• Keine Ergebnisse gefunden

AMathematicalModeloftheEconomicSystemwithDistributedPropertyRights Kolesnik,Georgiy MunichPersonalRePEcArchive

N/A
N/A
Protected

Academic year: 2022

Aktie "AMathematicalModeloftheEconomicSystemwithDistributedPropertyRights Kolesnik,Georgiy MunichPersonalRePEcArchive"

Copied!
15
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Volltext

(1)

Munich Personal RePEc Archive

A Mathematical Model of the Economic System with Distributed Property Rights

Kolesnik, Georgiy

15 May 2010

Online at https://mpra.ub.uni-muenchen.de/47680/

MPRA Paper No. 47680, posted 19 Jun 2013 19:24 UTC

(2)

!"

#$%&'(& ! "# $%& #' ()% %#+#", -.-(%" /,() ()% 01#0 %1(. 1,2)(- $,-

(1,34(%$ 5"#+2 -%6%15& 52%+(- ,- #+-,$%1%$ 7)% %8%( #' 01#0 %1(. 1,2)(-

5&&# 5(,#+#+()%%#+#",%9,%+.#'()%"5+52%"%+(-(15(%2,%-,--(4$,%$

*+,- , .

:

;<

/0123'4%.01#0 %1(. 1,2)(-#/+%1-),0#+(1#&%9,%+./%&'51%#"0 %(,

(,#++#+# #0 %15(,6%25"%=5-)%>4,&,31,4"

5 ,6

? :

@

A B

C D

:E DF GHHHE

I

½

A

?

:

?@ @ < @

J@ ;

< @

DK%+-%+L%M&,+2NOPQRL#1M%(5&NOSSRT)&%,'%1U,-)+.NOSQR

T(4&VNOSSE

?

XYDZ1#--"5+[51(NOSQR[51(L# #1%NOOHE?

C :

J@

:

½

!"#"

(3)

? DL \4&&%1 ]\51+%1.$ GHHNE

<

J

? @

J@ ?

< ; :

J@

: J@

^

: J@

_

DB

GHHSE;

@

? @ @

@ J@

` : : :

< J J

J :

?

J@ ?

< J

?

J@

_ < J@ ?

@

J

?

: B

?J

7 8 96

J @J

l

k

` a b

J@

J@

J@ J

:

Θ k × l

θ ij

(4)

JJ

i

j

c

θ •j = (θ 1j , . . . , θ kj )

j

J

θ i • = (θ i1 , . . . , θ il )

i

B J@

J d

W i (Θ, C) = l

j=1

θ ij C j = θ T i • C,

DNE

C = (C 1 , . . . , C l )

C F

:JD

:E

; : e @

J@

d DfgINGHHPE_

@

: :

: ? :

t

Φ j (t, ˜ a j , y)

@

˜ a j

<

y

`

J

J d

C j (˜ a j , y) = ∞

t=0

β t j Φ j (t, ˜ a j , y),

DGE

β j

C : J@

; J

J <

I J

j

A j

_

A = A 1 ⊗ · · · ⊗ A l

B

J :

W i

J

a i • = (a i1 , . . . , a il )

;

:

A k × l

I

j

::

A

a • j = (a 1j , . . . , a kj )

I <

j

`

<

: J@ :

a ij

< ^ :

¾

; <

:

θ ij

¾

$ !

& ' ( )*+,-(.+/"0+"011234

! 5 )* 06(. ,-",+"011-34 '

5"

(5)

@ J <

a •j

J

θ • j

<

˜ a j

d

˜

a j = R j (θ • j , a • j ).

DhE

` <

a ˜ j

<

y

J ^

@

: :

? J@ <

y

DGE <

J i<

<

˜ a = (˜ a 1 , . . . , ˜ a l )

C(˜ a)

F DNE

:J < d

W ( Θ, ˜ a ) = ΘC ( ˜ a ).

