The Remaining Income Method: A Direct, Single-Equation Approach to Infinitely Progressive Taxation

Traditionally, income tax has always been based on percentage of net income. The problem here is that it cannot be infinitely progressive because as tax rates approach 100% in a linear fashion, net income after taxes would peak and then eventually drop to zero. Furthermore, one first has to determine the tax rate before arriving at net income after taxes. Why not save a step and let net income before taxes itself directly determine net income after taxes?

Here we introduce the remaining income method, which will not only make it easier to determine net income after taxes, but will also provide an infinitely progressive scale of taxation. Simply put, net income after taxes will be provided be the following equation:

a = m + [√(bm) * ln (b/m)], where

• a = net income after taxes
• b = net income before taxes
• m = minimum wage

Because this equation is based on minimum wage, it is resistant to the effects of inflation and increase in cost of living, as explained here.

It can be seen in this equation that a minimum wage earner would have a net income of exactly minimum wage and would hence not pay any taxes. Furthermore, using calculus, it has been determined that the maximum slope of this equation is 1 and it occurs where a=b=m, which means everybody earning above minimum wage will pay taxes.

This equation assumes that everyone earns at least minimum wage. Otherwise, the equation would not work. In any case, nobody should earn below minimum wage.

A graph for this equation at a minimum wage of $10/day would look like this:

It can be seen it this graph that person who earns the minimum wage of $10 (x-axis) would keep all $10 (y-axis). On the other hand, someone earning $100 would keep over $80. Let us zoom out to include net incomes up to $1,000:

 

Now we see that the graph has become flatter. A $200 earner can keep just under $150; someone with a $500 wage keeps just under $300. At $1,000, you get to keep less than half of your wages, but that’s still nearly fifty times the minimum wage earner.

Let us now plot these values in Excel to better understand them:

As can be seen here, if you earn $100,000 or 10,000 times the minimum wage, you are taxed at around 91% and keep around $9,000, which is 900 times the minimum wage. Likewise, if you earn an insane 10 MILLION times the minimum wage PER DAY ($100,000,000), you are also taxed at an insane rate of 99.49%, yet you still get to keep $500,000, which is still 50,000 times the minimum wage. To put that into perspective, a minimum wage earner would have to work every day for 137 years to earn $500,000.

At higher income strata, tax deductions from income is no longer important. What matters is you have much more than enough to live a comfortable life many times over.

With this kind of taxation, governments in countries where income inequality is high may be able to collect more taxes, which can be used for projects that can alleviate poverty such as infrastructure, health and education. Consumption taxes may be lowered as well or even removed, further giving lower income families more purchasing power. Likewise, a highly equitable society would pay less taxes, and its government would not need to be too big.

Logarithmic Taxation Beyond 80%

Previously, we have tackled tax rates on individuals with income taxes up to ten thousand times the minimum wage. Pegging the maximum tax rate at 80% is crucial because further progressing would eventually lead to a tax rate of 100%. However, there is a way to continue making progressive taxation beyond ten thousand times the minimum wage.

For any person earning more than ten thousand times the minimum wage, the tax rate is automatically 80%. However, the remaining income would be subject to another tax described in this formula:

Second Tax Rate = [20 * log (A/2000M)]%, where

• A = net income after taxes
• M = minimum wage
• The calculated value is rounded down to the lower tenth of a percent

Here, 2000 was used as the coefficient for M because that would be the net income after tax of someone earning ten thousand times the minimum wage, which is the minimum value to obtain a tax rate of 80%.

Let us look at someone earning ten thousand times the minimum wage ($100,000 at $10)

Tax Rate = [20 * log (100000/10)]%

= (20 * log 10000)%

= (20 * 4)%

= 80%

Net Income After Tax = 100000 – (0.80 * 100000)

= $20,000.00

Second Tax Rate = [20 * log (20000/20000)]%

= (20 * log 1)%

= (20 * 0)%

= 0%

It can be seen that someone earning exactly ten thousand times the minimum wage will not be subject to the second tax rate. Let us now look at someone earning one million times the minimum wage ($10,000,000):

Tax Rate = 80%

Net Income After Tax = 10000000 – (0.80 * 10000000)

= $2,000,000.00

Second Tax Rate = [20 * log (2000000/20000)]%

= (20 * log 100)%

= (20 * 2)%

= 40%

Net Income After Second Tax = 2000000 – (0.40 * 2000000)

= $1,200,000.00

Even with an additional 40% second tax on top of the 80% tax, less than 90% of income was deducted, and a large sum of money is still kept.

