OPTIMAL PRODUCTION SALES STRATEGIES FOR A COMPANY AT CHANGING MARKET PRICE

Transcription

1 REVISA DE MAEMÁICA: EORÍA Y APLICACIONES (1) : CIMPA UCR ISSN: (PRIN), (ONLINE) OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY A CHANGING MARKE PRICE ESRAEGIAS ÓPIMAS DE VENAS Y PRODUCCIÓN PARA UNA COMPAÑÍA EN UN MERCADO CON PRECIOS CAMBIANES ELLINA V. GRIGORIEVA EVGENII N. KHAILOV Received: 23/Feb/214; Revised: 28/Aug/214; Acceped: 3/Oc/214 Deparmen of Mahemaics and Compuer Sciences, exas Woman s Universiy, Denon, X 7624, USA. egrigorieva@mail.wu.edu Deparmen of Compuaional Mahemaics and Cyberneics, Moscow Sae Lomonosov Universiy, Moscow, , Russia. khailov@cs.msu.su 89

2 9 E.V. GRIGORIEVA E.N. KHAILOV Absrac In his paper we consider a monopoly producing a consumer good of high demand. Is marke price depends on he volume of he produced goods described by he Cobb-Douglas producion funcion. A producionsales aciviy of he firm is modeled by a nonlinear differenial equaion wih wo bounded conrols: he share of he profi obained from sales ha he company reinvess ino expanding own producion, and he amoun of shor-erm loans aken from a bank for he same purpose. he problem of maximizing discouned oal profi on a given ime inerval is saed and solved. In order o find he opimal producion and sales sraegies for he company, he Ponryagin maximum principle is used. In order o invesigae he arising wo-poin boundary value problem for he maximum principle, an analysis of he corresponding Hamilonian sysem is applied. Based on a qualiaive analysis of his sysem, we found ha depending on he iniial condiions and parameers of he model, boh, singular and bangbang conrols can be opimal. Economic analysis of he opimal soluions is discussed. Keywords: nonlinear microeconomic conrol model; producion-sales sraegy; Ponryagin maximum principle; Hamilonian sysem. Resumen En ese arículo consideramos un monopolio produciendo un produco de consumo de gran demanda. Su precio de mercado depende del volumen de producción descrio por la función de producción de Cobb-Douglas. Una acividad de producción y venas de la firma es modelada por una ecuación diferencial no lineal con dos conroles de fronera: la paricipación en el resulado de las venas que la compañía reinviere para expandir su propia producción, y el mono de los présamos a coro plazo adquiridos del sisema bancario con el mismo propósio. Se planea y resuelve el problema de maximizar la ganancia oal desconada en un inervalo de iempo dado. Para enconrar las esraegias ópimas de producción y venas para la compañía, se usa el principio del máximo de Ponryagin. Para invesigar el problema de valores de dos punos de fronera que aparece para el principio del máximo, se aplica un análisis del sisema hamiloniano correspondiene. Basado en un análisis cualiaivo del sisema, enconramos que dependiendo de las condiciones iniciales y los parámeros del modelo, ano el conrol singular como el bang-bang pueden ser ópimos. Se discue un análisis económico de las soluciones ópimas. Palabras clave: modelo de conrol microeconómico no lineal; esraegia de producción y venas; principio del máximo de Ponryagin; sisema hamiloniano. Mahemaics Subjec Classificaion: 49J15, 58E25, 9A16, 93B3.

3 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 91 1 Inroducion Behavior of economic processes can be undersood and prediced using mahemaical models. Many well-known mulidimensional microeconomic models such as he Leonief model [11] or he Loon s model [12] are linear. While nonlineariy characerizes mos of economic processes and phenomena, analyical sudy of muli-dimensional nonlinear models is raher complex and laborious. Of course, one can experimen wih such models on a compuer, bu an analyical sudy of an economic model compared wih simulaion of he model on a compuer has an imporan advanage. I gives us he opporuniy o ge he whole picure of he sudied process or phenomenon for any values of he model s parameers, while he simulaion gives only a cerain fragmens of he general picure a he seleced parameer values. herefore, in order o sudy he long-erm rends, growh facors or assess he impac of various opions for managemen decisions, he sudy of he lower-secor nonlinear models are applied. One such model is he one-secor Solow-Swan model [15, 16]. In his model, he company produces a single produc, and sells i o obain profi. he profi from sales can be used for boh, consumpion and invesmen. he abiliy o manage such capial disribuion leads o he consideraion of various conrol models, as well as o he sudy of he corresponding opimal conrol problems [1, 7, 3, 4, 5, 8]. In his paper, he economic dynamics of a company is described by he Solow- Swan ype conrol model. Wha differs our work from he papers referred earlier is ha he marke (sales) price of he produced good depends on a cerain model of he marke. In Secion 2 we inroduce our conrol model alone wih he model-driven mechanism for he formaion of he marke price of he produc; he properies of he model are also sudied. Addiionally, in his secion we sae he corresponding opimal conrol problem and discuss he exisence of is opimal soluions. In order o analyze he opimal conrol problem, in Secion 3, we apply he Ponryagin maximum principle and sae he corresponding wo-poin boundary value problem for he maximum principle. In Secion 4 we sudy he possibiliy for he boundary value problem o have singular regimes. Secion 5 conains auxiliary resuls relaed o he subsequen analysis of he Hamilonian sysem, which is issued based on he boundary value problem of he maximum principle. Secion 6 presens an analysis of he Hamilonian sysem, based on he resuls obained in he previous secion. his analysis allows us o specify in Secion 7 all possible ypes of he opimal conrols ha can arise in he sudy of he opimal conrol problem. In Secion 8, we describe he numerical procedure ha helps us o invesigae he cases for which we canno predic analyically specific feaures of he opimal conrol, bu only is ype. Finally, in Secion 9 we describe possible direcions for furher research relaed o modificaion of he

