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We study the effects of the investment horizon on asset price volatility using a Learning to Forecast laboratory experiment. We find that, for short investment horizons, participants coordinate on self-fulfilling trend-extrapolating predictions. Price deviations are then reinforced and amplified, possibly leading to large bubbles and crashes in asset prices. For longer investment horizons such bubbles do not emerge and price volatility tends to be lower. This is due to the fact that, for longer horizons, there is more dispersion in participants’ forecasts, and participants extrapolate trends in past prices to a lesser extent. We also show that, independent of the investment horizon, if the initial history of asset prices is already relatively stable before participants start their prediction task, price volatility remains small, with prices close to their fundamental values for the duration of the experiment.

We extend the beauty contest game to two dimensions: each player chooses two numbers to be as close as possible to certain target values, which are linear functions of the averages of the two number choices. One of the targets depends on the averages of both numbers, making the choices interrelated. We report on an experiment where we vary the eigenvalues of the associated two-dimensional linear system and find that subjects can learn the Pareto-optimal Nash Equilibrium of the system if both eigenvalues are stable and cannot learn it if both eigenvalues are unstable. Interestingly, subjects can also learn it if the system has the saddlepath property – with one stable and one unstable eigenvalue — but only if the one unstable eigenvalue is negative. We show theoretically that our results cannot be explained by homogeneous level-k models where all agents apply the same level k depth of reasoning to their choices, including the naïve learning model. However, our results can be explained by a mixed cognitive-levels model, including the adaptive learning model. We also run a horserace between many models used in the literature with the winner being a simple mixed model with levels 0, 1, and equilibrium reasoning.

We analyze trading in a modified continuous double auction market. We study how more or less information about trading in a prior round affects allocative and informational efficiency. We find that more information reduces allocative efficiency in early rounds relative to less information but that the difference disappears in later rounds. Informational efficiency is not affected by the information differences. We complement the experiment with simulations of the Individual Evolutionary Learning model which, after modifications to account for the CDA, seems to fit the data reasonably well.

Highlights • Extended Sustainable Supply Chain includes heterogeneous consumers agents • Adaptive consumer behavior informs farmers expectations and organic farming adoption • Propagation of consumer organic preferences through agro-food supply chain is slow • Policies targeting social norms, not incomes, lead to more organic adoption • A transition to pro-environmental behavior is more volatile under social policies, The current intense food production-consumption is one of the main sources of environmental pollution and contributes to anthropogenic greenhouse gas emissions. Organic farming is a potential way to reduce environmental impacts by excluding synthetic pesticides and fertilizers from the process. Despite ecological benefits, it is unlikely that conversion to organic can be financially viable for farmers, without additional support and incentives from consumers. This study models the interplay between consumer preferences and socio-environmental issues related to agriculture and food production. We operationalize the novel concept of extended agro-food supply chain and simulate adaptive behavior of farmers, food processors, retailers, and customers. Not only the operational factors (e.g., price, quantity, and lead time), but also the behavioral factors (e.g., attitude, perceived control, social norms, habits, and personal goals) of the food suppliers and consumers are considered in order to foster organic farming. We propose an integrated approach combining agent-based, discrete-event, and system dynamics modeling for a case of wine supply chain. Findings demonstrate the feasibility and superiority of the proposed model over the traditional sustainable supply chain models in incorporating the feedback between consumers and producers and analyzing management scenarios that can urge farmers to expand organic agriculture. Results further indicate that demand-side participation in transition pathways towards sustainable agriculture can become a time-consuming effort if not accompanied by the middle actors between consumers and farmers. In practice, our proposed model may serve as a decision-support tool to guide evidence-based policymaking in the food and agriculture sector.

An asset pricing model with chartists, fundamentalists and trend followers is considered. A market maker adjusts the asset price in the direction of the excess demand at the end of each trading session. An exogenously given fundamental price discriminates between a bull market and a bear market. The buying and selling orders of traders change moving from a bull market to a bear market. Their asymmetric propensity to trade leads to a discontinuity in the model, with its deterministic skeleton given by a two-dimensional piecewise linear dynamical system in discrete time. Multiple attractors, such as a stable fixed point and one or more attracting cycles or cycles and chaotic attractors, appear through border collision bifurcations. The multi-stability regions are underlined by means of two-dimensional bifurcation diagrams, where the border collision bifurcation curves are detected in analytic form at least for basic cycles with symbolic sequences $${\hbox {LR}}^{n}$$ and $${\hbox {RL}}^{n}$$ . A statistical analysis of the simulated time series of the asset returns, generated by perturbing the deterministic dynamics with a random walk process, indicates that this is one of the simplest asset pricing models which are able to replicate stylized empirical facts, such as excess volatility, fat tails and volatility clustering.

