Appunti ed esercizi di metodi matematici per l'economia

~tOf\.it Q

0 \ 0)\)S\J \--1

f\ts\.~ ~:0 (\~te ~~l ~

la

\eaì9 teyt1'eç'drffi t l'./'\.1 1

1 Y)c

~\ ~(\J~\. \_'UGl\1()~ H:co~\-o

0 g

u::ri\SVfl\1 Ò\ (.,'t

~ f l . .

C ""

f'Q\'0 l ( OS..S:\è\

\.ffi ~l/, e~"-C\ 8~ ~

IIY' \-h (D10-nQ')i2

te

(Q

<no s'-

l,rrrure eq

s~ ~ro \8~-l~

hs

rfiB

la éìSS-oCJèl@

r c= a~+c J

è::- ~f'\..Y-oNè ~~\MLé ~S.·.J~

fiL

b : - \\0

fl0tO<"Ù

CON&J'«)

Y Y-~ ~() l ~r::tl

Ce b1 1L1

1'{'1

f'N\\$0\XP-1 \ D\

\R:sr~~) ~ J

0 8

U.S<ALE L

L ,LLM:TJR..~ ?

QJ\Ncu CN MocB.J_Q

È f\ffl OAS\/J:-

' , r~OrQ- ck~~ es~ eesm~

(!0ke68?

d,

\)ca e\ ~~()TE? \BC~ h

<:n-oeo (t

m--ed

l \-o

d m l0B

)e

v (Q .

Oì CC-.J(c\ \d

tav1~ \c~ ~f\~OX)' ~

d\

1J :::e Q

f\cc<X.t:-"

e>:::n':S "'

1èf'fr0

~~ e~ e~ d.Cnf\.•J.S.S\("5'('&~

tcll

CU\fà U1'B

è_tj

\€fu_. ' \:C"' -

~\e (nX}~U, h~r1 (d~s.erY\larro

Jro rrn

'

s~ ~

'o'6 rr&

lh1oclo ero

\-o ero IYì-f?

()e

cl &COJréì r{>él

VD

:1 yefl'O lro

&0c/Y)

b k2fY'

Q'8Y'Òo \ \&\or c'ell9

t . G'n

()d 1

''

\e

t

(m ~Nà. rne~\e

\-e

Su@21 fKCQJ ' .

)l (J21'(1

)

\ -- (

<?

\tt)d::~ N0.ì-\:Nn~ \NO\~--\- \è?\:;1~

\T

\

't>

f \I" '1Ylf>

OJ~\'W '\JJ..eì ~ OÒtJ d~

'li8o

\Pj) DJ-.nJJP

ìP

-( 'noo-e

i \~ ~~ ~a ~~\~wòòB ~J

d-Wçy ;

~(xx] ?0

- +Xoo+(,X <è cc.;;;·)(

x r \p -noo-c

èfO? 8{-wò<::~ÒÒeJ \~V

c),VJìw OJ \\UJ-Jt?

/-------1

&

0\N\\~ ~

,. Mf) @ 1\ lO

BJ'è

a

L \l.lbQ~ D~ ~ t2{ ·d(

l{Ò'çò \0 Wè}u.J

0\W

) 6

P 'oJ \.. .

'f

a>~ \WllUJ~v o ~

?

l ('G](}:

'(f)~ \7lf\~ f\ID)~QINl \

' /\'iLI

(O)(f /

CO\-Jb70J7\/)

Q 1

:fl-' 2

~~'db0'2.. ~7:/(ì~ ~fì t~ ~Od\

.

okVlSSGJ O.U

(O) 'C

=

Cr] :- u:

(r)

cz·)

+

-~ \··) \ z + <2

2

=(ù)c (<q- + OC CC+("L) ç;~

=- 'Js

~)é) -\- ~]

( :: - z(<è) ( + =-'c

06

7 (L}Q

\Gu..U~ ~Yf~ ~)'Q

?(

O)Q (f)C)

0 F 0 CG ·P,J

7 7 (

1 7 1

:Ba (JJ~ ~ ~o ~ ~ c:1~ ~s ~,w

'l) a w

0\ ì2ll

).tU) c}JJ

' <è .

