Abstract
We consider a learning mechanism where expected values of an economic variable in discrete time are computed in the form of a weighted average that exponentially discounts older data. Also adaptive expectations can be expressed as weighted sums of infinitely many past states, with exponentially decreasing weights, but these are not averages since the weights do not sum up to one for any given initial time. These two different kinds of learning, which are often considered as equivalent in the literature, are compared in this paper. The statistical learning dynamics with exponentially decreasing weights can be reduced to the study of a two-dimensional autonomous dynamical system, whose limiting sets are the same as those obtained with adaptive expectations. However, starting from a given initial condition, different transient dynamics are obtained, and consequently convergence to different attracting sets may occur. In other words, even if the two different kinds of learning dynamics have the same attracting sets, they may have different basins of attraction. This implies that local stability results are not sufficient to select the kind of long-run dynamics since this may crucially depend on the initial conditions. We show that the two-dimensional discrete dynamical system equivalent to the statistical learning with fading memory is represented by a triangular map with denominator which vanishes along a line, and this gives rise to particular structures of their basins of attraction, whose study requires a global analysis of the map. We discuss some examples motivated by the economic literature.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abraham, R., Gardini, L., & Mira, C. (1997). Chaos in discrete dynamical systems (A visual introduction in two dimensions). Springer.
Billings, L., & Curry, J. H. (1996). On noninvertible maps of the plane: Eruptions. CHAOS, 6, 108–119.
Billings, L., Curry, J. H., & Phipps, E. (1997). Lyapunov exponents, singularities, and a riddling bifurcation. Physical Review Letters, 79(6), 1018–1021.
Bischi, G.I., & Gardini, L. (1996) Mann iterations reducible to plane endomorphisms . In Quaderni di Economia, Matematica e Statistica, Facoltà di Economia (Vol. 36). Università di Urbino.
Bischi, G. I., & Naimzada, A. K. (1997). Global analysis of a nonlinear model with learning. Economic Notes, 26(3), 143–174.
Bischi, G. I., & Gardini, L. (1997). Basin fractalization due to focal points in a class of triangular maps. International Journal of Bifurcations & Chaos, 7(7), 1555–1577.
Bischi, G. I., Gardini, L., & Mira, C. (1999). Maps with denominator. Part 1: some generic properties. International Journal of Bifurcation & Chaos, 9(1), 119–153.
Bischi, G. I., Gardini, L., & Mira, C. (2003). Plane maps with denominator. Part II: Noninvertible maps with simple focal points. International Journal of Bifurcation & Chaos, 13(8), 2253–2277.
Bischi, G. I., Gardini, L., & Mira, C. (2005). Plane maps with denominator. Part III: Non simple focal points and related bifurcations. International Journal of Bifurcation & Chaos, 15(2), 451–496.
Bischi, G. I., Cavalli, F., & Naimzada, A. K. (2015). Mann iteration with power means. Journal of Difference Equations and Applications, 21(12), 1212–1233.
Bray, M. (1983) Convergence to rational expectations equilibrium. In R. Friedman & E. S. Phelps (Eds.), Individual forecasting and aggregate outcomes. Cambridge University Press.
Cavalli, F., & Naimzada, A. K. (2015). A tâtonnement process with fading memory, stabilization and optimal speed of convergence Chaos. Solitons & Fractals, 79, 116–129.
Chiarella, C. (1988). The cobweb model. Its instability and the onset of chaos. Economic Modelling, 5, 377–384.
Chiarella, C. (1991). The birth of limit cycles in Cournot oligopoly models with time delays. Pure Mathematics and Applications, 2, 81–92.
Cushing, J. M. (1978). Integrodifferential equations and delay models in population dynamics (Vol. 20). Lecture notes in biomathematics. Springer.
Deschamps, R. (1975). An algorithm of game theory applied to the duopoly problem. European Economic Review, 6, 187–194.
Dimitri, N. (1988) A short remark on learning of rational expectations. Economic Notes, 3.
Evans, G. W., & Honkapohja, S. (1995). Increasing social returns, learning and bifurcation phenomena. In A. Kirman & P. Salmon (Eds.), Learning and rationality in economics (pp. 216–235). Oxford: Basil Blackwell.
Foroni, I., Gardini, L., & Rosser, B, Jr. (2003). Adaptive and statistical expectations in a renewable resource market. Mathematics and Computers in Simulation, 63, 541–567.
Friedman, B. M. (1979). Optimal expectations and the extreme information assumption of rational expectations macromodels. Journal of Monetary Economics, 5(1), 23–41.
Fudenberg, D., & Levine, D. K. (1998) The theory of learning in games. The MIT Press.
