Simulation of a generalized asset exchange model with economic growth and wealth distribution

Although economies are complex systems that are difficult to understand [1], the consideration of simple models can provide insight if the questions are of a statistical nature and about the economy as a whole. One such question is whether economic growth can benefit all members of society [1, 3, 2]. Another question is to what degree is wealth accumulation a natural consequence of the way that wealth is exchanged and distributed. Whether these questions and others can be treated using methods appropriate to equilibrium systems is not settled [4].

In this paper we approach these questions by simulating an agent-based model that incorporates wealth exchange, economic growth, and its distribution. Agent-based wealth-exchange models exhibit a rich phenomenology [5, 6, 7, 8, 9, 10, 11, 12]. The agent-based asset-exchange model we generalize belongs to a class of wealth exchange models that has been of considerable interest and has provided insight into how economies function [13, 14, 15, 16, 17, 22, 23, 18, 19, 20, 21]. In these models the amount of wealth transfer is determined stochastically to represent the uncertainty of the value of an agent’s assets.

A particularly interesting agent-based model that incorporates wealth exchange is the Yard-Sale model [8, 10, 7, 14, 24, 25, 22, 21, 13, 19, 9, 17, 20, 18, 11, 12, 15, 16, 5, 6, 23]. In this model two agents are chosen at random and a fraction $f$ of the wealth of the poorer agent is transferred to the winning agent. The latter is determined at random with probability 1/2. If an equal amount of wealth is initially assigned to the $N$ agents, the result is that after many exchanges, the wealth is concentrated among increasingly fewer agents, culminating, in the limit of infinitely many wealth exchanges and $N\to\infty$, to a single agent holding a finite fraction of the total wealth [14, 5, 23, 22, 9, 15, 17, 21, 25].

To investigate the effect of economic growth on the wealth distribution, we generalize the Yard-Sale model so that the wealth $\mu W(t)$ is added to the system after $N$ exchanges, where $W(t)$ is the total wealth at time $t$ and the time $t$ is defined such that one unit of time corresponds to $N$ exchanges; $W(t)$ grows exponentially with the rate $\mu>0$. The motivation for this type of growth is the expected annual increase in the gross domestic product [26, 27].

To distribute the increase of the total wealth due to growth to individual agents, we introduce the distribution parameter $\lambda\geq 0$ and assign the added wealth to agent $i$ according to their wealth $w_{i}(t)$ at time $t$ as

The form of Eq. (1) implies that as $\lambda$ increases, the allocation of the added wealth is weighted more toward agents with greater wealth. We will refer to the model with the incorporation of exponential growth of the total wealth and the $\lambda$-dependent distribution mechanism in Eq. (1) as the Growth, Exchange, and Distribution (GED) model.

The motivation for the form of the wealth distribution in Eq. (1) is that in practice, not all agents benefit equally from economic growth, and that agents with more assets and resources are able to take more advantage of the growth of the economy. We argue in the Appendix that the allocation of growth according to Eq. (1) is consistent with economic data.

The distribution parameter $\lambda$, exchange parameter $f$, growth parameter $\mu$, and the number of agents $N$ determine the wealth distribution in the model. Our primary results can be grouped into two categories – the implications for economic systems and the implications for our understanding of the statistical mechanics of systems that are near-mean-field. We find that there is a phase transition at $\lambda=1$ such that for $\lambda<1$, all agents benefit from economic growth, there is economic mobility, and the system is in thermodynamic equilibrium. In contrast, for $\lambda\geq 1$, the system is non-stationary, there is no economic mobility, and there is wealth condensation as in the original Yard-Sale model. In the context of statistical mechanics we note that we can define an energy that satisfies the Boltzmann distribution for $\lambda<1$. The existence of the latter is consistent with the assumption that that the system is not just in a steady state, but is in thermodynamic equilibrium for $\lambda<1$.