J. ~

L-V\ O '?

~)~'J OUJO~rd

)[ m)()e\2; mJe

un

_q_

OJg] ·oR9 d

~'et) ~~ebo.AJ ~.· ~ìWJ'\J ·e,A\?~0)

?

? ) ():-- 1

loy)

~- ·e1J~G9

))\ : - ( = (<cOl ~')

fMé1" òJ

· \d( CJCd

d;:J

\ \ , l \?j \ IJOCI"Z

C'P '-'-'" l.lE>

<1\ ()kD<)

1

·*-co-c

~ NCn

-::n ttl<) \'18 N

b'l..N

Ol\1N CO.

0J

~( ~) ~\"-\

o o

?ll'J~CP CJf\1(\J~<)- C)~Ud2J11::J))t}

. 9'l.D013) 2

:;n~ w~

11 cor:J J

\- t=\

"lli\\~~ ~~vvf>J ~0\.tV'W'\

C::

\M.

- ''tld

(.f

o ~ ?l.lcN~ ..:\~

.

a l

= +X

, X) 0G ' · ' , '· . '.

C-

=(0),0. ~"

t-(o)Z-- QG-=O(_ -· , . .. : -- ..

v cc

l~)

' z) + =

( " - l tì

l ~'(Xl~-2X+w~

tèr x é[o,2:>J, -6-t?.o>a

~

-, l

l ~t ~

le

l\€1

l x:)

c\o tf"E'

>

o =

cm 1m o co l'(lstcif>r.:lJQ

t ':>

}a crQ .

dlYi'f"'Q ()l

n\\ n'

DeR\~ ~> c~ r

\a \)\

òecn\

~orm,,s02 Dn~]?r'' ~u

Sèc.DNt:P-.

-\ :::.) \,fR\?ft\ l

u-Y; le

qucrd., C:O'fr'<?

0BYJ~\\e ~"'~ d.'-J,;t/~;;w

~"()() ~Q_ _rr<2~

fo, (!c) !

O'' do

.c.

: - 2 o 1Lm

&Q_ 'h\ \<.D

=) CCYf'&)~rdt ~

\e ÒI...Jf'rn2

__

UiP.JI r.(HcifY'Q .

-~\~là ~·-~ d ce\

U1'G

COfòl rn0\Sc.e_

\.<n\\ l

1

~, ·1-81~ ~

c\c=e e(ckrdo. _

1rdm()O

1

<~'.oé:' ~~co 1

re

l l3 1a

!il vm cend>rn1.:1

Jcno t'E' IYlle d\ h, 'du~nYc\N?

fer0de'Yld ·

e. 6\ll

te

\3'

(X)Sih\0 \

chiìn\ n'"'v\.ScJ2

. l

) l _ ><. . .

èh

\l. l . .''. .· ' .

b

- \À ft)T~ l\~

R...- 1

li '

J ... \-- ,, i'\.0

u.:_. •

~~ter.e_ ck:rm~Go ~~eç ~Q(\ ~\O~)

C

&. 9(

~( 2,~ ) ~/ ?

S)

0-1fmESE e

K\Af\,0

" (J)'èf>, ~rttQme cf2

'-;> t\o\)

() ( ) - ,

2

·t' + d.orrn~,.h)\D e~t(~f?ot>ì.c

2o )(

+ 3 . . .

S .:.::• -

2, )ç -ì

8lQ.s)~ (2,S}~i- C2,s)+~ ~

- ui

w :-(to,'f5

en

~l~ ,01"\e~ ~\ ~ - \e LBrd,~ S'&~

fh"r8 Sc::JYO

X:O(ò

rcd ~\

Cl (J-==(. \ O!YIL

9

h. ~ ~ ~

-f- (

S) " > 8 ,

S '3.

