{'fraction_non_alphanumeric': 0.02214885128628422, 'fraction_numerical': 0.006328243224652634, 'mean_word_length': 4.978070175438597, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Emergent Statistical Wealth Distributions in Simple Monetary Exchange Models: A Critical Review

This paper reviews recent attempts at modelling inequality of wealth as an emergent phenomenon of interacting-agent processes. We point out that recent models of wealth condensation which draw their inspiration from molecular dynamics have, in fact, reinvented a process introduced quite some time ago by Angle (1986) in the sociological literature. We emphasize some problematic aspects of simple wealth exchange models and contrast them with a monetary model based on economic principles of market mediated exchange. The paper also reports new results on the influence of market power on the wealth distribution in statistical equilibrium. As it turns out, inequality increases but market power alone is not sufficient for changing the exponential tails of simple exchange models into Pareto tails.


Introduction
Since the days of Vilfredo Pareto, the frequency distribution of wealth among the members of a society has been the subject of intense empirical research. Recent research confirms that power-law behaviour with an exponent between 1 and 2 indeed seems to characterize the right tail of the distribution (Levy and Solomon, 1997;Castaldi and Milakovic, 2005). However, when applied to the entire shape of the empirical distribution, the power law would produce a rather mediocre fit and would be outperformed by other candidate processes like the lognormal or Gamma distributions. As it seems to emerge from the literature, a transition occurs in the data from an exponential shape to powerlaw behavior somewhere above the 90 percent quantile again.
These and other findings should give rise to modelling efforts explaining the remarkably similar wealth distribution of many developed countries. Unfortunately, economic theory has been quite silent on this topic for a long time. Until recently, one had to go back the to literature of the fifties and six-ties (e.g., Champernowne, 1953;Mandelbrot, 1961) to find stochastic models of wealth accumulation in modern societies. Recent advances in computer technology, however, open another avenue for analysis of the emergence of wealth distributions allowing this issue to be studied in a computational agentbased framework. Such a bottom-up approach could, in principle, be helpful in isolating the key mechanisms that apparently lead to a stratification of wealth in advanced economies. As it appears, this path has been pursued recently by physicists rather than economists (cf. Bouchaud and Mézard, 2000;Drǎgulescu and Yakovenko, 2000;Chakraborty and Chakrabarty, 2000;Silver, Slad and Takamoto, 2002, among others). However, it has been entirely overlooked in the pertinent publications that these models have an important predecessor in the sociological literature. Investigating essentially the same structures already almost twenty years ago, Angle, 1986, might be considered as the first contribution to agent-based analysis of wealth formation. In the following, I will shortly review Angle's interesting work as the prototypical agent-based model of wealth dynamics, based on particle-like microscopic interactions of agents. I will point out aspects of this class of models (covering most of the econophysics contributions mentioned above) that would be considered to be problematic by economists (section 2). As an alternative framework, I will, then, review the contribution by Silver et. al. (2002) which much better fits into standard economic reasoning, but nevertheless provides a similarly simple formalization of an agent-based exchange model (section 3). Section 4 presents some additional results expanding on the seminal framework of Silver et. al. Conclusions are in section 5.

Angle's Surplus Theory of Social Stratification and the Inequality Process
In a long chain of papers covering more than 15 years, sociologist John Angle has elaborated on a class of stochastic processes which he first proposed in 1986 as a generating mechanism for the universal emergence of inequality in wealth distributions in human societies. His starting point is evidence he attributes to archeological excavations that inequality among the members of a community is typically first found with the introduction of agriculture and the ensuing prevalence of food abundance: While simpler hunter/gatherer societies appear to be rather egalitarian, production of a "surplus" beyond subsistence level immediately seems to lead to a "ranked society" or some kind of "chiefdom" (Angle, 1986, p. 298).
So as soon as there is some excess capacity of food, processes seem to be set into motion from which inequality emerges. Angle, surveying earlier narrative work in sociology, sees this as the result of redistribution by which some members of society succeed in grabbing some of the surplus wealth of others. The relevant empirical observations are summarized as follows: "Proposition 1: Where people are able to produce a surplus, some of the surplus would be fugitive and would leave the possession of the people who produce it. ... Proposition 2: Wealth confers on those who possess it the ability to extract wealth from others. So netting out each person's ability to do this in a general competition for surplus wealth, the rich tend to take surplus away from the poor." (Angle, 1986, p. 298).
