The Effects of Taxes on Wealth Inequality in Artificial Chemistry Models of EconomicActivity

Wolfgang BanzhafDepartment of Computer Science and Engineering,Michigan State University, East Lansing, MI, USA

banzhafw@msu.edu

(Dated: July 8, 2020)

We consider a number of Artificial Chemistry models for economic activity and what consequencesthey have for the formation of economic inequality. We are particularly interested in what taxmeasures are effective in dampening economic inequality. By starting from well-known kineticexchange models, we examine different scenarios for reducing the tendency of economic activitymodels to form unequal wealth distribution in equilibrium.

I. INTRODUCTION

Today’s societies suffer, for the most part, from a formof economic inequality that seems very difficult to treator even only to attempt to remedy. Governments andeconomists have made repeated efforts to address thisproblem, realizing that it is connected to many otherserious survival problems of contemporary societies, likeclimate change [27], loss of biodiversity [21], racial andgender injustice [32, 35], health problems [18] and others.

Historians have studied the problem in a perspectiveacross times and generations, and found that it is part ofa complex of problems that cause societies to collapse [7].Simulation models have been formulated that corrobo-rate the causal relationship between economic inequalityand societal collapse [22]. In turn, some historians havedelved into a historical analysis of the reasons for theretreat of economic inequality, finding societal collapseamong them. Their more general findings point to otherprobably even more harmful and violent causes, catas-trophic events like epidemics, wars and revolutions that- together with societal collapse - are virtually the onlyreasons for a retreat of inequality (see the comprehensivestudy of Scheidel [31]). Economic inequality seems to beentrenched and potential non-harming remedies seriouslylacking.

Natural scientists have weighed in on this discussionas well, pointing out that situations of equality in bothnature and society can be compared and are ideal sym-metrical states, bound to disappear into non-symmetricalstates as soon as the pressure for equality is reduced [30].Studies have found that inequality of ”wealth” - properlydefined - also exists in the animal world [6], curiously ina similar distribution as among humans. The obviousquestion to ask then is, whether the unequal distributionin quantities like wealth or species abundance, in otherwords in stocks in both natural and social systems, andto a lesser extent the unequal distribution in quantitiesof flows like income or energy are expressions of universallaws that are acting in both the natural and the human-made (social) world. Econophysics is one of the fieldsthat asks such questions [20, 33] and our contribution

here will address some of the assumptions of such mod-els in the sections below.

Among economists and social scientists there is agree-ment about the status of economic inequality in societiesaround the globe, with the United States a particularlyegregious example [25, 28]. Governments have developedtools to address such issues, with taxes among the mostwidely used. As is well known, taxes actually play multi-ple roles in societies, and those roles should be conceptu-ally kept separate if one wants to apply those tools. Thethree major roles of taxes are (i) to generate income forgovernment entities on the national, state or local levelin order to offer services to the population; (ii) to redis-tribute wealth between segments of a society; and (iii) topenalize and discourage certain habits or uses among thepopulation. For clarity purposes, here we prefer to dis-cern the usage of this term and speak of government taxesand fees for purpose (i), redistributive taxes for purpose(ii) and penalty taxes for purpose (iii).

Among what is the general (and less discerning) usageof the term ”tax”, a number of different tools have beenapplied to generate cost-recovery for government services,like sales taxes, value added taxes, income taxes, wealthtaxes, inheritance taxes, luxury taxes, etc. Easiest to ad-minister are certainly those taxes that are applied dur-ing the event of a transaction, like a sales tax at themoment when an actual sale happens. Less easy to ad-minister are taxes that span a period of time, like taxesfor earnings or interest, to be collected when a transfer ofearning or interest happens periodically. Both of thesetaxes can be tied to relatively easily measurable flows.The most difficult taxes to administer are those that aretied to a stock, like wealth or inheritance taxes. Disputesquickly ensue about the measurability and comparabilityof stocks, their discountability etc, and, depending on theinterest of a party, minimization strategies are applied tocircumvent the tax duty following from a tax on stocksof any kind. An answer to such behavioural variationsis probably offered by game theory, but for the moment,we shall have to postpone consideration of tax avoidancestrategies and focus on the results of applying idealizedtaxing policies. Note also that we do not consider any

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other method for avoiding economic inequality in thispaper, though there might be other routes to improveequality like the creation or fostering of a gift economywhich, however, would require basic systemic changes toeconomic activity.

Among economists, there is an ongoing discussionabout the relation between taxation and productivity.Under the assumption that productivity and taxation areconflicting goals in a society, optimization methods seemto be an appropriate approach to reconcile those con-flicting goals. While economists formulated and studiedoptimal taxation theories [15, 16, 29], progress in tech-niques like computer simulation has allowed a whole setof models to be explored in this arena. The idea of suchmodels is that while they will often be very simple ab-stracted models of reality, their results can teach us alot about general taxation effects which can then be re-fined and underpinned by theoretical investigations. Theeconophysics models of wealth formation as well as thestudy of phenomena equivalent to wealth in ecologies aretwo examples already mentioned earlier.

