Papers by Gian-Italo Bischi
Journal of Difference Equations and Applications, 2015
In this paper we analyze a recurrence) (1 n n z f x = + , where n z is a weighted power mean of 0... more In this paper we analyze a recurrence) (1 n n z f x = + , where n z is a weighted power mean of 0 x ,…., n x. Such an iteration scheme has been proposed to model a class of non-linear forward-looking economic models (the state today is affected by tomorrow' s expectation) under bounded rationality; the agents employ a recursive learning rule to update beliefs using weighted power means of the past states. A proposition on the convergence of the dynamical system with memory, proven with a general weighted power mean, generalizes some results given in the literature, where only the arithmetic mean is considered. A power weighted mean with exponentially decreasing weights decreasing is proposed to simulate a fading memory. In this case the iteration scheme with memory is reduced to an equivalent two-dimensional autonomous map whose possible kinds of asymptotic behaviors are the same as those of a one-dimensional map. By this general technique it is proved, for a function f which maps a compact interval into itself, that the presence of a long memory has a stabilizing effect, in the sense that with a sufficiently strong memory convergence to a steady state is obtained even for an otherwise oscillating, or chaotic, dynamical system. In the appendix is considered an economic example from an overlapping generation models which leads to a harmonic mean.
We study a dynamic equilibrium model where agents have adaptive expectations and monetary authori... more We study a dynamic equilibrium model where agents have adaptive expectations and monetary authorities pursue an in‡ation target. We show how alternative monetary stabi- lization polices become more eective when …scal constraints on de…cits are implemented, although they are not binding at the equilibrium target. In particular, we show that the in‡ation target equilibrium can be locally, or even globally, stable for a large class of adaptive learning schemes. We also compare alternative stabilization policies in terms of their stabil- ity properties. Commonly postulated conditional Taylor-type rules tend to be dominated by other rules, such as an unconditional Friedman-type rule.
Physica A: Statistical Mechanics and its Applications, 2010
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit:
Macroeconomic Dynamics, 2006
In this paper we investigate the effects of herding on asset price dynamics during continuous tra... more In this paper we investigate the effects of herding on asset price dynamics during continuous trading. We focus on the role of interaction among traders, and we investigate the dynamics emerging when we allow for a tendency to mimic the actions of other investors, that is, to engage in herd behavior. The model, built as amean fieldin a binary setting (buy/sell decisions of a risky asset), is expressed by a three-dimensional discrete dynamical system describing the evolution of the asset price, its expected price, and its excess demand. We show that such dynamical system can be reduced to a unidirectionally coupled system. In line with therational herd behaviorliterature [Bikhchandani, S., Sharma, S. (2000), Herd Behavior in Financial Markets: A Review. Working paper, IMF, WP/00/48], situations of multistability are observed, characterized by strongpath dependence; that is, the dynamics of the system are strongly influenced by historical accidents. We describe the different kinds of ...
Journal of Mathematical Biology, 1988
We study a system of two integrodifferential equations which models the evolution of a biotic spe... more We study a system of two integrodifferential equations which models the evolution of a biotic species feeding on an abiotic resource. We also consider nutrient recycling with time delay. By Hopf bifurcation theory we prove the existence of stable oscillations for a range of values of the input of nutrients.
Journal of Economic Theory, 1999
We study forward-looking economic models assuming that agents take one step ahead expectations lo... more We study forward-looking economic models assuming that agents take one step ahead expectations looking back k time periods. We show that the dynamics of the economy with such an expectation function are characterized by the coexistence of perfect foresight and nonperfect foresight cycles. The stability of all these periodic solutions under bounded rationality is related to the stability of the perfect foresight cycles.
International Journal of Bifurcation and Chaos, 2006
Two-dimensional (Z1–Z3–Z1) maps are such that the plane is divided into three unbounded open regi... more Two-dimensional (Z1–Z3–Z1) maps are such that the plane is divided into three unbounded open regions: a region Z3, whose points generate three real rank-one preimages, bordered by two regions Z1, whose points generate only one real rank-one preimage. This paper is essentially devoted to the study of the structures, and the global bifurcations, of the basins of attraction generated by such maps. In particular, the cases of fractal structure of such basins are considered. For the class of maps considered in this paper, a large variety of dynamic situations is shown, and the bifurcations leading to their occurrence are explained.
