To an Axiomatic Model of Rate of Growth

Transcription

1 Appled Mahemacs, 3, 4, hp://dxdoorg/436/am34979 Publshed Onlne Sepember 3 (hp://wwwscrporg/journal/am To an Axomac Model of Rae of Growh Václav Sudený, Ivan Mezník Deparmen of Appled Mahemacs, Faculy of Economcs, and Admnsraon, Masaryk Unversy, Brno, Czech Republc Insue of Informacs, Faculy of Busness and Managemen, Brno Unversy of Technology, Brno, Czech Republc Emal: VclvS@gmalcom, meznk@fbmvubrcz Receved July 7, ; revsed January 6, 3; acceped January 3, 3 Copyrgh 3 Václav Sudený, Ivan Mezník Ths s an open access arcle dsrbued under he Creave Commons Arbuon Lcense, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal work s properly ced ABSTRACT In he paper an axomac approach o express raes of growh s presened The formula s gven of rae of growh a a pon as he lm case of rae of growh on an nerval and he nverse formula s derved o compue presen and fuure value of capal for an negrable rae of growh Incdenally some nconssences n currenly used formulas are poned ou Keywords: Ineres Rae; Inflaon Rae; Rae of Growh Inroducon The concep of an average change of an objecve funcon plays a crucal role n fnancal mahemacs Reflecng he objecve funcon f, s called an neres rae, an nflaon rae, and so on I s gven as he value of ( f ( + f ( f ( For a seady sae funcon he same resul may be obaned from he formula δ ( f ( + δ f ( f ( In macroeconomcs a smlar, bu nsananeous measure, relaed o a pon s needed Baro (3 employs he formula f ( f ( (see [] whch s n fac an average change of he frs dervave The relaon beween an average change on an nerval and an average change of s dervave has no been ackled n he leraure Ths leads o he problem of (( ( ( ( δ δ + Furher, s desr- lm f f f δ able o fnd a formula ha gves he fuure value of he objecve funcon ncludng he case when he rae of growh s neher consan nor pecewse consan funcon For a consan rae of growh funcon wh values ξ we have he formula f ( f ( ( + ξ, bu s d ( generalzaon ( e x ξ s s f (see [] among ohers does no work, because he subsuon of consan funcon wh value ξ( ξ f f + ξ Accordngly he ams of he paper are as follows To defne he concep of a rae of change by means of axoms (Secon To formulae he noon of a seady sae funcon o model exsng neres raes and o fnd correspondng compuaon formulas (Secon 3 3 To derve a lm verson of a rae of growh (Secon 4 4 To fnd he nverse formula ha enables o calculae he values of a sae funcon (Secons 5 and 6 5 To pon ou o some mpacs on currenly used formulas n fnancal mahemacs (Secon 7 Axoms does no yeld ( ( ( The symbol denoes he se of real numbers Consder a quany aanng values y, y for x, x respecvely A funcon : 4 s sad o be a generalzed rae of growh funcon (shorly rae of growh funcon f he followng Axoms A-A4 are sasfed: Axom A ( x, y, x, y ( x+, y, x +, y for any (nvarance wh respec o shf of me Axom A ( x, y, x, y ( x, y k, x, y k for any k (nvarance wh respec o homoees Axom A3 s ncreasng wh respec o he frs and fourh varables and decreasng wh respec o he Copyrgh 3 ScRes

