Applicability of Matrix Inverse in Simple Model of Economics An Analysis

Transcription

1 IOSR Journl of Mthemtic IOSR-JM e-issn: , p-issn: X. Volume, Iue 5 Ver. VI Sep-Oct. 4, PP 7-34 pplicility of Mtrix Invere in Simple Moel of Economic n nlyi Mr. nupm Srm Deprtment of Economic M.N.C.Blik Mhviyly Nlrim Country-Ini I. Introuction: The knowlege of mtrice i very importnt not only in mthemtic, ut lo in economic, ociology n moern pychology incluing inutril mngement. It houl e importnt to note tht the evolution of concept of mtrice i the reult of n ttempt to otin compct n imple metho of olving ytem of liner eqution. Mtrix lger i lo highly ignificnt it cn enle u to o mny thing. It provie informtion regring the teting of exitence of olution y evlution of eterminnt. While for olving ytem of liner eqution the role of mtrix invere metho i highly effective. Mtrix invere cn e etermine only when the given mtrix i non-ingulr in nture. In imple ene we cn y tht mtrix invere cn e etermine only uner the conition tht eterminnt vlue of the given mtrix i not ecome equl to zero. Mtrix invere, in ppliction i efine the rtio of joint mtrix n eterminnt vlue of tht mtrix. The joint mtrix i the trnpoe of cofctor mtrix where cofctor of ech n every element of the given mtrix i etermine on the i of minor of the mtrix. If i ume ny qure mtrix where then mtrix invere uully enote y - i efine j > i. We mut hve to follow certin importnt tep for uing mtrix invere which re follow. The eterminnt vlue of the given mtrix houl e foun out which houl not e zero, otherwie it cnnot perfectly etermine invere of the mtrix. The cofctor of ech n every element of the coefficient mtrix houl e etermine with their repective minor. The cofctor mtrix houl e foun out which i the rrngement of repective row n column of etermine cofctor. More importntly, the joint mtrix which i the trnpoe of the cofctor mtrix houl e clculte. Finlly, y uing the formul given in eqution i the invere of mtrix cn e etermine. Ojective: The mjor ojective of thi pper re To nlye the role of mtrix invere for olving liner mrket moel. To nlye the effectivene of invere of mtrix for etermining equilirium ntionl income, conumption expeniture n txe. nlying the role of the me rule for olution of IS-LM moel in economic. To tuy the role of mtrix invere uner two goo mrket moel. II. Methoology: The methoology ue in thi pper to tuy the ove mentione ojective i completely nlyticl n ecriptive in nture. The moel which re ue in thi pper re collecte from inirect ource like from vriou ook of mthemticl economic. III. Dicuion: In economic, mtrix lger i ue ominntly in vriou purpoe for etermining the equilirium vlue of ytem of liner economic eqution. The ppliction of mtrix invere i very much ignificnt in thi regr which re icue elow. Solution of Liner Mrket Moel: In liner mrket moel, quntity emn i function of price P of the prouct n quntity upply i n increing function of price P where mrket clering conition previl. We cn contruct the liner mrket moel 7 Pge

2 pplicility of Mtrix Invere in Simple Moel of Economic n nlyi = P, > = c + P c, > = Mrket clering conition The ove given moel cn e rrnge follow: P P c ii We cn rrnge eqution ii in mtrix form c P B B iii Now,, exit. Now, cofctor mtrix of i = j. = Trnpoe of cofctor mtrix Now, invere of mtrix i given y j. 8 Pge

3 pplicility of Mtrix Invere in Simple Moel of Economic n nlyi By utituting - in eqution iii, we hve X c c P c c The equilirium olution i c c c c c c P Solution of Simple Ntionl Income Moel: In economic, the imple ntionl income moel i given y Y C I G C Y T, T ty t Where Y i ntionl income, C i conumption expeniture, T i tx, t i rte of tx, I i government invetment n G i government expeniture. The ove given eqution cn e rrnge Y C I G iv Y C T v ty T vi When we rrnge eqution iv, v n vi in mtrix form, we hve Now, Y I G C t T B B vi 9 Pge

