

is,y

=6L

(3L

2+

4).

1W

eca

nuse

(6)

alon

gw

ith

(1),

(3),

(4)

and

(5).

dy dL

=6(3

L2+

4)+

6L(6

L)

=54

L2+

24

2A

lter

nat

ivel

y,not

eth

aty

=18

L3+

24L.

Now

apply

rule

s(3

),(4

)an

d(5

)to

obta

inth

esa

me

resu

ltas

abov

e.

Do

we

expec

tou

tput

togo

up

ordow

nas

we

use

mor

ela

bor

?Pro

bab

lyup.

This

assu

mption

rule

sou

tce

rtai

nfu

nct

ional

form

s.T

hat

is,we

thin

k,dy

dL

>0.

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iable

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Applica

tions

18

/47

Ano

ther

Eco

nom

icExa

mpl

e

Con

sider

cost

and

reve

nue

funct

ions.

We

know

the

profi

tsof

afirm

are

give

nby

reve

nue

min

us

cost

s.

Π(y

)=

R(y

)−c(y

)

What

wou

ldwe

expec

tth

ere

venue

and

cost

funct

ions

tolo

oklik

e?Pro

bab

lyth

efo

llow

ing: R

′ (y)

>0

and

c′ (

y)

>0

As

we

produce

mor

e,we

mak

em

ore

reve

nue.

How

ever

,as

we

produce

mor

e,co

sts

rise

.

Cle

arly,th

ere

isa

trad

e-off

.At

som

epoi

nt

mar

ginal

reve

nue

(R′ (

y))

will

equal

mar

ginal

cost

(c′ (

y))

.T

hat

isth

epr

ofit

max

imiz

ing

conditio

n.

We

will

revi

sit

this

idea

repea

tedly.

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ublin)

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gle

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iable

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ulu

sand

Applica

tions

19

/47

Cha

inRul

e

Suppos

enow

that

yis

afu

nct

ion

ofu

and

uitse

lfis

afu

nct

ion

ofx.

Such

afu

nct

ion

isca

lled

aco

mpos

ite

funct

ion.

As

anex

ample

,co

nsider

y=

(x2+

1)2

.H

ere,

u=

x2+

1an

dy

=u

2.

The

chai

nru

leis

use

dto

dea

lw

ith

such

case

s.Suppos

ewe

hav

ey

=f(u

)w

her

eu

=g(x

),so

that

y=

f(g

(x))

.T

he

chai

nru

leis

give

nby

,dy

dx

=f′ (

u)g

′ (x)

This

says

that

the

der

ivat

ive

ofy

with

resp

ect

tox

iseq

ual

toth

eder

ivat

ive

ofy

with

resp

ect

tou

tim

esth

eder

ivat

ive

ofy

with

resp

ect

tox.

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20

/47

Pro

ofof

the

Cha

inRul

e

Let

y=

f(g

(x))

.N

owle

tu

=g(x

)so

that

y=

f(u

).U

sing

the

tota

ldiff

eren

tial

,we

hav

e

dy

=f′ (

u)d

u(1

)

Sin

ceu

=g(x

),we

can

use

the

tota

ldiff

eren

tial

her

eto

get

du

=g′ (

x)d

x(2

)

Use

(2)

tosu

bst

itute

for

du

in(1

):

dy

=f′ (

u)d

u=

f′ (

u)[ g

′ (x)d

x] =

f′ (

u)g

′ (x)d

x(3

)

This

implie

sth

atdy

/dx

=f′ (

u)g

′ (x).

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21

/47

Exa

mpl

esus

ing

the

Cha

inRul

e

Inth

eab

ove

case

,we

hav

eu

=g(x

)=

x2+

1an

df(u

)=

u2.

Then

,f′ (

u)

=2u

and

g′ (

x)

=2x

,an

dby

the

chai

nru

le,we

hav

e,

dy

dx

=2u

×2x

=4x

(x2+

1)

Anot

her

exam

ple

:su

ppos

ey

=(x

1/2+

3x2)5

.T

hen

(show

for

yours

elf)

,

dy

dx

=5(x

1/2+

3x2)4

×[ (1

/2)x

−1/2+

6x]

Not

ice

the

diff

eren

ce.