2 V STUDENÝ, I MEZNÍK 37 second and hrd varables Axom A4 ( x y x y,,, for any y (nal condon- has zero value for consan funcons 3 Seady Sae Funcons 3 Defnon Le be a rae of growh funcon For a funcon f : he funcon ( ( ( ( Ff : x, x x, f x, x, f x s called a -rae of growh of f relaed o x, x A funcon f s called a -seady sae funcon f F f s a consan funcon For he smplcy we om f s clear from he conex Verbally, F f does no depend on he choce x, x 3 Lemma Every consan funcon s a -seady sae funcon for any rae of growh funcon Le be rae of growh funcon Then here exss a funcon f : such ha ( x, y, x, y λ x x, y y whch s decreasng wh respec o he frs varable, ncreasng wh respec o he second varable and holds λ ( x, Proof: The saemen follows mmedaely from Axom A4 Now, by Axom and by Axom ( ( x, y, x, y, y, x x, y (, y, x x, y,, x x, y y y y Pung λx x,,, x x, we ge y y ( The properes of λ are obvous and hence he saemn holds rue 33 Theorem Le f : be a connuous -seady sae funcon Then f s an exponenal funcon, e f : x Ae Bx for some consans A a B Proof: Le x a x x + h be gven Then here holds ( ( x, f ( x, x + h, f ( x + h ( x h, f ( x h, x h, f ( x h Snce f s a -seady sae funcon, from Defnon 3 follows ( x, f ( x, x + h, f ( x + h ( x, f ( x, x h, f ( x h + + and wh a vew o ( we ge ( ( + ( + ( f x + h f x h λ λ h, h, f x h f x As λ s njecve n any varable, holds and hence ( ( + ( ( ( h ( f x+ h f x + h f x+ f x h f x f x ( Furher, by nducon ( ( f ( x + h f ( x f x + h ( + f ( x f x h f x + nh f x + n h ( + f ( x f x h ( ( From here follows ha he values of f a all equdsan pons form a geomerc sequence Moreover, he mplcaon f ( x+ h f ( x+ h f ( x+ h f x f x + h f x ( n ( f ( x + h f ( x ( holds rue Therefore f aans he values of some exponenal funcon a all pons of he se, a, b b ah Snce hs se s dense n, he proof s compleed because of we obaned for all x we oban ( ( f x f x ( + f ( x f x h x x h and coosng for nsance ( ( ln( ( ( f x f e f f x x, h so f : x Ae Bx A f, B f f (For more deals abou solvng funconal equaons see, where ( ( ( ( ( Copyrgh 3 ScRes

3 38 V STUDENÝ, I MEZNÍK [3] 34 Theorem Le an exponenal funcon f : x Ae Bx be a -seady sae funcon Then here exss an ncreasng funcon φ wh he propery ( x, y, x, y y φ x x y ( Proof: From he assumpon for f follows ha here holds Bx Bx Bx3 Bx4 ( x, Ae, x, Ae ( x3, Ae, x4, Ae for all x < x, x3 < x4 Denong h x x we ge (n a vew of ( Bx Bx ( x, Ae, x, Ae Bx Bh λ Ae x x, (,e cons Bx λ h Ae h for all B Furher, pung B ln ( z ge and ( and usng ( we ( h h ln( hln z z λ hz, λ h,e λ,e λ, z h as requred 35 Noe,,,,,, ( x y x y y x x y y x x φ y In fnancal mahemacs he ranslaon φ : x x (3 s employed and consequenly he rae of growh funcon s of he form y x x ( x, y, x, y y whch s called a a compound neres (per un of me Besdes (more or less from hsorcal reasons also a smple neres (per un of me s used, gven by (4 ( x y x y ˆ,,, y y y x x ( where y s preseleced consan, usually he value n a predeermned nal me Ths rae does no sasfy Axom A, and hence here s no raonal reason o use Due o hs rae polynomals of he frs degree ( ˆ ( f : y + x, y, x, y 4 Infnesmal Verson In macroeconomcs an nsananeous measure of rae of growh s ofen needed Ths may for a funcon f be naurally gven by a lmng process as (see ( ν ( f ( x lm x x ( x, f ( x, x, f ( x f ( x x x f ( x f ( x lm x x φ φe f ( x The number ( ( ν f x s called a ν-rae of growh of f a pon x An d for (see (4 we have ν ( f ( x f ( x f ( x e In macroeconomcs a measure s used, denoed by ν obaned from (6 choosng φ ln, ν ( f ( x f f ( x ( x In an analogous way we may use he same funcon for he rae of growh on nerval x, x yeldng ( x, f ( x, x, f ( x (5 (6 (7 (8 f ( x x x ln ( f ( x ln ( f ( x (9 ln ( f x x x whch represens he relave change of he compose funcon ln f wh respec o he change of he argumen of he funcon Noce, ha he same lm has he smple n eres (see (5 leng x x ˆ 5 Consequence for he Ineres Rae Calculaons Usng (, he expresson ( x, f ( x, x, f ( x ( x ( x f φ f x x ( s he rae of growh of funcon f per un of me For Copyrgh 3 ScRes

