Mean-field theory of an asset exchange model with economic growth and wealth distribution

Agent-based asset exchange models have become useful aem1 ; aem2 ; krapiv ; aem3 ; aem4 ; aem5 ; oneofthefirst ; ijmp ; hayes ; Burda ; Chakraborti2 for studying the effects of chance on the distribution of wealth. These models consist of $N$ agents that can exchange wealth through pairwise encounters. Examples of the exchange mechanism include the transfer of a fixed amount of wealth and the exchange of a fixed percentage of the average of the wealth of the two agents. The common feature of these models is that the winner in the exchange is determined by chance.

Of particular interest is the Yard-Sale model aem2 ; oneofthefirst ; hayes ; popbogo ; ijmp ; Burda ; bogo ; bogo2 ; bogo3 ; devitt ; probyardsale ; Moukarzel ; rosser in which pairs of agents are chosen at random and one is designated as the winner with a probability usually taken to be 1/2. The winner receives a fraction $f$ of the wealth of the poorer agent. The result is that after many exchanges, one agent gains almost all of the wealth, a phenomena known as wealth condensation.

In this paper we study a generalization of the Yard-Sale model kang in which a fixed percentage $\mu$ of the total wealth is added to the system after $N$ exchanges. The added wealth is distributed according to

$$\Delta w_{i}(t)=\mu W(t)\frac{w^{\lambda}_{i}(t)}{\sum_{j=1}^{N}w^{\lambda}_{j}(t)},$$

(1)

where $w_{i}(t)$ is the wealth of agent $i$ at time $t$ and $\lambda\geq 0$ is the distribution parameter. The quantity $\mu W(t)$ is the change in the total wealth in the system at time $t$ due to economic growth, where $W(t)$ is the total wealth of the system at time $t$, and the parameter $\mu$ is the rate of growth. One unit of time corresponds to $N^{2}$ exchanges. This distribution mechanism is justified by economic data in the Appendix of Ref. kang, .

This model, which we denote as the growth, exchange and distribution (GED) model, was investigated numerically kang and shown to have a phase transition at $\lambda=1$. For $0<\lambda<1$ the wealth is not distributed uniformly, but wealth condensation is avoided. As $\lambda$ approaches 1 from below, the wealth distribution becomes more skewed toward the rich. However, there is economic mobility and poorer agents can become richer and richer agents can become poorer. In addition, every agent’s wealth increases exponentially as $e^{\mu t}$ due to economic growth as the system evolves with time. In contrast, for $\lambda\geq 1$ there is wealth condensation as was found in the original Yard-Sale model ($\mu=0$), and there is no economic mobility.

Numerical investigations indicate that the phase transition at $\lambda=1$ is continuous kang . The order parameter $\phi$ is defined as the fraction of the wealth held by all of the agents except the richest and goes to zero as $\lambda\rightarrow 1^{-}$ (for $N\rightarrow\infty$). Three exponents were introduced in Ref. kang, to characterize the behavior of various quantities as $\lambda\to 1^{-}$, including the order parameter $\phi\sim(1-\lambda)^{\beta}$ and the susceptibility $\chi\sim(1-\lambda)^{-\gamma}$ (the variance of the order parameter). As we will discuss in Sec. IV, we can define the total energy of the system and introduce the exponent $\alpha$ to characterize the critical behavior of the nonanalytic part of the mean energy as $(1-\lambda)^{1-\alpha}$. Similarly, we relate the specific heat to the variance of the energy and characterize its divergence as $(1-\lambda)^{-\alpha}$. Because there is no length scale in the GED model, there is no obvious way of defining a correlation length exponent.

Simulations at fixed values of $N$ in Ref. kang yield the estimates $\beta\approx 0$, $\gamma\approx 1$, $1-\alpha\approx-1$ and $\alpha\approx 2$. These values do not satisfy the scaling law scalingref

$$\alpha+2\beta+\gamma=2,$$

(2)

and do not appear to correspond to any known universality class.

In this paper we present a mean-field treatment of the GED model and find that the interpretation of the critical exponents is subtle. The theory shows that if the critical behavior is interpreted correctly, the exponents do satisfy Eq. (2) with $\beta=0$, $\gamma=1$ and $\alpha=1$. Moreover, we can define an energy and a Hamiltonian that allows us to obtain an equilibrium (Boltzmann) description of the GED model in the limit that the number of agents $N\to\infty$. The mean-field theory results are consistent with the simulations kang .

In addition to casting light on the nature of the critical point, the mean-field approach predicts that for $\lambda<1$, the wealth distribution can be made less skewed toward the rich by increased growth for fixed $N$, $\lambda$, and $f$. The mean-field approach also indicates that wealth inequality can be reduced for fixed $\lambda<1$ and fixed $N$ and $\mu$ by decreasing the value of $f$, corresponding to decreasing the magnitude of the noise. However, for $\lambda\geq 1$, economic growth does not avoid wealth condensation, and there is no economic mobility.

The structure of the remainder of the paper is as follows. In Sec. II we construct an exact differential equation for the GED model and then introduce the mean-field approximation to the equation. In Sec. III we show that there is a phase transition at $\lambda=1$ with critical slowing down, and obtain the values of the critical exponents $\beta$ and $\gamma$. In Sec. IV we introduce the Ginzburg parameter, define the total energy of the system, and determine the critical exponent $\alpha$. In Sec. V we compare the predicted mean-field exponents with the numerical estimates. We discuss the role of multiplicative noise in the GED model in Sec. VI and examine the relation between the GED model and the geometric random walk in Sec. VII. Finally in Sec. VIII, we discuss the implication of these results for critical phenomena in fully connected systems and systems with long but finite-range interactions, and discuss the implication of our results for the study of economic systems. Because the GED model is similar in several ways to the fully connected Ising model, we review some aspects of that model in the Appendix and discuss the Ginzburg criterion as a self-consistency check on the applicability of mean-field theory.

