ON A NEUMANN EQUILIBRIUM STATES IN ONE MODEL OF ECONOMIC DYNAMICS

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1 oral of re ad Appled Mathemats: Advaes ad Applatos Volme 8 Nmber ages Avalable at DO: ON A NEUMANN EQULBRUM STATES N ONE MODEL OF ECONOMC DYNAMCS Departmet of Mathematal Cyberets Ba State Uversty Ba Azerbaa e-mal: sabr88@yahoo.om Abstrat The model of eoom dyams of Nema type s osdered. Wth the help of the eqlbrm mehasm wthot loss a traetory of the model s ostrted. The eessary ad sffet odtos for the estee of the eqlbrm state wthot loss s derved.. trodto Let s trode the followg deotatos: ( B ) [ 0 ] { } s a preservato matr of the -th brah C ( C ) C > 0 { } oeffets where s a ost matr C s a mber of prodt of the -th brah eessary for the prodto of the t prodt of the -th brah. 200 Mathemats Sbet Classfato: 37N40 9B55 97N70. Keywords ad phrases: eqlbrm mehasms sperlear mappg eqlbrm state. Reeved Agst Setf Advaes blshers

2 88 F m C { } s a prodtve fto of the -th brah ( ) pre vetor µ mamal growth rate of the total wealth of the 2 -th brah. The we a defe the model Z by the prodtve [-4]: ~ ( ) ( ~ ~ a X X ) ( R ) ~ m ( ) [ 0 ] > 0 ( ). Here we desrbe the ostrto of the traetory of ths model by the help of eqlbrm mehasm wthot loss [4]. The worg of the eoomy as a dyam system s gve by the state y at tme t. Note that eqlbrm s a set ( y ) ad f the eqlbrm ests the the vetor ( ). est. s deftely proportoal to f y oe { the eqlbrm s obvosly does ot () f y oe { } the epadg t o the bass we obta γ moreover the vetor type of α ( ) the osmer problem. are soltos of (2) f oe may hoose µ ( ) by the way that the followg eqalty wold be satsfed: µ ( ) m m m 0 µ C C m

3 where ON A NEUMANN EQULBRUM STATES N ONE 89 removg the ests. C s ( ) ( ) matr obtaed from the matr C by -th olm ad -th row the eqlbrm wthot loss Thereafter aordg to ( ) va preservato matr B ad the prodto fto F yelds the followg vetor of dstrbted resores ~ y at tme t ad the the proess s repeated. 2. Ma art [7] eessary ad sffet odtos are obtaed for the estee of a eqlbrm state wthot loss (see Theorems 3 ad 4). Bt verfato of eessary ad sffet odtos for the estee of eqlbrm pres wthot loss s omplated eogh - a system of 2 lear eqaltes (24) or system of sperlear eqaltes (29) [7]. Therefore t s advsable to derve the eessary odtos for the estee of eqlbrm pres wthot loss. Cosder the speal ase whe all µ ( ) are the same that s the brahes grow wth the same mamal rate of the total wealth. Eqlbrm pres wthot loss are solto of (2) [7]. By Theorem Fa [8] the eessary odtos of absee of eqlbrm pres wthot loss a be formlated as the followg problem: By whh ( ) there est 0 ad ( ) satsfyg the eqalty for whh s vald () > 0. (2)

4 90 Let the pres ( ) are ormed by the relato ( ). 0 > (3) Cosder the left had sde of (2) tag to aot (3): [ ( )]. Ths the eqalty (2) taes the form [ ( )]. 0 > (4) Let s trode the set o dees () ma () ( ). m 2 (5) t taes plae Lemma. Let the odtos (3) ad the eqalty () () ( ) m ma > (6) are vald where () s a mber of elemets the set of dees (). ths ase there est the mbers 0 ad ( ) satsfyg () for whh (2) or (3) s flflled.

