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A contractive method for computing the stationary solution of the euler equation

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(1)

❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s

❊s❝♦❧❛ ❞❡ Pós✲●r❛❞✉❛çã♦

❡♠ ❊❝♦♥♦♠✐❛ ❞❛ ❋✉♥❞❛çã♦

●❡t✉❧✐♦ ❱❛r❣❛s

◆◦ ✹✺✻ ■❙❙◆ ✵✶✵✹✲✽✾✶✵

❆ ❈♦♥tr❛❝t✐✈❡ ▼❡t❤♦❞ ❢♦r ❈♦♠♣✉t✐♥❣ t❤❡

❙t❛t✐♦♥❛r② ❙♦❧✉t✐♦♥ ♦❢ t❤❡ ❊✉❧❡r ❊q✉❛t✐♦♥

❍✉♠❜❡rt♦ ▲✉✐③ ❆t❛✐❞❡ ▼♦r❡✐r❛✱ ❲✐❧❢r❡❞♦ ▲✳ ▼❛❧❞♦♥❛❞♦

❙❡t❡♠❜r♦ ❞❡ ✷✵✵✷

(2)

❖s ❛rt✐❣♦s ♣✉❜❧✐❝❛❞♦s sã♦ ❞❡ ✐♥t❡✐r❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❞❡ s❡✉s ❛✉t♦r❡s✳ ❆s

♦♣✐♥✐õ❡s ♥❡❧❡s ❡♠✐t✐❞❛s ♥ã♦ ❡①♣r✐♠❡♠✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛

❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s✳

❊❙❈❖▲❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❈❖◆❖▼■❆ ❉✐r❡t♦r ●❡r❛❧✿ ❘❡♥❛t♦ ❋r❛❣❡❧❧✐ ❈❛r❞♦s♦

❉✐r❡t♦r ❞❡ ❊♥s✐♥♦✿ ▲✉✐s ❍❡♥r✐q✉❡ ❇❡rt♦❧✐♥♦ ❇r❛✐❞♦ ❉✐r❡t♦r ❞❡ P❡sq✉✐s❛✿ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r

❉✐r❡t♦r ❞❡ P✉❜❧✐❝❛çõ❡s ❈✐❡♥tí✜❝❛s✿ ❘✐❝❛r❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❈❛✈❛❧❝❛♥t✐

▲✉✐③ ❆t❛✐❞❡ ▼♦r❡✐r❛✱ ❍✉♠❜❡rt♦

❆ ❈♦♥tr❛❝t✐✈❡ ▼❡t❤♦❞ ❢♦r ❈♦♠♣✉t✐♥❣ t❤❡ ❙t❛t✐♦♥❛r② ❙♦❧✉t✐♦♥ ♦❢ t❤❡ ❊✉❧❡r ❊q✉❛t✐♦♥✴ ❍✉♠❜❡rt♦ ▲✉✐③ ❆t❛✐❞❡ ▼♦r❡✐r❛✱ ❲✐❧❢r❡❞♦ ▲✳ ▼❛❧❞♦♥❛❞♦ ✕ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✿ ❋●❱✱❊P●❊✱ ✷✵✶✵

✭❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s❀ ✹✺✻✮ ■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛❢✐❛✳

(3)

! "#$ " % & ' ( )* +,

- . -/ & 0" $1#0"" % &

# ' 2 , * +,

2 3 4

2 ) + ) 2

2 , 2 & 2+ 2 2

5 6 7 ) 8 ( 2 5 090::

2 2 ) 2 ; +

) 2 5 00": , 2 , 2 & #

5 + + 2 : 6

/ #< 5 009: = 6 5 "" :

2 ; )+ > ( & ? 2 2 ( '

) (2 2 , ; ,

2 2 (+ @ 2 2 2 2 , 2

! " # $%%$ & ' " ( &

) *

& & +, - . /%%%01211%34!5

+, - .