The rate of change of the wealth of agent $i$ is given by a formally exact stochastic difference equation

	$\displaystyle\frac{\Delta w_{i}(t)}{1}$	$\displaystyle=$	$\displaystyle f\sum_{j}\Theta\big{[}w_{i}(t)-w_{j}(t)\big{]}\eta_{ij}(t)w_{j}(t)$		(3)
			$\displaystyle{}+f\sum_{j}\Big{\{}1-\Theta\big{[}w_{i}(t)-w_{j}(t)\big{]}\Big{\}}\eta_{ij(t)}w_{i}(t)+\mu W(t)\frac{w_{i}(t)^{\lambda}}{S(t)}.$

The denominator on the left-hand side of Eq. (3) is written as 1 to emphasize that Eq. (3) is a difference equation rather than a differential equation. Here

$$\Theta\big{(}w_{i}-w_{j}\big{)}=\begin{cases}1&\big{(}w_{i}\geq w_{j}\big{)}\\ 0&\big{(}w_{i}<w_{j}\big{)},\end{cases}$$

(4)

and

$$S(t)=\sum_{i}w^{\lambda}_{i}(t).$$

(5)

The parameter $f$ is the fraction of the poorer agents’s wealth that is exchanged, $\mu$ is the fraction of the total wealth that is added after $N$ exchanges, the parameter $\lambda$ determines the distribution of the added economic growth, and $\eta_{ij}(t)$ for $\i\neq j$ is a time-dependent random matrix element such that

$$\eta_{ij}(t)=\begin{cases}0&\mbox{agents $i$ and $j$ do not exchange wealth}\\ 1&\mbox{wealth is transferred from agent $j$ to agent $i$}\\ -1&\mbox{wealth is transferred from agent $i$ to agent $j$}.\end{cases}$$

(6)

($\eta_{ij}=0$ if $i=j$.) The matrix elements of $\eta$ can be chosen from any probability distribution with the constraint that if $\eta_{ij}=\pm 1$, then $\eta_{ji}=\mp 1$. This condition imposes the constraint that the exchange conserves the total wealth.

To obtain a differential equation we multiply and divide the denominator on the left-hand side of Eq. (3) by $N$, the number of agents. Because we will take the limit $N\rightarrow\infty$ and take one time unit to correspond to $N^{2}$ exchanges, we have that $1/N\rightarrow dt$. Note that in the simulations of Ref. kang , $N$ exchanges was chosen as the unit of time. In this case each agent will, on the average, exchange wealth with only one other agent and hence one exchange described by the difference equation would not take place in an infinitesimal amount of time. One exchange per agent does take place in an infinitesimal time if one time unit corresponds to $N^{2}$ exchanges during which each agent exchanges wealth with every other agent on the average.

The parameters $f$ and $\mu$ in Eq. (3) are the rates of exchange and growth, respectively, and are defined per $N$ exchanges to be consistent with the simulations. To obtain a consistent differential equation, these rates need to be scaled by $N$. We let

$$f=f_{0}/N\mbox{ and }\mu=\mu_{0}/N,$$

(7)

and assume that $f_{0}$ and $\mu_{0}$ are independent of $N$. We will see that these theoretical considerations imply that the values of the parameters $f$ and $\mu$ chosen in the simulations must be scaled with $N$ if the Ginzburg parameter is held fixed.

Because wealth is added to the system after every $N$ exchanges, the total wealth in the system at time $t$ is given by

$$W(t)=W(0)e^{\mu_{0}t}.$$

(8)

With these considerations Eq. (3) becomes

	$\displaystyle\frac{dw_{i}(t)}{dt}$	$\displaystyle={f_{0}}\sum_{j}\Theta\big{[}w_{i}(t)-w_{j}(t)\big{]}\eta_{ij}(t)w_{j}(t)$
		$\displaystyle{}+{f_{0}}\sum_{j}\Big{\{}1-\Theta\big{[}w_{i}(t)-w_{j}(t)\big{]}\Big{\}}\eta_{ij}(t)w_{i}(t)+{\mu_{0}}W(t)\frac{w_{i}(t)^{\lambda}}{S(t)}.$		(9)

To obtain a mean-field theory, we choose an agent whose wealth is $w(t)$ and let $w_{\rm mf}(t)$ be the mean wealth of the remaining agents. That is,

$$w_{\rm mf}(t)=\frac{W(t)-w(t)}{N-1}.$$

(10)

The mean-field version of Eq. (9) is

	$\displaystyle\frac{dw(t)}{dt}$	$\displaystyle=$	$\displaystyle f_{0}\Theta\big{[}w(t)-w_{\rm mf}(t)\big{]}\eta w_{\rm mf}(t)+f_{0}\Big{[}1-\Theta\big{[}w(t)-w_{\rm mf}(t)\big{]}\Big{]}\eta w(t)$		(11)
			$\displaystyle{}+\mu_{0}W(t)\frac{w(t)^{\lambda}}{S(t)}.$

The quantity $S(t)$ defined in Eq. (5) becomes

$$S(t)=w^{\lambda}(t)+(N-1)w_{\rm mf}^{\lambda}(t).$$

(12)

To obtain a mean-field description we have effectively coarse grained the exchanges between the chosen agent and the remaining $N-1$ agents in time, which implies a coarse graining of the noise associated with the coin flips that determine the exchange of wealth. By using the central limit theorem, we can take the noise in Eq. (11) to be random Gaussian. This assumption would not be valid if the chosen agent interacted with only one other agent in one unit of time. However, because the unit of time corresponds to $N^{2}$ exchanges, the chosen agent interacts with $N-1$ other agents and coarse graining in time makes sense. The coarse graining of the noise is another reason why it is necessary to choose $N^{2}$ exchanges to be one unit of time in the mean-field theory.

