Wealth Inequality and Wealth Effect*

Weihong HuangDivision of Economics

Nanyang Technological UniversitySingapore

AWHHuang@ntu.edu.sg

Yu ZhangDivision of Economics

Nanyang Technological UniversitySingapore

yzhang34@e.ntu.edu.sg

Abstract—In an artificial financial market without real growth, extreme income inequality produces extreme wealth inequality via accumulation. An agent-based model is built to study to what extent a strong tax policy can affect this process. Wealth effect is defined as a positive impact of current wealth on future wealth growth. When wealth effect exists, the appearance of extreme wealth inequality is inevitable.

I. INTRODUCTION

Recently, Joseph E. Stiglitz blames inequality for the 2008 financial crisis and for holding back the economic recovery in United States [1,2]. It is beyond doubt that the rising inequality is jeopardizing the whole economy. Both economists and politicians are eager to find a cure for it. Among all possible solutions, taxation is one of intensive discussion. Though a higher or more progressive tax can ease income inequality, we doubt its validity on controlling wealth inequality. Because wealth is accumulation of saving, which is close related to income, we conjecture that as long as the income inequality has not been diminished completely, extreme wealth inequality will appear eventually, sooner or later. That is to say, a tax policy can only postpone, but not avoid, the advent of extreme wealth inequality.

This study focuses on financial markets. Therefore, the income, wealth and tax discussed in this paper correspond respectively to financial gain or loss, the wealth accumulated by financial investments, and the tax imposed on financial investments, e.g. capital gain tax. As no consumption is involved in this situation, wealth is just accumulation of financial income. Because money always goes from one person to another in financial markets, the aggregate payoff can be treated approximately as a zero-sum payoff. The winner-take-all assumption, i.e. only one winner in every period, produces an extreme income inequality, which helps test the effectiveness of a tax, just like a stress text does. The method of agent-based modeling [3] is used to construct an artificial financial market filled with zero-intelligence agents [4, 5] who make random investment decisions. The assumption of zero-intelligence agents can be controversial. On one hand, with advantages for investigating markets, it has been widely used in agent-based computational economics and finance [6-8]. On the other hand, it is a strong simplification as in reality people are intelligent. Proper adoption of this assumption depends on the main focus of a study. It is a reasonable assumption usually when the study is on a macro level and irrelevant to individuals’ behavior or performance. This is consistent with

our study. Another reason for assuming zero-intelligence agents is that Levy and Levy [9] claim and we quote: "the main force driving inequality is the randomness of the investment process, rather than differential talent". Besides, this design allows us to focus on the mechanism of tax without being distracted by agents' strategies. Special attention is paid to the relation between tax rate and wealth inequality and to the wealth effect, which describes how current wealth affects future income distribution. The tax mentioned in this paper is a flat one, and the structure of tax is not discussed.

II. THE MODEL

A repeated number guessing mechanism is used to decide agents' gains and losses in every period. There are N agents with identical initial endowments1 w0. Every period, there is one winning number yt*, and all agents have to make their guesses. Agents have zero-intelligence in the sense that they pick random numbers from the interval (0,w0) as their guesses yi,t, i.e. yi,t U(0,w0), here i denotes a specific agent. The agent with a closest guess is the winner, and others losers. Losers lose some money in proportion to their guesses, like ayi,t, and the winner gets all the money losers lose. Before the end of one period, the government imposes a flat capital gain tax on the winner at a tax rate of TR and redistributes the tax among all active and inactive agents equally. No loan is allowed in this model, so an agent has to exit from the market when his wealth is lower than a minimum threshold bw0. However, this inactive agent can reenter, i.e. becoming active again, after his wealth exceeds the minimum threshold bw0 again by cumulating tax redistribution.