This paper offers an overview of the literature on the economic and financial applications of theory of nonlinear dynamics, especially bifurcation theory. After a short introductory discussion of the first nonlinear dynamic models in social sciences and the economic relevance of the zoo of bifurcations and complicated dynamics that such models can generate, we present an overview of the literature on nonlinear dynamic models in the areas of underdevelopment, environmental poverty traps, the management of common goods, industrial organization and financial markets. The review of the literature is enriched by reflections and ideas for future research.

In this paper we address the question of how individuals form expectations and invent, reinforce, and update their forecasting rules in a complex world. We do so by fitting a novel, parsimonious, and empirically validated genetic algorithm learning model with explicit heterogeneity in expectations to a set of laboratory experiments. Agents use simple linear first order price forecasting rules, adapting them to the complex evolving market environment with a Genetic Algorithm optimization procedure. The novelties are: (1) a parsimonious experimental foundation of individual forecasting behavior; (2) explanation of individual and aggregate behavior in three different experimental settings, (3) improved one- and 50-period ahead forecasting of experiments, and (4) characterization of the mean, median, and empirical distribution of forecasting heuristics. The median of the distribution of GA forecasting heuristics can be used in designing or validating simple Heuristic Switching Models.

© 2018 Elsevier B.V. The paper studies an oligopoly game, where firms can choose between price-taking and price-making strategies. On a mixed market price takers are always better off than price makers, though the profits of both types decline in the number of price takers. We investigate and confront two possibilities of firms’ decisions about their types: forward-looking equilibrium reasoning and backward-looking individual learning. We find that the Cournot outcome is the only equilibrium prediction and it is learnable if firms are sufficiently sensitive to profit differences. However, with a larger number of firms, a unilateral deviation from Cournot behavior becomes profitable. Under learning this incentive creates a space for permanent oscillations over different markets with a positive but low number of price takers.

© 2017, Springer-Verlag GmbH Germany. The moment of order submission plays an important role for the trading outcome in a Continuous Double Auction; submitting an offer at the beginning of the trading period may yield a lower profit, as the trade is likely to be settled at the own offered price, whereas late offers result in a lower probability of trading. This timing problem makes the order submission strategy more difficult. We extend the behavioral model of Individual Evolutionary Learning to incorporate the timing problem and study the limiting distribution of submission moments and the resulting offer function that maps submission moments to offers. We find that traders submit different offers at different submission moments the distribution of which uni-modal with a peak moving from late to early as the market size increases. This behavior exacerbates efficiency loss from learning. If traders evaluate profitability of their strategies over longer history, orders are submitted later with the same effect of market size.

We introduce a planar beauty contest game where agents must simultaneously guess two, endogenously determined variables, a and b. The system of equations determining the actual values of a and b is a coupled system; while the realization of a depends on the average forecast for a, as in a standard beauty contest game, the realization of b depends on both the average forecasts for a and for b. Our aim is to better understand the conditions under which agents learn the steady state of such systems and whether the eigenvalues of the system matter for the convergence or divergence of this learning process. We find that agents are able to learn the steady state of the system when the eigenvalues are both less than 1 in absolute value (the sink property) or when the steady state is saddlepath stable with the one root outside the unit circle being negative. By contrast, when the steady state exhibits the source property (two unstable roots) or is saddlepath stable with the one root outside the unit circle being positive, subjects are unable to learn the steady state of the system. We show that these results can be explained by either an adaptive learning model or a mixed cognitive levels model, while other approaches, e.g., naive or homogeneous level-k learning, do not consistently predict whether subjects converge to or diverge away from the steady state.

