Polterovich, Victor and Henkin, Gennadi (1998): A Differencedifferential Analogue of the Burgers Equation and Some Models of Economic Development.

PDF
MPRA_paper_21031.pdf Download (5MB)  Preview 
Abstract
The paper is devoted to investigation of a number of differencedifferential equations, among them the following one plays the central role: dFn/dt<=φ(Fn)(Fn1  Fn) (*) where, for every t, {Fn(t), n = 0, 1, 2, ...} is a probability distribution function, and φ is a positive function on [0, 1]. The equation (*) arose as a description of industrial economic development taking into account processes of creation and propagation of new technologies. The paper contains a survey of the earlier received results including a multydimensional generalization and an application to the economic growth theory.
If φ is decreasing then solutions of the Cauchy problem for (*) approach to a family of wavetrains. We show that diffusionwise asymptotic behavior takes place if φ is increasing. For the nonmonotonic case a general hypothesis about asymptotic behavior is formulated and an analogue of a Weinberger's (1990) theorem is proved. It is argued that the equation can be considered as an analogue of Burgers equation.
Item Type:  MPRA Paper 

Original Title:  A Differencedifferential Analogue of the Burgers Equation and Some Models of Economic Development 
Language:  English 
Keywords:  differencedifferential equations; Burgers equations; nonlinear diffusion; longtime asymptotic of Cauchy problem; evolution of industries; economic growth; innovation and imitation processes 
Subjects:  O  Economic Development, Technological Change, and Growth > O4  Economic Growth and Aggregate Productivity > O41  One, Two, and Multisector Growth Models O  Economic Development, Technological Change, and Growth > O3  Technological Change; Research and Development; Intellectual Property Rights > O33  Technological Change: Choices and Consequences; Diffusion Processes 
Item ID:  21031 
Depositing User:  Victor Polterovich 
Date Deposited:  04. Mar 2010 03:22 
Last Modified:  20. Feb 2013 23:13 
References:  Belenky, V., 1990a, A New Version of Evolutionary Model of Technology Diffusion. In: "Veroyatnostnye modeli mathematicheskoi economiki", CEMI Academy of Science of the USSR (in Russian), 1954 Belenky. V., 1990b, Diagram of growth of a monotonic function arid a problem of their reconstruction by the Diagram. Preprint. CEMI, Academy of Science of the USSR. Moscow, 144 (in Russian). Burgers, J.M., 1948, A mathematical model illustrating the theory of turbulence. Advances in Applied Mechanics, ed. R.V.Mises and T.V.Karman, v. 1, 171199. Davies, S., 1979, The diffusion of process innovations. Cambridge Univ. Press, Cambridge. Fisher, R.A., 1937, The wave of advance of advantageous genes. Amm. Eugen. 7. 355369. Gelman L.M., Levin M.I., Polterovich V.M., Spivak V.A., 1993, Modelling of Dynamics of Enterprises Distribution by Efficiency Levels for the Ferrous Metallurgy. Economic and Math. Methods, v. 29, # 3. 10711083 (in Russian). Glazjev, S.Y., Karimov, I.A., 1988, On a nontraditional model of technology substitution. In: Mathematical Modelling of Social, Economic and Demographic Processes. AllUnion Scientific Conference: Institute of Economics and forecasting. Erevan. 8992 (in Russian). Griliches, Z., 1957, Hybrid Corn: An Exploration in the Economics of Technological Change. Econometrica, 25. # 4. Henkin, G.M., Polterovich, V.M, 1991, Schumpeterian dynamics as a nonlinear wave theory. .J. Math. Econ., v. 20, 551590. Henkin, G.M., Polterovich. V.M.. 1994, A DifferenceDifferential Analogue of the Burgers Equation: Stability of the TwoWave Behavior. J. Nonlinear Sci.. v. 4, 497517. Hopf, E., 1950, The partial differential equation ut + uux = µuхх. Comm. on Pure and Appl. Math., v. 3, 201230. Iljin A., Olejnik О.A., I960, Asymptotic longtime behavior of the Cauchy problem for some quasilinear equation, Mat. Sbornic, v. 51, 191216 (in Russian). Iwai K., 1984a, Schumpeterian Dynamics, Part I: An evolutionary model of innovation and imitation, Journal of Economic Behavior and Organization, v. 5, 159190. Iwai K., 1984b, Schumpeterian Dynamics, Part II: Technological Progress. Form growth and "Economic Selection", Journal of Economic Behavior and Organization, v. 5, 287320. Kolmogoroff, A., Petrovsky, I., Piskunoff, N., 1937, Etude le'equation de la diffusion avec croissance de la quantite de matriere et son application a un probleme biologique. Bul. Univ. Moskau. Ser. Internet. Sect. A., v. 1. 125. Lax, P.D., 1954, Weak solutions of nonlinear hyperbolic equation and their numerical computation, Comm. Pure Appl. Math., v. 7, 159193. Levi D., Ragnisco 0., Brushi M., 1983, Continuous and Discrete Matrix Burgers' Hierarchies. II Nuovo Cimento. v. 74, # 1, 3351. Polterovich, V., Henkin, G., 1988a, An Evolutionary Model of the Interaction of the Processes of Creation and Adoption of Technologies. Economics and Mathematical Methods, v. 24, # 6, 10711083 (in Russian). Polterovich, V., Henkin, G., 1988b, Diffusion of Technologies and Economic Growth. Preprint. CEMI Academy of Sciences of the USSR, 144 (in Russian). Polterovich, V., Henkin, G., 1989, An Evolutionary Model of Economic Growth. Economics and Mathematical Methods, v. 25, # 3, 518531 (in Russian). Sato, K., 1975, Production functions and aggregation. NorthHolland, New York. Weinberger, H.F., 1990, Longtime behavior for a regulated scalar conservation law in the absence of genuine nonlinearity. Annale de l'Institute Henri Poincare, Analyse Nonlineaire, 407425. Whitham, G.B., 1974, Linear and nonlinear waves. Wiley, New York. Schumpeter, J.A., 1939, Business Cycles: A Theoretical, Historical and Statistical Analysis of the Capitalist Process, McGrawHill, New York. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/21031 