# Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

# Agent-Based Computational Economics

• Moshe Levy
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_6-7

## Keywords

microscopic simulation agent-based simulation financial markets bounded rationality heterogeneous expectations

## Definition of the Subject

Mainstream economic models typically make the assumption that an entire group of agents, e.g., “investors,” can be modeled with a single “rational representative agent.” While this assumption has proven extremely useful in advancing the science of economics by yielding analytically tractable models, it is clear that the assumption is not realistic: people are different one from the other in their tastes, beliefs, and sophistication, and as many psychological studies have shown, they often deviate from rationality in systematic ways.

Agent-based computational economics is a framework allowing economics to expand beyond the realm of the “rational representative agent.” Modeling and simulating the behavior of each agent and the interaction among agents agent-based simulation allows us to investigate the dynamics of complex economic systems with many heterogeneous and not necessarily fully rational agents.

The agent-based simulation approach allows economists to investigate systems that cannot be studied with the conventional methods. Thus, the following key questions can be addressed: How do heterogeneity and systematic deviations from rationality affect markets? Can these elements explain empirically observed phenomena which are considered “anomalies” in the standard economics literature? How robust are the results obtained with the analytical models? By addressing these questions the agent-based simulation approach complements the traditional analytical analysis and is gradually becoming a standard tool in economic analysis.

## Introduction

For solving the dynamics of two bodies (e.g., stars), with some initial locations and velocities and some law of attraction (e.g., gravitation), there is a well-known analytical solution. However, for a similar system with three bodies, there is no known analytical solution. Of course, this does not mean that physicists cannot investigate and predict the behavior of such systems. Knowing the state of the system (i.e., the location, velocity, and acceleration of each body) at time t allows us to calculate the state of the system an instant later, at time t + Δt. Thus, starting with the initial conditions, we can predict the dynamics of the system by simply simulating the “behavior” of each element in the system over time.

This powerful and fruitful approach, sometimes called “microscopic simulation,” has been adopted by many other branches of science. Its application in economics is best known as “agent-based simulation” or “agent-based computation.” The advantages of this approach are clear – they allow the researcher to go where no analytical models can go. Yet, despite of the advantages, perhaps surprisingly, the agent-based approach was not adopted very quickly by economists. Perhaps the main reason for this is that a particular simulation only describes the dynamics of a system with a particular set of parameters and initial conditions. With other parameters and initial conditions, the dynamics may be different. So economists may ask: what is the value of conducting simulations if we get very different results with different parameter values? While in physics the parameters (like the gravitational constant) may be known with great accuracy, in economics the parameters (like the risk-aversion coefficient or for that matter the entire decision-making rule) are typically estimated with substantial error. This is a strong point. Indeed, we would argue that the “art” of agent-based simulations is the ability to understand the general dynamics of the system and to draw general conclusions from a finite number of simulations. Of course, one simulation is sufficient as a counterexample to show that a certain result does not hold, but many more simulations are required in order to convince of an alternative general regularity.

This manuscript is intended as an introduction to agent-based computational economics. An introduction to this field has two goals: (i) to explain and to demonstrate the agent-based methodology in economics, stressing the advantages and disadvantages of this approach relative to the alternative purely analytical methodology, and (ii) to review studies published in this area. The emphasis in this entry will be on the first goal. While section “Some of the Pioneering Studies” does provide a brief review of some of the cornerstone studies in this area, more comprehensive reviews can be found in LeBaron (2000), Levy et al. (2000), Samanidou et al. (2007), and Tesfatsion (2001, 2002), in which part of section “Some of the Pioneering Studies” is based. A comprehensive review of the many entries employing agent-based computational models in economics will go far beyond the scope of this entry. To achieve goal (i) above, in section “Illustration with the LLS Model,” we will focus on one particular model of the stock market in some detail. Section “Summary and Future Directions” concludes with some thoughts about the future of the field.

## Some of the Pioneering Studies

### Schelling’s Segregation Model

Schelling’s (Schelling 1978) classical segregation model is one of the earliest models of population dynamics. Schelling’s model is not intended as a realistic tool for studying the actual dynamics of specific communities as it ignores economic, real-estate, and cultural factors. Rather, the aim of this very simplified model is to explain the emergence of macroscopic single-race neighborhoods even when individuals are not racists. More precisely, Schelling found that the collective effect of neighborhood racial segregation results even from individual behavior that presents only a very mild preference for same-color neighbors. For instance, even the minimal requirement by each individual of having (at least) one neighbor belonging to one’s own race leads to the segregation effect.

The agent-based simulation starts with a square mesh or lattice (representing a town) which is composed of cells (representing houses). On these cells reside agents which are either “blue” or “green” (the different races). The crucial parameter is the minimal percentage of same-color neighbors that each agent requires. Each agent, in his/her turn, examines the color of all his/her neighbors. If the percentage of neighbors belonging to his/her own group is above the “minimal percentage,” the agent does nothing. If the percentage of neighbors of his/her own color is less then the minimal percentage, the agent moves to the closest unoccupied cell. The agent then examines the color of the neighbors of the new location and acts accordingly (moves if the number of neighbors of his/her own color is below the minimal percentage and stays there otherwise). This goes on until the agent is finally located at a site in which the minimal percentage condition holds. After a while, however, it might happen that following the moves of the other agents, the minimal percentage condition ceases to be fulfilled and then the agent starts moving again until he/she finds an appropriate cell. As mentioned above, the main result is that even for very mild individual preferences for same-color neighbors, after some time the entire system displays a very high level of segregation.

A more modern, developed, and sophisticated reincarnation of these ideas is the Sugarscape environment described by Epstein and Axtell (1996). The model considers a population of moving, feeding, pairing, procreating, trading, warring agents and displays various qualitative collective events which their populations incur. By employing agent-based simulation, one can study the macroscopic results induced by the agents’ individual behavior.

### The Kim and Markowitz Portfolio Insurers Model

Harry Markowitz is very well known for being one of the founders of modern portfolio theory, a contribution for which he/she has received the Nobel Prize in economics. It is less well known, however, that Markowitz is also one of the pioneers in employing agent-based simulations in economics.

During the October 1987 crash, markets all over the globe plummeted by more than 20 % within a few days. The surprising fact about this crash is that it appeared to be spontaneous – it was not triggered by any obvious event. Following the 1987 crash, researchers started to look for endogenous market features, rather than external forces, as sources of price variation. The Kim-Markowitz (Kim and Markowitz 1989) model explains the 1987 crash as resulting from investors’ “Constant Proportion Portfolio Insurance” (CPPI) policy. Kim and Markowitz proposed that market instabilities arise as a consequence of the individual insurers’ efforts to cut their losses by selling once the stock prices are going down.

The Kim-Markowitz agent-based model involves two groups of individual investors: rebalancers and insurers (CPPI investors). The rebalancers are aiming to keep a constant composition of their portfolio, while the insurers make the appropriate operations to insure that their eventual losses will not exceed a certain fraction of the investment per time period.

The rebalancers act to keep a portfolio structure with (for instance) half of their wealth in cash and half in stocks. If the stock price rises, then the stocks weight in the portfolio will increase, and the rebalancers will sell shares until the shares again constitute 50 % of the portfolio. If the stock price decreases, then the value of the shares in the portfolio decreases, and the rebalancers will buy shares until the stock again constitutes 50 % of the portfolio. Thus, the rebalancers have a stabilizing influence on the market by selling when the market rises and buying when the market falls.

A typical CPPI investor has as his/her main objective not to lose more than (for instance) 25 % of his/her initial wealth during a quarter, which consists of 65 trading days. Thus, he/she aims to insure that at each cycle 75 % of the initial wealth is out of reasonable risk. To this effect, he/she assumes that the current value of the stock will not fall in one day by more than a factor of 2. The result is that he/she always keeps in stock twice the difference between the present wealth and 75 % of the initial wealth (which he/she had at the beginning of the 65-day investing period). This determines the amount the CPPI agent is bidding or offering at each stage. Obviously, after a price fall, the amount he/she wants to keep in stocks will fall and the CPPI investor will sell and further destabilize the market. After an increase in the prices (and personal wealth), the amount the CPPI agent wants to keep in shares will increase: he/she will buy and may support a price bubble.

