**1. Introduction**

Many forms of taxation evolved organically in different political and economic systems over human history [1]. A tax system deals with the flow of money from society to government, as tax revenues are collected and given back to society as governmen<sup>t</sup> spends tax revenue, including governmen<sup>t</sup> transfers in relation to wealth redistribution. Historically, the extent and purpose of taxation generally was not to benefit society as a whole [1]. Today, opinions on the extent and purpose of taxation are often politically charged, making it impossible to design a tax system that is acceptable to all positions. Arguments from different philosophical perspectives on relationships between society, economy, and public policy identify complex issues that need to be reconciled to achieve a rational tax system. The impetus of this work comes from many laudable debates in the United States (US) about the federal tax system regarding how to set tax rates, and on related issues that affect the nature of taxation that include governmen<sup>t</sup> spending, short-term deficits, long-term debt, governmen<sup>t</sup> dependency, poverty, income inequality, disproportionate tax burdens, and the diminishing wealth of the middle class. These ongoing debates indicate that the structure of a tax system has a significant impact on the economy and well-being of society. The aim of this work is to construct the foundation for a holistic tax system with universal appeal divorced from ideology, by taking a pragmatic approach to solving a wide range of problems faced by modern society.

#### *1.1. Motivations for a Holistic Tax System with Desirable Characteristics*

Based on the Gini index [2] from 1980 to the present, the middle class of the US is steadily shrinking, suggesting that the federal individual income tax could be changed to

**Citation:** Jacobs, D.J. Victory Tax: A Holistic Income Tax System. *Entropy* **2021**, *23*, 1492. https://doi.org/ 10.3390/e23111492

Academic Editors: Ryszard Kutner, H. Eugene Stanley and Christophe Schinckus

Received: 17 October 2021 Accepted: 8 November 2021 Published: 11 November 2021

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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

strengthen the middle class. Although a strong middle class fuels a consumer economy, the distribution of income (and wealth) in free markets is empirically found to be highly skewed toward a tiny percent of its population [3–5]. Remarkably, there is universality in heavy tailed distributions for income and wealth across a population in free markets, regardless of governmen<sup>t</sup> form and tax law. This work accepts the inevitability of highly skewed income dispersion in free-market economies, and then uses the free market and human nature to its advantage.

It is prudent to create incentives for economic growth, while discouraging governmen<sup>t</sup> dependency, which can lead to complacency and a less productive society. In this context, regressive and progressive taxation affects the economy and redistribution of wealth in different ways. When governmen<sup>t</sup> redistributes wealth, the effective tax rate (*ETR*) will vary between households. A (higher, lower) *ETR* makes it (more, less) difficult for a household to generate wealth due to a (higher, lower) tax burden. A pragmatic reason for the governmen<sup>t</sup> to redistribute income is that this practice creates an environment in which all segments of society can accumulate wealth.

A widely employed definition of a flat tax involves taxing labor income at a single marginal tax rate with an allowed deduction [6]. In this work, a flat tax is defined as having the *ETR* independent of income type and level. In practice, a flat tax is achieved by not allowing tax deductions or governmen<sup>t</sup> transfers, so that no redistribution of income is made. In the deductive approach taken here, the *ETR* dependence on income level is not a priori assumed. Rather, the *ETR* will be a result of a set of guiding principles concerned with fairness, maintaining long-term stability for society, and creating an incentive for personal economic growth at all income levels. There should be a simple way for the governmen<sup>t</sup> to collect tax revenues without runaway deficits and not impose excessive tax burdens on the population. If a tax system achieves these goals using a strange looking *ETR*, so be it.

A holistic tax system should create a net benefit to society while promoting individual interests, and therefore can be implemented by any political system provided the government is sincere about respecting human dignity and wants to maintain a stable free-market economy. Minimally, all persons in society should have equal opportunity to generate personal wealth from a market economy. The tax system should create incentives for individuals to generate wealth. The mathematical framework of the tax system should not be tied to specific policies. The tax system should have a structure independent of a population's income distribution. Moreover, it should be easy for a household to determine its tax burden, pay owed taxes, and receive governmen<sup>t</sup> assistance when needed. Likewise, it should be easy for the governmen<sup>t</sup> to manage administratively, deter runaway deficits, and adapt to society's needs. Applied across the income spectrum, the tax system should capitalize on the free market to create income growth opportunities, encourage self-reliance, and minimize governmen<sup>t</sup> dependence.

#### *1.2. Contributions from a Scientific Approach*

To my knowledge, the tax system developed herein is novel, although there are similarities with the concepts of a negative income tax [7] and basic income guarantee [8]. Specifically, the principles for a holistic tax system developed in Section 2.1 based on pragmatic considerations identify negative income tax and basic income guarantee as inadmissible. Positive and negative aspects of these other proposed tax systems have been extensively discussed [9–16]. The tax system developed here escapes the pitfalls of negative income tax [17]. The main criticism of a negative income tax is that it removes an incentive for people to work in low-wage jobs when it is more lucrative to obtain greater income when not working. This problem is generally referred to as a welfare trap [18]. Common criticisms against a basic income guarantee are that marginal tax rates become very high [19–22], it is too risky [19], and it is cost prohibitive on large scales [20,21]. Following five principles that are conceptually justified below, a simple holistic tax system emerges, and it is shown in Section 3 to be feasible and cost-effective.