The remainder of the paper is structured as follows: In Sec. II we show that the wealth distribution reaches a steady state and that wealth condensation is avoided for $\lambda<1$. In Sec. III we show that the GED model is effectively ergodic and that there is economic mobility for $\lambda<1$. In Sec. IV we find that we can define an energy that satisfies the Boltzmann distribution. We characterize the phase transition at $\lambda=1$ for a fixed number of agents in Sec. V.1 and estimate the critical exponents $\beta$, $\gamma$, and $\alpha$ associated with the behavior of the order parameter, its variance, and the energy, respectively. The result is that the exponents determined for fixed $N$ do not satisfy the usual scaling laws. The consequences of the system being describable by a mean-field theory [28] are discussed in Sec. V.2, where we introduce the Ginzburg parameter and show that scaling is restored if the transition is approached at fixed Ginzburg parameter rather than for fixed $N$. In Sec. VI we show that there is critical slowing down consistent with the mean-field theory predictions of Ref. [28]. We summarize and discuss our results in Sec. VII. In the appendix we argue from economic data that our method of distributing the growth is a reasonable zeroth order approximation.

Because the total wealth increases exponentially, it is convenient to introduce the rescaled wealth of an agent,

and consider the rescaled wealth distribution of the $N$ agents. That is, after the increased wealth due to growth is distributed to the agents, their wealth is scaled so that the total rescaled wealth equals $N$, the initial total wealth. In the following all references to the wealth of the agents will be to their rescaled wealth, and we will omit the tilde for simplicity.

The simulation of the GED model proceeds as follows:

1. 0.

   Usually, we assign the wealth of each agent at random and then rescale $w_{i}$ so that the total wealth $\sum_{i}w_{i}(t=0)$ is equal to $N$. The results for this section only are for $w_{i}(t=0)=1$.

2. 1.

   Choose agents $i$ and $j$ at random regardless of their wealth and determine the amount $f\min[w_{i}(t),w_{j}(t)]$ to be exchanged. Choose at random which agent gains and which agent loses.

3. 2.

   Implement step one $N$ times. Because the $N$ exchanges conserve the total wealth, the latter remains equal to $N$.

4. 3.

   Assign the additional wealth due to growth to the agents according to Eq. (1).

5. 4.

   Rescale $w_{i}(t)$ so that $W(t)=\sum w_{i}(t)=N$.

6. 5.

   Set $t=t+1$.

7. 6.

   Repeat steps 2–5 until a steady state wealth distribution is attained (for $\lambda<1$) and then determine the average values of the desired quantities of interest.

The simulations in this section are for $N=2500$, $f=0.1$, $\mu=0.001$, and various values of the distribution parameter $\lambda$. The qualitative results discussed here do not depend on the values of the number of agents $N$, the fraction of the poorer agents’s wealth that is exchanged $f$, and the growth parameter $\mu$.

We show in Fig. 1 the time dependence of the rescaled wealth distribution for $\lambda=0$, starting from the initial condition $w_{i}(t=0)=1$ for all $i$. The wealth disparity between richer and poorer agents initially increases until a steady state is established. Once a steady state is reached, the rescaled wealth distribution remains fixed, and the wealth in every rank increases as $e^{\mu t}$.

According to Eq. (1), the growth allocation is weighted more toward the richer agents as $\lambda\to 1^{-}$, thus leading to a less equal steady state rescaled wealth distribution (see Fig. 2). The time to reach a steady state increases as $\lambda$ approaches $1$, and as for $\lambda=0$, the wealth of all agents increases exponentially after a steady state has been reached. (The limit $\lambda\to 1$ will always be from below unless otherwise specified.) We find that for $0\leq\lambda<1$, “a rising tide lifts all boats” and all agents benefit from economic growth.

The time-dependence of the rescaled wealth distribution is shown for $\lambda=1$ in Fig. 3. A steady state is not reached, and the slope of the rescaled wealth distribution increases with time, corresponding to the accumulation of wealth by fewer and fewer agents until eventually a single agent gains almost all the wealth. Similar results are found for $\lambda>1$. The time for a single agent to dominate decreases as $\lambda$ increases for $\lambda>1$.