It will be convenient to write the growth term in Eq. (11) as

$$\mu_{0}W(t)\frac{w^{\lambda}(t)}{S(t)}=\mu_{0}W(t)\frac{[w(t)/W(t)]^{\lambda}}{[w(t)/W(t)]^{\lambda}+(N-1)^{1-\lambda}[1-w(t)/W(t)]^{\lambda}},$$

(13)

where we have used Eqs. (10) and (12) and divided the numerator and denominator by $W^{\lambda}(t)$.

To simplify Eq. (11), we first assume that $w(t)<w_{\rm mf}(t)$; that is, the wealth of the chosen agent is less than the mean wealth of the remaining $N-1$ agents. We use Eqs. (9) and (13) to obtain

$$\frac{dw(t)}{dt}=f_{0}\eta(t)w(t)+\mu_{0}W(t)\frac{[w(t)/W(t)]^{\lambda}}{[w(t)/W(t)]^{\lambda}+(N-1)^{1-\lambda}[1-w(t)/W(t)]^{\lambda}}.$$

(14)

We divide both sides of Eq. (14) by $W(t)$ and rewrite Eq. (14) as

$$\frac{d}{dt}\Big{(}\frac{w(t)}{W(t)}\Big{)}=f_{0}\eta(t)\frac{w(t)}{W(t)}+\mu_{0}\frac{[w(t)/W(t)]^{\lambda}}{[w(t)/W(t)]^{\lambda}+(N-1)^{1-\lambda}[1-w(t)/W(t)]^{\lambda}}-\mu_{0}\frac{w(t)}{W(t)},$$

(15)

where we have used the relation [see Eq. (8)]

$$\frac{1}{W(t)}\frac{dw(t)}{dt}=\frac{d}{dt}\Big{(}\frac{w(t)}{W(t)}\Big{)}+\mu_{0}\frac{w(t)}{W(t)}.$$

(16)

We next introduce the scaled wealth fraction

$$x(t)\equiv\frac{w(t)}{W(t)},$$

(17)

and rewrite Eq. (15) as

	$\displaystyle\frac{dx(t)}{dt}$	$\displaystyle=R(x,\eta,t)$		(18a)
	$\displaystyle R(x,\eta,t)$	$\displaystyle\equiv f_{0}\eta(t)x(t)+\mu_{0}\frac{x(t)^{\lambda}}{x(t)^{\lambda}+(N-1)^{1-\lambda}[1-x(t)]^{\lambda}}-\mu_{0}x(t).$		(18b)

Equation (18) expresses the time-dependence of the wealth of the chosen agent in contact with a mean-field representing the mean wealth of the remaining agents. Hence, the wealth of the chosen agent is not conserved.

For $\mu_{0}=0$, the total wealth $W$ is a constant because the noise $\eta(t)$ that determines the wealth transfer from the mean-field wealth to the chosen agent is the negative of the noise that governs the wealth transfer from the chosen agent to the mean field.

It is easy to show that for zero noise, $R(x,\eta=0,t)=0$ for $x=0$, 1, and $1/N$, and that these are the only fixed points of Eq. (18) for $\lambda\neq 1$. To determine the stability of the fixed points, we calculate the derivative $dR(x,\eta=0,t)/dx$ and obtain

$$\frac{dR(x,0,t)}{dx}=\mu_{0}\frac{\lambda x^{\lambda-1}}{x^{\lambda}+(N-1)^{1-\lambda}(1-x)^{\lambda}}-\mu_{0}\frac{x^{\lambda}[\lambda x^{\lambda-1}-\lambda(N-1)^{1-\lambda}(1-x)^{\lambda-1}]}{[x^{\lambda}+(N-1)^{1-\lambda}(1-x)^{\lambda}]^{2}}-\mu_{0},$$

(19)

where $x\equiv x(t)$. For $\lambda<1$, the derivatives at $x=0$ and $x=1$ are equal to $\infty$, which implies that these fixed points are unstable. The derivative at $x=1/N$ is equal to $\mu_{0}(\lambda-1)$, and hence the fixed point at $x=1/N$ is stable for $\lambda<1$. For $\lambda>1$ the derivative at $x=1/N$ is positive so that this fixed point is unstable. The derivative at $x=0$ and $x=1$ equals $-1$, and hence these fixed points are stable.

We next return to Eq. (9) and consider the case for which $w(t)>w_{\rm mf}(t)$. The growth term is the same as before. The exchange term in Eq. (3), $f_{0}\sum_{j}\Theta\big{(}w_{i}-w_{j}\big{)}\eta_{ij}w_{j}$, becomes $f_{0}\eta w_{\rm mf}$. We use Eq. (10) to write

$$\frac{dw(t)}{dt}=f_{0}\eta(t)\frac{W(t)-w(t)}{N-1}+\mu_{0}W(t)\frac{[w(t)/W(t)]^{\lambda}}{[w(t)/W(t)]^{\lambda}+(N-1)^{1-\lambda}[1-w(t)/W(t)]^{\lambda}}.$$

(20)

From Eq. (16) and the definition of $x(t)$ in Eq. (17) we have

$$\frac{dx(t)}{dt}=f_{0}\eta(t)\frac{1-x(t)}{N-1}+\mu_{0}\frac{x(t)^{\lambda}}{x(t)^{\lambda}+(N-1)^{1-\lambda}[1-x(t)]^{\lambda}}-\mu_{0}x(t).$$

(21)

Equation (18) for the poorer agent and Eq. (21) for the richer agent are the same except for the noise term, and hence the fixed points are the same. We again use Eq. (10) to rewrite Eq. (21) as

$$\frac{dx(t)}{dt}=f_{0}\eta(t){x}_{\rm mf}(t)+\mu_{0}\frac{x(t)^{\lambda}}{x(t)^{\lambda}+(N-1)^{1-\lambda}[1-x(t)]^{\lambda}}-\mu_{0}x(t),$$

(22)

where ${x}_{\rm mf}(t)=w_{\rm mf}(t)/W(t)$ is the fraction of the mean field agent’s rescaled wealth. Note that $x(t)$ is of order $1/N$ as is ${x}_{\rm mf}(t)$. Equation (22) will be used in Sec. III to discuss the phase transition and the critical exponents.

