Multi-variable Calculus and Optimization · 2019. 1. 3. · Multi-variable Calculus and...
26
Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus Reading 1 Chapter 7.4-7.6, 8 and 11 (section 6 has lots of economic examples) of CW 2 Chapters 14, 15 and 16 of PR Plan 1 Partial differentiation and the Jacobian matrix 2 The Hessian matrix and concavity and convexity of a function 3 Optimization (profit maximization) 4 Implicit functions (indifference curves and comparative statics) Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 2 / 51
Transcript of Multi-variable Calculus and Optimization · 2019. 1. 3. · Multi-variable Calculus and...
-
Mul
ti-v
aria
ble
Cal
culu
san
dO
ptim
izat
ion
Dudle
yCook
e
Trinity
Colleg
eD
ublin
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
1/
51
EC20
40Top
ic3
-M
ulti-v
aria
ble
Cal
culu
s
Rea
din
g
1Chap
ter
7.4-
7.6,
8an
d11
(sec
tion
6has
lots
ofec
onom
icex
ample
s)of
CW
2Chap
ters
14,15
and
16of
PR
Pla
n
1Par
tial
diff
eren
tiat
ion
and
the
Jaco
bia
nm
atrix
2T
he
Hes
sian
mat
rix
and
conca
vity
and
conve
xity
ofa
funct
ion
3O
ptim
izat
ion
(pro
fit
max
imiz
atio
n)
4Im
plic
itfu
nct
ions
(indiff
eren
cecu
rves
and
com
par
ativ
est
atic
s)
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
2/
51
-
Mul
ti-v
aria
ble
Cal
culu
s
Funct
ions
wher
eth
ein
put
consist
sof
man
yva
riab
les
are
com
mon
inec
onom
ics.
For
exam
ple
,a
consu
mer
’sutilit
yis
afu
nct
ion
ofal
lth
ego
ods
she
consu
mes
.So
ifth
ere
are
ngo
ods,
then
her
wel
l-bei
ng
orutilit
yis
afu
nct
ion
ofth
equan
tities
(c1,.
..,c
n)
she
consu
mes
ofth
en
goods.
We
repr
esen
tth
isby
writing
U=
u(c
1,.
..,c
n)
Anot
her
exam
ple
isw
hen
afirm
’spr
oduct
ion
isa
funct
ion
ofth
equan
tities
ofal
lth
ein
puts
ituse
s.So,
if(x
1,.
..,x
n)
are
the
quan
tities
ofth
ein
puts
use
dby
the
firm
and
yis
the
leve
lof
outp
ut
produce
d,th
enwe
hav
ey
=f(x
1,.
..,x
n).
We
wan
tto
exte
nd
the
calc
ulu
sto
ols
studie
dea
rlie
rto
such
funct
ions.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
3/
51
Par
tial
Der
ivat
ives
Con
sider
afu
nct
ion
y=
f(x
1,.
..,x
n).
The
par
tial
der
ivat
ive
off
with
resp
ectto
x iis
the
der
ivat
ive
off
with
resp
ect
tox i
trea
ting
allot
her
variab
les
asco
nst
ants
and
isden
oted
,
∂f ∂xi
orf x
i
Con
sider
the
follo
win
gfu
nct
ion:
y=
f(u
,v)
=(u
+4)(
6u+
v).
We
can
apply
the
usu
alru
les
ofdiff
eren
tiat
ion,now
with
two
pos
sibili
ties
:
∂f ∂u=
1(6
u+
v)+
6(u
+4)
=12
u+
v+
24
∂f ∂v=
0(6
u+
v)+
1(u
+4)
=u
+4
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
4/
51
-
Par
tial
Der
ivat
ives
-Eco
nom
icExa
mpl
es
Aga
in,su
ppos
ey
=f(x
1,.
..,x
n).
Ass
um
en
=2
and
x 1=
Kan
dx 2
=L;th
atis,y
=f(K
,L).
Then
assu
me
the
follo
win
gfu
nct
ional
form
:f(K
,L)
=K
0.2
5L
0.7
5;
i.e.
,a
Cob
b-D
ougl
aspr
oduct
ion
funct
ion.
The
par
tial
der
ivat
ives
are
give
nby
,
∂f ∂K=
0.25
K−0
.75L
0.7
5,
∂f ∂L=
0.75
K0.2
5L−0
.25
Alter
nat
ivel
y;su
ppos
eutilit
yis
U=
cα ac
1−α
b,w
her
ea
=ap
ple
s,b
=ban
anas
.W
efind:
∂U ∂ca
=α
( c b c a) 1−
α
,∂U ∂c
b=
(1−
α)( c
a c b
) αSo,
for
agi
ven
labor
input,
mor
eca
pital
raises
outp
ut.
For
give
nco
nsu
mption
ofap
ple
s,co
nsu
min
gm
ore
ban
anas
mak
esyo
uhap
pie
r.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
5/
51
Inte
rpre
tation
ofPar
tial
Der
ivat
ives
Mat
hem
atic
ally,th
epar
tial
der
ivat
ive
off
with
resp
ect
tox i
tells
us
the
rate
ofch
ange
when
only
the
variab
lex i
isal
lowed
toch
ange
.
Eco
nom
ical
ly,th
epar
tial
der
ivat
ives
give
us
use
fulin
form
atio
n.
Inge
ner
alte
rms:
1W
ith
apr
oduct
ion
funct
ion,th
epar
tial
der
ivat
ive
with
resp
ect
toth
ein
put,
x i,te
llsus
the
mar
ginal
product
ivity
ofth
atfa
ctor
,or
the
rate
atw
hic
had
ditio
nal
outp
ut
can
be
produce
dby
incr
easing
x i,hol
din
got
her
fact
ors
const
ant.
2W
ith
autilit
yfu
nct
ion,th
epar
tial
der
ivat
ive
with
resp
ect
togo
od
c ite
llsus
the
rate
atw
hic
hth
eco
nsu
mer
’swel
lbei
ng
incr
ease
sw
hen
she
consu
mes
additio
nal
amou
nts
ofc i
hol
din
gco
nst
ant
her
consu
mption
ofot
her
goods.
I.e.
,th
em
argi
nal
utilit
yof
that
good.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
6/
51
-
Der
ivat
ive
ofa
Lin
ear
Fun
ctio
n
As
with
funct
ions
ofon
eva
riab
le,to
gain
som
ein
tuitio
n,we
star
tw
ith
funct
ions
that
are
linea
r.
