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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 #$%#&. p01=20
p02=50
p03=100
k1=0.03
k2=0.05 t
k1't
k2't
20 50 100 0.03 0.05 10.03
0.05
20 50 100 0.03 0.05 20.0(
0.1
20 50 100 0.03 0.05 30.0)
0.15
20 50 100 0.03 0.05 *
0.
12
0.
220 50 100 0.03 0.05 5
0.15
0.25
20 50 100 0.03 0.05 (0.1+
0.3
20 50 100 0.03 0.05 0.21
0.35
20 50 100 0.03 0.05 +0.2*
0.*
20 50 100 0.03 0.05 )0.2
0.*5
20 50 100 0.03 0.05 10 0.3 0.5
p-t =p0'e/-k't-p01,k1
p-t =p0'e/-k't-p02,k1
p-t =p0'e/-k't-p03,k1
20.(0 20*1) 51.51+010* 103.03(020)21.232+*323 53.0+210+0( 10(.1(*21(121.+ * ( ) 5*.()3()1)+ 10).3+ 3+*22.5*1(+15 5(.35*203)3 112. 0+*0 )23.22(051 5 5+.0(512)3 11(.13025+
23.)311))5* 5).+2 ))++( 11).(55))2*.(5 55 (1.(**3+)** 123.2++ +)25.*0(3 0*1 (3.515)2(01 12 .031+522(.1 1313 (5.***2+2+3 130.+++5(52(.) 2* 3)) ( .*311+*) 13*.+(23()) p-t =p0'e/-k't-p01,k2
p-t =p0'e/-k't-p02,k2
p-t =p0'e/-k't-p03,k2
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21.02221*3( 52.555535) 21.02221*3(22.0)(( *+3 55.2*1(+ 0+ 22.0)(( *+323.22(051 5 5+.0(512)3 23.22(051 52*.*13151)3 (1.032+ )+2 2*.*13151)325.((0)25(5 (*.15231*13 25.((0)25(52(.) 2* 3)) ( .*311+*) 2(.) 2* 3))2+.35105(5 0.+ (*125 2+.35105(52).+000))35 *.5002*+3+ 2).+000))3531.323203+3 +.30+00)5 31.323203+332.)2*1552 +2.3103++1 32.)2*1552
2. odelul arrod Domar. olosind #$%#&, 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
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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 #$%#&
t &0 n 0 a s-1
-n< 't k-t1 100 0.00 100 0.0 0.3 1 0. 0.03 1)11.)351
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0 + 0 5 5 0 3 ))
21000
0.00+
1000
0.05
0.35
10
0.3 0.0 5*
1+*5.1(+)(
31000
0.00+
1000
0.05
0.35
10
0.3 0.1131
1 +1.(5*)0+
* 1000 0.00+ 1000 0.05 0.35 10 0.3 0.150+ 1 21.223(3*
51000
0.00+
1000
0.05
0.35
10
0.3 0.1++5
1((3. 1*))1
(1000
0.00+
1000
0.05
0.35
10
0.3 0.22(2
1(0+.) 5*(
1000
0.00+
1000
0.05
0.35
10
0.3 0.2(3)
155(.+(+0))
+1000
0.00+
1000
0.05
0.35
10
0.3 0.301(
150 .251225
)
100
0
0.00
+
100
0
0.0
5
0.3
5
1
0
0.
3 0.33)3
1*5).))++
*210
1000
0.00+
1000
0.05
0.35
10
0.3 0.3 1*1*.)+)+
%alculati traiectoria stocului total al capitalului pentru t=1 10 si !aceti ra!icul in #$%#&
.17-1
10
..-1-0.-
+
+++= nas
k en
ase Lt K t nnt
.-100.- 00+,0
t k et K t
=
4
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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
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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
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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
$
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! 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(
,
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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
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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
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=+= 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
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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
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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
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.t /!*)'/'t"1("/'t"2((
.t 0!0#)'1 g(+t
Ct!c
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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
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16
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%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$11
c Calculati venitul per capita si venitul total si faceti gra cele corespunzatoarein EXCEL
.-.- t ak t y =
.-.-.-.- 01 t k eaLt Lt aK t Y nt ==
1$
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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,
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= 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$6
56
1-
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53&3#,6$,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###
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