Ahmad Naimzada

• Professor (Full) at Università degli Studi di Milano-Bicocca

The purpose of this work is to provide a way to improve stability and convergence rate of a price adjustment mechanism that converges to a Walrasian equilibrium. We focus on a discrete tâtonnement based on a two-agent, two-good exchange economy, and we introduce memory, assuming that the auctioneer adjusts prices not only using the current excess d...

In this paper we propose a model in which the real side of the economy, de-scribed via a Keynesian good market approach, interacts with the stock market with heterogeneous speculators, i.e., optimist and pessimist fundamentalists. Employing analytical and numerical tools, we detect the mechanisms and the channels through which instabilities get tra...

We consider a simple evolutionary cobweb model with heterogeneous agents and propose a class of individuals aimed at addressing the limitations of perfectly rational agents and traditional fundamentalists. Fully rational expectations require complete knowledge of both the market structure and agents’ decisions. Conversely, traditional fundamentalis...

This study aims to provide a manageable symmetric two-players conflict model in which, defining measures for polarization and hostility, we investigate the effects of spillovers into the properties of the sets of equilibria, into the intensity of conflict, and into the endogenous changes in polarization and hostility. We show that, without spillove...

In the work, we prove the presence of chaotic dynamics, for suitable values of the model parameters, for the discrete-time system, composed of two coupled logistic maps, as formulated in Yousefi et al. [Discrete Dyn. Nat. Soc. 5, 161–177 (2000)], which describes two interdependent economies, characterized by two competitive markets each, with a dem...

The global recession triggered by the great financial crisis and the response to the COVID-19 emergency have renewed the interest in connecting business cycle dynamics to financial conditions. This paper proposes a simple macro-dynamic behavioural model of the interaction between the stock market (SM) and the economy's real sector (RS). We innovate...

We propose a model for exploring the feasibility and effectiveness of a green transition from dirty to clean technologies. It relies on an evolutionary framework for the technology selection that accounts for the environmental domain dynamics, in terms of pollution evolution. A regulator charges an environmental tax to the producers, and the agents...

We extend the dynamic Cournot duopoly framework with emission charges on output by Mamada and Perrings (Econ Anal Policy 66:370–380, 2020), which encompassed homogeneous products in its original formulation, to the more general case of differentiated goods, in order to highlight the richness in its static and dynamic outcomes. Each firm is taxed pr...

In the present work, starting from the 1D evolutionary Muthian cobweb model in [Hommes & Wagener, 2010], where the economy is populated by biased firms and unbiased fundamentalist firms, we investigate the effects generated by adaptively adjusted beliefs on the long-period price of the goods produced. As in [Hommes & Wagener, 2010], we focus on the...

We extend the evolutionary cobweb setting proposed in Hommes and Wagener [J. Econ. Behav. Organ. 75, 25-39 (2010)] in which the share updating mechanism is based on a comparison among the profits realized by the different kinds of agents, by assuming that the market is populated by rational producers, endowed with perfect foresight expectations abo...

We propose a model economy consisting of interdependent real, monetary and stock markets. The money market is influenced by the real one through a standard LM equation. Private expenditures depend on stock prices, which in turn are affected by interest rates and real profits, as these contribute to determine the participation level in the stock mar...

We study a financial market populated by heterogeneous agents, whose decisions are driven by “animal spirits”. Each agent may have either correct, optimistic or pessimistic beliefs about the fundamental value, which can change from time to time based on an evolutionary mechanism. The evolutionary selection of beliefs depends on a weighted evaluatio...

In this paper, we study a class of markets, among which we can mention agricultural and energy markets, characterized by seasonality, i.e., in which demand and/or supply conditions cyclically alternate with a precise and known periodicity. We propose a new theoretical framework based on a cobweb model with adaptive expectations, accordingly modifie...