Inth

efirs

tca

se,we

can

multip

lyou

ty

=(x

2+

1)2

=x

4+

2x2+

1an

duse

the

usu

alru

les.

Inth

ese

cond

case

,we

can

do

that

,but

itis

not

ago

od

idea

asit

bec

omes

very

mes

sy.

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22

/47

Eco

nom

icExa

mpl

e

Aga

in,co

nsider

the

tota

lre

venue

ofa

firm

,R

.It

also

dep

ends

onth

ete

chnol

ogy

(i.e

.,th

esh

ape

ofth

epr

oduct

ion

funct

ion)

and

labor

input.

That

is,

R=

k(y

);y

=f

(L)

⇒R

=k

(f(L

))

The

der

ivat

ive

ofth

isfu

nct

ion

is:

dR dl

=dR dy

df

dL

=k′ (

y)f

′ (L)

Eco

nom

icm

eanin

g:dR dy

ism

argi

nal

reve

nue

(as

abov

e).

df

dL

isth

em

argi

nal

product

ofla

bor

.

We

concl

ude

that

the

MRPL

iseq

ual

toth

eM

Rtim

esth

eM

PL.

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23

/47

Der

ivat

ive

ofth

eIn

vers

eFun

ctio

n

Con

sider

the

funct

ions

y=

f(x

)=

x2

and

x=

g(y

)=

√ y.

We

can

thin

kof

thes

etw

ofu

nct

ions

asin

vers

es.

Ifwe

take

xas

the

input,

apply

fto

itan

dth

enpas

sth

isou

tput

thro

ugh

the

funct

ion

g,we

get

bac

kx.

The

der

ivat

ives

ofin

vers

efu

nct

ions

are

rela

ted

toea

chot

her

.

Ify

=f(x

)an

dx

=g(y

)ar

ein

vers

efu

nct

ions,

we

hav

e,

g′ (

y)

=1 dy

dx

=1

f′ (

x)

Inth

eab

ove

case

,f′ (

x)

=2x

.T

her

efor

e,

g′ (

y)

=1

f′ (

x)

=1 2x

=1

2√ y

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24

/47

Exp

onen

tial

and

Log

arithm

icFun

ctio

nsI

Con

sider

the

follo

win

gfu

nct

ion: y

=b

x

yis

the

dep

enden

tva

riab

le,x

isin

dep

enden

t,an

db

isth

e(fi

xed)

bas

eof

the

expon

ent.

We

can

chan

geth

ebas

e.A

com

mon

bas

eis

e=

2.71

828.

...

This

isca

lled

the

nat

ura

lbas

e.

Why

do

we

care

abou

tth

is?

Bec

ause

ex

isit’s

own

der

ivat

ive

itis

easy

towor

kw

ith.

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25

/47

Exp

onen

tial

and

Log

arithm

icFun

ctio

nsII

Now

consider

the

rela

tion

ship

bet

wee

n8

and

64.

Cle

arly,82

=64

.

The

expon

ent

2is

the

loga

rith

mof

64to

the

bas

e8.

That

is,

log8(6

4)

=2.

Lik

ewise,

ifwe

hav

ey

=b

xth

enx

=lo

gb(y

).W

eal

sonot

eth

atif

y=

ex

then

x=

log e

(y)≡

ln(y

).A

gain

,th

ere

isa

use

fulre

sult,as

dln

(x)/

dx

=1

/x.

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26

/47

Rul

esfo

rExp

onen

tial

and

Log

arithm

icFun

ctio

ns

Log

arithm

ican

dex

pon

ential

funct

ions

are

very

com

mon

inec

onom

ics.

The

bas

icru

les

for

diff

eren

tiat

ing

expon

ential

and

loga

rith

mic

funct

ions

are

the

follo

win

g.

