dinamica tema-ana petrea

8/10/2019 Dinamica Tema-Ana Petrea

1/21

1. Determinati traiectoria de evolutie a populatiei pentru: P0=20, k=0,03 si k=0,05; P0=50, k=0,03 si k=0,05; P0=100, k=0,03 si k=0,05.

Reprezentati ra!icele cu a"utorul #$%#&. p01=20

p02=50

p03=100

k1=0.03

k2=0.05 t

k1't

k2't

20 50 100 0.03 0.05 10.03

0.05

20 50 100 0.03 0.05 20.0(

0.1

20 50 100 0.03 0.05 30.0)

0.15

20 50 100 0.03 0.05 *

0.

12

0.

220 50 100 0.03 0.05 5

0.15

0.25

20 50 100 0.03 0.05 (0.1+

0.3

20 50 100 0.03 0.05 0.21

0.35

20 50 100 0.03 0.05 +0.2*

0.*

20 50 100 0.03 0.05 )0.2

0.*5

20 50 100 0.03 0.05 10 0.3 0.5

p-t =p0'e/-k't-p01,k1

p-t =p0'e/-k't-p02,k1

p-t =p0'e/-k't-p03,k1

20.(0 20*1) 51.51+010* 103.03(020)21.232+*323 53.0+210+0( 10(.1(*21(121.+ * ( ) 5*.()3()1)+ 10).3+ 3+*22.5*1(+15 5(.35*203)3 112. 0+*0 )23.22(051 5 5+.0(512)3 11(.13025+

23.)311))5* 5).+2 ))++( 11).(55))2*.(5 55 (1.(**3+)** 123.2++ +)25.*0(3 0*1 (3.515)2(01 12 .031+522(.1 1313 (5.***2+2+3 130.+++5(52(.) 2* 3)) ( .*311+*) 13*.+(23()) p-t =p0'e/-k't-p01,k2

p-t =p0'e/-k't-p02,k2

p-t =p0'e/-k't-p03,k2

1


2/21

21.02221*3( 52.555535) 21.02221*3(22.0)(( *+3 55.2*1(+ 0+ 22.0)(( *+323.22(051 5 5+.0(512)3 23.22(051 52*.*13151)3 (1.032+ )+2 2*.*13151)325.((0)25(5 (*.15231*13 25.((0)25(52(.) 2* 3)) ( .*311+*) 2(.) 2* 3))2+.35105(5 0.+ (*125 2+.35105(52).+000))35 *.5002*+3+ 2).+000))3531.323203+3 +.30+00)5 31.323203+332.)2*1552 +2.3103++1 32.)2*1552

2. odelul arrod Domar. olosind #$%#&, determinati traiectoriile 4-t , -t , %-t ,cunoscand datele:

40=100 u.m., s=0.3,

=0. , 4-t =100' e( s ) t

, -t =6-t =s'4-t , %-t =

'4-t

t40 s n -s7n 't

4-t =40'e/8-s7n 't9

-t =6-t =s'4-t

%-t =n'4-t

1100

0.3

0. 0.*2+5 1*2) 153.305(+ *5.))1 0(3

10 .313)+1*

2100

0.3

0. 0.+5 1*2+5 235.02(33+ 0.50 )01(

1(*.51+*3 1

2


3/21

3100

0.3

0. 1.2+5 1*2+( 3(0.30+ ** 10+.0)2(23*

252.21(1213

*100

0.3

0. 1. 1*2+51* 552.3 3 )+ 1(5. 1213)(

3+(.((1(5)1

5

10

0

0.

3

0. 2.1*2+5

1*3 +*(.+20*505 25*.0*(1352

5)2. *3

15*(

100

0.3

0. 2.5 1*2+5 1 12)+.223)15 3+).*( 1 *5

)0+. 5(*0*

100

0.3

0. 3 1))0.2511 5) .0 533

13)3.1 5

+100

0.3

0. 3.*2+5 1*2) 3051.1(+135 )15.350**05

2135.+1()*

)100

0.3

0. 3.+5 1*2+5 *( .(1*2)1 1*03.2+*2+

32 *.33000*

10

100

0.3

0. *.2+5 1*2+( 1 1.0*+ 5* 2151.31*(2(

501). 3*12+

3. 6e cunosc datele:

&0=100, n=0.00+, 0=1000, =0.05, =0.35, a=10, s=0.3.