4 92 E.V. GRIGORIEVA E.N. KHAILOV conrol model so as o he corresponding opimal conrol problem. Secion 1 conains our conclusions. 2 Opimal conrol problem Consider a company ha produces a single consumer good of high demand on a sable marke. We assume ha he company is a monopoly in his marke. Moreover, we suppose ha he company sells all i produces a he marke price. Le x be he producion funds of he company. hen, Y = Π(x) is he amoun of he produced consumer good, where Π(x) is he Cobb-Douglas producion funcion given by he formula Π(x) = rx 1 σ x σ, σ (, 1]. Here x is he iniial producion funds, and r is he profiabiliy of he producion wih he iniial producion funds of x. Le p be he marke price of he produced consumer good, and p be is uni price. We assume ha he dynamics of he producion funds of he company obeys he Cauchy problem { ẋ() = I() δx(), [, ], x() = x, x >. (1) where δ is amorizaion facor ha shows depreciaion of he producion funds, and I() is oal invesmens made by he company o increase he producion funds. he main balance relaionship a ime [, ] is as follows p()y () = I 1 () + C(), (2) where he lef hand side is he oal profi obained from he marke sale of he consumer good in he amoun of Y () a he price p(). he righ hand side here is he sum of he own invesmens I 1 () and consumpion C(). We suppose ha he oal invesmen I() consiss of he own invesmen I 1 () and of he ouside invesmen I 2 (). he own invesmen I 1 () is he share of he profi u()p()y (), received from he sales of he consumer good on he marke. Invesmen I 2 () is he shor-erm loans K(). Here he funcions u(), K() are conrols, which saisfy he resricions: u() 1, K() K max. (3) From he formula (2) and previous argumens i is easy o see ha C() = (1 u())p()y ().

5 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 93 By he definiion of he funcion Y () and he equaion (1), we have he Cauchy problem { ẋ() = u()p()π(x()) + K() δx(), [, ], (4) x() = x, x >. By he model of he marke [18], we assume ha he marke price of he consumer good saisfies he following formula p() = p by () = p bπ(x()) >, [, ], (5) where p is he maximum possible marke price of he consumer good, and b is some given consan. Subsiuing he expression (5) ino he differenial equaion of he Cauchy problem (4), we rewrie i as follows { ẋ() = u() ( p bπ(x())) Π(x()) + K() δx(), [, ], (6) x() = x, x >. As he objecive funcion we choose he oal discouned profi of he company on he given ime inerval [, ] of he following ype I(u, K) = e ρ{ (1 u()) ( p bπ(x())) Π(x()) p Π(x()) } (1 + λ)k() d + e ρ x( ). I is required o maximize his objecive funcion on he se of admissible conrols, which consiss of all possible Lebesgue measurable funcions (u(), K()) saisfying he inequaliies (3) for almos all [, ]. In he expression (7), he value ρ is he discouning coefficien ha reflecs he rae of inflaion, and λ is he loan annual percenage rae. he erm e ρ x( ) is he discouned producion funds of he company a he final ime. Moreover, he soluion x() mus be posiive and saisfy he inequaliy from (5) for all [, ]. hus, we have he opimal conrol problem (7) subjec o (6) wih conrols saisfying (3). Nex, we consider he case of a linear Cobb-Douglas producion funcion, i.e. σ = 1. hen, he Cauchy problem (6) can be wrien as { ẋ() = u() ( p brx()) rx() + K() δx(), [, ], (8) x() = x, x >, and he objecive funcion (7) is ransformed as follows I(u, K) = e ρ{ (1 u()) ( p brx()) rx() p rx() } (1 + λ)k() d + e ρ x( ). (7) (9)