4 V STUDENÝ, I MEZNÍK 39 nsance, f f represens how he sae a dead accoun (neher deposs nor whdrawals depends on me assumng x, x are momens of me and he un of me s a year, hen ( x, f ( x, x, f ( x s he neres rae per a year, wh ereas f we choose n ( ( x x, φ we ge (denong he resulng funcon by ( x ( x x x f ( x, f ( x, x, f ( x f whch s a compound neres relaed o me segmen x x Besdes, holds + ( I s known, ha banks a he begnnng of he pas cenury (due o praccal reasons semmng from he nonexsence of compuers used o fnd he value for small he approxmaon by Taylor polynomal of he frs degree of funcon ( whch gves he resul ( + + O( (where O s Bachmann-Landau bg-o Consequenly, supposng neres rae was known for some me nerval (eg a monh, he neres rae for shorer nervals (eg a day was calculaed dvdng by 3 nsead of as he 3h roo To legalze hs naccuracy, he noon of an nerval of addng of neress was nroduced wh he clause, ha f he curren nerval was shorer han ha under assumpon, he neres wll be calculaed mulplyng only by a lnear par of he ncremen of he neres rae Hence funcon f represenng he sae of accoun beng n a seady sae was changed from exponenal o pecewse lnear havng wh he orgnal exponenal curve common only breakng pons Ths pracce s sll survvng, despe banks use sofware ha s defnely capable o calculae he roos The reason ress (probably wh he shorage of managemen heorecal compeence The dfference beween he exac value and s approxmaon, e an error of approxmaon s an ncreasng funcon when me approaches o nfny havng fne lm e because holds lm lm + + e ( Ths lm s employed n a number of books on fnancal mahemacs, alhough s nerpreaon s raher problemac When we calculae compound neres and manpulae wh a compound neres as wh a smple neres n such a way ha we dvde me nerval n equdsan subnervals and apply he neres ha s he lnear par of he approxmaon for hese subnervals, we oban he resul, whose lm for he number of subnervals approachng o nfny s gven by formula ( A magc appearance of Euler consan n hs calculaon gave brh he noon of connuous compoundng I may be smply verfed ha s n fac a compound neres, where n formula (4 he value ln ( + nsead of s appled The number ln ( + may be obaned as a rae of growh when pung φ ln n ( and hen by lmng we ge ν as n (8 6 Inverse Problem Le us use for he rae of growh formula (7 and denoe ν ν wh argumen n he sequel Then we have for a fxed ( f ( f ( f ( ν (3 e Supposng f s gven, hen (3 s he formule o fnd he rae of growh ν Alernavely, when ν s gven, hen (3 s a dfferenal equaon o ge he funcon f Ths equaon can be rearranged equvalenly o or wh he soluon where ν ( s ( ν ( + ( f ( ln ln ( ln ( ν ( ( f + f ( f ( ln ( ν ( s + d s ( e f, (4 s he neres rae per un of me a he momen s Performng he same calculaon for ν (see (8, we ge ν f f f wh he soluon ( ( ( ( ( f ( d e ν s s f ( Alhough he formula (5 s clearly smpler han (4, has dsadvanage, because yelds quanavely bad resuls For nsance f we subsue a consan neres rae n (5, we do no oban he formula for a compound neres! The followng example llusraes he use of formula (4 Example We assume ha he nflaon rae per a un of me (eg a year a me and me s known Sup- ha he nflaon rae per un of me a me s pose and a me Delberae on he nflaon rae on nerval, I s evden ha hs depends on he (5 Copyrgh 3 ScRes