We present a laboratory experiment designed to investigate the effect of the fee structure on mutual fund choice. We find that subjects tend to ignore periodic and small operating expenses fees and base their decisions on gross, instead of net, returns. A fee in the form of a, much larger, front-end load leads to lock-in into one of the funds. It is used by some subjects as a commitment device, but exacerbates the decision errors of other subjects. Although past returns do not convey information about future returns, return chasing helps explain subjects’ behavior. Ministry of Education (MOE) Nanyang Technological University We are grateful to the financial support from the FP7-EU-collaborative project Monetary and Fiscal Policy with heterogeneous agents(POLHIA) Grant no. 225048 and the Research Priority Area Behavioral Economics of the University of Amsterdam. Mikhail Anufriev and Jan Tuinstra acknowledge financial support from the Australian Research Council through Discovery Project DP140103501. Te Bao acknowledges the financial support from the Startup Grant of NTU (M4018740), Tier 1 Grant from MOE of Singapore (RG68/16) and National Natural Science Foundation of China (71803201, 71721001, 71773013 and 71873149). Angela Sutan acknowledges financial support for experiments from the PARI grant scheme from Bourgogne Franche Comté Region. Jan Tuinstra acknowledges financial support from Project BEAM (Behavioral and Experimental Analyses in Macro-finance) (NWO 464-15-143) which is partly financed by the Netherlands Organization for Scientific Research (NWO).

© 2016 Elsevier B.V. We run a laboratory experiment to study how human subjects switch between several profitable alternatives, framed as mutual funds, in order to provide a microfoundation for so-called heterogeneous agent models. The participants in our experiment have to choose repeatedly between two, three or four experimental funds. The time series of fund returns are exogenously generated prior to the experiment and participants are paid for each period according to the return of the fund they choose. For most cases participants’ decisions can be successfully described by a discrete choice switching model, often applied in heterogeneous agent models, provided that a predisposition toward one of the funds is included. The estimated intensity of choice parameter of the discrete choice model depends on the structure of the fund returns. In particular, it increases with correlation between past and future returns. This suggests human subjects do not myopically chase past returns, but are more likely to do so when past returns are more predictive of future returns, a feature that is absent in the standard heterogeneous agent models.

© 2018 Elsevier B.V. We present results from the first laboratory experiment on the seminal heuristic switching model introduced by Brock and Hommes (1997, 1998). Subjects choose between two alternatives, a sophisticated and stabilizing, but costly, heuristic, and a destabilizing, but cheap, heuristic, and are paid according to the performance of the chosen heuristic. Aggregate choices determine the evolution of a state variable and, consequently, the performance of both heuristics. Theoretically, an increase in the costs for the stabilizing heuristic generates instability and leads to endogenous fluctuations in both the state variable and the fraction of agents using that heuristic. We vary the costs of the stabilizing heuristic in the experiment and find that the predictions of the heuristic switching model are partially confirmed. For low costs the dynamics are stable. For high costs, the dynamics initially are unstable and exhibit the type of bubbles and crashes emblematic for the heuristic switching model. However, over time the pattern of bubbles and crashes disappears and the dynamics become more stable. We estimate a standard discrete choice model on aggregate choice data and observe that subjects have a tendency to become less sensitive to payoff differences when the environment is less stable, which has important implications for the application of heuristic switching models.

Recent turmoil on global financial markets has led to a discussion on which policy measures should or could be taken to stabilize financial markets. One such a measure that resurfaced is the imposition of short-selling constraints. It is conjectured that these short-selling constraints reduce speculative trading and thereby have the potential to stabilize volatile financial markets. The purpose of the current paper is to investigate this conjecture in a standard asset pricing model with heterogeneous beliefs. We model short-selling constraints by imposing trading costs for selling an asset short. We find that the local stability properties of the fundamental rational expectations equilibrium do not change when trading costs for short-selling are introduced. However, when the asset is overvalued, costs for short-selling increase mispricing and price volatility. © 2013 Elsevier B.V.

The presence of excess covariance in financial price returns is an accepted empirical fact: the price dynamics of financial assets tend to be more correlated than their fundamentals would justify. We advance an explanation of this fact based on an intertemporal equilibrium multi-assets model of financial markets with an explicit and endogenous price dynamics. The market is driven by an exogenous stochastic process of dividend yields paid by the assets that we identify as market fundamentals. The model is rather flexible and allows for the coexistence of different trading strategies. The evolution of assets price and traders' wealth is described by a high-dimensional stochastic dynamical system. We identify the equilibria of the model consistent with a baseline assumption of procedural rationality. We show that these equilibria are characterized by excess covariance in prices with respect to the dividend process. Moreover, we show that in equilibrium there is a positive expected marginal profit in choosing more risky portfolios. As a consequence, the evolutionary pressure generates a trend towards more remunerative strategies, which, in turn, increase the variance of prices and the dynamic instability of the system. © 2012 Elsevier B.V..