Here we suggest another route to explore taxationmodels, by virtue of very simple multi-agent systemscalled Artificial Chemistries [2, 8]. Artificial chem-istry models are based on an analogy between the sys-tem under consideration and a chemical reaction sys-tem. Stochastic interactions (”reactions”) happen be-tween agents called ”molecules” which obey predeter-mined rules. In contrast to many other MAS, however,the behavioural rules of these molecular agents are sim-ple and identical among agents of the same type, thusallowing the collective effects to emerge clearly from theset of chosen rules.

Here we are interested exclusively in distributive taxeffects. But before we go into details of such AC mod-els of wealth distribution, the major result of the currentinvestigation can be summarized as follows: These verysimple abstracted models of (economic) interaction be-tween agents result in a general outcome: income tax isoverrated as a means to achieve economic equality, ora remedy to at least reduce economic inequality. Evenincome taxes at the high end of the scale (progressive,high marginal rates) do only slightly dampen, but do noteliminate economic inequality. One needs to introduce awealth tax to tackle the problem. Due to the simplicityof the model, this result is general and can be expected tohold even if more complicated interactions or structuresare envisioned. We do not rule out, of course, that froma practical point of view, a combination of different taxesneeds to be employed, but a proper wealth tax needs tobe the major component of any system that hopes tosubstantially reduce economic inequality.

The rest of this paper is organized as follows: The nextsection (Section II) briefly reflects the current knowledgeon income and wealth distribution, Section III thendiscusses the kinetic exchange model of econophysicsand its key assumptions and results. Section IV explainsour artificial chemistry model of economic activity in

FIG. 1. Wealth distribution for 1996 (UK), top, reproducedfrom [10], and income distribution for 1997 (US), bottom,reproduced from [9].

detail. Section V then provides the simulation resultsof the model using different scenarios with and withoutdifferent types of taxes. Section VI discusses variouscounter-arguments against the primary result and somepractical implementation issues, Section VII concludesby putting the results in perspective. An Appendixgives more details on distributive effects of some of themodels studied.

II. INCOME AND WEALTH DISTRIBUTION

Income and wealth distribution in virtually all coun-tries of the world cannot be described other than unequal.Unfortunately, this is not a temporary situation broughtabout by some economic downturn or some unsuccessfulor incapable attempts of governments on behalf of theircitizens. Instead, it is a systemic, long-term problem ofvirtually all present and historic societies, the source ofmany of the problems that caused and still cause the col-lapse of their organization and of the well-being of theircitizenry. As Piketty has pointed out in his large-scalestudy [25] this problem will continue in the 21st centuryuntil we find a means to seriously fight its root causes.

The history of human economies provides a rich fieldfor learning about trends and tendencies. The anthro-pology/archaeology pair Kohler and Smith [17] providea larger overview of the history of inequality in humansocieties. The Italian economist and sociologist Pareto

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FIG. 2. Wealth distribution for top 1%, 90-99%, 50-90% andbottom 50% of population, development 1989-2020. [1]

studied wealth distribution in Europe already in the 19thcentury. He found that wealth distribution for the richersegments of a society follows a power law, today knownas the Pareto law [24]. This sector refers to the upperechelons of wealth and income, whereas the lower part ofthe distribution curve can be fit well with an exponen-tial or Gibbs distribution or a log-normal distribution.Yakovenko and Rosser [36], based on [11] offer a goodreview of data and models of income and wealth distri-bution. However, data on wealth are difficult to come byand often proxies have to be used. Figures from the UKon wealth (derived from inheritance) for 1996 and fromthe US on income distribution for 1997 produce a similarpicture (see Figure 1), though.

Commenting on income distribution, Chakrobarti et alwrite [4]:

”These observed regularities in income distri-bution may thus indicate a ’natural’ law ineconomics.”

For the United States, further evidence comes fromstatistics showing the development of wealth distributionacross percentiles of the population, Figure 2. As of early2020, the top 1% own more than 25% of the assets, whilethe bottom 50% have a share of approximately 5-7% ofthe assets.

Again, given that wealth seems to follow the same dis-tribution as income, it makes sense to use a universalexplanation for these tendencies, even if details might bedifferent.

III. THE KINETIC EXCHANGE MODEL

The kinetic exchange model for economic activitystarts with the hypothesis that the economic exchangeactivity of individual economic agents in the form oftrades can be compared to the movement and encounterof gas particles exchanging energy [5, 9]. In place of en-tropy maximization in the case of energy exchange they

would follow utility maximization principles in the caseof economic trades. Money would take the place of en-ergy. Benoit Mandelbrot [19] has succinctly summarizedthis idea:

”There is a great temptation to consider theexchanges of money which occur in economicinteraction as analogous to the exchanges ofenegy which occur in physical shocks betweenmolecules. In the loosest possible terms,both kinds of interactions ”should” lead to”similar’ states of equilibrium. That is, one”should” be able to explain the law of incomedistribution by a model similar to that usedin statistical thermodynamics ...”

