Tax Reform V.3
I have previously published my Tax Reform, Version 1 and Version 2, which both rely on the same basic mathematical functions. In this article I would like to go through the derivation of these functions in detail, stepbystep, with graphical pictures to show the methodology and the effects. At the same time, I will make some additional revisions to the second version. My intent is to show that we can generate a smooth tax function which depends on a few basic constants, and which does not require any intervention from yeartoyear, but instead automatically resets itself to correct for inflation and works to eliminate deficits, and specifically avoids violating the Flat Tax commitment.
First, we will start with a picture showing what raw gross income, GI, looks like in graph form with no exemptions, exceptions or loopholes:
Graph 1
Please observe that I have labeled both axes, horizontal and vertical, as just “Income”. Since income increases uniformly in both directions, we have a straight line with a slope of 1. I have not defined any units of income, just arbitrary numbers from 1 to 4 at this point.
Next let’s introduce the concept of “Median Income”, that point along the Gross Income line such that an equal number of people (“taxpayers” or “households” if we use the IRS terminology) falls on either side of this Median Income (MI) point. Now we can give more meaning to the scales I have used above: an MI of 1 on this graph is the annual Median Income for any given year – “this year” in my Tax Reform text. All of the plots in this version are in Monetary Units which are multiples of Median Income, not dollars. I will use the term MU for these units and will abbreviate that to the closest symbol I can find as a replacement for the upper case Greek letter mu, M. Thus, M = GI / MI.
My objective is to develop a simple way to convert the income line, in a series of steps, into a reasonable tax structure which closely matches the taxes paid by most people with today’s tax code, but using a single function that can be applied uniformly to everyone, to replace our current 70,000+ pages of specialinterestridden Tax Code.
To protect those with low incomes we must have some sort of “Living Cost Exemption”. For this I suggest a straight line starting at 1 (100%) at 0 income and declining to 0 at an income equivalent to M = 4 (blue dotted line). Then by multiplying GI by this exemption line, we can get another curve which is the M value of this Living Cost Exemption (the green dashed line):
Graph 2
As before, GI is the straight black line, the blue dotted line represents the Living Cost Exemption percentage and the green dashed line is the Exemption Value. Subtracting this Exemption value from Gross Income gives us the purple line, our Taxable Income. This sequence gives us a basic structure for computing Personal Income Tax which works quite well for about 85% of all taxpayers, up to 2*MI, or M = 2. For those below an M of about 0.4 some additional refinements will be included for Poverty protection in the form of a Tax Credit Fund, and above an M of 3.0 we will shift to a different curve to blend our Taxable Income function back onto the GI line to create a true Flat Tax over all higher incomes.
From Census data published by the government we know that our current population is now about 320 million people. Just under half of these are “taxpayers”, so at least 155 million people (normally) have jobs and report their incomes on tax returns each year. Thus, in the above graph, half of all this working population falls into the small square between zero and one in the bottom left quadrant; the other half of all income earners falls between 1 and some large number, over 100! Since the Median Income over the past four years has been close to $50,000, a M of 100 would be equivalent to about $5 Million. With 50% of all these people in that bottom square, it should be obvious that the distribution of the “rich”, above M = 4, must get very sparse very quickly. However, we also know that at least 1% of all these people do have incomes above a M of 10 (about $500,000), and that approximately a tenth of a percent have incomes above an M of 20 ($1,000,000) – annually! I will use these numbers to justify some thoughts about tax distribution, poverty and welfare (see comments below Graph 4, below).
In the first version of my Tax Reform, I shifted to a straight line above M = 2, extending the Exemption Value at a constant value of MI for all incomes greater than M = 2. This changed the Taxable Income curve to a straight line parallel to GI for all higher incomes. This appears as the purple line in graphs 4 and 5 below. The effect of this model is to give low income taxpayers a cost of living exemption that starts at $0 (100% of zero income) and increases rapidly as income increases, then more slowly as income approaches M = 2, then remains constant at the dollar equivalent of MI over all higher incomes. This poses two problems: 1) this model does not provide a mechanism for paying the cost of a Tax Credit for those with incomes below a Poverty Threshold, and 2) those with very much higher incomes do not need any costofliving exemption; including such an exemption at increasingly higher incomes results effectively in a regressive tax.