Gardini, L., Bischi, G. I., & Fournier-Prunaret, D. (1999). Basin boundaries and focal points in a map coming from Bairstow’s methods. CHAOS, 9(2), 367–380.
Gardini, L., Bischi, G. I., & Mira, C. (2007) Maps with vanishing denominators, 16970. www.scholarpedia.org, https://doi.org/10.4249/scholarpedia.3277.
Gu, E. G., & Hao, Y.-D. (2007). On the global analysis of dynamics in a delayed regulation model with an external interference. Chaos, Solitons & Fractals, 34(4), 1272–128.
Guesnerie, R., & Woodford, M. (1992) Endogenous fluctuations. In J. J. Laffont (Ed.), Advances in economic theory (Vol. II). Cambridge University Press.
Gumowski, I., & Mira, C. (1980). Dynamique Chaotique. Toulose: Cepadues editions.
Holmes, J. H., & Manning, R. (1988). Memory and market stability: The case of the cobweb. Economic Letters, 28, 1–7.
Hommes, C. (1991). Adaptive learning and roads to chaos. The case of the cobweb. Economic Letters, 36, 127–132.
Hommes, C. (1994). Dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand. Journal of Economic Behavior & Organization, 24, 315–335.
Hommes, C., Kiseleva, T., Kuznetsov, Y., & Verbic, M. (2012). Is more memory in evolutionary selection (de)stabilizing? Macroeconomic Dynamics, 16, 335–357.
Hommes, C. (2013). (2013) behavioral rationality and heterogeneous agents in complex economic systems. Cambridge University Press.
Jensen, R. V., & Urban, R. (1984). Chaotic price dynamics in a non-linear cobweb model. Economic Letters, 15, 235–240.
Lucas, R. E. (1986). Adaptive behavior and economic theory. Journal of Business, 59(4).
MacDonald, N. (1978) Time lags in biological models. Lecture notes in biomatemathics (Vol. 27). Springer.
Marimon, R., Spear, S. E., & Sunder, S. (1993). Expectationally driven market volatility: An experimental study. Journal of Economic Theory, 61, 74–103.
Marimon, R. (1997) Learning from learning in economics. In D. M. Kreps & K. F. Wallis (Eds.), Advances in economics and econometrics: Theory and applications, Vol. I. Cambridge University Press.
Matsumoto, A., & Szidarovszky, F. (2018) Dynamic oligopolies with time delays. Springer.
Matsumoto, A., & Szidarovszky, F. (2015). Dynamic monopoly with multiple continuously distributed time delays. Mathematics and Computers in Simulation, 108, 99–118.
Matsumoto, A. (2017) Love affairs dynamics with one delay in losing memory or gaining affection. In A. Matsumoto (Ed.), Optimization and dynamics with their applications. Springer.
Mira, C., Gardini, L., Barugola, A., & Cathala, J. C. (1996). Chaotic dynamics in two-dimensional noninvertible maps. World Scientific.
Naimzada, A. K., & Tramontana, F. (2009). Global analysis and focal points in a model with boundedly rational consumers. International Journal of Bifurcation & Chaos, 19(6), 2059–2071.
Nerlove, M. (1958). Adaptive expectations and cobweb phenomena. Quarterly Journal of Economics, 73, 227–240.
Pecora, N., & Tramontana, F. (2016) Maps with vanishing denominator and their applications. Frontiers in Applied Mathematics and Statistics. https://doi.org/10.3389/fams.2016.00011.
Radner, R. (1983) Comment to Convergence to rational expectations equilibrium by M. Bray. In R. Friedman & E. S. Phelps (Eds.), Individual forecasting and aggregate outcomes. Cambridge University Press.
Thorlund-Petersen, L. (1990). Iterative computation of Cournot equilibrium. Games and Economic Behavior, 2, 61–95.
Tramontana, F. (2016) Maps with vanishing denominator explained through applications in economics . Journal of Physics: Conference Series, 692, Conference 1.
Yee, H. C., & Sweby, P. K. (1994). Global asymptotic behavior of iterative implicit schemes. International Journal of Bifurcation & Chaos, 4(6), 1579–1611.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Bischi, G.I., Gardini, L., Naimzada, A. (2020). Path Dependence in Models with Fading Memory or Adaptive Learning. In: Szidarovszky, F., Bischi, G. (eds) Games and Dynamics in Economics. Springer, Singapore. https://doi.org/10.1007/978-981-15-3623-6_3
Download citation
DOI: https://doi.org/10.1007/978-981-15-3623-6_3
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-3622-9
Online ISBN: 978-981-15-3623-6
eBook Packages: Economics and FinanceEconomics and Finance (R0)