1f(x

)=

lnx⇒

f′ (

x)

=1

/x

2f(x

)=

lnu(x

)⇒

f′ (

x)

=u′ (

x)/

u(x

)3

f(x

)=

ex⇒

f′ (

x)

=ex

4f(x

)=

eu(x

)⇒

f′ (

x)

=eu(x

) u′ (

x)

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27

/47

Exa

mpl

esof

Fun

ctio

ns

Som

est

raig

htf

orwar

dex

ample

sof

expon

ential

and

loga

rith

mic

funct

ions

are:

f(x

)=

ln(x

α)⇒

f′ (

x)

=αx

(1/x

α)

=α

/x

f(x

)=

ln(x

2+

3x+

1)⇒

f′ (

x)

=(2

x+

3)/

(x2+

3x+

1)

f(x

)=

e5x⇒

f′ (

x)

=5e

5x

f(x

)=

5ex

2⇒

f′ (

x)

=10

xex

2

You

shou

ldbe

com

fort

able

with

allof

thes

e.

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28

/47

Log

arithm

icD

iffer

entiat

ion

Suppos

ewe

wan

tto

diff

eren

tiat

ey

=( x

2) x2

.W

epr

oce

edas

follo

ws:

1Tak

elo

gson

bot

hsides

: lny

=x

2ln

x2

=2x

2ln

x

2D

iffer

entiat

ebot

hsides

with

resp

ect

tox.

On

the

left

-hand

side,

byth

ech

ain

rule

,we

get

(1/y)(

dy

/dx).

On

the

right-

hand

side,

byth

epr

oduct

rule

,we

hav

e4x

lnx

+2x

.A

fter

re-a

rran

ging,

we

obta

in

dy

dx

=y(4

xln

x+

2x)

=(x

2)x

2[4

xln

x+

2x]

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29

/47

Eco

nom

icExa

mpl

e:Ela

stic

ity

ofD

eman

d

Suppos

ewe

hav

ea

dem

and

funct

ion

q=

D(p

).W

efe

elhap

pyas

sum

ing

D′ (

p)

<0;

that

is,th

edem

and

curv

eslop

esdow

nwar

ds.

How

ever

,how

does

quan

tity

chan

geas

pric

ech

ange

sal

ong

the

curv

e?

The

elas

tici

tyof

dem

and

isdefi

ned

as,

ε pq≡

dq q dp p

=p q

dq

dp

Itis

inte

rest

ing

tonot

eth

atwe

can

repr

esen

tth

eel

astici

tyof

dem

and

ina

mor

eco

nci

sem

anner

as ε pq

=d(ln[q

])d(ln[p

])

Thes

eex

pres

sion

sar

eeq

uiv

alen

t.

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Applica

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30

/47

Eco

nom

icExa

mpl

e:Ela

stic

ity

ofD

eman

d

To

see

this,not

eth

atby

using

the

chai

nru

le,we

hav

e

dln

[q]

dln

[p]

=d

ln[q

]q

×dq

dln

[p]

=d

ln[q

]dq

×dq

dp×

dp

dln

[p]

Sin

ced

ln[q

]/dq

=1

/q

and

dp

/d

ln[p

]=p

(by

the

inve

rse

rule

),th

ere

sult

follo

ws.

The

alte

rnat

ive

form

ula

for

the

elas

tici

tyca

nbe

use

fulin

som

eci

rcum

stan

ces.

Suppos

ewe

hav

eth

edem

and

funct

ion

q=

pa

wher

ea

<0

isa

const

ant.

(Why

must

we

hav

ea

<0?

)

We

can

com

pute

the

elas

tici

tyve

ryea

sily

asfo

llow

s.Sin

ceq

=p

a,

we

hav

eln

q=

aln

p.

Thus

dln

[q]/

dln

[p]=

ash

owin

gth

atth

eel

astici

tyof

dem

and

isa

const

ant

ever

ywher

e.

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31

/47

Hig

her

Ord

erD

eriv

ativ

es

We

call

f′ (

x)

the

firs

tder

ivat

ive

ofth

efu

nct

ion

f.

Sin

ceth

eder

ivat

ive

itse

lfis

afu

nct

ion,we

can

take

it’s

der

ivat

ive

–th

isis

calle

dth

ese

cond

der

ivat

ive

and

den

oted

d2y

dx

2or

f′′ (

x).