4 pplicility of Mtrix Invere in Simple Moel of Economic n nlyi t t t << n <t<, therefore n hence exit. Now, the cofctor mtrix of i 3 4 j. = 3 4 t t 4 5 t 5 6 t t t Trnpoe cofctor mtrix = t t t Now, invere of mtrix i given By utituting j t t t t. in eqution vii we hve t I G t t X t I G t Y ti G C t T t I G t. t 3 Pge

5 The olution of the moel i pplicility of Mtrix Invere in Simple Moel of Economic n nlyi I G Y t I G t C t I G t T t Generl Solution of IS-LM Moel: The IS-LM moel i relte to etermintion of equilirium rte of interetr n income Y where prouct mrket IS n money mrket LM imultneouly chieve equilirium. The liner eqution relte to prouct mrket i given Y C I C Y Where I r n n Where, gin, the mthemticl form of LM moel i given M M M Y r, n M i fixe money up ply Here, M n M re repectively money emn n money upply. We cn rrnge the IS-LM moel Y C I Y Y r Y Y r Y r viii gin, M = M Y r Y r ix When we rrnge eqution viii n ix in mtrix form, we hve Y r. X B X B x Now, 3 Pge

6 pplicility of Mtrix Invere in Simple Moel of Economic n nlyi 3 Pge exit, Now, cofctor mtrix of j Trnpoe of cofctor mtrix = Now, invere of mtrix i given j. When we utitute in eqution x, we hve r Y X The olution i r Y Solution uner Two Goo Mrket Moel: The two goo mrket moel i relte to two mrket proucing two commoitie which re ol in ech repective mrket. In uch mrket, the mtrix invere i ignificnt for etermining the elling price of commoitie t equilirium in ech mrket. The erivtion form of uch moel i hown follow. In mrket,,,, P mp m c P c Mrket clering conition

7 pplicility of Mtrix Invere in Simple Moel of Economic n nlyi Where n re repectively the quntity emn n upply in mrket, P n P re the price of mrket n repectively. gin, in mrket P np,, n In mrket, In mrket, P, Mrket clering conition We cn rrnge the eqution in oth mrket, S S P mp c P c P P mp P np P P mp c xi np P P np P xii When we rrnge eqution xi n xii in mtrix form, we hve Now, Now, cofctor mtrix of i m P c n P B B m nm n, exit 3 n 3 4 m n m j. Trnpoe of cofctor mtrix m = n Now, invere of mtrix i xiii 33 Pge

8 j. When we utitute X pplicility of Mtrix Invere in Simple Moel of Economic n nlyi m n nm in eqution xiii, we hve m n c. nm c m nm nm c m nm n c nm P P n c P P Thee re the equilirium price of commoitie in ech of the mrket. IV. Concluion: The concept of mtrix invere i very ueful in economic in olving imultneou eqution, in inputoutput nlyi n even in regreion nlyi. While for etermining the ectorl output in ttic n ynmic input-output nlyi, the ppliction of mtrix invere i very importnt. From our icuion, it i cler tht inverion of mtrix i poile if n only if two conition re tifie. Firtly, mtrix whoe invere i require i qure mtrix otherwie we cnnot e le to form the eterminnt of the mtrix. Seconly, the eterminnt of the mtrix houl not e zero which implie tht the mtrix whoe invere i require houl e non-ingulr. Reference: []. R.G.D llen, Mthemticl nlyi for Economic,.I.T.B.S. Puliher 8. []. Srinth Bruh, Bic Mthemtic & it ppliction in Economic, Mcmilln. [3]..C. Ching, Funmentl Metho of Mthemticl Economic, Mcgrw-Hill Interntionl Eition 984. [4]. E.T. Dowling, Introuction to Mthemticl Economic, Mcgrw-Hill. [5]. R..Lekhi & J. Singh, griculturl Economic: n Inin Perpective, lyni Puliher, Luhin. [6]. Meht- Mnni, Mthemtic for Economit, Sultn Chn & Son 7. [7]. nupm Srm, Mthemticl Economic & It ppliction, turi Printer & Puliher, Guwhti 4. [8]. R.N. Soni, Leing Iue in griculturl Economic, Vihl Pulihing Co., Delhi. 34 Pge