(4)

2 A 2 , 3 2 ( , ' 2 )

, (2 2 2 ) (2 A

; 2 5 % 5 009::+

2 , 7( 2 , 2 A A

(2 2 ( ) 2 ) + 2 #

2 & ; 5 00": B 2 , + <

6 / #< 5 """: ( 7 ( 2 2 , ( 2

2 B , 2 2 2

2+ 5 00" 00 : 2 2 ( 2 B #

) + @ 2 2 ,

A 2 C 2 5% 5 00 :: 2 B # ;

2 5= 5 099: = 8 B 5 009::+

2 2 2 , A

) + 2 ( 4

(2 2 2 2 ) 4; + = 2 2

( 2 + < 2 2

2 2 2 ) 2 , +

2 ( 7 2 , 2 A +

2 2 A (2 2 2 ) 2

2 ) 2 ) + 2 ) ,

2 2 + 6 ;

(2 2 4 2 +

2 ( + ( 2

2) 2 + 1 2 , 2 # A

+ ) ! 2 2 2 2

+ < 2 ;+

!" !#$% &

2 A , 2 ( ( 2 , ,) 2

( ' 2 2

, ) ' 2 (2 B 4 2 ) A , +

D ( 2 ( ) E , 2 2

4 2 +

8 , 2 A + 2 2 A

5 : ( , + ) 5 : F G +

< A , 2 A 5 : 2 2

5 : F "

) 5 : ' 2 2 '

5 5 : 5 :: F "

3 5

7

(5)

+

D ) 2 H 5H: F H+ D ( 7 2 ( '

"" #' H

5 : I 5 :

5 :

$#! &( 2 2 ; ) A ) A ,

2 2 ) , 2 B F "+ < )

3 2 '

5 : I 5 :

(2 2 ( 7 2 +

< , ; 2 2 ) , + ( 2

6 7 ) 8 ( 2 5 090:

5 : F 5 : I 5 :

(2 2 2 2 4

2 + 2 2 ) + ) )

2 2 # ) * 2

2 % , 2 ) +

+ ! $" ,

2 ( ( 2 )

2, 2 ) 2 5 :+ 2 2 4

2 ) A , A 2 ( 2 2

( 2 2 ) 2 4; +

- " " H

F 5 5H:: G 5H: F H 5 : 5 : 5H:

(2 5 5H:: 2 # 2 ) E , 5H:

+

4 2

F 5 :

8 H , 2 ( 2 2 + 2 5 H :

+ @, 2 ,) 2 4 2 5 2

4 : ) 2 H , 5 5H::+

38 8 85 ! 7 6

9 9 7 % 6 9

(6)

$##! + " " '

5 5 : 5 :: F " 5J:

5H:

-$ $# + " " '

5" :

2 2 1+ 2 ( 2 2 ' H H

A ) 2 4; + ) 2 2 4; )

5 :+ 2 2 2

2 2, 2 2 ) +

2 ( ) 2 ( 2 2 2 , 2 1+

2 ) 2, 2 (2 4 +

,,! . + + ! " H

" ' ! 5 : # 5H: "

#

/ -$ ,0 -# ! #$ !, 1!#',$"

2 ( ( , 2 2 2 1+ + & 4 ( (

2 2 2 +

8 2 ) , + 2 4 2 4 #

2 ( 2 A ' , '

4 ' ) ,)

5 5 :: I 5 5 : 5 5 ::: F "

2 ) 4; 2 A 5 5 ::

( (2 +" ( 2 2 2 A

& ? 2 + 2 4 2 ( A '

5 : F ;

# 5 : I 5 :

(2 2 , ( 2 2 , 5 : F 5 5 ::+

2 4 2 2 '

5 5 :: I 5 5 : 5 5 ::: F "

# #

(7)

(2 5 : F ; #

5 : I 5 :+ 2 4 2

A ) 2 F 5 :+ > 2 & 2 2

+ 2 2 A & ? +

& ; 5 00":: 2 2 7 B 2

(2 5 00" 00 : 2 +

8 5 009: 2 ' 2 4 2 2 1+

2 2 )+ ; , ( ( 2 ( 2

2 2 2 ( ' 5 : 2 (2 2

) 2 2 2 G 5 : 2 2 )

2 ) +

D ( , 2 2 2 (2 2

2 +$ 2 E ( 2 2 2 ,

2 ) 7 B 2 + = ) (

; 2 ) 2 +

8 ' , 5 H 2 ) :+ ;

+

'

5 5 :: F "

5 :

4 5 : + +

5 5 : 5 5 ::: F "

5 5 :: F "

4 5 :+

( 2 ( '

5 :

4 + 2 ( 2 2 A ,

A 2 ( A

(2 ( 2 +

; # * < - = #

* < - ! ! >

(8)

5 5 :: F "

4 5 :+

2 ( 2 2 5 : ) 5 : 2

( 2 7 ) B 2 +

!

2 ( 2 ( 2 ) '

5 : F 5 : F

(2 " 5 F 5 : F : " +

2 A ,)'

5 : F 5 : I 5 : F "

F F , 2 ; )

5 : F 5 : I I

2 ) 5 : F (2 2 4; +

2 2 ; , 2 , (

2 2 C 4 2 2 2 +

2 = / 2 ( 2 2 ( '

I I

(2 2 + 7 2 2 2 !