Note that $\lambda=1$ is a special case for which the increase in an agent’s wealth due to growth is proportional to the agent’s wealth. Hence, for $\lambda=1$, the ratio of the rescaled wealth of any two agents does not change after the distribution of the growth in wealth according to Eq. (1). Consequently, aside from the exponential growth of the total wealth, the entire dynamics of the system is driven only by the wealth exchange mechanism, and the evolution of the wealth distribution for $\lambda=1$ is identical to its evolution in the original Yard-Sale model; that is, the model with no economic growth.

The numerical results in Sec. II indicate that the GED model exhibits distinct behavior for $\lambda<1$ and $\lambda\geq 1$. In particular, for $\lambda<1$, all agents benefit from economic growth, whereas for $\lambda\geq 1$ only the richest agent becomes richer. In Sec. III.1 we show that the GED model is effectively ergodic for $\lambda<1$, but is not ergodic for $\lambda\geq 1$. In Sec. III.2 we find that the agents have nonzero economic mobility for $\lambda<1$, but have zero mobility for $\lambda\geq 1$. Unlike in Sec. II, we randomly assign the wealth of each agent at $t=0$ from a uniform distribution and then rescale the agents’ wealth so that the initial total wealth equals $N$.

We have seen that the GED model approaches a steady state and is effectively ergodic for $\lambda<1$. In the following, we will show that a reasonable definition of the total energy yields an energy distribution that is consistent with the Boltzmann distribution. The existence of the latter is consistent with the idea that that the system is not just in a steady state, but is in thermodynamic equilibrium for $\lambda<1$.

As discussed in Ref. [28] (following paper), a mean-field theory treatment of the GED model yields a quantity that can be interpreted as the total energy of the system at time $t$

Note that the energy is zero if all agents have the same wealth ($w_{i}=1$). Equation (8) yields a mean energy that is extensive, that is, $\langle E\rangle\propto N$ for fixed values of $\lambda$, $f$, and $\mu$. For example, for $\lambda=0.99$, $f=0.01$, and $\mu=0.1$, we find that $\langle E_{N=4000}\rangle/\langle E_{N=1000}\rangle=336.4/84.2=3.995\approx 4$.

The probability density $P(E)$ is shown in Fig. 6(a) for $N=5000$, $\lambda=0.8$, $f=0.01$, and $\mu=0.1$. As expected, $P(E)$ is fit well by a Gaussian. Fits of $P(E)$ to a Gaussian become less robust as $\lambda\to 1$ for fixed $N$, $f$, and $\mu$. We will discuss the behavior of $P(E)$ for larger values of $\lambda$ in Sec. V.2.

If the system is in thermodynamic equilibrium, we expect that the probability density $P(E,\beta)$ to be proportional to $g(E)e^{-\beta E}$, where $\beta$ is an effective inverse temperature that depends on $\lambda$, $f$, and $\mu$, and $g(E)$ is the density of states, which is independent of $\lambda$, $f$, and $\mu$ and hence independent of $\beta$. Because $g(E)$ is independent of $\beta$, the ratio $P(E,\beta_{1})/P(E,\beta_{2})$ is an exponential proportional to $\exp[-(\beta_{1}-\beta_{2})]E$ if the system is characterized by the Boltzmann distribution with the energy given by Eq. (8). The range of values of $E$ over which this ratio is nonzero and finite is limited by the overlap of the two probabilities, which becomes smaller as $N$ is increased. In Fig. 6(b) we see that the ratio $P(E,\beta_{2})/P(E,\beta_{1})$ is consistent with the Boltzmann distribution $e^{-(\beta_{2}-\beta_{1})E}$ for $N=5000$, $\lambda=0.8$, and $\mu=0.1$, with $\beta\propto f^{-1}$, $f_{2}=0.0095$ and $f_{1}=0.01$; a larger value of $f$ (for fixed values of $\lambda$ and $\mu$) corresponds to a smaller value of $\beta$ and hence a higher value of the effective temperature. The dependence of the effective temperature $\beta^{-1}$ on $f$ is consistent with the association of $f$ with the presence of both additive and multiplicative noise in the system [28]. Similar results hold for two similar values of $\lambda$ for fixed $f$ and $\mu$. The existence of an energy, and the observation that its probability is proportional to the Boltzmann distribution implies that the system can be considered to be in thermodynamic equilibrium, at least for $\lambda=0.8$.