5 33 V STUDENÝ, I MEZNÍK changes of he nflaon rae on, Consder he followng four cases of he nflaon rae:, f u < ι ( u, f u, u ι ( u +, u 3 ι ( u +, ι 4 ( u, f u, f u > Noce ha he frs and he las cases are rval he rae s consan and he nerval has a un lengh and hus he nflaon rae should be he same consan The general formula mus gve he same resul By (4 we have ( ( ln ( + ι ( u d u f f e (6 ( ( ( Applyng (6 we ge consecuvely (seng f ( ( ( ( ( ln ( du for : f e ( ln ( + u du for : f e ( ln ( + u du ( ln ( du for 3: f e for 4 : f e Now, applyng (5 we oban resuls du e 5798 u + du e 4638 u+ du e du e 4758 The resuls are surprsngly no equal (parcularly he frs and he las one whch s an evden falure Formula for he fuure value of he compound neres n case of consan neres rae s gven by ( ( ( f f + ξ (7 In case of pecewse consan neres rae, e f I are he values of consan neres rae per year on me nervals (, +,,, n, hen he neres rae per n, + s gven by ( 6 Theorem n ( ( + + I (8 Formula (7 s a specal case of formula (8 for a con- san neres rae and formula (8 s a specal case of formula (4 for a pecewse consan neres rae Proof: Assume ν ( I s consan Then here holds ( ln( + I du ( ln( + I ( ln( + I ( e e e + I and hence he frs par of he saemen holds rue Now le ν be pecewse consan possessng values I on nervals (, +,,, n and χ A be a characersc funcon of se A We have ι( χ(, I + and hence e ( ln ( + χ(, + Idu n 6 Theorem ( + ln( + I ln( + I n ( + ln( + I e e n ( ( ln n + I + ( I e + and he proof s compleed + Formula (4 s a lm case of formula (7 Proof: Frs we show, ha for every connuous funcon f defned on a closed nerval, here exss a sequence of pecewse consan funcons ξ wh he propery ξ f Le f be a connuous funcon Due o he as- sumpon Dom( f s he compac se Le be pos- x Dom( f we fnd a ve real number For every such h f O x O f x, Dam U s a dameer of U Consder a par- Dom f gven by n dsjon subnervals ( J n of he lengh δ In every subnerval J we choose a pon x, and denoe y f ( x Furher, defne ζ ( x y for all x J Then for every x Dom( f holds f ( x ζ ( x and for ξn ζ he above n neghborhood O( x a ( ( ( ( { O( x } forms a coverng of Dom ( f x Dom( f a fne subcovery Ω and defne δ mnu Ω ( Dam ( U where ( on of ( Choose propery s sasfed Snce he funconal ln ( + ι( s d s : e Φ ι s connuous n he opology of unform convergence, we ge ( ψ ( lmψ ( lm Φ Φ Φ Ψ and he p roof s compleed 7 Ineres Rae of Smple Compoundng As an mpac of he precedng consderaons le us pon o he ssue of smple compoudng Smple compoundng Copyrgh 3 ScRes

6 V STUDENÝ, I MEZNÍK 33 s a suaon n whch dependence of a quany on me s a polynomal of he frs degree (le us call he dependence of he quany on me a sae funcon In hs suaon, specal rae of growh s used (see (5 bu hs rae has fundamenal concepual flaws One of hem ress wh he mxng of dfferen ways of measurng he rae of growh From he above consderaons we can conclude, ha n all suaons he only one rae of growh s suffcen gven by ( In wha follows we compue he rae of growh of a quany, whch s smply compounded ( may be called compound neres rae of a smple compoundng I s evden f he rae of growh funcon s consan and posve, hen he sae funcon s ncreasng and convex Furher f he rae of growh funcon s posve