A biased dollar exchange model involving bank and debt with discontinuous equilibrium

Econophysics is a subfield of statistical physics that draw concepts and techniques from traditional physics and apply them to economics, finance, and related fields, and we refer the readers to the pioneer work [15] for many models motivated from econophysics. Among many important tasks in this area of research, a fundamental mission is to demystify how various (macroscopic) economical phenomena could be explained by (microscopic) laws in classical statistical physics under certain modeling assumptions, and we refer to [18, 25, 26, 27] for a general review on using kinetic theory for the modelling of wealth (re-)distributions.

Motivations for studying econophysics models are at least two-fold: from the perspective of a policy maker, how to exert influence on the growing wealth inequality in order to mitigate the alarming gap between rich and poor is a central issue to be dealt with. From a mathematical viewpoint, the underlying mechanisms behind the formation of macroscopic phenomena, for instance various possible wealth distributions arising from distinct agent-based dollar exchange models, remain to be thoroughly understood. For a given agent-based model, we aim at identifying a limit dynamics, which is fully deterministic, when we send the number of individuals/players to infinity, and then the deterministic system will be further analyzed with the hope of proving its convergence to equilibrium (if there is one) in the large time limit. This philosophy has been successfully implemented among the overabundant literatures across different fields of applied mathematics, see for instance [7, 2, 10, 24].

In this work, we consider a simple mechanism for money exchange involving a bank, meaning that there are a fixed number of agents (denoted by $N$) and one bank. We denote by $S_{i}(t)$ the amount of dollars the agent $i$ has at time $t$ and we suppose that $\sum_{i=1}^{N}S_{i}(0)=N\,\mu$ for some fixed $\mu\in\mathbb{N}_{+}$. Thus, each agent in this closed economic system has $\mu$ dollars on average. Moreover, we denote by $B_{*}:=B_{c}+B_{d}$ the initial amount of dollars in the bank, where $B_{c}$ and $B_{d}$ represent the amount of dollars owned by the bank in the form of “cash” and in the form of “debt” (borrowed by agents), respectively. Also, we introduce another parameter $\nu\in\mathbb{N}_{+}$, which measures the ratio of the bank’s initial wealth to the initial combined wealth of all the agents, and set $B_{*}=N\,\mu\,\nu$ to be the total amount of dollars put in the bank initially. We emphasize here that the parameters $\mu$ and $\nu$ are both dimensionless constants.

The model investigated in this work was a natural variant of the model proposed in [30] and revisited in several recent works [9, 21], which we now briefly recall first: at random times (generated by an exponential law), an agent $i$ (the “giver”) and an agent $j$ (the “receiver”) are picked uniformly at random. If either the “giver” $i$ has at least one dollar (i.e. $S_{i}\geq 1$) or if the central bank has “cash” (i.e. $B_{c}\geq 1$), then the receiver $j$ receives a dollar. Otherwise, when the giver $i$ has no dollar and the bank has no cash, then nothing happens. To further clarify the aforementioned model, we emphasize that dollars in the central bank are untouched when the giver $i$ has at least one dollar, and the giver $i$ will borrow a dollar from the bank (as long as the bank has money) if he/she has no dollar to give. The aforementioned model is termed as the unbiased exchange model with collective debt limit in [9] and the one-coin model with collective debt limit in [21]. Notice that when the bank gives a dollar to agent $j$, there is still one dollar withdrew from the giver $i$, i.e., the debt of agent $i$ increases. The debt of agent $i$ could be reduced once it will become a “receiver”. It is also important to notice that in this model the bank never loses money, it just transforms its “cash” $B_{c}$ into “debt” (and vice versa). Without the bank, agents can only give a dollar when they have at least one dollar (hence no debt is allowed).

We now introduce the model considered in the present manuscript as a biased version of the aforementioned model, i.e., we drop the rule of selecting the “giver” uniformly at random in favor of the requirement that the “giver” agent $i$ will be picked (at each jump time) at a rate proportional to $f(S_{i})$, where $f\colon\mathbb{R}\to(0,\infty)$ is a strictly positive function. From now on, we terms the model as the $f$-biased exchange model with collective debt limit, which can be represented by (1.1) below.

We emphasize that the unbiased exchange model with collective debt limit investigated in [9] is a special case of the model we introduced here where $f$ is a fixed positive constant (which can be normalized to $1$ without loss of generality). For now the function $f$ is an arbitrary positive function from $\mathbb{R}$ to $(0,\infty)$, but later we will impose appropriate constraints on $f$ for reasons to be further clarified in the subsequent development of our analysis.