In summary, the fixed points for all values of $\lambda$ are $x=0$, 1, and $1/N$ for the mean-field equations describing the wealth evolution of either the richer or poorer agent. For $\lambda<1$, the fixed points at 0 and 1 are unstable, and the fixed point at $1/N$ is stable, corresponding to all agents having an equal share of the total wealth on average. For $\lambda>1$, the fixed points at 0 and 1 are stable, and the fixed point at $x=1/N$ is unstable, which implies that if all the agents are assigned an equal amount of wealth at $t=0$, one agent will eventually accumulate all the wealth in a simulation of the model. Note that if we use the equation for which the chosen agent is richer than the “mean field” agent, then the stable fixed point reached when $\lambda>1$ is $x=1$; similarly, if we chose the equation for which the chosen agent is poorer than the mean field agent, the stable fixed point reached for $\lambda>1$ is $x=0$.

To analyze the phase transition at $\lambda=1$, we investigate Eq. (22), the mean-field differential equation for the richer agent, for $x\sim 1/N$ and $\lambda$ close to $1^{-}$. We let

$$x(t)=\frac{1}{N}-\delta(t),$$

(23)

assume $N\delta\ll 1$, and expand the second term on the right-hand side of Eq. (18b) to first order in $N\delta$. After some straightforward algebra we find that

$$\frac{d\delta(t)}{dt}=f_{0}\eta(t)x_{\rm mf}(t)-\mu_{0}(1-\lambda)\delta(t),$$

(24)

We multiply both sides of Eq. (24) by $N$ to obtain

$$\frac{dN\delta(t)}{dt}=f_{0}\eta(t){\tilde{w}}_{\rm mf}-\mu_{0}(1-\lambda)N\delta(t),$$

(25)

where ${\tilde{w}}_{\rm mf}=Nx_{\rm mf}$. We write ${\tilde{w}}_{\rm mf}=1-N\delta$, let

$$\phi=N\delta,$$

(26)

and rewrite Eq. (25) as

$$\frac{d\phi(t)}{dt}=f_{0}\eta(t)\big{[}1-\phi(t)\big{]}-\mu_{0}(1-\lambda)\phi(t).$$

(27)

As mentioned, we can assume the noise $\eta(t)$ to be associated with a random Gaussian distribution of coin flips. Note that $\eta$ is the average over $N$ coin flips and hence should scale as ${\sqrt{N}}/N\sim 1/{\sqrt{N}}$. Hence $\eta(t)$ in Eq. (27) is order $1/\sqrt{N}$, which implies that $\phi(t)\sim 1/\sqrt{N}$ and justifies our neglect of terms higher than first order. Simulations in Ref. kang show that the fluctuations are dominated by those near the $1/N$ fixed point.

Because $\phi(t)\sim 1/\sqrt{N}\ll 1$ for $N\gg 1$, we can ignore $\phi(t)$ compared to one in Eq. (27) and obtain

$$\frac{d\phi(t)}{dt}=f_{0}\eta(t)-(1-\lambda)\mu_{0}\phi(t).$$

(28)

The implications of neglecting the term $f_{0}\eta\phi(t)$ in Eq. (27), which generates multiplicative noise, are discussed in Sec. VI. Here we note that the multiplicative noise term vanishes if the limit $N\rightarrow\infty$ is taken before the critical point is approached, that is, if the mean-field limit is taken before $\lambda\rightarrow 1$. However, for finite $N$ the situation is more subtle.

The starting point for the derivation of Eq. (28) was Eq. (22), the mean-field equation for the richer chosen agent. If the chosen agent is poorer than the average of the other agents, similar arguments lead to the same equation as Eq. (28).

The form of Eq. (28) is identical to the linearized version of the Landau-Ginzburg equation ma ; kl-batrouni ; hh with $\phi$ as the fluctuatng part of the order parameter. Hence, $\lambda=1$ corresponds to a phase transition as was found in simulations of the GED model kang . As for the usual Landau-Ginzburg equation, the factor of $(1-\lambda)$ sets the time scale for $\mu_{0}\neq 0$. That is, as $\lambda\rightarrow 1^{-}$, there is critical slowing down, and the system decorrelates on the time scale

$$\tau\sim\frac{1}{\mu_{0}(1-\lambda)}.$$

(29)

Because the stable fixed point of the poorer agent is zero for $\lambda>1$ [see Eq. (18)] and is one for the richer agent [see Eq. (21)], the order parameter is constant for both $\lambda>1$ and $\lambda<1$, which indicates that there is a discontinuous jump in the order parameter at $\lambda=1$. Hence, the exponent $\beta$, which characterizes the way the order parameter approaches its value at the transition, is equal to zero.