0::, , 0Jrn"lta cr

=')

O , (TG

\a.

m\

C'r\

L~ ~ll"tco slc,~\1-~01'\-e

), ' -\è

tn jlll\"erval22a 1

c~ sCMnt-e ·

_) 0 (; F)

X 8

9 O)

l

!

l

l t

l G~

(X){) h

y Cl\

2

G

9 ~ ~ ~ ~ ~\

[ l

c

( ro

G f \

/ 1

o

x 2 h 'ò (Y\

g

l" ?,

\ l l - s

l\ l 11

(x) \ i.

t- 't\

l --! ;~

:

m~'tllt'l ;z>·.~ ~. "Q\J \}\W

:-.)JPCW\! X JòÒ .,

,j; 14

:-""~ ~· i-i>-)~0 l~ ~t

.:'.) . . Ni3.UJ\,W -\.W

ò

~ avQ~0 ~ ~ .d-JP~I)

-~

. r 2~~JJtt)

- --- · ·>

'

.. -, - E( ,~QJ\ \~~ ~\?l\ ~p ~

. .X VJì

l) ll

\à? '2\

~ 0:\UBJjj~ ~ ~

ò \é72j

d )ç:y<YJ0:\S. Oì

-~-- ~ ~ ~ '4-~LJ0\

)( c-=

() - \!

. , , - Q\ ' UD--W ()J,KJ

~ ~oo ~'S' ~~ es~~ ·

.

.J ò

-:-. \lj

~ ~Ol\,<..uf'Ò ~

t j

; 't?'(

'u1'J;f2Vl. Wl-i

·dJ

l

' ,. - . f Ò-{~ -~

.

.·: ·,\r

. / . .

· \I b \

~\0\ ~LABLWì& fJ~ì

(X) \W

~~ o;\.JJ \P

J& iYlH'-' 'L

;))'J\.)J\

r \P

1 JD\~ ">< 8--u

-~.:>l~ ~ \) .x Ovu.&lPJì :-

O>

' \;)S.().U()J P.)

(\JJJ l

: r:tfJ Ty :: (;

~

\)JJ 'L

X\) (O) ç- (

::

~\~Q e~\ò~~- V~ ~ ~g

'Q ' i·'\Ì) r l

~ o

OY:)W .ZS!

v-

~ Xl:~

(xJ

/d e

6~,A6~ ~ llCDf><9~,

h

0\(lì1ore

4 \0-dt??

~ ~ ~

J,

d

\0-lo-e d, s.Q

:J \

--........ ·fX1

rf)U\1\

lli)ì G\

~e r()~

CC'(Q)' c' de c\

bi?

va

\CT'Cr€ 2

x \!

Polc~- ~

h e

Ll x cc)co\,è_('{ì-() ·.

L)

COryO.s;:c,af"'{l()

Llb . : - ~

<o ::. -

(YYì:: .LJ.x 2 h(

-\ab?)~ b=

2.s

o)=

.Oa\}Q. / Qff 2S,

eu\

~lx-)-=-~)( +2s

\-a~lh ~w"'~k (~}cò ~~~\

):-St:RG+/0 2: \')

. .

)8 \

dOQrO~s , ~rn 'h C

mì'cl

.Q';Y"f'r-<.?rO a\. L\..) 1

~ "'~\'f

) l\

02.6 ob

'"K d ere\

' \8

febbrcl~O ~4, 1 ·+

Cr-t'f\J~rf)-() ~ mUli~ ~ '

d\

CC:rn

·~~cor.ç·, ~~ u.\c-ek)~\o ~ ).

(d (?:ìr))re c\\

rru~ QS(J~-Ss-a

l+'> ul-8Yìt1

e 1\

0

CùYì h

V0 0/'fl \ .

