**3. Results**

The adaptive nature of the victory tax requires *VTR* to be calculated each year. The procedure to calculate *VTR* from (10) favors tax revenue surpluses by assuming that all households take the maximum itemized deduction allowed. This situation yields the maximum *VTR*. Note that *VTR* must increase as itemized deductions increase to generate the same target revenue. The source of surplus is households that do not utilize all their itemized deductions. For analysis purposes, comparisons are made for a household that takes all or no itemized deductions to establish bounds for relevant quantities, such as *ETR* and *ATI*. Quantities are usually expressed as a function of the percentile of households. Percentile of households is calculated by ranking all households by net income, and then counting numbers of households at or below a certain income level to obtain a normalized scale from 0% to 100%.

#### *3.1. Parameter Exploration*

The shape of *ETR* as a function of the percentile of households depends on parameters {*BD*,*MD*,*MP*}, which are first explored as independent variables to elucidate how they affect the tax structure. Note that *MP* defines a percentage of total income, while *BD* and *MD* are in dollar amounts. However, when convenient, *BD* and *MD* will be specified in terms of the percentile of households. When stated *BD* = 20% and *MD* = 50%, this means *BD* and *MD* are respectively set to the income level at the 20 and 50 percentile of households. For example, for economy A, these percentiles translate to *BD* = \$16,331 and *MD* = \$44,612, which are the dollar amounts used in Figure 3.

**Figure 3.** Effective tax rate comparisons: For economy A the minimum *ETR* (blue) and maximum *ETR* (red) are shown for *BD* = 20% and *MD* = 50% in each of the panels with different *MP* given by: (**A**) 0% (red line covering blue line); (**B**) 30%; (**C**) 60%; (**D**) 90%.

Illustrated in Figure 3, as *MP* increases from 0 to 100% more deductions are allowed, leading to a gradual change in shape from a flat horizontal line to a "U"-shape. In Figure 3A, a flat tax appears when *MP* = 0, causing *ETR* = *VTR*. In general, *VTR* is the y-intercept of the *ETR* plots, and *ETR* < *VTR* for households with *N I* > 0. For *MP* values of 0%, 30%, 60%, 90%, the respective VTR are 11.97%, 15.49%, 18.35% and 20.37%. The maximum deduction implies *ETR* → *VTR* from below for high-income households. Note that *ETR* is regressive for low-income households until governmen<sup>t</sup> transfer is discontinued. This point occurs at an income level equal to the basic deduction, which creates a kink in the *ETR*. Thereafter, a flat *ETR* applies to all households until the maximum income a household can deduct is equal to the basic deduction. At this point, a bifurcation can occur where households can use itemized deductions. The red and blue lines correspond to taking only the basic deduction versus the maximum deduction. The last segmen<sup>t</sup> of percentile of households has a progressive *ETR*. As *MP* increases, more deductions are possible, causing *VTR* to increase and the range for a flat *ETR* to decrease. Only at *MP* = 1 will *ETR* = 0 at the bottom of the dip. The same analysis for economy B results in the same qualitative behavior as shown in Figure S5.

As shown in Figure 4, when *MP* = 0 the flat segmen<sup>t</sup> of *ETR* goes to 0%, starting at *BD* and ending at *MD* for households that take the maximum deduction. The red line tracks the maximum *ETR* for households without itemized deductions. As the basic deduction rises from 10% to 40% in steps of 10%, the *VTR* is 18.72%, 20.90%, 24.47% and 29.44%, respectively. The same analysis for economy B results in the same qualitative behavior as shown in Figure S6.

In Figure 5, *VTR* and average *ETR* are plotted for five different *BD* values, as a function of *MP* and *MD*. Panel A shows that *VTR* increases as more tax deductions are allowed. Panel C, on the other hand, shows that as more deductions become available, the average *ETR* decreases, where the average *ETR* reaches a minimum when *MP* = 100%. The flat plateaus observed in panels B and D, which extend longer for greater *BD*, appear because the maximum deduction cannot be less than the basic deduction. Generally, a low average *ETR* is the result of the greatest tax relief for the middle class. This result suggests that *MP* should be set to 100 percent for economies with large net income variations. The same qualitative behavior is shown in Figure S7 for economy B.

