math of optimization

8/9/2019 Math of Optimization

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THE MATHEMATICS OFOPTIMIZATION

From Nicholson and Snyder, Microeconomic Theory BasicPrinciples and Extensions, 10 th Edition, Chapter 2


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The Mathematics of Optimization

Many economic theories be in !ith theass"mption that an economic a ent is see#into fin$ the optima% &a%"e of some f"nction ' cons"mers see# to ma(imize "ti%ity ' firms see# to ma(imize profit


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Ma(imization of a F"nction of One )ariab%e

! Example" Pro#it maximi$ation

)(qf =

= f(q)

%&antity

*

q*

Maxim&m pro#its o# * occ&r at ' *


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Ma(imization of a F"nction of One )ariab%e

! The mana)er *ill li+ely try to ary q to see *herethe maxim&m pro#it occ&rs

an increase #rom q 1 to q 2 leads to a rise in

= f(q)

%&antity

*

q*

1

q1

2

q2

*>

q


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-

Ma(imization of a F"nction of One )ariab%e! .# o&tp&t is increased /eyond ' , pro#it *ill decline

an increase #rom q to q leads to a drop in

= f(q)

%&antity

*

q*

*

2

*q q

d dq

= < an$ * for 3

d q q

dq

< >

Therefore1 at q31d -dq m"st be $ecreasin


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Secon$ Or$er Con$ition

The secon$ or$er con$ition to represent a/%oca%0 ma(im"m is

6

6 33

7/ 0 *q q

q q

d f q

dq

=

== 9 %n

for any constant

x xd x da a a

dx x dx

a

= =

' a specia% case of this r"%e is de x-dx , e x


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1-

8"%es for Fin$in +eri&ati&es

: / 0 / 0;?9


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1


6

/ 0/ 0


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/ 0.*9

/ 0

ax axax axde de d ax e a ae

dx d ax dx= = =

Some e(amp%es of the chain r"%e inc%"$e

:%n/ 0; :%n/ 0; / 0 . ...9

/ 0d ax d ax d ax

adx d ax dx ax x

= = =

6 6 6

6 6

:%n/ 0; :%n/ 0; / 0 . 6.69 6

/ 0d x d x d x

xdx d x dx x x

= = =


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E(amp%e of Profit Ma(imization S"ppose that the re%ationship bet!een profit an$

o"tp"t is , .1*** q 5 >q6

The first or$er con$ition for a ma(im"m isd -dq , .1*** 5 .* q , *

q3 , .**

Since the secon$ $eri&ati&e is a%!ays 5.*1q , .** is a %oba% ma(im"m


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F"nctions of Se&era% )ariab%es

Most oa%s of economic a ents $epen$ on se&era%&ariab%es

' tra$e5offs m"st be ma$e The $epen$ence of one &ariab%e / y0 on a series of

other &ariab%es / x. 1 x61 1 xn0 is $enote$ by

. 6/ 1 19991 0n y f x x x=


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The partia% $eri&ati&e of y !ith respect to x. is$enote$ by

Partia% +eri&ati&es

. .. .

or or or x y f

f f x x

It is "n$erstoo$ that in ca%c"%atin the partia%$eri&ati&e1 a%% of the other xDs are he%$ constant


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A more forma% $efinition of the partia%$eri&ati&e is

Partia% +eri&ati&es

6

. 6 . 6

*. 19991

/ 1 19991 0 / 1 19991 0%im

n

n n

h x x

f x h x x f x x x f x h

+ =


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Ca%c"%atin Partia% +eri&ati&es6 6

. 6 . . 6 6

. . 6.

6 . 66

.9 If / 1 0 1 then

6 an$

6

y f x x ax bx x cx

f f ax bx

x

f f bx cx

x

= = + + = = + = = +

. 6

. 6 . 6

. 6

. 6. 6

69 If / 1 0 1 then

an$

ax bx

ax bx ax bx

y f x x e

f f f ae f be

x x

+

+ +

= = = = = =


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Ca%c"%atin Partia% +eri&ati&es

. 6 . 6

. 6. . 6 6

29 If / 1 0 %n %n 1 then

an$

y f x x a x b x

f a f b f f

x x x x

= = + = = = =


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Partia% +eri&ati&es

Partia% $eri&ati&es are the mathematica%e(pression of the ceteris paribus

ass"mption ' sho! ho! chan es in one &ariab%e affect someo"tcome !hen other inf%"ences are he%$constant


