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December 2024
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13 Reads
Instead of modelling an economic agent by a hysteron, we suggest a fluid–mechanical notion of rate-dependent hysteretic agents based on the theory of Poisson counters. It leads to a simple representation of assemblies of such agents. We discuss the properties of the new version of hysteresis and its advantages over classical models of hysteresis in economics.
April 2020
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82 Reads
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2 Citations
Mathematical Modelling of Natural Phenomena
We consider piecewise-linear, discrete-time, macroeconomic models that have a continuum of feasible equilibrium states. The non-trivial equilibrium set and resulting path-dependence are induced by stickiness in either expectations or the response of the Central Bank. For a low-dimensional variant of the model with one representative agent, and also for a multi-agent model, we show that when exogenous noise is absent from the system the continuum of equilibrium states is the global attractor and each solution trajectory converges exponentially to one of the equilibria. Further, when a uniformly bounded noise is present, or the equilibrium states are destabilized by an imperfect Central Bank policy (or both), we estimate the size of the domain that attracts all the trajectories. The proofs are based on introducing a family of Lyapunov functions and, for the multi-agent model, deriving a formula for the inverse of the Prandtl-Ishlinskii operator acting in the space of discrete-time inputs and outputs.
November 2018
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11 Reads
Quasi-equilibrium models for aggregate or emergent variables over long periods of time are widely used throughout finance and economics. The validity of such models depends crucially upon assuming that the system participants act both independently and without memory. However important real-world effects such as herding, imitation, perverse incentives, and many of the key findings of behavioral economics violate one or both of these key assumptions. We present a very simple, yet realistic, agent-based modeling framework that is capable of simultaneously incorporating many of these effects. In this paper we use such a model in the context of a financial market to demonstrate that herding can cause a transition to multi-year boom-and-bust dynamics at levels far below a plausible estimate of the herding strength in actual financial markets. In other words, the stability of the standard (Brownian motion) equilibrium solution badly fails a “stress test” in the presence of a realistic weakening of the underlying modeling assumptions.The model contains a small number of fundamental parameters that can be easily estimated and require no fine-tuning. It also gives rise to a novel stochastic particle system with switching and re-injection that is of independent mathematical interest and may also be applicable to other areas of social dynamics.KeywordsFinancial instabilityFar-from-equilibriumEndogenous dynamicsBehavioral economicsHerdingBoom-and-bust
November 2017
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1 Read
We analyze a simple macroeconomic model where rational inflation expectations is replaced by a boundedly rational, and genuinely sticky, response to changes in the actual inflation rate. The stickiness is introduced in a novel way using a mathematical operator that is amenable to rigorous analysis. We prove that, when exogenous noise is absent from the system, the unique equilibrium of the rational expectations model is replaced by an entire line segment of possible equilibria with the one chosen depending, in a deterministic way, upon the previous states of the system. The agents are sufficiently far-removed from the rational expectations paradigm that problems o indeterminacy do not arise. The response to exogenous noise is far more subtle than in a unique equilibrium model. After sufficiently small shocks the system will indeed revert to the same equilibrium but larger ones will move the system to a different one (at the same model parameters). The path to this new equilibrium may be very long with a highly unpredictable endpoint. At certain model parameters exogenously-triggered runaway inflation can occur. Finally, we analyze a variant model in which the same form of sticky response is introduced into the interest rate rule instead.
November 2017
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75 Reads
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1 Citation
Applications of Mathematics
We analyze a simple macroeconomic model where rational inflation expectations is replaced by a boundedly rational, and genuinely sticky, response to changes in the actual inflation rate. The stickiness is introduced in a novel way using a mathematical operator that is amenable to rigorous analysis. We prove that, when exogenous noise is absent from the system, the unique equilibrium of the rational expectations model is replaced by an entire line segment of possible equilibria with the one chosen depending, in a deterministic way, upon the previous states of the system. The agents are sufficiently far-removed from the rational expectations paradigm that problems o indeterminacy do not arise. The response to exogenous noise is far more subtle than in a unique equilibrium model. After sufficiently small shocks the system will indeed revert to the same equilibrium but larger ones will move the system to a different one (at the same model parameters). The path to this new equilibrium may be very long with a highly unpredictable endpoint. At certain model parameters exogenously-triggered runaway inflation can occur. Finally, we analyze a variant model in which the same form of sticky response is introduced into the interest rate rule instead.