Mandelbrot goes on to point out that actual incomedistribution is different, and considers models to accom-modate that difference. But the general idea of thesemodels is that if you have a population of agents that in-teract in a random fashion with each other, the distribu-tion of income and wealth approach certain equilibriumdistributions well known from physics.

Key assumptions of the simplest of these models ofincome and - by way of transfer - wealth distributionsare (i) a closed economic system in which the number ofeconomic agents and the total wealth of the system re-main constant, i.e. that trading exchange is the prevalentmode of economic activity; (ii) trading is restricted totwo-agent interactions; (iii) there is no negative wealth,thus an exchange cannot result in such and debt needsnot be considered; (iv) the exchange is symmetrical, inthat the basis of the exchange is normally a fixed amountor a percentage of the sum of the wealth of the partic-ipating agents; (v) the exchange conserves money, i.e.the amount given to one agent is taken from the other sothat the exchange does not change the total amount ofwealth in the system; (vi) an exchange process does notdepend on previous exchange processes, so the dynamicsis Markovian.

Key results of these models are that the income/wealthdistribution develops indeed unequally among the popu-lation of agents, and, regardless of what the initial con-ditions of the exchange dynamics are, tends to an equi-librium that has a distribution that looks similar to anexponential (Gibbs) or log-normal (Gibrat) distribution.The details vary based on model assumptions. For exam-ple, if there is an additional assumption of savings (re-served amounts of wealth not available for the economicexchange) then such models tend to reflect wealth dis-tribution in societies more closely. In particular, the tailof the previous distribution is now modified and followsa power law (Pareto law), just as empirically observed.Details can be found in [4, 36].

The kinetic exchange model has been criticized byeconomists as being unrealistic and even, to a degree,misleading [13]. One key criticism is

”The industrialised economies of the West,and increasingly of Asia, are emphatically not

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a conservative system: income is not, like en-ergy in physics, conserved by economic pro-cesses. Therefore, it is a fundamental fallacyto base economic models on a principle ofconservation. Yet this is an inevitable con-sequence of exchange-only models, since ex-change is a conservative process.”

We don’t think that this criticism is indeed the core ofthe problem because it is not a fundamental limitation ofexchange models. There are other kinetic exchange mod-els that allow for some of the money to go into taxation[3, 12, 14], thus reducing the amount that is availableto the agents subsequently. We show in this contribu-tion how easily such models can be formulated. In thesame vain one could imagine that money is gained in theprocess of exchange, thus allowing both agents to bene-fit from the exchange process and generating a growingeconomy [26, 34].

However, the issue is probably related to the fact thatkinetic model are often theoretical and ”... much of theeconophysics community appears to think that simplydoing good science is sufficient to have the work recog-nised, rather than relating to the motivations and incen-tives of policy makers ...” [23]. In the following, we in-tend to discuss an example of an easily formulated modelwith an exclusive focus on the purpose of the model: tounderstand distributional effects.

IV. USING AN ARTIFICIAL CHEMISTRY ASAN AGENT-BASED MODEL FOR ECONOMIC

ACTIVITY

Our agent-based model is based on an Artificial Chem-istry (AC) [2] with the idea that it is the interaction ofagents that is the most important driver of distributionof wealth and income. The detailed nature of that inter-action will determine the exact distribution of wealth ina population of agents, but for our modelling purposeshere, we study as simple a model as possible.

The baseline model assumes that all agents are ho-mogeneous in their behaviour, while possibly possessingdifferent amounts of wealth. Their interaction is basedon the following AC rules. In each iteration randomlychosen agents i and j, possessing amounts of respectivewealth of mi and mj encounter each other and exchangea good for an amount of money ∆m (see Figure 3). Interms of an AC, the population is in a well-mixed reactionvessel without inflow or outflow. We are not interestedin studying the flow of goods in the economy, only theflow of money. Thus, we assume something of value willflow in the counter direction of the money flow, with-out further specifying the nature of that flow. We arenot allowing negative wealth, so ∆m will be determinedas a random percentage of the smaller of the amountsmi and mj . Alternatively, we could define the attemptto exchange the larger amount of money as an ”elasticcollision” of the agents, but that would simply force us

FIG. 3. Agents with wealth m1,m2 at time t interact andexchange money ∆m in the process, resulting in agents withwealth m′

1,m′2 at time t + 1.

to draw another pair of agents, and therefore delay therelaxation of the system.