6 94 E.V. GRIGORIEVA E.N. KHAILOV In order o simplify he opimal conrol problem (3), (8), (9) we inroduce a new variable y = br p 1 x, and also following posiive consans α = pr, β = p r, where α > β. We also define a new conrol v() = br p 1 K(), where v() v max and v max = br p 1 K max. A las, le y = br p 1 x. Moreover, he posiiviy of he price p() in he formula (5) consiss of saisfying he condiion y() < 1 for all [, ]. hen, he Cauchy problem (8) and he objecive funcion (9) we rewrie as { ẏ() = αu()(1 y())y() + v() δy(), [, ], (1) y() = y, y (, 1), I(u, v) = p br ( e ρ{ α(1 u())(1 y())y() βy() } ) (1 + λ)v() d + e ρ y( ). (11) By he class of admissible conrols Ω( ) we undersand all possible Lebesgue measurable funcions (u(), v()) ha for almos all [, ] saisfy he resricions: u() 1, v() v max. (12) For convenience, we inroduce he new objecive funcion J(u, v) = e ρ{ α(1 u())(1 y())y() βy() } (1 + λ)v() d + e ρ y( ), (13) which is differen from he funcion (11) by he posiive facor p(br) 1. Maximizing his objecive funcion on he se Ω( ) of all admissible conrols, we will consider furher. Moreover, he soluion y() mus saisfy he inequaliies: < y() < 1, [, ]. (14) hus, we obain he opimal conrol problem (1), (12) (14). Now, we discuss he execuion of he resricions (14). Suppose ha in he subsequen argumens he inequaliy δ > v max is valid. I means ha he amoun of he loan K() is comparable o he amoun δx() of producion funds ougoing from producive aciviies. Noe ha if he righ inequaliy in (14) is violaed, hen i can lead o he excessive filling of he marke by he consumer good and reduce is marke price o zero, which is impossible. herefore, following saemen holds.

7 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 95 Lemma 1 For arbirary conrols (u( ), v( )) Ω( ) he soluion y() of he corresponding Cauchy problem (1) exiss on he enire inerval [, ] and saisfies he inequaliies (14). By Lemma 1 and resuls from [1], we see ha in he problem (1), (12) (14) here exiss he opimal decision consising of he opimal conrols (u (), v ()) and corresponding opimal soluion y (), [, ]. For heir analysis we apply he Ponryagin maximum principle [13]. he resuls of his analysis are presened in subsequen secions. 3 Ponryagin maximum principle We wrie he Hamilonian of he problem (1), (12) (14) as H(y, ψ, u, v, ) = ( ) αu(1 y)y+v δy ψ+e ρ( ) α(1 u)(1 y)y βy (1+λ)v, where ψ is he adjoin variable. hen, by he virue of he Ponryagin maximum principle [13], for he opimal conrols (u (), v ()) and corresponding opimal soluion y () here exis he nonrivial soluion ψ () of he adjoin sysem ( ) ψ () = αu ()(1 2y ()) δ ψ () ψ ( ) = e ρ, e ρ (α(1 u ())(1 2y ()) β ), [, ], (15) such ha he conrols (u (), v ()) maximize he Hamilonian H(y (), ψ (), u, v, ) wih respec o he values u, v for almos all [, ], and herefore saisfy he relaionships:, if α(1 y ())y ()(ψ () e ρ ) <, u () = u [, 1], if α(1 y ())y ()(ψ () e ρ ) =, 1, if α(1 y ())y ()(ψ () e ρ ) >, (16), if ψ () (1 + λ)e ρ <, v () = v [, v max ], if ψ () (1 + λ)e ρ =, v max, if ψ () (1 + λ)e ρ >. (17)

8 96 E.V. GRIGORIEVA E.N. KHAILOV Now, in hese relaionships and in he sysem (15), we will execue he subsiuion of he variable η () = e ρ ψ () 1. hen, he adjoin sysem (15) can be rewrien as ( ) η () = αu ()(1 2y ()) (δ + ρ) η () ( ) α(1 2y ()) (δ + ρ + β), [, ], (18) η ( ) =, and by posiiveness of values e ρ and α(1 y ())y (), he relaionships (16), (17) are ransformed as follows, if η () <, u () = u [, 1], if η () =, (19) 1, if η () >,, if η () < λ, v () = v [, v max ], if η () = λ, (2) v max, if η () > λ. herefore, based on sysems (1), (18) and relaionships (19), (2) we obain he wo-poin boundary value problem (PBVP) for he maximum principle of he ype { ẏ() = αu(η())(1 ( y())y() + v(η()) ) δy(), ( ) η() = αu(η())(1 2y()) (δ+ρ) η() α(1 2y()) (δ +ρ+β), where (21) y() = y, η( ) =, y (, 1), (22), if η <, u(η) = u [, 1], if η =, 1, if η >,, if η < λ, v(η) = v [, v max ], if η = λ, v max, if η > λ. (23) (24) Noe ha he opimal conrols (u (), v ()), opimal soluion y () and corresponding soluion η () of he adjoin sysem (18) saisfy his boundary value problem. Nex, we sudy he PBVP for he maximum principle (21) (24) in deails.