The concept of a basic income guarantee was discovered independently many times under different names [16]. In this paper, the similar concept of a need-based income guarantee is the result of the development of a pragmatic tax system based on a deductive approach. The critical difference to previously proposed schema [13] is in the way government transfer is handled. After constructing five guiding principles, which include a single tax rate with three types of tax deductions, the derived tax system provides a surprisingly low population average *ETR*, where the most tax relief goes to the middle class. The *ETR* for the ultra-rich is no higher than for the extreme poor. Moreover, low-income households are subjected to a regressive *ETR* that eliminates the welfare trap and encourages the poor to gain financial independence and move into the middle class. The tax system is unified with the welfare system, so that poverty and deficits can be virtually eliminated without imposing disproportionate tax burdens.

For the rest of this paper, Section 2 develops the tax system first conceptually and then mathematically. Next, several model economies are constructed to quantify the characteristics of the tax system. In Section 3, the parameters of the tax system are explored, leading to a set of parameters that maximize after-tax income for the vast majority of the population with minimal governmen<sup>t</sup> dependency. In this scenario, it turns out that the ETR takes the shape of a "V", where the middle class enjoys near zero effective tax rates. This V-signature inspires the name victory tax, which has been adopted because of its general applicability to widely varying economies. The victory tax system is then compared with a flat and linear progressive tax system. In Section 4, the benefits of a simple tax structure are discussed, as well as possible public policy decisions and future work. The conclusions of this paper are given in Section 5. The main conclusion is that the victory tax as a holistic income tax system is constrained in such a way that, with minimal governmen<sup>t</sup> involvement, households at all levels of income can reap benefits by using this tax system selfishly to gain wealth, which broadly helps society achieve a higher standard of living.

#### **2. Model and Methods**

The proposed tax system is deduced from five guiding principles that form the basis of a mathematical framework. This section starts with the conceptual framework, in which the rationale for the guiding principles is discussed. Recognizing that there are different perspectives, each principle is rationalized by questioning whether it has universal appeal. The goal is to transcend political biases as much as possible by rejecting what is not universal. An effort is made to distill an income tax system into essential elements. A mathematical framework is then developed with parameters that encapsulate a family of income tax systems that differ by the parameter values set by the state of the economy. This allows governments to adapt to society's needs over time.

### *2.1. Conceptual Framework*

In a bottom-up approach, five guiding principles for a holistic tax system are first listed. The rationale for each principle is then discussed as subsections. These principles shape the tax system by imposing constraints, which leads to a tax system that is easily parameterized within a mathematical framework. The principles are listed as:

	- (a) A basic deduction to offset living expenses.
	- (b) Itemized deductions that promote better standard of living.
	- (c) Capital loss deductions that promote economic growth.

#### 2.1.1. Income of All Types Is Taxed at the Same Rate, Independent of the Income Level

Taxing one type of income differently from another creates social discrimination. For example, it could be argued that income from work done by a teacher should be taxed twice as high as income from work done by a welder. Intrinsic to this argumen<sup>t</sup> is that the value of the work of a welder is more important than a teacher, or perhaps simply to offset the risks inherent in welding. However, an objective truth of this differentiation is not self-evident, as many teachers would argue otherwise. Many of these comparisons can be made, all of which end with arbitrary conclusions. Logic suggests that it is not the place of governmen<sup>t</sup> to judge the intrinsic value of income beyond the definition of legal and illegal activities. In practice, assigning different tax rates to different types of income creates a complicated system that leads to endless debate, because there is no universal truth for all cultures, all types of economy, all types of government, and certainly not a constant in time. The same logic is true when it comes to distinguishing income from labor versus investments or other forms of income, such as gifts, winnings, insurance or inheritance.

For free-market economies, the amount of income a person generates in terms of salary or return on investments determines the value society attaches to occupation and investment. Income is determined by tangible factors, such as supply and demand, investment decisions, the wealth potential of occupations, and the desire of an individual to be wealthy. For example, the income of a surgeon can be higher or lower than a professional athlete, depending on various factors. Therefore, the allocation of different tax rates on different types or income levels must be rejected. A household with orders of magnitude more income than another will pay proportionally that much more tax, which does not discriminate on income levels. Although sale taxes can coexist with this principle, no other form of taxation of a person's income is allowed. For example, this means that in the US, the separate payroll tax must be eliminated, as there can only be one tax rate on income, which is comprehensively taken care of by the income tax system within a holistic approach.

#### 2.1.2. A Single Tax Rate Adjusts to Ensure Government Fiscal Stability

A pertinent question is: What should the single tax rate be? A constant value (say 9%) could be argued as optimal, but this value is not self-evident. Indeed, any specific value would not be universally optimal for all cultures, economies, governments, and for all time, as the state of the economy fluctuates over time. Therefore, a variable tax rate that adapts to the revenue needed to cover projected governmen<sup>t</sup> spending is required. A dynamic tax rate allows a governmen<sup>t</sup> to control fiscal stability while adapting to shortand long-term economic conditions. Cycles of high and low tax rates induce an elastic response to balancing budgets (with limited liability), which ensures reliability in public services and mitigates the accumulation of long-term debt. Both of these attributes are necessary for long-term stability of society.