To obtain the critical exponent $\gamma$, we adopt an approach introduced by Parisi and Sourlas ps and note that the measure of a random Gaussian noise is given by ps

	$\displaystyle P(\{\eta_{j}\})$	$\displaystyle=\frac{\exp\Big{[}\!\int_{-\infty}^{\infty}-\beta\sum_{j}\eta_{j}^{2}(t)\,dt\Big{]}}{\!\int\!\prod_{j}\delta\eta_{j}\exp\Big{[}\!\int_{-\infty}^{\infty}-\beta\sum_{j}\eta_{j}^{2}(t)dt\Big{]}},$		(30)
	$\displaystyle P(\eta)$	$\displaystyle=\frac{\exp\Big{[}\!\int_{-\infty}^{\infty}-\beta N\eta^{2}(t)\,dt\Big{]}}{\!\int\delta\eta\exp\Big{[}\!\int_{-\infty}^{\infty}-\beta N\eta^{2}(t)dt\Big{]}}.$		(31)

The factor of $N$ in the argument of the exponential in Eq. (31) comes from the fact that $\eta_{j}(t)=\eta(t)$ for all $j$ in the mean-field approach. This factor of $N$ is consistent with the argument that $\eta(t)\sim 1/{\sqrt{N}}$. (In Ref. bigklein the factor of $N$ is not explicit, but is implicit in the integral over all space.)

We rewrite Eq. (28) as

$$\frac{1}{f_{0}}\frac{d\phi(t)}{dt}+\frac{(1-\lambda)}{f_{0}}\mu_{0}\phi(t)=\eta(t),$$

(32)

and replace $\eta(t)$ in Eq. (31) by the left-hand side of Eq. (32). This replacement requires a Jacobian, but in this mean-field case the Jacobian is unity kl-batrouni . Hence the, probability of $\phi$ is given by

$$P(\phi)=\frac{\exp\Big{\{}\!-\beta N\!\int_{-\infty}^{\infty}\Big{[}\dfrac{1}{f_{0}}\dfrac{d\phi(t)}{dt}+\dfrac{\mu_{0}(1-\lambda)}{f_{0}}\phi(t)\Big{]}^{2}\!dt\Big{\}}}{\!\int\!\delta\phi(t)\exp\Big{\{}\!-\beta N\!\int_{-\infty}^{\infty}\Big{[}\dfrac{1}{f_{0}}\dfrac{d\phi(t)}{dt}+\dfrac{\mu_{0}(1-\lambda)}{f_{0}}\phi(t)\Big{]}^{2}\!dt\Big{\}}}.$$

(33)

We now assume that the system is in a steady state so that $d\phi(t)/dt=0$ over a time scale of the order of $1/\mu_{0}(1-\lambda)$. Hence, the average $\langle\phi^{2}\rangle$ is given by

	$\displaystyle\langle\phi^{2}\rangle$	$\displaystyle=\frac{\!\int\delta\phi\,\phi^{2}\exp\Big{\{}-\beta N\!\int_{-\infty}^{\infty}dt\Big{[}\dfrac{\mu_{0}(1-\lambda)}{f_{0}}\phi\Big{]}^{2}\Big{\}}}{\!\int\delta\phi\,\exp\Big{\{}-\beta N\!\int_{-\infty}^{\infty}dt\Big{[}\dfrac{\mu_{0}(1-\lambda)}{f_{0}}\phi\Big{]}^{2}\Big{\}}}$		(34)
		$\displaystyle=\frac{\int\delta\phi\,\phi^{2}\exp\Big{[}-\beta N\dfrac{\mu_{0}(1-\lambda)}{f_{0}^{2}}\phi^{2}\Big{]}}{\int\delta\phi\,\exp\bigg{[}-\beta N\dfrac{\mu_{0}(1-\lambda)}{f_{0}^{2}}\phi^{2}\bigg{]}},$		(35)

where the range of integration over time is limited to the interval $1/\mu_{0}(1-\lambda)$.

Because we have assumed a steady state, the functional integral becomes a standard integral over $\phi$. We can take the limits of the integrals to be $\pm\infty$ because the factor of $N\gg 1$ in the exponential keeps $\phi$ of order $1/\sqrt{N}$. Hence, Eq. (35) now becomes

$$\langle\phi^{2}\rangle=\frac{\int_{-\infty}^{\infty}d\phi\,\phi^{2}\exp\Big{\{}\!-\beta N\dfrac{\mu_{0}(1-\lambda)}{f_{0}^{2}}\phi^{2}\!\Big{\}}}{\int_{-\infty}^{\infty}d\phi\,\exp\Big{\{}-\beta N\dfrac{\mu_{0}(1-\lambda)}{f_{0}^{2}}\phi^{2}\!\Big{\}}}.$$

(36)

By using simple scaling arguments we see that the second moment of the probability distribution diverges as

$$\langle\phi^{2}\rangle\sim\frac{f_{0}^{2}}{N\mu_{0}(1-\lambda)}.$$

(37)

The fluctuating part of the order parameter $\phi=N\delta$ is analogous to the fluctuating part of the order parameter $m=M/N$ of the fully connected Ising model, where $M$ is the total magnetization of the system and $N$ is the number of spins. To determine the susceptibility (per spin) of the Ising model, we need to multiply $[\langle m^{2}\rangle-\langle m\rangle^{2}]$ by $N$. Because $\langle\phi^{2}\rangle=f_{0}^{2}[N\mu_{0}(1-\lambda)]^{-1}$ [see Eq. (37)], the susceptibility (per agent) of the GED model is given by

$$\chi\sim\frac{f_{0}^{2}}{\mu_{0}(1-\lambda)}.$$

(38)

We conclude that the susceptibility diverges near the phase transition with the exponent $\gamma=1$.

Note that we can relate the variance of $\phi$ to the variance of the rescaled wealth. From the definition of $\delta(t)$ in Eq. (23) and the fact that $x(t)=w(t)/W(t)$ is the rescaled wealth [see Eq. (17)], we have

$$\phi(t)=1-Nx(t)=1-N\frac{w(t)}{W(t)}=1-N\tilde{w}(t).$$

(39)

We rescale the total wealth and hence the wealth of each agent so that $W(t)=N$ after the increased wealth due to economic growth has been assigned. Hence $\tilde{w}$ in Eq. (39) is the rescaled wealth of a single agent. Equation (39) will be useful in Sec. V where we compare the predictions of the theory to the results of the simulations in Ref. kang, .