Discrete Dynamics in Nature and Society, 2009
Binary choice games with externalities, as those described by Schelling (1973, 1978), have been r... more Binary choice games with externalities, as those described by Schelling (1973, 1978), have been recently modelled as discrete dynamical systems (Bischi and Merlone, 2009). In this paper we discuss the dynamic behavior in the case in which agents are impulsive; that is; they decide to switch their choices even when the difference between payoffs is extremely small. This particular case can be seen as a limiting case of the original model and can be formalized as a piecewise linear discontinuous map. We analyze the dynamic behavior of this map, characterized by the presence of stable periodic cycles of any period that appear and disappear through border-collision bifurcations. After a numerical exploration, we study the conditions for the creation and the destruction of periodic cycles, as well as the analytic expressions of the bifurcation curves.
Discrete & Continuous Dynamical Systems - B, 2009
Several discrete-time dynamic models are ultimately expressed in the form of iterated piecewise l... more Several discrete-time dynamic models are ultimately expressed in the form of iterated piecewise linear functions, in one-or two-dimensional spaces. In this paper we study a one-dimensional map made up of three linear pieces which are separated by two discontinuity points, motivated by a dynamic model arising in social sciences. Starting from the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point, and we give the analytic expressions of the bifurcation curves of the principal tongues (or tongues of first degree) for the family of maps considered, which depends on five parameters.
Decisions in Economics and Finance, 2013
Communications in Nonlinear Science and Numerical Simulation, 2010
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright
Chaos, Solitons & Fractals, 2014
Chaos, Solitons & Fractals, 2006
In this paper we describe some sequences of global bifurcations of attracting and repelling close... more In this paper we describe some sequences of global bifurcations of attracting and repelling closed invariant curves of two-dimensional maps that have a fixed point which may lose stability both via a supercritical Neimark bifurcation and a supercritical flip bifurcation. These bifurcations, characterized by the creation of heteroclinic and homoclinic connections or homoclinic tangles, are first described through qualitative phase diagrams and then by numerical examples.
The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Studies in Nonlinear Dynamics & Econometrics, 2012
Nonlinear Analysis: Real World Applications, 2012
This work moves from a recent paper by Antoci, Dei and Galeotti [1] where a dynamic model is prop... more This work moves from a recent paper by Antoci, Dei and Galeotti [1] where a dynamic model is proposed to describe an innovative method to improve environmental quality based on the exchange of …nancial activities, promoted by a Public Administration, between …rms and tourists in a given region. We extend their analysis in two directions: we …rst perform a global analysis of the basins of attraction to check the stability extents of the coexisting stable attractors of the model, and we show that some undesirable and sub-optimal stable equilibria always exist, whose basins may be quite intermingled with those of the optimal equilibrium; then we introduce a structural change of the model by assuming that the Public Administration, besides its action as an intermediary between visitors and polluting …rms, also performs a direct action for the pollution control. We show how the cost of this direct action of the Public Administration can be balanced by proper taxes and we prove that undesired equilibria can be ruled out by a suitable balance of …nancial instruments and direct actions of Public Administration for environmental remediation.
International Journal of Systems Science, 2008
In this article, single product Cournot oligopolies are considered, where the demand and cost fun... more In this article, single product Cournot oligopolies are considered, where the demand and cost functions are linear. While cost functions are completely known by all firms, they only partially know the demand function, as they misspecify the slope. At any stage of the repeated oligopoly game firms update the slope of their subjective demand functions on the basis of the discrepancy they observe between the expected price, computed according to believed demand, and the price they actually observe. This adjustment process has a unique steady state, where any subjective demand function coincides with the true demand function. If such steady state is stable, then the true slope of the demand function can be learned by all oligopolists, even if they start from misspecified initial guesses. Sufficient conditions for the stability of the steady state are given for n-firms oligopolies. In the particular case of a duopoly, an exact delimitation of the stability region in the parameters' space is given, and with the help of numerical simulations, the size and the shape of the basins of attraction is analysed, as well as the kinds of attracting sets that characterise the long-run dynamics of the learning process when the steady state is unstable.
Bifurcation Diagrams in the (δ R , τ R)-parameter plane129 8.3.3 1/n periodicity regions and thei... more Bifurcation Diagrams in the (δ R , τ R)-parameter plane129 8.3.3 1/n periodicity regions and their BCB boundaries 134 9 Appendix on the Myrber's map. 140 10 Bibliography.
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Papers by Gian-Italo Bischi