It is worth pointing out that policy makers have responsibility for developing debt accumulation or reduction plans. Regardless of the directions that policy makers decide, the adjustable tax rate makes tax revenue collecting responsive to governmen<sup>t</sup> policies. For example, if more funding is appropriated toward infrastructure or defense, the single tax rate will increase, and society can monitor and judge the benefits for increased taxes. In summary, a single tax rate offers transparency in governmen<sup>t</sup> spending, and in combination with other social-economic measures, the value that the governmen<sup>t</sup> attaches to society's well-being becomes transparent.

#### 2.1.3. Government Transfer Is Only Used to Establish a Minimum Standard of Living

For what reason, if any, should governmen<sup>t</sup> transfer be used to supplement household income? Arguably, governmen<sup>t</sup> transfers should not be used for anything other than to help a household achieve a minimal standard of living. This principle does not exclude governmen<sup>t</sup> support in other forms, such as tax deductions and public services. For example, governmen<sup>t</sup> spending on entitled health care, education, or other infrastructure does not constitute governmen<sup>t</sup> transfer. However, public services must be independent of the income level, void of any income qualification.

Providing public services to poor subpopulations is unnecessary, because the poorest households will live at the poverty line, which sets a minimum standard of living. Rather than designing social programs to help the poor, public services should be designed to help society. This paradigm shift of shared interest will ensure that social programs are of high quality. It is worth noting that public services will reduce poverty by reducing basic living costs. Conversely, the poverty line rises as free public services decrease. Importantly, this principle prohibits the use of governmen<sup>t</sup> transfers for unemployment or retirement compensation. Consequently, policy decisions will involve common interests in diverse and large segments of the population.

It is obvious that if governmen<sup>t</sup> transfers are used to subsidize households for anything other than supporting a minimum standard of living, an arbitrary number of good reasons to redistribute income will lead to a complex tax system that is not universal. Why, however, should governmen<sup>t</sup> redistribute income to set a minimum standard of living? Elimination of poverty is a singular case that appeals universally because of the innate human desire to live healthy and securely with dignity. This institutes the responsibility of the governmen<sup>t</sup> to provide the means for all individuals in society to live securely with dignity over countless generations.

In practice, the poverty line must be set to balance competing factors. The poverty line should not be set too low, because more productivity in the entire population will result if society as a whole has a functional standard of living. Conversely, if the poverty line is set too high, the tax rate will rise too high, stifling economic growth. As such, the minimum standard of living that society can tolerate sets the poverty line for households. Although the way policy makers define this poverty line is left open, it must be based on income (not savings or wealth). Government transfers supplement income to establish a minimum standard of living as a safety net. If a household starts with considerable assets and then unexpectedly finds itself without income, this household can survive at the poverty line, with basic needs fulfilled. Moreover, this household can use its savings, albeit a finite resource, to live a higher standard of living.

A consequence of having one specific reason for governmen<sup>t</sup> transfer is that the governmen<sup>t</sup> has minimal involvement in a free market. As another example, if a household with a large accumulation of debt suddenly loses its income, it is likely to lose its possessions if an agreemen<sup>t</sup> with its lenders cannot be reached. Responsibility and risk tolerances exist between lenders and households taking loans. Government transfer is used only to maintain a minimum standard of living, and this results in keeping the net amount of transfer to a minimum, and hence keeps the tax rate to a minimum.

The COVID-19 pandemic is an unfortunate example of a situation in which the victory tax system maintains a stable economy during a crisis. Households automatically receive governmen<sup>t</sup> transfers when their income falls below the poverty line due to job loss. Because of guaranteed basic income, the debate in the US on the scope of COVID-19 relief packages would be unnecessary since governmen<sup>t</sup> transfer creates a safety net of security. Nevertheless, financial losses from businesses and households would be expected. While many lenders would be eager to force foreclosure, other lenders would use unfortunate events as a growth opportunity to attract new (sound) customers by covering businesses from bankruptcy and households from personal losses. The free market would solve the vast majority of the problems, with governmen<sup>t</sup> regulations perhaps requiring debt collectors to exercise patience. As jobs reemerge, low-income households quickly increase their after-tax income, avoiding long-term economic stagnation.

2.1.4. Three Types of Tax Deductions Incentivize Wealth Accumulation

Tax deductions are used to reduce the tax burden on households for various reasons. For a certain amount of revenue to be collected, reducing the tax burden on a subset of households requires other households to pay disproportionately higher taxes. Of course,

different tax rates applied to different income levels or types cause disproportionate tax burdens. However, even if a single tax rate is applied to all income levels and types (e.g., nominally a flat tax), tax deductions create a non-flat *ETR* dependent on household income. Importantly, different types of tax deductions produce different relative benefits for households at different income levels. For example, the basic tax deduction that offsets minimum living expenses provides proportionately more benefit to low-income households. Itemized deductions predominately help households with middle-range income. Capital loss deductions on investment losses primarily benefit high-income households. In fact, low-income households cannot capitalize on capital loss tax deductions.