**Figure 4.** Effective tax rate comparisons: For economy A, the lower- and upper-bound *ETR*s are shown for *MD* = 50% and *MP* = 100% in each of the panels with different *BD* given by: (**A**) 10%; (**B**) 20%; (**C**) 30%; (**D**) 40%. At large *BD*, the signature "V" shape appears for the maximum *ETR*.

**Figure 5.** Tax rate comparisons: Trends in tax rates for economy A are explored. The legend applies to all panels, where different color lines represent a *BD* of 0%; 10%; 20%; 30%; 40%. The victory tax rate is shown as a function of (**A**) *MP*; (**B**) *MD*%. The average *ETR* is shown as a function of: (**C**) *MP*; (**D**) *MD*%. In panels A and C, *MD* = 50%, and in panels B and D *MP* = 100%.

In Figure 6, panels A and B, respectively, plot the minimum and maximum *ATI* as a function of percentile of households for different values of *BD*. Figure 6C compares the minimum and maximum *ATI* to the *ATI* from a flat tax. Since more tax revenue is needed for governmen<sup>t</sup> transfer when *BD* increases, *VTR* increases too. The advantage of increasing *BD* is that lower-income households receive considerably more ATI as *BD* increases. However, the gains in *ATI* decrease as household income increases until a point where *ATI* decreases for high-income households. As shown in Figure 6D, households that take the maximum deduction beyond the 80 percentile have less *ATI* compared to a flat tax. Clearly, tax deductions are paid for by the progressive nature of the victory tax on high-income households (especially ultra-high-income households). Figure S8 shows the same qualitative behavior in economy B.

**Figure 6.** After-tax income comparisons: Trends in *ATI* are explored for economy A. With *MD* = 50%; *MP* = 100%, and *BD* ranging from 10% to 40% the *ATI* as a function of household percentile is shown for the case: (**A**) minimum *ATI*; (**B**) maximum *ATI*. (**C**) Comparing a flax tax to the victory tax (V-tax), the minimum *ATI* (max V-tax), maximum *ATI* (min V-tax) and flat tax *ATI* are shown with *BD* = 30%, *MD* = 50%, *MP* = 100%. (**D**) For the same parameters used in panel C, the difference in min/max V-tax *ATI* relative to the *ATI* for a flax tax is shown.

Since *BD* = *PL*/(1 − *VTR*), and *PL* can be quantitatively measured, *MD* is the only parameter left to determine. A balanced approach must compromise the desire for a generous maximum itemized deduction compared to the desire for a low *VTR*. As living costs decrease, *PL* will decrease, suggesting that the maximum deduction should not be large, as purchasing power is strong. Conversely, the maximum deduction should increase with an increase in living costs to ensure that itemized deductions have a positive impact on households. This leads to a prototypical victory tax system in which the maximum deduction is proportional to the basic deduction, where *MD* = *kBD*. The choice of an effective value of *k* is examined in Figure 7. The minimum ETR is shown in Figure 7A,C when the poverty line is at \$16,814 and \$22,306 while considering three values for *k*. Summarizing many of these calculations, Figure 7B,D show the average *ETR* and *VTR* as a function of *k*. Taken together, *k* = 2 is a good compromise in tax structure. Moreover, for *k* < 2.5 the results are insensitive to *PL*. The same qualitative behavior is observed in Figure S9 for economy B.

**Figure 7.** Maximum deduction exploration: For economy A, the *ETR* as a function of percentile of households is shown in panels (**A**) and (**C**) for a poverty line of \$16,814 and \$22,306, respectively. The maximum deduction is set to be proportional to the poverty line, where different color lines show different proportionality constants set at 1.2, 2.6 and 4. As a function of *k* and for two different poverty levels, panel (**B**) shows the average *ETR* and panel (**D**) shows the victory tax rate.