From Eq. (33) we have that

$$P(\phi)=\frac{\exp\Big{\{}\!-\beta N\mu_{0}\dfrac{(1-\lambda)}{f_{0}^{2}}\phi^{2}\Big{\}}}{\!\int d\phi\exp\Big{\{}\!-\beta N\mu_{0}\dfrac{(1-\lambda)}{f_{0}^{2}}\phi^{2}\Big{\}}},$$

(40)

assuming that the system is in a steady state. From the expression of the action or Hamiltonian in Eq. (40), where $\phi^{2}$ is multiplied by $\beta N\mu_{0}(1-\lambda)/f_{0}^{2}$, we see that the Ginzburg parameter for the GED model is given by (up to numerical factors)

$$G=\frac{N\mu_{0}(1-\lambda)}{f_{0}^{2}}.$$

(41)

To understand why $G$ on Eq. (41) can be interpreted as the Ginzburg parameter, compare the form of Eq. (40) with the form of the Hamiltonian for the fully connected Ising model in Eq. (57) and the dependences of $G$ in Eqs. (41) and (58) on their respective parameters.

The inverse temperature $\beta$ (not to be confused with the order parameter critical exponent), which arises from the amplitude of the Gaussian noise, will be absorbed in the parameter $f_{0}$. The association of $\beta$ with $f_{0}$ is consistent with Eq. (32) in that we are relating the temperature to the amplitude of the noise and indicates that increasing the fraction of the poorer agent’s wealth transferred in an exchange is equivalent to increasing the amplitude of the noise.

The total energy for the GED model can be seen from the form of the action or the Hamiltonian in Eq. (40)

$$E=N\phi^{2},$$

(42)

in analogy with the Landau-Ginzburg-Wilson free field or Gaussian action for the fully connected Ising model lou . Equations (39) and (42) imply that the total energy of a system of $N$ agents is given by

	$\displaystyle E$	$\displaystyle=\sum_{i=1}^{N}(1-\tilde{w}_{i})^{2}$		(43a)
		$\displaystyle=-N+\sum_{i=1}^{N}\tilde{w}_{i}^{2},$		(43b)

where we have used that fact that $\sum_{i}\tilde{w}_{i}=N$.

The existence of a quantity that can be interpreted as an energy implies that the probability density of the energy is given by the Boltzmann distribution for $\lambda<1$. The latter is consistent with simulations of the GED model kang . The existence of the Boltzmann distribution also implies that the system is in thermodynamic equilibrium and is not just in a steady state for $\lambda<1$.

From Eq. (40) we find that $\langle\phi^{2}\rangle\sim f_{0}^{2}/[N\mu_{0}(1-\lambda)]$. Hence, we conclude from Eq. (42) that the mean energy per agent of the GED model scales as

$$\frac{\langle E\rangle}{N}\sim\frac{f_{0}^{2}}{N\mu_{0}(1-\lambda)}.$$

(44)

Equation (44) suggests that the mean energy per agent diverges as $(1-\lambda)^{-1}$ as $\lambda\rightarrow 1$ for fixed $N$, which is not physical. However, if we hold the Ginzburg parameter $G$ constant as $\lambda\to 1$, we find no divergence (the exponent is zero), which removes the apparent nonphysical behavior. That is,

$$\frac{\langle E\rangle}{N}\sim\begin{cases}(1-\lambda)^{-1}&\mbox{(fixed $N$)}\\ G^{-1}&\mbox{(constant $G$)}.\end{cases}$$

(45)

Equation (45) implies that the energy per agent is finite as we approach the critical point only if we hold $G$ constant.

Near the critical point the nonanalytic behavior of the mean energy per agent can be expressed as $(1-\lambda)^{1-\alpha}$, where $\alpha$ is the specific heat exponent. Equation (45) for $\langle E\rangle/N$ for constant Ginzburg parameter implies that $\alpha=1$. This result for $\alpha$ is what we would find if we require that $\beta$, $\gamma$, and $\alpha$ to satisfy the scaling relation in Eq. (2) with $\beta=0$ and $\gamma=1$.

We can also calculate $\alpha$ directly using the probability distribution in Eq. (40). To calculate the fluctuations in the total energy, we need to calculate the average of $\phi^{4}$. If we apply the probability in Eq. (40), we find that the fluctuations in the energy per agent, and hence the specific heat is proportional as $Nf^{4}[\mu_{0}(1-\lambda)]^{-2}$, where we have multiplied by $N$ as we did for the susceptibility per spin of the fully connected Ising model. Hence, the specific heat $C$ scales as

$$C\sim\frac{f_{0}^{4}}{N\mu_{0}^{2}(1-\lambda)^{2}},$$

(46)

and

$$C\sim\begin{cases}(1-\lambda)^{-2}&\mbox{(fixed $N$)}\\ (1-\lambda)^{-1}&\mbox{(constant $G$)}.\end{cases}$$

(47)

We see that if we keep the Ginzburg parameter constant, we find $C\sim f_{0}^{2}/[G\mu_{0}(1-\lambda)]$ and hence $\alpha=1$. Note that if we do not keep $G$ constant, we would find $\alpha=2$, which does not satisfy Eq. (2). As a consistency check, we can use Eqs. (44) and (46) to construct the Ginzburg parameter by comparing the fluctuations of the energy, that is, the heat capacity, to the mean energy:

$$\frac{NC}{{\langle E\rangle}^{2}}\propto\frac{f_{0}^{2}}{N\mu_{0}(1-\lambda)}=G^{-1}.$$

(48)