The process he designs as a formalisation of these ideas is a true interacting particle model: in a finite population, agents are randomly matched in pairs and try to catch part of the other's wealth. A random toss D t ∈ {0, 1} decides which of both agents is the winner of this conflict. Angle in various papers considers cases with equal winning probabilities 0.5 as well as others with probabilities being biased in favor of either the wealthier or poorer of both individuals. If the winner of this encounter is assumed to take away a fixed proportion of the other's wealth, ω, the simplest version of the "inequality process" leads to a stochastic evolution of wealth of individuals i and j who had bumped into each other according to: Time t is measured in encounters and one pair of agents from the whole population is chosen for this interaction in each period. Angle (1986) shows via simulations that this dynamics leads to a stationary distribution which can be reasonably well fitted by a Gamma distribution. Angle (1993) provides an argument for why the Gamma distribution approximates the equilibrium distribution of the process for empirically relevant values of its parameters. Later papers provide various extensions of the basic model. While the exponential decay of the Gamma distribution might not be in accordance with power law behavior at the high end of the richest individuals, Angle's model is the first agent-based approach matching several essential features of empirical wealth distributions which he carefully lists as desiderada (i.e. stylized facts) for a theory of inequality. Among other properties, he emphasizes the uni-modality with a mode above minimum income which could not be reproduced by a monotonic distribution function. Angle is also careful to point out that with binned data, realizations of his process would be hard to distinguish from realizations of Pareto random variables which he demonstrates via a few Monte Carlo runs.
Unfortunately, Angle's process might be hard to accept for economists as a theory of the emergence of inequality in market economies.
First, a glance at the list of the six mechanisms for appropriation of another agent's wealth might raise doubts about their relative importance in modern societies: for most countries of the world, "theft" should perhaps not be the most eminent mechanism for stratification of the wealth distribution. Note also that "genuinely voluntary exchange" is listed only at rank 5 and behind "exchange coerced by unequal power". However, voluntary exchange is at the heart of economic activity at all levels of development rather than being a minor facet.
However, despite being mentioned in the list of mechanisms of redistribution, voluntary exchange is not really considered in Angle's model in which an agent simply takes away part of the belongings of another. What is more, this kind of encounter would -in its literal sense -hardly be imaginable as both agents would rather prefer not to participate in this game of a burglar economy -at least if they possess a minimum degree of risk aversion. The model, thus, is not in harmony with the principle of voluntary participation of agents in the hypothesized process which economists would consider to be an important requirement for a valid theory of exchange activities. One should also note that another problem is the lack of consideration of the measurement of wealth (in terms of monetary units) and the influence of changes of the value of certain components of overall wealth.
Despite these problematic features from the viewpoint of economics, Angle's model deserves credit as the first contribution in which inequality results as an emergent property of an agent-based approach. A glance at the recent econophysics literature shows that the basic building blocks of practically all relevant contributions share the structure of the inequality process formalized by equation (1). The inequality process is, for example, practically identical to the process proposed by Bouchaud and Mézard (2000) and isomorphic to almost all other models mentioned above. This recent strand of research on wealth dynamics is, therefore, almost exemplary for the lack of coordination among research pursued on the same topic in different disciplines and for the unfortunate duplication of effort that comes along with it.
Interestingly, the above criticism concerning the structure of the exchange process had also been voiced in a review of monetary exchange models developed by physicists by Hayes (2002) who introduced the label of "theft and fraud" economies, but restricted it to variants in which the richer could lose more (in absolute value) than the poor. However, it is not clear why models which introduce a certain asymmetry to avoid this kind of exploitation should not also suffer from the lack of willingness of agents to participate in their exchange processes. It, therefore, appears that one might wish to reformulate the "burglar economies" in a way that brings elements of voluntary economic exchange processes into play. While the economics literature has not elaborated on wealth distributions emerging from exchange activities within a group of agents, a huge variety of approaches is available in economics that could be utilized for this purpose. An interesting start has been made in a recent paper by Silver, Slud and Takamoto (2002) which contains a two-good general equi-librium model of an economy with heterogenous agents. Somewhat ironically, the overall outcome of this model is the same as with the inequality process: the stationary wealth distribution turns out to be a Gamma distribution.