a %alculati traiectoria inzestrarii te nice a muncii pentru t=1 10 si !aceti ra!icul in #$%#&

t &0 n 0 a s-1

-n< 't k-t1 100 0.00 100 0.0 0.3 1 0. 0.03 1)11.)351

3


4/21

0 + 0 5 5 0 3 ))

21000

0.00+

1000

0.05

0.35

10

0.3 0.0 5*

1+*5.1(+)(

31000

0.00+

1000

0.05

0.35

10

0.3 0.1131

1 +1.(5*)0+

* 1000 0.00+ 1000 0.05 0.35 10 0.3 0.150+ 1 21.223(3*

51000

0.00+

1000

0.05

0.35

10

0.3 0.1++5

1((3. 1*))1

(1000

0.00+

1000

0.05

0.35

10

0.3 0.22(2

1(0+.) 5*(

1000

0.00+

1000

0.05

0.35

10

0.3 0.2(3)

155(.+(+0))

+1000

0.00+

1000

0.05

0.35

10

0.3 0.301(

150 .251225

)

100

0

0.00

+

100

0

0.0

5

0.3

5

1

0

0.

3 0.33)3

1*5).))++

*210

1000

0.00+

1000

0.05

0.35

10

0.3 0.3 1*1*.)+)+

%alculati traiectoria stocului total al capitalului pentru t=1 10 si !aceti ra!icul in #$%#&

.17-1

10

..-1-0.-

+

+++= nas

k en

ase Lt K t nnt

.-100.- 00+,0

t k et K t

=

4


5/21

c Calculati venitul per capita si venitul total si facetigra cele corespunzatoare in EXCEL

.-.- t ak t y =

.-.-.-.- 01 t k eaLt Lt aK t Y nt ==

5

t &0 k-t -t

11000

1)11.)35

2122).2(

21000

1+*5.1()

20*+ .)2

3 1000 1 +1.(55 1) +2.()

*1000

1 21.22*

1)111.()

51000

1((3.15

1+* 3.1*

(1000

1(0+.)+

1 +(5.3(

1000

155(.+(+

1 2+(.(

+

10

00

150 .2

51

1( 35.

+*)

1000

1*5).)))

1(211.1

10

1000

1*1*.))

15 11.*1


6/21

t n &0 a k-t>-t =a'k-t / 4-t =a'&0'e/-n't 'k-t

/

10.00+

1000

10

0.35

1)11.)351))

1*0. 1 +22 1*1+)).00)3

2

0.00

+

100

0

1

0

0.3

5

1+*5.1(+)( 13).031321

1 1*12((.+1(53

0.00+

1000

10

0.35

1 +1.(5*)0+

13 .33 2212 1*0((2.++*5

*0.00+

1000

10

0.35

1 21.223(3*

135.(++50*3 1*00+ .0 (

50.00+

1000

10

0.35

1((3. 1*))1

13*.0+*20(3 13)53).25*1

(0.00+

1000

10

0.35

1(0+.) 5*( 132.5233 13)01).2 53

0.00+

1000

10

0.35

155(.+(+0))

131.0050+01 13+52(.))(2

+ 0.00+ 1000 10 0.35 150 .251225 12).52+3)25 13+0(2.2 03

)0.00+

1000

10

0.35

1*5).))++*2

12+.0)2*0* 13 (2*.)*+

10

0.00+

1000

10

0.35 1*1*.)+)+

12(.()(2201 13 21*.+ 3

d Calculati punctele xe ale traiectoriei

01

=k

)(0,*32.17-1

2 = +=

san

k

6


7/21

e Calculati traiectoria de echili ru a stocului total de capital si a venitului deechili ru pentru t!1"1# si faceti gra cele EXCEL

= 20.- k e Lt K nt

= 1020 .-.-.-

nt nt e Lk e Lat Y

t a &0 n-t '=&0'e/-n't '*32.)( 4-t '=a'--&0'e/-n't 'k2' / ? @ A&0@e/An't? ?/ABC

10.35

10

1000

0.00+ *3(*2(.)1+ +*3 (.2))()

20.35

10

1000

0.00+ *3))21.5) 3 +5051.)*112

3 0.35 10 1000 0.00+ **3***.2(02 +5 32.))2 *

*0.35

10

1000

0.00+ **())5.130 +(*1).*) +(

50.35

10

1000

0.00+ *505 *.*3*+ + 111.5001(

(0.35

10

1000

0.00+ *5*1+2.* + +0).0*3(5

0.35

10

1000

0.00+ *5 +1).255) ++512.1 2 1

+

0.3

5

1

0

100

0

0.00

+ *(1*+5.233) +)220.)320()

