Some Quantitative Methods and Models in Economic Theory


Alexander V. Prasolov, PhD
Saint Petersburg State University, St. Petersburg, Russia

Series: Economic Issues, Problems and Perspectives
BISAC: BUS069030

This book takes an intermediate place between monographs and textbooks: on the one hand, it contains known, yet unusually portrayed facts, and on the other hand, the author brings his own results corresponding to the field of research. It is already obvious from the title that while reading the book, attention and concentration are required, as it is always necessary when studying books with mathematical content.

Mathematical models and methods in the economic theory are very various. They are as follows: econometrics, the game theory, operation research, nonlinear and chaotic dynamics and many other aspects as well. The book will be interesting only to those who are already familiar with corresponding tasks as well as to students at all levels specializing in economic dynamics, in decision-making methods, in forecasting effects of management and in the analysis of interaction of economic agents.

In terms of the most interesting and new models of economic dynamics, the authors emphasize multidimensional nonlinear systems of the differential equations of Lotka-Volterra type. These models have been constructed and analyzed, and scopes of their application and various methods of coefficients identification have been offered for them. The analysis of the competition between various economic agents (i.e. branches of economy, rival companies and sellers in the market) has been made. Another fact unusual to similar monographs is the inclusion of the theory of differential equations with the retarded argument. In economic theory, there are numerous examples of models being used with discrete time (they also have been given attention here) and with time lags (concentrated or distributed). Such an approach gives more adequate models without lags, but in the differential equations with continuous time, the introduction of delay complicates systems while the growth of delay the qualitative behavior of trajectories is changed. Additionally, there appear fluctuations such as stability being changed by instability, etc.
As the author has belonged to the St. Petersburg Mathematical School for more than thirty-five years, the list of references contains many Russian names which may be unknown to Western readers. However, the list also includes world classical scientists who devoted their works to mathematical methods in economics.

In this monograph, an attentive reader will find numerous points for further analysis which can become a subject of publications or theses. In some cases, the text is conducted in a polemic manner – that is, the author is always open for discussions and does not consider his work to be “the ultimate truth”. (Imprint: Nova)



Table of Contents


Chapter 1. Introduction

Chapter 2. Linear Dynamic Models

Chapter 3. Lotka-Volterra’s Models in Economics

Chapter 4. Various Dynamic Models

Appendix 1. Delay Differential Equations Theory (Brief Course)

Appendix 2. Optimal Size of Production

Appendix 3. Production Function




“In this important and timely book, Alexander V. Prasolov illustrates how dynamically rich Lotka-Volterra equations offer diverse and economically useful dynamic behavior for a wide range of economic problems. Splitting the theory into linear and non-linear parts, the importance and viability of nonlinear dynamics is illustrated convincingly. Using time-delays with Lotka-Volterra equations, dynamic macroeconomic models, sources of fluctuations in economic activity, capital and manufacturing decision models, models of international trade, models of management and advertising problems as well as the forecasting models are examined with exceptionally outstanding detail and insight. Additionally, the author brings forward unique list of references from Russian academic literature not commonly known in the western academic circles. I highly recommend this book to the scholars and students of economic dynamics and finance to delve into the rich world of the nonlinear apparatus being put forward in this exquisitely crafted scientific jewel.” – Ramo Gençay, Department of Economics, Simon Fraser University, Vancouver, Canada