The simulations reveal that even a relatively small fraction of CPPI investors (i.e., less than 50 %) is enough to destabilize the market, and crashes and booms are observed. Hence, the claim of Kim and Markowitz that the CPPI policy may be responsible for the 1987 crash is supported by the agent-based simulations. Various variants of this model were studied intensively by Egenter et al. (1999) who find that the price time evolution becomes unrealistically periodic for a large number of investors (the periodicity seems related with the fixed 65-day quarter and is significantly diminished if the 65-day period begins on a different date for each investor).

### The Arthur, Holland, Lebaron, Palmer, and Tayler Stock Market Model

Palmer et al. (1994) and Arthur et al. (1997) (AHLPT) construct an agent-based simulation model that is focused on the concept of coevolution. Each investor adapts his/her investment strategy such as to maximally exploit the market dynamics generated by the investment strategies of all others investors. This leads to an ever-evolving market, driven endogenously by the ever-changing strategies of the investors.

The main objective of AHLPT is to prove that market fluctuations may be induced by this endogenous coevolution, rather than by exogenous events. Moreover, AHLPT study the various regimes of the system: the regime in which rational fundamentalist strategies are dominating versus the regime in which investors start developing strategies based on technical trading. In the technical trading regime, if some of the investors follow fundamentalist strategies, they may be punished rather than rewarded by the market. AHLPT also study the relation between the various strategies (fundamentals vs. technical) and the volatility properties of the market (clustering, excess volatility, volume-volatility correlations, etc.).

In the first entry quoted above, the authors simulated a single stock and further limited the bid/offer decision to a ternary choice of: (i) bid to buy one share, (ii) offer to sell one share, or: (iii) do nothing. Each agent had a collection of rules which described how he/she should behave (i, ii, or iii) in various market conditions. If the current market conditions were not covered by any of the rules, the default was to do nothing. If more than one rule applied in a certain market condition, the rule to act upon was chosen probabilistically according to the “strengths” of the applicable rules. The “strength” of each rule was determined according to the rule’s past performance: rules that “worked” became “stronger.” Thus, if a certain rule performed well, it became more likely to be used again.

The price is updated proportionally to the relative excess of offers over demands. In Arthur et al. (1997), the rules were used to predict future prices. The price prediction was then transformed into a buy/sell order through the use of a Constant Absolute Risk-Aversion (CARA) utility function. The use of CARA utility leads to demands which do not depend on the investor’s wealth.

The heart of the AHLPT dynamics is the trading rules. In particular, the authors differentiate between “fundamental” rules and “technical” rules and study their relative strength in various market regimes. For instance, a “fundamental” rule may require market conditions of the type:
$$\mathrm{dividend}/\mathrm{current}\ \mathrm{price}>0.04$$
in order to be applied. A “technical” rule may be triggered if the market fulfills a condition of the type:
$$\mathrm{current}\ \mathrm{price}>10-\mathrm{period}\ \mathrm{moving}\ \mathrm{average}\ \mathrm{of}\ \mathrm{past}\ \mathrm{price}\mathrm{s}$$
The rules undergo genetic dynamics: the weakest rules are substituted periodically by copies of the strongest rules, and all the rules undergo random mutations (or even versions of “sexual” crossovers – new rules are formed by combining parts from two different rules). The genetic dynamics of the trading rules represent investors’ learning: new rules represent new trading strategies. Investors examine new strategies and adopt those which tend to work best. The main results of this model are:

## For a Few Agents, a Small Number of Rules, and Small Dividend Changes

• The price converges toward an equilibrium price which is close to the fundamental value.

• There are no bubbles, crashes, or anomalies.

• Agents follow homogeneous simple fundamentalist rules.

## For a Large Number of Agents and a Large Number of Rules

• There is no convergence to an equilibrium price, and the dynamics are complex.

• The price displays occasional large deviations from the fundamental value (bubbles and crashes).

• Some of these deviations are triggered by the emergence of collectively self-fulfilling agent price-prediction rules.

• The agents become heterogeneous (adopt very different rules).

• Trading volumes fluctuate (large volumes correspond to bubbles and crashes).

• The rules evolve over time to more and more complex patterns, organized in hierarchies (rules, exceptions to rules, exceptions to exceptions, and so on).

• The successful rules are time dependent: a rule which is successful at a given time may perform poorly if reintroduced after many cycles of market coevolution.

### The Lux and Lux and Marchesi Model

Lux (1998) and Lux and Marchesi (Levy et al. 1996) propose a model to endogenously explain the heavy-tailed distribution of returns and the clustering of volatility. Both of these phenomena emerge in the Lux model as soon as one assumes that in addition to the fundamentalists, there are also chartists in the model. Lux and Marchesi (Levy et al. 1996) further divide the chartists into optimists (buyers) and pessimists (sellers). The market fluctuations are driven and amplified by the fluctuations in the various populations: chartists converting into fundamentalists, pessimists into optimists, etc.

In the Lux and Marchesi model, the stock’s fundamental value is exogenously determined. The fluctuations of the fundamental value are inputted exogenously as a white noise process in the logarithm of the value. The market price is determined by investors’ demands and by the market clearance condition.

Lux and Marchesi consider three types of traders:
• Fundamentalists observe the fundamental value of the stock. They anticipate that the price will eventually converge to the fundamental value, and their demand for shares is proportional to the difference between the market price and the fundamental value.

• Chartists look more at the present trends in the market price rather than at fundamental economic values; the chartists are divided into:

• Optimists (they buy a fixed amount of shares per unit time)

• Pessimists (they sell shares)

Transitions between these three groups (optimists, pessimists, fundamentalists) happen with probabilities depending on the market dynamics and on the present numbers of traders in each of the three classes:
• The transition probabilities of chartists depend on the majority opinion (through an “opinion index” measuring the relative number of optimists minus the relative number of pessimists) and on the actual price trend (the current time derivative of the current market price), which determines the relative profit of the various strategies.

• The fundamentalists decide to turn into chartists if the profits of the latter become significantly larger than their own and vice versa (the detailed formulae used by Lux and Marchesi are inspired from the exponential transition probabilities governing statistical mechanics physical systems).

The main results of the model are:
• No long-term deviations between the current market price and the fundamental price are observed.

• The deviations from the fundamental price, which do occur, are unsystematic.

• In spite of the fact that the variations of the fundamental price are normally distributed, the variations of the market price (the market returns) are not. In particular the returns exhibit a frequency of extreme events which is higher than expected for a normal distribution. The authors emphasize the amplification role of the market that transforms the input normal distribution of the fundamental value variations into a leptokurtotic (heavy-tailed) distribution of price variation, which is encountered in the actual financial data.

• Clustering of volatility.

The authors explain the volatility clustering (and as a consequence, the leptokurticity) by the following mechanism. In periods of high volatility, the fundamental information is not very useful to insure profits, and a large fraction of the agents become chartists. The opposite is true in quiet periods when the actual price is very close to the fundamental value. The two regimes are separated by a threshold in the number of chartist agents. Once this threshold is approached (from below), large fluctuations take place which further increase the number of chartists. This destabilization is eventually dampened by the energetic intervention of the fundamentalists when the price deviates too much from the fundamental value. The authors compare this temporal instability with the on-off intermittence encountered in certain physical systems. According to Egenter et al. (1999), the fraction of chartists in the Lux Marchesi model goes to zero as the total number of traders goes to infinity, when the rest of the parameters are kept constant.

## Illustration with the LLS Model

The purpose of this section is to give a more detailed “hands-on” example of the agent-based approach and to discuss some of the practical dilemmas arising when implementing this approach, by focusing on one specific model. We will focus on the so-called LLS model of the stock market (for more details and various versions of the model, see Hellthaler (1995), Kohl (1997), Levy and Levy (1996), and Levy et al. (1994, 1996, 2000)). This section is based on the presentation of the LLS model in Chap. 7 of Levy et al. (2000).