Starting from a Muthian cobweb model, we extend the profit-based evolutionary setting in Hommes and Wagener (2010) by assuming that, in addition to pessimistic, optimistic and unbiased fundamentalists, the market is populated by rational producers, which correctly anticipate the next period price. Thanks to their introduction, we find that, differe...

We consider the overlapping generation model formulated in Dioikitopoulos (J Econ Dyn Control 93:260–276, 2018). Its innovative approach involves endogenous adaptations of the deficit/surplus to debt and income levels through an empirically estimated fiscal policy rule. We improve the analysis of the model in order to tackle the issue of debt susta...

We propose a discrete-time exchange economy evolutionary model with two groups of agents. In our setting the definition of equilibrium depends also on agents’ population shares, which affect the market clearing conditions. We prove that, despite such difference with the classical Walrasian framework, for all economies and population shares there ex...

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 t...

Starting from a Muthian cobweb model, we here extend the profit-based evolutionary settings in Hommes and Wagener (2010) and in Naimzada and Pireddu (2020a) by assuming that unbiased fundamentalists and several groups of biased fundamentalists face information costs that are inversely proportional to their bias. Like in those works, we deal with th...

In this paper we consider a nonlinear model for the real economy described by a multiplier–accelerator setup. The model comprises the government sector, which influences the output dynamics by means of the fiscal policy, and the money market, where the money supply depends upon the fluctuations in the economic activity. Through rigorous analytical...

We consider a Minskyan type model of a closed economy with autonomous public expenditure formulated in a discrete time framework, where an endogenous debt adjustment process is considered and where income variations account for real world physical and social constraints. The model is characterized by a unique nontrivial fixed point matching the eco...

We consider the competition among quantity setting players in a linear evolutionary environment. To set their outputs, players adopt, alternatively, the best response rule having perfect foresight or an imitative rule. Players are allowed to change their behavior through an evolutionary mechanism according to which the rule with better performance...

Founded on a Muthian cobweb model, this study extends the evolutionary setting in Hommes and Wagener (2010), by assuming that agents face heterogeneous information costs. We prove that the equilibrium, when globally eductively stable, may be unstable under evolutionary learning.

We extend the exchange economy evolutionary discrete-time model with heterogeneous agents introduced in ‘Fashion cycle dynamics in a model with endogenous discrete evolution of heterogeneous preferences’, by A. Naimzada and M. Pireddu, appeared in Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 055907 (2018), by considering generic bel...

We present a cobweb model to explain price adjustment in a competitive market with homogeneous firms based on assumptions from Prospect Theory. Price changes are evaluated with respect to a psychological reference price which enters directly into the demand function. Accordingly, firms face a downward-sloping demand curve that is kinked at the cons...

We propose a modelling approach to study Cournotian oligopolies of boundedly rational firms which continuously update production decisions on the basis of information collected periodically. The model consists of a system of differential equations with piecewise constant arguments, which can be recast into a system of difference equations. Consider...

We formulate an evolutionary oligopoly model where quantity setting players produce following either the static expectation best response or a performance-proportional imitation rule. The choice on how to behave is driven by an evolutionary selection mechanism according to which the rule that brought the highest performance attracts more followers....

Imitation-based behaviors are considered in economics with significant contributions in reference to homogeneous populations where players are characterized by the same decisional processes (see for example [42,48]). However, the presence of imitation behaviors is detected in experimental oligopolies coexisting with rational-like rules. This motiva...

We analyze a class of congestion games where agents use resources to send a finite amount of goods from an initial location to a terminal one. The resources are costly and costs are load dependent. In this context we concentrate on the heterogeneity because we not only assume that agents have limited computational capability but also they differ in...

In the rank of behavioral rules, imitation-based heuristics have received special attention in economics (see Vega-Redondo, 1997; Schlag, 1998). In particular, imitative behavior explains the evidence from experimental oligopolies where the Cournot–Nash equilibrium does not emerge as unique outcome and outputs are placed, for the most part, on fair...