Among the different types of tax deductions, a balance should be sought between benefits versus the disproportionate tax burdens created across the income spectrum of households. The structure of tax deductions should benefit society, as taxpayers seek to obtain the maximum after-tax income possible from their own interests. Tax deductions therefore offer specific redistribution mechanisms for governmen<sup>t</sup> support to motivate households to accumulate wealth, which in turn maintains a stable and growing economy. The rationale for the basic, itemized, and capital-loss tax deductions is discussed next, while key variables for the victory tax system are introduced.

*Basic tax deduction:* A minimum income is needed to live functionally in modern society. In the past, most people could live on natural resources or farm land. Unless free public services take the place of natural resources, job loss literally becomes life threatening. Hence, a basic deduction, *BD*, is incorporated to cover minimum living costs for a household. Although *BD* is a free parameter, it is appropriately related to the poverty line. Furthermore, *BD* should only depend on the number of dependents in a household and the cost for necessities (which is location dependent), as its sole purpose is to offset minimal living costs in the context of a social norm.

*Itemized tax deduction:* To incentivize financial independence, optional itemized deductions are allowed. Itemized deductions offer the governmen<sup>t</sup> flexibility in the tax code to encourage certain measures, such as buying a house, accumulating a retirement portfolio, compensating costs for professional training, education or medical needs, or making donations to charities. As such, itemized tax deductions create self-interest incentives for households to take measures that also benefit society as a whole. The net itemized deduction, *ID*, is incorporated into the general framework of the victory tax. The total income that can be deducted is capped at a maximum. Setting a maximum deduction prevents all households from not paying tax. Two methods are used to set the maximum total deduction. A maximum deduction, *MD*, and a maximum percentage, *MP*, of net income, *N I*. The total deduction, *TD*, allowed by a household is given by:

$$TD = \min(BD + ID, MD, MP \times NI) \tag{1}$$

For a household with a net income above the poverty line with no itemized deductions, its taxable income, *T I* is given as:

$$TI = \max(0, NI - TD) \quad \forall \ NI > BD \tag{2}$$

The equation for taxable income developed thus far is easy to understand. The combined total of basic and itemized deductions cannot exceed the maximum allowed deduction, nor a maximum percentage of income. Once the total deduction, *TD*, is determined, it will be used to reduce net income in order to achieve the taxable income. However, if the deduction is greater than the net income, *N I*, then the taxable income, *T I*, is set to zero, as it cannot be negative. The net income will be precisely defined after capital loss deductions are considered.

*Capital loss deduction:* To encourage households to increase their wealth through investments, capital loss deductions are used to mitigate risk. It is self-evident that governmen<sup>t</sup> cannot rescue all households from financial loss. Hence, under what circumstance, if any, should governmen<sup>t</sup> aid households to recover lost wealth? Imagine an individual

that invests \$100,000 in a company, and subsequently loses this investment due to the bankruptcy of that company. Another individual buys a \$100,000 painting, which is inadvertently destroyed in a fire. Should both scenarios be treated equally, or should a distinction between these two losses be made? The answer rests with public policy makers who create tax law, where the answer can range from no capital loss deduction for anything to almost everything. The concern addressed next is that if the tax burden for the wealthy is disproportionately reduced, the middle class has a much greater tax burden, as the poor contribute little to tax revenue. Therefore, to justify a capital loss deduction, it is prudent to make an analogy with government-run health care, which shares an equivalent concept of large-scale group insurance.

Capital loss tax deduction is similar to an insurance program managed by the government. Specifically, all household incomes are being taxed, but the governmen<sup>t</sup> only aids households suffering capital losses. A greater capital loss begets more governmen<sup>t</sup> aid. The practice of spreading investment risk across the entire population to cover only households that made poor investments is like a health insurance program. That is, sick and healthy individuals are taxed, but the governmen<sup>t</sup> only aids those suffering sickness. The greater the sickness begets more governmen<sup>t</sup> aid. Again, aid only goes to a subpopulation to keep people functional and productive, which is beneficial to society as a whole. Although only a subpopulation will benefit from the insurance, a priori it is unknown who will use it.

The arguments against universal health care (lack of resources, highly skewed redistribution of income and poor governmen<sup>t</sup> management) amplify against the rationale for capital loss tax deductions. The most troubling aspect is that only households with the highest income are predisposed to benefit. Thus, it is not self-evident that capital loss deductions should be included in a tax system. Nevertheless, creating opportunities that ensure the well-being of society must be the responsibility of government. When viewed as insurance, policy makers need only debate the scope of coverage. From this point of view, there is no universal answer for the scope of capital loss tax deduction or free health care services, since a weak economy cannot support the same level of coverage as a strong economy. As such, public policy debates will ultimately affect governmen<sup>t</sup> budgets, tax rates and poverty line. The proposed tax system is designed to support the outcome of these debates within the constraints inherent in the tax system.