### Background

Real-life investors differ in their investment behavior from the investment behavior of the idealized representative rational investor assumed in most economic and financial models. Investors differ one from the other in their preferences, their investment horizon, the information at their disposal, and their interpretation of this information. No financial economist seriously doubts these observations. However, modeling the empirically and experimentally documented investor behavior and the heterogeneity of investors is very difficult and in most cases practically impossible to do within an analytic framework. For instance, the empirical and experimental evidence suggests that most investors are characterized by Constant Relative Risk Aversion (CRRA), which implies a power (myopic) utility function (see Eq. 2 below). However, for a general distribution of returns, it is impossible to obtain an analytic solution for the portfolio optimization problem of investors with these preferences. Extrapolation of future returns from past returns, biased probability weighting, and partial deviations from rationality are also all experimentally documented but difficult to incorporate in an analytical setting. One is then usually forced to make the assumptions of rationality and homogeneity (at least in some dimension) and to make unrealistic assumptions regarding investors’ preferences, in order to obtain a model with a tractable solution. The hope in these circumstances is that the model will capture the essence of the system under investigation and will serve as a useful benchmark, even though some of the underlying assumptions are admittedly false.

Most homogeneous rational agent models lead to the following predictions: no trading volume, zero autocorrelation of returns, and price volatility which is equal to or lower than the volatility of the “fundamental value” of the stock (defined as the present value of all future dividends; see Shiller (1981)). However, the empirical evidence is very different:
• Trading volume can be extremely heavy (Admati and Pfleiderer 1988; Karpoff 1987).

• Stock returns exhibit short-run momentum (positive autocorrelation) and long-run mean reversion (negative autocorrelation) (Fama and French 1988; Jegadeesh and Titman 1993; Levy and Lim 1998; Poterba and Summers 1988).

• Stock returns are excessively volatile relative to the dividends (Shiller 1981).

As most standard rational-representative-agent models cannot explain these empirical findings, these phenomena are known as “anomalies” or “puzzles.” Can these “anomalies” be due to elements of investors’ behavior which are unmodeled in the standard rational-representative-agent models, such as the experimentally documented deviations of investors’ behavior from rationality and/or the heterogeneity of investors? The agent-based simulation approach offers us a tool to investigate this question. The strength of the agent-based simulation approach is that since it is not restricted to the scope of analytical methods, one is able to investigate virtually any imaginable investor behavior and market structure. Thus, one can study models which incorporate the experimental findings regarding the behavior of investors and evaluate the effects of various behavioral elements on market dynamics and asset pricing.

The LLS model incorporates some of the main empirical findings regarding investor behavior, and we employ this model in order to study the effect of each element of investor behavior on asset pricing and market dynamics. We start out with a benchmark model in which all of the investors are rational, informed, and identical, and then, one by one, we add elements of heterogeneity and deviations from rationality to the model in order to study their effects on the market dynamics.

In the benchmark model all investors are rational, informed, and identical (RII investors). This is, in effect, a “representative agent” model. The RII investors are informed about the dividend process, and they rationally act to maximize their expected utility. The RII investors make investment decisions based on the present value of future cash flows. They are essentially fundamentalists who evaluate the stock’s fundamental value and try to find bargains in the market. The benchmark model in which all investors are RII yields results which are typical of most rational-representative-agent models: in this model prices follow a random walk, there is no excess volatility of the prices relative to the volatility of the dividend process, and since all agents are identical, there is no trading volume.

After describing the properties of the benchmark model, we investigate the effects of introducing various elements of investor behavior which are found in laboratory experiments but are absent in most standard models. We do so by adding to the model a minority of investors who do not operate like the RII investors. These investors are efficient market believers (EMBs from now on). The EMBs are investors who believe that the price of the stock reflects all of the currently available information about the stock. As a consequence, they do not try to time the market or to buy bargain stocks. Rather, their investment decision is reduced to the optimal diversification problem. For this portfolio optimization, the ex-ante return distribution is required. However, since the ex-ante distribution is unknown, the EMB investors use the ex-post distribution in order to estimate the ex-ante distribution. It has been documented that in fact, many investors form their expectations regarding the future return distribution based on the distribution of past returns.

There are various ways to incorporate the investment decisions of the EMBs. This stems from the fact that there are different ways to estimate the ex-ante distribution from the ex-post distribution. How far back should one look at the historical returns? Should more emphasis be given to more recent returns? Should some “outlier” observations be filtered out? etc. Of course, there are no clear answers to these questions, and different investors may have different ways of forming their estimation of the ex-ante return distribution (even though they are looking at the same series of historical returns). Moreover, some investors may use the objective ex-post probabilities when constructing their estimation of the ex-ante distribution, whereas others may use biased subjective probability weights. In order to build the analysis step-by-step, we start by analyzing the case in which the EMB population is homogeneous, and then introduce various forms of heterogeneity into this population.

In the framework of the LLS model, we examine the effects of the EMBs’ deviations from rationality and their heterogeneity, relative to the benchmark model in which investors are informed, rational, and homogeneous. We find that the behavioral elements which are empirically documented, namely, extrapolation from past returns, deviation from rationality, and heterogeneity among investors, lead to all of the following empirically documented “puzzles”:
• Excess volatility

• Short-term momentum

• Longer-term return mean reversion

• Positive correlation between volume and contemporaneous absolute returns

• Positive correlation between volume and lagged absolute returns

The fact that all these anomalies or “puzzles,” which are hard to explain with standard rational-representative-agent models, are generated naturally by a simple model which incorporates the experimental findings regarding investor behavior and the heterogeneity of investors leads one to suspect that these behavioral elements and the diversity of investors are a crucial part of the workings of the market, and as such they cannot be “assumed away.” As the experimentally documented bounded-rational behavior and heterogeneity are in many cases impossible to analyze analytically, agent-based simulation presents a very promising tool for investigating market models incorporating these elements.

### The LLS Model

The stock market consists of two investment alternatives: a stock (or index of stocks) and a bond. The bond is assumed to be a riskless asset, and the stock is a risky asset. The stock serves as a proxy for the market portfolio (e.g., the Standard & Poor’s 500 index). The extension from one risky asset to many risky assets is possible; however, one stock (the index) is sufficient for our present analysis because we restrict ourselves to global market phenomena and do not wish to deal with asset allocation across several risky assets. Investors are allowed to revise their portfolio at given time points, i.e., we discuss a discrete time model.

The bond is assumed to be a riskless investment yielding a constant return at the end of each time period. The bond is in infinite supply and investors can buy from it as much as they wish at a given rate of r f . The stock is in finite supply. There are N outstanding shares of the stock. The return on the stock is composed of two elements:
1. (a)

Capital gain: If an investor holds a stock, any rise (fall) in the price of the stock contributes to an increase (decrease) in the investor’s wealth.

2. (b)

Dividends: The company earns income and distributes dividends at the end of each time period. We denote the dividend per share paid at time t by D t . We assume that the dividend is a stochastic variable following a multiplicative random walk, i.e., $${\tilde{D}}_t={D}_{t-1}\left(1+\tilde{z}\right)$$, where $$\tilde{z}$$ is a random variable with some probability density function f(z) in the range [z 1, z 2] (in order to allow for a dividend cut as well as a dividend increase, we typically choose: z 1 < 0, z 2 > 0).

The total return on the stock in period t, which we denote by R t is given by
$${\tilde{R}}_t=\frac{{\tilde{P}}_t+{\tilde{D}}_t}{P_{t-1}},$$
(1)
where $${\tilde{P}}_t$$ is the stock price at time t.
All investors in the model are characterized by a von Neumann-Morgenstern utility function. We assume that all investors have a power utility function of the form:
$$U(W)=\frac{W^{1-\alpha }}{1-\alpha },$$
(2)
where α is the risk-aversion parameter. This form of utility function implies Constant Relative Risk Aversion (CRRA). We employ the power utility function (Eq. 2) because the empirical evidence suggests that relative risk aversion is approximately constant (e.g., see Friend and Blume (1975), Gordon et al. (1972), Kroll et al. (1988), and Levy (1994), and the power utility function is the unique utility function which satisfies the CRRA condition. Another implication of CRRA is that the optimal investment choice is independent of the investment horizon (Samuelson 1989, 1994). In other words, regardless of investors’ actual investment horizon, they choose their optimal portfolio as though they are investing for a single period. The myopia property of the power utility function simplifies our analysis, as it allows us to assume that investors maximize their one-period-ahead expected utility.