An economic environment is a feedback system, where the dynamics of aggregate variables depend on individual expectations and vice versa. The type of feedback mechanism is crucial for the aggregate outcome. Experiments with human subjects (Heemeijer et al., J Econ Dyn Control 33:1052-1072, 2009) have shown that price converges to the fundamental level in a negative feedback environment but fails to do so under positive feedback. We present an explanation of these experimental results by means of a model of evolutionary switching between heuristics. Active heuristics are chosen endogenously, on the basis of their past performance. Under negative feedback an adaptive heuristic dominates explaining fast price convergence, whereas under positive feedback a trend-following heuristic dominates resulting in persistent price deviations and oscillations. © 2011 The Author(s).

We analyze the endogenous price formation mechanism of a pure exchange economy with two assets, riskless and risky. The economy is populated by an arbitrarily large number of traders whose investment choices are described by means of generic smooth functions of past realizations. These choices can be consistent with (but not limited to) the solutions of expected utility maximization problems. Under the assumption that individual demand for the risky asset is expressed as a fraction of individual wealth, we derive a complete characterization of equilibria. It is shown that irrespectively of the number of agents and of their behavior, all possible equilibria belong to a one-dimensional “Equilibrium Market Curve”. This geometric tool helps to illustrate the possibility of different phenomena, as multiple equilibria, and can be used for comparative static analysis. We discuss the relative performances of different strategies and the selection principle governing market dynamics on the basis of the stability analysis of equilibria.

We study a model in which individual agents use simple linear first order price forecasting rules, adapting them to the complex evolving market environment with a smart Genetic Algorithm optimization procedure. The novelties are: (1) a parsimonious experimental foundation of individual forecasting behaviour; (2) an explanation of individual and aggregate behavior in four different experimental settings, (3) improved one-period and 50-period ahead forecasting of lab experiments, and (4) a characterization of the mean, median and empirical distribution of forecasting heuristics. The median of the distribution of GA forecasting heuristics can be used in designing or validating simple Heuristic Switching Model.

We consider a pure exchange economy where one risky and one riskless security are traded in discrete time. Individual demands are expressed as fractions of individual wealth and depend on traders' forecasts about future price movement. Introducing the 'Equilibrium Market Line' as the locus of all possible equilibrium returns, we show that, irrespectively of the number of traders and of their investment behavior, the economy possesses isolated equilibria where a single agent dominates the market and continuous manifolds of equilibria where many agents hold finite wealth shares. Moreover, we prove that no global dominance order relation among strategies can be defined. © 2006 Elsevier B.V. All rights reserved.

This paper stresses the importance of heterogeneity in learning rules. We introduce an evolutionary competition between different learning rules and demonstrate that, though these rules can coexist, their convergence properties are strongly affected by heterogeneity. We consider a Bertrand oligopoly with differentiated goods. Firms do not have full information about the demand structure and they want to maximize their perceived one-period profit by applying one of two different learning rules: OLS learning and gradient learning. We analytically show that the stability of gradient learning depends on the distribution of learning rules over firms. In particular, as the number of gradient learners increases, gradient learning may become unstable. We study evolutionary competition between the learning rules by means of computer simulations and illustrate that this change in stability for gradient learning may lead to cyclical switching between the rules. Stable gradient learning typically gives higher average profit than OLS learning, making firms switch to gradient learning. This however, destabilizes gradient learning which, because of decreasing profits, makes firms switch back to OLS learning. This cycle may repeat itself indefinitely.

We consider an analytically tractable asset pricing model describing the trading activity in a stylized market with two assets. Traders are boundedly rational expected utility maximizers with different beliefs about future prices and different investment horizons. In particular, we analyze the effects of the latter source of heterogeneity on the dynamics of price. We find that in the case with homogeneous agents, longer investment horizons lead to more stable dynamics. This is not true, however, in the case of a mixed population of traders, when the increase of heterogeneity in the investment horizons can introduce instability in the system. Furthermore, the role of heterogeneity turns out to be different for different trading behaviors and its effect on the aggregate dynamics depends on the whole ecology of agents' beliefs. Copyright © 2012 De Gruyter.