A cobweb model, characterized by boundedly rational producers with a production adjustment mechanism based on the gradient rule, is described by a nonlinear discrete time dynamical system of the plane. Firms do not have a complete knowledge of the demand function and try to infer how the market will respond to their production changes by an empiric...

We propose an exchange economy evolutionary model with discrete time, in which there are two groups of agents characterized by different structures of preferences. The share updating mechanism depends in a monotone manner on the goods’ consumption, described in terms of the calorie intakes. In such framework we investigate the existence of equilibr...

The boundedly rational heterogeneous agent literature can be considered to have properly started with a number of contributions in the early 90s, with the impressive contribution by Carl Chiarella who, amongst his huge field of research, developed several milestone models of heterogeneous interacting agents that are able to generate a number of dyn...

We propose two exchange economy evolutionary models with heterogeneous agents, in which the share updating mechanisms depend on the goods’ consumption, described in terms of the calorie intakes. In the first setting we assume that the share updating rule is monotone in the calorie intake, while in the second framework we suppose that it is non-mono...

We extend the model of Dieci and Westerhoff [J. Evol. Econ. 22(2), 303–329 (2012)], where the authors analyse a speculative housing market populated by heterogeneous interacting agents described by a two dimensional nonlinear discrete time dynamical system. They show the emergence of complicated dynamics through the occurrence of bifurcations for p...

We propose a discrete-time exchange economy evolutionary model, in which two groups of agents are characterized by different preference structures. The reproduction level of a group is related to its attractiveness degree, which depends on the social visibility level, determined by the consumption choices of the agents in that group. The attractive...

In the present paper, a model of a market consisting of real and financial interacting sectors is studied. Agents populating the stock market are assumed to be not able to observe the true underlying fundamental, and their beliefs are biased by either optimism or pessimism. Depending on the relevance they give to beliefs, they select the best perfo...

We consider the competition among quantity setting players in a deterministic nonlinear oligopoly framework characterized by an isoelastic demand curve. Players are characterized by having heterogeneous decisional mechanisms to set their outputs: some players are imitators, while the remaining others adopt a rational-like rule according to which th...

In the present paper a model of a market consisting of real and financial interacting sectors is studied. Agents populating the stock market are assumed to be not able to observe the true underlying fundamental, and their beliefs are biased by either optimism or pessimism. Depending on the relevance they give to beliefs, they select the best perfor...

We analyze a class of congestion games where two agents must send a finite amount of goods from an initial location to a terminal one. To do so the agents must use resources which are costly and costs are load dependent. In this context we assume that the agents have limited computational capability and they use a gradient rule as a decision mechan...

We propose an exchange economy evolutionary model with discrete time, in which there are two utility-maximizing groups of agents which differ in the preference structure. Assuming an evolutionary mechanism based on the relative utility values realized by the two kinds of agents, we analytically and numerically investigate the existence of equilibri...

The simple pure exchange model with two individuals and two goods by Day and Pianigiani (1991), extensively analyzed by Day (1994) and taken up again by Mukherjy (1999), is discussed and extended with the purpose of showing that chaos in a discrete tâtonnement process of this kind can be controlled if the auctioneer uses a smooth, nonlinear formula...

We propose a financial market model with optimistic and pessimistic fundamentalists who, respectively, overestimate and underestimate the true fundamental value due to ambiguity in the stock market. We assume that agents form their beliefs about the fundamental value through an imitative process, considering the relative ability shown by optimists...

The dynamics of a pure exchange overlapping generations model with endogenous money growth rule is investigated. We consider a nonlinear monetary policy rule which, in each period, bounds the money growth rate so that money is determined by the deviation of the inflation rate from its target. More precisely, we introduce such a mechanism through a...

In the present paper, we investigate the dynamics of a model in which the real part of the economy, described within a multiplier-accelerator framework, interacts with a financial market with heterogeneous speculators, in order to study the channels through which the two sectors influence each other. Employing analytical and numerical tools, we inv...