We model two different types of investors: rational, informed, and identical (RII) investors and efficient market believers (EMB). These two investor types are described below.

#### Rational Informed Identical (RII) Investors

RII investors evaluate the “fundamental value” of the stock as the discounted stream of all future dividends and thus can also be thought of as “fundamentalists.” They believe that the stock price may deviate from the fundamental value in the short run, but if it does, it will eventually converge to the fundamental value. The RII investors act according to the assumption of asymptotic convergence: if the stock price is low relative to the fundamental value, they buy in anticipation that the underpricing will be corrected and vice versa. We make the simplifying assumption that the RII investors believe that the convergence of the price to the fundamental value will occur in the next period; however, our results hold for the more general case where the convergence is assumed to occur some T periods ahead, with T > 1.

In order to estimate next period’s return distribution, the RII investors need to estimate the distribution of next period’s price, $${\tilde{P}}_{t+1}$$, and of next period’s dividend, $${\tilde{D}}_{t+1}$$. Since they know the dividend process, the RII investors know that $${\tilde{D}}_{t+1}={D}_t\left(1+\tilde{z}\right)$$ where $$\tilde{z}$$ is distributed according to f(z) in the range [z 1, z 2]. The RII investors employ Gordon’s dividend stream model in order to calculate the fundamental value of the stock:
$${p}_{t+1}^f=\frac{E_{t+1}\left[{\tilde{D}}_{t+2}\right]}{k-g},$$
(3)
where the superscript f stands for the fundamental value, $${E}_{t+1}\left[{\tilde{D}}_{t+2}\right]$$ is the dividend corresponding to time t + 2 as expected at time t + 1, k is the discount factor or the expected rate of return demanded by the market for the stock, and g is the expected growth rate of the dividend, i.e.,
$$g=E\left(\tilde{z}\right)={\displaystyle {\int}_{z_1}^{z_2}f(z)z\mathrm{d}z}.$$
The RII investors believe that the stock price may temporarily deviate from the fundamental value; however, they also believe that the price will eventually converge to the fundamental value. For simplification we assume that the RII investors believe that the convergence to the fundamental value will take place next period. Thus, the RII investors estimate P t + 1 as
$${P}_{t+1}={P}_{t+1}^f.$$
The expectation at time t + 1 of $${\tilde{D}}_{t+2}$$ depends on the realized dividend observed at t + 1:
$${E}_{t+1}\left[{\tilde{D}}_{t+2}\right]={D}_{t+1}\left(1+g\right).$$
Thus, the RII investors believe that the price at t + 1 will be given by
$${P}_{t+1}={P}_{t+1}^f=\frac{D_{t+1}\left(1+g\right)}{k-g}.$$
At time t, D t is known, but D t+1 is not; therefore P t+1 f is also not known with certainty at time t.

However, given D t , the RII investors know the distribution of $${\tilde{D}}_{t+1}$$:

where $$\tilde{z}$$ is distributed according to the known f(z). The realization of $${\tilde{D}}_{t+1}$$ determines P t+1 f. Thus, at time t, RII investors believe that P t+1 is a random variable given by
$${\tilde{P}}_{t+1}={\tilde{P}}_{t+1}^f=\frac{D_t\left(1+\tilde{z}\right)\left(1+g\right)}{k-g}.$$
Notice that the RII investors face uncertainty regarding next period’s price. In our model we assume that the RII investors are certain about the dividend growth rate g, the discount factor k, and the fact that the price will converge to the fundamental value next period. In this framework the only source of uncertainty regarding next period’s price stems from the uncertainty regarding next period’s dividend realization. More generally, the RII investors’ uncertainty can result from uncertainty regarding any one of the above factors or a combination of several of these factors. Any mix of these uncertainties is possible to investigate in the agent-based simulation framework, but very hard, if not impossible, to incorporate in an analytic framework. As a consequence of the uncertainty regarding next period’s price and of their risk aversion, the RII investors do not buy an infinite number of shares even if they perceive the stock as underpriced. Rather, they estimate the stock’s next period’s return distribution and find the optimal mix of the stock and the bond which maximizes their expected utility. The RII investors estimate next period’s return on the stock as
$${\tilde{R}}_{t+1}=\frac{{\tilde{P}}_{t+1}+{\tilde{D}}_{t+1}}{P_t}=\frac{\frac{D_t\left(1+\tilde{z}\right)\left(1+g\right)}{k-g}+{D}_t\left(1+\tilde{z}\right)}{P_t},$$
(4)
where $$\tilde{z}$$, the next year’s growth in the dividend, is the source of uncertainty. The demands of the RII investors for the stock depend on the price of the stock. For any hypothetical price P h , investors calculate the proportion of their wealth x they should invest in the stock in order to maximize their expected utility. The RII investor i believes that if he/she invests a proportion x of his/her wealth in the stock at time t, then at time t + 1 his/her wealth will be
$${\tilde{W}}_{t+1}^i={W}_h^i\left[\left(1-x\right)\left(1+{r}_f\right)+x{\tilde{R}}_{t+1}\right],$$
(5)
where $${\tilde{R}}_{t+1}$$ is the return on the stock, as given by Eq. 1, and is the wealth of investor W h i at time t given that the stock price at time t is P h .
If the price in period t is the hypothetical price P h , the t + 1 expected utility of investor i is the following function of his/her investment proportion in the stock, x:
$$EU\left({\tilde{W}}_{t+1}^i\right)= EU\left({W}_h^i\left[\left(1-x\right)\left(1+{r}_f\right)+x{\tilde{R}}_{t+1}\right]\right).$$
(6)
Substituting $${\tilde{R}}_{t+1}$$ from Eq. 4, using the power utility function (Eq. 2), and substituting the hypothetical price P h for P t , the expected utility becomes the following function of x:
$$EU\left({\tilde{W}}_{t+1}^i\right)=\frac{{\left({W}_h^i\right)}^{1-\alpha }}{1-\alpha }{\displaystyle \underset{z_1}{\overset{z_2}{\int }}{\left[\left(1-x\right)\left(1+{r}_f\right)+x\left(\frac{\frac{D_t\left(1+z\right)\left(1+g\right)}{k-g}+{D}_t\left(1+z\right)}{P_h}\right)\right]}^{1-\alpha }f(z)\mathrm{d}z},$$
(7)
where the integration is over all possible values of z. In the agent-based simulation framework, this expression for the expected utility, and the optimal investment proportion x, can be solved numerically for any general choice of distribution f(z). For the sake of simplicity, we restrict the present analysis to the case where $$\tilde{z}$$ is distributed uniformly in the range [z 1, z 2]. This simplification leads to the following expression for the expected utility:
$$EU\left({\tilde{W}}_{t+1}^i\right)=\frac{{\left({W}_h^i\right)}^{1-\alpha }}{\left(1-\alpha \right)\left(2-\alpha \right)}\frac{1}{\left({z}_2-{z}_1\right)}\left(\frac{k-g}{k+1}\right)\frac{P_h}{x{D}_t}\left\{{\left[\left(1-x\right)\left(1+{r}_f\right)+\frac{x}{P_h}\left(\frac{k+1}{k-g}\right){D}_t\left(1+{z}_2\right)\right]}^{\left(2-\alpha \right)}\right.\left.-{\left[\left(1-x\right)\left(1+{r}_f\right)+\frac{x}{P_h}\left(\frac{k+1}{k-g}\right){D}_t\left(1+{z}_1\right)\right]}^{\left(2-\alpha \right)}\right\}.$$
(8)
For any hypothetical price P h , each investor (numerically) finds the optimal proportion x h which maximizes his/her expected utility given by Eq. 8. Notice that the optimal proportion, x h , is independent of the wealth, W h i . Thus, if all RII investors have the same degree of risk aversion, α, they will have the same optimal investment proportion in the stock, regardless of their wealth. The number of shares demanded by investor i at the hypothetical price P h is given by
$${N}_h^i\left({P}_h\right)=\frac{x_h^i\left({P}_h\right){W}_h^i\left({P}_h\right)}{P_h}.$$
(9)

#### Efficient Market Believers (EMB)

The second type of investors in the LLS model is EMBs. The EMBs believe in market efficiency – they believe that the stock price accurately reflects the stock’s fundamental value. Thus, they do not try to time the market or to look for “bargain” stocks. Rather, their investment decision is reduced to the optimal diversification between the stock and the bond. This diversification decision requires the ex-ante return distribution for the stock, but as the ex-ante distribution is not available, the EMBs assume that the process generating the returns is fairly stable, and they employ the ex-post distribution of stock returns in order to estimate the ex-ante return distribution.