[1] Abraghimov A.P. etc. (1988) Research of dynamics of factor elasticity of production functions. Moscow.[2] Allen R. G. D. (1960) Mathematical Economics. Second edition. London. McMillan and Co LTD. New York. St. Martin’s Press.[3] Anderson T.W. (1971) The statistical analysis of time series. – John Wiley & Sons, Inc. NY. 1971.[4] Arrowsmith D.K., Place C.M.(1982) Ordinary differential equations. Qualitative approach with applications. N.Y. Chapman and Hall. 1982.[5] Ashmanov S.A. (1980) Mathematical models and methods in economics. – Moscow. Moscow university press. 1980.[6] Ashmanov S.A. (1987) Mathematical models in economics. – Moscow. Moscow university press. Part 2. 1987.[7] Bagwell K, Staiger RW (1990) A Theory of Managed Trade. The American Economic Review, Vol. 80, No. 4, pp. 779-795.[8] Bagwell K, Staiger RW (1999) An Economic Theory of GATT. The American Economic Review, Vol. 89, No. 1, pp. 215-248.[9] Baldwin R (1987) Politically Realistic Objective Functions and Trade Policy PROFs and Tariffs. Economics Letters,24: 287-290.[10] Barbashin Ev.A.(1967)Introduction to stability theory. — Moscow. ‘Nauka.’ 1967.[11] Barkalov N.B. (1981) Production functions in models of economic growth. – Moscow. Moscow university press. 1981.[12] Baxter, M.W., Rennie, A.J.O., (1997) Financial Calculus. Introduction to Derivative Pricing. Cambridge University Press. 1997.[13] Bellman, R., Cooke K. (1963) Differential-Difference Equations. – Academic Press, NY. 1963.[14] Beltrami E. (2002) Mathematical models for Society and Biology. – Academic Press, NY. 2002.[15] Berezin I. S., Zhidkov N.P. (1959) Calculation methods: In 2 vols. Moscow. Fizmatgiz, 1959.[16] Bergstrom, A.R. (1967). The Construction and Use of Economic Models. London: English University Press.[17] Bickerdike CF (1906) The Theory of Incipient Taxes. Economic Journal, 16(64): 529-535.[18] Bradford S (2006) Protection and unemployment. Journal of International Economics 69: 257–271.[19] Brander JA, Spencer BJ (1981) Tariffs and the Extraction of Foreign Monopoly Rents under Potential Entry. The Canadian Journal of Economics /Revue Canadienned’ Economique Vol. 14, No. 3, pp. 371-389.[20] Bulavski V.A. (1995) Management of dynamics of economic system through external consumption/Economics and mathematical methods. 1995, №2.[21] Carter, R.A.L. and Zellner, A. (2004) The ARAR Error Model for Univariate Time Series and Distributed Lag. Studies in Nonlinear Dynamics & Econometrics, 8(1), article 2. 2004.[22] Chizhova O. N. (2000) On non-local extendability of system with non-bounded delay. “Problems of mechanics and control processes.” SPbSU. Iss. 18. 2000.[23] Dean J. (1969). Pricing Pioneering Products. The Journal of Industrial Economics 17 (July).[24] Dean J. (1976). Pricing Policies for New Products. Harvard Business Review.[25] Deng S., Yano C. A. (2006). Joint Production and Pricing Decisions with Setup Costs and Capacity Constraints. Management Science 52 (5)[26] Dixit A. Norman V. (1980) International Trade Theory.— Cambridge University Press, 1980. 339 pp.[27] Dolan R. J., Jeuland A. P. (1981). Experience Curve and Dynamic Demand Models: Implication for Optimal Pricing Strategies. Journal of Marketing, 45.[28] Domar E.D. (1957) Essays in the theory of economic growth. New York, 1957.[29] D’Otum A., Sharaev Yu. (1998) Education and endogenous economic growth: Lucas model. Moscow: 1998.[30] Drunchenko V. P. (1996) The theorem on existence and uniqueness of Couchy solution for a differential equation with delay and a piecewise continues initial//Vestnik SPbU.-1996.-Iss.1.- . 25-30. [31] Dubovskii S. V. (1977) Mathematical models of economic processes: survey. Moscow, 1977.