9 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 97 4 Singular regimes Now, we will sudy he presence of singular regimes in he PBVP (21) (24). his means sudying he occurrence of he cases in which he soluion η() can be equal o or λ on some finie subinervals of he segmen [, ], or which means ha he relaionships (23), (24) canno uniquely deermine he conrols u and v, respecively. Analyzing he formulas (23), (24) for he conrols u(η), v(η) we see ha singular regimes are possible only in he following wo siuaions. Le η() = for all [, ]. Since η() = < λ, hen obviously ha he conrol v(η()) =. From he equaliy η() = on he inerval, we find ha η() =. Subsiuing hese equaliies ino he second equaion of he PBVP (21) (24) we find he expression α(1 2y) (δ + ρ + β) =. By his equaliy, we inroduce he value y = α δ ρ β. (25) 2α herefore, a he poin (y, ) of he plane (y, η), he singular regime is possible only if y >, ha also can be wrien as α > δ + ρ + β. If we have he opposie inequaliy α δ + ρ + β, hen he singular regime is no available. Furher, we suppose ha α > δ + ρ + β. By he formula (25), we have y (, 1). Le us find he conrol u on his singular regime. From he firs equaion of he PBVP (21) (24) we find he required formula u sing = δ α(1 y ). I is easy o check ha his conrol is admissible, i.e. u sing (, 1). Le η() = λ for all [, ]. Since η() = λ >, hen obviously ha he conrol u(η()) = 1. From equaliy η() = λ on he inerval we find ha η() =. Subsiuing hese equaliies ino he second equaion of he PBVP (21) (24) we find he expression ( ) ( ) α(1 2y) (δ + ρ) λ + α(1 2y) (δ + ρ + β) =.

10 98 E.V. GRIGORIEVA E.N. KHAILOV By his equaliy, inroduce he value y = α δ ρ β(1 + λ) 1. (26) 2α herefore, a he poin (y, λ) of he plane (y, η), he singular regime is possible only if y >, or for α > δ + ρ + β(1 + λ) 1. If we have he opposie inequaliy α δ + ρ + β(1 + λ) 1, hen he singular regime is no possible. Furher, we suppose ha α > δ +ρ+β(1+λ) 1. By he formula (26), we have y (, 1). Le us find he conrol v on his singular regime. From he firs equaion of he PBVP (21) (24) we find he required formula v sing = αy 2 (α δ)y. I is easy o check ha v sing <, and hence his conrol is no admissible. herefore, he singular regime: u = u sing, v =, y = y, η = (27) is possible, if he inequaliy α > δ + ρ + β holds. 5 Auxiliary resuls Le us consider he sysem of he equaions (21) and he relaionships (23), (24), which form he corresponding Hamilonian sysem. We will invesigae his sysem in deails. Firs, we will sudy he curves in he plane (y, η) on which ẏ = and η =. Le ẏ =. From he firs equaion of he sysem (21) we have αu(η)(1 y)y + v(η) δy =. (28) Nex, we consider he possible cases for he conrols u(η) and v(η). Le u(η) =. his is possible, when η <. hen, η < λ, and herefore v(η) =. Hence, we obain he line y =. Le u(η) = 1. his is possible, when η >. In his case here are wo possible siuaions.

11 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 99 Le η > λ. hen, we have v(η) = v max. herefore, he equaion (28) can be wrien as αy 2 (α δ)y v max =. he discriminan of his quadraic equaion is posiive, and so i has wo disinc roos: ỹ = (α δ) + (α δ) 2 + 4αv max, 2α ỹ = (α δ) (α δ) 2 + 4αv max. 2α I is easy o check ha ỹ (, 1) and ỹ <. hen, we have he line y = ỹ. Le < η < λ. hen, we have v(η) =. herefore, he equaion (28) can be wrien as αy 2 (α δ)y =, from which we find wo roos y = and y = α 1 (α δ). Inroduce he value ȳ = α δ α. he locaion of ȳ on he axis y depends on he raio beween α and δ. I is easy o see ha for α > δ we have ȳ (, 1); for α = δ we have ȳ = ; for α < δ we have ȳ <. Noe immediaely ha for all values of he parameers he inequaliy ȳ < ỹ holds. Finally, we obain he line y = for α δ, and he lines y =, y = ȳ for α > δ. Combining he previous resuls for he curve ẏ =, we have he relaionships: { y =, if η < λ, α δ : y = ỹ, if η > λ, y =, if η < λ, (29) α > δ : y = ȳ, if < η < λ, y = ỹ, if η > λ.