The fundamental problem of interest is the derivation/idenfication of the limiting money distribution among the agents as the total number of agents $N$ and time $t$ become large. We propose two analytical approaches for addressing this problem with illustration provided by figure 1 below. The first approach, detailed in section 2, relies on the study of the limiting money distribution for all (fixed) values of the number $N$ of agents as time goes to infinity and then simplification of the probability that a typical agent has $n$ dollars at (“thermodynamic”) equilibrium in the large population limit.

The second approach is a mean-field approach carried out in the spirit of the recent work [9], which will be presented in section 3. The first step in the mean-field approach lies in the derivation of its limit dynamics as the number of agents goes to infinity (i.e., $N\to\infty$). With this goal, we introduce the probability distribution of dollars:

with $p_{n}(t)=\{``\text{probability that a typical agent has }n\text{ dollars at time}~{}t"\}$. The evolution of ${\bf p}(t)$ will be given by a system of (deterministic) nonlinear ordinary differential equations. However, to rigorously justify the transition from a stochastic interacting agents systems into a deterministic set of ODEs, one needs the so-called propagation of chaos [28]. Heuristically speaking, the propagation of chaos property in the specific context of our model says that the (random) agent $i$ and agent $j$ picked to engage in the binary trading activity (1.1) are statistically independent as $N\to\infty$. The rigorous derivation of the mean-field ODE system (1.3) from its underlying stochastic $N$-agents dynamics is out of the scope of the present paper, but the derivation has been fully justified in various models arising from econophysics, see for instance [4, 5, 6, 8, 14]. The major difficulty in the mean-field argument lies in the fact that the evolution of ${\bf p}(t)$ is split into two phases. Indeed, the evolution of ${\bf p}(t)$ changes at the first time when there is no more cash in the bank. We will denote by $t_{*}$ the time at which such event occur, i.e., at $t_{*}$ the bank is empty for the first time. We will show that the evolution equation for ${\bf p}(t)$ takes the following form

where the exact expressions for the operator $Q_{1}$ and $Q_{2}$ are given by (3.5) and (3.17), respectively. Once the large population limit has been carried out and a limit (deterministic) dynamics has been uncovered, one naturally attempts to investigate the asymptotic behavior of the probability mass function ${\bf p}(t)$ as $t\to\infty$, with the intent of proving convergence to the unique equilibrium distribution (if such unique equilibrium exists). Unfortunately, we do not aim to address the problem of the convergence to equilibrium for the two-phase ODE system (1.3) analytically, and we will be content with numerical evidence on such convergence behavior.

Although we will only investigate a specific binary dollar exchange model in the current work, other exchange rules can also be imposed and studied, leading to other econophysics models. To name a few, the so-called immediate exchange model introduced in [17] assumes that pairs of agents are randomly and uniformly picked at each random time, and each agent transfers a random fraction of their money to the other agent, where these fractions are independent and uniformly distributed on $[0,1]$. The so-called uniform reshuffling model investigated in [5, 15, 20, 23] proposes that the total amount of money of two randomly and uniformly picked agents possessed before interaction is uniformly redistributed among the two agents after interaction. The so-called repeated averaging model studied for instance in [6] studies the mechanism where two randomly selected agents share half of their wealth with each other at each binary exchange. The binomial reshuffling model proposed in a recent work [3] is a variant of the uniform reshuffling mechanism in which the agents’ combined wealth is redistributed according to a binomial distribution. For models with saving propensity, models involving debts, or other econophysics models, we refer the readers to [1, 11, 12, 21, 9, 13, 16, 29] and the references therein.

In this section, we will tackle the problem of finding the limiting money distribution (as $N,\,t\to\infty$) using the techniques employed in [19, 20, 21, 22]. Define $\mathcal{C}(N,M,B_{*})$ to be the set of possible configurations with $N$ agents having a total initial wealth of $M=N\,\mu$ dollars, where the initial amount of dollars in the bank equals $B_{*}=N\,\mu\,\nu$, i.e.,

Let us fix any $\xi\in\mathcal{C}(N-1,M-1,B_{*})$ and introduce $\xi^{i}\in\mathcal{C}(N,M,B_{*})$ defined via

Then $\{S(t)\}_{t\geq 0}$ is a continuous-time Markov chain having state space $\mathcal{C}(N,M,B_{*})$. Note also that all transitions of the process are of the form $\xi^{i}\to\xi^{j}$ for some $i\neq j$.