The question of how the prototypical victory tax system responds to extreme variation in the economy is addressed by considering a series of six log-normal test economies, characterized by a Gini index ranging from 0.05 to 0.72 (see Figures S1–S3). Figure 8A,C show the minimum *ETR* for these test economies, with poverty lines of \$16,418 and \$22,306, respectively. As the middle class expands, the *ETR* flattens, resulting in the lowest possible *VTR*. This flattening occurs because *MP* = min(*<sup>r</sup>*, 1). For the 6 test economies from highest to lowest Gini index, the coefficient of variations (i.e., *r* = *σ*/*μ*) are, respectively, 304%, 152%, 76%, 38%, 19% and 9%. As income dispersion increases, *VTR* increases, making *ETR* very low for middle-class households. Next, Figure 8B plots *VTR* and the average *ETR* for the test economies as a function of Gini index. In Figure 8D the ratio defined by the total governmen<sup>t</sup> transfer for eradicating poverty to the total tax revenue collected is plotted against the Gini index. For a Gini index of 0.56 (modeling the 2003 US economy), the total governmen<sup>t</sup> transfer amounts to 23.50% or 42.30% of the total tax revenue collected when the poverty line is \$16,418 or \$22,306, respectively (recall 56% was used in 2003 from IRS data).

Based on the above exploration of parameters {*BD*,*MD*,*MP*}, henceforth the prototypical victory tax system will have: *BD* = *PL*/(1 − *VTR*), *MD* = 2*PL* and *MP* = min(*σ*/*μ*, 1). The objective measures of the economy dynamically alter the victory tax structure, where it becomes flatter as the middle class becomes stronger. When the middle class shrinks as dispersion of net income increases, the victory tax increases *VTR* and lowers *ETR* for the middle class as it morphs into the V-signature. The steepness of V increases as the dispersion in net income increases. A steep regressive tax at low incomes provides the poor with the means to move upwards into the middle class. The progressive segmen<sup>t</sup> of the victory tax on high-income households provides the additional tax revenue necessary to form the V. These results show that the victory tax system is a type of governor to maintain a strong middle class.

**Figure 8.** Systematic variation in net income dispersion: Trends in *ETR* with respect to income dispersion are shown in panels (**A**) and (**C**) for six economies described by a log-normal distribution with standard deviations ranging from \$7000 to \$229,000 about a mean income of \$75,300. The panel (**C**) legend also applies to panel (**A**). The *ETR* for the \$7000 standard deviation is not shown because if plotted, it is flat and hidden under the magenta line. As a function of Gini index, panel (**B**) plots the victory tax rate (vtr1 or vtr2) and average *ETR* (etr1 or etr2) for cases 1 and 2 corresponding to a poverty line of \$16,814 and \$22,306. For the same two cases, panel (**D**) plots total governmen<sup>t</sup> transfer divided by total tax collected (gtr1/rev or gtr2/rev).

#### *3.2. Flat, Linear Progressive and Victory Tax Comparisons*

Here, the victory tax is compared with a flat and linear progressive tax under identical conditions (e.g., the same income distribution, total tax revenue and poverty line). The flat tax is a victory tax with *MP* = 0. A linear progressive tax system is defined when *ETR* is a linear function of the percentile of households with *ETR* = 0 for a household with no self-generated income. The progressive tax rate, *PTR*, sets a maximum tax rate adjusted to collect the desired amount of tax revenue. For example, households at 20 and 50 percentiles will have *ETR* = 0.2*PTR* and *ETR* = 0.5*PTR*, respectively. Note that the linear progressive tax system does not satisfy the guiding principles 1 and 4. The *ETR* for each tax system is shown in Figures S10 and S11 for poverty lines \$16,814 and \$22,306, respectively. For a poverty line of \$22,306, the flat tax rate, FTR, is 14.80%, while PTR is 18.60% and *VTR* is 27.74%, whereas the population average *ETR* is 14.80%, 9.30% and 11.76%, respectively.