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Partia% +eri&ati&es

e m"st be concerne$ !ith ho! &ariab%esare meas"re$

' if q represents the "antity of aso%ine$eman$e$ /meas"re$ in bi%%ions of %iters0 an$ p represents the price in $o%%ars per %iter1 then q- p !i%% meas"re the chan e in $eman$ /in

bi%%ions of %iters per year0 for a $o%%ar per %iterchan e in price


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E%asticity

E%asticities meas"re the proportiona% effectof a chan e in one &ariab%e on another ' "nit free

The e%asticity of y !ith respect to x is

1 y x

y

y x y x ye x x y x y

x

= = =


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E%asticity an$ F"nctiona% Form S"ppose that

y = a + bx + other terms

In this case1

1 y x

y x x xe b b

x y y a bx= = = + +

e y,x is not constant ' it is important to note the point at !hich the

e%asticity is to be comp"te$


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E%asticity an$ F"nctiona% Form

S"ppose that

y = ax b

In this case1

.1

b y x b

y x xe abx b

x y ax

= = =


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E%asticity an$ F"nctiona% Form S"ppose that

%n y = %n a G b %n x

In this case1

1

%n%n y x

y x ye b

x y x = =

E%asticities can be ca%c"%ate$ thro" h%o arithmic $ifferentiation


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o"n Ds Theorem

n$er enera% con$itions1 the or$er in !hich partia% $ifferentiation is con$"cte$ to e&a%"atesecon$5or$er partia% $eri&ati&es $oes notmatter

ij ji f f =


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se of Secon$5Or$er Partia%s Secon$5or$er partia%s p%ay an important ro%e

in many economic theories

One of the most important is a &ariab%eDso!n secon$5or$er partia%1 f ii ' sho!s ho! the mar ina% inf%"ence of xi on

y/ y- xi0 chan es as the &a%"e of xi increases

' a &a%"e of f ii J * in$icates $iminishin mar ina%effecti&eness


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Ma(imizationB Se&era% )ariab%es

S"ppose that y , f / x. 1 x61 1 xn0 If a%% xDs are &arie$ by a sma%% amo"nt1 the tota%

effect on y !i%% be

. 6. 6

999 nn

f f f dy dx dx dx

x x x = + + +

. . 6 6 999 n ndy f dx f dx f dx= + + + This e(pression is the tota% $ifferentia%


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First5Or$er Con$ition for a

Ma(im"m /or Minim"m0 A necessary con$ition for a ma(im"m /or minim"m0 of the

f"nction f / x. 1 x61 1 xn0 is that dy , * for any combination ofsma%% chan es in the xDs

The on%y !ay for this to be tr"e is if

. 6 999 *n f f f = = = = A point !here this con$ition ho%$s is ca%%e$ a critica% point


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Fin$in a Ma(im"m

S"ppose that y is a f"nction of x. an$ x6 y , 5 / x. 5 .06 5 / x6 5 606 G .*

y , 5 x.6

G 6 x. 5 x66

G = x6 G > First5or$er con$itions imp%y that

..

66

6 6 *

6 = *

y x

x y

x x

= + =

= + =

O8 3.

36

.

6

x

x

=

=


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Imp%icit F"nction Theorem

It may not a%!ays be possib%e to so%&e imp%icitf"nctions of the form g / x1 y0,* for "ni "ee(p%icit f"nctions of the form y , f / x0 ' mathematicians ha&e $eri&e$ the necessary

con$itions ' in many economic app%ications1 these con$itions are

the same as the secon$5or$er con$itions for ama(im"m /or minim"m0


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+eri&ati&es of imp%icit f"nctions

Imp%icit f"nctionB / 1 0 *

Tota% $ifferentia%B *

Hence1 the imp%icit $eri&ati&e can be fo"n$

as the ne ati&e of the ratio of partia% $eri&ati&esof the imp%icit f"nction9

x y

x

y

f x y

f dx f dy

f dydx f

dydx

=

= +

=


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Pro$"ction Possibi%ity Frontier