This paper considers a pure exchange overlapping generations model in which the money-growth rate is endogenous and follows a feedback rule. Different specifications for the monetary policy rule are analyzed, namely a so-called current, forward, or backward-looking feedback rule, depending on whether the monetary authority uses the actual, expected...

We study a family of monopoly models for markets characterized by time-varying demand functions, in which a boundedly rational agent chooses output levels on the basis of a gradient adjustment mechanism. After presenting the model for a generic framework, we analytically study the case of cyclically alternating demand functions. We show that both t...

In the rank of behavioral rules, imitation-based heuristics has received special attention in economics (see [14] and [12]). In particular, imitative behavior is considered in order to understand the evidences arising in experimental oligopolies which reveal that the Cournot-Nash equilibrium does not emerge as unique outcome and show that an import...

In this paper we study a simple model based on the cobweb demand-supply framework with costly innovators and free imitators. The evolutionary selection between technologies depends on a performance measure which is related to the degree of memory. The resulting dynamics is described by a two-dimensional map. The map has a fixed point which may lose...

In this paper, we show how a rich variety of dynamics may arise in a simple multiplier–accelerator model when a nonlinearity is introduced in the investment function. A specific sigmoidal functional form is used to model investments with respect to the variation in national income, in order to bound the level of investments. In fact, due to obvious...

The purpose of this paper is to propose a symmetric two player general contest model in order to study the relationship between equilibria and crucial structural parameters of the model. In particular, given a general specification of the players’ set of possible entries, of the agents’ utility functions, and of the rules that presides over outcome...

In this work we study oligopoly models in which firms adopt decision mechanisms based on best response techniques with different rationality degrees. Firms are also assumed to face resource or financial constraints in adjusting their production levels, so that, from time to time, they can only increase or decrease their strategy by a bounded quanti...

We analyze a simple model based on the cobweb demand-supply framework with costly innovators and free imitators and study the endogenous dynamics of price and firms’ fractions in a homogeneous good market. The evolutionary selection between technologies depends on a performance measure in which a memory parameter is introduced. The resulting dynami...

We present a dynamic model for a boundedly rational monopolist who, in a partially known environment, follows a rule-of-thumb learning process. We assume that the production activity is continuously carried out and that the costly learning activity only occurs periodically at discrete time periods, so that the resulting dynamical model consists of...

We develop a model with intra-generational consumption externalities, based on the overlapping generation model by Diamond (1965). More specifically, we consider a two-period lived overlapping generation economy, assuming that the utility of each consumer depends also on the average consumption level by the consumers in the same generation. We supp...

In this paper we investigate the dynamics of a Cournot duopoly game with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update the quantity produced in each period. As in Ahmed et al. (2015), the demand functions are derived from an underlying CES utility function. The present analysis reveals that a...

In this paper, we investigate the dynamic properties of an overlapping generations' model with capital accumulation, in which agents work in both periods of life. We compare three different expectation mechanisms: perfect foresight, myopic foresight, and adaptive expectations, focusing, in particular, on this last one. We show that the steady...

We propose an exchange economy evolutionary model with agents heterogeneous in the structure of preferences. Assuming that the share updating mechanism is non-monotone in the calorie intake, we find multistability phenomena involving equilibria characterized by the coexistence of heterogeneous agents.

The simple pure exchange model with two individuals and two goods by Day and Pianigiani (1991), extensively analyzed by Day (1994, Ch. 10) and taken up again by Mukherjy (1999), is discussed and extended with the purpose of showing that chaos in a discrete tâtonnement process of this kind can be controlled if the auctioneer uses a smooth, nonlinear...

We investigate the dynamics of a cobweb type model with nonlinear demand and supply curves in which producers make forecasts on future prices with a backward looking expectation formation mechanism: the expected price for the next period is obtained by a weighted average of the prices observed in the last two periods. The study herewith presents ai...