[32] Dubovskii S. V., Osipov S. N. (1994) Problems of structural stability in a model of economic growth and cycles. Avtomat. i Telemekh., 1994, no. 9, 134–140.[33] Economic process modeling. (1973) Under edition of Dadayan V.S. Moscow. Economics. 1973.[34] Edgeworth FY (1908) Appreciations of Mathematical Theories. Economic Journal 18(72): 541-556.[35] Efimov M. N., Movshovich S. M. (1973) The analysis of the balanced growth in dynamic model of a national economy/Economics and mathematical methods. 1973, №1.[36] El’sgol’ts L.E. (1954) Stability of solutions of difference-differential equations. Moscow. Uspekhi matematicheskih nauk. 1954. Vol. 9. Iss. 4. P. 95—112.[37] Ermakov S. M., Zhiglyavskii A. A. (1987) Mathematical theory of the optimal experiment. – Moscow. Nauka, 1987.[38] Fermi E. (1956) Thermodynamics. Dover books on physics. 1956.[39] Ferrer G., Swaminathan J. M., (2006) Managing New and Remanufactured Products, Management Science 52 (1).[40] Foundations of optimal control. (1990)Under edition of Krotov V.F. Moscow. Vysshay shkola, 1990.[41] Gale D. (1956). A closed linear model of production. In: H. W. Kuhn et al. (eds.), Linear Inequalities and Related Systems, Princeton University Press, pp. 285–303.[42] Galperin VM, Ignatiev SM, Morgunov VI (2000) Microeconomics: 2 volumes/General edition by Galperin VM. SPb, Economic School.[43] Gantmacher, F.R. (1959) The Theory of Matrices. Vols. 1-2, Chelsea, 1959.[44] Gershenson M.A. (1975) The analysis of the simplified dynamic models of interindustry balance. Novosibirsk. Nauka, 1975.[45] Goodwin R.M. (1951) The nonlinear accelerator and the persistence of business cycles//Econometrica 19, 1951.[46] Granberg A.G. (1985) Dynamical models of national economy. Moscow: Economics, 1985.[47] Huang, J.Z. and Yang, L. (2004) Identification of Non-Linear Additive Autoregressive Models. J. Royal Statistical Society, (Series B), 66, Part 2, 463-477. 2004.[48] Hull, John C. (1997) Options, Futures, and Other Derivatives. Prentice-Hall, Inc. 3 ed. 1997.[49] Intriligator M. (1971) Mathematical optimization and economic theory. Prentice-Hall, N-Y. 1971.[50] Ivanilov Yu.P. etc. (1983) Public production function. Moscow, 1983.[51] Ivanilov Yu.P., Lanets S.A. (1984) Analysis and creation of production functions with variable elasticity of substitution on resources. Moscow, 1984.[52] Ivanilov Yu.P., Lotov A.V. (1979) Mathematical models in economics. -Moscow: Nauka, 1979.[53] Ivanov Yu.N., Tokarev V.V., Uzdemir A.P. (1994) Mathematical description of economics elements. Moscow: Fizmathlit, 1994.[54] Ivanov, V. and Kilian, L. (2005) A Practitioner’s Guide to Lag Order Selection for VAR Impulse Response Analysis. Studies in Nonlinear Dynamics & Econometrics, 9(1), article 2. 2005.[55] Joldybaeva S.M., Lotov A.V. (1989) Aggregated production functions of linear models, 1989.[56] Junius T. Remarks on models and methods in growth theory//V symposium of operations research. Part II: sections 3–7. Königstein. Ts 1981.[57] Kalman R.E., Falb P. Lp, Arbib M.A. (1969) Topics in mathematical system theory. MC Graw-Hill Book c. N.Y. 1969.[58] Kalish S. (1985). A New Product Adoption Model with Price, Advertising, and Uncertainty. Management Science 31 (12) [59] Kantorovich L.V. (1939) Mathematical methods of organizing and planning of production. Leningrad., 1939.[60] Kantorovich L.V. etc. (1974) Modern mathematical tools of economic control. Sverdlovsk, 1974.[61] Karlin S. (1992) Mathematical Methods and Theory in Games, Programming, and Economics. — Dover Publications, 1992. [62] Kato J. (1971) On Liapunov-Razumikhin type theorems. — Lect. Notes Math. 1971. 243. 54-65.[63] Keen M, Wildasin D (2004) Pareto-Efficient International Taxation. The American Economic Review, Vol. 94, No. 1, 259-275.