Social Dynamics aims at providing a unifying view on the evolution of social systems, including s... more Social Dynamics aims at providing a unifying view on the evolution of social systems, including social conflicts and dilemmas, interactions between populations and the environment, financial markets and the broader macroeconomy, by looking at the interplay between individual behavior at the micro-level and collective behaviors at the macro-level. This view emphasizes the role of boundedly rational behavior based on local information and conditioned by psychological biases, the interactions among heterogeneous agents and the resulting aggregate behaviors which, in turn, provide feedback to individual decisions.....
We consider a standard short-run Kaleckian macromodel developed by Charles (2010), where the auth... more We consider a standard short-run Kaleckian macromodel developed by Charles (2010), where the author, according to well-known Post-Keynesian models such as Delli Gatti, Gallegati and Gardini (1993), proposes different specifications for an endogenous propensity to invest to analyzes the short-run dynamics of the model, and shows that when firms' managers adopt abnormal behaviours, due to pressures from shareholders on the propensity to invest, the system exhibits bifurcation phenomena and complex behaviour. The analysis emphasizes that the behaviour of stockholders (and institutions that own portfolios) may be destabilizing for a financial capitalist economy. The endogenous propensity to invest proposed in Charles is a function of the current capacity utilization of the economic system. In this paper we extend the approach by considering an endogenous propensity to invest with memory: not only the current but also the past utilization capacity define, through a weighted average, the propensity to invest. In this way, a two-dimensional discrete dynamical system is obtained and an analysis of the dynamical behaviours is conducted. We show that memory has a stabilizing role for weighted average close to an uniform distribution, otherwise, when the weighted average takes extreme values, the system is destabilized through flip or Neimark-Saker bifurcations.
We consider a learning mechanism where expected values of an economic variable in discrete time a... more 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.
The paper gives an elementary introduction to the concept of bifurcation in the context of dynami... more The paper gives an elementary introduction to the concept of bifurcation in the context of dynamical systems, providing a historical background as well as examples of applications in real life.
Nel corso del Novecento l’uso dei modelli matematici, che si era già rivelato così utile in fisic... more Nel corso del Novecento l’uso dei modelli matematici, che si era già rivelato così utile in fisica e ingegneria, è stato introdotto, talvolta con difficoltà non trascurabili, anche in discipline tradizionalmente considerate poco adatte ad un simile approccio, quali l’economia, la sociologia, la biologia. Un percorso che inizia prendendo a prestito molte delle idee e dei modelli della fisica, tanto che nel discorso inaugurale dell’anno accademico 1901-1902 all’Università di Roma il grande fisico matematico Vito Volterra (1860- 1940) pronunciava le seguenti parole: «è intorno a quelle scienze nelle quali le matematiche solo da poco tempo hanno tentato d’introdursi, le scienze biologiche e sociali, che è più intensa la curiosità, giacché è forte il desiderio di assicurarsi se i metodi classici, i quali hanno dato così grandi risultati nelle scienze meccanico-fisiche, sono suscettibili di essere trasportati con pari successo nei nuovi ed inesplorati campi che si dischiudono loro dinanzi». In effetti l’economia arriverà, nel corso della prima metà del secolo, a un’elegante formulazione assiomatico-deduttiva della teoria dell’equilibrio economico, con l’utilizzo di metodi matematici eleganti e sofisticati. Ma le ipotesi di base, che coinvolgono concetti legati alle scelte degli individui influenzate dal loro livello di razionalità, influenzate da componenti psicologiche e interazioni sociali, condizionano fortemente i risultati ottenuti, e tuttora molti ritengono che il fatto che i metodi matematici si siano rivelati così utili in fisica non implica che lo siano anche per l’economia e le scienze sociali. La formalizzazione sempre più astratta di tali modelli, insieme alla loro difficoltà a spiegare e prevedere alcuni fenomeni economici e sociali osservati, ha portato a frequenti polemiche sulla reale opportunità di trasformare le discipline sociali in teorie matematiche, con strumenti che talvolta sembrano impiegati come fine a se stessi. In questo articolo, dopo aver brevemente delineato la storia della progressiva matematizzazione dell’economia, ci si concentrerà soprattutto sull’utilizzo in economia dei modelli dinamici non lineari, anche questi sviluppati inizialmente in fisica. Si tratta di modelli deterministici utilizzati per prevedere, ed eventualmente controllare, l’evoluzione temporale di sistemi reali. Basati su equazioni di evoluzione, espresse mediante equazioni differenziali o alle differenze a seconda che si consideri il tempo continuo o discreto, il loro studio qualitativo permette di ottenere informazioni sul tipo di comportamento che emergerà nel lungo