Comparisons of *ATI* in Figure 9A show that each tax system ensures the lowest possible *ATI* is at *PL*, but this occurs at different percentiles and with different trends. The linear progressive tax creates a welfare trap. That is, starting with no employment and full dependency on governmen<sup>t</sup> transfer, the initial *ATI* is above *PL*, and then *ATI* decreases as employment increases until *AT I* = *PL*, and thereafter *ATI* starts increasing. Clearly, 30% of the lowest income households are better off not working than to work for any amount of time in a low-wage job. Although the flat tax does not penalize part-time employment and/or working low-wage jobs, it does not offer any advantage for individuals to work in a low-wage job. Quite clearly, the victory tax provides an incentive for low-income households to seek employment, where they will gain significant wealth as they move into the middle class. Consider, for example, two households living on the poverty line, the first fully dependent on governmen<sup>t</sup> transfers and the second fully self-supporting. The second household enjoys \$8565 more *ATI* (a 38% increase) due to the basic tax deduction.

**Figure 9.** Tax system comparisons for economy A with a \$22,306 poverty line: (**A**) After-tax income for flat, linear progressive and victory tax systems are shown in different colors; (**B**) The Lorenz curves for tax revenue are shown. The dashed vertical lines of corresponding color to the tax system indicate the percentile of households from which point onward the collected tax revenue is sufficient to pay for all governmen<sup>t</sup> transfer needed to eliminate poverty.

For each tax system, a Lorenz curve is shown in Figure 9B for tax revenues. The Gini index for these Lorenz curves quantifies how uniform the tax burden is across income levels. A proportionate tax burden will show a similar Gini index for tax revenue as that for income received. For economy A, the Gini index for income received is 56% (see Figure S1). A flat tax without taxing governmen<sup>t</sup> transfers yields an identical Gini index of 56% for tax revenues. However, with governmen<sup>t</sup> transfers taxed, the flat tax gives a Gini index of 49% on tax revenues, with a higher tax burden on low-income households. The linear progressive tax system has a Gini index of 68% for tax revenues, which puts the greatest tax burden on high-income households. The victory tax reduces the tax burden on high-income households with a Gini index of 62% on tax revenues. Thus, a victory tax maintains a reasonably balanced tax burden over all income levels, where the flat tax is a special limit of the victory tax. The Lorenz curve for the victory tax is generally not monotonic, showing that low and high-income households inherit the greatest tax burden, which is why a low *ETR* is possible for the middle class.

For a (flat, progressive, victory) tax system, revenue from (7.7%, 3.3%, 5.2%) of households with the highest income corresponds to (35.2%, 30.7%, 42.3%) of the total tax revenue redistributed as governmen<sup>t</sup> transfer. Clearly the victory tax reinvests the most back into society, which is responsible for removing the welfare trap [18] through its regressive *ETR* for low-income households. In general, as governmen<sup>t</sup> transfers decrease due to an expanded middle class, the degree of disproportionate tax burden decreases. Figure S12 shows the same qualitative behaviors with a lower poverty line.

Now the question of how these three tax systems respond to the poverty line is addressed. Tax rates as a function of PL for the flat, linear progressive and victory tax systems are shown in Figure 10A–C, respectively. This analysis is applied to the series of six log-normal economies described in Section 2.3.2 to elucidate the effect of income dispersion for a fixed mean income. Here, the characteristic tax rate of a tax system is tracked as the poverty line is varied for different income dispersion scenarios. The lowest dispersion of \$7000 represents the situation in which the vast majority of households fall into the middle class, with only rare cases of poor or rich households. As income dispersion increases, the middle class shrinks, and the percentage of extreme poor significantly increases. In contrast, only a small increase in ultra-rich households occurs. At high income dispersion, wealthy households pay much more tax revenue, because most of the taxable income generated across the population comes from wealthy households. Moreover, as income dispersion increases, more households fall below the poverty line, which increases tax revenues that must be collected to cover the increase in governmen<sup>t</sup> transfer. As a reference point, the final value of the considered standard deviation (\$229,000) is approximately twice the dispersion found in the 2003 US economy. Although income dispersion in the US economy has steadily increased since 2003, it is still lower than the largest dispersion considered here.

In the victory tax system (including the special case of flat tax), it is seen that *VTR* increases as *PL* increases. In particular, as the poverty line is increased, *VTR* must increase more rapidly, as population dispersion is larger. Consequently, a large middle class keeps the *VTR* low, and the progressive part of the victory tax on high-income households will be most shallow.