We analyze the recurrence x n+1 = f (zn), where zn is a weighted power mean of x 0 ,. .. , xn, which has been proposed to model a class of non-linear forward-looking economic models with bounded rationality. Under suitable hypotheses on weights, we prove the convergence of the sequence xn. Then, to simulate a fading memory, we consider exponentiall...

We introduce and study a family of discrete-time dynamical systems to model binary choices based on the framework proposed by Schelling in 1973. The model we propose uses a gradient-like adjustment mechanism by means of a family of smooth maps and allows understanding and analytically studying the phenomena qualitatively described by Schelling. In...

In the present paper we propose a model in which the real side of the economy, described via a Keynesian good market approach, interacts with the stock market with heterogeneous speculators, i.e., optimistic and pessimistic fundamentalists, that respectively overestimate and underestimate the reference value due to a belief bias. Agents may switch...

In this paper we study some dynamical features of electricity markets modelling demand and supply by means of a nonlinear cobweb model. We consider a demand function periodically perturbed to take into account the real world seasonalities (daily, weekly and yearly) while supply function can include a stochastic term that is able to encompass possib...

In the present paper, we consider a nonlinear financial market model in which, in order to decrease the complexity of the dynamics and to achieve price stabilization, we introduce a price variation limiter mechanism, which in each period bounds the price variation so that the current price is forced to belong to a certain interval determined by the...

We study a monopolistic market characterized by a constant elasticity demand function, in which the firm technology is described by a linear total cost function. The firm is assumed to be boundedly rational and to follow a gradient rule to adjust the production level in order to optimize its profit. We focus on what happens on varying the price ela...

In this paper, we propose and compare three heterogeneous Cournotian duopolies, in which players adopt best response mechanisms based on different degrees of rationality. The economic setting we assume is described by an isoelastic demand function with constant marginal costs. In particular, we study the effect of the rationality degree on stabilit...

In this paper, we propose a financial market model with heterogeneous speculators, i.e., optimistic and pessimistic fundamentalists that, respectively, overestimate and underestimate the true fundamental value due to ambiguity in the stock market, which prevents them from relying on the true fundamental value in their speculations. Indeed, we assum...

We study complex phenomena arising from a simple optimal choice consumer model, starting from the classical framework in Benhabib and Day (Rev Econ Stud 48(3):459–471, 1981). We introduce elements of increasing complexity (dynamic adjustment processes, nonlinear social interdependence, agents heterogeneity, agents local interaction) and we investig...

We study a heterogeneous duopolistic Cournotian game, in which the firms, producing a homogeneous good, have reduced rationality and respectively adopt a “Local Monopolistic Approximation” (LMA) and a gradient-based approach with endogenous reactivity, in an economy characterized by isoelastic demand function and linear total costs. We give conditi...

In this paper we study oligopolies of generic size consisting of heterogeneous firms, which adopt best response adjustment mechanisms with either perfect foresight (rational firms) or static expectations (naive firms). Assuming an isoelastic demand function and possibly different marginal costs for the two groups of firms, we focus on the local sta...

We study heterogeneous Cournot oligopolies of variable sizes and compositions, in which the firms have different degrees of rationality, being either rational firms with perfect foresight or naive best response firms with static expectations. Each oligopoly can be described using its size and composition, that is, the fraction of firms that are rat...

We study a triopoly game with heterogeneous players. The market is characterized by a nonlinear (isoelastic) demand function and three competitors. The main novelty is the double route to complex dynamics that we find and is quite rare in heterogeneous triopoly models. We show that the two routes have important implications for the economic interpr...

We develop a macroeconomic behavioral model in order to analyze the interactions between product and financial markets. The real subsystem is represented by a simple Keynesian income–expenditure model, while the financial subsystem is represented by an equilibrium stock market with heterogeneous speculators, i.e., chartists and fundamentalists. The...