periodo, e di come questo è influenzato dai principali parametri. La scoperta che modelli dinamici non lineari (che sono la regola nei sistemi sociali, caratterizzati da interazioni e meccanismi di feed-back) possono esibire comportamenti denotati col termine di caos deterministico per la proprietà di amplificare in modo difficilmente prevedibile perturbazioni arbitrariamente piccole (la cosiddetta sensitività rispetto alle condizioni iniziali, o “effetto farfalla”) ha suscitato un certo imbarazzo e nel contempo creato nuove possibilità. L’imbarazzo è dovuto al fatto che, come descriveremo meglio in seguito, la presenza di caos deterministico rende insostenibile l’ipotesi di agente economico razionale, ovvero capace di prevedere correttamente. Le nuove possibilità sono legate al fatto che quei sistemi economici e sociali caratterizzati da fluttuazioni in apparenza casuali potrebbero essere governati da leggi del moto deterministiche (anche se non lineari). In ogni caso, gli studi sui sistemi dinamici non lineari hanno portato a distinguere fra la rappresentazione matematica deterministica e la prevedibilità. L’attuale crisi economica ha senz’altro contribuito a riaccendere il dibattito sul modo di studiare i sistemi economici e sociali e la capacità di spiegare e prevedere. Le scienze sociali, e in particolare l’economia, sono davvero una scienza? Come può una scienza non prevedere e non accorgersi di quello che sta succedendo? Si tratta, come vedremo nel seguito, di domande ricorrenti, e anche in questa occasione qualcuno ha detto che, inseguendo i formalismi matematici, si sta perdendo di vista la realtà, mentre altri sostengono che il problema sta negli specifici formalismi adottati, che quelli usati sono superati, legati ad una matematica “vecchia”, magari mutuata in modo acritico da altre discipline. La speranza è allora che la crisi economica comporti un cambio di paradigma anche nella modellizzazione matematica
Journal of Economic Interaction and Coordination, 2020
Why is populism emerging now in Europe? Why is it present in USA and Latin America? What model of... more Why is populism emerging now in Europe? Why is it present in USA and Latin America? What model of political choice may explain these facts? Our paper addresses these questions by building an evolutionary game with two groups of players that decide whether to support a populist party by weighting demand for redistribution and demand for tough policy against immigration. Fundamentally, it is assumed that agents care about immigration the more they fear it and the higher number of other people care about it. Overall, positive shifts in the fear of immigration and increases in inequality drive citizens to converge toward supporting populists. The stability of the equilibria depends on the crucial parameters of the model, namely: fear of immigrants, the effect that the population type (the number of citizens supporting populism) have on individual preferences, economic inequality. Different equilibria represent different cases of populism: South-American left-wing populism and European r...
The previous chapters have already dealt with the behavior of boundedly rational firms in an olig... more The previous chapters have already dealt with the behavior of boundedly rational firms in an oligopoly. Although the firms know the true demand relationship, we have assumed that they do not know their competitors’ quantity choices. Instead they form expectations about these quantities and they base their own decisions on these beliefs. In particular, we have focused on several adjustment processes that firms might use to determine their quantity selections and we have investigated the circumstances under which such adjustment processes might lead to convergence to the Nash equilibrium of the static oligopoly game. However, the information that firms have about the environment may be incomplete on several accounts. For example, players may misspecify the true demand function or just misestimate the slope of the demand relationship, the reservation price, or the market saturation point. However, if firms base their decisions on such wrong estimates, they will realize that their beliefs are incorrect, since the market data they observe (for example, market prices or quantities) will be different from their predictions. Obviously, firms will try to update their beliefs on the demand relationship and this will give rise to an adjustment process. In other words, firms will try to learn the game they are playing. Following this line of thought, in this chapter we study oligopoly models under the assumption that firms either use misspecified price functions (Sect. 5.1) or do not know certain parameters of the market demand (Sect. 5.2). The main questions we want to answer are the following. If we understand an equilibrium in a game as a steady state of some non-equilibrium process of adjustment and “learning,” what happens if the players use an incorrect model of their environment? Does a reasonable adaptive process (for example, based on the best response) converge to anything? If so, to what does it converge? Is the limit that can be observed when the players play their perceived games (close to) an equilibrium of an equilibrium of the underlying true model? Is the observed situation consistent with the (limit) beliefs of the players?