Since the capital loss deduction benefits society as a growth mechanism, it is included in the victory tax system to ensure generality. Nevertheless, since only a subset of households reaps the benefits, it is prudent to limit this redistribution of wealth to prevent a higher tax burden on households with much lower income. This limitation is similar to a maximum coverage limit in an insurance policy. Note that governmen<sup>t</sup> transfer to the poor is limited by the poverty line, and a limit to the maximum itemized deduction was also introduced. In the same spirit, a cap on capital loss deductions is set by not counting capital losses that exceed capital gains within a given year. From a consistency point of view, paying tax on income must be over the same time period regardless of the income type or income level of a household. To roll over capital losses to future years, the same time period must apply to all income types. A tax system in which taxes are due annually seems reasonable, compared to alternatives such as every four months or every four years. For the prototype tax system constructed and demonstrated in this paper, capital losses are limited to one-year windows, which correspond to taxes collected annually. Although there is a maximum deduction for capital losses per year, no lifetime limit for capital losses is set. Likewise, there are annual limits, but no lifetime limit to governmen<sup>t</sup> transfer or itemize deductions.

The definition of net income within the victory tax system is given as:

$$NI = E + \max(0, CG - CL) \tag{3}$$

where *E* defines earnings from employment, *CIG* defines capital income gained, and *CIL* defines capital income lost. Note that *CIG* includes all types of income that are not earnings from employment or governmen<sup>t</sup> transfers. Upon inspection of (3) within a given year, the governmen<sup>t</sup> will maximally allow a household to deduct as much capital loss as gained. The amount of risk assumed is therefore determined by the skill of the investor. If an investor incurs more capital loss in a given year than capital gains, the governmen<sup>t</sup> will not allow this excess loss to be deducted on the grounds that the investor creates too much risk for society to absorb. This restriction strengthens a free market: investors will exercise prudent judgments to ensure that gains are greater than losses when governmen<sup>t</sup> assistance is limited. In practice, the tax system encourages investors to focus on fundamentals and long-term investments, spreading losses over multiple years. Placing annual limits on capital loss deductions (in fact all types of tax deductions) minimizes governmen<sup>t</sup> dependency.

#### 2.1.5. Tax Deductions Cannot Be Applied on Government Transfer

This principle is self-evident, because income from governmen<sup>t</sup> transfers is at the expense of all taxpayers who make a productive contribution to society through earnings and/or capital gains. Note that allowing deductions on governmen<sup>t</sup> transfer would amplify governmen<sup>t</sup> assistance. This guideline minimizes governmen<sup>t</sup> dependency.

#### 2.1.6. Unique Property of the Victory Tax System

To emphasize the unique properties of the victory tax system compared to other basic income guarantee tax systems, an important consequence follows when the first, third and fifth principles are combined. Since governmen<sup>t</sup> transfer income is taxed, but deductions cannot be applied to this part of household income, a regressive *ETR* emerges as a function of income percentile for the poor. The regressive *ETR* enables low-income households receiving governmen<sup>t</sup> aid to become self-reliant and achieve higher income levels without a welfare trap (the analog to a nucleation barrier). The formation of a low-income regressive *ETR* will become clear in the results section.

#### *2.2. Mathematical Framework*

The five principles examined above are now considered axioms to construct the general mathematical framework of the victory tax system. Relevant variables for a household to calculate tax liability are described in Table 1 for convenient reference.


**Table 1.** Alphabetically ordered list of variables for the victory tax system and their description.

In addition to (3) defining net income, the victory tax formulas are given as:

$$GTI = \max(0, BD - NI) \tag{4}$$

$$TD = \min(BD + ID, \max(BD, MD), MP \times NI) \tag{5}$$

*T I* = *GTI* + max(0, *N I* − *TD*) (6)

$$TAX = VTR \times TI \tag{7}$$

$$ATI = GTI + NI - TAX \tag{8}$$

$$ETR = \frac{TAX}{GTI + NI} \tag{9}$$

The variables {*E*,*CIG*,*CIL*} quantify income characteristics of a household. Notice that the net income of a household can never be negative. The maximum income for a household eligible for governmen<sup>t</sup> transfer is determined by the basic deduction, *BD*. The word "eligible" underscores the restriction that households with incomes above *BD* cannot receive governmen<sup>t</sup> transfer. Otherwise, *GTI* defined in (4) covers the income deficiency of a low-income household in order to put it on the poverty line. The parameters {*BD*,*MD*,*MP*} define the maximum limits on allowed deductions. After the total tax deduction is determined from (5), the taxable income is calculated by (6). The victory tax rate, *VTR*, multiplies the taxable income of a household to obtain tax liability (7). The after-tax income is calculated from (8), which adds governmen<sup>t</sup> transfer to net income minus paid tax. The *ETR* is defined in (9) as tax paid divided by the total income. Since no deductions can be applied to governmen<sup>t</sup> transfer, it works out that *ETR* → *VTR* when *N I* → 0. Although counterintuitive, the poorest households pay the highest effective tax rate among the entire population.

Within the victory tax system, *VTR*, depends on the target tax revenue, *TTR*, deduced from projected budget needs for the next year set by policy makers. In addition, *BD*, should be proportional to the poverty line, *PL* which is the income required to maintain a minimum living cost. Reflecting the state of the economy, *BD* will change annually to ensure that the least possible after-tax income, *AT I*, corresponds to *PL*. From Equations (3) and (4) a household with *N I* = 0 will have a taxable income equal to *BD* from governmen<sup>t</sup> transfer, and after-tax income will be given by *AT I* = *BD* − *VTR* × *BD*. This shows the insightful relationship that *BD* = *PL*/(1 − *VTR*), which indicates that as *VTR* increases, the basic tax deduction increases at a higher rate. When public policy makers propose to increase *VTR* for building infrastructure or defense, *BD* will automatically increase even if the poverty line remains constant.