We
mot
ivat
edth
enot
ion
ofa
der
ivat
ive
bysa
ying
that
itwas
the
slop
eof
the
line
whic
h“l
ook
edlik
eth
efu
nct
ion
arou
nd
the
poi
nt
x 0.”
When
we
hav
en
variab
les,
the
nat
ura
lnot
ion
ofa
“lin
e”is
give
nby
y=
a 0+
a 1x 1
+..
.+a n
x n
When
ther
ear
etw
ova
riab
les,
x 1an
dx 2
,th
efu
nct
ion
y=
a 0+
a 1x 1
+a 2
x 2is
the
equat
ion
ofa
pla
ne.
Inge
ner
al,th
efu
nct
ion
y=
a 0+
a 1x 1
+..
.+a n
x nis
refe
rred
tosim
ply
asth
eeq
uat
ion
ofa
pla
ne.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
7/
51
Two
Var
iabl
eLin
ear
Fun
ctio
n
Con
sider
the
pla
ne
y=
a 0+
a 1x 1
+a 2
x 2.
How
does
the
funct
ion
beh
ave
when
we
chan
gex 1
and
x 2?
Cle
arly,if
dx 1
and
dx 2
are
the
amou
nts
byw
hic
hwe
chan
gex 1
and
x 2,we
hav
e,
dy
=a 1
dx 1
+a 2
dx 2
Not
efu
rther
mor
eth
atth
epar
tial
sar
e,
∂y ∂x1
=a 1
and
∂y ∂x2
=a 2
We
can
then
write
,
dy
=∂y ∂x
1dx 1
+∂y ∂x
2dx 2
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
8/
51
-
Jaco
bian
Mat
rix
We
can
take
this
resu
lt(i.e
.,th
atdy
=∂y ∂x
1dx 1
+∂y ∂x
2dx 2
)an
dre
cast
itin
mat
rix
(her
e,ve
ctor
)fo
rm.
dy
=[ ∂y ∂x
1
∂y ∂x2
]︸
︷︷︸
≡J
[ dx1
dx 2
]
We
call
Jth
eJa
cobia
n.
Tak
eou
rpr
evio
us
exam
ple
.T
he
vect
or[ ∂y ∂x
1
∂y ∂x2
] =[ a 1
a 2] is
what
we
can
thin
kof
asth
eder
ivat
ive
ofth
epla
ne.
This
vect
orte
llsus
the
rate
sof
chan
gein
the
direc
tion
sx 1
and
x 2.
The
tota
lch
ange
isth
eref
ore
com
pute
das
a 1dx 1
+a 2
dx 2
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
9/
51
Gen
eral
Fun
ctio
ns
Now
consider
am
ore
gener
altw
ova
riab
lefu
nct
ion,y
=f(x
1,x
2).
With
age
ner
alfu
nct
ion
y=
f(x
1,x
2),
the
idea
isto
find
apla
ne
whic
hlo
oks
loca
llylik
eth
efu
nct
ion
arou
nd
the
poi
nt
(x1,x
2).
Sin
ceth
epar
tial
der
ivat
ives
give
the
rate
sof
chan
gein
x 1an
dx 2
,it
mak
esse
nse
topic
kth
eap
prop
riat
epla
ne
whic
hpas
ses
thro
ugh
the
poi
nt
(x1,x
2)
and
has
slop
es∂f
/∂x
1an
d∂f
/∂x
2in
the
two
direc
tion
s.
The
der
ivat
ive
(or
the
tota
lder
ivat
ive)
ofth
efu
nct
ion
f(x
1,x
2)
at
(x1,x
2)
issim
ply
the
vect
or[ ∂y ∂x
1
∂y ∂x2
] wher
eth
epar
tial
der
ivat
ives
are
eval
uat
edat
the
poi
nt
(x1,x
2).
We
can
inte
rpre
tth
eder
ivat
ive
asth
eslop
esin
the
two
direc
tion
sof
the
pla
ne
whic
hlo
oks
“lik
eth
efu
nct
ion”
arou
nd
the
poi
nt
(x1,x
2).
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
10
/51
-
An
Exa
mpl
e
When
we
hav
ea
funct
ion
y=
f(x
1,.
..,x
n),
the
der
ivat
ive
at
(x1,.
..,x
n)
issim
ply
the
vect
or[ ∂y ∂x
1..
.∂y ∂x
n
] .Tak
ea
mor
eco
ncr
ete
exam
ple
.Suppos
e,
y=
8+
4x2 1+
6x2x 3
+x 3
The
Jaco
bia
nm
ust
be,
[ 8x1
6x3
1] =
[ ∂y ∂x1
∂y ∂x2
∂y ∂x2
]So
we
can
write
,
dy
=[ 8x
16
1]⎡ ⎣
dx 1
dx 2
dx 3
⎤ ⎦
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
11
/51
An
Eco
nom
icExa
mpl
e
Con
sider
the
utilit
yfu
nct
ion
from
bef
ore;
U=
cα ac
1−α
b,w
her
ea
=ap
ple
s,b
=ban
anas
.T
he
Jaco
bia
nis
ave
ctor
ofth
epar
tial
s(w
hic
hwe
alre
ady
com
pute
d).
The
chan
gein
utilit
yis
ther
efor
egi
ven
byth
efo
llow
ing.
dU
=[ ∂U ∂c
a
∂U ∂cb
][dc a
dc b
]
=[ α
( c b c a)1−α
(1−
α)( c
a c b
) α][ dc
a
dc b
]
The
sam
eis
true
for
the
Cob
b-D
ougl
aspr
oduct
ion
funct
ion,
f(K
,L)
=K
0.2
5L
0.7
5.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
12
/51
-
Sec
ond-
Ord
erPar
tial
Der
ivat
ives
We
now
mov
eon
tose
cond-o
rder
par
tial
der
ivat
ives
.O
ur
mot
ivat
ion
will
be
econ
omic
prob
lem
s/in
terp
reta
tion
.
Giv
ena
funct
ion
f(x
1,.
..,x
n),
the
seco
nd-o
rder
der
ivat
ive
∂2f
∂xix
jis
the
par
tial
der
ivat
ive
of∂f ∂x
iw
ith
resp
ect
tox j
.
The
abov
em
aysu
gges
tth
atth
eor
der
inw
hic
hth
eder
ivat
ives
are
take
nm
atte
rs:
inot
her
wor
ds,
the
par
tial
der
ivat
ive
of∂f ∂x
iw
ith
resp
ect
tox j
isdiff
eren
tfrom
the
par
tial
der
ivat
ive
of∂f ∂x
jw
ith
resp
ect
tox i
.