Different EMB investors may disagree on the optimal number of ex-post return observations that should be employed in order to estimate the ex-ante return distribution. There is a trade-off between using more observations for better statistical inference and using a smaller number of only more recent observations, which are probably more representative of the ex-ante distribution. As in reality, there is no “recipe” for the optimal number of observations to use. EMB investor i believes that the m i most recent returns on the stock are the best estimate of the ex-ante distribution. Investors create an estimation of the ex-ante return distribution by assigning an equal probability to each of the m i most recent return observations:
$${\mathrm{Prob}}^i\left({\tilde{R}}_{t+1}={R}_{t-j}\right)=\frac{1}{m^i}\mathrm{for}\ j=1,\dots, {m}^i.$$
(10)
The expected utility of EMB investor i is given by
$$EU\left({W}_{t+1}^i\right)=\frac{{\left({W}_h^i\right)}^{1-\alpha }}{1-\alpha}\frac{1}{m^i}{\displaystyle \sum_{j=1}^{m^i}{\left[\left(1-x\right)\left(1+{r}_f\right)+x{R}_{t-j}\right]}^{1-\alpha },}$$
(11)
where the summation is over the set of m l most recent ex-post returns, x is the proportion of wealth invested in the stock, and as before W h i is the wealth of investor i at time t given that the stock price at time t is P h . Notice that W h i does not change the optimal diversification policy, i.e., x. Given a set of m i past returns, the optimal portfolio for the EMB investor i is an investment of a proportion x *i in the stock and (1−x *i ) in the bond, where x *i is the proportion which maximizes the above expected utility (Eq. 11) for investor i. Notice that x *i generally cannot be solved for analytically. However, in the agent-based simulation framework, this does not constitute a problem, as one can find x *i numerically.

#### Deviations from Rationality

Investors who are efficient market believers, and are rational, choose the investment proportion x* which maximizes their expected utility. However, many empirical studies have shown that the behavior of investors is driven not only by rational expected utility maximization but by a multitude of other factors (e.g., see Samuelson (1994), Thaler (1993, 1994), and Tversky and Kahneman (1981, 1986)). Deviations from the optimal rational investment proportion can be due to the cost of resources which are required for the portfolio optimization – time, access to information, computational power, etc. – or due to exogenous events (e.g., an investor plans to revise his/her portfolio, but gets distracted because his/her car breaks down). We assume that the different factors causing the investor to deviate from the optimal investment proportion x* are random and uncorrelated with each other. By the central limit theorem, the aggregate effect of a large number of random uncorrelated influences is a normally distributed random influence, or “noise.” Hence, we model the effect of all the factors causing the investor to deviate from his/her optimal portfolio by adding a normally distributed random variable to the optimal investment proportion. To be more specific, we assume
$${x}^i={x}^{*i}+{\tilde{\varepsilon}}^i,$$
(12)
where $${\tilde{\varepsilon}}^i$$ is a random variable drawn from a truncated normal distribution with mean zero and standard deviation σ. Notice that noise is investor specific; thus, $${\tilde{\varepsilon}}^i$$ is drawn separately and independently for each investor.

The noise can be added to the decision-making of the RII investors, the EMB investors, or to both. The results are not much different with these various approaches. Since the RII investors are taken as the benchmark of rationality, in this entry we add the noise only to the decision-making of the EMB investors.

#### Market Clearance

The number of shares demanded by each investor is a monotonically decreasing function of the hypothetical price P h (see Levy et al. 2000). As the total number of outstanding shares is N, the price of the stock at time t is given by the market clearance condition: P t is the unique price at which the total demand for shares is equal to the total supply, N:
$${\displaystyle \sum_i{N}_h^i\left({P}_t\right)}={\displaystyle \sum_i\frac{x_h\left({P}_t\right){W}_h^i\left({P}_t\right)}{P_t}=N,}$$
(13)
where the summation is over all the investors in the market, RII investors as well as EMB investors.

#### Agent-Based Simulation

The market dynamics begin with a set of initial conditions which consist of an initial stock price P 0, an initial dividend D 0, the wealth and number of shares held by each investor at time t = 0, and an initial “history” of stock returns. As will become evident, the general results do not depend on the initial conditions. At the first period (t = 1), interest is paid on the bond, and the time 1 dividend $${\tilde{D}}_1={D}_0\left(1+\tilde{z}\right)$$ is realized and paid out. Then investors submit their demand orders, N h i (P h ), and the market clearing price P 1 is determined. After the clearing price is set, the new wealth and number of shares held by each investor are calculated. This completes one time period. This process is repeated over and over, as the market dynamics develop.

We would like to stress that even the simplified benchmark model, with only RII investors, is impossible to solve analytically. The reason for this is that the optimal investment proportion, x h (P h ), cannot be calculated analytically. This problem is very general and it is encountered with almost any choice of utility function and distribution of returns. One important exception is the case of a negative exponential utility function and normally distributed returns. Indeed, many models make these two assumptions for the sake of tractability. The problem with the assumption of negative exponential utility is that it implies Constant Absolute Risk Aversion (CARA), which is very unrealistic, as it implies that investors choose to invest the same dollar amount in a risky prospect independent of their wealth. This is not only in sharp contradiction to the empirical evidence but also excludes the investigation of the two-way interaction between wealth and price dynamics, which is crucial to the understanding of the market.

Thus, one contribution of the agent-based simulation approach is that it allows investigation of models with realistic assumptions regarding investors’ preferences. However, the main contribution of this method is that it permits us to investigate models which are much more complex (and realistic) than the benchmark model, in which all investors are RII. With the agent-based simulation approach, one can study models incorporating the empirically and experimentally documented investors’ behavior and the heterogeneity of investors.

### Results of the LLS Model

We begin by describing the benchmark case where all investors are rational and identical. Then we introduce to the market EMB investors and investigate their affects on the market dynamics.

#### Benchmark Case: Fully Rational and Identical Agents

In this benchmark model all investors are RII: rational, informed, and identical. Thus, it is not surprising that the benchmark model generates market dynamics which are typical of homogeneous rational agent models.

##### No Volume
All investors in the model are identical; they therefore always agree on the optimal proportion to invest in the stock. As a consequence, all the investors always achieve the same return on their portfolio. This means that at any time period, the ratio between the wealth of any two investors is equal to the ratio of their initial wealths, i.e.,:
$$\frac{W_t^i}{W_t^j}=\frac{W_0^i}{W_0^j}.$$
(14)
As the wealth of investors is always in the same proportion, and as they always invest the same fraction of their wealth in the stock, the number of shares held by different investors is also always in the same proportion:
$$\frac{N_t^i}{N_t^j}=\frac{\frac{x_t{W}_t^i}{P_t}}{\frac{x_t{W}_t^j}{P_t}}=\frac{W_t^i}{W_t^j}=\frac{W_0^i}{W_0^j}.$$
(15)
Since the total supply of shares is constant, this implies that each investor always holds the same number of shares and there is no trading volume (the number of shares held may vary from one investor to another as a consequence of different initial endowments).
##### Log Prices Follow a Random Walk

In the benchmark model all investors believe that next period’s price will converge to the fundamental value given by the discounted dividend model (Eq. 3). Therefore, the actual stock price is always close to the fundamental value. The fluctuations in the stock price are driven by fluctuations in the fundamental value, which in turn are driven by the fluctuating dividend realizations. As the dividend fluctuations are (by assumption) uncorrelated over time, one would expect that the price fluctuations will also be uncorrelated. To verify this intuitive result, we examine the return autocorrelations in simulations of the benchmark model.