and decreases suffcenly quckly, hen he sae funcon s ncreasng bu concave Now we are lookng for he rae of growh funcon, whch makes he sae funcon affne, e has he form of a polynomal of he frs degree To fnd, we consder he sae funcon (see (4 ( Is dervave s gven by ln ( +ξ ( s d s f e (9 ( ( + ξ ( ln ( + ξ ( d ln e s s f and second dervave by f ( ( ( e ln +ξ s d s ( ( ξ( ( ξ + ( ln + + ( ln ( + ξ( ξ( + ξ ( Snce he sae funcon s polynomal of he frs degree s second dervave mus be equal o zero If f ( and ξ ( >, hen second dervave s equal o zero for such ξ (, ha are soluon of he dfferenal equaon ( ( ( ( ( ( ( ( ξ + ln + ξ + ln + ξ ξ ( The soluon of ( s ξ ( C for any c C For from ( C ln e ( onsan he gven value ( and hence he paral soluon s ( + ξ ( ξ we ge ξ ( + ln( + ξ ( e ( ξ ( ξ ( ln( If we subsue hs rae no (4, we ge ( f ( e f ( ( ξ ( f ln ( ( ξ s ln( + ξ ( + d s + ( ln + + ( (3 whch s really an affne funcon, e a sae funcon of a smple compoundng Applyng (5, subsung x and x arbrary and seng corespondng y f ( x, y f ( x and y f ( due o (3, we oban formula for he rae of smple compoundng of (3 ( ξ( ι ln + whle rae of growh funcon of (3 s 8 Conclusons ι + ι e In he arcle we presened an explc formula for all possble raes of growh possessng naural properes (descrbed by Axoms A-A4 (see ( Furher we derved he new formula for he rae of growh a a pon by lmng process Ths formula enables o assgn o sae funcon s rae of growh (see (7 Moreover formula s gven o fnd a sae funcon on condon s rae of growh funcon a any pon s known (see (4 Alhough he choce of Axoms A-A4 seems o be naural, he condon ha any exponenal funcon s a seady sae funcon s of crucal mporance I s an open problem of fndng a smpler condon or o show ha hs condon may be derved from he axoms REFERENCES [] R J Barro and X Sala--Marn, Economc Growh, nd Edon, 3 [] J Dupačová, J Hur and J Š epán, Sochasc Modelng n economcs and Fnance, Kluwer Academc Publshers, [3] J Aczél, Lecures on Funconal Equaons and Ther Applcaons, Academc Press, New York, 966 [4] V Sudený, Funconal Equaon of he Rae of Inflaon, e-prn Archve of Coronell Unversy, 3 The frs usage of he new way and new formulas can reader fnd n [4] The usual approach o he problem problemac, as shown n hs arcle can be seen n [5-8] For more deals of mahemacal analyss used here see [9] Copyrgh 3 ScRes

7 33 V STUDENÝ, I MEZNÍK hp://arxvorg/abs/mah/37395 [5] J C Van Horne, Fnancal Markes, Raes and Flows, Prence Hall, Englewood Clfs, 978 [6] R C Meron, Connuous Tme Fnance, Blackwell, Cambrdge, 99 [7] I Karazas and S Shreve, Mehods of Mahemacal Fnance, Sprnger, New York, 998 [8] T Cpra, Mahemacs of Secures, HZ, Praha, [9] J Deudonné, Trease on Analyss, Vol III, Pure and Appled Mahemacs, Vol -III, Academc Press, New York, London, 97 Classfcaon Codes: MSC (Mahemacs Subjec Classfcaon 6A, Rae of growh of funcons, orders of nfny, slowly varyng funcons 6P, Applcaons o economcs 39B, Equaons for real funcons 9B, Fundamenal opcs (basc mahemacs, mehodology; applcable o economcs n general 9B8, Fnance, porfolos, nvesmen 9B4, Prce heory and marke srucure 34A5, Explc soluons and reducons JEL (Journal of Economc Leraure E4, Money and Ineres Raes: General (ncludes measuremen and daa E43, Deermnaon of Ineres Raes; Term Srucure of Ineres Raes C, Mahemacal Mehods C43, Index Numbers and Aggregaon C63, Compuaonal Technques C65, Mscellaneous Mahemacal Tools Copyrgh 3 ScRes