0.35

10

1000

0.00+ *(51+0.5( 2 +))35.3(( +

10

0.35

10

1000

0.00+ *(+)05.*)0 )0(55.52232

$


8/21

! Calculati investitiile rute si consu%ul pentru t!1"1# in %ari%i per capita si%ari%i totale si faceti gra cele&i't(!s)a)*'t(+c't(!'1"s()a)*'t(+

s a k-ti-t =s'a'k-t / c-t =-1

s 'a'k-t /0.3

10

0.35

1)11.)351)) *2.23153*( )+.5*02* 5

0.3

10

0.35

1+*5.1(+)( *1. 0)3)(3* ) .321)2* )

0.3

10

0.35

1 +1.(5*)0+ *1.2011((3 )(.13(05*+(

0.3

10

0.35

1 21.223(3* *0. 0(5512) )*.)+1)5301

0.3

10

0.35

1((3. 1*))1 *0.2252(1++ )3.+5+)**3)

0.3

10

0.35

1(0+.) 5*( 3). 5 0131 )2. ((3(3)1

0.3

10

0.35

155(.+(+0)) 3).30152*03 )1. 0355(0(

0.3

10

0.35

150 .251225 3+.+5+51 ( )0.(()+ * +

0.3

10

0.35

1*5).))++*2 3+.*2 21*2 +).((*(+331

0.3

10

0.35 1*1*.)+)+ 3+.00++((0* ++.(+ 35*0)

1 2 3 4 5 6 $ , - 1##

2#

4#

6#

,#

1##

12#

14#

16#

c't(i't(

.'t(!s/'t(C't(!/'t(".'t(

,


9/21

0nalizati efectele cresterii ratei econo%iilor de s#!#&3 la s1!#&350supra traiectoriei de echili ru

*2)!''##, #( #&35)1#(+'1 '#&35"1((!54,&,4#6#1-

k2'-t '=&0'e/-n't '5*+.+*0(01) 4-t '=a'--&0'e/-n't 'k2' / ? @ A&0@e/An't? ?/ABC

-

s 4-t -t =s'4-t%-t =4-t-t

0.31*1+)).00)3 *25(). 02 ) ))32).30(51

0.3

1*12((.+1(

5 *23+0.0**)5 )+++(. 1550.3

1*0((2.++*5 *21)+.+(53* )+*(*.01)1*

0.31*00+ .0 ( *202(.123 )+0(0.)53(

0.313)53).25*1 *1+(1. (23 ) ( .* +(

0.313)01).2 53 *1 05. +25) ) 313.*)2

0.313+52(.))(2 *155+.0)++ )()(+.+) 3(

0.3 13+0(2.2 03 *1*1+.(+10+ )((*3.5+)1+0.3 13 (2*.)*+ *12+ .*+** )(33 .*(3(

0.313 21*.+3 *11(*.*(31) )(050.*1*1


10/21

5*+.+*0(01) 553235.*31* )1( ).01*355*+.+*0(01) 55 ((5.*52* )2*13.132125*+.+*0(

01) 5(2130.)*( )3153.12+315*+.+*0(01) 5(((32.1)+3 )3+)).050015*+.+*0(01) 5 11().*)3 )*(50.)**(5*+.+*0(01) 5 5 *3.1213 )5*0+.+(015*+.+*0(01) 5+0353.3 21 )(1 2.+**535*+.+*0(01) 5+5000.53)* )()*2.)*(5*

5*+.+*0(01) 5+)(+*.)1+) ) 1).215135*+.+*0(01) 5)**0(.+0+* )+501.())(+serva% ca o %odi care a ratei de crestere a eono%iilor duce la o crestere a lui

*2) cu aproape 116 fapt ce duce la du larea valorilor de echili ru 7't() si /'t()&

*. 8entru %odelul lui 9althaus considera% datele 7!1&5 8#!1###& Calculatipopulatia pentru t!1"1# faceti gra cul calculati punctul x si analizatista ilitatea& 8opulatia pt!*)p't"1( :: pt!'1 *()p't"1( :: solutia analitica

pt!'1 *(+t)p#

8unctul fix p)!'1 *(p) :: p)!# ;ta ilitatea

1#

t p0 k pt=-1


11/21

=+= 0

.1-limlim pk p t t t t

6istem asimptotic insta il, punct !i repelor

5. odelul arrod Domar, varianta discreta

6e dau datele: 40=1000, =0.25, s=0.3.