Suppose mi < mj , and a percentage p flows in thedirection mj . Then each agent updates their ”wealth” initeration t + 1 with the following equations:

mi(t + 1) = mi(t)−∆m

mj(t + 1) = mj(t) + ∆m(1)

with

∆m = p ·mi (2)

Note that only one agent, randomly chosen, receivesmoney, the other is supposed to receive a correspond-ing good. The total amount of money is, of course, un-changed by this operation, but the operation is asym-metrical. Since the smaller amount determines the ac-tual flow, if one of the agents is getting very poor witha wealth close to 0, the amount flowing is getting corre-spondingly smaller, and the exchange creeps to a halt.We do not modulate the probability of exchanges basedon their size in this baseline model.

As has been pointed out above, the kinetic exchangemodels of econophysics assume a symmetrical exchange,based on the sum of both agents’ wealth, and havingmoney flowing in both directions. This makes sensewhen thinking about energy flows in a physical sys-tem, but here we are discussing a different system wherethe assumption of symmetry is not justified. Artificialchemistries are no stranger to asymmetrical processes, infact, it is their general case.

The result of the kinetic exchange model is that itequilibrizes much faster and to a less unequal distribu-tion. Our model’s equilibrium state is extreme, with oneagent of the population holding all the wealth, and allothers holding none, even when starting from a fullyequal wealth distribution at the outset. The relax-ation times for approaching this equilibrium are, how-ever, much longer, so that in a reasonable simulation itis never reached.

Figure 4 compares our AC model with the kinetic ex-change model of Dragulescu and Yakovenko [9] at differ-ent iterations. That model determines the exchange ofmoney as a percentage of the average wealth of the two

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FIG. 4. Wealth distribution in population of 1,000 agents (10bins, a.u.). Comparison of baseline economic exchange model(left) and kinetic exchange model (right) at t = 1, 000; 10, 000;and 100, 000. Initial distribution was equal among all 1,000agents.

agents:

∆m = p · (mi + mj)/2 (3)

We can see clearly that the inequality resulting fromthe asymmetrical agent model is even higher and form-ing faster than the unequal wealth distribution from thekinetic exchange model. Note that both simulationsstarted from an equal distribution of wealth among all1,000 agents. But regardless of initial state, both mod-els develop heavy inequality. Many other model vari-ants can be formulated that lead to qualitatively similarresults: Even from an ideal state of equal distributionamong agents, wealth inequality develops as if it were anatural law.

A classical measure of economic inequality is providedby the Gini coefficient g, 0 ≤ g ≤ 1, defined as:

g =1

2N2m

∑i,j

‖mi −mj‖ (4)

Figure 5 shows the Gini coefficient for both simulationsand how it develops over time. As expected, for the ki-netic model approaching an exponential distribution, theGini coefficient fluctuates around 1/2 which signals a sub-stantial inequality in wealth distribution among agents.For our baseline model, the Gini approaches 1, the mostunequal Gini coefficient possible. Generally speaking,Gini coefficients below 0.25 indicate high equality in adistribution.

FIG. 5. Gini coefficients in population of 1,000 agents for thebaseline economic and the kinetic exchange model. Initialdistribution was equal among all 1,000 agents.

V. RESULTS

In this section we shall discuss our simulation resultson the following different scenarios:

1. A flat income tax with regular (30%), high (60%)and low (5%) values

2. A progressive income tax between 30% and 75%

3. A wealth tax of medium, high and low value

For some of these tax regimes, we also study differentredistribution cases.

I Redistribution to all tax payers

II Redistribution to a select group of tax payers (thosewith negative income, or lower half of wealth distri-bution)

As emphasized earlier, in these simulations we do notconsider tax as income for local, state or federal govern-ments. This is an important, but different considerationfor tax usage. Rather, we are interested only in the dis-tributional effects of very simple interaction rules. Whenwe show distributional effects, we also do not show ab-solute values, we rather bin into deciles of wealth, i.e.agents relative to each other.

A. A Flat Income Tax

In the flat income tax regime we apply a given tax rater to the income of each agent, as it develops over periodsof time. In the current simulations, we apply the taxevery 10 iterations to the difference of wealth an agenthas accumulated over this period. Thus, we adjust eachagent i’s wealth by the following formula:

mi(t) = mi(t)− r[mi(t)−mi(t− 10)] (5)

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FIG. 6. Wealth distribution in population of 1,000 agents (10bins, a.u.). Flat income tax regime, with regular flat tax of30%. Comparison of case l (left) and case II (right) for re-distribution policy at t = 1, 000; 10, 000; and 100, 000. Initialdistribution was equal among all 1,000 agents. Difference isvirtually not visible.

provided its earnings over the period, mi(t)−mi(t−10),are positive. Otherwise, we do nothing.

Once tax from all agents has been collected in the pe-riod, resulting in an amount T (t), we distribute it backto the agents according to (I) equal redistribution; or (II)select redistribution. This updates the wealth of agent iagain to:

mi(t) = mi(t) +T (t)

N∀i ∈ N (Case I)

mi(t) = mi(t) +T (t)

|Nn|∀i ∈ Nn (Case II)

(6)

where Nn is the set of all those who loose in the trans-actions, i.e. those agents with negative income over theperiod mi(t)−mi(t− 10) < 0.