DjE

F J

d :

J

R j

< :

C j

g

: < : I

Θ

g DhE DjE

W ( Θ, A ) = W ( Θ, R ( Θ, A )) = ΘC ( R ( Θ, A )),

DkE

R(Θ, A)) = (R 1 (θ •1 , a •1 ), . . . , R l (θ •l , a •l ))

? DkE J

k

:

Γ( Θ )

Θ

B

W i ( Θ, A )

J@

<

a i • ∈ A

`

A − i

C

i

I <

i

Θ

a i• ( A − i , Θ )

d

a i ( A − i , Θ ) = arg max

a i

• ∈ A W i ( Θ, A ).

? <

a i• (A − i , Θ)

J J@@

;<

¿

A ( Θ )

¿

7 '! &

"$

R

& '' '

'

"$

"

(6)

F

Θ

< J @

˜

a(Θ) = R(Θ, A (Θ)),

DQE

C

W(Θ) = W(Θ, A (Θ)).

F J

I

x = (x 1 , . . . , x l )

J d

x j = k

i=1

θ ij , j = 1, . . . , l.

DPE

I

@ d

U (˜ a ) = k

i=1

W i ( Θ, ˜ a ) = x T C (˜ a ).

DSE

I

U (Θ)

A (Θ)

J@ J

Θ

d

U (Θ) = U (˜ a(Θ)) = k

i=1

W i (Θ),

a(Θ) ˜

DQE

c

Θ

U (Θ)

I

: @

; < :

:

: aC b :

J@ @ `

: lJ

: :

J J :

? J :J

d

C J

@? <

:

:

g: :

W i

:

(7)

: <

:

B :

θ ij = r i θ 1j , ∀ i = 2, . . . , k, ∀ j = 1, . . . , l,

DOE

r i > 0

C: : @

`

Θ 0 i

d

W i ( Θ 0 , A ) = r i W 1 ( Θ 0 , A ) ∀ i > 1,

:

W i (Θ 0 , A)

:

DOE J

Θ 0

(l + k − 1)

d

θ = (θ 1 , . . . , θ l )

: :

(r 2 , . . . , r k )

d

Θ 0 = rθ T ,

DNHE

r = (r 1 , r 2 , . . . , r k )

D

r 1 = 1

E

g J@

d

Θ 0 = ρx T ,

DNNE

x = (x 1 , . . . , x l )

C

J DPER

ρ = (ρ 1 , . . . , ρ k )

C : :

k i=1

ρ i = 1

_

Θ 0

ρ

J@ d

ρ i = r i k m=1

r m

.

DNHE d

Θ 0 = k

i=1

r i ρθ T .

DNGE

gDOE J

j

x j =

k

i=1

θ ij = θ j k

i=1

r i ,

θ = x

k i=1

r i

.

DNhE

(8)

^ DNNE

<

: `

:

J

U

_

:

W(Θ, ˜ a)

DkE

U (˜ a)

DSE DNNE J

<

˜ a

d

W(Θ 0 , ˜ a) = Θ 0 C(˜ a) = ρx T C(˜ a) = ρ U (˜ a).

DNjE

` <

R

J

`

R

J d J

<

<

˜ a

J@

J

U( ˜ a )

g :

A

a i • = ˜ a

i = 1, . . . , k

F

˜

a = R(Θ 0 , A ).

_ J

Θ 0

:

A

;<

Γ(Θ 0 )

`

i

J J :

A

<

˜ a

d

˜

a = R(Θ 0 , A ).

FDNjEd

W i (Θ 0 , A ) = W i (Θ 0 , ˜ a ) = ρ i U (˜ a ) ≥ ρ i U(˜ a ) = W i (Θ 0 , ˜ a ) = W i (Θ 0 , A ),

A = A 0 )

F:

U ( Θ )

U (Θ 0 ) = max

Θ

U (Θ),

DNkE

J

DSE

? DNkE `

a

J@ bJ

:,;<6 5

k

W j (Θ)

l

!

"

9

!'"$ &

! "

(9)

Θ 0

<

Θ

J

i = 1, . . . , k

W i (Θ) ≥ W i0 ),

DNQE

g DNQE

U (Θ) > U (Θ 0 ),

JDNkE

g

J` J _

Θ

I<

Θ 0

J@ <J

Θ

` :

d

<

˜ a(Θ 0 )

<

Θ 0

; J

Θ

Θ 0

J@

< d

W(Θ 0 , ˜ a(Θ)) = W(Θ).