Yet at sufficient quantities, the second tax rate can approach 80% and it has to be pegged there again. A third tax rate can be introduced in the same manner:

Third Tax Rate = [20 * log (A/4000000M)]%, where

• A = net income after taxes
• M = minimum wage
• The calculated value is rounded down to the lower tenth of a percent

Let us look at a person earning an improbable one billion times the minimum wage ($10,000,000,000):

Tax Rate = 80%

Net Income After Tax = 10000000000 – (0.80 * 10000000000)

= $2,000,000,000.00

Second Tax Rate = 80%

Net Income After Second Tax: 2000000000 – (0.80 * 2000000000)

= $400,000,000.00

Third Tax Rate = [20 * log (400000000/40000000)]%

= (20 * log 10)%

= (20 * 1)%

= 20%

Net Income After Third Tax = 400000000 – (0.20 * 400000000)

=$320,000,000

Here, so much money has been taxed, but $320,000,000 was still kept, which is 32,000,000 times the minimum wage.

If a person earns a googol or even a googolplex times the minimum wage, this recursive pegging at 80% and constant reapplying of the logarithmic formula will work because nobody is every taxed at 100%. All net incomes after all taxes will always increase. Hence, this system of taxation is progressive until infinity.

 

Logarithmic Taxation on Corporations

Logarithmic taxation based on minimum wage may apply not only to individuals but to corporations as well. Because corporations earn a lot more than individuals, corporations would tend to have a larger tax rate than individuals given the same equation, which would unduly cripple corporations of necessary capital. However, an equation more suited for corporations would not look so different than the one originally used for individuals:

Tax Rate = [5 * log (W/M)]%, where

• W = net corporate income before taxes
• M = minimum wage
• The calculated value is rounded down to the lower tenth of a percent

Using this formula, every ten-fold increase in income above the minimum wage would result in a 5% increase in tax rate instead of 20% seen in individuals.

For example, we look at a corporation earning ten thousand times the minimum wage (at $10 minimum wage, that would be $100,000):

Tax Rate = [5 * log (100000/10)]%

= (5 * log 10000)%

= (5 * 4)%

= 20%

Net Income After Tax = 100000 – (0.20 * 100000)

= $80,000.00

In a previous example, an individual with the same net income would be taxed at 80% and keep only $20,000.

For a corporation to be taxed at 80%, it would have to earn a staggering ten quadrillion times the minimum wage ($100,000,000,000,000,000):

Tax Rate = [5 * log (100000000000000000/10)]%

= (5 * log 10000000000000000)%

= (5 * 16)%

= 80%

 

At this point, it would be impractical to compute for the net income after taxes because no corporation would have a net income of ten quadrillion times the minimum wage.

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How will this affect policy making? Whenever there is an increase in the minimum wage, corporations suffer because they are able to hire less workers, resulting in higher unemployment rates. However, using this method, increasing the minimum wage would effectively lower corporate taxes, increasing their capability to hire more employees. Consequently, increasing the minimum wage would have a smaller impact on unemployment.

 

 

The Effects of Minimum Wage on Taxation

One advantage of logarithmic taxation based on minimum wage is that it automatically accommodates changes in minimum wage due to rising cost of living and inflation. Increasing the minimum wage will decrease the tax rate across all income strata.

Let us look at a previous example where a person earns $100 in a place with a minimum wage of $10:

Tax Rate = [20 * log (100/10)]%

= (20 * log 10)%

= (20 * 1)%

= 20%

Net Income After Tax = 100 – (0.20 * 100)

= $80.00

Let us now increase the minimum wage to $12:

Tax Rate = [20 * log (100/12)]%

= (20 * 0.9208)%

= 18.4%

Net Income After Tax = 100 – (0.184 * 100)

= $81.60

$1.60 has been saved, which may seem insignificant. However, cumulative inflation over decades may eventually lead to a minimum wage three times the original value. Let us see how this same person would be taxed if the minimum wage is $30:

Tax Rate = [20 * log (100/12)]%

= (20 * 0.5228)%

= 10.4%

Net Income After Tax = 100 – (0.104 * 100)

= $89.60

It is clear in these examples that continuous increases in minimum wage will eventually lead to significant deductions in tax rate. Using a logarithmic approach to taxation based on minimum wage ensures a tax rate formula that is stable over time.

