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Revista de Matemática Teoría y Aplicaciones
Print version ISSN 1409-2433
Rev. Mat vol.22 n.1 San José Jan./Jun. 2015
Optimal production–sales strategies for a company at changing market price
Estrategias óptimas de ventas y producción para una compañía en un mercado con precios cambiantes
Estrategias óptimas de ventas y producción para una compañía en un mercado con precios cambiantes
Abstract
In this paper we consider a monopoly producing a consumer good of high demand. Its market price depends on the volume of the produced goods described by the Cobb-Douglas production function. A productionsales activity of the firm is modeled by a nonlinear differential equation with two bounded controls: the share of the profit obtained from sales that the company reinvests into expanding own production, and the amount of short-term loans taken from a bank for the same purpose. The problema of maximizing discounted total profit on a given time interval is stated and solved. In order to find the optimal production and sales strategies for the company, the Pontryagin maximum principle is used. In order to investigate the arising two-point boundary value problem for the máximum principle, an analysis of the corresponding Hamiltonian system is applied. Based on a qualitative analysis of this system, we found that depending on the initial conditions and parameters of the model, both, singular and bangbang controls can be optimal. Economic analysis of the optimal solutions is discussed.
Keywords: nonlinear microeconomic control model; production-sales strategy; Pontryagin maximum principle; Hamiltonian system.
Resumen
En este artículo consideramos un monopolio produciendo un producto de consumo de gran demanda. Su precio de mercado depende del volumen de producción descrito por la función de producción de Cobb-Douglas. Una actividad de producción y ventas de la firma es modelada por una ecuación diferencial no lineal con dos controles de frontera: la participación en el resultado de las ventas que la compañía reinvierte para expandir su propia producción, y el monto de los préstamos a corto plazo adquiridos del sistema bancario con el mismo propósito. Se plantea y resuelve el problema de maximizar la ganancia total descontada en un intervalo de tiempo dado. Para encontrar las estrategias óptimas de producción y ventas para la compañía, se usa el principio del máximo de Pontryagin. Para investigar el problema de valores de dos puntos de frontera que aparece para el principio del máximo, se aplica un análisis del sistema hamiltoniano correspondiente. Basado en un análisis cualitativo del sistema, encontramos que dependiendo de las condiciones iniciales y los parámetros del modelo, tanto el control singular como el bang-bang pueden ser óptimos. Se discute un análisis económico de las soluciones óptimas.
Palabras clave: modelo de control microeconómico no lineal; estrategia de producción y ventas; principio del máximo de Pontryagin; sistema hamiltoniano.
Mathematics Subject Classification: 49J15, 58E25, 90A16, 93B03.
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References
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[18] Weitzman, M.L. (2003) Income, Wealth, and Maximum Principle. Harvard University Press, Cambridge, UK. [ Links ]
[2] Aseev, S.M.; Kryazhimskii, A.V. (2007) “The Pontryagin maximum principle and optimal economic growth problems", Proceedings of the Steklov Institute of Mathematics 257(1): 1–255. [ Links ]
[3] Caputo, M.R. (2005) Foundations of Dynamic Economic Analysis: Optimal Control Theory and Applications. Cambridge University Press, Cambridge, UK. [ Links ]
[4] Carlson, D.A.; Haurie, A.B.; Leizarowitz, A. (1991) Infinite Horizon Optimal Control: Deterministic and Stochastic Systems. Springer–Verlag, Berlin–Heidelberg–New York. [ Links ]
[5] Demin, N.S.; Kuleshova, E.V. (2008) “Control of a one–sector economy on finite time interval with account of tax deductions", Journal of Computer and Systems Sciences International 47(6): 918–929. [ Links ]
[6] Grigorieva, E.V.; Khailov, E.N. (2007) “Optimal control of a nonlinear model of economic growth", Discrete and Continuous Dynamical Systems, supplement volume: 456–466. [ Links ]
[7] Intriligator, M.D. (2002) Mathematical Optimization and Economic Theory. SIAM, Philadelphia. [ Links ]
[8] Khelifi, A. (2010) “Explicit solution to optimal growth models", International Journal of Economics and Finance 2(5): 116–121. [ Links ]
[9] Kryazhimskii, A.; Watanabe, C. (2004) Optimization of Technological Growth. Gendaitosho, Japan. [ Links ]
[10] Lee, E.B.; Marcus, L. (1967) Foundations of Optimal Control Theory. John Wiley & Sons, New York. [ Links ]
[11] Leontief, W.W. (1966) Input-Output Economics. Oxford University Press, New York. [ Links ]
[12] Loon, P.A. (1983) Dynamic Theory of the Firm: Production, Finance, and Investment. Lecture Notes in Economics and Mathematical Systems, vol. 218. Springer–Verlag, Berlin–Heidelberg–New York. [ Links ]
[13] Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mishchenko, E.F. (1962) Mathematical Theory of Optimal Processes. John Wiley & Sons, New York. [ Links ]
[14] Robinson, R.C. (2004) An Introduction to Dynamical Systems: Continuous and Discrete. Pearson Prentice Hall, New Jersey. [ Links ]
[15] Solow, R.M. (1956) “Contribution to the theory of economic growth", Quarterly Journal of Economics 70(1): 65–94. [ Links ]
[16] Swan, T.W. (1956) “Economic growth and capital accumulation", Economic Record 32(2): 334–361. [ Links ]
[17] Vasil’ev, F.P. (2002) Optimization Methods. Factorial Press, Moscow. [ Links ]
[18] Weitzman, M.L. (2003) Income, Wealth, and Maximum Principle. Harvard University Press, Cambridge, UK. [ Links ]
*Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX 76204, USA. E-Mail: egrigorieva@mail.twu.edu
†Department of Computational Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992, Russia. E-Mail: khailov@cs.msu.su Received: 23/Feb/2014; Revised: 28/Aug/2014; Accepted: 3/Oct/2014