As indicated in the introduction, the rates of transition from $\xi^{i}$ to $\xi^{j}$ with $i\neq j$ for the process $S(t)$ are given by

We first show that $\{S(t)\}_{t\geq 0}$ is a finite, irreducible and aperiodic Markov chain, whence it admits a unique stationary distribution.

We are now ready to state the following general convergence result, which follows readily from Lemmas 2.1 and 2.2.

From now on, we will take a “carefully crafted” nonlinear function $f$, which is provided in the following definition.

The motivation to consider and treat this particular (“strange looking”) nonlinear $f$ is at least two-fold and will be elaborated in section 3 when we employ a mean-field approach. For the moment, we proceed with this specific choice of $f$ to deduce the following corollary.

We now state some elementary identities from combinatorics, whose proofs are standard.

Now we can specialize the content of Theorem 1 to deduce the following convergence result.

In this section, we carry out a formal mean-field argument of the stochastic agent-based dynamics (1.1) as the number of agents $N$ goes to infinity, following the procedure detailed in [9].

The biased exchange model with collective debt limit can be written in terms of a system of stochastic differential equations, thanks to the framework set up in [4, 9] for the study of the basic unbiased exchange model as well as some of its variants. Introducing $\{\mathrm{N}_{t}^{(i,j)}\}_{1\leq i,j\leq N}$, which represents a collection of independent Poisson processes with intensity $\frac{f(S_{i})}{N}$, the evolution of each $S_{i}$ is given by:

where we use the notation $\delta_{0}(B_{c}):=\mathbbm{1}\{B_{c}=0\}$. By the obvious symmetry, we can focus on the case when $i=1$ and notice that whenever $B_{c}\geq 1$, the SDE for $S_{1}$ simplifies to

If we introduce

then the two Poisson processes $\mathrm{\bf N}^{1}_{t}$ and $\mathrm{\bf M}^{1}_{t}$ are of intensities $f(S_{1})$ and $\frac{1}{N}\sum_{j=1}^{N}f(S_{j})$, respectively. Motivated by (3.2), we give the following definition of the limiting dynamics of $S_{1}(t)$ as $N\rightarrow\infty$ from the process point of view, providing that $B_{c}\geq 1$.

Its time evolution is given by the following dynamics:

with

The distribution ${\bf p}(t)$ naturally preserves its mass (since it is a probability mass function) and the mean value (as the total amount of money in the whole system is preserved), thus if we introduce the affine subspaces:

and

then it is clear that the unique solution ${\bf p}(t)$ of (3.4)-(3.5) with ${\bf p}(0)\in\mathcal{S}^{+}_{\mu}$ satisfies ${\bf p}(t)\in\mathcal{S}_{\mu}$ for all $t>0$. Moreover, we will introduce a class $\mathcal{G}$ of functions $f\colon\mathbb{R}\to(0,\infty)$ by setting

If we define the average amount of “debt” per agent as

then this quantity will be non-decreasing for all $f\in\mathcal{G}$ since

in which the last inequality follows directly from the assumption that $f\in\mathcal{G}$.

From now on, we will always impose the condition $f\in\mathcal{G}$ on the function $f$ unless otherwise stated. Since the average amount of debt each agent can sustain in the underlying stochastic $N$-agents system is at most $\mu\,\nu$, we therefore terminate the evolution of Phase I (3.4)-(3.5) until $t=t_{*}$, where

At the level of the agent-based system, after the first time when there is no cash in the bank, i.e., when $t\geq t^{\textrm{stoc}}_{*}:=\min\{\tau>0\mid B_{c}(\tau)=0\}$, the analysis is much more involved so heuristic reasoning plays a major role in this manuscript. We notice that we have the following basic relations for all time $t\geq 0$:

and $B_{c}\geq 0$. Therefore, the evolution of $B_{c}$ is much faster than the evolution of each of the $S_{i}$’s, indicating that (3.12) is really a “fast” dynamics compared to (3.1). These observations motivate the next definition:

We again denote by ${\bf p}(t)=\left(\ldots,p_{-n}(t),\ldots,p_{-1}(t),p_{0}(t),p_{1}(t),\ldots,p_{n}(t),\ldots\right)$ the law of the process $\widetilde{S}_{1}(t)$ for $t\geq t_{*}$, i.e. $p_{n}(t)=\mathbb{P}\left[\widetilde{S}_{1}(t)=n\right]$ for $t\geq t_{*}$. Its time evolution is given by the following dynamics:

with