12 1 E.V. GRIGORIEVA E.N. KHAILOV Le η =. From he second equaion of he sysem (21) we have ( ) ( ) αu(η)(1 2y) (δ + ρ) η + α(1 2y) (δ + ρ + β) =. (3) Nex, we consider he possible cases for he conrol u(η). Le u(η) = 1. his is possible, when η >. Inroduce he value ŷ = α ρ δ. 2α hen, from he expression (3) we find he formula η = 1 β 2α(y ŷ). (31) he graph of he funcion (31) defines he increasing hyperbola wih he line y = ŷ as is verical asympoe. he locaion of ŷ on he axis y depends on he raio beween α and (ρ + δ). I is easy o see ha for α > ρ + δ we have ŷ (, 1); for α = ρ + δ we have ŷ = ; for α < ρ + δ we have ŷ <. Noe immediaely ha he inequaliy ŷ < ỹ holds for all values of he parameers. Moreover, i is easy o see ha for α δ he inequaliy ŷ < ȳ is valid. he graph of (31) is also characerized by he poin (y, ), in which his graph inersecs wih he axis y. From he formula (25) we find ha he locaion of y on he axis y depends on he raio beween α and (ρ + δ + β). I is easy o see ha for α > ρ + δ + β we have y (, 1); for α = ρ + δ + β we have y = ; for α < ρ + δ + β we have y <. From he analysis of he formula (31) we obain ha for α ρ + δ here is no curve η = a he considered region η >. On he conrary, for α > ρ + δ here exiss he curve ha is defined on he inerval (max{; y }, ŷ) and given by he formula (31). Le u(η) =. his is possible, if η <. hen, from he expression (3) we find he formula η = 2α δ + ρ (y y ), (32)

13 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 11 which defines he decreasing linear funcion. As in he previous case, a he poin (y, ) he graph of he funcion (32) inersecs wih he axis y. herefore, we obain he curve η = ha is defined on he inerval (max{; y }, 1) and given by he formula (32). Combining he previous resuls for he curve η =, we have he relaionships: α ρ + δ : η = 2α δ + ρ (y y ), if y (, 1), (33) η = δ 2α + ρ (y y ), if y (, 1), ρ+δ < α ρ+δ+β : β η = 1 2α(y ŷ), if y (, ŷ), (34) η = δ 2α + ρ (y y ), if y (y, 1), α > ρ + δ + β : η =, if y = y, (35) β η = 1 2α(y ŷ), if y (y, ŷ). Secondly, we define he signs of derivaives ẏ and η in he plane (y, η). Le us consider he derivaive ẏ. For his, we ransform he firs equaion of he sysem (21) as follows ẏ = αu(η)y 2 + (αu(η) δ)y + v(η). (36) From he analysis of he expression (36), we see ha a η < he equaliies u(η) =, v(η) = and he relaionship ẏ = δy < hold. he line y = iself is a rajecory of he Hamilonian sysem (21), (23), (24); a η > λ( he equaliies )( u(η) ) = 1, v(η) = v max and he relaionship ẏ = α y ỹ y ỹ hold. From his expression we conclude ha o he righ of he line y = ỹ he inequaliy ẏ < holds, and o he lef of i he inequaliy ẏ > is valid. he line y = ỹ iself is a rajecory of he Hamilonian sysem (21), (23), (24); a < η < λ he equaliies u(η) = 1, v(η) = and he relaionship ẏ = αy(y ȳ) hold. From his expression we conclude ha for α δ he inequaliy ẏ < is valid. For α > δ we obain ha o he righ of he line y = ȳ he inequaliy ẏ < holds, and o he lef of i he inequaliy ẏ > is valid. he lines y =, y = ȳ hemselves are he rajecories of he Hamilonian sysem (21), (23), (24).

14 12 E.V. GRIGORIEVA E.N. KHAILOV Le us consider he derivaive η. For his, we will ransform he second equaion of he sysem (21) for η < o he ype for η > o he ype ( η = (δ + ρ) η + 2α ) δ + ρ (y y ), (37) ( ) β η = 2α(y ŷ) η (38) 2α(y ŷ) Analyzing he signs of he erms in he righ hand sides of he equaions (37), (38) in he regions η <, η >, we have he following conclusions: a α δ + ρ above he line (33) we see ha η >, and below his line we have η < ; a ρ + δ < α ρ + δ + β o he lef of he curves (34) we see ha η <, and o he righ of hese curves we have η > ; a α > ρ + δ + β o he lef of he curve (35) we see ha η <, and o he righ of his curve we have η >. 6 Hamilonian sysem Now, we will analyze he Hamilonian sysem (21), (23), (24) depending on he iniial condiion y and he parameers α, δ, ρ, β. he following hree cases will be considered. Case 1. Le α < δ + ρ + β. I follows from he argumens of Secion 5 ha he Hamilonian sysem (21), (23), (24) has no res poins in he region η >. A η < he sysem (21) akes he form of he linear sysem { ẏ = δy, η = 2αy + (δ + ρ) ( η 2αy ). δ + ρ (39) Analysis of his sysem shows ha here exiss a res poin (, 2αy ), y <. (4) δ + ρ