While
this
can
hap
pen
,it
turn
sou
tth
atif
the
funct
ion
f(x
1,.
..,x
n)
iswel
l-beh
aved
then
the
order
ofdiff
eren
tiat
ion
does
not
mat
ter.
This
resu
ltis
calle
dYou
ng’
sT
heo
rem
.
We
shal
lon
lybe
dea
ling
with
wel
l-beh
aved
funct
ions
(i.e
.,You
ng’
sT
heo
rem
will
alway
shol
d).
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
13
/51
Eco
nom
icExa
mpl
e
For
the
product
ion
funct
ion
y=
K0.2
5L
0.7
5,we
can
eval
uat
eth
e
seco
nd-o
rder
par
tial
der
ivat
ive,
∂2y
∂K∂L
,in
two
diff
eren
tway
s.
First
,since
∂y ∂K=
0.25
K−0
.75L
0.7
5,ta
king
the
par
tial
der
ivat
ive
ofth
isw
ith
resp
ect
toL,we
get,
∂ ∂L
∂y ∂K=
3 16K
−0.7
5L−0
.25
And
since
∂f ∂L=
0.75
K0.2
5L−0
.25,we
hav
e,
∂ ∂K
∂y ∂L
=3 16
K−0
.75L−0
.25
This
illust
rate
sYou
ng’
sT
heo
rem
:no
mat
ter
inw
hic
hor
der
we
diff
eren
tiat
e,we
get
the
sam
ean
swer
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
14
/51
-
Eco
nom
icIn
terp
reta
tion
The
seco
nd-o
rder
par
tial
der
ivat
ives
can
be
inte
rpre
ted
econ
omic
ally.
Our
exam
ple
product
ion
funct
ion
isy
=K
0.2
5L
0.7
5.
Ther
efor
e,∂2
y
∂K2
=−
3 16K
−1.7
5L
0.7
5
This
isneg
ativ
e,so
long
asK
>0
and
L>
0.
This
tells
us
that
the
mar
ginal
product
ivity
ofca
pital
dec
reas
esas
we
add
mor
eca
pital
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
15
/51
Eco
nom
icIn
terp
reta
tion
As
anot
her
exam
ple
,co
nsider
,
∂2y
∂K∂L
=3 16
K−0
.75L−0
.25
This
ispos
itiv
e,so
longa
sK
>0,
L>
0.
This
says
that
the
mar
ginal
product
ivity
ofla
bou
rin
crea
ses
asyo
uad
dm
ore
capital
.
Ital
sosu
gges
tsa
sym
met
ry.
The
abov
eca
nbe
inte
rpre
ted
assa
ying
that
the
mar
ginal
product
ivity
ofca
pital
incr
ease
sw
hen
you
add
mor
ela
bor
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
16
/51
-
Con
cave
and
Con
vex
Fun
ctio
ns
Our
inte
rest
inth
ese
SO
Cs
ism
otiv
ated
byth
elo
calve
rsus
glob
alm
ax/m
inof
funct
ions.
Pre
viou
sly,
we
rela
ted
this
toco
nca
vity
and
conve
xity
.
Inth
eca
seof
one
variab
le,we
defi
ned
afu
nct
ion
f(x
)as
conca
veif
f′′ (
x)≤
0an
dco
nve
xif
f′′ (
x)≥
0.
Not
ice
that
inth
esingl
e-va
riab
leca
se,th
ese
cond-o
rder
tota
lis,
d2y
=f′′ (
x)(
dx)2
Hen
ce,we
can
(equiv
alen
tly)
defi
ne
afu
nct
ion
ofon
eva
riab
leto
be
conca
veif
d2y≤
0an
dco
nve
xif
d2y≥
0.T
he
adva
nta
geof
writing
itin
this
way
isth
atwe
can
exte
nd
this
defi
nitio
nto
funct
ions
ofm
any
variab
les.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
17
/51
Sec
ond-
Ord
erTot
alD
iffer
ential
Suppos
ey
=f(x
1,x
2).
The
firs
tor
der
diff
eren
tial
isdy
=f x
1dx 1
+f x
2dx 2
.
Thin
kof
dy
asa
funct
ion.
It’s
diff
eren
tial
is,
d(d
y)
=[f x
1x 1
dx 1
+f x
1x 2
dx 2
]dx 1
+[f x
1x 2
dx 1
+f x
2x 2
dx 2
]dx 2
Col
lect
ing
term
s,
d2y
=f x
1x 1
(dx 1
)2+
2fx 1
x 2dx 1
dx 2
+f x
2x 2
(dx 2
)2
The
seco
nd-o
rder
tota
ldiff
eren
tial
dep
ends
onth
ese
cond-o
rder
par
tial
der
ivat
ives
off(x
1,x
2).
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
18
/51
-
Sec
ond-
Ord
erTot
alD
iffer
ential
Coro
llary
:If
we
hav
ea
gener
alfu
nct
ion
y=
f(x
1,.
..,x
n),
one
can
use
asim
ilar
proce
dure
toge
tth
efo
rmula
for
the
seco
nd-o
rder
tota
ldiff
eren
tial
.
This
isa
little
mor
eco
mplic
ated
but
itca
nbe
writt
enco
mpac
tly
as,
d2y
=n ∑ i=1
n ∑ j=1
f xix
jdx i
dx j
E.g
.,y
=f(x
1,x
2,x
3)
the
expr
ession
bec
omes
,
d2y
=f x
1x 1
(dx 1
)2+
f x2x 2
(dx 2
)2+
f x3x 3
(dx 3
)2
+2f
x 1x 2
dx 1
dx 2
+f x
1x 3
dx 1
dx 3
+f x
2x 3
dx 2
dx 3
Aga
in,we
care
abou
tth
isbec
ause
afu
nct
ion
y=
f(x
1,.
..,x
n)
will
be
conca
veif
d2y≤
0an
dco
nve
xif
d2y≥
0.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
19
/51
Max
ima/
Min
ima/
Sad
dle
Poi
nts
As
inth
esingl
eva
riab
leca
se,we
are
real
lyaf
ter
max
ima
and
min
ima.
The
firs
tor
der
conditio
ns
alon
eca
nnot
distingu
ish
bet
wee
nlo
cal
max
ima
and
loca
lm
inim
a.