In recent `learning to forecast' experiments with human subjects (Hommes, et al. 2005), three different patterns in aggregate price behavior have been observed: slow monotonic convergence, permanent oscillations and dampened fluctuations. We show that a simple model of individual learning can explain these different aggregate outcomes within the same experimental setting. The key idea of the model is the evolutionary selection among heterogeneous expectation rules, driven by the relative performance of the rules. Out-of-sample predictive power of our switching model is higher compared to the rational or other homogeneous expectations benchmarks. Our results show that heterogeneity in expectations is crucial to describe individual forecasting behavior as well as aggregate price behavior.

We present an experiment that investigates the effect of the fee structure and past returns on mutual fund choice. We find that subjects pay too little attention to the (periodic and small) operation expenses fee, but the more salient front-end load is used as a commitment device and leads to lock-in into one of the funds. In addition, even when subjects know that future returns are independent of past returns, these past returns are an important determinant of subjects' investment choices.

The dynamics of a financial market with heterogeneous agents are analyzed under different market architectures. We start with a tractable behavioral model under Walrasian market clearing and simulate it under different trading protocols. The key behavioral feature of the model is the switching by agents between simple forecasting rules on the basis of a fitness measure. By analyzing the dynamics under order-driven protocols we show that the behavioral and structural assumptions of the model are closely intertwined. The high responsiveness of agents to a fitness measure causes excess volatility, but the frictions of the order-driven markets may stabilize the dynamics. We also analyze and compare allocative efficiency and time series properties under different protocols.Keywords: Asset pricing model; Heterogeneous beliefs; Learning; Trading protocols; Market architectureJEL classification codes: G12; D44; D61; C62

The recent macroeconomic literature stresses the importance of managing heterogeneous expectations in the formulation of monetary policy. We use a simple frictionless DSGE model to investigate inflation dynamics under alternative interest rate rules when agents have heterogeneous expectations and update their beliefs based on past performance as in Brock and Hommes (1997). The stabilizing effect of different monetary policies depends on the ecology of forecasting rules (i.e., the composition of the set of predictors), on agents' sensitivity to differences in forecasting performance and on how aggressively the monetary authority sets the nominal interest rate in response to inflation. In particular, if the monetary authority only responds weakly to inflation, a cumulative process with rising inflation is likely. On the other hand, a Taylor interest rate rule that sets the interest rate more than point for point in response to inflation stabilizes inflation dynamics, but does not always lead the system to converge to the rational expectations equilibrium as multiple equilibria may persist.

This special issue of the Journal of Economic Dynamics and Control is a collection of 10 papers presented at a workshop on Complexity in Economics and Finance, held at the Lorentz center in Leiden, the Netherlands, on October 22-27, 2007. The benchmark approach to economic and financial economic theory is to assume the existence of a fully rational, representative agent that exists inside an economy with complete markets. In dynamic settings, this approach typically assumes that the representative agent maximizes her expected lifetime utility subject to a (perceived) budget constraint. One implication is that optimizing agents will be forward-looking, and hence, there will be a crucial role for expectations in individual decisions. It follows that depending on the nature of expectations, macroeconomic and financial outcomes will be impacted by the beliefs of agents. Although expectations play a primary role in macroeconomic and financial outcomes, there is no consensus on how agents actually form expectations. The Rational Expectations (RE) paradigm, still the benchmark model in economics, assumes that agents’ subjective expectations are consistent with the actual stochastic process generated by those beliefs. A large literature questions the RE hypothesis on theoretical and empirical grounds. Theoretically, it can be argued that RE are not realistic because of the strong cognitive and computational assumptions required for agents’ beliefs to be rational. To actually compute rational beliefs, agents would need to know the precise structure and laws of motion for the economy, even though this structure depends on agents’ beliefs (c.f. Evans and Honkapohja (2001)). Moreover, rational expectations models often may possess multiple equilibria, on which individual agents must coordinate. Empirical studies have shown that survey data and macroeconomic data are not consistent with rational expectations. Mankiw, Reis, and Wolfers (2003) and Branch (2004) show that survey data on inflation expectations exhibit a wide degree of time-varying heterogeneity that can not be generated by a representative rational expectation. Milani (2007) finds that a simple New Keynesian model where rational expectations are replaced by an adaptive learning rule is preferred by the data to the best fitting model solved under the assumption of rational expectations. The standard economic approach also fails to explain many important features of economic systems, e.g., the emergence of speculative bubbles and crashes in financial markets, fat tails, long memory and clustered volatility in the returns distribution of

