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Discrete and Continuous Dynamical Systems - B

November 2021 , Volume 26 , Issue 11

Preface: Special issue on nonlinear dynamical systems in economic modeling

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Preface: Special issue on nonlinear dynamical systems in economic modeling
Laura Gardini and Iryna Sushko
2021, 26(11): i-iv doi: 10.3934/dcdsb.2021241 +[Abstract](8095) +[HTML](54) +[PDF](82.05KB)
Firms, technology, training and government fiscal policies: An evolutionary approach
Elvio Accinelli, Filipe Martins, Humberto Muñiz, Bruno M. P. M. Oliveira and Alberto A. Pinto
2021, 26(11): 5723-5754 doi: 10.3934/dcdsb.2021180 +[Abstract](708) +[HTML](158) +[PDF](494.88KB)

In this paper we propose and analyze a game theoretical model regarding the dynamical interaction between government fiscal policy choices toward innovation and training (I&T), firm's innovation, and worker's levels of training and education. We discuss four economic scenarios corresponding to strict pure Nash equilibria: the government and I&T poverty trap, the I&T poverty trap, the I&T high premium niche, and the I&T ideal growth. The main novelty of this model is to consider the government as one of the three interacting players in the game that also allow us to analyse the I&T mixed economic scenarios with a unique strictly mixed Nash equilibrium and with I&T evolutionary dynamical cycles.

Environmental degradation and indeterminacy of equilibrium selection
Angelo Antoci, Marcello Galeotti and Mauro Sodini
2021, 26(11): 5755-5767 doi: 10.3934/dcdsb.2021179 +[Abstract](541) +[HTML](138) +[PDF](325.3KB)

This paper analyzes an intertemporal optimization problem in which agents derive utility from three goods: leisure, a public environmental good and the consumption of a produced good. The global analysis of the dynamic system generated by the optimization problem shows that global indeterminacy may arise: given the initial values of the state variables, the economy may converge to different steady states, by choosing different initial values of the control variable.

An optimal control problem of monetary policy
Andrea Bacchiocchi and Germana Giombini
2021, 26(11): 5769-5786 doi: 10.3934/dcdsb.2021224 +[Abstract](682) +[HTML](113) +[PDF](447.22KB)

This paper analyses an optimal monetary policy under a non-linear Phillips curve and linear GDP dynamics. A central bank controls the inflation and the GDP trends through the adjustment of the interest rate to prevent shocks and deviations from the long-run optimal targets. The optimal control path for the monetary instrument, the interest rate, is the result of a dynamic minimization problem in a continuous-time fashion. The model allows considering various economic dynamics ranging from hyperinflation to disinflation, sustained growth and recession. The outcomes provide useful monetary policy insights and reveal the dilemma between objectives faced by the monetary authority in trade-off scenarios.

Water taxes and fines imposed on legal and illegal firms exploiting groudwater
Marta Biancardi, Lucia Maddalena and Giovanni Villani
2021, 26(11): 5787-5806 doi: 10.3934/dcdsb.2021008 +[Abstract](1049) +[HTML](344) +[PDF](1356.83KB)

This paper uses a differential game approach to investigate a model that represents the exploitation of groundwater, taking into account the strategic and dynamic interactions among users of the resource and public authority. Agents' behaviour may influence their gains but also the overexploitation of the aquifer. The effects of legal and illegal firms' actions and the contribution of taxes and penalties imposed by public authorities, are analysed by studying Feedback equilibria in order to capture the problem of non-compliance with resource management regimes and to discuss policy options in a non-cooperative and cooperative context. We show that illegal extractions can be a significant stumbling block on the path towards implementing of better management and environmental policies and we explain how, in order to fight this phenomenon, the public authority must increase controlled activity rather than taxation, but also encourage cooperation between legal firms under appropriate conditions.

Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs
Serena Brianzoni and Giovanni Campisi
2021, 26(11): 5807-5825 doi: 10.3934/dcdsb.2021116 +[Abstract](884) +[HTML](299) +[PDF](3960.98KB)

It is well known that regulation and efficiency are two important issues on banking literature. The goal of the paper is to analyse them through a banking duopoly model with heterogeneous expectations. To this purpose, we consider two scenarios. In the first one, we focus on regulation effects. In particular, empirical literature on Italian banks finds evidence on the asymmetry of the costs of regulation that penalize small banks with respect to the large ones. In this direction, we analyse a duopoly model where small banks and large banks have different forecasting rules and we capture the differences of the regulations' effects assuming asymmetry in the cost functions. We introduce linear cost function for small banks and quadratic cost function for large banks. In the second scenario, we study the relation between regulation and bank efficiency highlighting empirical results showing that large banks register higher level of inefficiency than small banks. Moreover, in order to stress new evidences and to confirm empirical results on banking regulation and efficiency, we conduct an analytical and numerical analysis.

Population dynamics and economic development
Andrea Caravaggio, Luca Gori and Mauro Sodini
2021, 26(11): 5827-5848 doi: 10.3934/dcdsb.2021178 +[Abstract](773) +[HTML](156) +[PDF](621.43KB)

This research develops a continuous-time optimal growth model that accounts for population dynamics resembling the historical pattern of the demographic transition. The Ramsey model then becomes able to generate multiple determinate or indeterminate stationary equilibria and explain the process of the transition from a state with high fertility and low income per capita to a state with low fertility and high income per capita. The article also investigates the emergence of damped or persistent cyclical dynamics.

