Mathematical model of Unemployment- an analysis with delay

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1 Globl Jounl of Mthmtcl Scncs: Thoy nd Pctcl. ISSN 97- Volum 9 Numb 7 pp. 5-7 Intntonl Rsch Publcton Hous Mthmtcl modl of Unmploymnt- n nlyss wth dly Gulbnu Pthn P.H.Bhthwl Sttstcl Assstnt Dctot of Economcs nd Sttstcs Gndhng Ind Rtd. Pof. & Hd Dptmnt of Mthmtcs VNSGU Sut Ind Abstct Ths wo psntd nd nlyzd mthmtcl modl fo unmploymnt usng fou dynmc vbls. In ths modl w nlyzd n ffct of th cton of th govnmnt nd pvt scto to contol unmploymnt wthout ny dly nd lso obsv th ffct of psnc of dly. Also w nlyzd th contbuton of slf-mploymnt fo dcs unmploymnt. W found th non-ngtv qulbum pont of th systm to chc th stblty. At lst Numcl smulton s gvn to comp wth nlytcl sult. Ky wods: Employd psons unmployd psons slf-mploymnt nwly ctd Vcncs psnt obs dly.. INTRODUCTION: Unmploymnt s on of th mo poblm fo th wold. Unmploymnt s not n ssu fo th ptcul ttoy but t s th sous poblm fo whol wold. Fo hlthy conomy of ny county hgh unmploymnt t s l b. Nolopoulos nd Tznts [] dvlopd nd nlyzd modl fo housng llocton of homlss fmls du to ntul dsst. Bsd on ths concpt Ms nd Sngh [ ] psntd nonln mthmtcl modl fo unmploymnt. G.N.Pthn nd P.H.Bhthwl [7] dvlopd mthmtcl modl fo unmploymnt wth ffct of slf-mploymnt bsd on concpt of bov pps. N.Sgh M.Nmtu nd D.Dc psntd n [] nonln dynmc modl usng

2 Gulbnu Pthn nd P.H. Bhthwl fou vbls: Numb of unmployd psons numb of mployd psons numb of psnt obs n th mt nd numb of nwly ctd vcncs. In [] M. Nmtu psntd modl fo unmploymnt bsd on som concpt of [ ] wth ddng nw vbl numb of mmgnts. In [9] G.N.Pthn nd P.H.Bhthwl dvlopd mthmtcl modl fo unmploymnt usng fou dynmc vbls wthout ny dly. Usng concpt of ths pps w dvlopd dynmc mthmtcl modl fo unmploymnt wth fou vbls: Numb of unmployd psons numb of mployd psons numb of psnt obs n th mt nd numb of nwly ctd vcncs. W ntoduc n ffct of slf-mploymnt wth th ssumpton tht govnmnt nd pvt scto both td to ct nw vcncs wthout ny dly nd ft tht w nlyzd th mpct of dly n ctng nw vcncs by govnmnt nd pvt scto. Th pp s ognzd s follows: Scton dscbs Modl fo unmploymnt Scton dscbs n qulbum nlyss Scton dscbs th stblty of qulbum pont Numcl smulton dscbs n scton 5 nd Concluson s gvn n scton.. MATHEMATICAL MODEL: In ths pocss w ssum tht ll ntnts of th ctgoy unmploymnt fully qulfd to do ny ob t ny tm t. Numb of unmployd psons U t ncss wth constnt t. Th t of movmnt fom unmployd clss to mployd clss s ontly popotonl to U t nd P t V t E t. Wh P t dnotd th psnt obs n th mt vlbl by govnmnt nd pvt scto. Govnmnt nd pvt scto ty to ct nw vcncs dnotd by V t nd numb of mployd psons dnotd E t Mgton s wll s h of unmployd psons s popotonl to th numb wth th t mny tms mployd pson lv th ob bcus of dsstsfcton o fd fom th ob nd ont unmployd clss wth th t. Unmployd psons hv to ct chncs fo slf mploymnt to suvv. Unmployd pson who stt th own ndpndnt wo nd bcom slf mployd s popotonl to ts numb wth th t 5. s th t of h nd tmnt of mployd pson. Th vton n th psnt ob s popotonl to c nd dpcton t n psnt obs s c. nd t of nwly ctd vcncs nd dmnuton of nwly ctd vcncs. dscbs th dly n ctng nw vcncs. du U P V E U E U 5

3 Mthmtcl modl of Unmploymnt- n nlyss wth dly 7 de dp dv U P V E E U E c U c P 5 U t V Lmm : Th st ={ U E P V : c U E P V } c wh mn s gon of ttcton fo th systm nd t ttcts ll solutons nttng n th nto of th postv octnt. Poof: Fom quton w gt d U t E t U t E t Whch gvs d U t E t U t E t Wh mn. By tng lmt supmum lm sup U t E t t fom w hv dp cu t cp t dp c cp t