[64] Kharitonov V.L. (1982) On determination of the most delay in stability tasks. Differential equations. 1982. Vol. 18. Iss 4.[65] Kim J.G., Menzefricke U., Feinberg F.M., (2007) Capturing Flexible Heterogeneous Utility Curves: A Bayesian Spline Approach. Management Science, vol. 53, no 2, February.[66] Kim K.H, Roush F.W (1988) Strategic Tariff Equilibrium and Optimal Tariffs. Mathematical Social Sciences 15: 105-131.[67] Kleiner G.B. (1986) Production functions: theory, methods, application. Moscow, 1986.[68] Kleiner G.B. (1980) Methods of production function analysis. Moscow, 1980.[69] Kolmogorov A.N. (1972) Quality study of mathematical models of population dynamics. Problems of cybernetics, iss. 25. Moscow. Nauka. 1972.[70] Kondratiev N.D. (1993) Special opinion. Selected works in 2 volumes. Moscow. Nauka. 1993.[71] Kossov V.V. (1973) Interbranch models. Moscow: Economics, 1973.[72] Kowalczyk C., Skeath S.E. (1994)Pareto ranking optimal tariffs under foreign monopoly. Economics Letters 45: 355-359.[73] Krasovskii N.N. (1963) Stability of Motion. Stanford University Press. 1963. [74] Krichevskii I.R. (1970) Concepts and fundamentals of thermodynamics. Moscow. Chemistry, 1970. [75] Kuzutin V., Zenkevich N., Eremeev V. (2003) Geometry. SPb, Lan, 2003.[76] Lahiri S, et al. (2002) Optimal foreign aid and tariffs. Journal of Development Economics 67: 79–99.[77] Lancaster K. (1968) Mathematical Economics, Macmillan, 1968.[78] Leontief V. etc. (1953) Studies in the structure of the American economy. N.Y., 1953.[79] Leontief V. (1966) Essays in economics. Theories and theorizing. N.Y., 1966.[80] Lindert P.H. (1986) International economics. Eighth Edition. IRWIN 1986.[81] Malinvaud E. (1972) Lectures on Microeconomic Theory. North-Holland, 1972.[82] Matrosov V.M. (1973) The method of vector Lyapunov functions in analysis of interconnected distributed systems. Automation and Remote Control, 1973, 34:1, 1–15.[83] Mayer W (1984) Endogenous Tariff Formation. The American Economic Review Vol. 74, No. 5, pp. 970-985.[84] McConnell, C. R., Brue, S. L. (1993) Economics: principles, problems and policies. McGraw-Hill. N.Y. 17 edit. 2008.[85] McDonalds N. (1978) Lecture Notes in Biomathematics. Time lags in Biological Models. – Springer-Verlag, Berlin. 1978.[86] McDonalds N. (1988) Biological Delay Systems: Linear Stability Theory. – Cambridge University Press, Cambridge. 1988.[87] Mednitskii V.G. etc. (1998) Forms of dynamical equilibrium of closed economy. Moscow. Economics and mathematical methods. 1998, №2.[88] Mendeleev D.I. (1950) Sensible tariff or research about development of the Russian industry in connection with its common customs tariff. Moscow. Complete works of Mendeleev, vol. 19, 1950, pp. 230-937.[89] Mendeleev D.I. (1952) Methods of natural sciences in studying of the prices. Moscow. Complete works of Mendeleev, vol. 21, 1952, pp 33—42.[90] Methods of construction and use of macroeconomic and branch production functions. Moscow. CEMI, 1979. [91] Microeconomics. (1998) Moscow – SPb. Poisk. Under edition of Yakovleva E.B. 1998.[92] Mikolajska Z. (1967) Remaque sur la stabilite d’une solution du systeme d’eqs.diff.a parametre. Collog. math.,18(1967) 59-66.[93] Milovanov V.P. (1994) On one approach to modeling of a pricing mechanism. Economics and mathematical methods. 1994. Vol. 30. No.1. Pp. 137-147.[94] Mityagin B.S. (1972) Noyes on mathematical economics. Moscow. Uspekhi matematicheskikh nauk. 1972, №3.[95] Modern economic ideas. (1981) Under edition of Vaintraub S. Moscow. Progress. 1981.[96] Modern bourgeois theories of economic growth and cycle. (1979) Under edition of Entov R.M. etc. Moscow. Nauka. 1979.[97] Morishima M. (1964) Equilibrium, stability and growth. A multi-sectorial analysis. Oxford. 