**Figure 10.** Tax rate dependence on poverty line: A series of six log-normal economies with income dispersion characterized by standard deviation ranging from \$7000 to \$229,000 about a mean income of \$75,300. ( **A**) flat; (**B**) linear progressive; ( **C**) victory tax. ( **D**) The tax rate for a flat, linear progressive and victory tax are shown together for economy A, which is used to mimic the 2003 US economy. The average effective tax rate over the population for the victory tax (denoted as vt-etr) is also shown as a function of poverty line.

Figure 10B shows that a linear progressive tax system requires an increase in PTR as the middle class expands. Interestingly, PTR is generally insensitive to the poverty line. Only when there is a wide income gap will the dependence on the poverty line become substantial. As the middle class shrinks, the tax rate rises quickly, but unfortunately there is no mechanism for the middle class to reexpand. Contrary to the linear progressive tax, as shown in Figure 10A,C, the flat and victory tax have the intuitive rank ordering that *VTR* decreases as the strength of the middle class increases.

For economy A shown in Figure 10D, *VTR* rapidly increases as a function of *PL*, while the average *ETR* remains markedly insensitive to *PL*. A low average *ETR* can be obtained even when *VTR* is high, because most of the population substantially benefits from itemized tax deductions, except for households for which the maximum itemized deduction is dwarfed relative to their net income. As the middle class shrinks, the tax

base also shrinks, and the *VTR* increases. This creates an elastic response to income gaps, which means the tax structure makes it easier for the middle class to expand when market pressures act to shrink it. Conversely, when the middle class expands, the V flattens, encouraging accumulation of wealth in households above the middle class level. In summary, the victory tax has desirable elasticity [24] to create stability in the economy, where the sharpness of the V-shape increases as income dispersion becomes extreme, and flattens as the middle class expands, fueling a consumer-based economy.

#### *3.3. Distribution of Government Assistance*

Government transfers and tax deductions are two forms of governmen<sup>t</sup> assistance. However, the deductions on capital losses were not reflected in the above analyses, as net income distributions account for capital gains and losses. To quantify the degree of governmen<sup>t</sup> assistance in terms of optional deductions, estimates of the average amount of itemized and capital loss deductions must first be modeled. Simple models are presented to quantify the total itemized and capital deductions in the population as a function of the percentile of households. These models are not part of the victory tax system. However, they are useful in characterizing how governmen<sup>t</sup> assistance is distributed across households. The qualitative estimates made in this subsection are only intended for illustration and discussion purposes.

The net itemized deduction, *NID*, is modeled as: *NID* = *f* 2 2 max(0, *N I* − *BD*) where *f* is the percentile of households. For economy A with a poverty line of \$22,306, the basic deduction is \$30,871 and the maximum deduction is \$44,612. The net itemized deduction for a household at the (20, 40, 60, 80) percentile, with corresponding NI of (\$16,331, \$33,878, \$57,787, \$101,844), produce a mean itemized deduction of (\$0, \$240, \$4844, \$22,711). The factor of *f* 2 models the qualitative trend that the more income a household has, the more likely it will utilize itemized deductions up to *MD*.

The total capital loss, *TCL*, is modeled as: *TCL* = *NIf* <sup>4</sup>*CL*/(1 − *CL*). The factor of *NIf* 4 models household capital gains, where it rapidly tends to zero as *f* → 0, and tends to *N I* as *f* → 1 to reflect the empirical observation that high-income households have larger portions of their income from investments. The parameter *CL* is a ratio of capital losses to capital gains for the tax year. Since *CL* reflects an average over the population, the range from 0.1 to 0.7 suffices to quantify the fraction of governmen<sup>t</sup> assistance applied to capital loss deductions. Illustrating this qualitative model: For *CL* = 20% and for households at the (20, 40, 60, 80) percentiles, *TCL* is modeled as (\$7, \$216, \$1877, \$10,429). For example, a household with gross income of \$112,273 and capital loss of \$10,429 has a net income of \$101,844.