This paper is devoted to the study of some global dynamical properties and bifurcations of two-di... more This paper is devoted to the study of some global dynamical properties and bifurcations of two-dimensional maps related to the presence, in the map or in one of its inverses, of a vanishing denominator. The new concepts of focal points and of prefocal curves are introduced in order to characterize some new kinds of contact bifurcations specic to maps with denominator. The occurrence of such bifurcations gives rise to new dynamic phenomena, and new structures of basin boundaries and invariant sets, whose presence can only be observed if a map (or some of its inverses) has a vanishing denominator. 1.
In this paper, we propose a unitary formulation for evolutionary oligopoly models with memory. In... more In this paper, we propose a unitary formulation for evolutionary oligopoly models with memory. In particular, we consider behavioral rules that are stationary at the Nash equilibria so that we can study the stability of the oligopolistic model with memory with generic strategies for determining quantities. Although the introduction of memory does not affect the local stability properties of equilibria, we show how the presence of memory impacts the global dynamics of the system and how the question about the role of memory does not always provide a clear answer in terms of model complexity.
Journal of Economic Interaction and Coordination, 2020
We propose an oligopoly model where players can choose between two kinds of behaviors, denoted as... more We propose an oligopoly model where players can choose between two kinds of behaviors, denoted as cooperative and aggressive, respectively. Each cooperative agent chooses the quantity to produce in order to maximize her own profit as well as the profits of other agents (at least partially), whereas an aggressive player decides the quantity to produce by maximizing his own profit while damaging (at least partially) competitors’ profits. At each discrete time, players face a binary choice to select the kind of behavior to adopt, according to a proportional imitation rule, expressed by a replicator equation based on a comparison between accumulated profits. This means that the behavioral decisions are driven by an evolutionary process where fitness is measured in terms of current profits as well as a weighted sum of past gains. The model proposed is expressed by a nonlinear two-dimensional iterated map, whose asymptotic behavior describes the long-run population distribution of coopera...
Social Dynamics aims at providing a unifying view on the evolution of social systems, including s... more Social Dynamics aims at providing a unifying view on the evolution of social systems, including social conflicts and dilemmas, interactions between populations and the environment, financial markets and the broader macroeconomy, by looking at the interplay between individual behavior at the micro-level and collective behaviors at the macro-level. This view emphasizes the role of boundedly rational behavior based on local information and conditioned by psychological biases, the interactions among heterogeneous agents and the resulting aggregate behaviors which, in turn, provide feedback to individual decisions.....
We consider a standard short-run Kaleckian macromodel developed by Charles (2010), where the auth... more We consider a standard short-run Kaleckian macromodel developed by Charles (2010), where the author, according to well-known Post-Keynesian models such as Delli Gatti, Gallegati and Gardini (1993), proposes different specifications for an endogenous propensity to invest to analyzes the short-run dynamics of the model, and shows that when firms' managers adopt abnormal behaviours, due to pressures from shareholders on the propensity to invest, the system exhibits bifurcation phenomena and complex behaviour. The analysis emphasizes that the behaviour of stockholders (and institutions that own portfolios) may be destabilizing for a financial capitalist economy. The endogenous propensity to invest proposed in Charles is a function of the current capacity utilization of the economic system. In this paper we extend the approach by considering an endogenous propensity to invest with memory: not only the current but also the past utilization capacity define, through a weighted average, the propensity to invest. In this way, a two-dimensional discrete dynamical system is obtained and an analysis of the dynamical behaviours is conducted. We show that memory has a stabilizing role for weighted average close to an uniform distribution, otherwise, when the weighted average takes extreme values, the system is destabilized through flip or Neimark-Saker bifurcations.
We consider a learning mechanism where expected values of an economic variable in discrete time a... more 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.
The paper gives an elementary introduction to the concept of bifurcation in the context of dynami... more The paper gives an elementary introduction to the concept of bifurcation in the context of dynamical systems, providing a historical background as well as examples of applications in real life.