Let us turn to the simulation of the model. We first describe the parameters and initial conditions used in the simulation and then report the results. We simulate the benchmark model with the following parameters:
• Number of investors = 1,000.

• Risk-aversion parameter α = 1.5. This value roughly conforms with the estimate of the risk-aversion parameter found empirically and experimentally.

• Number of shares = 10,000.

• We take the time period to be a quarter, and accordingly we choose:

• Riskless interest rate r f = 0.01.

• Required rate of return on stock k = 0.04.

• Maximal one-period dividend decrease z 1 = −0.07.

• Maximal one-period dividend growth z 2 = 0.10.

• $$\tilde{z}$$ is uniformly distributed between these values. Thus, the average dividend growth rate is g = (z 1 + z 2)/2 = 0.015.

Initial conditions: Each investor is endowed at time t = 0 with a total wealth of $1,000, which is composed of 10 shares worth an initial price of$50 per share and $500 in cash. The initial quarterly dividend is set at$0.5 (for an annual dividend yield of about 4 %). As will soon become evident, the dynamics are not sensitive to the particular choice of initial conditions.

Figure 1 shows the price dynamics in a typical simulation with these parameters (simulations with the same parameters differ one from the other because of the different random dividend realizations). Notice that the vertical axis in this figure is logarithmic. Thus, the roughly constant slope implies an approximately exponential price growth or an approximately constant average return.
The prices in this simulation seem to fluctuate randomly around the trend. However, Fig. 1 shows only one simulation. In order to have a more rigorous analysis, we perform many independent simulations and employ statistical tools. Namely, for each simulation we calculate the autocorrelation of returns. We perform a univariate regression of the return in time t on the return on time tj:
$${R}_t={\alpha}_j+{\beta}_j{R}_{t-j}+\varepsilon,$$
where R t is the return in period t and j is the lag. The autocorrelation of returns for lag j is defined as
$${\rho}_j=\frac{\operatorname{cov}\left({R}_t,{R}_{t-j}\right)}{{\widehat{\sigma}}^2(R)},$$
and it is estimated by $$\widehat{\beta}$$. We calculate the autocorrelation for different lags, j = 1,…40. Figure 2 shows the average autocorrelation as a function of the lag, calculated over 100 independent simulations. It is evident both from the figure that the returns are uncorrelated in the benchmark model, conforming with the random-walk hypothesis.
##### No Excess Volatility
Since the RII investors believe that the stock price will converge to the fundamental value next period, in the benchmark model, prices are always close to the fundamental value given by the discounted dividend stream. Thus, we do not expect prices to be more volatile than the value of the discounted dividend stream. For a formal test of excess volatility, we follow the technique in Shiller (1981). For each time period we calculate the actual price, P t , and the fundamental value of discounted dividend stream, p t f , as in Eq. 3. Since prices follow an upward trend, in order to have a meaningful measure of the volatility, we must detrend these price series. Following Shiller, we run the regression:
$$\ln {P}_t= bt+c+\underset{t}{\varepsilon },$$
(16)
in order to find the average exponential price growth rate (where b and c are constants). Then, we define the detrended price as: $${p}_t={P}_t/{e}^{\widehat{b}t}$$. Similarly, we define the detrended value of the discounted dividend stream p t f and compare σ(p t ) with σ(p t f ). For 100- to 1,000-period simulations, we find an average σ(p t ) of 22.4 and an average σ(p t f ) of 22.9. As expected, the actual price and the fundamental value have almost the same volatility.

To summarize the results obtained for the benchmark model, we find that when all investors are assumed to be rational, informed, and identical, we obtain results which are typical of rational-representative-agent models: no volume, no return autocorrelations, and no excess volatility. We next turn to examine the effect of introducing into the market EMB investors, which model empirically and experimentally documented elements of investors’ behavior.

#### The Introduction of a Small Minority of EMB Investors

In this section we will show that the introduction of a small minority of heterogeneous EMB investors generates many of the empirically observed market “anomalies” which are absent in the benchmark model and indeed, in most other rational-representative-agent models. We take this as strong evidence that the “nonrational” elements of investor behavior which are documented in experimental studies and the heterogeneity of investors, both of which are incorporated in the LLS model, are crucial to understanding the dynamics of the market.

In presenting the results of the LLS model with EMB investors, we take an incremental approach. We begin by describing the results of a model with a small subpopulation of homogeneous EMB believers. This model produces the abovementioned market “anomalies”; however, it produces unrealistic cyclic market dynamics. Thus, this model is presented both for analyzing the source of the “anomalies” in a simplified setting and as a reference point with which to compare the dynamics of the model with a heterogeneous EMB believer population.

We investigate the effects of investors’ heterogeneity by first analyzing the case in which there are two types of EMBs. The two types differ in the method they use to estimate the ex-ante return distribution. Namely, the first type looks at the set of the last m1 ex-post returns, whereas the second type looks at the set of the last m2 ex-post returns. It turns out that the dynamics in this case are much more complicated than a simple “average” between the case where all EMB investors have m1 and the case where all EMB investors have m 2. Rather, there is a complex nonlinear interaction between the two EMB subpopulations. This implies that the heterogeneity of investors is a very important element determining the market dynamics, an element which is completely absent in representative-agent models.

Finally, we present the case where there is an entire spectrum of EMB investors differing in the number of ex-post observations they take into account when estimating the ex-ante distribution. This general case generates very realistic-looking market dynamics with all of the abovementioned market anomalies.

#### Homogeneous Subpopulation of EMBs

When a very small subpopulation of EMB investors is introduced to the benchmark LLS model, the market dynamics change dramatically. Figure 3 depicts a typical price path in a simulation of a market with 95 % RII investors and 5 % EMB investors. The EMB investors have m = 10 (i.e., they estimate the ex-ante return distribution by observing the set of the last 10 ex-post returns). σ, the standard deviation of the random noise affecting the EMBs’ decision-making, is taken as 0.2. All investors, RII and EMB alike, have the same risk-aversion parameter α = 1.5 (as before). In the first 150 trading periods, the price dynamics look very similar to the typical dynamics of the benchmark model. However, after the first 150 or so periods, the price dynamics change. From this point onwards the market is characterized by periodic booms and crashes. Of course, Fig. 3 describes only one simulation. However, as will become evident shortly, different simulations with the same parameters may differ in detail, but the pattern is general: at some stage (not necessarily after 150 periods), the EMB investors induce cyclic price behavior. It is quite astonishing that such a small minority of only 5 % of the investors can have such a dramatic impact on the market.

In order to understand the periodic booms and crashes, let us focus on the behavior of the EMB investors. After every trade, the EMB investors revise their estimation of the ex-ante return distribution, because the set of ex-post returns they employ to estimate the ex-ante distribution changes. Namely, investors add the latest return generated by the stock to this set and delete the oldest return from this set. As a result of this update in the estimation of the ex-ante distribution, the optimal investment proportion x* changes, and EMB investors revise their portfolios at next period’s trade. During the first 150 or so periods, the informed investors control the dynamics and the returns fluctuate randomly (as in the benchmark model). As a consequence, the investment proportion of the EMB investors also fluctuates irregularly. Thus, during the first 150 periods, the EMB investors do not effect the dynamics much. However, at point a, the dynamics change qualitatively (see Fig. 3). At this point, a relatively high dividend is realized, and as a consequence, a relatively high return is generated. This high return leads the EMB investors to increase their investment proportion in the stock at the next trading period. This increased demand of the EMB investors is large enough to effect next period’s price, and thus a second high return is generated. Now the EMB investors look at a set of ex-post returns with two high returns, and they increase their investment proportion even further. Thus, a positive feedback loop is created.