The mean-field theory predictions for the exponents $\alpha=1$, $\beta=0$, and $\gamma=1$ are consistent with the simulation results reported in Ref. kang, for fixed $G$. As discussed in Sec. III, mean-field theory also predicts that there is only one time scale near the phase transition and that the time scale diverges as ($1-\lambda)^{-1}$ for fixed Ginzburg parameter, an example of critical slowing down [see Eq. (29)]. This prediction is consistent with the simulation results for the mixing time associated with the wealth metric kang and the energy decorrelation time, which were both found to diverge as $(1-\lambda)^{-2}$ for fixed $G$. The apparent discrepancy between the $(1-\lambda)^{-2}$ divergence found in the simulations and the $(1-\lambda)^{-1}$ divergence predicted by Eq. (29) is due to the difference in the choice of the unit of time in the simulation ($N$ exchanges) and in the mean-field theory ($N^{2}$ exchanges). To account for the difference in time units, we need to divide the simulation result by $N$ with the result that $N^{-1}(1-\lambda)^{-2}\sim(1-\lambda)(1-\lambda)^{-2}=(1-\lambda)^{-1}$, where we have used the relation $N\propto(1-\lambda)^{-1}$ for fixed $G$ [see Eq. (41)].

The simulations for fixed $G$ indicate that the energy per agent approaches a constant as $(1-\lambda)\rightarrow 0$. This behavior is associated with the nonanalytic part of the energy per agent. This result for the $\lambda$-independence of the nonanalytic part of the energy per agent is inconsistent with the relation between the energy per agent and the specific heat, $C\propto\partial\langle E(\lambda)\rangle/\partial\lambda$. The $(1-\lambda)^{-1}$ dependence of the specific heat for fixed $G$ near $\lambda=1$ suggests that the mean energy per agent could include a logarithmic dependence on $\lambda$. For example, the form, $\langle E\rangle/N\sim a_{0}+a_{L}/\log(1-\lambda)$, where $a_{0}$ and $a_{L}$ are independent of $\lambda$, implies that the specific heat scales as $C\sim[\log(1-\lambda)]^{-2}(1-\lambda)^{-1}$, thus yielding $\alpha=1$ with logarithmic corrections, which standard mean-field theory cannot predict and are very difficult to detect in simulations.

There is also agreement between the exponents predicted by mean-field theory and those determined in the simulations when the measurements are done at fixed $N$. From Eq. (44) we see that if $N$ is held constant, the mean energy per agent is predicted to diverge as $(1-\lambda)^{-1}$, which is consistent with the simulations kang , although this divergence, is unphysical because it implies that the mean energy per agent would become infinite. The exponent $\alpha$ is predicted to be equal to 2 for fixed $N$, which is also in agreement with the simulations kang .

A sensitive test of whether the system is in equilibrium is given by the form of the wealth distribution of the agents. The derivation of the Gaussian form of the wealth distribution [see Eq. (40)] assumes that the system is in a steady state and that $G\to\infty$ and implies that the distribution of the energy is a Boltzmann distribution. The wealth distribution $P(w)$ in Eq. (40) is predicted to depend only on the value of $G$ and not on the parameters $\lambda$, $f_{0}$, and $\mu_{0}$ separately. Figure 1 shows the distribution of wealth for fixed $G=10^{6}$ and different values of $\lambda$ and $N$. Although the distributions are similar, we see that the wealth distribution is not invariant with respect to changes of $\lambda$ for fixed $G$, even though both distributions are well fit by a Gaussian. Similar changes in $P(w)$ are found for changes in the other parameters for fixed $G$.

To understand this behavior, we return to Eq. (27), the mean-field equation for the evolution of the wealth near the $1/N$ fixed point, which we repeat here for convenience:

$$\frac{d\phi(t)}{dt}=f_{0}\eta(t)\big{[}1-\phi(t)\big{]}-\mu_{0}(1-\lambda)\phi(t).$$

(27)

In Sec. III we argued that the multiplicative noise term $f_{0}\eta\phi$ can be neglected because $\phi$ is assumed to be much less than one. However, we retained the “driving” term $\mu_{0}(1-\lambda)\phi$ and did not consider whether the multiplicative noise term was small compared to the driving term. To determine if this condition holds, we recall that the (average) noise $\eta$ is assumed to be random Gaussian. Our assumption that the amplitude of the Gaussian noise is proportional to $1/\sqrt{N}$ is consistent with the dependence of the Gaussian noise in the mean-field limit of thermal models such as the Ising model (see, for example, Ref. bigklein, ).

Because the amplitude of the Gaussian noise is of order $1/\sqrt{N}$, we can neglect it compared to the driving term in Eq. (27) if $f_{0}/\sqrt{N}\ll\mu_{0}(1-\lambda)$, or

$$M\equiv\frac{{\sqrt{N}}\mu_{0}(1-\lambda)}{f_{0}}\gg 1.$$

(49)

Equation (49) defines the parameter $M$. The condition $M\gg 1$ for the neglect of the multiplicative noise term, as well as the condition $G\gg 1$ has several implications.

1. 1.

   Both $G$ and $M$ diverge in the mean-field limit for which first $N\to\infty$ and then the critical point at $\lambda=1$ is approached kac . If these limits are taken in this order, the mean-field treatment neglecting the multiplicative noise is exact (see Ref. bigklein and references therein).

2. 2.

   A smaller value of $f_{0}$ makes the system more describable by a mean-field treatment, which explains the better agreement of the exponents determined from the simulations for finite values of $N$ with the exponents calculated from a theory that neglects the multiplicative noise.

3. 3.

   A large value of $G$ does not necessarily imply a large value of $M$; that is, as $\lambda\to 1$, the multiplicative noise can become important even though $G$ is still much greater than one.

4. 4.