Ear%ier e(amp%eB6 x6 G y6 , 66> Can be re!rittenB f / x1 y0 , 6 x6 G y6 5 66> , * Keca"se f x , = x an$ f y , 6 y, the opport"nity cost tra$e5off

bet!een x an$ y is= 6

6 x

y

f dy x xdx f y y

= = =


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The En&e%ope Theorem

The en&e%ope theorem concerns ho! theoptima% &a%"e for a partic"%ar f"nction chan es

!hen a parameter of the f"nction chan es This is easiest to see by "sin an e(amp%e


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S"ppose that y is a f"nction of x

y , 5 x6 G ax For $ifferent &a%"es of a1 this f"nction

represents a fami%y of in&erte$ parabo%as If a is assi ne$ a specific &a%"e1 then y

becomes a f"nction of x on%y an$ the &a%"e of x that ma(imizes y can be ca%c"%ate$


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6 . .2 2-6 L-== 6 => >-6 6>-=? 2 L

Optimal Values of x and y for alternative values of a


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As a increases,the maximal valuefor y ( y*) increases

The relationshipbetween a and yis quadratic

y 5#6a7


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S"ppose !e are intereste$ in ho! y3 chan esas a chan es

There are t!o !ays !e can $o this ' ca%c"%ate the s%ope of y $irect%y ' ho%$ x constant at its optima% &a%"e an$ ca%c"%ate

y- a $irect%y


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To ca%c"%ate the s%ope of the f"nction1 !e m"st so%&efor the optima% &a%"e of x for any &a%"e of a

S"bstit"tin 1 !e et

6 *

36

dy x a

dxa

x

= + =

=

6 6

6 6 6

3 / 30 / 30 / - 60 / - 60

3 - = - 6 - =

y x a x a a a

y a a a

= + = +

= + =


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Therefore1

dy3 /da , 6 a-= , a-6 , x3

K"t1 !e can sa&e time by "sin the en&e%opetheorem ' for sma%% chan es in a1dy3-da can be comp"te$ by

ho%$in x at x3 an$ ca%c"%atin y- a $irect%y from y


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The En&e%ope TheoremB S"mmary

The en&e%ope theorem states that the chan e inthe optima% &a%"e of a f"nction !ith respect to a

parameter of that f"nction can be fo"n$ by

partia%%y $ifferentiatin the ob ecti&e f"nction!hi%e ho%$in x /or se&era% xDs0 at its optima%&a%"e

3 O 3/ 0Pdy y x x ada a

= =


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The En&e%ope Theorem B Many )ariab%es

The en&e%ope theorem can be e(ten$e$ to thecase !here y is a f"nction of se&era% &ariab%es

y , f / x. 1 xn1a0

Fin$in an optima% &a%"e for y !o"%$ consist ofso%&in n first5or$er e "ations

y- xi , * / i , .1 1 n0


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Optima% &a%"es for these xDs !o"%$ be$etermine$ that are a f"nction of a

3 3. .

3 36 6

3 3

, / 01, / 01

/ 09n n

x x a x x a

x a=


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S"bstit"tin into the ori ina% ob ecti&e f"nctionyie%$s an e(pression for the optima% &a%"e of y / y30

y3 , f : x. 3/ a01 x63/ a01 1 xn3/ a01a;

+ifferentiatin yie%$s

. 6

. 6

3 999 nn

dxdx dxdy f f f f da x da x da x da a

= + + + +


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6 6 3 3. 6 . 6

6 6. 6 . 6

. 6

E(amp%eB

/ .0 / 60 .* .1 61 3 .*

Instea$ of .*1 "se the parameter

/ 1 1 0 / .0 / 60In this case1 the optima% &a%"es of an$ $o not $epen$

on 9 So

3

3 .

y x x x x y

a

y f x x a x x a x x

a

y a

dyda

= + = = =

= = +

=

=


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Constraine$ Ma(imization

hat if a%% &a%"es for the xDs are not feasib%eQ ' the &a%"es of x may a%% ha&e to be positi&e '

a cons"merDs choices are %imite$ by the amo"ntof p"rchasin po!er a&ai%ab%e

One metho$ "se$ to so%&e constraine$ma(imization prob%ems is the Ra ran ianm"%tip%ier metho$


