AgentBased Computational Economics
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Keywords
microscopic simulation agentbased simulation financial markets bounded rationality heterogeneous expectationsDefinition 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.
Agentbased 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 agentbased simulation allows us to investigate the dynamics of complex economic systems with many heterogeneous and not necessarily fully rational agents.
The agentbased 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 agentbased 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 wellknown 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 “agentbased simulation” or “agentbased 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 agentbased 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 riskaversion coefficient or for that matter the entire decisionmaking rule) are typically estimated with substantial error. This is a strong point. Indeed, we would argue that the “art” of agentbased 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 agentbased computational economics. An introduction to this field has two goals: (i) to explain and to demonstrate the agentbased 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 agentbased 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, realestate, and cultural factors. Rather, the aim of this very simplified model is to explain the emergence of macroscopic singlerace 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 samecolor 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 agentbased 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 samecolor 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 samecolor 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 agentbased 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 agentbased 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 KimMarkowitz (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 KimMarkowitz agentbased 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 65day 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 agentbased 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 65day quarter and is significantly diminished if the 65day 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 agentbased 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 everevolving market, driven endogenously by the everchanging 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, volumevolatility 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 RiskAversion (CARA) utility function. The use of CARA utility leads to demands which do not depend on the investor’s wealth.
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.

Trading volume is low.

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 selffulfilling agent priceprediction 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 heavytailed 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.

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)

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).

No longterm 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 (heavytailed) 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 onoff 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 “handson” example of the agentbased 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 socalled 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
Reallife 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.

Trading volume can be extremely heavy (Admati and Pfleiderer 1988; Karpoff 1987).

Stock returns exhibit shortrun momentum (positive autocorrelation) and longrun 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 rationalrepresentativeagent 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 rationalrepresentativeagent models, such as the experimentally documented deviations of investors’ behavior from rationality and/or the heterogeneity of investors? The agentbased simulation approach offers us a tool to investigate this question. The strength of the agentbased 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 rationalrepresentativeagent 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 exante return distribution is required. However, since the exante distribution is unknown, the EMB investors use the expost distribution in order to estimate the exante 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 exante distribution from the expost 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 exante return distribution (even though they are looking at the same series of historical returns). Moreover, some investors may use the objective expost probabilities when constructing their estimation of the exante distribution, whereas others may use biased subjective probability weights. In order to build the analysis stepbystep, we start by analyzing the case in which the EMB population is homogeneous, and then introduce various forms of heterogeneity into this population.
An important issue in market modeling is that of the degree of investors’ rationality. Most models in economics and finance assume that people are fully rational. This assumption usually manifests itself as the maximization of an expected utility function by the individual. However, numerous experimental studies have shown that people deviate from rational decisionmaking (Thaler 1993, 1994; Tversky and Kahneman 1981, 1986, 1992). Some studies model deviations from the behavior of the rational agent by introducing a subgroup of liquidity or “noise” traders. These are traders that buy and sell stocks for reasons that are not directly related to the future payoffs of the financial asset – their motivation to trade arises from outside of the market (e.g., a “noise trader’s” daughter unexpectedly announces his/her plans to marry, and the trader sells stocks because of this unexpected need for cash). The exogenous reasons for trading are assumed random and thus lead to random or “noise” trading (see Grossman and Stiglitz 1980). The LLS model takes a different approach to the modeling of noise trading. Rather than dividing investors into the extreme categories of “fully rational” and “noise traders,” the LLS model assumes that most investors try to act as rationally as they can, but are influenced by a multitude of factors causing them to deviate to some extent from the behavior that would have been optimal from their point of view. Namely, all investors are characterized by a utility function and act to maximize their expected utility; however, some investors may deviate to some extent from the optimal choice which maximizes their expected utility. These deviations from the optimal choice may be due to irrationality, inefficiency, liquidity constraints, or a combination of all of the above.

Excess volatility

Shortterm momentum

Longerterm return mean reversion

Heavy trading volume

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 rationalrepresentativeagent 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 boundedrational behavior and heterogeneity are in many cases impossible to analyze analytically, agentbased 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.
 (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.
 (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., \( {\tilde{D}}_t={D}_{t1}\left(1+\tilde{z}\right) \), 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).
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.
However, given D _{ t }, the RII investors know the distribution of \( {\tilde{D}}_{t+1} \):
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 exante return distribution for the stock, but as the exante distribution is not available, the EMBs assume that the process generating the returns is fairly stable, and they employ the expost distribution of stock returns in order to estimate the exante return distribution.
Deviations from Rationality
The noise can be added to the decisionmaking 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 decisionmaking of the EMB investors.
Market Clearance
AgentBased 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 twoway interaction between wealth and price dynamics, which is crucial to the understanding of the market.
Thus, one contribution of the agentbased 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 agentbased 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
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.

Number of investors = 1,000.

Riskaversion parameter α = 1.5. This value roughly conforms with the estimate of the riskaversion 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 oneperiod dividend decrease z _{1} = −0.07.

Maximal oneperiod 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.
No Excess 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 rationalrepresentativeagent 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 rationalrepresentativeagent 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 exante return distribution. Namely, the first type looks at the set of the last m_{1} expost returns, whereas the second type looks at the set of the last m_{2} expost 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 m_{1} 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 representativeagent models.
Finally, we present the case where there is an entire spectrum of EMB investors differing in the number of expost observations they take into account when estimating the exante distribution. This general case generates very realisticlooking market dynamics with all of the abovementioned market anomalies.
Homogeneous Subpopulation of EMBs
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 exante return distribution, because the set of expost returns they employ to estimate the exante 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 exante 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 expost 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 a–b in Fig. 3).
When the price is at the high level (segment b–c), 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 expost 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 c–d). 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 expost 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 expost 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 boomcrash cycle is initiated. This cycle repeats itself over and over almost periodically.
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.
Heavy Trading Volume
As explained above, shares change hands continuously between the RII investors and the EMB investors. When a “boom” starts, the RII investors observe higher expost 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
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 1,000period 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
The twoEMBpopulation 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 twoEMBpopulation 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 expost observations which he/she utilizes to estimate the exante 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).
The heterogeneous EMB population model generates the following empirically observed market phenomena:
Return Autocorrelation: Momentum and Mean Reversion
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 exante 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 expost 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 exante 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 shortterm 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.
Discussion of the LLS Results
The LLS model is an agentbased 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.

Shortterm momentum

Longerterm mean reversion

Excess volatility

Heavy trading volume

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 boundedrational 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 presentday knowledge in economic theory. With the agentbased 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 agentbased 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 agentbased simulations. Similarly, when results emerge from agentbased 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 agentbased 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 agentbased approach will prove as fruitful in economics as it has been in so many other branches of science.
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