2 + 2 #

, , ) +

4 ( 2 ( 2 2 (

4 2 ' F 0$ K "$G L+ 2

5 : F H +

4 5 : F 5 : F 1! 2 ) 2

2 " 5(2 2 : 2 5 2

, ( % 1!0 " &:+ 4 F F $ 2 2

) ; ) 7 ( ( ) 2 ( 2 2 2 %

5 2 , ( "9 " %:+

(9)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4

0.6 0.8 1 1.2 1.4 1.6 1.8 2

k t

kt+1

Growth Model with Logarithmic Utility Function

'()*+ ' ( 2 ( 2 5 : F F $

F 1! F 0$ F "+

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32

k t

kt+1

Growth Model with Power Utility Function

'()*+ ' ( 2 ( 2 F F $

F $ F 0$ F "+

(10)

" ! ! #$ # %%&''

8 5 009: ) ( 2 ,) "5 : F 5 :

5 " F : " + 2 A , A

2 ,)'

5# # # : F# I$ #

# # F "

(2 # 2 5 : 2 ) % $5 : F

< ) 4 ,) ' K" L K" L 2 2

$ 5 :

5 : F 5 :

(2 2 ) A , ' H#F 5 I$5 ::

@ 2 ( , 2 '

" $5 : $5 : $5 : 5 ":

" $5 : $5 : "

2 2 A 8 5 009: )

) 2 7+ = 2 2 # ) 5 2 #

,) 8 5 009::+

1 2 ( ) ; 2 ( '

F ! F 00 2 K "$G L+ D (2

2 2 2 5 2 2 2 1M " ,

:+

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

pt

pt+1

Monetary Model of Li (1998)

'()*+ 1' 6 ) 2 ) 8 5 009:

(11)

( ! #) * # %%+''

& 2 5 00!: )B ( # ( 2 ( 2 , ; )+

2 ) 2 2 2 B ,)'

5 : F 5 : 5 :

(2 2 2 ; 2

7 (2 2 ; ) 2 ,

, 5" :+ 2 2 A , A ,)'

5 : F 5 : I 5 : F "

2 A )

H F 5 :5 :

5 :5 : I 5 :

2 '

I 5 I : I I

(2 2 2 2 2 ; ) 5 2

7 :+

! 2 ( 2 ; 2 ) 2 )

2 ) + 2 ' F $ F 0$

F $ F $ 2 K "$G L+ D (2 2

2 2 5 2 2 2 11 " &:+

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

xt xt+1

Two−Sector Growth Model of Boldrin and Rustichini (1994)

'()*+ !' 6 ) 2 ( # ( 2

& 2 5 00!: ( 2 F F F $ F 0$ F "+

(12)

2 , " "

2 ( 2 ; 2 ) 2

A + C E

2 2 2 # ) #

, + 2 2 ) E , ) 2

) +

2 A 2) 2 ( 2 2 4 2

A 2 ) + 2 2) 2 2 )

2 2 ) 2 ) +

< 2 2 2 2 2 2 2

2 B 2 ( ;

2 + @ 2 ,) ;

2 +

3

4 4 $##! + ' < 2 2 ; " 2 2 5H H H:

I 5&:

5&&:

8 , 2 2 5 : 5 5 :: 4

2, 2 5H H: ( ) E ,

5H H 5H:: F 5H H H: F "

5 5 :: 4 ; 5 : 2, 2

5H H:+ &) 2 2 2 ; ' ' 5" : )

E , ' 5H: 5H: 5 F : 2 2

5 5 :: F " 5H: (F F 5 :

= 2 5H:

5 : F 5 5 : 5 ::K 5 5 : 5 :: I 5 5 : 5 :: 5 5 :: 5 :L

5 : I 5H:

, 5&: 2 2 5 : 5H:+

@, 2

5 : F H I 5H I%5 H::5 H:)% 5H:

(13)

5 : H

- .