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--

Ra ran ian M"%tip%ier Metho$

S"ppose that !e !ish to fin$ the &a%"es of x. 1 x61 1 xn that ma(imize

y , f / x. 1 x61 1 xn0 s"b ect to a constraint that permits on%y

certain &a%"es of the xDs to be "se$

g / x. 1 x61 1 xn0 , *


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Ra ran ian M"%tip%ier Metho$ The Ra ran ian m"%tip%ier metho$ starts

!ith settin "p the e(pression

L , f / x. 1 x61 1 xn 0 G g / x. 1 x61 1 xn0 !here is an a$$itiona% &ariab%e ca%%e$ aRa ran ian m"%tip%ier

hen the constraint ho%$s1 L , f beca"se g / x. 1 x61 1 xn0 , *


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Ra ran ian M"%tip%ier Metho$ First5Or$er Con$itions

. . .

6 6 6

. 6

- *

- *

- *

R- / 1 19991 0 *n n n

n

x f g

x f g

x f g

g x x x

= + = = + =

= + = = =


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Ra ran ian M"%tip%ier Metho$ The first5or$er con$itions can enera%%y be

so%&e$ for x. 1 x61 1 xn an$

The so%"tion !i%% ha&e t!o propertiesB ' the xDs !i%% obey the constraint ' these xDs !i%% ma#e the &a%"e ofL /an$ therefore

f 0 as %ar e as possib%e


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Ra ran ian M"%tip%ier Metho$ At the optima% choices for the xDs1 the ratio of

the mar ina% benefit of increasin xi to themar ina% cost of increasin xi sho"%$ be thesame for e&ery x is the common cost5benefit ratio for a%% ofthe (Ds

mar ina% benefit ofmar ina% cost of

i

i

x x

=


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Ra ran ian M"%tip%ier Metho$ If the constraint !as re%a(e$ s%i ht%y1 it !o"%$ not

matter !hich x is chan e$ The Ra ran ian m"%tip%ier pro&i$es a meas"re of

ho! the re%a(ation in the constraint !i%% affect the&a%"e of y

pro&i$es a sha$o! price to the constraint


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Ra ran ian M"%tip%ier Metho$

A hi h &a%"e of in$icates that y co"%$ be increase$s"bstantia%%y by re%a(in the constraint ' each x has a hi h cost5benefit ratio

A %o! &a%"e of in$icates that there is not m"ch to be aine$ by re%a(in the constraint ,* imp%ies that the constraint is not bin$in


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+"a%ity

Any constraine$ ma(imization prob%em hasassociate$ !ith it a $"a% prob%em in

constraine$ minimization that foc"sesattention on the constraints in the ori ina% prob%em


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+"a%ity

In$i&i$"a%s ma(imize "ti%ity s"b ect to a b"$ et constraint ' $"a% prob%emB in$i&i$"a%s minimize the

e(pen$it"re nee$e$ to achie&e a i&en %e&e% of"ti%ity

Firms minimize the cost of inp"ts to pro$"ce

a i&en %e&e% of o"tp"t ' $"a% prob%emB firms ma(imize o"tp"t for a i&en

cost of inp"ts p"rchase$


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S"ppose a farmer ha$ a certain %en th of fence/ ! 0 an$ !ishe$ to enc%ose the %ar est possib%erectan "%ar shape

Ret x be the %en th of one si$e Ret y be the %en th of the other si$e Prob%emB choose x an$ y so as to ma(imize the

area / " , x#y0 s"b ect to the constraint that the perimeter is fi(e$ at ! , 6 x G 6 y


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Since y-6 , x-6 , 1 x m"st be e "a% to y ' the fie%$ sho"%$ be s "are ' x an$ y sho"%$ be chosen so that the ratio of

mar ina% benefits to mar ina% costs sho"%$ be thesame

Since x , y an$ y , 6 1 !e can "se the

constraint to sho! that x , y , ! -=

, ! -


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Interpretation of the Ra ran ian m"%tip%ier ' if the farmer !as intereste$ in #no!in ho! m"ch

more fie%$ co"%$ be fence$ by a$$in an e(tra yar$

of fence1 s" ests that he co"%$ fin$ o"t by$i&i$in the present perimeter / ! 0 by

' th"s1 the Ra ran ian m"%tip%ier pro&i$esinformation abo"t the imp%icit &a%"e of theconstraint