To tackle the first problem, I estimated what fraction of MI would yield a minimal living standard for single taxpayers and for taxpayers with one, two, three and four dependents. In keeping with my observation in Tax Reform V.1 that family size is the result of choices that people make – and should not be a burden to other taxpayers – I put an upper limit cap of Poverty protection at four dependents. I then created five Tax Credit curves to provide this protection (see Graph 3).
Finally, to cover the cost of this Tax Credit Fund, I now continue the initial Taxable Income curve (the upward curving parabola) out to an income equivalent to M = 3, and then shift to an inverted parabola which is tangent to the first at M = 3 and tangent to the GI line at M = 6 (see Graph 4). Thus TI is a continuous smooth “S” curve between M = 0 and 6, then equals GI over all higher incomes. Assuming a Flat Tax of 15%, the actual effective tax rate is less than FT at incomes below M = 6 and is a true Flat Tax from that level upward. The difference at high incomes between this new tax model and the V.1 model is highlighted in gray in Graph 5. I have highlighted the maximum Poverty threshold Tax Credit zone in pink at the bottom left corner of this same graph, and the actual source of Tax Credit funds along the bottom edge in green. As noted in my Tax Reform V.2, those who need to make use of their available Tax Credit are expected to repay the Fund when they are able to do so. However, no tax or interest penalty will be charged for nonrepayment. Not all poverty cases will be able to repay what they borrow – and some will choose not to do so – but the loss of credit worthiness that attaches to nonrepayment is expected to minimize this deficiency.
Graph 3
In graph 3 I have shown only the very lowest income range, out to M = 0.5, or the 2008 equivalent of about $25,000 Gross Income. The dark blue curve, starting at .2 MI (or $10,000), is below today’s minimum wage, but it applies only to single taxpayers with no dependents, so it should be adequate over the short term and is designed to provide the equivalent of minimum welfare or unemployment insurance. For married taxpayers with no children the next level of Threshold, light blue, is intended to provide adequate short term income for two people, and so on up to the dark green curve, at the top of the sequence, for four dependents. The orange curve along the lower edge of the graph represents the Taxable Income over this range, matching the purple curve in Graph 2, above. Each Threshold curve exactly parallels the Taxable Income curve for reasons I explain later and an alternative choice of intermediate thresholds will also be offered.
The zone below each of these Poverty Threshold curves but above the Gross Income line is the Tax Credit amount a taxpayer can access, depending on his/her personal Gross Income and number of dependents. Of course, only those who actually have income (or are recently unemployed and actively looking for work), and who are not claimed as a dependent on another person’s tax return, are eligible for this Credit amount. The reason I designed these Threshold curves to parallel TI is that my original Tax Reform (Version 1) specifically requires that everyone pay the Flat Tax on Taxable Income, and I also specified that every tax payer must put a portion of his/her gross income into a TaxDeferred Savings Account, or PSA (for healthcare costs, education costs and ultimately for retirement pension). In that first version I defined the minimum contribution to this PSA at 5% of GI, but limited to no more than TI (TI = 5% of GI at GI = .2 MI, so the PSA contribution will always be less than 5% below that income level).
After reconsidering these rules I have concluded that a better choice would be to require that both the employer and employee contribution should be 10% (still with the TI upper limit on the employee portion) so that a large enough fund will accumulate in the PSA during an employee’s working life to cover any normal medical and educational costs, and still leave enough after retirement to live on comfortably. The implication of this change suggests that employers should judge the net worth of each employee, then deposit 10% of that value into the employee’s PSA, and pay the employee the remaining 90% as his Gross Income. The employee must then deposit 10% of this GI into his PSA on his own. This sequence serves several purposes: 1) the employer’s contribution to each PSA is greater than the amount currently mandated, but no further benefits, such as health or unemployment insurance, will be expected or needed; 2) it results in more than the employee’s 10% going into the PSA automatically (the employee’s contribution is 10% of his 90% reported gross income, or only 9% of his worth to the employer), and 3) it serves to impress on the employee the need to actively participate in the accumulation of savings for later needs, just as today’s “Payroll Tax” does.