15 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 13 he eigenvalues ( δ), (δ + ρ) of he sysem (39) are real and have opposie signs. herefore, he res poin (4) is a saddle poin. he line y = from he relaionships (29) corresponds o he posiive eigenvalue, and he line η = 2α ( y 2δ + ρ ) 2δ + ρ δ + ρ y o he negaive eigenvalue. Moreover, boh of hese lines are he rajecories of he sysem (39). he behavior of he rajecories similar o his sysem is presened, for example, in [14]. Considering he signs of he derivaives ẏ and η i is easy o see ha he rajecories of he Hamilonian sysem (21), (23), (24) wih growh of will approach he line η = only from he region η <, inersec i, and hen go away from i in he region η >. By he second iniial condiion (22), he soluion of he PBVP (21) (24) mus end on he line η =, and hence, by he relaionships (23), (24), he corresponding conrols (u(η), v(η)) have he ype u(η) =, v(η) =. (41) In he considered case a α > δ + ρ + β(1 + λ) 1, in he region η > here is one more res poin (, y ), y <. (42) ŷ Simple calculaions show ha he marix of he linear sysem corresponding o he sysem (21) has he eigenvalues (α δ), ( 2αŷ) ha are real and have opposie signs. herefore, he res poin (42) is unsable. he behavior of rajecories of he Hamilonian sysem (21) in he neighborhood of he poin (42) does no affec heir behavior in he neighborhood of he line η =. In his siuaion he phase porrai of he Hamilonian sysem (21), (23), (24) is presened in Figure 1. Case 2. Le α = δ + ρ + β. he difference from he previous case is only ha in his case, he res poins (4), (42) are convered ino one poin which is locaed a he origin. he behavior of he rajecories of he Hamilonian sysem (21), (23), (24) in he neighborhood of he line η = is similar o Case 1. herefore, he conrols (u(η), v(η)) have he ypes (41). Case 3. Le α > δ+ρ+β. We see ha y >. I follows from he argumens of Secion 5 ha o he righ of he curve (35), he behavior of he rajecories of he Hamilonian sysem (21), (23), (24) is he same as in he previous wo cases. herefore, a y > y wih growh of, he rajecories approach he line η = from he region η <, inersec i, and hen go away from i in he region η >. So, he conrols (u(η), v(η)) have he ypes (41). Moreover, he singular regime (27) is possible if and only if he moion occurs along he line (32) and ends a he poin (y, ). (In Figure 2 such a rajecory is shown as a hick curve ha

16 14 E.V. GRIGORIEVA E.N. KHAILOV 4. η η= η= y Figure 1: Phase porrai of he Hamilonian sysem a α < δ + ρ + β. sars on he righ of y and a negaive value of η). he corresponding conrol u(η) swiches from he value o he value u sing, and he conrol v(η) remains equal o he value. A y = y he conrols (u(η), v(η)) have he singular regime (27), i. e. he conrol (u(η), v(η)) have he ypes u(η) = u sing, v(η) =. A las, we can show ha a he region η > here exiss he rajecory of he Hamilonian sysem (21), (23), (24) saring from he poin (y, ) and moving o he axis η as is decreasing. herefore, o he lef of he curve (35), he behavior of he rajecories of he sysem (21) is opposie. Namely, a y < y wih he growh of, he rajecories approach he line η = from he region η >, inersec i, and hen go away from i in he region η <. herefore, by he relaionship (23), we obain u(η) = 1, and he conrol v(η) will be eiher or i will swich from he value v max o. Also, when he moion occurs alone he menioned rajecory and ends a he poin (y, ), hen he singular regime (27) may occur. he corresponding conrol v(η) remains a he value, and he conrol u(η) swiches from he value 1 o he value u sing. In his case he phase porrai of he Hamilonian sysem (21), (23), (24) is presened in Figure 2.

17 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY η η= 1 y * y Figure 2: Phase porrai of he Hamilonian sysem a α > δ + ρ + β. Remark 1 Similar analysis of Hamilonian sysems for economic growh models is presened in [9, 2, 6]. 7 ypes of he opimal conrols From he argumens of Secion 6 we can make he following conclusions abou he ypes of he opimal conrols (u (), v ()), [, ]. Depending on he iniial condiion y and he parameers α, δ, ρ, β he following siuaions are possible. Le α δ + ρ + β. hen, he opimal conrols (u (), v ()) are he funcions of he ypes u () =, v () =, [, ]. (43) Le α > δ + ρ + β. hen, if y > y, he opimal conrols (u (), v ()) can be he funcions

