An agent-based model of the observed distribution of wealth in the United States

The Pareto distribution was first discovered by Vilfredo Pareto in the late 19th century (Chipman 1976; Persky 1992). He found that, regardless of how an economy is structured, the wealth and income distributions among the upper tier of the population (typically, the upper 10% of a population is examined) follow a power law, which came to be known as a Pareto distribution (Jones 2015). The Pareto distribution is given by the relationwhere \(\omega _m\) is some minimum wealth and \(\alpha \) is the Pareto index. Equation 1 states that the probability of someone’s wealth \(\omega ^*\) being greater than \(\omega \) is proportional to \(\omega ^{-\alpha }\). Thus, it is a power law, and, in most economies, \(\alpha \) is between 1 and 3 (Ghosh et al. 2016; Gabaix 2009; Chipman 1976).

Rapid improvements in the availability of data on wealth have reinvigorated empirical and theoretical investigation of the Pareto distribution. One recent discovery is that the lower 90% of the population does not follow a power law, but rather follows an exponential distribution¹ (Drăgulescu and Yakovenko 2001), which is defined by the probability density functionwhere \(T_{\omega }\) is the “wealth temperature” and c is a normalizing constant. These exponential and Pareto distributions fit to observed wealth distributions in the United Kingdom (UK) are illustrated in Fig. 1, which is taken from Drăgulescu and Yakovenko (2001) and Yakovenko and Rosser (2009).

A common measurement of inequality in the distribution of wealth is the Gini coefficient. If we arrange the individuals in an economy by their wealth in m distinct groups such that \(\{\omega _i\}_{i=1}^{m}\) is non-decreasing (\(\omega _{i+1} \ge \omega _i\) for all i) and \(\varOmega =\sum _{i}^{m}{\omega _i}\) is the total wealth of the economy, then the Gini coefficient, G, is defined as:There have been many attempts to describe why wealth inequality arises (Yakovenko and Rosser 2009). The macro-economic perspective has shifted to Piketty’s findings published in his Capital in the Twenty-First Century (Piketty 2013). The model that emerges from Piketty’s work generates a Pareto distribution through the combined effects of several factors, some of most significant being birthrate, growth rate, and return rate (Jones 2015).

While Piketty’s work focuses almost exclusively on the Pareto distribution of wealth, a variety of econophysical models, which rely extensively on descriptions of stock market transactions (Yakovenko and Rosser 2009), either produce a distribution characteristic of the exponential distribution, or a power law, but not both. Moreover, by relying on specific economic data and processes to generate wealth inequality, these models are intrinsically tailored to a specific type of economy. Consequently, these models can explain inequality within a specific economic system, but they do not shed light on Pareto’s assertion that inequality is intrinsic to all economic systems.

In contrast, we propose a model that fits the entire wealth distribution without modeling any specific economic process other than growth. The fundamental assumption in this model is that individuals who possess greater wealth derive from it a greater ability to secure more wealth. We quantify this advantage and show that the resulting dynamic in a growing economy leads to a Pareto distribution of the wealthy minority and exponential distribution of the less wealthy majority. The model’s free parameter is calibrated to reproduce the observed distribution of wealth in the United States (US) from the period between 1988 and 2012. Projections with this model to 2088 produce a striking separation between the shrinking minority of super-rich and everyone else.

We assume an exponentially growing economywhere \(\varOmega (t)\) is the sum of the wealth of all individuals at time t and \(\lambda \) is the growth rate of the economy. In our model, the economy does not grow continuously, but instead grows in small, fixed increments \(\delta \omega \). Hence, over a time T, the number of increments n will beEach increment is given to exactly one individual. Let \(\omega _{i,k}\) be the wealth possessed by the ith individual following the kth distribution, with \(k=0\) indicating the initial wealth of i. The probability \(\varPsi (i,k+1)\) that i will receive increment \(k+1\) iswhere \(p(\omega _{i,k})\) is a “wealth power.” In our model, we letValues for \(\lambda \), \(\varOmega (0)\), and the initial \(\omega _{i,0}\) can be determined directly from data. We assume \(\beta \) to be a parameter intrinsic to the economy, and with it we can directly control the change in inequality. This is quickly explainable by Eq. 7. For \(0<\beta <1\), the power of wealthier individuals is diminished much more than the power of poorer individuals. For \(\beta =1\), the model gives all individuals a proportional amount of power, and for \(\beta >1\), the model gives a disproportionate amount of power to the richer individuals in the market. We will choose \(\beta \) by fitting economic trajectories produced by the model to historical data.

It is unclear how to select \(\delta \omega \), which is primarily a computational artifact and has no obvious analog in the real world. If \(\delta \omega \) is on the order of \(\omega _{i,0}\), then the first individual to win an increment accrues a tremendous advantage. This circumstance is unlikely to produce a realistic outcome and suggests that \(\delta \omega \) should be small. If \(\delta \omega \) is much smaller than \(\omega _{i,0}\), then those who win wealth increments gain a negligible advantage. Presumably, a very small \(\delta \omega \) would eventually approach the same distribution as a slightly larger \(\delta \omega \), but our simulations show that the distribution does not correctly form within a reasonable amount of simulated time, and in addition, the computational time implied by Eq. 5 puts a practical lower limit on \(\delta \omega \). We have therefore chosen a value of \(\delta \omega \) that is small in proportion to \(\omega _{i,0}\), but large enough that the individual who receives a wealth packet gains a marginal advantage. A more comprehensive study of how \(\delta \omega \) affects outcomes will be reserved for future work.