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First5or$er con$itionsB L + - x = 6 5 + U y , *

L + - y = 6 5 + U x , *

L + - + = " $ x U y , *

So%&in 1 !e et

x , y , " .-6

The Ra ran ian m"%tip%ier / + 0 , 6 " 5.-6


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En&e%ope Theorem V Constraine$

Ma(imization S"ppose that !e !ant to ma(imize

y , f / x. 1 1 xn%a)

s"b ect to the constraint

g / x. 1 1 xnWa0 , *

One !ay to so%&e !o"%$ be to set "p theRa ran ian e(pression an$ so%&e the first5or$er con$itions


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Ine "a%ity Constraints

In some economic prob%ems the constraintsnee$ not ho%$ e(act%y

For e(amp%e1 s"ppose !e see# to ma(imize y= f / x. 1 x60 s"b ect to

g / x. 1 x60 *1

x.

*1 an$ x6 *


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One !ay to so%&e this prob%em is to intro$"cethree ne! &ariab%es / a1b1 an$ c0 that con&ertthe ine "a%ities into e "a%ities

To ens"re that the ine "a%ities contin"e toho%$1 !e !i%% s "are these ne! &ariab%es toens"re that their &a%"es are positi&e


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g / x. 1 x60 5a 6 , *W

x. 5 b6 , *W an$

x6 5 c6

, * Any so%"tion that obeys these three e "a%ity

constraints !i%% a%so obey the ine "a%ity

constraints


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e can set "p the Ra ran ian

L , f / x. 1 x60 G . : g / x. 1 x60 5a 6; G 6: x. 5 b6; G 2: x6 5 c6;

This !i%% %ea$ to ei ht first5or$er con$itions


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L - x. , f . G . g . G 6 , *

L - x6 , f . G . g 6 G 2 , *

L - a , 56a . , *

L - b , 56 b 6 , *

L - c , 56 c 2 , *

L - . , g(x . ,x60 5a 6 = * L - 6 , x. 5 b6 = *

L - 2 , x6 5 c6 = *


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Accor$in to the thir$ con$ition1 either a or . , * ' if a , *1 the constraint g / x. 1 x60 ho%$s e(act%y

' if . , *1 the a&ai%abi%ity of some s%ac#ness ofthe constraint imp%ies that its &a%"e to theob ecti&e f"nction is *

Simi%ar comp%emetary s%ac#nessre%ationships a%so ho%$ for x. an$ x6


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These res"%ts are sometimes ca%%e$ X"hn5T"c#er con$itions ' they sho! that so%"tions to optimization

prob%ems in&o%&in ine "a%ity constraints !i%%$iffer from simi%ar prob%ems in&o%&in e "a%ityconstraints in rather simp%e !ays

' !e cannot o !ron by !or#in primari%y !ithconstraints in&o%&in e "a%ities


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Secon$ Or$er Con$itions 5

F"nctions of One )ariab%e Ret y , f / x0 A necessary con$ition for a ma(im"m is that

$ y-$ x , f D/ x0 , *

To ens"re that the point is a ma(im"m1 y m"st be $ecreasin for mo&ements a!ay from it


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F"nctions of One )ariab%e The tota% $ifferentia% meas"res the chan e in y

dy , f D/ x0 dx

To be at a ma(im"m1 dy m"st be $ecreasinfor sma%% increases in x

To see the chan es in dy1 !e m"st "se thesecon$ $eri&ati&e of y


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F"nctions of One )ariab%e

Note that d 6 y J * imp%ies that f / x0dx6 J * Since dx& m"st be positi&e1 f / x0 J *

This means that the f"nction f m"st ha&e aconca&e shape at the critica% point

6 6:


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F"nctions of T!o )ariab%es S"ppose that y , f / x. 1 x60 First or$er con$itions for a ma(im"m are

y- x. , f . , *

y- x6 , f 6 , *

To ens"re that the point is a ma(im"m1 y m"st$iminish for mo&ements in any $irection a!ayfrom the critica% point