Lik
ewise,
the
firs
tor
der
conditio
ns
cannot
iden
tify
whet
her
aca
ndid
ate
solu
tion
isa
loca
lor
glob
alm
axim
a.W
eth
us
nee
dse
cond
order
conditio
ns
tohel
pus.
For
apoi
nt
tobe
alo
calm
axim
a,we
must
hav
ed
2y
<0
for
any
vect
orof
smal
lch
ange
s(d
x 1,.
..,d
x n);
that
is,y
nee
ds
tobe
ast
rict
lyco
nca
vefu
nct
ion.
Sim
ilarly,
for
apoi
nt
tobe
alo
calm
inim
a,we
must
hav
ed
2y
>0
for
any
vect
orof
smal
lch
ange
s(d
x 1,.
..,d
x n);
that
is,y
nee
ds
tobe
ast
rict
lyco
nve
xfu
nct
ion.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
20
/51
-
Hes
sian
Mat
rix
-T
wo
Var
iabl
eCas
e
As
thin
gsst
and,non
eof
this
isnot
par
ticu
larly
use
fulsince
itis
not
clea
rhow
togo
abou
tve
rify
ing
that
the
seco
nd-o
rder
tota
ldiff
eren
tial
ofa
funct
ion
ofn
variab
les
isnev
erpos
itiv
eor
nev
erneg
ativ
e.
How
ever
,not
ice
that
we
can
write
the
seco
nd
order
diff
eren
tial
ofa
funct
ion
oftw
ova
riab
les
(that
is,
d2y
=f 1
1(d
x 1)2
+2f
12dx 1
dx 2
+f 2
2(d
x 2)2
)in
mat
rix
form
inth
efo
llow
ing
way
. d2y
=[ dx
1dx 2
][ f11
f 12
f 12
f 22
][dx 1
dx 2
]
The
mat
rix
[ f 11
f 12
f 12
f 22
] isca
lled
the
Hes
sian
mat
rix,
H.
Itis
sim
ply
the
mat
rix
ofse
cond-o
rder
par
tial
der
ivat
ives
.
Whet
her
d2y
>0
ord
2y
<0
lead
sto
spec
ific
rest
rict
ions
onth
eH
essian
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
21
/51
Hes
sian
Mat
rix
-G
ener
alCas
e
Inth
ege
ner
alca
se,th
ese
cond
order
diff
eren
tial
can
be
writt
enas
,
d2y
=[ dx
1dx 2
...
dx n
]⎡ ⎢ ⎢ ⎣f 1
1f 1
2..
.f 1
n
f 12
f 22
...
f 2n
...
...
...
...
f 1n
f 2n
...
f nn
⎤ ⎥ ⎥ ⎦⎡ ⎢ ⎢ ⎢ ⎣dx 1
dx 2 . . .
dx n
⎤ ⎥ ⎥ ⎥ ⎦If
we
wan
tto
know
som
ethin
gab
out
d2y
allwe
nee
dto
do
isev
aluat
eth
eH
essian
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
22
/51
-
Hes
sian
Mat
rix
-Eco
nom
icExa
mpl
e
Ret
urn
ing
toth
eCob
b-D
ougl
asutilit
yca
se;U
=c
α ac
β b,w
her
ea
=ap
ple
s,b
=ban
anas
.T
he
firs
t-or
der
diff
eren
tial
was
,
dU
=[ αc
α−1
ac
β bβc
α ac
β−1
b
][dc a
dc b
]
The
seco
nd-o
rder
diff
eren
tial
is,
d2U
=[ dc
adc b
][∂2
U∂c
a∂c
a
∂2U
∂ca∂c
b∂2
U∂c
b∂c
a
∂2U
∂cb
∂cb
] [dc a
dc b
]
wher
e, [ ∂2U
∂ca∂c
a
∂2U
∂ca∂c
b∂2
U∂c
b∂c
a
∂2U
∂cb
∂cb
] =[ α
(α−
1)c
α−2
ac
β bα
βc
α−1
ac
β−1
b
αβc
α−1
ac
β−1
bβ
(β−
1)c
α ac
β−2
b
]
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
23
/51
Pos
itiv
eD
efini
tean
dN
egat
ive
Defi
nite
Mat
rice
s
The
Hes
sian
mat
rix
issq
uar
e.W
eca
nsa
yso
met
hin
gab
out
squar
em
atrice
s:
Asy
mm
etric,
squar
em
atrix,
An,is
said
tobe
pos
itiv
edefi
nite
iffo
rev
ery
colu
mn
vect
orx�=
0(d
imen
sion
1×
n)
we
hav
ex
TA
x>
0.
Asy
mm
etric,
squar
em
atrix
An
isneg
ativ
edefi
nite
iffo
rev
ery
colu
mn
vect
orx�=
0(d
imen
sion
1×
n),
we
hav
ex
TA
x<
0.
To
chec
kfo
rco
nca
vity
/con
vexi
tywe
eval
uat
eth
eH
essian
.B
ut
the
Hes
sian
(for
our
purp
oses
)ca
nbe
PD
orN
D.
Ifth
eH
essian
isN
Dth
efu
nct
ion
isco
nca
ve.
Ifth
eH
essian
isPD
the
funct
ion
isco
nve
x.A
llwe
nee
dto
be
able
todo
then
isdet
emin
ew
het
her
agi
ven
mat
rix
isN
Dor
PD
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
24
/51
-
Che
ckin
gfo
rPD
-nes
san
dN
D-n
ess
Let
A=
⎡ ⎢ ⎢ ⎣a 11
a 12
...
a 1n
a 12
a 22
...
a 2n
...
...
...
...
a 1n
a 2n
...
a nn
⎤ ⎥ ⎥ ⎦bea
n×
nsy
mm
etric
squar
em
atrix.
The
prin
cipal
min
ors
ofA
are
the
ndet
erm
inan
ts,
a 11,∣ ∣ ∣ ∣a
11
a 12
a 12
a 22
∣ ∣ ∣ ∣,∣ ∣ ∣ ∣ ∣ ∣a 1
1a 1
2a 1
3
a 12
a 22
a 23
a 13
a 23
a 33
∣ ∣ ∣ ∣ ∣ ∣,...,|A|
The
mat
rix
Ais
pos
itiv
e-defi
nite
ifan
don
lyif
allth
epr
inci
pal
min
ors
are
pos
itiv
e.