4 8 Gulbnu Pthn nd P.H. Bhthwl By tng lmt supmum whch lds to c lm sup P t t c fom w hv dv U t V t dv V t By tng lmt supmum whch lds to lm sup V t t Ths povs th lmm.. EQUILIBRIUM ANALYSIS: Th modl systm - hs only on non ngtv qulbum pont E U* E* P* * whch obtnd by solvng th followng st of lgbc V qutons. U P V E U E 5U 5 U P V E E U E 5 c U cp 7 U V 8 Tng n ddton of quton 5 nd U E E U 9 Fom 7 cu P c

5 Mthmtcl modl of Unmploymnt- n nlyss wth dly 9 Fom 8 U V U P V E Wh c c Put vlus of quton 9 nd n 5 w gt A U AU A Wh A A 5 A. Fom quton h U AU AU A Snc A = ll postv nd numb of chngs n sgns of quton s only on. So by Dsct s ul quton hs only on postv soluton sy U *. So w gt th non-ngtv qulbum pont of modl wth coodnts: E* c U P* c U * * U * V* So E U* E* P* * s qud non ngtv soluton of th Modl. V

6 Gulbnu Pthn nd P.H. Bhthwl. STABILITY ANALYSIS: Stblty of qulbum pont wthout ny dly: To chc th locl stblty fo t qulbum pont E U* E* P* * V clcult th vtonl mtx M of th modl systm cospondng to E U* E* P* V*. q q l q q l M c c Wh l P V E q l U l l q l 5 l q l 5 q l Th chctstc quton of bov mtx s d d d d 5 Wh d q q d c q q c q c c q q c l l d q[ q c c ] cq q[ q c cl l] c l c q l q d qcq q[ qc l c c ] cql lqc w Snc d d d d postv thn ll coffcnts of quton 5 postv nd som lgbc mnpulton convy tht dd d nd ddd d d d. So by Routh Huwtz ct ll oots of quton 5 ngtv o hvng ngtv l pt. Thfo qulbum pont E U* E* P* * s loclly symptotclly stbl. V

7 Mthmtcl modl of Unmploymnt- n nlyss wth dly Stblty of qulbum pont wth dly: To chc th locl stblty fo t qulbum pont E U* E* P* * V clcult th vtonl mtx M nd M of th modl systm cospondng to E U* E* P* V*. dx Mx t M x t Wh T x t u t t p t v t w u t t pt nd v t smll ptubtons ound th qulbum pont E M q q c q q l l c l l M Wh l P V E q l l U q l 5 q l q l 5 Th chctstc quton of systm s Wh q q c 7 q q c q c c q q c l q l c [ q c c ] cq q[ q c c ] l q ] q c q q [ q c l c c q l l l q c q lc q q lc.

8 Gulbnu Pthn nd P.H. Bhthwl Now to chc th stblty of Eq. 7 w should not dctly us Routh-Huwtz cton. W chc tht Hopf -bfucton occus nd fo tht w hv to show tht Eq. 7 hs p of puly mgny oots. Fo ths w substtut n Eq. 7 nd w gt 8 sn cos sn cos cos sn 9 Sptng l nd mgny pt of Eq. 9 w gt sn cos cos sn By squng nd ddng Eq. nd Eq. By tng xpnson of ths 8 Substtutng n bov Eq. thn w hv f Wh

9 Mthmtcl modl of Unmploymnt- n nlyss wth dly If ll = nd stsfs Routh-Huwtz cton thn th s no postv oot of Eq... ll oots of Eq. ngtv o hvng ngtv l pt. So by Routh-Huwtz cton qulbum E s symptotclly stbl fo ll dly. Conty f ll dos not stsfy th Routh-Huwtz cton thn th s t lst on postv oot xst of Eq. fo. Fom ths w gt tht snc so. Whch gvs th condton fo th xstnc of p of puly mgny oots of Eq. 7. c P V E U } U c c 5 { 5 Fom Eq. nd Eq. w gt tn Fo postv w hv cospondng s gvn by n n tn n=.. By Butl s lmm w cn sy tht qulbum E mns stbl fo. Now to chc tht Hopf- bfucton occus t w hv to chc tht stsfs th tnsvslty condton. Lmm : Tnsvslty condton s dr sgn d

10 Gulbnu Pthn nd P.H. Bhthwl Poof: By dffnttng Eq. 7 wth spct to w hv Now d d dr sgn d dr sgn d d = sgn R d = sgn 8 m m m H m m m Snc condton 5 s stsfd thn w hv postv nd fo tht Tnsvslly condton s stsfs. Ths shows tht f condton 5 stsfs thn qulbum E s symptotclly stbl fo.. [ nd unstbl fo. Th condton of Hopfbfucton s stsfd so podc soluton occus whn psss th fo qulbum E. 5. NUMERICAL SIMULATION: Fo th Numcl smulton usng MATLAB 7.. w consd th followng c. c... Th qulbum vlus of th modl : U * 77 P * 598 E * 87 V * 77.