Nel corso del Novecento l’uso dei modelli matematici, che si era già rivelato così utile in fisic... more Nel corso del Novecento l’uso dei modelli matematici, che si era già rivelato così utile in fisica e ingegneria, è stato introdotto, talvolta con difficoltà non trascurabili, anche in discipline tradizionalmente considerate poco adatte ad un simile approccio, quali l’economia, la sociologia, la biologia. Un percorso che inizia prendendo a prestito molte delle idee e dei modelli della fisica, tanto che nel discorso inaugurale dell’anno accademico 1901-1902 all’Università di Roma il grande fisico matematico Vito Volterra (1860- 1940) pronunciava le seguenti parole: «è intorno a quelle scienze nelle quali le matematiche solo da poco tempo hanno tentato d’introdursi, le scienze biologiche e sociali, che è più intensa la curiosità, giacché è forte il desiderio di assicurarsi se i metodi classici, i quali hanno dato così grandi risultati nelle scienze meccanico-fisiche, sono suscettibili di essere trasportati con pari successo nei nuovi ed inesplorati campi che si dischiudono loro dinanzi». In effetti l’economia arriverà, nel corso della prima metà del secolo, a un’elegante formulazione assiomatico-deduttiva della teoria dell’equilibrio economico, con l’utilizzo di metodi matematici eleganti e sofisticati. Ma le ipotesi di base, che coinvolgono concetti legati alle scelte degli individui influenzate dal loro livello di razionalità, influenzate da componenti psicologiche e interazioni sociali, condizionano fortemente i risultati ottenuti, e tuttora molti ritengono che il fatto che i metodi matematici si siano rivelati così utili in fisica non implica che lo siano anche per l’economia e le scienze sociali. La formalizzazione sempre più astratta di tali modelli, insieme alla loro difficoltà a spiegare e prevedere alcuni fenomeni economici e sociali osservati, ha portato a frequenti polemiche sulla reale opportunità di trasformare le discipline sociali in teorie matematiche, con strumenti che talvolta sembrano impiegati come fine a se stessi. In questo articolo, dopo aver brevemente delineato la storia della progressiva matematizzazione dell’economia, ci si concentrerà soprattutto sull’utilizzo in economia dei modelli dinamici non lineari, anche questi sviluppati inizialmente in fisica. Si tratta di modelli deterministici utilizzati per prevedere, ed eventualmente controllare, l’evoluzione temporale di sistemi reali. Basati su equazioni di evoluzione, espresse mediante equazioni differenziali o alle differenze a seconda che si consideri il tempo continuo o discreto, il loro studio qualitativo permette di ottenere informazioni sul tipo di comportamento che emergerà nel lungo periodo, e di come questo è influenzato dai principali parametri. La scoperta che modelli dinamici non lineari (che sono la regola nei sistemi sociali, caratterizzati da interazioni e meccanismi di feed-back) possono esibire comportamenti denotati col termine di caos deterministico per la proprietà di amplificare in modo difficilmente prevedibile perturbazioni arbitrariamente piccole (la cosiddetta sensitività rispetto alle condizioni iniziali, o “effetto farfalla”) ha suscitato un certo imbarazzo e nel contempo creato nuove possibilità. L’imbarazzo è dovuto al fatto che, come descriveremo meglio in seguito, la presenza di caos deterministico rende insostenibile l’ipotesi di agente economico razionale, ovvero capace di prevedere correttamente. Le nuove possibilità sono legate al fatto che quei sistemi economici e sociali caratterizzati da fluttuazioni in apparenza casuali potrebbero essere governati da leggi del moto deterministiche (anche se non lineari). In ogni caso, gli studi sui sistemi dinamici non lineari hanno portato a distinguere fra la rappresentazione matematica deterministica e la prevedibilità. L’attuale crisi economica ha senz’altro contribuito a riaccendere il dibattito sul modo di studiare i sistemi economici e sociali e la capacità di spiegare e prevedere. Le scienze sociali, e in particolare l’economia, sono davvero una scienza? Come può una scienza non prevedere e non accorgersi di quello che sta succedendo? Si tratta, come vedremo nel seguito, di domande ricorrenti, e anche in questa occasione qualcuno ha detto che, inseguendo i formalismi matematici, si sta perdendo di vista la realtà, mentre altri sostengono che il problema sta negli specifici formalismi adottati, che quelli usati sono superati, legati ad una matematica “vecchia”, magari mutuata in modo acritico da altre discipline. La speranza è allora che la crisi economica comporti un cambio di paradigma anche nella modellizzazione matematica
Journal of Economic Interaction and Coordination, 2020
Why is populism emerging now in Europe? Why is it present in USA and Latin America? What model of... more Why is populism emerging now in Europe? Why is it present in USA and Latin America? What model of political choice may explain these facts? Our paper addresses these questions by building an evolutionary game with two groups of players that decide whether to support a populist party by weighting demand for redistribution and demand for tough policy against immigration. Fundamentally, it is assumed that agents care about immigration the more they fear it and the higher number of other people care about it. Overall, positive shifts in the fear of immigration and increases in inequality drive citizens to converge toward supporting populists. The stability of the equilibria depends on the crucial parameters of the model, namely: fear of immigrants, the effect that the population type (the number of citizens supporting populism) have on individual preferences, economic inequality. Different equilibria represent different cases of populism: South-American left-wing populism and European r...