For

mal

ly,

d2y

dx

2=

d(d

ydx)

dx

Suppos

ef(x

)=

x5.

Then

,f′ (

x)

=5x

4an

df′′ (

x)

=20

x3.

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32

/47

Hig

her

Ord

erD

eriv

ativ

es

Sin

ceth

ese

cond

der

ivat

ive

isal

soa

funct

ion,we

can

also

take

its

der

ivat

ive.

This

isca

lled

the

third

der

ivat

ive

and

den

oted

f′′′

(x)

toin

dic

ate

that

this

funct

ion

isfo

und

byth

ree

succ

essive

oper

atio

ns

ofdiff

eren

tiat

ion,

star

ting

with

the

funct

ion

f.

One

can

continue

this

proce

ssbut

we

will

typic

ally

not

gobey

ond

the

seco

nd

der

ivat

ive.

Exa

mple

continued

:Suppos

ef(x

)=

x5.

Then

,f′ (

x)

=5x

4,f

′′ (x)

=20

x3,f

′′′(x

)=

60x

2.

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33

/47

Con

cave

and

Con

vex

Fun

ctio

ns

Afu

nct

ion

f(x

)is

calle

dco

nca

veif

f′′ (

x)≤

0at

allpoi

nts

ofits

dom

ain.

An

econ

omic

exam

ple

ofa

conca

vefu

nct

ion

isth

epr

oduct

ion

funct

ion

(usu

ally

).

Afu

nct

ion

f(x

)is

calle

dco

nve

xif

f′′ (

x)≥

0at

allpoi

nts

ofits

dom

ain.

Ifth

ein

equal

itie

sar

est

rict

,th

enth

efu

nct

ion

isca

lled

strict

lyco

nca

veor

strict

lyco

nve

x.

Not

e:Con

cavi

tyan

dco

nve

xity

do

not

say

anyt

hin

gab

out

the

firs

tder

ivat

ive

(incr

easing

ordec

reas

ing

funct

ions)

.

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34

/47

Exa

mpl

esof

Con

cavi

tyan

dCon

vexi

ty

The

funct

ion

f(x

)=

x2

isco

nve

xon

the

dom

ain

x≥

0bec

ause

f′′ (

x)

=2

>0

for

allx≥

0.

The

funct

ion

g(x

)=

lnx

isco

nca

veon

the

dom

ain

x>

0bec

ause

g′′ (

x)

=−1

/x

2<

0fo

ral

lx

>0.

Afu

nct

ion

may

be

eith

erco

nca

venor

conve

xon

agi

ven

dom

ain.

For

exam

ple

,co

nsider

f(x

)=

−(2

/3)x

3+

10x

2+

5xon

the

dom

ain

x≥

0.

Inth

isca

se,f′′ (

x)

=−4

x+

20;so

,

1f′′ (

x)≥

0if

x≤

5(c

onve

x)2

f′′ (

x)

<0

ifx

>5

(con

cave

)

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Applica

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35

/47

Eco

nom

icExa

mpl

esof

Con

cavi

tyan

dCon

vexi

ty

Con

sider

the

follo

win

gpr

oduct

ion

funct

ion:

y=

lα.

We

know

,y′ =

αlα−1

>0.

But

y′′

=(α

−1)α

lα−2

.T

his

ison

lypos

itiv

ew

hen

α<

1.T

hat

isco

nsist

ent

with

dim

inishin

gm

argi

nal

retu

rns

tola

bor

.

Con

sider

the

reve

nue

and

cost

sfu

nct

ions

from

bef

ore.

Now

we

mig

ht

suppos

eth

efo

llow

ing.

R′ (

y)

>0;

R′′(y

)<

0;c′ (

y)

>0;

c(y

)′′>

0

As

we

produce

mor

e,we

mak

em

ore

reve

nue,

but

aspr

oduct

ion

rise

sev

erhig

her

,th

ead

ditio

nal

unit

gener

ate

less

reve

nue

than

the

last

.

How

ever

,as

we

produce

mor

e,co

sts

rise

,an

das

we

continue

topr

oduce

,ea

chex

tra

unit

ism

ore

cost

lyth

atth

ela

st.