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How will this affect policy making? Because an increase in minimum wage will lead to less taxation, there is incentive among lawmakers to keep the minimum wage reasonably high so that everybody would be taxed less. This will not only benefit the higher income strata who would lose less of their earnings from taxes, but more importantly, minimum wage earners would be happier due to their raw increase in net income.

The disadvantage with a high minimum wage is that the government would be gathering less taxes every time the minimum wage is increased. Furthermore, corporations may be able to hire less employees due to high wages, hence promoting unemployment. However, the latter concern may be addressed if corporate tax was also based on minimum wage. This will be tackled in a later article.

 

Primer to Logarithmic Taxation Based on Minimum Wage

One of the problems with income tax is the use of income brackets to determine how of a person’s wages should be taxed. Although progressive, these brackets are rigid, arbitrary, and vulnerable to inflation, discouraging people to pursue greater incomes for themselves in order to pay less taxes. Furthermore, such system of taxation requires constant updating to keep up with costs of living and inflation.

Logarithmic taxation seeks to address all these problems by eliminating income brackets altogether and providing a single, smooth, non-linnear, progressive increase to income tax as personal income increases. To make such a system favourable for minimum wage earners, the minimum wage itself would be an integral variable in determine the tax rate.

First of all, the equation for determining the tax rate must be formulated:

Tax Rate = [20 * log (N/M)]%, where

• N = Net income before taxes
• M = minimum wage
• The calculated value is rounded down to the lower tenth of a percent

Using this formula, every ten-fold increase in income above the minimum wage would result in a 20% increase in tax rate.

For purposes of illustration, let us say that the minimum wage is $10 per day. The tax rate of a person earning minimum wage would be as follows:

Tax Rate = [20 * log (10/10)]%

= (20 * log 1)%

= (20 * 0)%

= 0%

Hence, a minimum wage earner has no income tax. Let us now look at a person earning twice the minimum wage ($20):

Tax Rate = [20 * log (20/10)]%

= (20 * log 2)%

= (20 * 0.3)%

= 6%

Net Income After Tax = 20 – (0.06 * 20)

= $18.80

Here, it is shown that a person earning twice the minimum wage loses less than $2. Now, let us see a person earning ten times the minimum wage ($100):

Tax Rate = [20 * log (100/10)]%

= (20 * log 10)%

= (20 * 1)%

= 20%

Net Income After Tax = 100 – (0.20 * 100)

= $80.00

Here, we see a significant portion of income lost ($20), which is the entire wage of the second example. However, the person earning ten times the minimum wage still keeps eight times more money than the minimum wage earner, and he can still afford a comfortable lifestyle.

Now, let us look at someone earning one hundred times the minimum wage ($1,000):

Tax Rate = [20 * log (1000/10)]%

= (20 * log 100)%

= (20 * 2)%

= 40%

Net Income After Tax = 1000 – (0.40 * 1000)

= $600.00

In this example, up to $400 is lost to taxation, forty times the minimum wage, yet the person earning $1,000 still keeps sixty times more money than a minimum wage earner.

Let us turn things up a notch and see how someone earning ten thousand times the minimum wage ($100,000) is taxed:

Tax Rate = [20 * log (100000/10)]%

= (20 * log 10000)%

= (20 * 4)%

= 80%

Net Income After Tax = 100000 – (0.80 * 100000)

= $20,000.00

Here, an astounding sum of money is taxed, and the person still keeps two thousand times more money than the minimum wage earner. To put that into perspective, the minimum wage earner has to work every day for nearly five-and-a-half years to earn that $20,000.

It would be prudent to peg the maximum tax rate at 80% for now because using the same formula indefinitely would eventually result into a 100% income tax, which is absurd. However, the same logarithmic system can be applied recursively to the remaining income after 80% taxation. This will be addressed in a later article.