An Exchange Economy with Changing Preferences
Unlike the framework reviewed in the previous section, the setting of Silver et al. is an extremely familiar one for economists. Their economy consists of two goods, denoted x and y which necessitate the introduction of a relative price p being defined as the current value of a unit of good y in units of good x. Note that with this assumption, considerations of revaluation of wealth components come into play which are altogether neglected in the sociological/physical models. All agents of the economy have their preferences formalized by a so-called Cobb-Douglas utility function: ( Here, i and t are indices for the individuals and time, respectively. x i,t and y i,t are, therefore, the possessions of good x and y by individual i at time t and f i,t ∈ [0, 1] is a preference parameter which might differ among individuals and, for one and the same individual, might also change over time. U i,t , then, is utility gained by individual i at time t. Individuals start with a given endowment in t = 0 and try to maximize their utility via transactions in a competitive market where one good is exchanged against the other. Given their possessions of both goods at some time t − 1, it is a simple exercise to compute their demands for goods x and y at time t given the current preference parameter f i,t : In (3), we have used the standard assumption that agents take the price as given in a competitive market. Note that this market, therefore, dispenses with any assumption of unequal exchange or even exploitation which is so central to the microscopic process of the previous chapter.
Summing up demand and supply by all our agents, we can easily calculate the equilibrium price which simultaneously clears both markets: After meeting in the market, each agent possesses a different bundle of goods and his wealth can be evaluated as: The driving force of the dynamics of the model by Silver et al. is simply the assumption of stochastically changing preferences: all f i,t are drawn anew in each period independently for all individuals. In the baseline scenario, the f i,t are simply drawn from a uniform distribution over [0, 1], but other distributions lead to essentially the same results. The dynamics is, thus, generated via the agents' needs to rebalance their possessions in order to satisfy their new preference ordering. With all agents attempting to change the composition of their "wealth", price changes are triggered because of fluctuations in the overall demand for x and y. This leads to a revaluation of agents previous possessions, x i,t−1 and y i,t−1 , and works like a capital gain or loss.
To summarize, we have a model in which all agents are identical except for their random preference shocks and no market or whatsoever power is attributed to anyone. The resulting inequality (illustrated as the benchmark case p m = 0 in Fig. 1) is, therefore, the mere consequence of the eventualities of the history of preference changes and ensuing exchanges of goods. We, therefore, do not have to impose any type of "power" in order to endogeneously generate a stratification of the wealth distribution that -like the model of section 2 -is able to capture all except the very end (the Pareto tail) of the empirical data.

Some Extensions of the Monetary Exchange Model
The model by Silver et al. demonstrates that stratification of wealth can result from an innocuous exchange dynamics without agents robbing or fleecing each other. It should, therefore, be a promising avenue to supplement the simpler dynamic models in the previous section. In some extensions, we, therefore, tried to explore the sensitivity of this approach to certain changes of its underlying assumptions. Among the many sensitivity tests we could imagine, we started with the following variations of the basic framework: • replacement of market interaction by pairwise exchange, • introduction of agents with higher bargaining power so that the outcome of pairwise matches could differ from a competitive framework, • introduction of natural differences among agents of some kind: here we assumed that for part of the population, preference changes are less pronounced than for others, • introduction of savings via a framework which allows for money as an additional component in the utility function.
Due to space limitations, we will not provide detailed results on all of these experiments, but will rather confine ourselves to one particularly interesting variant: the introduction of market power. Introducing market power of some sort is certainly interesting in light of the focus of the sociological and physics-inspired literature on issues of power of some individuals over others. Different avenues for implementing market power seem possible. Here, for the sake of a first exploration of this issue, we chose a very simple and extreme one. We assume that part of the population can act as monopolists in pairwise encounters: if they are matched with an agent from the complementary subset of non-monopolists, they can demand the monopoly price. If two non-monopolists are matched, we compute the competitive solution. We do the same when two monopolists meet each other assuming that their potential monopolistic power cancels out. Although this is an almost trivial insight in economics, it should be noted that the monopolist is not entirely free in dictating any price/transaction combination, but has to observe the constraint that the other agent has to voluntarily participate in the transaction. Since the option to not agree on the transaction would leave the monopolist with a zero gain as well, even in this extreme market scenario "exploitation" is much more limited than in a world of "theft and fraud". Note also that although one could perhaps speak of exploitation (when comparing the monopoly setting with the competitive price), no expropriation is involved whatsoever since even the non-monopolist will still increase his utility by his transaction with the more "powerful" monopolist.