6crieti traiectoria de evolutie a venitului, calculati punctul !i , analizati sta ilitatea -tipul d

punct !i si !aceti ra!icul traiectoriei pentru t=1 10.

Eraiectoria de evolutie a venitului:

6t=s'4t

t= -4t 4-t 1

6t= t

Din relatiile de mai sus o tinem ecuatia:

1

= t t Y

s

Y

#cuatia de mai sus are solutia:

0Y sY

t

t

=

Punctul !i :

0= = Y Y sY

6ta ilitatea:

11

t 40 s 4t=--v7v s /t '40 1

1000

0.25

0.3 5000

21000

0.25

0.3 25000

31000

0.25

0.3 125000

* 1000 0.25 0.3 (25000

51000

0.25

0.3 3125000

(1000

0.25

0.3 15(25000

1000

0.25

0.3 +125000

+1000

0.25

0.3 3)0(25000

)

100

0

0.2

5

0.

3 1)5312500010

1000

0.25

0.3 ) (5(25000


12/21

Deoarece F 7- s F= F0.257-0.25 0.3 F= F0.257- 0.05 F= F 5F= 5G1 spunem ca sistemul este i

(. odelul lui 6olloH in timp discret reprezentare in #$%#& a e ercitiului rezolvat la

seminar

1 2 3 4 5 6 $ , - 1##

5

1#

15

2#

25

*t

I servam ca de a lun ul timpului, inzestrarea

te nica a muncii scade.

. odelul ciclului comercial al lui icks

6e dau datele: cJ=0. 5, k=2.5, =0.1, K0=100,

40=100, 4 1=50. 6crieti ecuatia de dinamica a venitului, calculati valorile indicatorilor di

ta el pentru t=1 10 si !aceti ra!icele.

4t=%t< t

%t=cJ4-t 1

t= t,4< t,K

t,4=k-4-t 1 4-t 2

t,K=K0-l< /t

12

t % ktPkt=%' /t


13/21

6u stituind, o tinem ecuatia de dinamica a venitului:.-.1- 2101 +++= t t

t t t Y Y k g AY cY

t * 0# g /'t"1( c< /'t"1("/'t"2(

12&5

1##

#&1 1##

#&$5 5#

22&5

1##

#&1 1##

#&$5 5#

32&5

1##

#&1 1##

#&$5 5#

42&5

1##

#&1 1##

#&$5 5#

52&5

1##

#&1 1##

#&$5 5#

62&5

1##

#&1 1##

#&$5 5#

$2&5

1##

#&1 1##

#&$5 5#

,2&5

1##

#&1 1##

#&$5 5#

-2&5

1##

#&1 1##

#&$5 5#

1#

2&5

1##

#&1 1##

#&$5 5#

13


14/21

.t /!*)'/'t"1("/'t"2((

.t 0!0#)'1 g(+t

Ct!c


15/21

t n 7# a s gL#

0#

1##,

1###

#

#&35

1#

#&

3

#

1##

5#

2##,

1###

#

#&35

1#

#&

3

#

1##

5#

3##,

1###

#

#&35

1#

#&

3

#

1##

5#

4##,

1###

#

#&35

1#

#&

3

#

1##

5#

5##,

1###

#

#&35

1#

#&

3

#

1##

5#

6##,

1###

#

#&35

1#

#&

3

#

1##

5#

$##,

1###

#

#&35

1#

#&

3

#

1##

5#

,##,

1###

#

#&35

1#

#&

3

#

1##

5#

-##,

1###

#

#&35

1#

#&

3

#

1##

5#

1#

##,

1###

#

#&35

1#

#&

3

#

1##

5#

*2! ''n g =( as( +'1 '>"1((

*'t(!'7#"*2 ( e+?"'1">('n g =(@t *2

-263&245,26 145,-263&245,26 1,-#&6$4446-263&245,26 22--&335-41-263&245,26 26,5&34533$-263&245,26 3#4-&-5,23--263&245,26 33-4&36#655-263&245,26 3$1-&6$2,52-263&245,26 4#26&-52----263&245,26 431$&2##614-263&245,26 45-1&35-,#,

15


16/21

16


17/21

%alculati traiectoria stocului total al capitalului pentru t=1 10 si !aceti ra!icul in #$%#& K (t )= L0 e nt k (t )

