April 1, 2013
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Mathematically elegant progressive taxation
Writing fiction often gives us opportunities to explore different ways the world might be. I devised a tax system for the Terlaroni that suits their social-democratic and mathematically-minded approach to the world… and discovered that it actually seems like a pretty good idea.
The current US tax system is progressive, but not very much. We still require people below the poverty level to pay taxes, capital gains rates are much lower than ordinary income rates, and there is basically no difference in tax rate between an income of $1 million and an income of $1 billion.
I have devised a mathematically elegant tax system that solves these problems. The basic principle is a proportional tax on luxury, that is, the tax claims a fixed proportion of the extra utility you receive from wealth above subsistence.
The utility of wealth is widely believe to be logarithmic, such that $100,000 is as much better than $10,000 as $10,000 is than $1,000. While this tends to break down near the subsistence level, it works quite well above that, which is where we’ll be using it anyway. Normalize so that the subsistence level yields utility of 0.
For a subsistence level s and income x, this yields utility U:
U = b ln[(x-s)/s]
Now, if the government takes a proportional tax on luxury, this defines a parameter k that decides the tax rate for any given income level (though be careful not to confuse it with the tax rate itself). Let x2 be after-tax income and x1 be before-tax income.
U(x2) = k U(x1)
b ln [(x2-s)/s] = k b ln[(x1-s)/s]
The parameter b becomes irrelevant (it’s basically a measurement unit), and we can solve this explicitly for x2:
x2 = s (1 + (x1/s – 1)^k)
This results in a progressive tax rate, which is 0 at x1 = 2s, increases in x1, and approaches 1 as x1 approaches infinity. Note that higher k indicates more take-home pay, that is, a lower tax rate.
There is one other wrinkle; as written, this would mean a net negative tax rate on incomes between 1s and 2s, and even weirder, an undefined tax rate for incomes below s. We can solve this problem easily, by saying that incomes below 2s (that is, below twice the subsistence level) are untaxed. If the subsistence level is set at half the median income (as it is in Europe), then this means that incomes below the median are untaxed.
I added another feature as well: incomes less than subsistence receive a subsidy, which raises them up to the subsistence level. This could essentially eliminate the need for all other means-tested welfare programs.
Note that there are no “tax brackets” in this system, just a smooth progression of tax rates; furthermore, an increase in gross income will never result in a decrease in net income. In the regime x1 < s, net income remains constant at s due to the subsidy; at s < x1 < 2s, net income is just gross income; and then at 2s < x1, net income is a monotonically increasing function of gross income which increases at a strictly decreasing rate. Due to rounding to the nearest cent, we could say that income is actually a step function, but for realistic amounts of wealth, the steps are quite short; even into the millions you get another cent of net income at about every 4 cents of gross income.
Moreover, there is no maximum net income; a gross income of 10^20 s still produces a larger net income than a gross income of 10^19 s, even though these are ludicrously huge amounts of wealth better attributed to gods than men. (Assuming a subsistence level of $10,000 per year, 10^19 s would be about a billion times the world’s present GDP; these individuals own galaxies.)
I created a computer simulation of this tax system using Monte Carlo methods. I randomly generated 1 million different incomes according to a Pareto distribution (with some noise added), and simulated the resulting revenue from using this system with different values of the rate parameter k.
I even included an adjustment for the Laffer effect (higher taxation slows economic growth); this adjustment was admittedly not very precise, but this is hardly my fault, as the Laffer effect has never been pinned down quantitatively. We all agree that it should have some effect; the debate is about the size of the effect. For the model, I used a rather strong effect, as will be apparent from the results. I don’t believe the effect is actually this strong, but I wanted to be fair to those who do.