DNPE

? DNNEF

x

Θ 0

ρ

` DNjEd

W(Θ 0 , ˜ a(Θ)) = ρ U (˜ a(Θ)) = ρ U (Θ).

DNSE

DNPE DNSE :

ρ

DNNE

Θ 0

ρ = W(Θ)

U (Θ) .

DNOE

_ :

A 0 )

Θ 0

< J

W(Θ)

_ DNkEDNPEDNSE J

i

d

W i (Θ) = W i0 , ˜ a(Θ)) = ρ i U(Θ) ≤ ρ i U (Θ 0 ) = W i0 ).

DGHE

J

i

Θ 0

Θ

F J@

:,;<6 7 #

Θ

"

(10)

U (Θ)

DGHE

`

F <

J : J

B J :

J@

@ :

? ;9;. = @8A;

`J < :

@J `

NHHm

θ

(1 − θ)

:

Θ

Θ =

1 θ 0 1 − θ

.

` J B

_

c

f::J d

P (˜ a) = 1 − ˜ a 1 − ˜ a 2 ,

˜ a = (˜ a 1 , ˜ a 2 )

C J@ e :

f: J

Π j (˜ a) = (P (˜ a) − c)˜ a j .

< e

˜ a j ≥ 0

J Ie

˜

a 1

˜

a 1 = a 11 .

?

˜ a 2

<

J@ d

˜

a 2 = θa 12 + (1 − θ)a 22 ,

a 12

a 22

C e

` :

j Φ j (t, ˜ a )

J

Π j ( ˜ a )

` : DGE

j

: d

C j ( ˜ a ) = Π j ( ˜ a )

1 − β .

(11)

J

W 1 (Θ, ˜ a) = Π 1 (˜ a) + θΠ 2 (˜ a), W 2 ( Θ, ˜ a ) = (1 − θ)Π 2 ( ˜ a ).

:

?:aC b J@

θ = 0

B:

A

d

a 11 = a 22 = 1 − c

3 ,

DGNE

d

Π j = 1 − c

3 2

j = 1, 2.

_

θ = 1

: J

: <d

W 1 ( ˜ a ) = Π 1 ( ˜ a ) + Π 2 ( ˜ a ) → max

˜ a∈ A .

:

A

a 11 + a 22 = 1 − c

2 .

DGGE

F Y

:

W 1 ( Θ, ˜ a )

W 2 ( Θ, ˜ a )

J

θ < 1

a 12 ( Θ ) = 0

e

: Ji

J d

a 11 ( Θ ) = (1 − θ)(1 − c)

3 − θ ,

DGhE

a 22 (Θ) = 1 − c

(1 − θ)(3 − θ) .

DGjE

;

θ = 0

e J DGNE `

θ → 1 ˜ a 1 (Θ) → 0

˜

a 2 (Θ) = (1 − θ)a 22 (Θ) → 1 − c 2 ,

e

:! !

"

(12)

`

Π 1 (Θ) = (1 − θ)(1 − c) 2

(3 − θ) 2 , Π 2 (Θ) = 1 − c

3 − θ 2

,

DGkE

DNE

W 1 ( Θ ) = 1 − c

3 − θ 2

, W 2 ( Θ ) = (1 − θ)(1 − c) 2 (3 − θ) 2 .

f:

W 1

θ

J@ _

e

θ

< J

_ :d

θ

< <

W 2

θ

ge

Q( Θ ) = ˜ a 1 ( Θ ) + ˜ a 2 ( Θ ) = (2 − θ)(1 − c) (3 − θ) .

f:

Q

θ

DGE :

:

θ → 1

F

a C Cb :

Y

$

J@ J :J

(13)

Gdi e

Q

Θ

` DNNEDNOE

DGkE

Θ

d

Θ 0 = 1

2 − θ 1 2 − θ 1−θ 2−θ

1−θ 2−θ

.