1964.[98] Moskalenko A.I. (1999) Optimal control by models of economic dynamic. Novosibirsk. Nauka. 1999.[99] Nascimento F., Vanhonacker W.R. (1988). Optimal Strategic Pricing of Reproducible Consumer Products Management Science, vol. 34, no. 8, August.[100] Nemchinov V.S. (1965) Economics and mathematics. Moscow. 1965.[101] Ng, S. and Perron, P. (2001) Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power,” Econometrica, 69, no 6, p. 1519. 2001.[102] Nikaido H. (1968) Convex structures and economic theory. Academic press, N.Y. 1968.[103] Noble P. M., Gruca T. S. (1999). Industrial Pricing: Theory and Managerial Practice. Marketing Science, vol. 18, no. 3.[104] Noghin VD (2005) Decision making in multicriteria environment: a numerical approach (2d edition). Moscow, FIZMATLIT.[105] Noghin VD (2008) Pareto set reducing problem: approaches to solution,Artificial intelligence and decision making 1: 98-112.[106] Noghin V.D., Prasolov A.V. (2011) The quantitative analysis of trade policy: a strategy in global competitive conflict. Int. J. Business Continuity and Risk Management, 2011. Vol. 2. № 2. P. 167-182.[107] Ortega J. M., Poole W. G. (1981) An introduction to numerical methods for differential. Pitman Publ. Inc. 1981.[108] Petrosyan L.A., Zakharov V.V. (1997) Mathematical models in ecology. SPb. SPbSU Press, 1997.[109] Pomanskii A.B., Trofimov G. Yu. (1989) Mathematical models in economic cycle theory. Moscow. Economics and mathematical methods. 1989.- Vol. 25. Iss. 5. Pp. 825-840.[110] Prasolov A.V. (1991) Mathematical models of control. Leningrad. LGU Press, 1991.[111] Prasolov A.V. (1995) Analytical and numerical methods of dynamic processes investigation. SPb. SPbSU Press. 1995.[112] Prasolov A.V., Stepanov A.V. (1995) Application of the local modeling method to economic problems. SPb. SPbSU Press. 1995.[113] Prasolov A.V., Churkin A.V. (1998) Production functions and their use to firm theory. SPb. SPbSU Press. 1998.[114] Prasolov AV (1999) On one possible approach to the analysis of protectionism. Moscow. Economics and Mathematical Methods 2: 153-156.[115] Prasolov A.V. (2000) Mathematical models of dynamics in economics. SPb. Publ. by SPb University of economics and finance. 2000.[116] Prasolov A. V., Wei K. C. (2000) On Forecast of Exchange Rate of a Foreign Currency. IEEE International Conference on Control Applications & IEEE International Symposium on Computer-Aided Control Systems Design. September 25-27, 2000, Anchorage Hilton, Anchorage, Alaska, USA.[117] Prasolov A.V. (2005) Dynamic Competitive Analysis in Automotive Industry //Proc. International Conference on Stability and Control Processes, St. Petersburg, 2005.[118] Pugel TA, Lindert PH (2000) International Economics. IRWIN, McGraw-Hill. [119] Puu, T. (1989) Nonlinear Economic Dynamics, Springer-Verlag, Berlin, Heidelberg. 1989.[120] Rao, Vithala. (1984). Pricing Research in Marketing: The State of the Art, The Journalof Business. 57.[121] Razumikhin B.S. (1956) On stability of systems with delay. Moscow. Applied math.and mechenics. 1956. Vol. 20. Iss. 4. Pp. 500—512.[122] Reichlin, P. (1996) Endogenous Cycles in Competitive Models: An Overview. Studies in Nonlinear Dynamics & Econometrics, 1(4), article 1. 1996.[123] Ricardo D. (1817) On the principles of political economy and taxation. London.[124] Rogovskii E.A. (1984) On macroeconomic production functions. Moscow. CEMI AS. 1984.[125] Samuelson P.A., Nordhaus W.D. (1995) Economics Fifteenth Edition. McGraw-Hill Com. 1995.[126] Sasieni, M.W. (1989) Optimal advertising strategies.-Marketing Science. Vol. 8, no. 4, Fall 1989, pp. 358-372. [127] Schogalev I.R. (1971) Analysis of some models of economic growth. Moscow. Energy. Cybernetics to communism support. Vol. 6. 