The two parameters {*MD*,*MP*} determine the shape of the ETR. In practice, *MD* and *MP* will be functions of the number of dependents in a household, denoted as *d*. For simplicity, *MP* is considered independent of *d*. Although *MD* can be set with considerable latitude through tabulation, a sound approach is to set *MD* proportional to *PL*. By assuming *MD* = *k* × *PL*(*d*), the parameterization details for *MD* as a function of dependents are inherited from *PL*(*d*). For the US, the poverty line for a household with a certain number of dependents is publicly available in tables [23]. More generally, an objective economic measure will be used to define *PL*(*d*). Although not considered here to keep the analyses clear, the poverty line will generally depend on location (region within a country), since all regions do not have the same cost of living.

Parameters for the victory tax system are explored in Section 3. It is found that when *MP* = 0 and *ID* = 0, a V-shape signature emerges for the *ETR* as a function of income percentile, with *ETR* = 0 marking the bottom of the V. Importantly, (5) determines the allowed tax deduction after taking into account basic and itemized deductions and maximum percent of *N I*. Since itemized plus basic deductions increase total deduction, the term max(*BD*, *MB*) that appears in (5) is needed to enforce consistency. In particular, when *MD* is set below *BD* as an independent variable, the basic deduction is still offered, but itemized deductions are no longer allowed. However, *MP* can reduce the maximum allowed deduction below *BD* without inconsistency, because *MP* × *N I* is a competing restriction on tax deductions. Note that a flat tax corresponds to the limit *MP* → 0, where *ETR* is constant, independent of income percentile.

As *MP* gradually changes from 1 to 0, the V-shape morphs into a U-shape as the "U" becomes shallower, with the minimum *ETR* increasing as *VTR* decreases. These shape changes create an elastic tax system [24]. At *MP* = 0 a flat tax emerges as a special case of a victory tax system, where *BD* sets the threshold income in which governmen<sup>t</sup> transfer is no longer received. A prototypical victory tax system considered in this work is defined by: *MD* = *kPL* and *MP* = min(*<sup>r</sup>*, <sup>1</sup>), where *r* is the coefficient of variation in net income over the population. Specifically, *r* is the ratio of the standard deviation in *N I* to the mean *N I*, which serves as a convenient objective measure of the economy. The application of objective measures updates *PL* and *r* each year, allowing the victory tax system to respond dynamically to changes in the free market and public policy.

The basic deduction is determined in the prototypical victory tax once parameter *k* is specified together with the poverty line, *PL*(*d*). By combining equations of the victory tax system, the tax liability of a household with *d* dependents is given as *TAX* = *VTR* × *fd*(*x*|*ID*, *VTR*) from Equation (7), and its taxable income is expressed as:

$$\begin{split} f\_d(\mathbf{x}|ID, VTR) &= \max\left[0, \frac{PL(d)}{1 - VTR} - \mathbf{x}\right] \\ &+ \max\left\{0, \mathbf{x} - \min\left[\left(\frac{PL(d)}{1 - VTR} + ID\right), \max\left(kPL(d), \frac{PL(d)}{1 - VTR}\right), r\mathbf{x}\right] \right\} \end{split} \tag{10}$$

where *x* is the net income (replacing *N I* for simpler mathematical notation) and the subscript *d* denotes the fact that the poverty line depends on the number of dependents in a household. Equation (10) is the result of substituting all the relevant variables described above in the arguments of Equation (6). Note that *fd*(*x*|*ID*, *VTR*) depends on *VTR*, which is the dependent variable to be determined. In addition, *VTR* = *TTI*/*TTR* where *TTI* is the total taxable income over the population, and *TTR* is the total tax revenue to be collected over the population. To calculate *VTR* = *TTI*/*TTR*, we must have *TTI*, which is given by the net sum of *fd*(*x*|*ID*, *VTR*) over all households in the population. However, to calculate *TTI*, we must have *VTR* because *fd*(*x*|*ID*, *VTR*) depends on *VTR*. Despite this circular dependence, it is straightforward to numerically solve for *VTR* iteratively. Uncertainties in *VTR* will primarily arise from estimates in *TTR* that will lead to surpluses or deficits at the end of a tax year when governmen<sup>t</sup> spending deviates from budgeted allocations. These calculations are not technically difficult. The tax agency will have all income data from previous years, and *TTI*, based on the latest tax records, can be calculated with all details from the tax code. However, the aim of this paper is to analyze the general characteristics of the victory tax system, rather than focus on nuanced details.