As in Version 1, the employee may still deposit additional income (up to his TI) to increase the amount available later. However, the upper limit of total PSA contribution remains at 25% of MI, so those with incomes greater than MI cannot put the entire amount of their Taxable Income into their PSA; they must pay the appropriate amount of tax on the difference. Plus, as I noted in Version 1, all of the money deposited into a PSA, including the employer portion, will be taxed on withdrawal so there will no longer be a state of affairs such as we have today where 47% of taxpayers pay no tax at all.
The upward curving Thresholds are designed to assure that increasing one’s earnings will not result in loss of spendable income due to the increasing PSA contribution. Since anyone below the Poverty Line clearly does not have a Taxable Income reaching 10% of GI, all of the Taxable Income of these taxpayers must go into their PSA account, thus deferring all taxes until those Savings need to be used, with no diminution of their total spendable income from the Tax Credit plus actual Gross Income.
In Graph 4, below, the red line is the standard Gross Income, the blue curve is the upright parabola portion of TI which provides the low income Living Cost Exemption, the green curve is the inverted parabola which smoothly reconnects TI back onto the GI line, and the purple line is the GI – MI line described in Version 1.
Graph 4
It is the tax on the portion of income in the gap between the TI and purple curves which I rely on to cover the cost of the Tax Credit fund. Since the per person Tax Credit is very small (less than .4 * MI) but the number of users likely to be quite large, the per person contribution by higher income taxpayers necessarily is significantly greater to offset the smaller number of such contributors. The comparison should become clearer in Graph 5 below which enlarges the view above out to only M = 12 and adds in the actual tax curves for both versions (V.1 and V.3) of my Tax Reform. Although Graph 5 shows this detail for a smaller range on incomes, it clearly shows the relatively greater area of contribution to the Tax Credit fund (extended green area at the bottom right) as compared with the Tax Credit user area at the bottom left corner (pink). This user area shows the credit accessible by those with two dependents, about the average family size expected for Tax Credit borrowers. A majority of the Credit fund users will have at most two dependents, and most of those will need only a small portion of the credit allowed because they will be employed and have some income of their own.
Graph 5
The portion of the total spectrum of incomes that falls within the first piece of this graph (up to M = 3) covers almost 90% of all taxpayers, so it is within this area that most people will be interested in comparing what they pay the IRS under our current Tax Code with what they would pay under this proposed scheme. From those who have given me enough information to compare, I have found that most would pay the same or slightly less under my plan, while a very few would pay up to about 10% more than they do now. In other words, this plan does a surprisingly good job of matching the effect of the present Code, but does so with no Special Interest exemptions or constant yearly adjustments by Congress. The only values that change from year to year are MI and FT (Flat Tax rate), and both of these adjustments are designed to be automatic. MI is calculated annually by the GAO or IRS or the Census Bureau – or by all three – and made available to the public within a few months after the end of each fiscal year, certainly before the April 15 Tax Return date of the following year.
FT could be automated as described in my Tax Reform V.1, but I would like to suggest a better way in this version:
Rather than putting the full burden of an annual deficit on the back of the taxpayer, I suggest increasing FT by no more than 1% in any given year and at the same time cutting the salaries of ALL Federal employees by the same 1%. This most emphatically includes all elected officials and members of all Executive Departments. In addition, whenever there is a deficit, all such salaries should be capped at that new level for as many years as are required to eliminate the deficit. Each year a deficit continues, FT should again be increased by 1% and Federal salaries again reduced by 1%. At some point (I recommend at FT = 20% maximum), the FT increases should be waived, but the Federal salary cuts should continue until our government finds a way to eliminate their wasteful spending and correct the problems which actually cause the deficits. Note that implementation of the other parts of my Tax Reform V.1, in particular the privatization of Social Security (phased in over about 20 years) and replacement of it and healthcare costs with my PSA accounts, will cause Federal spending to automatically fall by about half as compared with today’s outlays, so current deficits ought to disappear naturally.