This paper demonstrates how both quantitative and qualitative results of general, analytically tractable asset-pricing model in which heterogeneous agents behave consistently with a constant relative risk aversion assumption can be applied to the particular case of ``linear'' investment choices. In this way it is shown how the framework developed in Anufriev and Bottazzi (2005) can be used inside the classical setting with demand derived from utility maximization. Consequently, some of the previous contributions of the agent-based literature are generalized. In the course of the analysis of asymptotic market behavior the main attention is paid to a geometric approach which allows to visualize all possible equilibria by means of a simple one-dimensional curve referred as the Equilibrium Market Line. The case of linear (particularly, mean-variance) investment functions thoroughly analyzed in this paper allows to highlight those features of the asymptotic dynamics which are common to all types of the CRRA-investment behavior and those which are specific for the linear investment functions.

In this paper we propose an explanation of the findings of a recent laboratory market forecasting experiment. In the experiment the participants were asked to predict prices for 50 periods on the basis of past realizations. Three different aggregate outcomes were observed in an identical environment: slow monotonic price convergence, persistent price oscillations, and oscillatory dampened price fluctuations. Individual predictions exhibited a high degree of coordination, although the individual forecasts were not commonly known. To explain these findings we propose an evolutionary model of reinforcement learning over a set of simple forecasting heuristics. The key element of our model is the switching between heuristics on the basis of their past performance. Simulations show that such evolutionary learning can reproduce the qualitative patterns observed in the experiment. © 2008 Springer-Verlag Berlin Heidelberg.

The paper analyzes the dynamics in a model with heterogeneous agents trading in simple markets under different trading protocols. Starting with the analytically tractable model of [4], we build a simulation platform with the aim to investigate the impact of the trading rules on the agents' ecology and aggregate time series properties. The key behavioral feature of the model is the presence of a finite set of simple beliefs which agents choose each time step according to a fitness measure. The price is determined endogenously and our focus is on the role of the structural assumption about the market architecture. Analyzing dynamics under such different trading protocols as the Walrasian auction, the batch auction and the 'order-book' mechanism, we find that the resulting time series are similar to those originating from the noisy version of the model [4]. We distinguish the randomness caused by a finite number of agents and the randomness induced by an order-based mechanisms and analyze their impact on the model dynamics. © 2006 Springer-Verlag Berlin Heidelberg.

In this paper we analyze a dynamic, asset pricing model where an arbitrary number of heterogeneous, procedurally rational investors divide their wealth between two assets. Both fundamental dividend process and behavior of traders are modeled in a very general way. In particular, agents' choices are described by means of the generic smooth functions defined on a commonly available information set. The choices are consistent with (but not limited to) the solutions of the expected utility maximization problems. As a natural rest point of the corresponding dynamics we propose the notion of the Behavioral Consistent Equilibria (BCE) where the aggregate dynamics are consistent with the agents' investment choices. We show that provided that the dividend process is given, all possible equilibria of the system can be characterized by means of one-dimensional Equilibrium Market Line (EML). This geometric tool allows to separate the effects of dividend process and agents' behaviors on the aggregate dynamics. Namely, the precise shape of this line depends on the character of the dividend process, but the realized equilibrium, i.e.~a point on the line, is determined by the ecology of agents' behaviors. We argue that the EML can be useful in investigation of the questions of existence, multiplicity and stability of the BCE and provide corresponding examples. The EML also allows to make the comparative static exercises in a framework with heterogeneous agents and discuss the relative performances of different strategies. The notion of BCE can be considered as a generalization of the Rational Expectations Equilibrium on the framework with heterogeneous traders. It can be, therefore, useful also in other fields of economics where heterogeneity of actors plays an important role for the aggregate outcome

This paper analyzes to what extent and how the trading activity of a group of heterogeneous agents can be described, in the aggregate, as the result of the investment decision of a single “representative” agent. We consider a two-asset pure exchange economy populated by CRRA traders whose individual demands are functions of the past market history. If individual choices are expressed as noisy versions of a common behavior, and the number of agents is large, one can consider the Large Market Limit of the economy and reduce the model to a low-dimensional stochastic system. We investigate the goodness of this approximation under different market conditions and different agents ecologies. The results of the analysis can be used in the study of the general case with an arbitrary number of heterogeneous agents.