C'\'f\, l l

11 1

0 ·

~ 121

s

o ~ LOJI~>o

. l J

\ 2 3 G

l S l O

't

seol

o no

9W

? 758 l

=t

· ) +) \ ~\-l]_O!'({) d?~~ ·i-

c\$ç()\re lE?

robS8rfr0 \Brldb\

u

dò"&

G~ _0~<'1!)\ ~ ~~ ~ O ~

·i- lt ( l

d d. b '?J ,

dS.Sod3 \ 1, S

'2, ?.:>i

o{ C , r 1

t4- .

\co o \é:' 0

Cf'f\ \[0.

'-'(Y\

~~'~\lE ~ ~S\) ~~~~

È W

R\CA\ARç. \)1\t

) Q flWo~'}t4l:\O f\).j~RO U7~1 2l!()'1-?

Oe-2 ()) 1'\.k(. u1o~ ?'

l -\--3

' l

~ci-~ ~ ~ ~&Saf'\'lk ~~

0 k .

G:xr:)) OJNì h ( :() e

QE?\ l

C

4. eX>).

l .

l .

. ·_x-

yj : ~

·\f- X1

\.f~- ~z-

Y1 ' Xj

- . y - ~ ~

:t _. - 2 ~-

=-~ X-:2

'Cj 2

2:> - .

+ L{-

\00- 2 l

~') l~~ ~)<

l~ ~~x- l ~~

2\j- = 8b

-

1y x -

co~ l~

t

\(56

c '? 0

qaw \à?

~\'0\ ~èYLUJC\W ,~~

\çSOJOJ\?

U{J s :.

~JUJo.ucò ~.x

\ç ò.Jvòl.l-\0

i::_cx:f'C h - : <es

~

(<G)

ffi '38- ·c;/

"G

~~l c:MDL\~\J\.~~~d0\7 (\\~

6->SèDo.ld _Q

\'':> ,

\7

O.ò'd~LN_s:- 2"JOl\-~ d~~~

? è,g\OC:·

7t

~1l<C}\SçQ:) ~(:)VJ ~\ç·\fB1:3 ~

Wn

4) 'P

v; ' ?e olo.A.0~à \.LOJ 0\SCDcxJò

UJì

Ott\f10ò

)...<$:1 ~ ::x'\l .

~f)g-1) x~ ~- hov ~o"FèJ~--· x\J~

0 "ar o

= XÌJ<0

hOJ -\- ù)CJ

, -.

=A m yt")g

x +

Q\

o ::: +-><'"10 ~ ~))

0\

d c;lì

\'-\)<~}t ~

( ::: f)C) -\ 0:C

} -· G<(. ( t)

0}

)J !Jg

~( :\"IJC'H(S'ì i'é±.

S\ Ol

J (o J (

+

rJ9 :: b8

~

) 1J 9

O'tì l

iJ --1_

0\

( -K :: 1-x·

c:~

s. r

1J

h 'C~- g1j~X8ÌJ

=

n X~))= ~Z~+8ìJ- <1\-X~\J~{)

CJ\-f8D

~(t)J =

J (Sl} )2-'0 9 C?JS

~ \~C;,

Y

. ott'O~ ~~ C'\~$ ~

\1 \ì

0\\g,} yç\;y _Q

)or

-\ \?

op.;, 1J

t\

,, IJJO

\\U)

i 1-

l9 :d

\UJJO~ p

0 \j

7

0\l.ll ~J Or

C(:G

0\c{)()$ a$

~\t)) +cDS t- fJ7X

70

è))

t

} GDC\ -: C'OI)C- ~ ~7X7'l

(11 0)) ( "(OP

~ oo~,

(O) (CD8

CDS -t òlléJ?

\81\J \1

O~~l '~ o~ (3CX1s ò

OJé).JJJ0W

èE~\ d\~<?\\ -·'~~ lvç~ ~Ja

1_\wo.sd-0 "(;()J<

m (D)OO$ r

J (h])

~~(DJ 008u

OOJ

C;: c d?