The formation of the winning number in every period is critical. It follows

, , , 1 , , 1t i t i t i t i t i ti i

y y s w s w (1)

where wi,t-1 is agent i's wealth at the end of period t-1, and si,t is the logical value of agent i's state at the beginning of period t, an active agent has a state of 1 and an inactive agent has a state of 0. This equation shows that the winning number is a mean of active agents' guesses weighted by their wealth. That is, a rich agent has a larger impact on the winning criterion than a poor agent. As a result, wealthy agents enjoy higher probabilities of winning than poor agents. We refer to this phenomenon as the

* We acknowledge the helpful comments of anonymous reviewers.

1. This assumption can be relaxed to a normal distribution, which is more reasonable. But such modification will not change our qualitative conclusions. The reason why we reserve a uniform distributed initial endowment is that it shows the emergence of inequality from perfect equality better.

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Fig. 1. The evolvement of wealth inequality under different tax rates. For each curve, from top to bottom, the tax rate is increasing.

wealth effect.

If we use it* to denote the winner in period t, agents' wealth accumulation and states can be formulized as

,

, , 1 ,

,

, , ,

,

,1,

, 0, 1

, 0

1,

1,

0

10

i t t

i t i t i t

t t t i t t

i t i t t i t t

t i t

t i ts i i

i ti t

i t

w w w

Y TR Y TR Y N s i i

w ay TR Y N s i i

TR Y N s

Y a y

if w bws

if w bw

(2)

where i,t is the wealth change of agent i in period t , and Yt is the winner's pretax income in period t.

At a glance, this number guessing mechanism may look like Keynes' beauty contest [10], but they are different in many aspects. In Keynes' beauty contest game, participants have intelligence and information about the winning criterion. They know that the person with a guess closest to the mean of all guesses will win. And they include this information into their strategies. While in our model, though the winner is decided in

a similar way, agents have neither intelligence nor information. In a larger sense, our number guessing mechanism falls outside the scope of game theory, because there is no strategy at all in it. The concept of zero-intelligence precludes agents' strategies, motivations and rationalities, so that we can focus on wealth accumulation, which is realized here via this repeated number guessing mechanism.

III. WEALTH INEQUALITY

This model is intentionally designed to produce extreme income inequality. The zero-sum payoff and winner-take-all mechanism all serve this goal well. Will this extreme income inequality naturally leads to extreme wealth inequality? Because of the randomness in agents' guesses and their multiple exits and reentries, this problem is not analytically tractable. So we turn to the method of agent-based modeling, where numerical simulation makes it easy to track every detail in the process of wealth accumulation and in the evolvement of wealth inequality.

Gini coefficient [11] is adopted as the measure of wealth inequality. Its value varies from 0 to 1, and a higher value represents a higher wealth inequality. Fig. 1 shows the evolvement of Gini coefficient under the following parameters N=100, w0=100 and a=b=0.2. Assigning identical value to aand b dispels the possibility of negative individual wealth which can be caused by unaffordable guesses. Each curve represents the average of 50 simulations.

423

Fig. 2. The evolvement of wealth inequality with wealth effect. For each curve, from top to bottom, the tax rate is increasing.

In Fig. 1, all curves start from 0, because the initial wealth follows a uniform distribution. As time goes by, the values of Gini coefficient under every tax rate increase steadily. From perfect equality, wealth inequality emerges and becomes worse. With a tax rate lower than 0.5, the values of Gini coefficient eventually converge to 0.88. A higher tax rate only slows down the speed of convergence. According to parameters used, the extreme situation, that one agent seizes almost all wealth and drives others out of market, produces a Gini coefficient about 0.891. In our numerical simulation, the convergence of Gini coefficient to a value so close to our theoretical prediction indicates the advent of extreme wealth inequality. However, when the tax rate is higher than 0.5, the values of Gini coefficient stay at certain levels. And the higher the tax rate, the lower the final level of Gini coefficient. Here, these final levels are time-constrained by our simulations. What will happen next? Will these values stay at their final levels forever? Or will they further increase and converge to the extreme wealth inequality level as curves with tax rates lower than 0.5 do? The later situation seems more likely to happen, based on the behavior of the curve with a tax rate 0.5. Till the end of our simulation, it is halfway through its convergence to a higher value. However, because of too much randomness involved, it is a little early to make a conclusion for sure. In the next section, after removing some randomness, a clearer view will be revealed.