For clarity and without loss of generality, the characteristics of the victory tax system are analyzed using an average household size and with the subscript *d* suppressed. This allows *TTI* to be expressed through the simple function *<sup>N</sup>*(*x*), which gives the income distribution over households of the average size. Defining *No* to be the number of such households, and *p*(*x*) the probability density function quantifying how income is distributed over these households, *<sup>N</sup>*(*x*) = *No p*(*x*). Assuming all households will take the maximum itemized deduction based on (Equation (10)), a lower-bound estimate for *TTI* is obtained. With *fd*(*x*|*IDmax*, *VTR*) → *f*(*x*|*VTR*) from the simplifying assumptions, the total taxable income of the entire population is given by:

$$TTI = N\_o \int\_0^\infty f(\mathbf{x}|VTR) \ p(\mathbf{x}) \, d\mathbf{x} \,\,\,. \tag{11}$$

#### *2.3. Test Economies*

The victory tax system will be characterized by a series of test economies. A test economy is modeled by the income distribution of the population and poverty line. In this paper, the characteristics of the US economy are used as a starting template and for comparison in discussing the importance of the results. However, the main interest is on investigating general trends that show how the victory tax system responds to dramatic changes in income distribution. Therefore, several test economies are considered that systematically deviate from the US economy, where the size of the middle class gradually shifts from the largest segmen<sup>t</sup> of society to the smallest. In particular, the standard deviation in income distribution is used to expand and shrink the middle class, which respectively decreases and increases the income gap between the low middle class and the wealthiest households. Quantitative comparisons are made between a series of test economies in which the variance in household income is systematically varied as the average household income is fixed.

#### 2.3.1. Income Distribution

Accurate modeling of income distribution over a population has received much attention [3–5]. The income distribution of a population is represented as a probability density function (PDF) denoted as *p*(*x*), where *x* is net income. Income distributions are modeled by the *κ*-generalized statistics [3–5] and log-normal statistics. Although the log-normal PDF is a qualitatively adequate model for free markets, it underestimates population density with very high or very low incomes. The *κ*-generalized PDF provides a more accurate model description. In particular, the empirically observed Pareto power law tail [25] is recovered for high incomes (ultra-rich) and also more statistical weight is given to the extreme poor (both effects take away statistical weight from the middle class). The *κ*-generalized PDF is defined as:

$$p(x) = \frac{a\beta\left(\frac{x}{\mu}\right)^{a-1} \exp\_{\mathbf{x}}\left[-\beta\left(\frac{x}{\mu}\right)^{a}\right]}{\sqrt{1 + \kappa^{2}\beta^{2}\left(\frac{x}{\mu}\right)^{2a}}}\tag{12}$$

$$\beta = \frac{1}{2\kappa} \left[ \frac{\Gamma\left(\frac{1}{\alpha}\right) \Gamma\left(\frac{1}{2\overline{\kappa}} - \frac{1}{2\overline{\alpha}}\right)}{\kappa + \alpha \Gamma\left(\frac{1}{2\overline{\kappa}} + \frac{1}{2\overline{\alpha}}\right)} \right]^d \tag{13}$$

$$
\Gamma(y) = \int\_0^\infty t^{y-1} \, e^{-t} \, dt \tag{14}
$$

$$\exp\_{\kappa}(y) = \left(\sqrt{1 + \kappa^2 y^2} + \kappa y\right)^{\frac{1}{\kappa}}\tag{15}$$

The two parameters {*<sup>α</sup>*,*<sup>κ</sup>*} are adjusted to fit to empirical data. Although it is not prohibitively difficult to evaluate the *κ*-generalized PDF and other properties from *κ*-generalized statistics [5], the form of this distribution is not convenient to create a systematic series of test economies. Log-normal statistics are therefore used to quantitatively analyze systematic trends in a number of economies that range from a strong to a weak middle class.

The simpler log-normal PDF is defined as:

$$p(\mathbf{x}) = \frac{1}{\mathbf{x}\gamma\sqrt{2\pi}} \exp\left[\frac{-(\ln x - \lambda)^2}{2\gamma^2}\right] \tag{16}$$

$$\gamma = \sqrt{\ln \left[ 1 + \left( \frac{\sigma}{\mu} \right)^2 \right]} \tag{17}$$

$$
\lambda = \ln(\mu) - \frac{\sigma^2}{2} \tag{18}
$$

Again, only two parameters {*λ*,*γ*} characterize the PDF. However, they easily relate to the mean, *μ*, and standard deviation, *σ*, of income of the population. The mean household income is defined by the total net income from all households, divided by the total number of households. Both log-normal and *κ*-generalized distributions use the same mean

household income, but they require different standard deviations to best fit the empirical adjusted gross income data for the US, as explained below.

#### 2.3.2. Test Economy Parameterization

The IRS data [26] from 1979 to 2009 was invoked to mimic the income distribution of the US. The IRS method of statistical weighting normalized the data to produce an effective number of dependents per household, and the reported income was based on 2009 dollars after adjusting for inflation. Year 2003 is chosen as an illustration. The US economy had just recovered after a market correction and began to grow over the next four years. The snapshot of the 2003 economy (in 2009 dollars) reflects a time of stability and the beginning of economic growth. The reported IRS data deals with adjusted gross income (AGI), which is analogous to net income (NI). At the level of analysis presented in this work, IRS data [26] (and NIH data [23]) are invoked to obtain realistic parameters while providing context for discussions. However, because AGI and NI are not the same, the test economies constructed in this work are best viewed as hypothetical examples.