Rate Constant: 0.1 Revenue: 788033
GDP: 1.96501e+006 Per capita: 1.96501
Population: 1000000 Taxable: 105886 Subsidized: 732728
Taxable fraction: 0.105886 Subsidized fraction: 0.732728
Revenue fraction: 0.401032
Rate Constant: 0.2 Revenue: 961836
GDP: 2.25564e+006 Per capita: 2.25564
Population: 1000000 Taxable: 160220 Subsidized: 595526
Taxable fraction: 0.16022 Subsidized fraction: 0.595526
Revenue fraction: 0.426414
Rate Constant: 0.3 Revenue: 1.47999e+006
GDP: 2.89757e+006 Per capita: 2.89757
Population: 1000000 Taxable: 204407 Subsidized: 483638
Taxable fraction: 0.204407 Subsidized fraction: 0.483638
Revenue fraction: 0.510769
Rate Constant: 0.4 Revenue: 1.6487e+006
GDP: 3.21021e+006 Per capita: 3.21021
Population: 1000000 Taxable: 243337 Subsidized: 397248
Taxable fraction: 0.243337 Subsidized fraction: 0.397248
Revenue fraction: 0.51358
Rate Constant: 0.5 Revenue: 1.9732e+006
GDP: 3.71347e+006 Per capita: 3.71347
Population: 1000000 Taxable: 278129 Subsidized: 329426
Taxable fraction: 0.278129 Subsidized fraction: 0.329426
Revenue fraction: 0.531363
Rate Constant: 0.6 Revenue: 2.14879e+006
GDP: 4.12788e+006 Per capita: 4.12788
Population: 1000000 Taxable: 309204 Subsidized: 275932
Taxable fraction: 0.309204 Subsidized fraction: 0.275932
Revenue fraction: 0.520555
Rate Constant: 0.7 Revenue: 1.91983e+006
GDP: 4.22614e+006 Per capita: 4.22614
Population: 1000000 Taxable: 338792 Subsidized: 231781
Taxable fraction: 0.338792 Subsidized fraction: 0.231781
Revenue fraction: 0.454277
Rate Constant: 0.8 Revenue: 2.13355e+006
GDP: 4.98527e+006 Per capita: 4.98527
Population: 1000000 Taxable: 368044 Subsidized: 193993
Taxable fraction: 0.368044 Subsidized fraction: 0.193993
Revenue fraction: 0.42797
Rate Constant: 0.9 Revenue: 1.56116e+006
GDP: 5.37341e+006 Per capita: 5.37341
Population: 1000000 Taxable: 394985 Subsidized: 162370
Taxable fraction: 0.394985 Subsidized fraction: 0.16237
Revenue fraction: 0.290533
Rate Constant: 1 Revenue: -26892.7
GDP: 5.48033e+006 Per capita: 5.48033
Population: 1000000 Taxable: 420554 Subsidized: 135556
Taxable fraction: 0.420554 Subsidized fraction: 0.135556
Revenue fraction: -0.00490713
Due to the randomization, different runs of the model will produce slightly different results, but the model appears to be fairly stable against such variation. Only take the first two decimal places literally.
Notice the Laffer effect: Per capita GDP at k=0.1 (a tax that takes 90% of luxury) is only 1.97 times subsistence, while per capita GDP at k=0.9 (a tax taking only 80% of luxury) is 5.37 times subsistence. At s=$10,000, these would be per capita GDPs of $19,700 and $53,700 respectively. Surely no one can claim a stronger Laffer effect than that; otherwise how would Norway have a per capita GDP PPP higher than that of the US despite taxing at almost twice the rate?
Note that in fact the revenue as a fraction of GDP is lower under k=0.1 even though the tax rates are all higher; this is due to the effect of subsidies. At very high tax rates, the strong Laffer effect yields an economy in such poor condition that a large portion of the population falls below subsistence and needs to be subsidized. Since my model calculates net revenue for the government (after these transfers), it results in a smaller fraction of GDP as government revenue.
I’ve also included k=1 for comparison, which is a system in which there is a subsidy up to subsistence but no taxation at all. As you might imagine, the government runs a deficit in this scenario, though the deficit is not all that large because the assumed Laffer effect leads to economic growth that lifts most people out of poverty anyway.
The highest revenue is obtained at k=0.6 and k=0.8, and since k=0.8 has less poverty and overall higher GDP, it’s obviously to be preferred. This results in government revenue of 37% of GDP (comparable to Norway). You might think that this would yield a massive, draconian tax rate; on the contrary, most of the population would less than they presently do.
The following table compares gross income, net income, and effective tax rate:
Gross Net Tax rate
1.00 1.00 0.00%
2.00 2.00 0.00%
3.00 2.74 8.63%
4.00 3.41 14.79%
5.00 4.03 19.37%
6.00 4.62 22.94%
7.00 5.19 25.81%
8.00 5.74 28.21%
9.00 6.28 30.24%
10.00 6.80 32.00%
20.00 11.54 42.28%
30.00 15.79 47.37%
40.00 19.74 50.64%
50.00 23.50 53.00%
100.00 40.49 59.51%
200.00 70.04 64.98%
300.00 96.62 67.79%
400.00 121.44 69.64%
500.00 145.04 70.99%
1000.00 251.99 74.80%
2000.00 438.17 78.09%
3000.00 605.76 79.81%
4000.00 762.31 80.94%
5000.00 911.14 81.78%
10000.00 1585.77 84.14%
100000.00 10000.92 90.00%
1000000.00 63096.68 93.69%For incomes up to 5 times subsistence (i.e. $50,000), the rate is actually lower than what one would pay under the current US system. For incomes up to 10 times subsistence ($100,000), it is about the same
You’ll notice that the rate becomes exceedingly high at the end of the table; at the bottom row, you’re paying over 93% of your income, which reminds of “one for you, nineteen for me” in the Beatles song “Tax Man”, which would be a rate of 95%. (This was in fact the marginal rate on very high incomes in Britain at the time, and the Beatles were new millionaires unaccustomed to tax avoidance.)
You should also note the gross income at the bottom of the table: one million times subsistence, that is, about $10 billion per year. The only people who would pay this rate are the likes of Warren Buffett and Bill Gates. Moreover, they would still be left with 63,000 times subsistence, that is $63 million per year.
If implemented in the US, this might yield a flight of the very rich to tax havens (which totally doesn’t happen at all right now, right?); but if it could be implemented in most of the world, it would actually leave the ordinal distribution of wealth exactly the same, affecting only the cardinal distribution. Anyone who obtains their sense of self-worth from being the richest person they know would still be the richest person they know. And the people hurt the most would be precisely those with the most to lose.
Occupy says “tax the rich!”; they may be onto something here.
