A Mathematical Model for Unemployment-Taking an Action without Delay

Transcription

1 Advnes in Dynmil Systems nd Applitions. ISSN Volume Number (7) pp. -8 Reserh Indi Publitions A Mthemtil Model for Unemployment-Tking n Ation without Dely Gulbnu Pthn Diretorte of Eonomis nd Sttistis Gndhingr Gujrt-38 Indi. P.H. Bhthwl Retd. Prof. & Hed Deprtment of Mthemtis Veer Nrmd South Gujrt University Surt Gujrt-3957 Indi. Abstrt The present work desribed nd nlyzed mthemtil model for unemployment using system of dynmi differentil equtions. In this model we nlyzed n effet of the tion of the government nd privte setor to ontrol unemployment without ny dely. Also we observed the effet of ttempt of self-employment mde by unemployed persons. The model is desribed by system of differentil equtions nd lso found the nonnegtive equilibrium point of the system to hek the stbility. At lst Numeril simultion is given to ompre with nlytil result. Keywords: Employed persons unemployed persons self employment newly reted vnies present jobs.. INTRODUCTION Unemployment is the serious problem for the whole world. Unemployment effets eonomilly soilly nd mny times mentlly to the person. This problem is not limited to the prtiulr person but it thes the whole fmily nd slowly-slowly

2 Gulbnu Pthn nd P.H. Bhthwl the whole ountry espeilly young genertion. Although they re full of life with so mny drems they get depressed from long period of unemployment. Nikolopoulos nd Tznetis ([]) developed nd nlyzed model for housing llotion of homeless fmilies due to nturl disster. Bsed on this onept Misr nd Singh ([]) presented nonliner mthemtil model for unemployment. G.N.Pthn nd P.H.Bhthwl ([7]) developed mthemtil model for unemployment with effet of self employment bsed on onept of bove ppers. N.Sirghi M.Nemtu nd D.De presented in ([3]) nonliner dynmi model using four vribles: Number of unemployed persons number of employed persons number of present jobs in the mrket nd number of newly reted vnies. In ([]) M. Nemtu presented model for unemployment bsed on some onept of ([3]) with dding new vrible number of immigrnts. Using onept of these ppers we developed dynmi mthemtil model for unemployment with four vribles: Number of unemployed persons number of employed persons number of present jobs in the mrket number of newly reted vnies nd we introdue n impt of self employment with the ssumption tht government nd privet setor both tried to rete new vnies without ny dely. The pper is orgnized s follows: Setion desribes Model for unemployment Setion 3 desribes n equilibrium nlysis Setion desribes the stbility of equilibrium point Numeril simultion desribes in setion 5 nd Conlusion is given in setion.. MATHEMATICAL MODEL: In this proess we ssume tht ll entrnts of the tegory unemployment re fully qulified to do ny job t ny time t. Number of unemployed persons U ( inreses with onstnt rte. The rte of movement from unemployed lss to employed lss is jointly proportionl to U ( nd ( P( V( E( ). Where P ( denoted the present jobs in the mrket vilble by government nd privte setor. Government nd privet setor try to rete new vnies without dely denoted by V ( nd number of employed persons denoted E ( Migrtion s well s deth of unemployed persons is proportionl to their number with the rte 3 mny times employed person leve the job beuse of disstisftion or fired from their job nd joint unemployed lss with the rte. Unemployed persons hve to rete hnes for self employment to survive. Unemployed person who strt their own independent work nd beome self employed is proportionl to its number with the rte 5. is the rte of deth nd retirement of employed person. The vrition in the present job is proportionl to nd depreition rte in present jobs is. nd re rte of newly reted vnies nd diminution of newly reted vnies.

3 A Mthemtil Model for Unemployment-Tking n Ation without Dely 3 du de dp dv U( P V E U E U ) 3 5 () U( P V E E U E () U ) P U V Lemm: The set ={( U E P V ) : 5 (3) () U E P V }where min( 3 ) is region of ttrtion for the system () () nd it ttrts ll solutions inititing in the interior of the positive otnt. Proof: From eqution () () we get d ( U( E( ) 3U ( E( Whih gives d ( U( E( ) ( U( E( ) Where min( 3 ). By tking limit supremum lim sup( U( E( ) t from (3) we hve dp U ( P( dp P( By tking limit supremum whih leds to lim sup P( t

4 Gulbnu Pthn nd P.H. Bhthwl from () we hve dv U( V ( dv V ( By tking limit supremum whih leds to lim sup V ( t ) t This proves the lemm. 3. EQUILIBRIUM ANALYSIS The model system () - () hs only one non negtive equilibrium point E ( U* V*) whih obtined by solving the following set of lgebri equtions. ( 3 5 U( P V E) E 5U E U P V E) U E U (5) () U P (7) U V (8) Tking n ddition of eqution (5) nd () 3U E From (7) From (8) U P E U 3 (9) () U V ()