'P éfl1

\J(;

éJ\J v~<:;.

\ ç:,' CfJJOJ a

\1 d

OJa-wNJ

~ ~ òfb>~ \t~dJÒ ~~~~ ~

\CD

JS E?>ò

\d<;(j

1 .. "OQ?S6

l S)

( (.: )OO<G' o

J

) 9) ~· ~

( C:

0- S OO\D

o

'f)

\df'J \(? ò~cO Òo~S' -- ...

SCh<òl \) GV()..U)fìUJ

~a CX:>\D

(C\ tOGJS\

-=

ool)s CD'IJc;-

t

- \ OQ'ò ~ OO"Cf.;- - OOÌJG

\--G>QlìS

t~1 oo"r)c

--c-

~1 ~ ~ ~~\ò)

·sG

01t-D?3~{ ~d 0\

(\ç

~CBLW.t OlJOQ~~ Ocn~() :

c... o

s'

<.;::

cSCJ'rìG

C'S t<òC)ç

oe;~- ~l; ~ ~-/::...

OUJf''C{J'jòOE? 0/r: '(:;

tìS \'

ooag-c;

gç, ) S

05 *- l:CG ~ ~OÌJ

.:

'!::.\)

l S

, \OCJ't

\CD "(CIS 'SLOlì

):_ OS' O OOOJZ)

8C/ - ")1)2

t

) OOj(i

)j_ jp

~

=

t Cè t

cx:XJS9 1S1 3

) OOOS9

OS

( .:: 'sZS<(.l 'D - CJS-k(ò9 (

r:v

o

~'~Dd:si ~l)

t.

b ~ ~C(: \ ~ ~ ~- ~~~f"B.

(d) - \-d)

ç·oz; ( IG

VCN~

b ~CN~ ~Sci'r\

· #

<._-:: N

llli11J Nì

\ 1

p

0 ~ ~(:?èlc) N9 _\11)

\ (\$)

\Q O

*

(e td os ·

a

, a;

)_('QJ;\'1 2

=

d

b

1 ~ yO y

{ gCG \ G (

- ) -\--OS

- z ro'j :: CXJg2BE

:' ~Q\CJ<G

9 ~ ~e~

( .:tr 1- 0 s

·

a; Ò c::=

ll.C\CN ?FJ\-::;,

21" 2

b ~(dì _(ç:+O)- .:- OOG\g<t:

CG \9 \OZ 'LWa

0\:8~ò ~'d ~M~

J NO'J

\"l

( \N "kD

l N 71J b-W9\3 Cl

Or\\ \O ·olLVld

~ ~\ oc~ ~èSOJ;y) ~-:J70

C) - dj

b =

(

l ~

( :: () S

Ò \- -ç =

g \ \OG O

( e,

--

Gd- a

g~ ~dG"

J;-

\ \o-z.

dc:-

è.à- o::ro"Z,~<t,+

~dr ~"è ~ ~ ~ c..~

· · - "P9

ooc,"\

,. """ o / - z

à ;.:c;>'Gc CY\J

z. .- 1

l\:ç.- (cozfYèlCv.J ::

~ ~:ÌJ_ lJ

ò - .z +

~ ~:~

d : - G lO'Z\ OOCJ

-=

lr l- 13

(-f

0 2

z. oog -~

/ L

l

0\';:J~çf' çy..

9JJ

<00 '\2 ex-p

\_L' 0-i~W ~\W l~U.{fl\\1.1\i~ 0\..AJ~

~~{\\~(]Q ~csonw1 1\~è>-UJ ~ ·.BQ~dV ~

\J-:1' 98'"0

l

·.