F

1−θ 2−θ

N @ N

(1 − θ) 2 2−θ

J

G @ Gg< : :

< J DGkE

`

Θ 0

<

: d

U (˜ a) = Π 1 (˜ a) + Π 2 (˜ a).

B<

e DGGE

1−c

2

2

c

W 1 (Θ 0 ) = 1 2 − θ

1 − c 2

2

, W 2 (Θ) = 1 − θ 2 − θ

1 − c 2

2

.

? e

J : D hE

:

B C *+

(14)

J

`

J@

J <

:

g

<

_ J@

:

?

J@ J :J : _

` J

? J

(15)

` <

:

` :

<

nN℄ B ? ?@ g_ ` p Y

:

@ e qq? Frg a`

bCGHHSC=#NCgPPCSQ

nG℄ F s d :<

?GFN Cg`d ^ <GHHH

nh℄ f : aI@ : :

J: DfgINEb r `Y

GH JGHHP=#GkQ

nj℄ Z1#--"5+T[51(t 7)%#-(-5+$3 %+%u(-#'#/+%1-),0d5()%#1.#'6%1(,5&5+$

&5(%15&,+(%215(,#+ qqK#'v#&,(,5&w#+#".CNOSQCU#&OjCv QONCPNO

nk℄ [51( t L# #1% K v1#0 %1(. x,2)(-5+$ ()%=5(41% #' ()%y,1" qq K #'v#&,(,5&

w#+#".CNOOHCU#&OSC=#QCv NNNOCNNkS

nQ℄ K%+-%+ L L%M&,+2 ] 7)%#1.#' ()% y,1"dL5+52%1,5&z%)56,#1 !2%+. {#-(

5+${50,(5&T(14(41% qqK#'y,+5+,5&w#+#",-CNOPQCU#& hC=#jCv

hHkChQH

nP℄ L#1M xT)&%,'%1! U,-)+. x L5+52%"%+(t/+%1-),0 5+$ L51M%( U5&45(,#+d

5+w"0,1,5&!+5&.-,- qqK#'y,+5+,5&w#+#",-CNOSSCU#&GH C=#NG C

vGOhChNk

nS℄ L \4&&%1[L]\51+%1.$| }+-,$%6-t4(-,$% t/+%1-),0d!v#&,(,5&7)%#1.#'()%

y,1"qq x!=~K#'w#+#",-CGHHNCU#&hGCvkGPCkjN

nO℄ T)&%,'%1!U,-)+.x 512%T)51%)#&$%1- 5+${#10 #15(% {#+(1#& qqK#'v#&,(,5&

w#+#".CNOSQCU#&OjC=#hCv jQNCjSS

nNH℄ T(4&Vx L5+52%1,5&{#+(1#& #' U#(,+2 x,2)(-dy,+5+,+2v#&,,%- 5+$ ()%L51M%(

'#1{#10 #15(%{#+(1#& qqK#' y,+5+,5&w#+#",-CNOSSCU#&GH C=#NG C

vGkCkj

Referenzen

ÄHNLICHE DOKUMENTE

Driouchi, Ahmed and Kadiri, Molk and Alaoui Belghiti, Moulay Abdelaziz. Institute of Economic Analysis Prospective Studies, Al

Таким образом, учет влияния распределения акционерного капитала компании на цену пакетов акций позволяет объяснить наблюдаемый значительный разрыв между

The considered scheme of behaviour of pollution zone boun- daries around the coastal fXJint source of pollution is valid in the case of interaction between one leading

Kiselev, A., A Systems Approach to Health Care, RM-75-31, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1975.. Fleissner, P., An Integrated Model of

a trajectory in the love space (see Figure 4) representing the predicted time evolution of the involvement of Scarlett and Rhett, eleven chronologically ordered short segments of

5.1.2 Agrarian Land Reform of the Russian Federation 31 5.1.3 Land Reform and the Regulation of Land Markets 33 5.1.4 The System of Russian Land Law in Transition 35 5.2 Types of

The paper presents a mathematical analysis of an endogenous growth model for two economies with absorptive capacities.. The model was proposed by Gernot

Математическое моделирование заслуживает особого внимания, поскольку сегодня требуется подготовка специалистов, владеющих не только специальностью, но