We find that various time scales increase rapidly as $\epsilon\to 0$, which limits how close the simulations can approach the transition. One time scale of interest is $\tau_{\rm ra}$, the mean lifetime of the richest agent. We expect that if the mobility of the agents is nonzero, then the richest agent at $t=0$ will no longer be the richest after some time has elapsed. We define $\tau_{\rm ra}$ as the mean time that a particular agent remains the richest and assume that $\tau_{\rm ra}$ is a simple measure of the decorrelation time of the wealth of individual agents.

Another time scale of interest is the mixing time $\tau_{m}$ associated with the time-dependence of the wealth metric; $\tau_{m}$ is related to the inverse slope of the wealth metric as

We also computed the time-displaced energy autocorrelation function given by

where $E(t)$ is the value of the energy of the system at time $t$. We find that $C_{E}(t)$ relaxes exponentially, and hence we can extract the energy decorrelation time $\tau_{E}$. Our simulation results for these various times are for fixed Ginzburg parameter ($G=10^{6}$).

For fixed $G$ the simulation results for the $\epsilon$-dependence of $\tau_{\rm ra}$ are shown in Fig. 13(a) and are consistent with the power law $\epsilon^{-1}$. To obtain accurate results for the wealth metric $\Omega(t)$, we averaged $\Omega(t)$ over ten origins and found that the linear dependence of $\Omega(0)/\Omega(t)$ holds over a wide range of $t$ and yields robust values of $\tau_{m}$. The exponential dependence of $C_{E}(t)$ holds for $t\lesssim 5\tau_{E}$, yielding some uncertainty in the fitted values of $\tau_{E}$. The $\epsilon$-dependence of $\tau_{m}$ and $\tau_{E}$ are shown in Fig. 13(b). We see that both $\tau_{m}$ and $\tau_{E}$ increase rapidly as $\epsilon\to 0$ and that their $\epsilon$-dependence is consistent with $\epsilon^{-2}$.

Although the mean-field theory [28] makes no direct predictions for the mixing time, the $\epsilon$-dependence of $\tau_{m}$ can be understood by noting that the metric measures the time it takes for the average wealth of each agent to equal the global average. Because the time it takes for the richest agent to cease being the richest and for another agent to assume that role diverges as approximately $\epsilon^{-1}$, the time for a system of $N$ agents to mix is $N\epsilon^{-1}$. Because $N\propto\epsilon^{-1}$ for fixed $G$ [see Eq. (16)], we find that $\tau_{m}\sim\epsilon^{-2}$ in agreement with the simulations.

The mean-field theory of Ref. [28] predicts that the decorrelation time for fixed $G$ scales as

The reason for the apparent discrepancy between the $\epsilon$-dependence predicted by Eq. (23) and our simulation results for $\tau_{m}$ and $\tau_{E}$ is that the time unit in the simulations corresponds to $N$ exchanges, during which one agent exchanges wealth with only one agent on the average. In contrast, the applicability of mean-field theory requires that in one time unit, one agent exchanges wealth with $N$ agents on the average. Hence, the simulation and mean-field time units differ by a factor of $N$ or $1/\epsilon$ for fixed Ginzburg parameter.

The divergent behavior of $\tau_{m}$ and $\tau_{E}$ are examples of critical slowing down, which is associated with a cooperative effect and is not a property of a single agent. In contrast, $\tau_{\rm ra}$ is a property of a single agent rather than of the system as a whole and becomes independent of $\epsilon$ if we define the time as required by the applicability of mean-field theory.