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F"nctions of T!o )ariab%es The tota% $ifferentia% of y is i&en by

dy , f . dx . G f 6 dx6

The $ifferentia% of that f"nction is

d 6 y , / f .. dx. G f .6 dx60dx. G / f 6. dx. G f 66dx60dx6

d 6 y , f .. dx. 6 G f .6 dx6dx. G f 6. dx. dx6 G f 66dx66 Ky o"n Ds theorem1 f .6 , f 6. an$

d 6 y , f .. dx. 6 G 6 f .6 dx. dx6 G f 66dx66


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F"nctions of T!o )ariab%esd 6 y , f .. dx. 6 G 6 f .6 dx. dx6 G f 66dx66

For this e "ation to be "nambi "o"s%y ne ati&e

for any chan e in the (Ds1 f .. an$ f 66 m"st bene ati&e

If dx6 , *1 then d 6 y , f .. dx. 6

' for d 6 y J *1 f .. J *

If dx. , *1 then d 6 y , f 66 dx66

' for d 6 y J *1 f 66 J *


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F"nctions of T!o )ariab%esd 6 y , f .. dx. 6 G 6 f .6 dx. dx6 G f 66dx66

If neither dx. nor dx6 is zero1 then d & y !i%% be"nambi "o"s%y ne ati&e on%y if

f .. f 66 5 f .6 & Y *

' the secon$ partia% $eri&ati&es / f .. an$ f 660 m"st bes"fficient%y ne ati&e so that they o"t!ei h any

possib%e per&erse effects from the cross5partia%$eri&ati&es / f .6 , f 6. 0

C t i $ M (i i ti


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S"ppose !e !ant to choose x. an$ x6 toma(imize

y , f / x. 1 x60

s"b ect to the %inear constraintc 5 b . x. 5 b6 x6 , *

e can set "p the Ra ran ianL , f / x. 1 x60 G /c 5 b . x. 5 b6 x60

C t i $ M (i i ti


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The first5or$er con$itions are

f . 5 b . , *

f 6 5 b6 , *c 5 b . x. 5 b6 x6 , *

To ens"re !e ha&e a ma(im"m1 !e m"st"se the secon$ tota% $ifferentia%

d 6 y , f .. dx. 6 G 6 f .6 dx. dx6 G f 66dx66



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On%y the &a%"es of x. an$ x6 that satisfy theconstraint can be consi$ere$ &a%i$ a%ternati&esto the critica% point

Th"s1 !e m"st ca%c"%ate the tota% $ifferentia% ofthe constraint

5b . dx. 5 b6 dx6 , *

dx6 , 5/ b . -b60dx.

These are the a%%o!ab%e re%ati&e chan es in x. an$ x6



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Keca"se the first5or$er con$itions imp%y that f . - f 6 , b . -b61 !e can s"bstit"te an$ et

dx6 , 5/ f . - f 60 dx.

Sinced 6 y , f .. dx. 6 G 6 f .6 dx. dx6 G f 66dx66

!e can s"bstit"te for dx6 an$ etd 6 y , f .. dx. 6 5 6 f .6 / f . - f 60dx. 6 G f 66/ f . 6- f 660dx. 6



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Combinin terms an$ rearran ind 6 y , f .. f 66 5 6 f .6 f . f 6 G f 66 f . 6 :dx. 6- f 66;

Therefore1 for d 6 y J *1 it m"st be tr"e that f .. f 66 5 6 f .6 f . f 6 G f 66 f . 6 J *

This e "ation characterizes a set of f"nctionsterme$ "asi5conca&e f"nctions ' any t!o points !ithin the set can be oine$ by a

%ine containe$ comp%ete%y in the set

Conca&e an$ 4"asi5Conca&e


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4

Conca&e an$ 4 asi5Conca&eF"nctions

The $ifferences bet!een conca&e an$ "asi5conca&e f"nctions can be i%%"strate$ !ith thef"nction

y = f / x. 1 x60 , / x. x60'

!here the xDs ta#e on on%y positi&e &a%"es an$' can ta#e on a &ariety of positi&e &a%"es

Conca&e an$ 4"asi5Conca&e


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4(

Conca&e an$ 4 asi5Conca&eF"nctions

No matter !hat &a%"e ' ta#es1 this f"nction is"asi5conca&e

hether or not the f"nction is conca&e$epen$s on the &a%"e of ' ' if ' J *9>1 the f"nction is conca&e ' if ' Y *9>1 the f"nction is con&e(