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Applica

tions

36

/47

Intr

odu

ctio

nto

Unc

onst

rain

edO

ptim

izat

ion

We

can

also

rela

tese

cond-o

rder

conditio

ns

tonec

essa

ryan

dsu

ffici

ent

conditio

ns

for

max

imiz

atio

npr

oble

ms.

That

is,to

find

the

max

imum

,it

isnot

enou

ghto

“set

the

firs

tder

ivat

ive

toze

ro”.

We

nee

dto

chec

kse

cond

order

conditio

ns

explic

itly.

Two

fam

iliar

exam

ple

s:

1Suppos

eth

ata

mon

opol

ist

face

sa

dem

and

curv

ep

=10

0−

qan

dhas

product

ion

cost

sgi

ven

byc(q

)=

q2.

Ifth

em

onop

olist

isin

tere

sted

inm

axim

izin

gpr

ofits

,how

much

shou

ldhe

produce

?

2Suppos

eth

ata

firm

has

fixe

dco

stof

100

and

variab

leco

sts

give

nby

v(q

)=

q2.

Ifth

efirm

wan

tsto

min

imiz

eav

erag

eco

sts

ofpr

oduct

ion,

how

much

shou

ldit

produce

?

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iable

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tions

37

/47

Nec

essa

ryCon

dition

sfo

rO

ptim

izat

ion

Ifa

diff

eren

tiab

lefu

nct

ion

f(x

)re

aches

its

max

imum

orm

inim

um

atx∗

then

f′ (

x∗ )

=0.

Rea

son:

Con

sider

the

tota

ldiff

eren

tial

dy

=f′ (

x)d

x

Ifth

efu

nct

ion

reac

hes

am

axim

um

orm

inim

um

atx∗

then

itm

ust

be

impos

sible

toin

crea

seor

dec

reas

eth

eva

lue

ofth

efu

nct

ion

bysm

all

chan

ges

inx.

How

ever

,if

f′ (

x∗ )

�=0,

then

itis

alway

spos

sible

tom

ake

dy

larg

eror

smal

lerby

mak

ing

(sm

all)

appr

opriat

ech

ange

sin

x.

Ther

efor

e,we

must

hav

ef′ (

x∗ )

=0

ata

max

imum

orm

inim

um

.

This

isca

lled

anec

essa

ryco

nditio

nbec

ause

itca

nnot

guar

ante

eth

atx

isin

dee

da

max

imum

orm

inim

um

.It

isen

tire

lypos

sible

that

f′ (

x∗ )

=0

but

x∗

isnei

ther

am

axim

um

nor

am

inim

um

.

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

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gle

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iable

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ulu

sand

Applica

tions

38

/47

Glo

balan

dLoca

lM

axim

a

Apoi

nt

x∗

isca

lled

agl

obal

max

imum

ofth

efu

nct

ion

f(x

)if

f(x

∗ )≥

f(x

)fo

ral

lx

inth

edom

ain

off.

Apoi

nt

x∗

isca

lled

alo

calm

axim

um

ofth

efu

nct

ion

f(x

)if

ther

eis

a“s

mal

lin

terv

al”

cente

red

atx∗

such

that

f(x

∗ )≥

f(x

)fo

ral

lx

inth

esm

allin

terv

al.

Not

eth

enec

essa

ryco

nditio

non

lyid

entifies

alo

calm

axim

um

orm

inim

um

,but

not

agl

obal

max

imum

orm

inim

um

.

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

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iable

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ulu

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Applica

tions

39

/47

Sec

ond

Ord

eror

Suffi

cien

tCon

dition

sfo

rO

ptim

izat

ion

We

also

hav

eth

efo

llow

ing

conditio

ns:

1If

f′ (

x∗ )

=0

and

f′′ (

x∗ )

<0,

then

x∗

isa

loca

lm

axim

um

off(x

).2

Iff′ (

x∗ )

=0

and

f′ (

x∗ )

>0,

then

x∗

isa

loca

lm

inim

um

off(x

).