Although the results of our simulations are consistent with the $\epsilon$-dependence in Eq. (23) for constant Ginzburg parameter, Eq. (23) also predicts that $\tau_{E}$ is independent of the values of $f$ and $N$. As discussed in Ref. [28], the derivation of Eq. (23) neglects the effects of both additive and multiplicative noise. The weak dependence of $\tau_{m}$ and $\tau_{E}$ on $N$ and $f$, as well as their dependence on $\mu$, is discussed in Ref. [28].

We have generalized the Yard-Sale model to incorporate economic growth and its distribution according to the wealth of the agents as determined by Eq. (1) and the parameter $\lambda$. Our numerical results suggest that there are two phases. For $\lambda<1$ the system reaches a steady state with economic mobility, is effectively ergodic, and can be considered to be in thermodynamic equilibrium. In contrast, for $\lambda\geq 1$ there is no economic mobility, the system does not reach a steady state, and in the limit $t\to\infty$, there is condensation of a finite fraction of the system’s wealth in a vanishingly small number of agents. In addition, the system is not ergodic and shares some of the characteristics of the geometric random walk which is also not ergodic and cannot be treated by equilibrium methods [4, 38].

It is remarkable that it is possible to define a thermodynamic energy for a system that involves wealth and has no obvious energy analogue. The interpretation of the energy and its variance is subtle and thermodynamic consistency is achieved only if the mean-field limit is taken appropriately.

We showed in Sec. IV that $P(E)$, the probability density of the energy of the system, is well fit by a Gaussian function for $N=5000$ and $\lambda=0.8$. Simulations for $N=5000$ and values of $\lambda$ much closer to one show departures from a Gaussian, even though the wealth fluctuation metric still indicates that the system is effectively ergodic. The deviation of $P(E)$ from a Gaussian for fixed $N$ (and fixed $\mu$ and $f$) is due to the fact that $G$ decreases as $\lambda\to 1$ and eventually becomes too small for mean-field theory to be applicable. Simulations for fixed Ginzburg parameter begin to show deviations from a Gaussian distribution for $\lambda$ much closer to one. Although the Ginzburg parameter is large and fixed, the importance of multiplicative noise increases as $\lambda\to 1$, and eventually the effect of the multiplicative noise can no longer be ignored [28].

Our numerical values of the various exponents are consistent with the mean-field theory of Ref. [28], but their estimated numerical values must be viewed with caution because they are obtained by extrapolation over a limited range of $\lambda$ and for finite values of $G$ and $N$. Much larger values of $G$ would be needed to obtain more accurate numerical results for $\lambda$ closer to one.

The transition at $\lambda=1$ is from a system in thermodynamic equilibrium for $\lambda<1$ to a system that undergoes wealth condensation for $\lambda\geq 1$. In Ref. [28] it is shown that the evolution of the model for $\lambda\geq 1$ is the same as unstable state evolution in the fully connected Ising model for model A dynamics [39]. In Fig. 14 we show the evolution of the wealth of the richest agent after the value of $\lambda$ has been changed from $\lambda=0.8$ to $\lambda=1.01$ and to $\lambda=1.05$. We see that the wealth of the richest agent initially increases exponentially. We also find that the duration of exponential growth decreases for larger values of $\lambda$ after the change (not shown).

Our results have possible important economic implications. As $\lambda$ is increased, the benefits of growth are weighted more toward the wealthy, and wealth inequality increases. Nevertheless, as long as $\lambda<1$, the wealth of agents of all ranks grows at the same rate once a steady state is reached, and all agents benefit from economic growth. However, if the benefits of growth are skewed too much toward the wealthy ($\lambda\geq 1$), poor and middle rank agents no longer benefit from economic growth, and wealth condensation occurs. For $\lambda=1$ the model reduces to the geometric random walk with resulting wealth condensation [4], as shown in the following paper [28].

There is some question whether economic systems can be treated as being in equilibrium or even exhibit effective ergodicity [4, 38, 10]. Our results suggest that ergodicity and equilibrium may depend on various system parameters. Because parameters such as $\lambda$ and $\mu$ are not temporal constants in real economies, our results also suggest that the applicability of equilibrium methods may be situational and vary with time.

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