Figure 6 shows the results of applying a flat income taxof r = 30% and subsequent equal redistribution of the re-sulting taxes to all (case I) or the agents that have lostincome in the transaction (case II). While in the earlystages of the interactions there is some difference visi-ble, that difference seems to become smaller and smalleras the number of interactions increases. We show theGini coefficients for these flat tax experiments in the Ap-pendix. The vast majority of agents has become verypoor, despite the application of an income tax, and de-spite starting out with an exact equal distribution ofwealth.

One other way of looking at the distributional effectsof taxes is to depict the development of wealth in certain

FIG. 7. Wealth development in different quantiles of the pop-ulation of 1,000 agents. Flat income tax regime, with regularflat tax of 30%. Comparison of income tax application withredistribution to all, case I (above) and baseline case, untaxed(below). Initial distribution was equal among all 1,000 agents.Quantiles shown: Bottom half of the population vs top 10%and top 1% wealthiest agents.

quantiles of the population. Figure 7 compares these dis-tributional effects of the 30% flat tax rate with the un-taxed system. So, there actually is an effect of a flatincome tax policy on distribution of wealth (note thedifference in scales) for different quantiles of the pop-ulation, with the untaxed system approaching 100% ofwealth possession rather quickly, but this tendency isonly dampened to a degree in the case of a flat incometax. A higher flat tax rate does further dampen the in-equality, see Figure 8, but even at 60% flat tax rate verysubstantial inequality ensues.

We present a final set of simulations on a flat incometax by moving in the opposite direction: A low flat taxrate of 5% (Figure 9) which again demonstrates the in-fluence of income tax rates on wealth inequality, thougha weak one in this case.

B. A Progressive Income Tax

Now we present simulations on a progressive incometax, applied with a rate of between 15% and 45%, 60%and 75%. The lowest amount is applied for income abovea certain tax-free threshold. While this is arbitrary, inlight of the starting wealth of each agent, we set the low-est rate to begin taxing at incomes of $150. In order

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FIG. 8. Wealth development in different quantiles of the pop-ulation of 1,000 agents. Flat income tax regime, with highflat tax of 60%. Initial distribution was equal among all 1,000agents. Quantiles shown: Bottom half of the population vstop 10% and top 1% wealthiest agents.

FIG. 9. Wealth development in different quantiles of the pop-ulation of 1,000 agents. Flat income tax regime, with highflat tax of 5%. Initial distribution was equal among all 1,000agents. Quantiles shown: Bottom half of the population vstop 10% and top 1% wealthiest agents.

to keep things simple, we linearly increase the tax ratebetween this minimum and the maximum being reachedat $850, $1,200 and $1,550, respectively, for the abovementioned maximal tax rate. So there is some amountof income free of taxes, but the tax rate quickly rises tothe maximum value.

Figure 10 shows the development of the distributionof wealth in the population over 100,000 iterations. Theeffects of this tax regime are quite similar to those of aflat tax of 30%. Figure 11 (top) shows the application ofa 60% marginal tax rate, and Figure 11 (bottom) showsthe application of a serious 75% marginal tax rate.

While it is obvious that there is an effect of the highermarginal tax on the development of wealth, the effect isnot as serious as one might expect from a marginal taxrate of 75%! In fact, the redistributive effect is quitedisappointing, despite an effort that sounds very serious.The top 1% have gained a share of 10% of the assets whilethe bottom half has lost most of its half of the assets andstands at about 8% of assets after 100,000 iterations.

FIG. 10. Wealth development in different quantiles of thepopulation of 1,000 agents. Progressive income tax regime,with tax rate of between 15% and 45%, linearly growing inthe income interval from $150 to $850. Initial distribution wasequal among all 1,000 agents. Quantiles shown: Bottom halfof the population vs top 10% and top 1% wealthiest agents.

In summary, a very simple Artificial Chemistry modelof economic activity that allows a starting state of allagents having equal wealth – a condition no society canhope to start from – will always quickly develop economicinequality under an income tax regime intended to dis-tribute wealth. We are not saying that income taxes donot have an effect, in fact, it can be seen that largermarginal tax rates do have effects, but these effects aretiny compared to the goal of keeping economic equalityin a society. If one further adds the consideration thatno society starts with perfect equality, but would have tostart from a situation of economic inequality at the out-set, the outlook for income tax remedies is even worse.

C. A Wealth Tax

We now turn to another tax model that is not based onincome, but on accumulated wealth. This tax is appliedto the total of an agent’s wealth, but the frequency of itsapplication is reduced to a tenth. That is approximatelythe relation between applying a tax every month vs oncea year. Here we are not concerned about the practicalityof such a tax, but its mere ”theoretical” application andeffects.