Not

eth

atit

ispos

sible

that

apoi

nt

may

be

am

axim

um

orm

inim

um

and

yet

not

satisf

yth

esu

ffici

ent

conditio

nfo

rm

axim

izat

ion

orm

inim

izat

ion.

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

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gle

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iable

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ulu

sand

Applica

tions

40

/47

Eco

nom

icExa

mpl

e

We

now

retu

rnto

the

econ

omic

exam

ple

sab

ove.

Suppos

eth

em

onop

olist’s

profi

tfu

nct

ion

isgi

ven

by,

Π(y

)=

py−

c(y

)=

(100

−y)y

−y

2

The

nec

essa

ryco

nditio

nim

plie

sth

atΠ

′ (y)

=10

0−

2y−

2y=

0or

y=

25.

The

firm

’sav

erag

eco

stis

give

nby

AC

(y)

=10

0/y

+y.

The

nec

essa

ryco

nditio

nAC

′ (y)

=0

give

s−1

00/y

2+

1=

0or

y2−

100

=0.

Ther

efor

e,y

=10

ory

=−1

0.Sin

ceq

=−1

0is

not

pos

sible

,we

must

hav

ey

=10

.

To

chec

kth

atth

ese

are

indee

dth

em

axim

um

and

min

imum

,we

can

eith

erplo

tth

etw

ofu

nct

ions,

orch

eck

the

seco

nd

order

conditio

ns

for

optim

izat

ion.

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

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gle

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iable

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ulu

sand

Applica

tions

41

/47

Eco

nom

icExa

mpl

eCon

tinu

ed...

Inth

em

onop

olist’s

prob

lem

,not

eth

atth

ese

cond

der

ivat

ive

Π′′ (

y)

=−4

for

ally,(in

par

ticu

lar,

fory

=25

)an

dhen

cey

=25

isa

loca

lm

axim

um

.In

the

firm

’spr

oble

m,AC

′′ (y)

=20

0/q

3;

subst

ituting

y=

10gi

ves

AC

′′ (10

)=

200

/10

00=

1/5

>0.

Hen

ce,

y=

10is

alo

calm

inim

um

.

We

do

not

know

whet

her

thes

ear

elo

calor

glob

alm

inim

a.T

he

impor

tance

ofco

nca

vean

dco

nve

xfu

nct

ions

stem

from

the

fact

that

the

loca

lm

axim

aof

conca

vefu

nct

ions

are

also

glob

alm

axim

a;th

elo

calm

inim

aof

conve

xfu

nct

ions

are

also

glob

alm

inim

a.

Inth

em

onop

olist’s

prob

lem

,th

epr

ofit

funct

ion

isco

nca

ve;hen

cey

=25

isa

glob

alm

axim

um

.In

the

firm

’spr

oble

m,th

eAC

funct

ion

isco

nve

xon

the

dom

ain

qy

>0

and

hen

ceit

isa

glob

alm

inim

um

.

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

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gle

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iable

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ulu

sand

Applica

tions

42

/47

Ano

ther

Exa

mpl

e:Sup

ply,

Dem

and,

and

Tax

es

Suppos

ewe

hav

ea

mar

ket

with

dem

and

funct

ion

qd

=a 0

−a 1

pan

dsu

pply

funct

ion

qs=

b0+

b1p.

The

gove

rnm

ent

wan

tsto

impos

ean

exci

seta

xt

onth

ism

arke

t.W

hat

rate

shou

ldth

ego

vern

men

tch

oos

eif

itwan

tsto

max

imiz

eta

xre

venue?

Withou

tta

xation

,m

arke

teq

uili

briu

mis

det

erm

ined

byth

eco

nditio

nq

d=

qs.

Thus,

a 0−

a 1p

=b

0+

b1p,w

hic

hgi

ves,

p∗

=a 0

−b

0

a 1+

b1

and

q∗

=a 0

b1+

a 1b

0

a 1+

b1

When

ata

xt

isim

pos

ed,th

eeq

uili

briu

mco

nditio

ns

require

that

(i)

qd

=q

san

d(ii)

pb

=p

s+

t.

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

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gle

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iable

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ulu

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Applica

tions

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/47

Sup

ply

and

Dem

and

Con

tinu

ed...