The
mat
rix
Ais
neg
ativ
e-defi
nite
ifan
don
lyif
its
prin
cipal
min
ors
alte
rnat
ein
sign
,st
arting
with
aneg
ativ
e.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
25
/51
Exa
mpl
esof
PD
and
ND
Mat
rice
s
One
can
see
how
this
wor
ksif
we
look
atth
em
atrice
sI n
and−I
n.
The
prin
cipal
min
ors
ofI
are
unifor
mly
1;on
the
other
han
dth
epr
inci
pal
min
ors
of−I
star
tw
ith−1
and
then
alte
rnat
ein
sign
.T
hus,
the
mat
rice
sI n
and−I
nar
etr
ivia
llypos
itiv
e-defi
nite
and
neg
ativ
e-defi
nite.
Con
sider
,
[ 11
14
] .One
can
easily
show
that
the
mat
rix
is
pos
itiv
e-defi
nite.
How
?W
ell,
the
prin
ciple
min
ors
are
1an
d3.
The
mat
rix
[ −1
11
−4] is
neg
ativ
e-defi
nite
asth
epr
inci
ple
min
ors
are−1
and
3.
Am
atrix
may
be
nei
ther
pos
itiv
e-defi
nite
nor
neg
ativ
e-defi
nite.
For
inst
ance
,co
nsider
the
mat
rix
[ −1
11
4
] .D
udle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
26
/51
-
ND
and
PD
Rel
atio
nto
Con
cavi
tyan
dCon
vexi
ty
We
defi
ned
afu
nct
ion
tobe
conca
veif
d2y≤
0fo
ral
lx
and
tobe
conve
xif
d2y≥
0fo
ral
lx.
Ifth
efu
nct
ion
satisfi
esa
stro
nge
rco
nditio
n–
d2y
<0
for
allx
–th
enit
will
be
said
tobe
strict
lyco
nca
ve.
Anal
ogou
sly,
ifd
2y
>0
for
allx,th
enit
will
be
said
tobe
strict
lyco
nve
x.
We
can
also
rela
teth
enot
ions
ofpos
itiv
e-defi
niten
ess
and
neg
ativ
e-defi
niten
ess
tost
rict
conca
vity
and
strict
conve
xity
.
1A
funct
ion
fis
strict
lyco
nca
veif
the
Hes
sian
mat
rix
isal
way
sneg
ativ
edefi
nite.
2A
funct
ion
fis
strict
lyco
nve
xif
the
Hes
sian
mat
rix
isal
way
spos
itiv
edefi
nite.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
27
/51
Exa
mpl
e
Con
sider
the
follo
win
g.Is
f(x
,y)
=x
1/4y
1/2
conca
veor
conve
xov
erth
edom
ain
(x,y
)≥
0?
Com
pute
the
seco
nd
order
par
tial
s.E.g
.,f x
y(x
,y)
=f y
x(x
,y)
=(1
/8)x
−3/4y−1
/2
and
f xx(x
,y)
=−
(3/4)(
1/4)x
−7/4y
1/2.
So,
H=
[ −(3
/4)(
1/4)x
−7/4y
1/2
(1/8)x
−3/4y−1
/2
(1/8)x
−3/4y−1
/2
−(1
/4)x
1/4y−3
/2
]T
he
prin
ciple
min
ors
are,
−(3
/4)(
1/4)x
−7/4y
1/2
<0
and,
∣ ∣ ∣ ∣−(3
/4)(
1/4)x
−7/4y
1/2
(1/8)x
−3/4y−1
/2
(1/8)x
−3/4y−1
/2
−(1
/4)x
1/4y−3
/2
∣ ∣ ∣ ∣=
(3/64
)x−7
/4y
1/2x
1/4y−3
/2−
(1/64
)x−3
/4y−1
/2x−3
/4y−1
/2
=(1
/64
)[ 2x−6
/4] /y
>0
We
concl
ude
x1
/4y
1/2
isco
nve
xov
erth
edom
ain
(x,y
)≥
0.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
28
/51
-
Unc
onst
rain
edO
ptim
izat
ion
with
Mul
tipl
eVar
iabl
es
We
now
link
conca
vity
and
conve
xity
toop
tim
izat
ion
prob
lem
sin
econ
omic
s.
Con
sider
the
gener
alm
axim
izat
ion
prob
lem
,
max
f(x
1,.
..,x
n)
The
firs
tor
der
conditio
ns
for
max
imiz
atio
nca
nbe
iden
tified
from
the
conditio
nth
atth
efirs
tor
der
diff
eren
tial
shou
ldbe
zero
atth
eop
tim
alpoi
nt.
That
is,a
vect
orof
smal
lch
ange
s(d
x 1,.
..,d
x n)
shou
ldnot
chan
geth
eva
lue
ofth
efu
nct
ion.
We
thus
hav
e
dy
=f x
1dx 1
+f x
2dx 2
+..
.+f x
ndx n
=0
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
29
/51
First
Ord
erCon
dition
s
Suppos
eth
atwe
hav
ea
funct
ion
of2
variab
les
y=
f(x
1,x
2).
Then
,th
efirs
tor
der
conditio
ndy
=0
implie
sth
atf x
1dx 1
+f x
2dx 2
=0.
Ther
efor
e,th
eon
lyway
we
can
hav
ef x
1dx 1
+f x
2dx 2
=0
isw
hen
f x1=
0an
df x
2=
0.
By
the
sam
elo
gic,
for
age
ner
alfu
nct
ion
y=
f(x
1,.
..,x
n),
the
firs
tor
der
conditio
ns
are
f x1=
0,f x
2=
0,..
.,f x
n=
0
Not
eth
atth
ese
conditio
ns
are
nec
essa
ryco
nditio
ns
inth
atth
eyhol
dfo
rm
inim
izat
ion
prob
lem
sal
so.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
30
/51
-
Rel
atio
nto
SO
Cs
for
Opt
imiz
atio
n
We
can
now
sum
mar
ize
the
firs
tan
dse
cond
order
conditio
ns
for
anop
tim
izat
ion
prob
lem
inth
efo
llow
ing
way
:
If(x
∗ 1,..
.,x∗ n)
isa
loca
lm
axim
um
,th
enth
efirs
tor
der
conditio
ns
imply
that
f i(x
∗ 1,..
.,x∗ n)
=0
for
i=
1,..
.,n.
The
seco
nd
order
conditio
nim
plie
sth
atth
eH
essian
mat
rix
eval
uat
edat
(x∗ 1,
...,
x∗ n)
must
be
neg
ativ
edefi
nite.