1971.[128] Shapiro L.D. etc. (1987)Economical-mathematical modeling. Tomsk. 1987.[129] Shimanov S.N. (1960) On instability of system motion with time lag. Moscow. Applied mathematics and mechanics. 1960. Vol. 24. Iss. 1.[130] Solow R.M. (1994). Perspectives on Growth Theory. Journal of Economic Perspectives, American Economic Association, 1994, vol. 8(1), pages 45-54.[131] Stepanov A.V. (1996) Mathematical methods and algorothms of of numerical modeling of dynamic processes. Theses. SPbSU. 1996.[132] Stoléru L. (1975) Economic equilibrium and growth. Amsterdam: North-Holland Pub. Co. 1975.[133] Sverezhev Yu. M., Logofet D.O. (1978) Stability of biologic societies. Moscow. Nauka. 1978.[134] Sviridenko K.S., Fadeev B.A. (1988)Calculation of differential characteristics of two-factorial production dependences. Moscow. 1988.[135] Swami S, Khairnar P.J. (2006). Optimal Normative Policies for Marketing of Products with Limited Availability. Annual Operation Research, 143.[136] Tarasevich LS, Grebennikov PI, Leusskiy AI (2006) Microeconomics. Moscow, Publ. “Yurite.”[137] Tax Collection of the Russian Federation (2002) Section VIII: Federal Taxes; Chapter 21: Value-added tax.[138] Thompson G.L., Teng J-T. (1984). Optimal Pricing and advertising Policies for New Product Oligopoly Models. Marketing Science, vol. 3, no 2, Spring.[139] Tinbergen J., Bos H. (1962) Mathematical models for economic growth. N.Y. McGraw-Hill publishing Co. 1962.[140] Tkachenko D.I. (2000) On conditions of production growth in one model with delay. SPbSU, Control processes and stability. Proceedings of the XXXI international conference. 2000.[141] То, Кам. (1988) On some methods of production function construction. Moscow. 1988.[142] Tsypkin Ya.Z. (1946) Stability of systems with delayed feedback. Automatica and telemekhanica. 1946. Vol. 7, Iss. 2.[143] Vechkanov G.S., Vechkanova G.R., Pulyaev V.T. (1998) Brief economic encyclopedia. – SPb.’Petropolice.’ 1998.[144] Vinokurov V.R. (1967) Asymptotic behavior of solutions of one class of the integro-differential Volterra’s equations//Differential equations. 1967.- Vol. 3.- №7.- Pp. 1095-1099.[145] Vishnev S.M. (1968) Economic parameters. Introduction to the index theory of economic systems and models. Moscow, 1968.[146] Volterra V (1931) Lec¸ons sur la the´orie mathe´matique de la lutte pour la vie. Gauthier-Villars, Paris.[147] Vorkuev B.L. (1999) Models of macro- and microeconomics. – Moscow. Teis, 1999.[148] Voronovitskii M.M. (1994) Inflation model in an economy in transition//Moscow.Economics and mathematical methods. 1994. Vol. 30. -Iss. 3.- Pp. 117-128.[149] Winker, P. (2000) Optimized Multivariate Lag Structure Selection: Using the Global Optimization Heuristic Threshold Accepting. Computational Economics, 16, 87-103. 2000.[150] Witte S. Yu. (1912) The lectures about a national and public economy, SPb. 1912.[151] Zaslavskii A. Ya. (1989) Theorems of the turnpike in models with variable technology//Economics and mathematical methods. 1989, №1.[152] Zellner, A. (2002) My Experiences with Nonlinear Dynamic Model in Economics. Studies in Nonlinear Dynamics & Econometrics, 6(2), article 1. 2002.[153] Zissimos B (2009) Optimum tariffs and retaliation: How country numbers matter. Journal of International Economics 78: 276–286.[154] Zubov V.I. (1975) Lecture on the control theory.- Moscow: Nauka, 1975.[155] Zubov V.I. Petrosyan L.A. (1982) Mathematical methods in planning.- L-d: Leningrad university press. 1982.[156] Zubov V.I. (1989) Oscillations and waves. –L-d.: Leningrad university press. 1989.

The book is written for students, postgraduates, researchers who apply mathematical methods in economical dynamics.

Additional information