Two other assumptions made in Tax Reform V.1 are essential in order for this system to work properly. First, we must tax all real Capital Gains (see Capital Gains Revisited, below); second, we must avoid any form of double taxation and we must not tax business profits. Today we routinely tax “Corporate Profits” at an extremely high rate (as of the date of this revision the rates have gone up to almost 40% – in France I understand the business profit tax may go up to 75%!). Taxing business profits at all is so clearly counterproductive that I cannot comprehend why we do it. Taxing businesses at rates this high accomplishes only one thing: it guarantees that all businesses that can do so will move elsewhere and all of their profit taxes will be lost to any government stupid enough to consider it. The fact is that all businesses survive and attract investment by distributing as much of their profits as they reasonably can to their employees (higher incomes and more jobs), to their management and executive people (the best companies buy better managers), to their investors in the form of dividends and interest, and if they are truly profitable, in the form of Capital Gains, the ultimate source of the huge incomes of the exceptionally wealthy. What the wealthy will lose by paying tax on all of their investment Gains – at the same flat rate (15% envisioned here) as all of their ordinary income – they can expect to win back many times over from the increased value and growth of the businesses in which they invest. With almost every country in the world competing for jobcreating business, but then turning around and taxing away a large portion of the resulting profits, any government wise enough to completely eliminate the Corporate Tax will soon realize the benefits in the form of more profitable and fastergrowing businesses within their own borders, resulting in more jobs, higher incomes and far more income tax revenue.
It was my intent to add to this discussion a comparison between the simple tax method above with some form of display of our current tax system. After numerous attempts to create a meaningful picture, I have concluded that there simply is no way to show how today’s system works in graphic form. There is no correlation between Gross Income and Taxable Income because of the many thousands of Special Interest exemptions, deductions, exceptions and loopholes. With the current tax code, no two people in the country, with the same Gross Income, have the same Taxable Income or owe the same amount of tax.
Capital Gains Revisited
In both Versions V.1 and V.2 I recommended the same method for moderating the tax on Capital Gains by multiplying the Gain value by MI (previous year) / MI (current year). After careful review, I believe that there is a better, fairer, and simpler way to accomplish the desired results. My initial concern was to persuade investors to report and pay taxes on their unrealized gains annually rather than waiting until their investments were sold – sometimes many years later. All too often, tax considerations dictate holding stocks longer than is prudent; other times the effect of tax rules compels investors to sell before the optimum time. If inflation effects and tax regulations could be removed from this calculation, most investors could make more intelligent decisions as to when to buy and sell their holdings.
I believe I was on the right track in adjusting the gains using relative MI values from yeartoyear but my original method was unnecessarily complicated and did not adequately offset the effects of GDP growth over longer holding periods. Clearly, if one invests X dollars today, holds that investment for more than a year, then sells it for the same dollar amount as was originally invested, one has lost money! GDP and MI both grow yearoveryear by about the same amount. Over the past half century or more this annual growth rate has averaged over 4.5%, so for example an investment made 50 years ago would, today, have to be worth more than 9 times the cost basis in order to break even (1.045 raised to the 50^{th} power = 9.033). Graph 6 below shows a smoothed average of the MI growth since 1948. Interestingly, if I had picked the price of a gallon of gasoline in 1948 and showed its growth over the 60 intervening years, that curve would track the MI growth curve quite closely. In the early 1950s, when I started driving, a gallon cost about 30 cents. In the mid1970s, the price was around 60 cents. Today it is just above $3 per gallon. These prices fit almost perfectly along this same curve, so gas prices, in real terms, have not changed significantly in 60 years. This same observation applies, to a close approximation, for most basic goods, commodities and services.
Using this knowledge we can use the growth in MI from “last year” to “this year” each full year an investment is held to determine the breakeven selling price. If the stock value “this year” exceeds this breakeven point, there is a real gain. This real gain times FT should be the tax owed. If the stock value is less than this breakeven value but greater than the cost basis, there is neither a gain nor a loss; the gap between the two values is what I call the “risk zone” for any investment (see Graph 7, below).