Our first wealth tax, which simply takes each agent’swealth and subjects it to a flat tax we call regular is ap-plying a rate of 30%. Thus, at a frequency 10 times lowerthan the previous income tax which taxed the differencein wealth in a given period, we now tax the entire posses-sion of the agent. Figure 12 shows the distributional ef-fect of such a wealth tax after 1, 000; 10, 000 and 100, 000iterations. The total of the tax is redistributed equallyto all agents, our previous case I.

Figure 13 shows how the quantiles develop under such atax. As is clearly visible, after an early relaxation phase,wealth distribution of certain quantiles of the popula-tion are quite stable in a band. The bottom half of the

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FIG. 11. Wealth development in different quantiles of thepopulation of 1,000 agents. Progressive income tax regime,with tax rate of between 15% and 60%, linearly growing in theincome interval from $150 to $1,200 (top) and between 15%and 75%, linearly growing in the income interval from $150to $1,550 (bottom). Initial distribution was equal among all1,000 agents. Quantiles shown: Bottom half of the populationvs top 10% and top 1% wealthiest agents.

population quickly looses around 10% of its share, but re-mains stable thereafter. The top 10% of the populationgain around 5-6% of additional wealth while the top 1%moves up to a share of approximately 2.5%. While theseare big numbers still, they are in no way comparable tothe quantile development with an income tax. The rea-son is that always the full amount of wealth accumulatedby an agent is the basis of taxation. This, in combi-nation with the systematic redistribution of all proceedsfrom the tax allows the agents to develop their wealth inonly a tiny band. Figure 12 actually shows a healthy dis-tribution, with most agents in the middle bin (”middleclass”), and a more or less symmetric, but quickly fallingoccupation of bins higher and lower in wealth.

We also examined a high wealth tax regime of taxingwealth at 60% and a low regime of taxing it at 5%, seeFigures 14 and 15. What is striking is that a wealthtax in general is able to quickly stabilize a distributionof wealth in the population, of course at different levelsdepending on the severity of the tax. We can see fromthe graphs that a 60% wealth tax does not add muchdistributional utility compared to only half of that rate(the regular case). A low wealth tax (here 5%) as it isconventionally discussed in economic circles as a max-

FIG. 12. Development of wealth distribution in populationof 1,000 agents (10 bins, a.u.). Flat wealth tax regime, withregular flat tax of 30% at t = 1, 000; 10, 000; and 100, 000 iter-ations. Initial distribution was equal among all 1,000 agents.Very effective redistribution of wealth, keeping the middleclass dominant.

imum measure (and normally discarded), on the otherhand, leads to a more familiar distribution of wealth inthe population. If we choose an even lower tax rate forsuch a flat wealth tax, say a 1% rate, we can see effectssimilar to a strong income tax (see Appendix).

These results seem at first sight somewhat counter-intuitive. Why would a wealth tax have so much strongerdistributive effects than an income tax? After all, anincome tax is supposed to tax the changes in wealth, soshouldn’t it have the same effect as an admittedly smaller

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FIG. 13. Wealth development in different quantiles of thepopulation of 1,000 agents. Initial distribution was equalamong all 1,000 agents. Quantiles shown: Bottom half ofthe population vs top 10% and top 1% wealthiest agents.

FIG. 14. Wealth distribution in population of 1,000 agents (10bins, a.u.). Flat wealth tax regime, with high (60%) and low(5%) flat tax of 30%. Comparison of high (left) and low (right)for redistribution policy at t = 1, 000; 10, 000; and 100, 000.Initial distribution was equal among all 1,000 agents. Highwealth tax increases wealth of middle class, low wealth taxreduced it in favour of a few high net-worth individuals.

tax on the entire wealth? The answer is ”yes”. But oneneeds to keep in mind that income is only a tiny portionof overall wealth of an individual agent, and its influencein our model shrinks as economic inequality grows larger(since exchange is determined by the agent with smallerwealth). Thus, an income tax becomes progressively lesseffective in curbing the differential effects of wealth ina population. From our simulations, we can see that awealth tax of approximately 1% corresponds to a high flatincome tax of 60%, a close to two orders of magnitudedifference in effectiveness!

FIG. 15. Wealth development in different quantiles of thepopulation of 1,000 agents. Different flat wealth tax regime,with tax rate of of 60% (top) and 5% (bottom). Initial dis-tribution was equal among all 1,000 agents. Quantiles shown:Bottom half of the population vs top 10% and top 1% wealth-iest agents. Distributions quickly stabilize after a short relax-ation period.

Note that the assumption of an asymmetric role ofagents in an exchange is crucial. Absent the ability togo into debt, an agent can only afford and pay for goodsand services up to the value of their wealth (but nor-mally even to a smaller amount). Thus, the larger thedifference in wealth between two agents, the smaller theamount of exchange in relation to the wealth of the agentwith larger wealth, and therefore, the smaller the effect oftheir income tax. In other words, as economic inequalitygrows, so does the inefficiency of income taxes.