We propose a framework to analyse the dynamical process of decision and opinion formation of two economic homogeneous and boundedly rational agents that interact and learn from each other over time. The decisional process described in our model is an adaptive adjustment mechanism in which two agents take into account the difference between their ow...

We analyze a duopolistic Cournotian game with firms producing a homogeneous good, isoelastic demand function and linear total cost functions. In this economic setting, the traditional dynamic adjustment based on the classical best reply mechanism is very demanding in terms of rationality and information set. Therefore, in the competition we study,...

In this paper, we show how a rich variety of dynamical behaviors can emerge in the standard Keynesian income-expenditure model when a nonlinearity is introduced, both in the cases with and without endogenous government spending. A specific sigmoidal functional form is used for the adjustment mechanism of income with respect to the excess demand, in...

In contrast with the canonical models, Naimzada and Ricchiuti (2008, 2009) show that the interaction of groups of agents who have the same trad-ing rule but present different beliefs about the fundamental value could be a source of instability in financial markets. Differently from Naimzada and Ricchiuti (2008, 2009), we assume the market maker emp...

In this paper we propose a model describing the dynamical process of decision and opinion formation of two economic homogeneous interacting and boundedly rational agents. The decisional process represented in our model is given by an adaptive adjustment mechanism in which two agents take into account the difference between their own opinion and the...

We study the existence of equilibrium trajectories under least squared learning that are not those arising from rational expectations. In partic-ular we introduce the notion of "Relevant Learning Equilibria". Bullard (1994) [1] shows the existence of equilibria under learning, called "Learn-ing Equilibria", among which it is possible to …nd equilib...

The aim of this paper is to investigate the dynamics of the fashion cycle as originally described by Simmel (1904). The theoretical models used in the more recent economic literature (Coelho and McClure, 1993; Corneo and Jeanne, 1997; Karni and Schmeidler, 1990; Matsuyama, 1992; Stigler and Becker, 1977) have the undeniable advantage of making the...

We develop a model with intra-generational consumption externalities, based on the overlapping generation version of Diamond (1965) model. More specifically, we consider a two-period lived overlapping generation economy, assuming that the utility of each consumer depends also on the average level of consumption by other consumers in the same genera...

In this paper we propose a framework in order to analyze the dynamical process of decision and opinion formation of two economic homogeneous and boundedly rational agents that interact and learn from each other over time. The decisional process described in our model is an adaptive adjustment mechanism in which two agents take into account the diff...

We develop a macroeconomic behavioral model in order to analyze the interac-tions between real and financial markets. The real subsystem is represented by a simple Keynesian income-expenditure model, while the financial subsystem is represented by an equilibrium stock market with heterogeneous speculators, i.e., chartists and fundamentalists. The i...

a b s t r a c t This paper examines the interactions between economic activity and consumption externalities in an overlapping generations model. Existence of multiple steady states is studied from a global point of view, and possible mechanisms producing cycles and chaotic behavior are analyzed. Wealth-sensitive positionality is found to be able t...

In the last decade an increasing number of theoretical works that analyzed the dynamics of financial markets have given a clear demonstration that the representative agent "is not simply an analytical convenience as often explained, but is both unjustified and leads to conclusions which are usually misleading and often wrong" [kirman, 1992]. In the...

a b s t r a c t This note is an answer to a previous model on conformity in public goods contributions developed by Carpenter (2004), where a population evolution is allowed according to the standard replicator dynamic (Taylor and Jonker, 1978; Maynard Smith, 1982). To confirm his theoretical prediction, Carpenter devel-oped an experiment showing t...

In this paper we consider a Cournot–Bertrand duopoly model with linear demand and cost functions and with product differentiation. We propose a dynamic framework for the study of the stability properties of this kind of mixed oligopoly game, a rather neglected topic in the existing literature despite its relevance. In particular, in this paper we h...