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4-

Homo eneo"s F"nctions

A f"nction f / x. 1 x61 xn0 is sai$ to behomo eneo"s of $e ree ' if

f / tx. 1tx61 txn0 , t '

f / x. 1 x61 xn0 ' !hen a f"nction is homo eneo"s of $e ree one1 a

$o"b%in of a%% of its ar "ments $o"b%es the &a%"eof the f"nction itse%f

' !hen a f"nction is homo eneo"s of $e ree zero1a $o"b%in of a%% of its ar "ments %ea&es the &a%"eof the f"nction "nchan e$


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4

Homo eneo"s F"nctions

If a f"nction is homo eneo"s of $e ree ' 1 the partia% $eri&ati&es of the f"nction !i%% be

homo eneo"s of $e ree ' 5.


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43

E"%erDs Theorem

E"%erDs theorem sho!s that1 for homo eneo"sf"nctions1 there is a $efinite re%ationship

bet!een the &a%"es of the f"nction an$ the

&a%"es of its partia% $eri&ati&es


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100

Homothetic F"nctions

For both homo eneo"s an$ homotheticf"nctions1 the imp%icit tra$e5offs amon the&ariab%es in the f"nction $epen$ on%y on the

ratios of those &ariab%es1 not on their abso%"te&a%"es


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101


S"ppose !e are e(aminin the simp%e1 t!o&ariab%e imp%icit f"nction f / x1 y0 , *

The imp%icit tra$e5off bet!een x an$ y for at!o5&ariab%e f"nction is

dy-dx , 5 f x- f y

If !e ass"me f is homo eneo"s of $e ree ' 1its partia% $eri&ati&es !i%% be homo eneo"s of$e ree ' 5.


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102


The imp%icit tra$e5off bet!een x an$ y is.

.

/ 1 0 / 1 0

/ 1 0 / 1 0

' x x

'

y y

t f tx ty f tx tydy

dx t f tx ty f tx ty

= =

If t , .- y1

< 1. 1.

< 1. 1.

x x

y y

x x ( f f

y ydydx x x

( f f y y

= =

h


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10


The tra$e5off is "naffecte$ by the monotonictransformation an$ remains a f"nction on%y ofthe ratio x to y


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Important Points to NoteB


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10-


+eri&ati&es are often "se$ in economics beca"se economists are intereste$ in ho!mar ina% chan es in one &ariab%e affect

another ' partia% $eri&ati&es incorporate the ceteris

paribus ass"mption "se$ in most economicmo$e%s



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10


The mathematics of optimization is animportant too% for the $e&e%opment ofmo$e%s that ass"me that economic a ents

rationa%%y p"rs"e some oa% ' the first5or$er con$ition for a ma(im"m

re "ires that a%% partia% $eri&ati&es e "a% zero



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10


Most economic optimization prob%emsin&o%&e constraints on the choices thata ents can ma#e

' the first5or$er con$itions for a ma(im"ms" est that each acti&ity be operate$ at a%e&e% at !hich the ratio of the mar ina%

benefit of the acti&ity to its mar ina% cost



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103


The Ra ran ian m"%tip%ier is "se$ to he%pso%&e constraine$ ma(imization prob%ems ' the Ra ran ian m"%tip%ier can be interprete$ as

the imp%icit &a%"e /sha$o! price0 of theconstraint



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104


The imp%icit f"nction theorem i%%"stratesthe $epen$ence of the choices that res"%tfrom an optimization prob%em on the

parameters of that prob%em



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110


The en&e%ope theorem e(amines ho!optima% choices !i%% chan e as the

prob%emDs parameters chan e

Some optimization prob%ems mayin&o%&e constraints that are ine "a%itiesrather than e "a%ities



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111


First5or$er con$itions are necessary b"tnot s"fficient for ens"rin a ma(im"m orminim"m

' secon$5or$er con$itions that $escribe thec"r&at"re of the f"nction m"st be chec#e$



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Certain types of f"nctions occ"r in manyeconomic prob%ems ' "asi5conca&e f"nctions obey the secon$5

or$er con$itions of constraine$ ma(im"mor minim"m prob%ems !hen the constraintsare %inear

' homothetic f"nctions ha&e the property thatimp%icit tra$e5offs amon the &ariab%es$epen$ on%y on the ratios of these &ariab%es

math of optimization

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