   The Ginzburg parameter $G$ controls the level of mean field and $M$ controls the influence of the multiplicative noise. It is necessary to keep both parameters constant to obtain results consistent with the mean-field theory. Because we cannot keep both parameters constant simultaneously, there will always be some inconsistency of the results for finite values of $N$ and $G$. These inconsistencies can be minimized for sufficiently large $N$ by increasing $\mu_{0}$ or decreasing $f_{0}$. The point is that we need to be careful in interpreting the results of simulations. An example of the limitations of the mean-field theory and the neglect of both the additive and multiplicative noise terms is shown in Table 1. We see that both $\tau_{E}$, the energy decorrelation time, and $\tau_{m}$, the mixing time, depend weakly on $f$ in contrast to Eq. (29) which predicts that these times are independent of $f$. The dependence of $\tau_{E}$ and $\tau_{m}$ on $f$ reflects the possible importance of the multiplicative noise.

For either zero growth, $\mu_{0}=0$, or for the critical point, $\lambda=1$, the mean-field equation for the rescaled wealth, Eq. (18), reduces to

$$\frac{dx(t)}{dt}=f_{0}\eta(t)x(t).$$

(50)

If we use the Ito interpretation for the effect of the multiplicative noise in Eq. (50), the solution for $x(t)$ is

$$x(t)=x(t=0)\exp\Big{[}-\frac{f_{0}^{2}}{2}t+f_{0}W_{t}\Big{]},$$

(51)

where $W_{t}$ is a Brownian noise or Wiener process and is given by

$$W_{t}=\!\int^{t}\!\eta(t^{\prime})dt^{\prime}.$$

(52)

Because Eq. (50) results from either setting $\lambda=1$ or $\mu_{0}=0$, Eq. (50) implies that the mean-field treatment of the GED model for $\mu_{0}=0$ and $\lambda\neq 1$ results in the same distribution as the geometric random walk without the drift term peters ; peters-klein . For $\lambda=1$ and $\mu_{0}\neq 0$, the solution is $e^{\mu_{0}t}x(t)$, where $x(t)$ is the solution with $\mu_{0}=0$, and Eq. (51) describes the distribution of the geometric random walk with the drift or growth term peters ; peters-klein .

This result, which follows from the analysis of the mean-field equation, Eq. (18), is not applicable if $N$ is held constant because $G=0$ for $\mu_{0}=0$ or $\lambda=1$, and hence the mean-field approach does not apply. If we keep $G$ constant, Eq. (50) is applicable because the condition $G\gg 1$ is compatible with either $\mu_{0}\approx 0$ or $\lambda\approx 1^{-}$. To show numerically that the GED model reduces to the geometric random walk at the critical point involves fixing the value of $G$ and determining the form of the wealth distribution for $\lambda\neq 1$ and $\mu_{0}>0$ and then extrapolating the wealth distribution in the limit $\lambda\to 1$ or $\mu_{0}\to 0$. Such an extrapolation would be a difficult and time consuming process.

We have investigated a simple agent-based model of the economy in which two agents are chosen at random to exchange a fraction of the poorer agent’s wealth. Economic growth is distributed according to the parameter $\lambda$. The larger the value of $\lambda$, the greater the fraction of the growth that is distributed to the agents at the higher end of the wealth distribution. The model, which we call the GED model, was treated theoretically with a mean-field approach and was shown to have a critical point at $\lambda=1$, consistent with simulations kang . The critical exponents are consistent with scaling and the simulations if the Ginzburg parameter is large and held constant as the critical point is approached.

The agreement of the mean-field theory with the simulations implies that for finite but large $G$ and $M$, the GED model can be characterized as near-mean-field bigklein . That is, the system is mean-field in the limit $N\rightarrow\infty$, and is well approximated by mean-field theory if both $N$ and $M\gg 1$, provided that the Ginzburg parameter $G\gg 1$ and is held constant as the transition is approached.

The mean-field theory and the simulations raise some interesting questions about the relation between growth, uncertainty and wealth inequality and the applicability of statistical physics. The questions concerning statistical physics include the following:

• •

  The inclusion of distribution and growth allows the system to be treated by the methods of equilibrium statistical mechanics, but only if the distribution parameter $\lambda<1$ and in the limit that the number of agents $N\rightarrow\infty$. A similar result holds for models of earthquake faults for long-range stress transfer bigklein ; rundle . It is unclear how many driven dissipative non-equilibrium systems become describable by equilibrium methods in the mean-field limit.

• •

  We used equilibrium methods to calculate the critical exponents in agreement with the simulations, but the exponent $\alpha$ associated with the specific heat is thermodynamically consistent only if the Ginsburg parameter is held constant. Similar results were found for the fully connected Ising model lou ; kangspecificheat . Insight into why holding $G$ constant is necessary will be discussed in detail in a future publication kangspecificheat .

• •

  A subtle feature of using a mean-field approach to treat the GED model for $N\gg 1$ but finite is the presence of multiplicative as well as additive noise. The effect of the multiplicative noise is controlled by the parameter $M$ defined in Eq. (49). From the agreement of the theory with the simulations, we conclude that the neglect of the multiplicative noise in the theory is a good approximation for $M\gg 1$. The role of multiplicative noise is of particular interest for models of the economy in light of the non-ergodicity of the geometric random walk, which includes multiplicative noise peters ; peters-klein .

• •

  To obtain an equilibrium description of critical point behavior, we defined an order parameter and then obtained the order parameter exponent $\beta$ and the susceptibility exponent $\gamma$. To obtain the specific heat exponent $\alpha$, we defined an energy, which also allowed us to obtain the $\lambda$ dependence of the energy as $\lambda$ approaches its critical value. The definitions of the order parameter and the energy generate a thermodynamically consistent set of exponents that characterize the critical point. Is our choice of order parameter and energy unique, or are there other definitions that would lead to another set of thermodynamically consistent exponents?