Employing the models for NID and *TCL*, governmen<sup>t</sup> assistance, GA, is given as:

$$GA = GTR \times (1 - VTR) + VTR \times (TD + TCL) \tag{19}$$

where *TD* is total allowed deduction from basic and itemized deductions. In addition to governmen<sup>t</sup> transfer, reducing the tax burden on households through tax deductions contributes to governmen<sup>t</sup> assistance. The average tax deductions made by households at the (20, 40, 60, 80) percentiles are estimated to be (\$16,338, \$31,329, \$37,592, \$55,042). The average of (*TD* + *TCL*) over the population gives the average tax deduction, which can exceed *MD* because *TCL* is included. Due to *TCL*, a plot of governmen<sup>t</sup> assistance versus percentile is not informative, because governmen<sup>t</sup> assistance to the top 0.1% of households dwarfs all other forms of assistance on an absolute scale. For example, with *CL* = 20%, the average tax deduction at the 99.9 percentile is \$436,583. However, relative comparisons in governmen<sup>t</sup> assistance can be made. Here, *GA* is divided by *GTR* + *N I* to define a fraction of assistance that shows how governmen<sup>t</sup> assistance is distributed over the percentile of households. Note that the assistance fraction for the extreme poor is not 100% for the victory tax. For a household with *N I* = 0, which receives all its income from governmen<sup>t</sup> transfers, the assistance fraction is equal to *GA*/*GTR* = 1 − *VTR*. For economy A, and *PL* of \$22,306, recall *VTR* is 27.74%, meaning 0.7226 is the fraction of governmen<sup>t</sup> assistance when *N I* = 0. The relative governmen<sup>t</sup> assistance decreases if only basic and itemized deductions are considered.

The percentage of governmen<sup>t</sup> aid as a function of percentile of households for different *CL* ranging from 10% to 70% is shown in Figure 11A,B. The deviation of CL considered causes "fanning" in the tail of the assistance fraction. Qualitative characteristics and general trends are not sensitive to the models used for tax deductions. High-income households with high profit-to-loss ratios have the smallest assistance fractions, but they receive much more absolute assistance than poor households. Middle-class households have too much income to benefit from governmen<sup>t</sup> transfers, and too little income to ge<sup>t</sup> much government assistance from tax deductions. However, in the victory tax system, middle-class households have the lowest *ETR*. As the poverty line increases, the assistance fraction increases for households at all income levels, not just low-income households. Together, these trends allow the victory tax system to be relatively balanced, without creating serious disproportionate tax burdens. In short, the extreme poor and ultra-wealthy pay the highest tax rates, but also receive the greatest governmen<sup>t</sup> assistance through various mechanisms.

**Figure 11.** Government Assistance Distribution: Analysis of tax deductions for economy A: ( **A**,**C**) consider a \$16,814 poverty line and (**B**,**D**) consider a \$22,306 poverty line. Panels ( **A**,**B**) show the assistance fraction as a function of percentile of households. Panels ( **C**,**D**) show the Lorenz curves for governmen<sup>t</sup> assistance from ( **A**,**B**), respectively. The diagonal line colored light grey in panels ( **C**,**D**) is drawn to guide the eye.

Figure 11C,D show Lorenz curves for governmen<sup>t</sup> assistance. The associated Gini index of these Lorenz curves will be 0 for proportionate governmen<sup>t</sup> assistance at all income levels. It is noticed that the Gini index depends on the average ratio of population loss to profit. For (10%, 30%, 50%, 70%), the Gini index for governmen<sup>t</sup> assistance is (0.00, 0.17, 0.34, 0.53). The relatively low Gini index values indicate an adequate balance in the way governmen<sup>t</sup> assistance is distributed across income levels. Since the Lorenz curve is mostly below the diagonal line (when *CL* > 10%), ultra-high-income households receive the most governmen<sup>t</sup> assistance on an absolute scale. This results sugges<sup>t</sup> the ultra-rich will significantly increase governmen<sup>t</sup> dependency in economic downturns, where massive losses occur (i.e., CL large). This is another sign of elasticity to maintain social-economic stability, because governmen<sup>t</sup> aid to the wealthy is strongest at the time when the risk of investing is highest.