5 ON A NEUMANN EQULBRUM STATES N ONE 9 roof. trode the sets ( ) ad 2 ( ) by the formlas (5). Let the odtos (3) ad () be satsfed. The defe as follows: ad ( ) ε f 2() ( ) 0 f () 2 ε f () () 0 f () () ( ) where ε > 0 s some mber ( ). Sbstttg the vales ad ( ) to the odto () t s easy to see that t s satsfed. Mltplyg both sdes of the eqalty (6) by ε ( ε > 0) smmg over we get ad ε ma > ε m( ). (7) () () From the other had sbstttg the vales (4) (or (2)) osderg (3) we get (7). ad ( ) to Lemma s proved. Theorem. Let the mbers 0 > 0 ( ) be sh that ma > 0 ad µ ( ). f eqlbrm pres wthot loss by gve followg type eqalty trs tre: ad some > ( ) 0 est the the

6 92 ma m () () m( ) 0. The proof mmedately follows from Fa [8] ad Lemma. Cosder the Nema-Gale model gve by prodto mappg a: ~ a( X ) X ( ~ ~ ) ( R ) 0 ~ B where ( F ( ) F ( )) ( ) { 2 } B s a dagoal matr the ma dagoal of whh has a form ( ) [ 0 ] ; F ( ) m > 0. We defe the Nema eqlbrm state ths model Z [ 5 6]. Reall that the set ( α X ) s a Nema eqlbrm state f the followg odtos are satsfed: α X a( X ) (8) [ Y ] α [ X ] ( Y a( X )) (9) [ X ] > 0 (0) where a s a prodto mappg of the osdered model Z. Note that the pres of the prodts do ot deped o the brahes where they are osdered.e. ( ) where ( ). Note that odegeeratg ase meas that

7 ON A NEUMANN EQULBRUM STATES N ONE 93. m α () De to the fat that the system () oly strt eqaltes shold be satsfed therefore we are dealg wth eqlbrm wthot a loss.e.. t follows from ths. σ (2) Assme that. µ α (3) Note that µ s a mamal growth rate of the total wealth of th - brah [4 9 0]: ( ) [ ] ma 0 µ U where ( ). m U De () ad (3) we have ( ) [ ] ( ) [ ] α U U (4) where ( ) ( ). X The system of relatos (4) oe may obta also from the odto (9) tag. X X

8 94 Nema eqlbrm state ( α X ) s determed p to a fator as the mltplato of ay of the relatos (8)-(0) by a postve fator does ot hage these relatos (reall that α s a sperlear mappg). Therefore t s stable to tae σ the formla (2). From (2) follows Nema eqlbrm vetor X s defed by the parameters σ σ. De to above osderatos we get that Nema eqlbrm state ( α X ) s defed by 2 varablesç 2 2 şeş by α σ σ. Note that m σ. The sbstttg (2) to () ad (4) we obta a system of eqatos wth respet to α σ σ : σ σ σ σ σ σ σ σ ( ); σ α. (5) addto to ths system these varables shold satsfy the eqalty (0) whh de to (2) taes the form σ > 0. (6) Ths fat the followg theorem s proved. Theorem 2. Nema eqlbrm state that satsfes the Eqato () a be ostrted sg the eqlbrm model wthot loss by µ ( ) where the eqlbrm a be fod from the α system (5) ad (6).

9 ON A NEUMANN EQULBRUM STATES N ONE 95 Note: Let Nema pres ( ) are ormed by the odto [ ] ( ). ths ase σ [ ]. (7) Sbstttg (7) to (5) we obta a system wth respet to varables α. Referees [] V. L. Maarov ad A. M. Rbov Mathematal Theory of the Eoomal Dyams ad Eqlbrm Mosow Naa p. [2] A. M. Rbov Mathematal Models of the Epaded Reprodto Models Legrad Naa 983. [3] A. M. Rbov Sperlear Mltvaled Mappqs ad ther Applatos to Eoomal-Mathematal roblems Legrad Naa 980. [4] A. M. Rbov Eqlbrm Mehasms for Effetve ad Developmet of Dyam Models of rodto ad Ehage Tehal Cyberets 968. [5] G. B. Kleer rodto Ftos Mosow 986. [6]. A. Krass The Models of Eoom Dyams Mosow 976. [7] S.. Hamdov O a osmer problem re ad Appled Mathemats oral 5(6) (206) [8] Fa O the Systems of Lear eqaltes the boo Lear eqaltes ad Close roblems Mosow L 969. [9] R. M. Romer Mathess the theory of eoom growth The Amera Eoom Revew 05(5) (205). [0] V. F. Demyaov ad L. V. Vaslyev No-dfferetable Optmzato Mosow Naa 98. g