Notice that as the price goes up, the informed investors realize that the stock is overvalued relative to the fundamental value P f and they decrease their holdings in the stock. However, this effect does not stop the price increase and break the feedback loop because the EMB investors continue to buy shares aggressively. The positive feedback loop pushes the stock price further and further up to point b, at which the EMBs are invested 100 % in the stock. At point b, the positive feedback loop “runs out of gas.” However, the stock price remains at the high level because the EMB investors remain fully invested in the stock (the set of past m = 10 returns includes at this stage the very high returns generated during the “boom” – segment ab in Fig. 3).

When the price is at the high level (segment bc), the dividend yield is low, and as a consequence, the returns are generally low. As time goes by and we move from point b toward point c, the set of m = 10 last returns gets filled with low returns. Despite this fact, the extremely high returns generated in the boom are also still in this set, and they are high enough to keep the EMB investors fully invested. However, 10 periods after the boom, these extremely high returns are pushed out of the set of relevant ex-post returns. When this occurs, at point c, the EMB investors face a set of low returns, and they cut their investment proportion in the stock sharply. This causes a dramatic crash (segment cd). Once the stock price goes back down to the “fundamental” value, the informed investors come back into the picture. They buy back the stock and stop the crash.

The EMB investors stay away from the stock as long as the ex-post return set includes the terrible return of the crash. At this stage the informed investors regain control of the dynamics and the stock price remains close to its fundamental value. Ten periods after the crash, the extremely negative return of the crash is excluded from the ex-post return set, and the EMB investors start increasing their investment proportion in the stock (point e). This drives the stock price up, and a new boom-crash cycle is initiated. This cycle repeats itself over and over almost periodically.

Figure 3 depicts the price dynamics of a single simulation. One may therefore wonder how general the results discussed above are. Figure 4 shows two more simulations with the same parameters but different dividend realizations. It is evident from this figure that although the simulations vary in detail (because of the different dividend realizations), the overall price pattern with periodic boom-crash cycles is robust.

Although these dynamics are very unrealistic in terms of the periodicity, and therefore the predictability of the price, they do shed light on the mechanism generating many of the empirically observed market phenomena. In the next section, when we relax the assumption that the EMB population is homogeneous with respect to m, the price is no longer cyclic or predictable, yet the mechanisms generating the market phenomena are the same as in this homogeneous EMB population case. The homogeneous EMB population case generates the following market phenomena.

As explained above, shares change hands continuously between the RII investors and the EMB investors. When a “boom” starts, the RII investors observe higher ex-post returns and become more optimistic, while the EMB investors view the stock as becoming overpriced and become more pessimistic. Thus, at this stage the EMBs buy most of the shares from the RIIs. When the stock crashes, the opposite is true: the EMBs are very pessimistic, but the RII investors buy the stock once it falls back to the fundamental value. Thus, there is substantial trading volume in this market. The average trading volume in a typical simulation is about 1,000 shares per period, which are 10 % of the total outstanding shares.

##### Autocorrelation of Returns
The cyclic behavior of the price yields a very definite return autocorrelation pattern. The autocorrelation pattern is depicted graphically in Fig. 5. The autocorrelation pattern is directly linked to the length of the price cycle, which in turn are determined by m. Since the moving window of ex-post returns used to estimate the ex-ante distribution is m = 10 periods long, the price cycles are typically a little longer than 20 periods long: a cycle consists of the positive feedback loop (segment a, b in Fig. 3) which is about 2–3 periods long, the upper plateau (segment b, c in Fig. 3) which is about 10 periods long, the crash that occurs during one or two periods, and the lower plateau (segment d, e in Fig. 3) which is again about 10 periods long, for a total of about 23–25 periods. Thus, we expect positive autocorrelation for lags of about 23–25 periods, because this is the lag between one point and the corresponding point in the next (or previous) cycle. We also expect negative autocorrelation for lags of about 10–12 periods, because this is the lag between a boom and the following (or previous) crash and vice versa. This is precisely the pattern we observe in Fig. 5.
##### Excess Volatility

The EMB investors induce large deviations of the price from the fundamental value. Thus, price fluctuations are caused not only by dividend fluctuations (as the standard theory suggests) but also by the endogenous market dynamics driven by the EMB investors. This “extra” source of fluctuations causes the price to be more volatile than the fundamental value P f.

Indeed, for 100- to 1,000-period independent simulations with 5 % EMB investors, we find an average σ(p t ) of 46.4 and an average σ(p t f ) of 30.6; That is, we have excess volatility of about 50 %.

As a first step in analyzing the effects of heterogeneity of the EMB population, in the next section we examine the case of two types of EMB investors. We later analyze a model in which there is a full spectrum of EMB investors.

#### Two Types of EMBs

One justification for using a representative agent in economic modeling is that although investors are heterogeneous in reality, one can model their collective behavior with one representative or “average” investor. In this section we show that this is generally not true. Many aspects of the dynamics result from the nonlinear interaction between different investor types. To illustrate this point, in this section we analyze a very simple case in which there are only two types of EMB investors: one with m = 5 and the other with m = 15. Each of these two types consists of 2 % of the investor population, and the remaining 96 % are informed investors. The representative agent logic may tempt us to think that the resulting market dynamics would be similar to that of one “average” investor, i.e., an investor with m = 10. Figure 6 shows that this is clearly not the case. Rather than seeing periodic cycles of about 23–25 periods (which correspond to the average m of 10, as in Fig. 3), we see an irregular pattern. As before, the dynamics are first dictated by the informed investors. Then, at point a, the EMB investors with m = 15 induce cycles which are about 30 periods long. At point b there is a transition to shorter cycles induced by the m = 5 population, and at point c there is another transition back to longer cycles. What is going on?
These complex dynamics result from the nonlinear interaction between the different subpopulations. The transitions from one price pattern to another can be partly understood by looking at the wealth of each subpopulation. Figure 7 shows the proportion of the total wealth held by each of the two EMB populations (the remaining proportion is held by the informed investors). As seen in Fig. 7, the cycles which start at point a are dictated by the m = 15 rather than the m = 5 population, because at this stage the m = 15 population controls more of the wealth than the m = 5 population. However, after 3 cycles (at point b), the picture is reversed. At this point the m = 5 population is more powerful than the m = 15 population, and there is a transition to shorter boom-crash cycles. At point c the wealth of the two subpopulations is again almost equal, and there is another transition to longer cycles. Thus, the complex price dynamics can be partly understood from the wealth dynamics. But how are the wealth dynamics determined? Why does the m = 5 population become wealthier at point b, and why does it lose most of this advantage at point c? It is obvious that the wealth dynamics are influenced by the price dynamics; thus, there is a complicated two-way interaction between the two. Although this interaction is generally very complex, some principle ideas about the mutual influence between the wealth and price patterns can be formulated. For example, a population that becomes dominant and dictates the price dynamics typically starts underperforming, because it affects the price with its actions. This means pushing the price up when buying, and therefore buying high, and pushing the price down when selling. However, a more detailed analysis must consider the specific investment strategy employed by each population. For a more comprehensive analysis of the interaction between heterogeneous EMB populations, see Levy et al. (1996).

The two-EMB-population model generates the same market phenomena as did the homogeneous population case: heavy trading volume, return autocorrelations, and excess volatility. Although the price pattern is much less regular in the two-EMB-population case, there still seems to be a great deal of predictability about the prices. Moreover, the booms and crashes generated by this model are unrealistically dramatic and frequent. In the next section we analyze a model with a continuous spectrum of EMB investors. We show that this fuller heterogeneity of investors leads to very realistic price and volume patterns.

#### Full Spectrum of EMB Investors

Up to this point we have analyzed markets with at most three different subpopulations (one RII population and two EMB populations). The market dynamics we found displayed the empirically observed market anomalies, but they were unrealistic in the magnitude, frequency, and semipredictability of booms and crashes. In reality, we would expect not only two or three investor types, but rather an entire spectrum of investors. In this section we consider a model with a full spectrum of different EMB investors. It turns out that more is different. When there is an entire range of investors, the price dynamics become realistic: booms and crashes are not periodic or predictable, and they are also less frequent and dramatic. At the same time, we still obtain all of the market anomalies described before.