The previous chapters have already dealt with the behavior of boundedly rational firms in an olig... more The previous chapters have already dealt with the behavior of boundedly rational firms in an oligopoly. Although the firms know the true demand relationship, we have assumed that they do not know their competitors’ quantity choices. Instead they form expectations about these quantities and they base their own decisions on these beliefs. In particular, we have focused on several adjustment processes that firms might use to determine their quantity selections and we have investigated the circumstances under which such adjustment processes might lead to convergence to the Nash equilibrium of the static oligopoly game. However, the information that firms have about the environment may be incomplete on several accounts. For example, players may misspecify the true demand function or just misestimate the slope of the demand relationship, the reservation price, or the market saturation point. However, if firms base their decisions on such wrong estimates, they will realize that their beliefs are incorrect, since the market data they observe (for example, market prices or quantities) will be different from their predictions. Obviously, firms will try to update their beliefs on the demand relationship and this will give rise to an adjustment process. In other words, firms will try to learn the game they are playing. Following this line of thought, in this chapter we study oligopoly models under the assumption that firms either use misspecified price functions (Sect. 5.1) or do not know certain parameters of the market demand (Sect. 5.2). The main questions we want to answer are the following. If we understand an equilibrium in a game as a steady state of some non-equilibrium process of adjustment and “learning,” what happens if the players use an incorrect model of their environment? Does a reasonable adaptive process (for example, based on the best response) converge to anything? If so, to what does it converge? Is the limit that can be observed when the players play their perceived games (close to) an equilibrium of an equilibrium of the underlying true model? Is the observed situation consistent with the (limit) beliefs of the players?
This paper is devoted to the study of some global dynamical properties and bifurcations of two-di... more This paper is devoted to the study of some global dynamical properties and bifurcations of two-dimensional maps related to the presence, in the map or in one of its inverses, of a vanishing denominator. The new concepts of focal points and of prefocal curves are introduced in order to characterize some new kinds of contact bifurcations specic to maps with denominator. The occurrence of such bifurcations gives rise to new dynamic phenomena, and new structures of basin boundaries and invariant sets, whose presence can only be observed if a map (or some of its inverses) has a vanishing denominator. 1.
In this paper, we propose a unitary formulation for evolutionary oligopoly models with memory. In... more In this paper, we propose a unitary formulation for evolutionary oligopoly models with memory. In particular, we consider behavioral rules that are stationary at the Nash equilibria so that we can study the stability of the oligopolistic model with memory with generic strategies for determining quantities. Although the introduction of memory does not affect the local stability properties of equilibria, we show how the presence of memory impacts the global dynamics of the system and how the question about the role of memory does not always provide a clear answer in terms of model complexity.
Journal of Economic Interaction and Coordination, 2020
We propose an oligopoly model where players can choose between two kinds of behaviors, denoted as... more We propose an oligopoly model where players can choose between two kinds of behaviors, denoted as cooperative and aggressive, respectively. Each cooperative agent chooses the quantity to produce in order to maximize her own profit as well as the profits of other agents (at least partially), whereas an aggressive player decides the quantity to produce by maximizing his own profit while damaging (at least partially) competitors’ profits. At each discrete time, players face a binary choice to select the kind of behavior to adopt, according to a proportional imitation rule, expressed by a replicator equation based on a comparison between accumulated profits. This means that the behavioral decisions are driven by an evolutionary process where fitness is measured in terms of current profits as well as a weighted sum of past gains. The model proposed is expressed by a nonlinear two-dimensional iterated map, whose asymptotic behavior describes the long-run population distribution of coopera...
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