5H I%5 H:: H '

2 ( 2 'F' F' +

D 2 ( 7 'F + 8 F ' "G 4 5H: + 6

2 + 2 5H: 2 ; A 5H: 2 2

5 5 :: F "+ @ 2 ( 2 ; N F N ) 2

A )+ 4 ' K" L ,)

5%: F 5 N : 5 I%5 N : 5 ::

2 5": F 5 : F " 5%: F 5 N : 5 N : "+ 6 5

5 I%5 N : 5 ::: 4 ; 2 +

8 , 2 , 5H:+ 8 5 : , A 5H:

+ 6 5 5 :: A H5H: 5 : 2 ; , A

5H:+ &) 2 ) 5 5 :: 5 5 :: F "+

> ( A ) ) 2 2 A 5 5 ::

+ D ) 2 2 5 5 :: 2

A + 2 2 , 5H: ( ; ,

H (2 2 5 ): 5H:+ 2 2 4 +

( ( 2 2 "+ 8 2 '

5 : F K 5 : I 5 : 5 5 :: 5 :L

K 5 5 ::5 5 :: I 5 5 :: 5 :L

5 5 5 : 5 ::+ 7 2 2 2 5 : (

2 &

5 : I 5 I : I

(2 " " 5,) 5&&::+ 2 3 ) 2 2 I +

4 4 -$ $# + ' 8 , 2 , + 6 2 ; "

2 2 2

O ' 5H H H:

4 ,)

O5 ! ! : F 5 :K 5 : I 5 :! ! L

8 2 B + + 2 ; * " 2 2 '

O5 ! : O5 ! : *

/

I+ K ! ! I ! ! L 5&&&:

!

! 0 0

(14)

(2 ( F 5 : 5H: ! F 5! ! : ( F

+ F +

2 2 5 : 2 ( 2 " 2

2 + + ) ( 2 2 ) 8 1+ '

5* I+ :5 I :

* F

5 2 + " (2 ":+

D ( 2 2 ' 4 ,) 5 : F #

5 4 8 1+ :+

- ,) 8 1+ (F 5H:

5 : F O5 5 : 5 : 5 5 :: 5 ::

@, 2

F 5 : (F

5 : 5 : F5 5 : 5 ::

I 5 5 : 5 :: I 5 :5 :

2

I I 5 /:

,) 2 4 2 +

2 5&&&: 5&,:

*5 I 5 I : : I 5 I :+

6 ' + ) 2 ( ; )

H+ 8 2 ; 2 ( , H + &) 2

& 2 ; 2 2 ; H 2 2 F

5 : 5 :

4 4 ,,! . + +' ; " - " , 2 2 5 I -: 5

- : 2 5H: + 5(: 2 2

- 2 2 H 5 : 5H: 5H:+ 8 F

5H: F 5H: 2 # (F " )+

5H: ( 2 '

(15)

* - I I- I

I+ K I L I+

4 ( 2 2 ( A '

I K- I L I G

5 I : G I- G

2 2 A 2 4 ( '

5 *-: K* -I+ L5 I -:

IK* -5 I : I* 5 I I : I+ 5 I :L

4

F K* -I+ L5 I -:5 *-:

F K* -5 I : I* 5 I I : I+ 5 I :L5 *-:

( 2 2

I

&)

I

/

(2

F = ; 5= ; :

6 + (2 - " 2 - " 2 ( 4

2 2 # +

!1 $ 5 5 6 ! 7 %$ - " ) 889* P6 2 6 #

2 - ( 2 = ,) #6 #6 # A < 2Q %

& R 6 9 0# +

, 5 ! " - ) 88/* P- ( 2 ) = ( 2

; Q M 5 : 1 1#! +

(16)

- " ! 5 6 ) 889* P6 2 2 ( 2 ,) #A

; ,) Q % & 6 #

9 1# M+

,$#! 5 : 6 ) 889* P6 2 6 2 - ( 2 = ,) )#

Q % & R 6 9 # 0+

,$#! 5 : 6 ) 88 * P A , ) ( 2

;Q $0 "0 # "!+

6 5 7 ) 88 * P C = 2 6 < - ( 2 = Q

% 2 ) $9 ! "#$ +

6 5 7 ) 88;* P = 2 Q 2 = ,

= 2 +

5 3 6 ) 88;* P < ) @ A <

= ) = Q % ) 11$#$ +

!, ! 5 : ! <! $ ) 99 * P@ 2 ) 2 )

2 & 2 Q & / + 1 + $ #9+

! $ 5 ) 8;;* P6 # 2 ,) B ; #

Q D 7 = )+

! $ 5 ! $ = ) 88;* P B ; < 2G

6 Q ( 7 +

! "5 ! 6 0 > 0 ! ) 88;* P< ) )

< 2 < = Q MM !"0# M+

&$.5 ! !" )% - $" * ) 8;8* P = 2

) Q > ) , = + 8 +

!., 5 6 ! -, 0 ) 889* P6 6 2 - ( 2 = ' <

< 6 = 2 Q % & R 6

9 # +

Referências

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