We provide explanations for the results of the Levy, Levy and Solomon model, a recent simulation model of financial markets. These explanations are based upon mathematical analysis of a dynamic model of a market with an arbitrary number of heterogeneous investors allocating their wealth between two assets. The investors' choices are endogenously modeled in a general way and, in particular, consistent with the maximization of an expected utility. We characterize the equilibria of the model and their stability and discuss implications for the market return and agents' survival. These implications are in agreement with the results of previous simulations. Thus, our analytic approach allows to explore the robustness of the previous analysis and to expand its spectrum. © 2006 Springer-Verlag Berlin Heidelberg.

An agent-based model of a simple financial market with arbitrary number of traders having relatively general behavioral specifications is analyzed. In a pure exchange economy with two assets, riskless and risky, trading takes place in discrete time under endogenous price formation setting. Traders' demands for the risky asset are expressed as fractions of their individual wealths, so that the dynamical system in terms of wealth and return is obtained. Agents' choices, i.e. investment fractions, are described by means of the generic smooth functions of an infinite information set. The choices can be consistent with (but not limited to) the solutions of the expected utility maximization problems. A complete characterization of equilibria is given. It is shown that irrespectively of the number of agents and of their behavior, all possible equilibria belong to a one-dimensional "Equilibrium Market Line". This geometric tool helps to illustrate possibility of different phenomena, like multiple equilibria, and also can be used for comparative static analysis. The stability conditions of equilibria are derived for general model specification and allow to discuss the relative performances of different strategies and the selection principle governing market dynamics.

We consider a simple pure exchange economy with two assets, one riskless, yielding a constant return, and one risky, paying a stochastic dividend, and we assume trading to take place in discrete time inside an endogenous price formation setting. Traders demand for the risky asset is expressed as a fraction of their individual wealth and is based on future prices forecast obtained on the basis of past market history. We describe the evolution of price and wealth distribution in the general case where any number of heterogeneous traders is allowed to operate in the market and any smooth function which maps the infinite information set to the present investment choice is allowed as agent's trading strategy. We give a complete characterization of equilibria and derive stability conditions analyzing a dynamical system of arbitrary large dimension. We show that this system can only possess isolated generic equilibria where a single agent dominates the market and continuous manifolds of non-generic equilibria where many agents hold finite wealth shares. Irrespectively of agents number and of their behavior, we show that all possible equilibria returns belong to a one dimensional ``Equilibria Market Line''. Our general result extends previous contributions and allows a better understanding of the selection principle governing the asymptotic market dynamics

We consider a simple pure exchange economy with two assets, one riskless, yielding a constant return on investment, and one risky, paying a stochastic dividend. Trading takes place in discrete time and in each trading period the price of the risky asset is fixed through the market clearing condition. Individual demands are expressed as fractions of traders wealth and depend on traders forecasts about future price movement. Under these assumptions, we derive the stochastic dynamical system that describes the evolution of price and wealth. We study the cases in which one or two agents operate in the market, identifying the possible equilibria and discussing their stability conditions. The main novelty of this paper rests in the abstraction from the precise characterization of agents' beliefs and preferences. In this respect our results generalize several previous contributions in the field. In particular, we show that, irrespectively of agents' behavior, the system can only possess isolated generic equilibria where a single agent dominates the market and continuous manifolds of non-generic equilibria where heterogeneous agents hold finite shares of the aggregate wealth. Moreover, we show that all possible equilibria belong to a one dimensional "Equilibria Market Line". Finally we discuss the role of different parameters for the stability of equilibria and the selection principle governing market dynamics.