18 16 E.V. GRIGORIEVA E.N. KHAILOV eiher of he ypes u () =, v () =, [, ], (44) or u () = {, if θ, u sing, if θ <, v () =, [, ], (45) where θ (, ) is he momen of swiching; if y = y, he opimal conrols (u (), v ()) are he funcions of he ypes u () = u sing, v () =, [, ]; (46) if y < y, he opimal conrols (u (), v ()) can be he funcions eiher of he ypes u () = 1, v () =, [, ], (47) or u () = 1, [, ], v () = { v max, if τ,, if τ <, (48) or u () = { 1, if θ, u sing, if θ <, v () =, [, ], (49) or u () = { { 1, if θ, u sing, if θ <, v v max, if τ, () =, if τ <, (5) where θ, τ (, ) are he momens of swiching. In Figures 3 9 he graphs of he opimal conrols (u (), v ()) are presened for he siuaion α > δ + ρ + β a differen values of y, respecively.

19 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 17 1 u * 1 u * u sing θ * v * v * v max v max Figure 3: Opimal conrols (u (), v ()) for y > y. Figure 4: Opimal conrols (u (), v ()) for y > y. 1 u sing u * v * v max Figure 5: Opimal conrols (u (), v ()) for y = y.

20 18 E.V. GRIGORIEVA E.N. KHAILOV 1 u * 1 u * v * v * v max v max τ * Figure 6: Opimal conrols (u (), v ()) for y < y. Figure 7: Opimal conrols (u (), v ()) for y < y. 1 u sing u * 1 u sing u * θ * θ * v * v * v max v max τ * Figure 8: Opimal conrols (u (), v ()) for y < y. Figure 9: Opimal conrols (u (), v ()) for y < y.

21 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 19 8 Numerical soluion of he opimal conrol problem Now, we describe he mehod of solving he opimal conrol problem (1), (12) (14) for differen cases. Le α > δ + ρ + β and y > y. For an arbirary value θ [, ] we define he conrol u θ () as {, if θ, u θ () = u sing, if θ <. he conrols (u θ (), v ()), where v () =, include all possible ypes (44), (45) of he opimal conrols (u (), v ()). hen, we subsiue he conrols (u θ (), v ()) ino he sysem (1) and inegrae his sysem over he inerval [, ]. his yields he funcion y θ (), which corresponds o he conrols (u θ (), v ()), and which we hen subsiue ino he funcional (13). he resul is he funcion of one variable F (θ) = J(u θ, v ), θ [, ]. Le α > δ + ρ + β and y < y. For arbirary values θ, τ [, ] we define he conrols u θ (), v τ () as { { 1, if θ, v max, if τ, u θ () = v τ () = u sing, if θ <,, if τ <. he conrols (u θ (), v τ ()) include all possible ypes (47) (5) of he opimal conrols (u (), v ()). hen, we subsiue he conrols (u θ (), v τ ()) ino he sysem (1) and inegrae his sysem over he inerval [, ]. his yields he funcion y θ,τ (), which corresponds o he conrols (u θ (), v τ ()), and which we hen subsiue ino he funcional (13). he resul is he funcion of wo variables G(θ, τ) = J(u θ, v τ ), θ, τ [, ]. hus, he opimal conrol problem (1), (12) (14) is reduced o he problems of consrained minimizaion: for α > δ + ρ + β and y > y ; G(θ, τ) F (θ) min θ [, ] min (θ,τ) [, ] [, ]

22 11 E.V. GRIGORIEVA E.N. KHAILOV for α > δ + ρ + β and y < y. he mehods for numerical soluion o hese problems are well-known [17]. In he siuaions α δ + ρ + β, α > δ + ρ + β and y = y, he opimal conrols (u (), v ()) are he consan funcions of he ypes (43), (46) respecively, and herefore, hey do no require numerical calculaions. 9 Discussion of fuure research Furher invesigaions of he model presened in Secion 2 can be conduced no only by considering he Cobb-Douglas producion funcion of he general form, i.e. σ (, 1), bu also i can proceed in he following wo ways. Firs, we can consider a nonlinear model of he marke, if insead of he demand formula (5) we sudy he dependence of he ype p = p ( 1 ( by p ) 1 γ ), (51) where γ >, γ 1 is a given consan. Obviously, he expression (5) occurs, if γ = 1. he formula (51) reflecs he effecs of price dropping or sharp price increase of he consumer good on he marke. Secondly, we can consider a siuaion, when no all produced consumer good Y is sen o he marke, bu only is par, wy. Here w [, 1]. he remaining par (1 w)y is sen o he company s warehouse for sorage. In his case, i is possible o conrol of hese pars. herefore, here is anoher conrol w(), [, ] saisfying he resricions w() 1. hen he Cauchy problem (6) and he objecive funcion (7) will have he following ypes: { ẋ() = u()w() ( p bw()π(x())) Π(x()) + K() δx(), [, ], x() = x, x >. I(u, w, K)= e ρ{ (1 u())w() ( p bw()π(x())) Π(x()) p Π(x()) } q(1 w())π(x()) (1 + λ)k() d + e ρ x( ). where q is sorage cos of a uni of he consumer good. 1 Conclusion In his paper we obained imporan mahemaical resuls regarding opimal producion and sales aciviy of a monopoly. We found ha he opimal sraegies