5 A Mthemtil Model for Unemployment-Tking n Ation without Dely 5 U P V E () Where 3 Put vlues of eqution (9) nd () in (5) we get A U AU A Where A A 3( ) 5 A ( ). (3) From eqution (3) h ( U) AU AU A () Sine Ai i = ll re positive nd number of hnges in signs of eqution () is only one. So by Desrt s rule eqution () hs only one positive solution sy U *. So we get the non-negtive equilibrium point of model with oordintes: E* U P* 3U * * U * V* So E ( U* *) is required non negtive solution of the Model. V. STABILITY ANALYSIS To hek the lol stbility of equilibrium point E( U* V*) we lulte the vritionl mtrix M of the model system () () orresponding to E ( U* *) V

6 Gulbnu Pthn nd P.H. Bhthwl q q l q q l M Where l ( P V E q ) l U l l q l 3 5 l q l 5 q l The hrteristi eqution of bove mtrix is d d d d 3 3 (5) Where d q q d q ( q ) q ( q q l l ) d3 q[ q( ) ] q q[ q( ) l l] l ( q ) l( q ) d qq q[ q l( )] ql lq Sine d d d3 d re positive then ll oeffiients of eqution (5) re positive nd some lgebri mnipultion onvey tht dd d3 nd dd3 d3 d d d. So by Routh Hurwitz riteri ll roots of eqution (5) re negtive or hving negtive rel prt. Therefore equilibrium point E ( U* *) is lolly symptotilly stble. V 5. NUMERICAL SIMULATION For the Numeril simultion using MATLAB 7.. we onsider the following The equilibrium vlues of the model re: U * 9 P * 7 E * 835 V * 7.

7 A Mthemtil Model for Unemployment-Tking n Ation without Dely 7 The eigenvlues of the vritionl mtrix orresponding to the equilibrium point E ( U* V*) of model system () - () re: i. All eigenvlues re either negtive or hving negtive rel prt. So equilibrium E ( U* *) is lolly symptotilly stble. V Using bove Fig. nd Fig. represent the grph of vritions in the number of unemployed persons with respet to time with differene vlues of nd 5 respetively. Fig. shows tht if rte of unemployed persons to join employed lss is inreses thn number of unemployed person dereses. Fig. indites tht if rte of self employment goes higher thn number of unemployed person goes lower. It lso shows tht for lower unemployed rte needs very high self employment rte. 5 =. =. =. Unemployment U( Time ( Fig =.7 5 =7 5 =5 Unemployment U( Time ( Fig.

8 8 Gulbnu Pthn nd P.H. Bhthwl CONCLUSION The pper proposed nd nlyzed nonliner mthemtil model for unemployment using four dynmi vribles: Number of unemployed persons number of present jobs in the mrket number of employed persons nd newly reted vnies. We find tht equilibrium point is lolly symptotilly stble. Theoretil lultion is verified by Numeril simultion whih is done using MATLAB 7.. From bove lultions nd grphs we n see tht to derese number of unemployed person the rte of unemployed persons who joined employed lss should be higher tht mens the government nd privte setor both try to rete higher number of new vnies. We lso observe tht if present jobs vries then in this sitution rte of self employed should be higher to redue unemployed. From fig. it n be seen tht despite differene of 5 (rte of self employmen is very high grph of unemployed person with respet to time is very lose. It suggests tht if present job vries then rte of self employed should be high to ontrol unemployment. REFERENCES [] A.K.Misr A.K.Singh A mthemtil model for unemployment IJTPC 3 pp [] A.K.Misr A.K.Singh A Dely mthemtil model for the ontrol of unemployment Differ Equ Dyn Syst (3) 3pp [3] N.Sirghi M.Nemtu D.De A dynmi model for unemployment ontrol with distributed dely Mthemtil Methods in finne nd business Administrtion Proeeding of the Interntionl Business Administrtion onferene Tenerife Spin pp.-8 [] M.Nemtu A dynmi model for unemployment ontrol with migrtion nd distributed dely Mthemtil Methods in finne nd business Administrtion Proeeding of the Interntionl Business Administrtion onferene Tenerife Spin pp.3-3. [5] N. Sirghi M.Nemtu Deterministi nd stohsti dvertising diffusion model With dely WSEAS Trnstion on system nd ontrol (8) 3pp [] C.V.Nikolopoulos DE Tznetis A model for housing llotion of homeless popultion due to nturl disster Nonliner Anl.3pp [7] G.N.Pthn P.H.Bhthwl A Mthemtil Model of unemployment with effet of self employment IOSR- JM () 5 pp [8] M. Nemtu M.Pirte G. Mire D. Opris Stohsti Fuzzy Hybrid delyed Dynmis Heterogeneous ompetitions with Produt Differentition WSEAS Trnstion on mthemtis 7(9) pp