If the value drops below the cost basis, there is a real loss and this loss may be used (if the stock is sold at this time) to offset an equal amount of real gain in other investments or to claim a tax refund on the current year’s loss against gain taxes paid in previous years on the same stock – or to offset a portion of normal income if other gains or previous tax payments are not adequate. Current tax rules limit the deduction of such losses to $3000 per year, an arbitrary and meaningless choice since MI has grown by a considerable amount since this rule was put in place. A more reasonable choice would be to limit the loss offset to 15% of the lesser of: a) the net loss (after offsetting other gains) or b) the available Gross Income. For losses below the dollar equivalent of MI, the offset should be limited to simply 15% of GI. Obviously, such an offset would be applied to GI before calculating TI for tax purposes.
As I recommended in both earlier versions, this growth adjustment would be available to the investor only if he chooses to declare gains annually and pay the accrued Flat Tax on all gains (realized and unrealized) every year. Stock prices vary enough over long holding periods that it would be expected that many holdings would vary from gain to null (risk zone) to loss and back fairly frequently. When any given stock is in a null or loss position but is expected to recover, using the rules above should make the choice to hold or sell at an optimal point much simpler and wiser. Selling while a stock is in the risk (null) zone would result in no gain and no tax (any tax paid at an earlier position of real gain would be refunded), while a real loss would result in the gainloss offset options described above. If a stock is sold at a real gain which is less than the position a year earlier, then the tax paid the previous year would constitute an overpayment and the excess would be refunded. Since the MI growth can be calculated relative to the purchase date, (see Appendix, below) the original cost basis can be used, along with that total growth to calculate the correct tax owed each year and any final tax at the time of actual sale. In any case, the same Flat Tax rate applied to normal “earned” income will be applied to the net real gain realized at the time of sale and adjusted for gain taxes paid in prior years.
Here I have plotted the performance of a hypothetical investment:
Graph 7
The red “+” symbols represent the annual stock values of an investment with a “basis” of 1 at year 0. The red line indicates this basis multiplied by an annual growth of 4.6% per year. In this example, the stock increases in value enough each year so that there is never a real loss condition, but in years 3 and 4 there was also no real gain. A tax would have been paid on the real gains in years 1 and 2, then all of that tax would have been refunded at the end of year 3. In years 5 and 6, additional tax would be owed, but a portion of that would be refunded in year 7. Years 8 and 9 would incur small amounts of additional tax. Note that at the end of year 6, the real gain is greater than in any later year. If the holding had been sold at that time, the investor would have realized his maximum real gain and would have paid the maximum amount of tax on that gain.
A Closer Look at Poverty Thresholds
Current law mandates a Minimum Wage for most workers of $7.25 per hour. This gives an annual income of $7.25 times 40 (hours / week) times 52 (weeks) = $15,080 per year. One of the disadvantages of this minimum wage is that many new hires fresh out of high school (or worse, high school dropouts) with no previous work record are viewed as poor risks by potential employers, with the result that they find it extremely hard to find jobs. The unemployment rate for wouldbe workers in the 1924 age range right now is above 20%! One of the primary justifications for my Tax Credit Fund is that these young people can still get a job, at whatever wage an employer deems fair, and then apply for a Tax Credit equal to the difference between that annualized wage and the appropriate Threshold curve. Once this worker has a record that justifies a better wage, the need for continuing Tax Credit usage diminishes. Thus the Minimum Wage would no longer be needed and its disadvantages would disappear.
Although I have made no allowance for family size in my primary Taxable Income structure (above the Poverty Zone), it is clear that at these very low Poverty income ranges family size is a major consideration in how much Tax Credit is likely to be needed. In the simplistic set of curves shown in Graph 3 above, I have simply divided the zeroincome Threshold range, from 20% of MI to 40% of MI, into four equally spaced values for one, two, three and four dependents. A more realistic spacing would allow for the fact that the first dependent is generally more expensive (to feed, clothe and shelter) than additional dependents. A better choice of spacing might be 20% (single), 28% (one dependent), 34% (two), 38% (three) and 40% (four or more dependents). Since our current Minimum Wage was predicated on preventing abject poverty for a worker with one or two dependents, this range of Tax Credits produces very similar results but is actually slightly more generous, and it caters to family size while the Minimum Wage is a onesizefitsall choice that would be inadequate for a larger family and is overly generous for single workers. In addition, it automatically adjusts to CostofLiving growth as MI grows. With today’s roughly $50,000 MI, these modified percentages would equate to annual incomes of $10,000 (single), $14,000 (one), $17,000 (two), $19,000 (three) and $20,000 (four or more). I have seen various estimates of what constitutes a reasonable poverty line, ranging from $15,000 to as high as $25,000. These estimates vary with location because costofliving varies with location. It stands to reason then that the Federal Threshold should err on the low side to minimize cost, and any shortfall in areas with higher living costs should be dealt with by State or Local governments in those areas.