If(x
∗ 1,..
.,x∗ n)
isa
loca
lm
inim
um
,th
enth
efirs
tor
der
conditio
ns
imply
that
f i(x
∗ 1,..
.,x∗ n)
=0
for
i=
1,..
.,n.
The
seco
nd
order
conditio
nim
plie
sth
atth
eH
essian
mat
rix
eval
uat
edat
(x∗ 1,
...,
x∗ n)
must
be
pos
itiv
edefi
nite.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
31
/51
Eco
nom
icExa
mpl
e:Pro
fit
Max
imiz
atio
n
To
bet
ter
under
stan
dal
lof
this,we
can
look
ata
firm
’sop
tim
izat
ion
prob
lem
.
Suppos
ea
firm
can
sell
it’s
outp
ut
at$1
0per
unit
and
that
it’s
product
ion
funct
ion
isgi
ven
byy
=K
1/4L
1/2.
What
com
bin
atio
nof
capital
and
labou
rsh
ould
the
firm
use
soas
tom
axim
ize
profi
tsas
sum
ing
that
capital
cost
s$2
per
unit
and
labou
r$1
per
unit?
The
firm
’spr
ofits
are
give
nby
Π(K
,L)
=10
y−
2K−
L︸
︷︷︸
reve
nue
min
us
cost
s
=10
K1
/4L
1/2−
2K−
L
This
isa
prob
lem
ofunco
nst
rain
edop
tim
izat
ion
with
multip
leva
riab
les.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
32
/51
-
Eco
nom
icExa
mpl
eCon
tinu
ed
We
can
use
the
firs
tor
der
conditio
ns
toob
tain
pote
ntialca
ndid
ate
sfo
rop
tim
izat
ion,as
with
one
variab
le.
For
the
firm
we
hav
eth
efo
llow
ing
profi
tm
axim
izat
ion
prob
lem
:
Π(K
,L)
=10
K1
/4L
1/2−
2K−
L
The
firs
tor
der
conditio
ns
are
∂Π ∂K
=(5
/2)K
−3/4L
1/2−
2=
0
∂Π ∂L
=5K
1/4L−1
/2−
1=
0
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
33
/51
Eco
nom
icExa
mpl
eCon
tinu
ed
Div
idin
gth
efirs
tfirs
t-or
der
conditio
nby
the
seco
nd
firs
t-or
der
conditio
ngi
ves
us
(1/2)L
/K
=2,
or,
L=
4K
Subst
ituting
this
into
the
seco
nd
firs
t-or
der
conditio
n,we
get
5K1
/4(4
K)−
1/2
=1,
orK
−1/4
=2
/5,
or,
K=
(5/2)4
=62
5/16
Fin
ally,we
hav
e,L
=62
5/4
This
isou
rca
ndid
ate
solu
tion
for
am
axim
um
.B
ut
tove
rify
that
itre
ally
isth
em
axim
um
we
nee
dto
chec
kth
ese
cond-o
rder
conditio
ns.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
34
/51
-
Eco
nom
icExa
mpl
eCon
tinu
ed
The
candid
ate
solu
tion
sar
e,K
∗=
625
/16
and
L∗
=62
5/4.
The
Hes
sian
mat
rix
atan
y(K
,L)
isgi
ven
by
H=
[ −15 8K
−7/4L
1/2
5 4K
−3/4L−1
/2
5 4K
−3/4L−1
/2
−5 2K
1/4L−3
/2
]N
ote
that
since
K>
0,L
>0.
f 11
=−
15 8K
−7/4L
1/2
<0
|H|=
−( −
5 2
) 15 8K
−7/4+
1/4L−3
/2+
1/2−
( 5 4)2
K−3
/4−3
/4L−1
/2−1
/2
=(7
5/16
)K−3
/2L−1
−(2
5/16
)K−3
/2L−1
=(5
0/16
)K−3
/2L−1
>0
This
show
sth
atth
ese
cond
order
conditio
ns
are
satisfi
edat
(K∗ ,
L∗ )
,w
hic
hsh
ows
itis
alo
calm
axim
um
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
35
/51
Impl
icit
Fun
ctio
nsan
dCom
para
tive
Sta
tics
So
far,
we
hav
eco
nsider
the
inte
rpre
tation
offu
nct
ions
with
mor
eth
anon
eva
riab
leth
rough
the
Jaco
bia
nan
dth
eH
essian
.
We
hav
eal
sore
late
dth
eH
essian
toth
enot
ions
ofco
nca
vity
and
conve
xity
and
max
ima
and
min
ima,
via
anop
tim
izat
ion
prob
lem
.
How
ever
,w
hen
we
dea
lw
ith
funct
ions
ofm
ore
than
one
variab
le,we
also
run
into
other
com
plic
atio
ns.
Bas
ical
ly,we
tend
toth
ink
ofth
eLH
Sva
riab
leof
afu
nct
ion
asen
dog
enou
san
dth
eRH
Sva
riab
le(s
)as
exog
enou
s.B
ut
som
etim
esex
ogen
ous
variab
les
and
endog
enou
sva
riab
les
cannot
be
separ
ated
and
we
nee
dto
diff
eren
tiat
eim
plic
itly.
Implic
itdiff
eren
tiat
ion
isal
souse
fulw
hen
thin
king
abou
tutilit
yfu
nct
ions
(indiff
eren
cecu
rves
)an
dpr
oduct
ion
funct
ions
(iso
quan
tcu
rves
).
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
36
/51
-
Impl
icit
Fun
ctio
nT
heor
em
An
exam
ple
ofan
implic
itfu
nct
ion,i.e.
,on
ew
her
eth
eex
ogen
ous
and
endog
enou
sva
riab
les
cannot
be
separ
ated
,is,
F(x
,y)
=x
2+
xyey+
y2x−
10=
0
Itis
not
clea
rhow
tose
par
ate
out
yfrom
xor
even
whet
her
itca
nbe
don
e.H
owev
er,we
may
still
wan
tto
find
out
how
the
endog
enou
sva
riab
lech
ange
sw
hen
the
exog
enou
sva
riab
lech
ange
s.
Under
cert
ain
circ
um
stan
ces,
ittu
rns
out
that
even
when
we
cannot
separ
ate
out
the
variab
les,
we
can
non
ethel
ess
find
the
der
ivat
ive
dy
/dx
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
37
/51
Impl
icit
Fun
ctio
nT
heor
em
The
follo
win
gre
sult
enab
les
us
tofind
the
der
ivat
ives
for
implic
itfu
nct
ions.