We will develop two closely related simulators for the proposed model: one which repeatedly distributes a single wealth packet and updates \(\varPsi (\cdot ,k+1)\) as described in Sect. 2, and one that distributes multiple packets between updates to \(\varPsi (\cdot ,k+1)\). The single-packet model will be used to reproduce the trajectory of wealth shares in the US. The multi-packet model is shown to produce trajectories similar to the single-packet model, but we leaves for future work an investigation of the larger populations enabled by the multi-packet model.

The value of \(\beta \) is central to the dynamics of our model, as experimentation with it on the presupposed exponential and Pareto distributions in Sect. 4.2 has shown that it controls the change in the wealth inequality in an economy. This is concurrent with the predicted role of \(\beta \) in Sect. 2. For \(0<\beta <1\), we observed that the value of the Pareto index \(\alpha \) would increase, implying less inequality in the population. For \(\beta =1\), the model would not alter the initial exponential and Pareto distributions for an extended period of time, maintaining its initial value of \(\alpha \). Lastly, with \(1<\beta <2\), we observed the distribution moving to lower values of \(\alpha \), suggesting more inequality. For any \(\beta \ge 2\), we found that the model would quickly diverge from the expected distributions, resulting in a “winner takes all” situation, where there was only one individual far wealthier than the rest of the population. For \(1<\beta <2\), we observed similar behavior, but the divergence was much slower.

Simulations have shown that \(\beta = 1\) is not sufficient to create an exponential and Pareto distributions with \(1<\alpha <3\) when the model is initialized with a uniform distribution. In most cases, \(\alpha >3\) for \(\beta =1\) when run for a long period of time. Lastly, when projecting from historical data, we found that the part of the population described by the Pareto distribution decreased in size as \(\alpha \rightarrow 1\) over time. We attribute this to the divergence of the model, and the closer \(\alpha \) was to 1, the more quickly a single individual became extremely wealthy, which is not in accordance with the exponential and Pareto distributions (for \(\beta >1\)).

The model we propose mirrors empirical wealth distributions in the US. The underlying assumption of the model is in accordance with Pareto’s view that human nature, and in particular our innate tendency to use our resources to improve our own position, is sufficient to generate an uneven distribution of wealth. Indeed, we may hypothesize that economic growth and the tendency for wealth to breed more wealth are central to simultaneously forming the exponential and Pareto distributions of wealth inequality.

The proposed model could also be used to generate income inequality under other appropriate assumptions. In particular, if we assume an increase in GDP is reflected as an increase in income for individuals across the economy, and if the wealth power function is reinterpreted as an income power function, then the model would distribute increases in income as GDP grows. Since there is no change in the dynamics, the model would mimic the exponential and Pareto distributions of income inequality as well.

The historical data considered here span a short period for two reasons. First, there is a scarcity of data quantifying the growth of inequality at other times. Second, from the data that was available to us, the period from 1988 to 2012 was the best example of sustained economic growth. Because sustained growth is a central assumption of the model, it was natural to focus our first validation study on that period. More extensive validation with an extended model that can account for greater economic volatility will be the subject of future research.

A more substantial limitation of the model when trying to reproduce long historical trajectories is its inability to account for social, economic, and geopolitical factors that induce very sudden returns to more equitable conditions. Prime examples of such upheaval in the United States are the Great Depression and the end of the Second World War. It is notable that there are no modern examples of wealth concentrating as our model anticipates for 2088, and this is inspite of the short 76 years that the projection covers (from 2012–2088). This fact seems to limit the model to relatively short spans between events that alter the dynamics of the economy in some significant way.

Nontheless, there is a need to understand the parameter \(\beta \) in more depth, as it seems to dictate the rate at which inequality grows or diminishes. One weakness in our approach to selecting \(\beta \) was a focus on fitting the Pareto part of the empirical data. The exponential growth of the economy combined with the preferential attachment of new wealth to already wealthy individuals is expected to produce an essentially exponential distribution of wealth; that is, the exponential part of the distribution is a given. Indeed, the Pareto portion of the simulated distributions shrink with time. Hence, our approach is centered on getting a good fit to the most transient part of the data. An interesting direction for future work would be to reverse this procedure by fitting \(\beta \) to the exponential majority of the data and then seeing how accurately the Pareto minority is matched by the model.

Another direction for future work is to explore what specific economic or human tendencies underlay \(\beta \). Of the many possibilities, at least two are intuitively appealing. One of these is that the rate at which an individual can consume wealth to maintain a standard of living is almost certainly limited (a point made by Hanauer (2014) when he stated “I earn about 1000 times the median American annually, but I don’t buy thousands of times more stuff. My family purchased three cars over the past few years, not 3000.”). At the lower end of the wealth spectrum, we might expect the vast majority of an individual’s wealth to be expended on basic needs. Hence, these individuals may have less opportunity - less excess wealth - with which to improve their position. At the upper end, the rate limit is reached and excess wealth accumulates, leaving vast reserves to seize new opportunities. Another closely related effect is the redistribution of wealth that occurs through economic activity (e.g., the purchase of a good or service has the effect of transferring wealth from buyer to seller), taxation, charities, and other mechanisms.

Exploring these types of possibilities requires a more refined model of the population to account for the statistical distribution of consumption behavior, engagement in wealth transferring activities, and other factors that vary, possibly substantially, from individual to individual. The scalable algorithm presented in Sect. 3.2 will be an important tool for these future investigations.