VI. DISCUSSION

The natural question to ask, then, is how would oneimplement such a wealth tax on a larger scale and inreality? This question is especially important, given thecounter-arguments that will be recruited to discourageany attempt at this. In the following we shall discussthese aspects in particular in light of

• The attempts at tax avoidance

• The availability of funds for a wealth tax (the liq-uidity problem)

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• The redistribution vs. government support role oftaxes

• Issues of enforcement and practical measures

• The possibility of a combinations of a wealth taxwith a guaranteed basic income or other ideas

Before we address these implementation questions,we would like to discuss some of the obvious reserva-tions raised by opponents of a wealth tax. A non-comprehensive list of counter-arguments from economistscould include the following arguments:

• The model builds a closed system, with no agentscoming and going. — This is a correct observation,but in contrast to many other models of the econ-omy, in principle, we can randomly introduce andremove agents from this system (akin to birth anddeath processes). This can be examined, but it willnot bring substantially different other aspects intoconsideration, except that one might want to studythe effects of an inheritance tax. But the nice thingabout a wealth is that an inheritance is, sooner orlater, appearing as a contribution to wealth, andalready covered by a wealth tax, if only with a de-lay.

• Another argument, frequently brought up againstkinetic exchange models, and transferable to thismodel is that production is actually the main driverof an economy, not the exchange between partic-ipants. Any production or gain in productivitywould not be reflected here. — Again, this ob-servation is correct, but this is not the point ofthe model. In fact, absolute values of wealth arenot studied here, we instead only look at the rela-tive position of agents in the wealth hierarchy. Forthe study of distribution, which is relative amongagents, the influence by the absolute growth of theeconomy is not of interest. We could even add anamount akin to production gains to the amounts tobe distributed, but that will not make distributivechanges.

• The model does not accurately reflect the wealthdistribution found empirically. — Again, this is by-and-large correct, but not the point of this modeleither. This model studies the rough distributionaleffects of taxation, not the exact outcome. Thereare other models that attempt much closer accu-racy when modelling, see, e.g., savings efforts andtheir effects on the long tail of the distribution inkinetic exchange models [4].

• All of this is well known since the 1960s. — Per-haps yes, but given the critical importance of awealth tax as a remedy against economic inequal-ity, it seems that it is not worked on properly forserious implementation purposes. We thus would

argue for more research and a closer look at rela-tively high wealth taxes and their implementation.

• Wealth in general is difficult to measure, and there-fore it is difficult to tax. — As a general observa-tion, this is certainly true. But property taxes (onprivate homes) are an example of how wealth (in acertain branch of the economy) can be measured.The message is again one of relative versus absolutemeasurements. As long as the same principles areapplied to wealth of the same type, their relativedifferences will go into the tax calculation. Thatis enough to have distributive effects. The actualweight of a type of wealth and the rate of redistri-bution is then subject to political considerations.

• The real purpose of taxes is not redistribution, butstate income. — The aspect of state funding is cer-tainly in need to be addressed. This can be doneby removing part of the redistribution amounts andtransferring them to the state. It is again an abso-lute, not relative amount we are talking about here,and as such, it cannot have distributive effects.

Let us return now to the implementation questionsposed above.

Attempts at tax avoidance

Any tax system will have to deal with a certain degreeof tax cheating and systematic tax avoidance schemes.As for the cheating, one cannot avoid that, but it willprobably remain in the same proportion as our currenttax systems. If anything, a tax system that is perceivedas more just will likely be at the lower end of the cheat-ing proportion. As for systematic tax avoidance, this canonly be addressed by a certain degree of auditing. A taxsystem based on a wealth tax is not fundamentally dif-ferent in this regard from a tax system based on incometaxes. If anything, a simple flat wealth tax will prob-ably be easier to administer and audit (in particular ifthere can be specialist audit teams formed for differenttypes of wealth, that work together to determine overallamounts). As we indicated earlier, a likely candidate forexamining avoidance effects is game theory.

Liquidity

This is indeed a problem for high net-worth individ-uals that might be taxed substantial amounts withoutbeing able to mobilize the necessary liquidity from theassets taxed. It is likely best to keep an income tax inplace, and implement the calculation based on a wealthtax with the annual tax declaration which then wouldconsider the income tax as an advance payment on thefinal wealth tax amounts. To be clear: We envision a tax

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system entirely based on a wealth tax, but contempora-neously relying on an income tax as a source of liquidity.This would include dividend and interest payments taxedat source (a withholding tax). Illiquid assets, however,would likely have to be sold, at least partially to procureenough liquidity for tax payments. A pragmatic delayfor required payments could be installed, also taking ac-count of the amounts redistributed to everyone from theoverall tax revenue. As for the amounts returned to tax-payers, these could be based on an estimate of the taxvolume expected in any given year, and corrected in asubsequent year with the actual amounts.

Government Support versus Redistribution

Federal, state and local governments draw income fromtaxes and fees, but taxes are their primary source forproviding services. We have not modelled such a pur-pose of taxes in this contribution. However, it can beeasily accommodated if the system is opened to the out-side world and production is introduced as a source ofadditional income. Our assumption then would be thatthe government’s share of income cannot be larger (andshould preferably be smaller) than production income.

