Time-Sace Moel of Business Flucuaions Aleei Kouglov*, Mahemaical Cene 9 Cown Hill Place, Suie 3, Eobicoke, Onaio M8Y 4C5, Canaa Email: Aleei.Kouglov@SiconVieo.com * This aicle eesens he esonal view of auho.
ABSTRACT Hee auho mae an aem o een he Coninuous-Time Moel of Business Flucuaions on he sace omain. Reseach mehoology is base on Time-Sace Moel of Wave Poagaion eveloe by auho o escibe flucuaion ocesses in hysics. Jounal of Economic Lieaue Classificaion Numbes: E 3. Keywos: Business Flucuaions
. Inoucion Recen Coninuos-Time Moel of Business Flucuaions (e.g. see [, 3]) escibes he naue of business flucuaions in he ime omain only. Lae auho eveloe (see [4]) he Time-Sace Moel of Wave Poagaion, which successfully escibes some flucuaion oblem in hysics. Auho uae he Business Flucuaions moel hee o escibe such flucuaions boh in ime an in sace.. Moel Assumions The following is hel in he ime omain. A each oin in sace I assume: (a) The eivaive of commoiy s ice wih esec o ime is iecly ooional o he amoun of commoiy s efici a his ime (whee efici means he commoiy s eman minus is suly). (b) The secon eivaive of commoiy s oucion wih esec o ime is iecl ooional o he eivaive of commoiy s ice wih esec o ime. (c) The eivaive of commoiy s eman wih esec o ime is invesel ooional o he eivaive of commoiy s ice wih esec o ime. The following is hel in he sace omain. A each oin in ime I assume: () The eivaive of commoiy s ice wih esec o iecion is iecl ooional o he amoun of commoiy s efici a his lace. (e) The secon eivaive of commoiy s oucion wih esec o iecion is iecly ooional o he eivaive of commoiy s ice wih esec o iecion. 3
(f) The eivaive of commoiy s eman wih esec o iecion is invesel ooional o he eivaive of commoiy s ice wih esec o iecion. Fo simliciy, I will escibe he one-imensional sace. 3. Moel Desciion Le me consie an economical moel wih one commoiy. I enoe ( ) volume of commoiy s eman in oin a ime, an ( ) commoiy s oucion in oin a ime. V, he volume of I assume ha sysem was a es unil ime =, i.e. oucion ae V ) ) = coincie wih eman ae ( ) an in sace, an whole commoiy was consume. Theefoe ( ) = ) cons = ) V, he V, = in evey oin in ime,, () ) = V ) = V, () fo < < an fo all. oin =. A ime =, I assume ha commoiy s eman ae has been incease in he Hence (,) = (, ) +. (3) 4
Le s look how he iniial commoiy s efici ae sace an ime. will be oagae in he Accoing o ou assumions we can wie he following eessions. In he ime omain, (, ) P V (, ) V (, )) = λ ( V, (4) ( ) P( ),, = µ, (5) ( ) P( ) V,, = ν, (6) whee,, µ, ν > a ime. ( ) V f, λ ae consans, an P (, ) is commoiy s ice in oin We can ewie equaions (4) (6) o escibe ynamics of commoiy s efici in ime, V f (, ) V (, ) f (, ) = + ν λ + µ λv f, (7) whee V ) = V ) V ( ), ( ) f, f ) V f, =. Equaion (7) is he oinay iffeenial equaion of he secon oe, an can be esolve by sana mehos (see [, 5]). Noe ha he value (, ) In he sace omain, ) P V f when + ) V )) fo all. = λ ( V, (8) > 5
V V ) P ) = µ, (9) ) P( ) = ν, (), whee, λ, µ, ν > ae consans. We also can ewie equaions (8) () o escibe ynamics of commoiy s efici ( ) 5]). V f, in sace, V f ) V ) f ) = + ν λ + µ λv f. () Equaion () is also he oinay iffeenial equaion of he secon oe (see [, Hee again he value ( ), when + fo all >. V f I sai befoe abou inenion o eesen value ( ) V f which oagae fom iniial value of commoiy s efici ae Le me show how he value ( ) V f, of commoiy s efici,., is obaine fo > an >. If I enoe c he velociy of efici seaing, hen commoiy s efici (sulus) in oin sas o aea a ime Theefoe, if =. c V f,. hen ( ) If = + (whee > ) we may fin a fis (, ) V f fom equaion (7), an iniial values V f (,) = an f (,) =. Then we fin V f (, ) fom equaion (), an iniial values (, ) V f an (, ) =. This (, ) V f V f is he 6
commoiy s efici in oin a ime fo ieal insan velociy of efici seaing ha is equal o he commoiy s efici in oin a ime ( + ) fo efici seaing wih eal veloci c. 3. Dynamics of Pices Similaly o (7), we can eess he ynamics of commoiy s ice in ime, P (, ) P(, ) (, ) + = + ν + λ µ P C whee C λ (, ) P(, )) iniial ime. λ, () ( µ =. Noe ( ), ( ), P, ae values in oin a Theefoe, P(, ) (, ) (, ) P µ when + fo all. > An i follows he ynamics of commoiy s ice in sace, ) P ) P ) + = + ν + λµ P C whee C λ (, ) P(, )) iniial oin. λ, (3) ( µ =. Noe ( ), ( ), P, ae values a ime in Theefoe, P ) (, ) (, ) P µ when + fo all. > 7
4. Conclusive Remaks I eose hee only some feaues (which ae alicable o he henomenon of business flucuaions) of he Time-Sace Moel of Wave Poagaion. Fo moe insigh ino he oblem, one shoul aess he oiginal aicle [4] o subsequen aes. Refeences. V. I. Anol, Oinay Diffeenial Equaions, 3 eiion, Singe Velag, Belin; New Yok, 99.. A. Kouglov, Dynamics of Business Flucuaions in he Leonief-ye Economy, Woking Pae ew-mac/9877, Washingon Univesiy in S. Louis, July 998 (available a h://econwa.wusl.eu). 3. A. Kouglov, Mahemaical Moel of Comeiive Imacs beween Business Eniies, Woking Pae ew-mic/9933, Washingon Univesiy in S. Louis, Mach 999 (available a h://econwa.wusl.eu). 4. A. Kouglov, Dual Time-Sace Moel of Wave Poagaion, Woking Pae hysics/9994, Los Alamos Naional Laboaoy, Seembe 999 (available a h://.lanl.gov). 5 I.G. Peovski, Oinay Diffeenial Equaions, Penice Hall, Englewoos Cliffs, NJ, 966. 8