Theo
rem
:Let
f(x
,y)
=0
be
anim
plic
itfu
nct
ion
whic
his
continuou
sly
diff
eren
tiab
lean
d(x
0,y
0)
be
such
that
f(x
0,y
0)
=0.
Suppos
eth
at∂f ∂y
(x0,y
0)�=
0.T
hen
,
dy
dx
∣ ∣ ∣ (x 0,y0)
=−
∂f ∂x(x
0,y
0)
∂f ∂y(x
0,y
0)
So,
withou
tsp
ecifyi
ng
xan
dy
we
can
find
dy
dx
ata
give
npoi
nt,
her
e(x
0,y
0).
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
38
/51
-
Abs
trac
tExa
mpl
eof
Impl
icit
Fun
ctio
nT
heor
em
Con
sider
f(x
,y)
=x
2+
xyey+
y2x−
10=
0.Suppos
ewe
wan
tto
eval
uat
eth
eder
ivat
ive
atso
me
poi
nt
(x0,y
0)
such
that
f(x
0,y
0)
=0.
Using
the
rule
sof
diff
eren
tiat
ion,
∂f ∂y=
xey+
xyey+
2xy
This
can
be
zero
for
som
ex
and
ybut
assu
me
that
the
poi
nt
(x0,y
0)
isnot
one
ofth
em.
Now
diff
eren
tiat
ew
ith
resp
ect
tox,
∂f ∂x=
2x+
yey+
y2
The
implic
itfu
nct
ion
theo
rem
allo
ws
us
tow
rite
the
follo
win
g.
dy
dx
∣ ∣ ∣ (x 0,y0)
=−
2x0+
y 0ey 0
+y
2 0
x 0ey 0
+x 0
y 0ey 0
+2x
0y 0
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
39
/51
Eco
nom
icExa
mpl
eof
Impl
icit
Fun
ctio
nT
heor
em
One
use
ofth
eim
plic
itfu
nct
ion
theo
rem
inEco
nom
ics
isto
find
the
slop
esof
indiff
eren
cecu
rves
and
isoquan
ts.
Con
sider
the
product
ion
funct
ion
use
dea
rlie
r.
y=
K0.2
5L
0.7
5
Ifwe
fix
y=
y 0,th
enth
eco
mbin
atio
nof
all(K
,L)
whic
hgi
ves
y 0units
ofou
tput
isca
lled
anisoquan
t.
Suppos
e(K
0,L
0)
ison
esu
chco
mbin
atio
n.
We
wan
tto
find
the
slop
eof
the
isoquan
t,th
atis
dK dL
∣ ∣ ∣ (K 0,L
0).
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
40
/51
-
Eco
nom
icExa
mpl
eCon
tinu
ed
Set
ting
y=
y 0,we
can
write
the
product
ion
funct
ion
inim
plic
itfu
nct
ion
form
as,
F(K
,L)
=K
0.2
5L
0.7
5−
y 0=
0
Using
the
valu
esfo
rth
epar
tial
der
ivat
ives
found
earlie
r,we
hav
e,
dK dL
∣ ∣ ∣ (K 0,L
0)
=−
0.75
K0.2
50
L−0
.25
0
0.25
K−0
.75
0L
0.7
50
=−3
K0
L0
Not
eth
atK
0�=
0,L
0�=
0.
This
equat
ion
tells
us
the
slop
eof
the
isoquan
tat
(K0,L
0).
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
41
/51
Indi
ffer
ence
Cur
ves
As
afinal
exam
ple
,co
nsider
the
follo
win
gutilit
yfu
nct
ion:
U=
[ αcη a+
βc
η b
] 1/η,w
her
ea
=ap
ple
s,b
=ban
anas
.T
his
isca
lled
aco
nst
ant
elas
tici
tyof
subst
itution
utilit
yfu
nct
ion.
When
η=
0we
get
bac
kto
the
Cob
b-D
ougl
asutilit
yfu
nct
ion.
Hol
din
gutilit
yfixe
d,we
hav
eth
efo
llow
ing:
[ αcη a+
βc
η b
] 1/η−
U0
=0.
Now
use
the
implic
itfu
nct
ion
theo
rem
.
dc a
dc b
∣ ∣ ∣ (c a,0,c
b,0
)=
−∂U ∂c
a(c
a,0,c
b,0)
∂U ∂cb(c
a,0,c
b,0)
=−
α β
( c a c b) η−
1
Suppos
ewe
look
atth
eM
RS
atth
epoi
nt(t
c a,t
c b).
The
indiff
eren
cecu
rves
hav
eth
esa
me
slop
eat
(ca,c
b)
and
at(t
c a,t
c b)
for
any
t>
0.T
he
CES
utilit
yfu
nct
ion
isan
exam
ple
ofpr
efer
ence
sth
atar
ehom
othet
ic.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
42
/51
-
Com
para
tive
Sta
tics
and
Impl
icit
Rel
atio
ns:
Sup
ply
and
Dem
and
One
ques
tion
we
also
ofte
nnee
dto
answ
eris
how
under
lyin
gpar
amet
ers
affec
tth
eeq
uili
briu
mof
asu
pply
and
dem
and
syst
em.
Suppos
eth
atwe
hav
ea
mar
ket
wher
eth
edem
and
and
supply
funct
ions
are
give
nby
the
follo
win
g.
D=
a−
bp
+cy
,S
=α
+βp
The
par
amet
ers
a,b,c
,α,β
are
allpos
itiv
eco
nst
ants
.
The
equili
briu
mpr
ice
is,
p∗
=a−
α+
cy
b+
β
We
can
now
chec
khow
the
under
lyin
gpar
amet
ers
(a,b
,c,y
,α,β
)aff
ect
the
equili
briu
mpr
ice.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
43
/51
Com
para
tive
Sta
tics
:Sup
ply
and
Dem
and
Inou
rsim
ple
exam
ple
,we
can
get
the
par
tial
der
ivat
ive,
∂p∗ /
∂a=
1/(b
+β)
All
other
thin
gsre
mai
nin
gco
nst
ant,
anin
crea
sein
a,in
crea
ses
the
equili
briu
mpr
ice.
We
can
sim
ilarly
find
the
impac
tof
chan
ges
inot
her
par
amet
ers
onth
eeq
uili
briu
mpr
ice.