23 OPIMAL PRODUCION SALES SRAEGIES FOR A COMPANY 111 depend on he comparison of he values of he model s parameers, and ha depending on wheher α is greaer or less han or equal o he value of (δ + ρ + β), differen opimal conrols can be used in order o maximize profi. I is ineresing ha for some iniial condiions and model s parameer, singular conrol can be also opimal. Le us rewrie he key comparison α vs (δ + ρ + β) in he following equivalen form ( p p )r vs (δ + ρ). On is lef, here is he profi of he monopoly obained from he sales of he consumer good on he marke a he maximum possible marke price p produced by he uni of he producion funds. On is righ, here is he oal loss of he company caused by he amorizaion of he producion funds (δ) and by rae of inflaion (ρ). hus, if he profi ( p p )r is greaer han he oal loss (δ + ρ), hen depending on he iniial condiion y, he company can do boh, reinves par of he obained profi ino expansion of he producion, and even ake loans. Opposie, if ( p p )r δ +ρ, hen he monopoly mus be advised o produce and sale wihou aking any loans or planning expansion of he producion. Finally, invesigaion of he modified model wih nonlinear producion funcion and wih addiional invenory conrol will be conduced in our fuure research. Acknowledgemens he auhors would like o hank Prof. S.M. Aseev for his suggesions regarding he saemen of he problem and Dr. S.N. Avvakumov for his help in creaing illusraions for his paper. References [1] Arrow, K.J. (1968) Applicaion of conrol heory o economic growh", in: Lecures in Applied Mahemaics, Mahemaics of he Decision Sciences, par 2, 12, AMS, Providence, RI: [2] Aseev, S.M.; Kryazhimskii, A.V. (27) he Ponryagin maximum principle and opimal economic growh problems", Proceedings of he Seklov Insiue of Mahemaics 257(1): [3] Capuo, M.R. (25) Foundaions of Dynamic Economic Analysis: Opimal Conrol heory and Applicaions. Cambridge Universiy Press, Cambridge, UK. [4] Carlson, D.A.; Haurie, A.B.; Leizarowiz, A. (1991) Infinie Horizon Opimal Conrol: Deerminisic and Sochasic Sysems. Springer Verlag, Berlin Heidelberg New York.

24 112 E.V. GRIGORIEVA E.N. KHAILOV [5] Demin, N.S.; Kuleshova, E.V. (28) Conrol of a one secor economy on finie ime inerval wih accoun of ax deducions", Journal of Compuer and Sysems Sciences Inernaional 47(6): [6] Grigorieva, E.V.; Khailov, E.N. (27) Opimal conrol of a nonlinear model of economic growh", Discree and Coninuous Dynamical Sysems, supplemen volume: [7] Inriligaor, M.D. (22) Mahemaical Opimizaion and Economic heory. SIAM, Philadelphia. [8] Khelifi, A. (21) Explici soluion o opimal growh models", Inernaional Journal of Economics and Finance 2(5): [9] Kryazhimskii, A.; Waanabe, C. (24) Opimizaion of echnological Growh. Gendaiosho, Japan. [1] Lee, E.B.; Marcus, L. (1967) Foundaions of Opimal Conrol heory. John Wiley & Sons, New York. [11] Leonief, W.W. (1966) Inpu-Oupu Economics. Oxford Universiy Press, New York. [12] Loon, P.A. (1983) Dynamic heory of he Firm: Producion, Finance, and Invesmen. Lecure Noes in Economics and Mahemaical Sysems, vol Springer Verlag, Berlin Heidelberg New York. [13] Ponryagin, L.S.; Bolyanskii, V.G.; Gamkrelidze, R.V.; Mishchenko, E.F. (1962) Mahemaical heory of Opimal Processes. John Wiley & Sons, New York. [14] Robinson, R.C. (24) An Inroducion o Dynamical Sysems: Coninuous and Discree. Pearson Prenice Hall, New Jersey. [15] Solow, R.M. (1956) Conribuion o he heory of economic growh", Quarerly Journal of Economics 7(1): [16] Swan,.W. (1956) Economic growh and capial accumulaion", Economic Record 32(2): [17] Vasil ev, F.P. (22) Opimizaion Mehods. Facorial Press, Moscow. [18] Weizman, M.L. (23) Income, Wealh, and Maximum Principle. Harvard Universiy Press, Cambridge, UK.