The best fit for the IRS 2003 AGI data using the *κ*-generalized distribution yields *α* = 1.50 and *κ* = 0.56 with a relative fit error of ±5% across all income brackets. This result gives a market income distribution (synonymous with AGI) compared to total income, which includes governmen<sup>t</sup> transfer and other forms of US subsidies, such as food stamps, etc. Henceforth, this market income distribution will be associated with the economy A, shown in Figure 1. At the far right end of the tail, 10 households with an average income of approximately \$113,000,000 per year are captured. The mean household income over the entire population is \$75,300 with a standard deviation of \$171,712. The median income is \$44,612 indicating that ultra high-income households skew the distribution, causing the mean to be considerably larger than the median. As Figure 1 shows, the most probable market income (the mode) is approximately \$10,000 per year.

The log-normal distribution, which fits well to the 2003 IRS income data, has the same mean income and a smaller standard deviation of \$114,475. This log-normal distribution is employed as another test economy, henceforth referred to as economy B. A smaller standard deviation in market income indicates a larger middle class, since more households picked at random are likely to be closer to the mean income. Differences in market income between economy A and B are shown in Figure 2 on a log–log plot. The *κ*-generalized and lognormal distributions describe similar market incomes between \$4000 and \$2,000,000, but the peak in the log-normal distribution shifts upward near \$12,000 to compensate for depleting probability from extreme income ranges. The Lorenz curves for these two test economies, shown in Figure S1, appear virtually identical.

A series of six log-normal economies, all with the same mean income of *μ* = \$75,000 is also considered, with standard deviations spaced by approximately powers of 2, such that *σ* = \$7000, \$14,000, \$28,600, \$57,240, \$114,475 and \$229,000 where the second largest standard deviation is economy B. In Figure S2, the six distributions are compared on a log–log plot, and their Lorenz curves are compared in Figure S3. All six log-normal economies have equal total adjustable income generated by the population, enabling a systematic method to study the dependence of a tax system on the strength of the middle class.

**Figure 1.** Market income distribution for economy A: The household AGI probability density ranging from \$0 to \$200,000 is shown. Population quintiles are marked with light brown dashed vertical lines at 20%, 40% 60%, and 80%. Median and mean incomes are respectively marked as blue and red dashed vertical lines. (**inset**) The same distribution is shown without cutting out data from high income households. The horizontal line at the bottom of the graph highlights the heavy tailed distribution, indicating only a tiny number of households reach this level of income.

**Figure 2.** Market income distribution comparison: On a log–log scale the *κ*-distribution defining economy A and log-normal distribution defining economy B are compared. The *κ*-distribution puts more statistical weight for the ultra-rich and extreme-poor subpopulations as reflected in the wings of the distribution. The empirical median and mean incomes for the US 2003 economy are shown as blue and red vertical dashed lines.

#### 2.3.3. Poverty Line, Tax Revenue and Government Transfer

Other information characterizes an economy, beyond income distribution, such as the number of people in the population and the poverty line. For the US economy in 2003, the number of tax-paying households was approximately \$112,100,000 and the population among those households was approximately 290 million, yielding an effective household size of approximately 2.6 people. For illustration, and without altering the conclusions of the analysis, the poverty line for a mean household size of 2.6 is estimated by interpolating data from the 2009 Department of Health and Human Services data [23], yielding \$16,814.

Unlike the victory tax system, which is the only source of governmen<sup>t</sup> transfer to a household, there are many state and federal aid programs for the poor in the US to offset housing, energy, food, education, health care, and so on. Once other forms of governmen<sup>t</sup> assistance are eliminated, a better estimate for the poverty line is \$22,306. Examples are juxtaposed with both estimates to quantify how much tax burden increases when the poverty line is raised, which takes into account removal of public services for low-income households. In the victory tax system, there can be public services, but without income qualifications. As another point of reference, the poverty line in 2021 is listed as \$21,960 for a household of 3 [23].

The target net tax revenue is set at \$854,182,445,000 for all test economies, which corresponds to tax revenue collected by the IRS in 2003 after all governmen<sup>t</sup> transfers were distributed. To compare all tax systems and economies, the total governmen<sup>t</sup> transfer is calculated and added to the target net tax revenue of this hypothetical budget. When the income distribution has (more, less) households below the poverty line, the total tax revenue, TTR, to be collected will (increase, decrease). It is worth noting that approximately 35 percent of US federal tax revenue was derived from payroll tax, and 19 percent came from other sources [27]. Payroll tax and all other forms of individual taxation at the federal level outside of individual income tax are eliminated in the victory tax system. Although corporate tax can coexist with a victory tax system, no corporate tax is considered for simplicity of analysis in this paper. Here, all tax revenues come from individual income tax, making the VTR of the test economies an upper bound.

For comparison, it is insightful to examine how US tax revenues were redistributed in 2003. Total tax revenues collected were \$1,952,929,045,000, of which \$1,098,746,600,000 was then redistributed to households through governmen<sup>t</sup> transfers. As such, more than 56% of the tax revenue collected was redistributed to tax payers, as captured in Figure S4. Despite the enormous amount of tax revenue redistributed to households, unfortunately approximately 15% of households live in poverty in the US [23]. In the US, tax revenues are redistributed to households at all income levels, including high-income brackets, creating both complexity and inefficiency.