In this model each investor has a different number of ex-post observations which he/she utilizes to estimate the ex-ante distribution. Namely, investor i looks at the set of the m i most recent returns on the stock, and we assume that m i is distributed in the population according to a truncated normal distribution with average $$\overline{m}$$ and standard deviation σ m (as m ≤ 0 is meaningless, the distribution is truncated at m = 0).

Figure 8 shows the price pattern of a typical simulation of this model. In this simulation 90 % of the investors are RII, and the remaining 10 % are heterogeneous EMB investors with $$\tilde{m}=40$$ and σ m = 10. The price pattern seems very realistic with “smoother” and more irregular cycles. Crashes are dramatic, but infrequent and unpredictable.

The heterogeneous EMB population model generates the following empirically observed market phenomena:

##### Return Autocorrelation: Momentum and Mean Reversion
In the heterogeneous EMB population model, trends are generated by the same positive feedback mechanism that generated cycles in the homogeneous case: high (low) returns tend to make the EMB investors more (less) aggressive, this generates more high (low) returns, etc. The difference between the two cases is that in the heterogeneous case, there is a very complicated interaction between all the different investor subpopulations, and as a result there are no distinct regular cycles, but rather, smoother and more irregular trends. There is no single cycle length – the dynamics are a combination of many different cycles. This makes the autocorrelation pattern also smoother and more continuous. The return autocorrelations in the heterogeneous model are shown in Fig. 9. This autocorrelation pattern conforms with the empirical findings. In the short run (lags 1–4), the autocorrelation is positive – this is the empirically documented phenomenon known as momentum: in the short run, high returns tend to be followed by more high returns, and low returns tend to be followed by more low returns. In the longer run (lags 5–13), the autocorrelation is negative, which is known as mean reversion. For even longer lags the autocorrelation eventually tends to be zero. The short-run momentum, longer-run mean reversion, and eventual diminishing autocorrelation create the general “U shape” which is found in empirical studies (Fama and French 1988; Jegadeesh and Titman 1993; Poterba and Summers 1988) and which is seen in Fig. 9.
##### Excess Volatility

The price level is generally determined by the fundamental value of the stock. However, as in the homogeneous EMB population case, the EMB investors occasionally induce temporary departures of the price away from the fundamental value. These temporary departures from the fundamental value make the price more volatile than the fundamental value. Following Shiller’s methodology we define the detrended price, p, and fundamental value, p f . Averaging over 100 independent simulations, we find σ(p) = 27.1 and σ(p f ), which is an excess volatility of 41 %.

##### Heavy Volume

As investors in our model have different information (the informed investors know the dividend process, while the EMB investors do not) and different ways of interpreting the information (EMB investors with different memory spans have different estimations regarding the ex-ante return distribution), there is a high level of trading volume in this model. The average trading volume in this model is about 1,700 shares per period (17 % of the total outstanding shares). As explained below, the volume is positively correlated with contemporaneous and lagged absolute returns.

##### Volume Is Positively Correlated with Contemporaneous and Lagged Absolute Returns

Investors revise their portfolios as a result of changes in their beliefs regarding the future return distribution. The changes in the beliefs can be due to a change in the current price, to a new dividend realization (in the case of the informed investors), or to a new observation of an ex-post return (in the case of the EMB investors). If all investors change their beliefs in the same direction (e.g., if everybody becomes more optimistic), the stock price can change substantially with almost no volume – everybody would like to increase the proportion of the stock in his/her portfolio, this will push the price up, but a very small number of shares will change hands. This scenario would lead to zero or perhaps even negative correlation between the magnitude of the price change (or return) and the volume. However, the typical scenario in the LLS model is different. Typically, when a positive feedback trend is induced by the EMB investors, the opinions of the informed investors and the EMB investors change in opposite directions. The EMB investors see a trend of rising prices as a positive indication about the ex-ante return distribution, while the informed investors believe that the higher the price level is above the fundamental value, the more overpriced the stock is and the harder it will eventually fall. The exact opposite holds for a trend of falling prices. Thus, price trends are typically interpreted differently by the two investor types and therefore induce heavy trading volume. The more pronounced the trend, the more likely it is to lead to heavy volume and, at the same time, to large price changes which are due to the positive feedback trading on behalf of the EMB investors.

This explains not only the positive correlation between volume and contemporaneous absolute rates of return but also the positive correlation between volume and lagged absolute rates of return. The reason is that the behavior of the EMB investors induces short-term positive return autocorrelation, or momentum (see above), that is, a large absolute return this period is associated not only with high volume but also with a large absolute return next period and therefore with high volume next period. In other words, when there is a substantial price increase (decrease), EMB investors become more (less) aggressive and the opposite happens to the informed traders. As we have seen before, when a positive feedback loop is started, the EMB investors are more dominant in determining the price, and therefore another large price increase (decrease) is expected next period. This large price change is likely to be associated with heavy trading volume as the opinions of the two populations diverge. Furthermore, this large increase (decrease) is expected to make the EMB investors even more optimistic (pessimistic) leading to another large price increase (decrease) and heavy volume next period.

In order to verify this relationship quantitatively, we regress volume on contemporaneous and lagged absolute rates of return for 100 independent simulations. We run the regressions:
$$\begin{array}{cc}\hfill {V}_t\hfill & \hfill =\alpha +{\beta}_C\left|{R}_t-1\right|+{\varepsilon}_t\kern1em \mathrm{and}\hfill \\ {}\hfill {V}_t\hfill & \hfill =\alpha +{\beta}_L\left|{R}_{t-1}-1\right|+{\varepsilon}_t,\hfill \end{array}$$
(17)
where V t is the volume at time t and R t is the total return on the stock at time t and the subscripts C and L stand for contemporaneous and lagged. We find an average value of 870 for $${\widehat{\beta}}_C$$ with an average t-value of 5.0 and an average value of 886 $${\widehat{\beta}}_L$$ for with an average t-value of 5.1.

#### Discussion of the LLS Results

The LLS model is an agent-based simulation model of the stock market which incorporates some of the fundamental experimental findings regarding the behavior of investors. The main nonstandard assumption of the model is that there is a small minority of investors in the market who are uninformed about the dividend process and who believe in market efficiency. The investment decision of these investors is reduced to the optimal diversification between the stock and the bond.

The LLS model generates many of the empirically documented market phenomena which are hard to explain in the analytical rational-representative-agent framework. These phenomena are:
• Short-term momentum

• Longer-term mean reversion

• Excess volatility

• Positive correlation between volume and contemporaneous absolute returns

• Positive correlation between volume and lagged absolute returns

• Endogenous market crashes

The fact that so many “puzzles” are explained with a simple model built on a small number of empirically documented behavioral elements leads us to suspect that these behavioral elements are very important in understanding the workings of the market. This is especially true in light of the observations that a very small minority of the nonstandard bounded-rational investors can have a dramatic influence on the market and that these investors are not wiped out by the majority of rational investors.

## Summary and Future Directions

Standard economic models typically describe a world of homogeneous rational agents. This approach is the foundation of most of our present-day knowledge in economic theory. With the agent-based simulation approach, we can investigate a much more complex and “messy” world with different agent types, who employ different strategies to try to survive and prosper in a market with structural uncertainty. Agents can learn over time, from their own experience and from their observation about the performance of other agents. They coevolve over time and as they do so, the market dynamics change continuously. This is a worldview closer to biology than it is to the “clean” realm of physical laws which classical economics has aspired to.

The agent-based approach should not and cannot replace the standard analytical economic approach. Rather, these two methodologies support and complement each other: When an analytical model is developed, it should become standard practice to examine the robustness of the model’s results with agent-based simulations. Similarly, when results emerge from agent-based simulation, one should try to understand their origin and their generality, not only by running many simulations but also by trying to capture the essence of the results in a simplified analytical setting (if possible).

Although the first steps in economic agent-based simulations were made decades ago, economics has been slow and cautious to adopt this new methodology. Only in recent years has this field begun to bloom. It is my belief and hope that the agent-based approach will prove as fruitful in economics as it has been in so many other branches of science.

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