November 2017
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27 Reads
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1 Citation
We consider piecewise linear discrete time macroeconomic models, which possess a continuum of equilibrium states. These systems are obtained by replacing rational inflation expectations with a boundedly rational, and genuinely sticky, response of agents to changes in the actual inflation rate in a standard Dynamic Stochastic General Equilibrium model. Both for a low-dimensional variant of the model, with one representative agent, and the multi-agent model, we show that, when exogenous noise is absent from the system, the continuum of equilibrium states is the global attractor. Further, when a uniformly bounded noise is present, or the equilibrium states are destabilized by an imperfect Central Bank policy (or both), we estimate the size of the domain that attracts all the trajectories. The proofs are based on introducing a family of Lyapunov functions and, for the multi-agent model, deriving a formula for the inverse of the Prandtl-Ishlinskii operator acting in the space of discrete time inputs and outputs.
June 2016
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151 Reads
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12 Citations
Mathematica Bohemica
Certain financial market strategies are known to exhibit a hysteretic structure similar to the memory observed in plasticity, ferromagnetism, or magnetostriction. The main difference is that in financial markets, the spontaneous occurrence of discontinuities in the time evolution has to be taken into account. We show that one particular market model considered here admits a representation in terms of Prandtl-Ishlinskii hysteresis operators, which are extended in order to include possible discontinuities both in time and in memory. The main analytical tool is the Kurzweil integral formalism, and the main result proves the well-posedness of the process in the space of right-continuous regulated functions.
May 2016
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226 Reads
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14 Citations
SIAM Journal on Applied Dynamical Systems
We consider a piecewise linear two-dimensional dynamical system that couples a linear equation with the so-called stop operator. Global dynamics and bifurcations of this system are studied depending on two parameters. The system is motivated by modifications to general-equilibrium macroeconomic models that attempt to capture the frictions and memory-dependence of realistic economic agents.
May 2016
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4 Reads
We consider a piecewise linear two-dimensional dynamical system that couples a linear equation with the so-called stop operator. Global dynamics and bifurcations of this system are studied depending on two parameters. The system is motivated by modifications to general-equilibrium macroeconomic models that attempt to capture the frictions and memory-dependence of realistic economic agents.
November 2015
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86 Reads
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13 Citations
Discrete and Continuous Dynamical Systems - B
We consider a system of operator equations involving play and Prandtl-Ishlinskii hysteresis operators. This system generalizes the classical mechanical models of elastoplasticity, friction and fatigue by introducing coupling between the operators. We show that under quite general assumptions the coupled system is equivalent to one effective Prandtl-Ishlinskii operator or, more precisely, to a discontinuous extension of the Prandtl-Ishlinskii operator based on the Kurzweil integral of the derivative of the state function. This effective operator is described constructively in terms of the parameters of the coupled system. Our result is based on a substitution formula which we prove for the Kurzweil integral of regulated functions integrated with respect to functions of bounded variation. This formula allows us to prove the composition rule for the generalized (discontinuous) Prandtl-Ishlinskii operators. The composition rule, which underpins the analysis of the coupled model, then establishes that a composition of generalized Prandtl-Ishlinskii operators is also a generalized Prandtl-Ishlinskii operator provided that a monotonicity condition is satisfied.