Appendix – Tax Computation Formulas

Normal Tax calculation using this version of Tax Reform:
a. Divide GI by MI to get M: M = GI / MI
b. From M = 0 to 3: TI = M² / 4 (upright parabola)
c. From M = 3 to 6: TI = 9 – (12 – M)² / 12 (inverted parabola)
d. Above M = 6: TI = M
e. Tax (in dollars) = (TI * FT) * MI at all levels of M.
Note that this sequence can easily be formulated into a computer program or into an “APP” for any handheld device. Such a program could download “this year’s” MI value from the internet and calculate the correct tax for whatever input value of GI is entered by the user.

Capital Gain tax calculation (after first full year), assuming annual gain tax payment:
Assume stock Value at purchase (cost basis), V_{0} = $1000
a) Obtain MI for year 0 (purchase year) from internet: MI_{0 }
b) Obtain MI for year 1 (end of next tax year): MI_{1 }
c) “Zero Gain Level” = Original cost basis times (MI_{1} / MI_{0})
For example, if MI0 was $51000 and MI1 is $53000, then the corrected basis for year 1 would be 1000*1.0392 = $1039.20. (53000 / 51000 = 1.0392 is the MI growth rate for year 1; as a percent this is the same as 3.92% growth)
d) If the stock Value, V_{1} , is greater than $1039 at the end of the first full year after purchase, then the real gain will be (V_{1} – $1039). This value (combined with any other gains and losses and assuming a net positive value) will then be added to the taxpayer’s M in the tax calculation (section 1. above) to compute his tax.

Capital Gain after first year: the same process should be followed except that MI_{n} is used in place of MI_{1} for each year “n” in steps 2. b and 2. c to calculate a yearly “zero gain level”, and any prior tax on real gain for this stock holding must be deducted from the tax on the current year’s total real gain. Refer to the section above on Capital Gains Revisited for instructions on treating overall gains and losses.
One final note: An assumption that stocks are bought and sold at the beginning or end of a year appears to be implied above since the MI_{n} values above refer to annual values downloaded from the internet, but since purchase and sale dates seldom occur right at yearends, it may be necessary to interpolate midyear MI values between the annual values. For this purpose it is perfectly valid to use linear interpolation, rather than the more precise exponential interpolation, on either a monthly or daily basis. (The error either way is small enough that it is inconsequential.) As an example, if a purchase is made, say, on May 15, the (linearly) interpolated MI_{0} would be MI_{n }+ (MI_{m} – MI_{n}) * (135 / 365) where May 15 is the 135^{th} day of a normal year and m and n represent the start and end of the purchase year. Since MI growth is quite slow, the analogous monthly interpolation would be fairly similar. For very large investments on any midyear dates either interpolation method can make a difference in one’s tax bill. In the example above, the purchase date MI_{0} value would be 51000 + (53000 – 51000) * (135 / 365) = $51,740 using daily interpolation (or * (5 / 12) = $51,830 using monthly interpolation; the IRS would issue rules regarding how to implement this choice). Similar calculations would apply to the MI for midyear sale dates. This degree of refinement probably will not be allowed for “shortterm” holdings, less than a full year, but for very longterm holdings all of the above calculations are likely to make many thousands of dollars difference over time to the investor who agrees to pay his “unrealized” gain tax annually.
The calculations described above may seem complex to those not mathematically inclined, but please realize that when compared with the abstruse complexities of our current tax code, all of this is the ultimate in simplicity. Sadly, it will remove the need for employment of many tax lawyers, accountants and programmers (once the first “APPs” have been written), but the savings, both to taxpayers and to our government, will accumulate for centuries.