IV. WEALTH EFFECT

The essence of wealth effect is the impact of current wealth on future change of wealth. We try to argue that the appearance of extreme wealth inequality is inevitable when there is wealth effect.

In the above model, the winning number is the mean of agents' guesses weighted by their wealth, as shown in (1). A wealthy agent affects the formation of the winning number more, and therefore increases his chance to win. But his impact on his probability of winning is not proportional to his wealth. As agents guess randomly, it is possible that a poor agent gets a closest guess out of pure luck. This possibility obscures the evolvement of wealth inequality. Follow the previous model design, except that now the winner is randomly picked with probabilities proportional to agents' wealth, the development ofGini coefficient is shown in Fig. 2. All curves converge to a level of Gini coefficient 0.88, except the one with tax rate 1. A higher tax rate only postpones, but not avoids, the advent of extreme wealth inequality. The speed of convergence is accelerated under each tax rate. As for the curve with tax rate 1, perfect equality fails to maintain because of the randomness in majority's losses. In the end, the Gini coefficient stays at a level about 0.395.

424

Fig. 3. The evolvement of wealth inequality without wealth effect. For each curve, from top to bottom, the tax rate is increasing.

Assuming the probability of winning proportional to current wealth has pushed our model to one direction. Besides, we can push it to another direction, supporting our argument by counterexamples. If there is no wealth effect, that is, removing wi,t-1 from (1) so that the winning number is just a mean of allactive agents' guesses, the evolvement of wealth inequality is presented in Fig. 3. Without tax, i.e. the curve with tax rate 0, the extreme wealth inequality happens as always. All other curves with non-zero tax rate have not reached extreme inequality till the end of simulation. Instead, their values of Gini coefficient stay at certain levels, and a higher tax rate corresponds to a lower level of Gini coefficient. That is to say, when the wealth effect is absent, the extreme wealth inequality disappears. This example shows that the wealth effect is critical in the formation of extreme wealth inequality.

One peculiar phenomenon in Fig. 3 is that those rather “smooth” curves become “erratic” once their corresponding final levels are reached. This is an illusion caused by the logarithmic time axis. In fact, those curves fluctuate withconstant amplitude and frequency. Similar things also happened in Fig. 1 and Fig. 2.

Last but not the least, the result of our model is robust under substantial modifications. A higher value of parameter a, the proportion of loss to guess, accelerates the process ofconvergence. A larger parameter b, the minimum wealth proportion for an agent to be active, lowers the level of Gini coefficient under extreme wealth inequality. When a is larger

than b, wealth inequality can be worse as some agents mayhave negative wealth. The distribution of initial endowment affects the starting points of all curves in Fig. 1, e.g. when wi,0 N(100, 20) the initial Gini coefficient is about 0.112, but not their further development or final states.

V. CONCLUSION

In this paper, we use a repeated number guessing mechanism to study wealth inequality and important factors during its formation in financial markets. The model is intentionally designed to produce extreme income inequality, and then extreme wealth inequality arises. We identify the wealth effect, i.e. the positive impact of current wealth on future wealth growth, as a key link in this process. When wealth effect works, a flat tax, no matter how high the tax rate is, can only postpone, but not avoid, the advent of extreme wealth inequality. We support our argument with examples under reasonable modifications.

As a purely hypothetical model, because of strong assumptions we made, it is too early to compare our work with empirical ones at this stage. Besides, most available empirical data on inequality are about economic inequality rather than financial markets. For future work, some basic assumptions, e.g. zero-sum payoff or winner-take-all assumption, will be properly relaxed so that we can expand this model to more economic scenarios, or even get support from empirical work.

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