Equili

briu

mnow

requires

a 0−

a 1p

b=

b0+

b1(p

b−

t).

This

give

s

p∗ b

=a 0

−b

0+

b1t

a 1+

b1

,p∗ s

=a 0

−b

0−

a 1t

a 1+

b1

,q∗ t

=a 0

b1+

a 1b

0−

a 1b

1t

a 1+

b1

The

tax

reve

nue

asa

funct

ion

ofth

eta

xra

tet

is:

T(t

)=

tq∗ t

=t

[ a 0b

1+

a 1b

0−

a 1b

1t

a 1+

b1

]

We

wan

tto

find

tto

max

imiz

eT

(t).

First

,we

iden

tify

tsu

chth

atdT

/dt

=0.

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yCooke

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ublin)

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gle

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iable

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ulu

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tions

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/47

Sup

ply

and

Dem

and

Con

tinu

ed...

Diff

eren

tiat

ing

T(t

)an

deq

uat

ing

toze

rogi

ves

dT dt

=[ a 0

b1+

a 1b

0−

2a1b

1t

a 1+

b1

] =0

This

give

sus

auniq

ue

valu

eof

t:

t∗=

a 0b

1+

a 1b

0

2a1b

1

We

now

hav

eto

chec

kw

het

her

t∗co

rres

pon

ds

toa

max

imum

ornot

.W

ehav

e

T′′ (

t)=

−2a

1b

1

a 1+

b1

Thus,

the

seco

nd

der

ivat

ive

isa

const

ant,

and

since

a 1>

0,b

1>

0it

isal

way

sneg

ativ

e.T

his

show

sth

atth

eta

xre

venue

funct

ion

T(t

)is

conca

ve,an

dhen

cet∗

isnot

only

alo

calbut

also

agl

obal

max

imum

.

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

Sin

gle

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iable

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ulu

sand

Applica

tions

45

/47

Rem

ark

onth

eN

eces

sary

Con

dition

for

Opt

imiz

atio

n

The

conditio

nf′ (

x)

=0

ata

max

imum

orm

inim

um

isva

lidon

lyif

xis

inth

e“i

nte

rior

”of

the

dom

ain

ofth

efu

nct

ion.

Bas

ical

ly,if

we

look

atth

ear

gum

ent

give

nfo

rsh

owin

gth

atif

x∗

isa

max

imum

orm

inim

um

,th

enf′ (

x)

=0,

itsh

ould

be

not

iced

that

the

argu

men

tdep

ended

onth

eab

ility

tom

ake

smal

lch

ange

sin

x.

How

ever

,at

a“b

oundar

ypoi

nt”

we

cannot

mak

ece

rtai

nch

ange

s.For

inst

ance

,if

the

funct

ion

isdefi

ned

for

allx

inth

ein

terv

al[a

,b],

then

ata,

we

can

only

incr

ease

x;sim

ilarly

atb,we

can

only

dec

reas

ex.

Thus,

the

argu

men

tth

atf′ (

x)

=0

isnot

true

ifth

em

axim

um

occ

urs

ata

orb.

We

can

modify

the

nec

essa

ryco

nditio

nf′ (

x)

=0

toac

count

for

thes

e“b

oundar

ypoi

nts

”but

we

will

dea

lw

ith

such

situ

atio

ns

when

we

discu

ssco

nst

rain

edop

tim

izat

ion.

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

Sin

gle

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iable

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ulu

sand

Applica

tions

46

/47

Rou

ndup

You

shou

ldnow

be

able

todo

the

follo

win

g:

1D

iffer

entiat

efu

nct

ions

(inc.

log

and

exp.)

.

2Chec

kfo

rco

nca

vity

/con

vexi

ty.

3A

pply

thes

eid

eas

tom

icro

econ

omic

prob

lem

s.

4U

nder

stan

dth

eim

por

tance

ofnec

essa

ryve

rsus

suffi

cien

tco

nditio

ns.

Dudle

yCooke

(Trinity

Colleg

eD

ublin)

Sin

gle

Var

iable

Calc

ulu

sand

Applica

tions

47

/47