Transitions between metastable long-run consumption behaviors in a stochastic peer-driven consumer network
Jochen Jungeilges, Trygve Kastberg Nilssen, Tatyana Perevalova and Alexander Satov
2021, 26(11): 5849-5871 doi: 10.3934/dcdsb.2021232 +[Abstract](585) +[HTML](71) +[PDF](637.07KB)

We study behavioral change - as a transition between coexisting attractors - in the context of a stochastic, non-linear consumption model with interdependent agents. Relying on the indirect approach to the analysis of a stochastic dynamic system, and employing a mix of analytical, numerical and graphical techniques, we identify conditions under which such transitions are likely to occur. The stochastic analysis depends crucially on the stochastic sensitivity function technique as it can be applied to the stochastic analoga of closed invariant curves [14], [1]. We find that in a moderate noise environment increased peer influence actually reduces the complexity of observable long-run consumer behavior.

Public debt dynamics under ambiguity by means of iterated function systems on density functions
Davide La Torre, Simone Marsiglio, Franklin Mendivil and Fabio Privileggi
2021, 26(11): 5873-5903 doi: 10.3934/dcdsb.2021070 +[Abstract](1059) +[HTML](274) +[PDF](1759.51KB)

We analyze a purely dynamic model of public debt stabilization under ambiguity. We assume that the debt to GDP ratio is described by a random variable, and thus it can be characterized by investigating the evolution of its density function through iteration function systems on mappings. Ambiguity is associated with parameter uncertainty which requires policymakers to respond to such an additional layer of uncertainty according to their ambiguity attitude. We describe ambiguity attitude through a simple heuristic rule in which policymakers adjust the available vague information (captured by the empirical distribution of the debt ratio) with a measure of their ignorance (captured by the uniform distribution). We show that such a model generates fractal-type objects that can be characterized as fixed-point solutions of iterated function systems on mappings. Ambiguity is a source of unpredictability in the long run outcome since it introduces some singularity features in the steady state distribution of the debt ratio. However, the presence of some ambiguity aversion removes such unpredictability by smoothing out the singularities in the steady state distribution.

Stability switching and its directions in cournot duopoly game with three delays
Akio Matsumoto and Ferenc Szidarovszky
2021, 26(11): 5905-5923 doi: 10.3934/dcdsb.2021069 +[Abstract](873) +[HTML](290) +[PDF](1679.39KB)

A three-delay duopoly is considered where the firms have identical implementation delays with different information delays. The equilibrium is locally asymptotically stable without delays however this stability is lost with increasing values of the delays. The stability properties of the equilibrium depend on the common implementation delay of the firms and on the sum of the two information delays. The stability switching curves are first analytically characterized and illustrated, and then the direction of the stability switching is determined at each point of the curves. The possibility of multiple pure imaginary eigenvalues is also discussed when the directions of the stability switches cannot be determined. Simulation examples illustrate the theoretical results.

On dynamics in a medium-term Keynesian model
Hiroki Murakami and Rudolf Zimka
2021, 26(11): 5925-5940 doi: 10.3934/dcdsb.2021145 +[Abstract](842) +[HTML](319) +[PDF](674.49KB)

This paper rigorously examines the (in)stability of limit cycles generated by Hopf bifurcations in a medium-term Keynesian model. The bifurcation equation of the model is derived and the conditions for stable and unstable limit cycles are presented. Numerical simulations are performed to illustrate the analytical results.

Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure
Anastasiia Panchuk and Frank Westerhoff
2021, 26(11): 5941-5964 doi: 10.3934/dcdsb.2021117 +[Abstract](1144) +[HTML](253) +[PDF](4057.6KB)

We study a simple financial market model with interacting chartists and fundamentalists that may give rise to multiband chaotic attractors. In particular, asset prices fluctuate erratically around their fundamental values, displaying a significant bull and bear market behavior. An in-depth analytical and numerical study of our model furthermore reveals the emergence of a new bifurcation structure, a phenomenon that we call a bandcount accretion bifurcation structure. The latter consists of regions associated with chaotic dynamics only, the boundaries of which are not defined by homoclinic bifurcations, but mainly by contact bifurcations of particular type where two distinct critical points of certain ranks coincide.

A viral transmission model for foxes-cottontails-hares interaction: Infection through predation
Simona Viale, Elisa Caudera, Sandro Bertolino and Ezio Venturino
2021, 26(11): 5965-5997 doi: 10.3934/dcdsb.2021158 +[Abstract](660) +[HTML](196) +[PDF](661.27KB)

The Eastern cottontail Sylvilagus floridanus is a lagomorph native to North America, introduced in Italy since the 1960s. In Central and Northern Italy, the cottontail overlaps its range with the native European hare Lepus europaeus and affects the predator-prey dynamics of native hares and foxes. Field data indicate that the cottontail is susceptible to infection by the European brown hare syndrome (EBHS) virus. Although the real role of cottontails and native foxes in the spreading of EBHS viruses is yet uncertain, we present a cottontail-hare-fox model including possible effects of EBHS, imported by foxes, through environmental contamination. A rather complete map of the possible system equilibria and their mutual relationship and transition is established.

A three-country Kaldorian business cycle model with fixed exchange rates: A continuous time analysis
Rudolf Zimka, Michal Demetrian, Toichiro Asada and Toshio Inaba
2021, 26(11): 5999-6015 doi: 10.3934/dcdsb.2021143 +[Abstract](703) +[HTML](403) +[PDF](960.49KB)

This paper analyses a three-country, fixed exchange rates Kaldorian nonlinear macroeconomic model of business cycles. The countries are connected through international trade, and international capital movement with imperfect capital mobility. Our model is a continuous time version of the discrete time three-country Kaldorian model of Inaba and Asada [22]. Their paper provided numerical studies of the dynamics of the three countries under fixed exchange rates. This paper provides analytical examinations of the local stability of the model´s equilibria, and of the existence of business cycles. The results are illustrated by numerical simulations.

2020 Impact Factor: 1.327
5 Year Impact Factor: 1.492
2020 CiteScore: 2.2




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