... Under cyclic loading, one can naturally obtain limit circles with barodesy [22,23]. For mathematical modeling of granular and multiphase media, we cite [15][16][17][28][29][30], and for well-posedness and stability analysis of the relevant differential equations, we refer to [11,12,14,33]. Our study relies on a simplified version of the hypoplastic constitutive relation that was introduced independently by Bauer [5] and Gudehus [18]. ...
April 2020
Mathematical Modelling of Natural Phenomena
... The following result is proved in [13,Lemma 4.4] as a special case of the so-called Brokate's identity. ...
November 2017
... This theory is a research topic of great interest in the scientific community because it offers certain advantages. It is related to many problems not only theoretical but also practical in Statistics, Financial Mathematics, and Particle Physics, see, e.g., [10], [25], [33]. ...
June 2016
Mathematica Bohemica
... Systems with hysteresis have been investigated for a long time (see [1][2][3][4][5][6] and references therein). Such nonlinear systems are of interest for theoretical and applied problems (see, e. g., [7][8][9][10][11][12][13][14][15]). Hysteresis occupies one of the key positions in the current research on nonlinear phenomena and processes, in particular, in the present-day nonlinear control theory (see [16][17][18][19][20][21][22]). ...
May 2016
SIAM Journal on Applied Dynamical Systems
... so that the formula (2.3) is meaningful by virtue of [11]; while the assumptions on both the initial value function and the input function ensure that we integrate over a bounded interval. Unlike in [11], though, the integral in (2.3) can be interpreted as the Lebesgue integral thanks to the regularity of ϕ and the fact that the left derivative of p r coincides with its distributional derivative almost everywhere. ...
November 2015
Discrete and Continuous Dynamical Systems - B
... A change in the psychology of workers, a change in labour laws, the strength of union labour bargaining and minimum wages are exogenous factors that determine the rate of structural unemployment over a certain time period (Guichard & Rusticelii, 2010), where the economic sense of the unemployment hysteresis phenomenon is revealed. Another example of the hysteresis phenomenon is when the employment status of workers is determined by the bargaining power over the wages of currently employed workers, who determine the propensity of the private sector to hire workers at set wages, where in the time interval studied by the influence of an external factor, the unemployment rate is formed, such a hysteresis phenomenon is referred to as "insider-outsider" (Gali, 2020;Grinfeld et al., 2009). ...
February 2009
... The classical theory deals with input functions which are either continuous or piecewise constant. More recent applications in modeling economic processes, where jump discontinuities may spontaneously occur during the evolution, as for example in [3,4,5,6,11,12], require to consider a more general class of inputs. A good candidate seems to be the space G(0, T ) of regulated functions, that is, functions f : [0, T ] → R that admit both one-sided limits f (t−), f (t+) with the convention f (0−) = f (0), f (T +) = f (T ). ...
March 2013
Discrete and Continuous Dynamical Systems - B
... Insufficient study of the issue on the simultaneous impact of the technological structure of the economics and the pattern of consumer demand on the degree of market mechanisms stability should be noted. In the paper (Cross et al., 2013), the hypothesis that mathematical modelling and numerical results obtained at describing financial markets (Prudnikov et al., 2010) correctly reflect the essence of the processes occurring only through the use of hysteresis quantizers, was proved. One more result relates to the study of the dependency of the economic decline and the potential output (Cross et al., 2012). ...
December 2013
Journal of Economic Methodology
... Some typical applications have been to model friction for the force-displacement relation or to model elasticity and hardening in different applications of the stress-strain relation in rheology or material strength theory. Apart from the applications in mechanics [13], it has also been used in finance and electronics [13,14]. ...
September 2013
Physical Review E
... Introduction. In a series of publications [11,9,12,20,10,19] we have developed a model of financial markets that, in common with a number of other models, reproduces the most important observed statistics of real-world financial systems. Such heterogeneous agent models are very convenient as they make explicit the assumptions on the psychology and motivations of traders (so that the plausibility of the assumptions can be compared across models) and they are also relatively easy to program. ...
January 2010