As it turns out, allowing for monopoly power indeed changes the resulting wealth distribution. Fig. 1 shows the pdf for (fixed) fractions of monopolists. Varying the proportion of monopolists from 0 (the former competitive scenario with pair-wise transactions) to 0.4 we see a slight change in the shape of the distribution. As it happens all distributions still show pronounced exponential decline and can be well fitted by Gamma distributions. However, the estimated parameters of the Gamma distribution show a systematic variation. In particular, the slope parameter decreases with the fraction of monopolists, p m . A closer look at the simulation results also shows that the average wealth of monopolists exceeds that of other agents but the difference decreases with increasing p m . Note that the Gini dispersion ratio (G) is a negative function of λ for the Gamma distribution: G = Γ (λ+0.5) π 2 Γ (λ+1) , so that the increasing inequality would also be indicated by this popular statistics. The result that monopoly power is not neutral with respect to the distribution of wealth is certainly reassuring. However, we may also note that its introduction in the present framework does not lead to a dramatic change of the shape of the distribution. In particular, it does not seem to lead to anything like a Pareto tail in place of the exponential tail of the more competitive society. Since we have already chosen the most extreme form of market power in the above setting it seems also unlikely that one could obtain widely different results with milder forms of bargaining power.

Conclusions and Outlook to Future Research
What kind of conclusions can be drawn from this review of different approaches to agent-based models of wealth stratification? First, it is perhaps obvious that this author would like to advocate an approach in line with standard principles of economic modelling. If one is not willing to follow the emphasis of the sociological literature on all types of exertion of power, and if one tends to the view that wealth is influenced more by legal economic activity than by illegal theft and fraud, economic exchange should be explicitly incorporated in such models. This would also help to identify more clearly the sources of the changes of wealth. Note that despite the voluntary participation of agents in the exchange economy and the utility-improving nature of each trade, a change in the distribution of wealth comes with it. The difference to earlier models is that the changes in wealth are explained by deeper, underlying economic forces while they are simply introduced as such in the models reviewed in sec. 2. Market exchange models also allow to consider changes of monetary evaluation of goods and assets as a potentially important source of changes in an individual's nominal wealth.
Unfortunately, monetary exchange so far does not provide an explanation of the power-law characterizing the far end of the distribution. As we have shown above, even an extremely unequal distribution of market power within the population seems not sufficient to replicate this important empirical feature. Following recent proposals in the literature one could try additional positive feedback effects that give agents with an already high level of wealth an additional advantage (West, 2005;Sinha, 2005).
In the above model, one could argue that the more wealthy agents would also acquire more bargaining power together with their higher rank in the wealth hierarchy. Whether this would help to explain the outer region, remains to be analyzed. However, there are perhaps reasons to doubt that the Pareto feature might be the mere result of clever bargaining. A glance at the Forbes list of richest individuals (analyzed statistically by Levy andSolomon, 1997, andCastaldi andMilakovic, 2005) reveals that the upper end of the distribution is not populated by smart dealers who in a myriad of small deals succeeded to outwit their counterparts. Rather, it is the founders and heirs of industrial dynasties and successful companies operating in new branches of economic activity whom we find there 1 . The conjecture based on this anecdotal evidence would be that the upper end of the spectrum has its roots in risky innovative investments. Few of these succeed but the owners behind the succeeding ones receive an overwhelming reward. This would suggest that models without savings and investments should lack a mechanism for a power law tail. One would, therefore, have to go beyond such conservative models and combine their exchange mechanism (which works well for the greater part of the distribution) with an economically plausible process for the emergence of very big fortunes.