Issues of Enforcement and Practical Measures

Enforcing the payment of a tax is best done by collect-ing it at source. This cannot easily be done with a wealthtax, so it is better to tax every income at an appropri-ate rate, and then to compensate the year after. Certaincountries are better equipped for enforcing tax payments,especially given the potential that wealth moves to an-other country, but the general approach would be a col-laboration between nations.

Combinations of a Wealth Tax with a Guaranteed BasicIncome or other Ideas

We do not exclude the possibility that the best courseof action would be a combination of the wealth tax withsome sort of guaranteed basic income. However, for nowwe have not included that or other modifications into ourtaxation scheme.

VII. CONCLUSION

The simulations reported here use a very simple modelof economic activity. Basing a study of economic activityon artificial chemistries opens a different way of examin-ing collective effects in economic models. Notably, we are

completely free to define the interaction rules betweenagents based on what we believe is important and ig-noring supposedly unimportant features, which includesbreaking symmetries or conservation laws or other prin-ciples that play a role in other disciplines.

The results shine a harsh light on the idea that fiddlingwith income tax systems can rectify the highly unequaldistribution of economic assets that exists today in mostsocieties. Income taxes are by definition only appliedto changes in wealth and normally vanish for very smallor negative incomes. They, therefore, cannot correct asituation that is unequal from the outset, at least notwithout a substantial redistribution beyond the revenuean income tax can generate, something like a stable basicincome or a large basic personal deduction that can bemonetized if not taken in as agent income. In our systemthis was demonstrated by the fact that income taxes wereapplied only to agents that engaged in economic activity.

It is worth mentioning that the unequal distributionof wealth today is actually a good starting point for in-troducing a wealth tax. The reason is that effective taxrates are close to zero for most agents, and nonlinearlyincrease to both sides of the wealth distribution. Thatis due to the redistribution of revenue from this tax asit was introduced here. In a situation where most agentsare at the lower end of the wealth spectrum a compara-tively even smaller percentage of agents will have to paysubstantial amounts of tax. It can be safely assumed,that the scenario also allows for more mobility betweensegments of the society (at least as far as wealth is con-cerned).

Real economies are perhaps positioned between theasymmetric and symmetric cases of economic activity dis-cussed here, with the symmetric case (kinetic exchange)more benign by having less pressure toward inequalitythan the asymmetric case used as a baseline in thismanuscript. This would allow for some flexibility in re-gard to the actual rate of a wealth tax. However, if so-cieties want to address their natural tendency to createinequality and avoid harmful ways of redistribution ofunequal wealth, a wealth tax looks to be the single mosteffective tool to achieve progress toward a more just dis-tribution.

ACKNOWLEDGMENTS

W.B. gratefully acknowledges support from the JohnR. Koza endowment fund to Michigan State University.Computer simulations were executed with Julia v1.0.2under MacOS 10.15.5.

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APPENDIX

Figure 16 shows the Gini coefficients for the flat incometax regime of Section V A, while Figure 17 shows the Ginifor a progressive income tax scenario (see Section V B).

FIG. 16. Gini coefficient for different flat income tax regimesat 30%, 60% and 5% with a population of 1,000 agents, start-ing from an equal distribution of wealth. Starting at ”0”, Giniindicates substantial inequality in all cases.

As is clearly visible from the application of income taxschemes at different levels, their effect on wealth distri-bution is much smaller than the natural pressure for un-equal distribution of wealth from normal economic ac-tivity. What is striking is that a progressive income taxin the current model does not even have the same effectas a flat tax, likely due to the small spread in incomedistribution through individual economic transactions inour model.

FIG. 17. Gini coefficient for different progressive income taxregimes with marginal tax rate of 45%, 60% and 75% with apopulation of 1,000 agents, starting from an equal distributionof wealth. Gini indicates even more substantial inequalitythan flat tax rate in all cases.

Turning finally to the effects of wealth taxes on eco-nomic inequality, we show Gini coefficients for the threewealth regimes we have considered in Section V C. Fig-ure 18 shows the Gini for the three rates of a flat wealthtax we considered in Section V C. After quickly growingfrom ideal equality at ”0”, the Gini remains in a narrowband for all wealth tax regimes with an average of around0.35 for the lowest wealth tax. If one were to apply a 1%wealth tax only, though, the resulting wealth distributionwould develop the classical exponential distribution, onpar approximately with a 60% flat income tax.

FIG. 18. Gini coefficient for different wealth tax regimes witha tax rate of 30%, 60% and 5% with a population of 1,000agents, starting from an equal distribution of wealth. Giniindicates good economic equality, with fluctuations in bandsof 0.01 to 0.05, with a maximum value of the Gini coefficientof 0.35 calculated for the lowest wealth tax rate.