We consider a simple pure exchange economy with two assets, one riskless, yielding a constant return, and one risky, paying a stochastic dividend, and we assume trading to take place in discrete time inside an endogenous price formation setting. Traders demand for the risky asset is expressed as a fraction of their individual wealth and is based on future prices forecast obtained on the basis of past market history. The general case is studied in which an arbitrary large number of heterogeneous traders operates in the market and any smooth function which maps the infinite information set to the present investment choice is allowed as agent's trading behavior. A complete characterization of equilibria is given and their stability conditions are derived. We find that this economy can only possess isolated generic equilibria where a single agent dominates the market and continuous manifolds of non-generic equilibria where many agents hold finite wealth shares. We show that irrespectively of agents number and of their behavior, all possible equilibria belong to a one dimensional "Equilibria Market Line". Finally we discuss the relative performances of different strategies and the selection principle governing market dynamics.

In this paper we study the dynamics of a simple asset pricing model describing the trading activity of heterogeneous agents in a "stylized" market. The economy in the model contains two assets: a bond with risk-less return and a dividend paying stock. The price of the stock is determined through market clearing condition. Traders are speculators described as expected utility maximizers with heterogeneous beliefs about future stock price and with heterogeneous estimation of risk. In particular, we consider traders who base their investment decision on different time horizons and we analyze the effect of these differences on the price dynamics. Under suitable parameterization, the stock no-arbitrage "fundamental" price can emerge as a stable fixed point of the model dynamics. For different parameterizations, however, the market shows cyclical or chaotic price dynamics with speculative bubbles and crashes. We find that the sole heterogeneity of agents with respect to their time horizons is not enough to guarantee the instability of the fundamental price and the emergence of non-trivial price dynamics. However, if different groups of agents are characterized by different trading behaviors, the introduction of heterogeneous investment horizons can help to decrease the stability region of the "fundamental" fixed point. The role of time horizons turns out to be different for different trade behaviors and, in general, depends on the whole ecology of agents' beliefs. We demonstrate this effect discussing a case in which the increase of fundamentalists time horizons can lead to cyclical or chaotic price behavior, while the same increase for the chartists helps to stabilize the fundamental price.

In this paper we study the dynamics of a simple asset pricing model describing the trading activity of heterogeneous agents in a "stylized" market. The economy in the model contains two assets: a bond with risk-less return and a dividend paying stock. The price of the stock is determined through market clearing condition. Traders are speculators described as expected utility maximizers with heterogeneous beliefs about future stock price and with heterogeneous estimation of risk. In particular, we consider traders who base their investment decision on different time horizons and we analyze the effect of these differences on the price dynamics. Under suitable parameterization, the stock no-arbitrage "fundamental" price can emerge as a stable fixed point of the model dynamics. For different parameterizations, however, the market shows cyclical or chaotic price dynamics with speculative bubbles and crashes. We find that the sole heterogeneity of agents with respect to their time horizons is not enough to guarantee the instability of the fundamental price and the emergence of non-trivial price dynamics. However, if different groups of agents are characterized by different trading behaviors, the introduction of heterogeneous investment horizons can help to decrease the stability region of the "fundamental" fixed point. The role of time horizons turns out to be different for different trade behaviors and, in general, depends on the whole ecology of agents' beliefs. We demonstrate this effect discussing a case in which the increase of fundamentalists time horizons can lead to cyclical or chaotic price behavior, while the same increase for the chartists helps to stabilize the fundamental price.

This discussion paper led to a publication in 'Macroeconomic Dynamics' (2013). Vol. 17(8), pp. 1574-1604. The recent macroeconomic literature stresses the importance of managing heterogeneous expectations in the formulation of monetary policy. We use a stylized macro model of Howitt (1992) to investigate inflation dynamics under alternative interest rate rules when agents have heterogeneous expectations and update their beliefs based on past performance as in Brock and Hommes (1997). The stabilizing effect of different monetary policies depends on the ecology of forecasting rules, on agents' sensitivity to differences in forecasting performance and on how aggressively the monetary authority sets the nominal interest rate in response to inflation. In particular, if the monetary authority only responds weakly to inflation, a cumulative process with rising inflation is likely. On the other hand, a Taylor interest rate rule that sets the interest rate more than point for point in response to inflation stabilizes inflation dynamics, but does not always lead the system to converge to the rational expectations equilibrium as multiple equilibria may persist, even when a fully rational, but costly, expectations rule is part of the ecology of forecasting strategies.