*'t(!'7#"*2 ( e+?"'1">('n g =(@t *2 t n L#

7't(!L#e+'nt( *'t(

145, 1##,

1## 146-$#&6$2

1,-#&6$4446 2##,

1## 1-21#$&4$46

22--&335-41 3##,

1## 2355#1&5#$1

26,5&34533$ 4##,

1## 2$$23-&534,

3#4-&-5,23- 5##,

1## 31$4#4&1-66

33-4&36#655 6##,

1## 356#$4&2#2-

3$1-&6$2,52 $##,

1## 3-3324&5236

4#26&-52--- ,##,

1## 42-226&56$6

431$&2##614 -##,

1## 463,4,&3532

45-1&35-,#,1#

##,

1## 4-$254&6$11

c Calculati venitul per capita si venitul total si faceti gra cele corespunzatoarein EXCEL

.-.- t ak t y =

.-.-.-.- 01 t k eaLt Lt aK t Y nt ==

1$


18/21

d Calculati punctele xe ale traiectoriei

Ain conditia de stationaritate

n+g+

sa k

Deducem k 1=0 si k 2 =(n+g+

as )

1

1 . %um k 2 l am calculat anterior, k1'=0,

k2'= -263&245,26&

e Calculati traiectoria de echili ru a stocului total de capital si a venitului deechili ru pentru t!1"1# si faceti gra cele EXCEL

= 20.- k e Lt K nt

1,


19/21

= 1020 .-.-.-

nt nt e Lk e Lat Y

7't()!L#)e+'n)t()*'t(+

/'t()!a)'L#)e+'n)t()*2)(+ )'L#)e+'n)t((+'1" (

12-#&5$1,21 2465#&$-1631424&$65442 24,4,&1,232153$&-,5#46 25#4$&153611636&,35621 2524$&$1,161$25&12-232 2544-&,,,$31,#5&2-35#- 25653&6$,1,

1,$,&-,33- 25,5---461-4$&3,5#4$ 26#66&165652#11&3,2#4$ 262$4&,,--22#$1&652-$6 264,5&2,555

! Calculati investitiile rute si consu%ul pentru t!1"1# in %ari%i per capita si%ari%i totale si faceti gra cele&i't(!s)a)*'t(+ c't(!'1"s()a)*'t(+ .'t(!s/'t( C't(!/'t(".'t(

i't(!s)a)*'t(+

c-t =-1 s 'a'k-t / -t =s4-t %-t =4-t -t

3,&4#-5-#65 ,-&6223$,1,3,$1&$15

46-#34#2$

4,

42$16 -,&155333$342$4&2-6

33--$3&35,#

-4

45,6,35, 1#5&1135-54613&-55

141#$65&,-5

32

4$&56322622 11#&-,#,6124-1#&5#6

,61145$&,4-

35

4-&$3#64$,- 116,1$,451$5&3,$

6-12#$5&-#4

62

51&62,14,36 12#&4656$-55415&,,#

531263$6

56

1-


20/21

53&3#,6$,35 124&3,6-1625636&-5#

1$13152&,,3

$3

54&,1#41##3 12$&,-#-56$5,42&155

1413631&6-5

33

56&161-314$ 131#6,6#34&146

1414#$-&6$4

33

5$&3,53#,35 133&,--#52,6214&-5,

-3145#1&5$#

,3

1 2 3 4 5 6 $ , - 1##

5###

1####

15###

2####

25###

C't(.'t(c't(i't(

0nalizati efectele cresterii ratei econo%iilor de la s!#&3 la s1!#&35 asupratraiectoriei de echili ru&

k2' k1-t -t ' 4-t '

22$&-,45--,

-5$&2#$2##6

1113&,22122

6$41-,661

22$&-,45--,

16,6&42-,#2

136,&,,322,

6$-5$$,66

22$&-,45--,

2415&6524#2

1564&$,#112

6,4-&4,-3#,

22$&-,45--,

3144&,$5##3

1$2-&,,#,--

6-#4&33644,

22$&-,45--,

3,$4-$6#4

1,$5&$6516,

6-5-&622$$6

22$&-,45--,

46#3&32#2#5

2##,&43342$

$#15&351,#,

22$&-,45--,

5332&542,#6

2131&44#-3,

$#$1&52$#,-

22$&-,45--,

6#61&$654#$

224$,61$,

$12,&1521-2

22$&-,45--,

6$-#&-,,##,

2356&-4-$65 $1,5&23#$2

22$&-,45- $52#&21#6# 2462&1$##5 $242&$663#

2#


21/21

dinamica tema-ana petrea

Documents