11 Mthmtcl modl of Unmploymnt- n nlyss wth dly 5 Th gnvlus of th vtonl mtx cospondng to th qulbum pont E U* P* E* * of modl systm - fo : V.5. 9 nd.. All gnvlus ngtv. So qulbum E U* P* E* * s loclly symptotclly stbl. V Usng bov Fg. nd Fg. psnt th gph of vtons n th numb of unmployd psons wth spct to tm wth dffnc vlus of nd 5 spctvly. Fg. shows tht f t of unmployd psons to on mployd clss s ncss thn numb of unmployd pson dcss. Fg. ndcts tht f t of slf-mploymnt gos hgh thn numb of unmployd pson gos low but w lso obsv tht fo vy hgh t of slf mploymnt w gt only lmtd dcmnt n unmploymnt. Usng bov ll vlus ncludng vlus of qulbum pont n Eq. w gt =.55. Put ths vlu of n Fg. Eq. w hv ctcl vlu of s Unmploymnt Ut Numb of Psons.9 x =.5 =. = Tm t Y Fg.

12 Gulbnu Pthn nd P.H. Bhthwl Unmploymnt Ut Numb of Psons. x =.7 5=5 5= Tm t Y. CONCLUSION: Th pp poposd nd nlyzd nonln mthmtcl modl fo unmploymnt usng fou dynmc vbls: Numb of unmployd psons numb of psnt obs n th mt numb of mployd psons nd nwly ctd vcncs. W fnd tht qulbum pont s loclly symptotclly stbl wthout ny condton n bsnc of dly. But n psnc of dly qulbum pont s not lwys stbl. In psnc of dly qulbum pont s stbl wth som condton.. f condton 5 stsfs thn w gt such tht qulbum pont s stbl fo nd unstbl fo. Thotcl clculton s vfd by Numcl smulton whch s don usng MATLAB 7.. Fom bov clcultons w cn s tht n bsnc of dly w gt th qulbum pont wthout ny condton. Tht s unmploymnt cn ducd by mpovng mo nwly ct vcncs nd wth hgh t of slf-mploymnt. Vton of psnt obs lso ffcts th unmploymnt postvly nd lso ngtvly ccodng wth ts ncsng o dcsng t spctvly. In psnc of dly w gt stblty wth som condtons. It mns t s mo tough to contol unmploymnt n psnc of

13 Mthmtcl modl of Unmploymnt- n nlyss wth dly 7 dly n comp of bsnc n dly. Fom Fg. t cn b obsv tht wth hgh slf mploymnt t th s lmtd dcmnt n unmploymnt. Thfo to dcs unmploymnt nds vy hgh slf mploymnt t nd lso good ffots of govnmnt nd pvt scto n ctng nw vcncs wthout ny dly. 7. REFERENCES: [] A.K.Ms A.K.Sngh A mthmtcl modl fo unmploymnt IJTPC pp.-8. [] A.K.Ms A.K.Sngh A Dly mthmtcl modl fo th contol of unmploymnt Dff Equ Dyn Syst pp.9-7. [] N.Sgh M.Nmtu D.Dc A dynmc modl fo unmploymnt contol wth dstbutd dly Mthmtcl Mthods n fnnc nd busnss Admnstton Pocdng of th Intntonl Busnss Admnstton confnc Tnf Spn pp.-8 [] M.Nmtu A dynmc modl fo unmploymnt contol wth mgton nd dstbutd dly Mthmtcl Mthods n fnnc nd busnss Admnstton Pocdng of th Intntonl Busnss Admnstton confnc Tnf Spn pp.-. [5] N. Sgh M.Nmtu Dtmnstc nd stochstc dvtsng dffuson modl Wth dly WSEAS Tnscton on systm nd contol 8 pp. -5. [] C.V.Nolopoulos DE Tznts A modl fo housng llocton of homlss populton du to ntul dsst Nonln Anl.pp [7] G.N.Pthn P.H.Bhthwl A Mthmtcl Modl of unmploymnt wth ffct of slf mploymnt IOSR- JM 5 pp.7-. [8] M. Nmtu M.Pt G. Mc D. Ops Stochstc Fuzzy Hybd dlyd Dynmcs Htognous compttons wth Poduct Dffntton WSEAS Tnscton on mthmtcs 79 pp [9] G.N.Pthn P.H.Bhthwl Unmploymnt-Dscusson wth Mthmtcl Modl IJBMER pp.9-7. [] G.N.pthn P.H.Bhthwl A mthmtcl modl fo Unmploymnt- Tng n cton wthout dly ADSA 7 pp.-8.

14 8 Gulbnu Pthn nd P.H. Bhthwl