.. ,. \8)-::>

psr e

a ,·

ç;, ~~ -oo~ ~~ O\~ ~o

\_t? \llÀ0J

\jJ

1

.s

1) ·èJ<è

++\;))-è G

W ( ==-

ce -\-7

0 8

O

w

)C1d

\'MQO 0\N 7 2

9 \l.UJ~!J ~t>UdC\Jc)~

2

( -\-· O<:_ L.UJ 0\<.Al

~L\)J30 WJ~)\'Sò ~fc:l'alÒ

C'$;

0\\l Vì"J

1\'!31

ctJ(U)

1

'lr;; "l~~~' ~~u....t.:i ~ ~ ~ ~,) ~)

\)è)WJ\J

VJ) <JJJ

\à

~0-T(ì;-) ~ '9.{\

è{)W)

OltOd~<;~ \~ ~biJ ~~ er

s v

l

e t

o + d--(J

~ ~

~ òti)~Wld

z

t) <J

-\- t-

l)

l i'(:U~3

( ( '\çjCJW_J o

w

(9 ~(8)L,'a=(8)'3 =<JH~2 % 8

08ç+

<:=8\;CòY 1

'PJ \\)

P'\

l<..w

-Ut9\)~')

:1s=; C\

~ ~v)

0\.t_\)j

~b)) ~boO\

+oOO-c

=

b ~o s~ol ~D' \~S'S'~~<;'~ ~

::- <. \:t72.::::o \-r=>

\ ìtJ1?

.J l N 7\

'o 7'\J

d~ )79 \'A CY\J'èP\9

\).:b\l

( J (=:(b) 0~0] O't)~\7\:1\tt)O\~ \p

(& \tV

Oy dJd

~o~\~ \~:) 1\

\72 \0

OJWJ \.0

c:NtD\9 .

~ ~(b)) ~(O~JCOF"i-OoOC: c:o:~f6t IJ~Ìoy

oy=·b jE>

~j(Jì~ ·

9

( r e

s l? w

01dd ~

El

L\ wouJ..JlJ..lP

l('\

1

IS>;n QJ-;

~(b)S boos (b)N

~~OJ \(\~) ~

t'J '{ò

B)OJ' d-uo40Jl ·

tUJJ

\' ·

~ pl\;)~, )~ ~ ~~~ ~b \W'l~dr ~~

b

}Eò

r lP

,

'l'~ ò~?

m \)l.ld6

.

'P l"'

\b

-( s

~ ~bex;v+

bJ bcos:-

=- ( = cxm; jp

c

b 00) ~ )00~ (DG DC:06

) (Oz

oos

\-

( OC0-G -c:. =

J

Ol~ ~t! Y)

Dèfl5=f. ]::_ N . ).":

<4

= 1/

'o'l.l.2'Jè3~1N

= OOC,

OLSO/ ":S1:! 0 1

<-7w) OGSy

.x:N

G)~t;\-\ ~

0..U

O:L<;QY) :::- ")d "'-\J-.1

-=

\-X}

J 9. ;:- ooçf ~zç

- -+x'-UJ +-x

tr --

o)_çGJ

~O'in10Vc)

\O

(e ~±) "è:

I\:(((P1l3 o;:N) 1J C5'%' f)..?JQ!

\a

OL<;'O)

(_<i

)J ~ t(~)

OOSy :: C0}.:.5

OO?v

o :::- Qr\tXD\<Jf\D

~ ~LC1.0jON\7Id 0.1-~ =-.=1.~0'\JV\1

O)_<;OJ r

o =

2>7

~CfJ ~(jtf),.( ) oo~

'dV

~S1 'OtJ O~· CO Pkt) -J.:_'r)

ocs ( <a

TI CA1Y) 'CJZ\) \\ CDO \

. . ) O

,. l

tJ

'\w

0-AD~~\J f\.~ ~ ~ ~

Gf\'J Y t\\ o-z.b O

UÒ Cb) OOS (

\t

(ì 1

,

1

=aN~<} \:N V !

" ..:: l-) ~)- (X)SO'- .:: 56

UJj - l\