Impor
tantly,
we
can
also
,in
man
yca
ses,
conduct
the
com
par
ativ
est
atic
sw
ithou
tso
lvin
gfo
rth
eeq
uili
briu
mva
lues
ofth
een
dog
enou
sva
riab
les.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
44
/51
-
Gen
eral
Fun
ctio
nExa
mpl
eof
Sup
ply
and
Dem
and
Con
sider
the
follo
win
gsy
stem
mor
ege
ner
aldem
and
and
supply
syst
em.
qd
=D
(p,T
),q
s=
S(p
,T)
Her
e,T
isth
ete
mper
ature
ona
give
nday
.W
eas
sum
eth
atD
p<
0,D
T>
0,S
p>
0,S
T<
0.
Equili
briu
mre
quires
that
D(p
,T)
=S(p
,T)
but
we
cannot
com
pute
the
equili
briu
mpr
ice
explic
itly.
Den
ote
the
equili
briu
mpr
ice
p∗ (
T):
obse
rve
that
itis
afu
nct
ion
ofth
epar
amet
erT
.
Inse
rtin
gp∗ (
T)
into
the
equili
briu
mco
nditio
n,we
hav
e,
D(p
∗ (T
),T
)=
S(p
∗ (T
),T
)
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
45
/51
Gen
eral
Fun
ctio
nExa
mpl
eof
Sup
ply
and
Dem
and
We
diff
eren
tiat
ebot
hsides
with
resp
ect
toT
whic
hgi
ves,
Dpdp∗
dT
+D
T=
Spdp∗
dT
+S
T
Rea
rran
ging,
we
get
dp∗
dT
=S
T−
DT
Dp−
Sp
Not
eth
atsince
ST
<0,
DT
>0,
Dp
<0,
Sp
>0
itfo
llow
sth
atdp∗ /
dT
>0.
Hen
ce,ev
enw
ithou
tbei
ng
able
toco
mpute
p∗ (
T)
explic
itly,we
can
still
say
that
the
equili
briu
mpr
ice
incr
ease
sfo
llow
ing
anin
crea
sein
the
tem
per
ature
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
46
/51
-
Exa
mpl
eCon
tinu
ed
Can
we
say
anyt
hin
gab
out
the
equili
briu
mquan
tity
?
Not
eth
atth
eeq
uili
briu
mquan
tity
must
satisf
y,
q∗ (
T)
=D
(p∗ (
T),
T)
Diff
eren
tiat
ing
with
resp
ect
toT
we
get
dq∗
dT
=D
pdp∗
dT
+D
T=
DpS
T−
SpD
T
Dp−
Sp
Inth
isca
se,th
eden
omin
ator
ispos
itiv
ebut
we
cannot
say
anyt
hin
gab
out
the
num
erat
orunle
sswe
hav
em
ore
spec
ific
info
rmat
ion
abou
tth
em
agnitudes
ofD
p,S
p,D
Tan
dS
T.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
47
/51
Gen
eral
For
mul
atio
n
Suppos
ewe
hav
ea
gener
alse
tof
equili
briu
mco
nditio
ns
give
nby
f1(x
1,.
..,x
n;α
1,.
..,α
m)
=0
f2(x
1,.
..,x
n;α
1,.
..,α
m)
=0
...=
...
fn(x
1,.
..,x
n;α
1,.
..,α
m)
=0
Let
us
suppos
eth
atwe
can
solv
eth
isse
tof
equat
ions
toge
t(x
∗ 1,..
.,x∗ n)
.N
ote
that
each
x∗ i
will
be
afu
nct
ion
ofal
lth
eunder
lyin
gpar
amet
ers
(α1,.
..,α
n).
Inco
mpar
ativ
est
atic
s,we
typic
ally
are
inte
rest
edin
know
ing
how
(x∗ 1,
...,
x∗ n)
chan
gew
hen
ther
eis
asm
allch
ange
inon
eof
the
par
amet
ers,
say
αi.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
48
/51
-
Gen
eral
For
mul
atio
n
We
proce
edby
diff
eren
tiat
ing
each
ofth
eeq
uili
briu
mco
nditio
ns
with
resp
ect
toα
i.T
his
give
s
f1 1
dx∗ 1
dα
i+
f1 2
dx∗ 2
dα
i+
...+
f1 n
dx∗ n
dα
i=
−f1 αi
f2 1
dx∗ 1
dα
i+
f2 2
dx∗ 2
dα
i+
...+
f2 n
dx∗ n
dα
i=
−f2 αi
. . .=
. . .
fn 1
dx∗ 1
dα
i+
fn 2
dx∗ 2
dα
i+
...+
fn n
dx∗ n
dα
i=
−fn αi
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
49
/51
Gen
eral
For
mul
atio
n
Inm
atrix
not
atio
nth
isca
nbe
writt
enas
⎡ ⎢ ⎢ ⎢ ⎣f1 1
f1 2
...
f1 n
f2 1
f2 2
...
f2 n
. . .. . .
. . .. . .
fn 1
fn 2
...
fn n
⎤ ⎥ ⎥ ⎥ ⎦⎡ ⎢ ⎢ ⎣dx∗ 1
dα
idx∗ 2
dα
i. . .d
x∗ n
dα
i
⎤ ⎥ ⎥ ⎦=⎡ ⎢ ⎢ ⎢ ⎣−
f1 αi
−f2 αi
. . . −fn αi
⎤ ⎥ ⎥ ⎥ ⎦H
owdo
we
solv
efo
rdx∗ j/
dα
i?U
sing
the
usu
alm
ethods
(lik
eCra
mer
’sru
le).
All
ofth
isgi
ves
us
ave
rypow
erfu
lto
olm
ore
anal
yzin
gre
lative
lyco
mplic
ated
econ
omic
sm
odel
sin
asim
ple
way
.
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
50
/51
-
Rou
ndup
You
shou
ldnow
be
able
todo/
under
stan
dth
efo
llow
ing:
1Par
tial
diff
eren
tiat
ion
and
the
Jaco
bia
nm
atrix
2T
he
Hes
sian
mat
rix
and
conca
vity
and
conve
xity
ofa
funct
ion
3O
ptim
izat
ion
(pro
fit
max
imiz
atio
n)
4Im
plic
itfu
nct
ions
(indiff
eren
cecu
rves
and
com
par
ativ
est
atic
s)
Dudle
yCooke
(Trinity
Colleg
eD
ublin)
Multi-va
riable
Calc
ulu
sand
Optim
ization
51
/51
