Formula to approximate current US income tax rates

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conradchaffee

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Hello all,

As I'm sure you know, current US income tax rates are broken into six brackets with different marginal tax rates for each bracket. The 2012 rates are:

"10% on taxable income from $0 to $8,700, plus
15% on taxable income over $8,700 to $35,350, plus
25% on taxable income over $35,350 to $85,650, plus
28% on taxable income over $85,650 to $178,650, plus
33% on taxable income over $178,650 to $388,350, plus
35% on taxable income over $388,350"

I've come up with an equation that approximates these rates fairly well (within 1.5%) for incomes up to $500,000. Let's call it the "Chaffee tax," to differentiate it from the real income tax. The function I propose is:

Chaffee tax = (Income^1.2845)/133

Here is how the "Chaffee tax" compares to current real income tax rates.

IncomeTotal real taxTotal Chaffee taxDifferenceEffective real taxEffective Chaffee taxDifference
$8,700$870$864-$610.00%9.93%-0.07%
$35,350$4,868$5,231$36313.77%14.80%1.03%
$85,650$17,443$16,302-$1,14120.36%19.03%-1.33%
$178,650$43,483$41,913-$1,57024.34%23.46%-0.88%
$388,350$112,684$113,633$95029.02%29.26%0.24%
$500,000$151,761$157,208$5,44730.35%31.44%1.09%
$29,188,238$10,192,644$29,188,238$18,995,59434.92%100.00%65.08%

As you can see, there is one major problem with my proposed formula: it's an exponential function, so tax as a percentage of income keeps rising indefinitely. At the $29,188,238 income level, the Chaffee tax would be 100% of income. Above that, people would owe more than they earned (i.e. the income tax rate would be more than 100%). This is clearly absurd.

How can I alter my "Chaffee tax" formula so that it maintains or improves its fit to current rates, but then levels off and asymptotically approaches an effective tax rate of 35% (or any other arbitrary maximum tax rate)?
 
conradchaffee said:
Hello all,

As I'm sure you know, current US income tax rates are broken into six brackets with different marginal tax rates for each bracket. The 2012 rates are:

"10% on taxable income from $0 to $8,700, plus
15% on taxable income over $8,700 to $35,350, plus
25% on taxable income over $35,350 to $85,650, plus
28% on taxable income over $85,650 to $178,650, plus
33% on taxable income over $178,650 to $388,350, plus
35% on taxable income over $388,350"

I've come up with an equation that approximates these rates fairly well (within 1.5%) for incomes up to $500,000. Let's call it the "Chaffee tax," to differentiate it from the real income tax. The function I propose is:

Chaffee tax = (Income^1.2845)/133

Here is how the "Chaffee tax" compares to current real income tax rates.

Income
Total real tax
Total Chaffee tax
Difference
Effective real tax
Effective Chaffee tax
Difference
$8,700
$870
$864
-$6
10.00%
9.93%
-0.07%
$35,350
$4,868
$5,231
$363
13.77%
14.80%
1.03%
$85,650
$17,443
$16,302
-$1,141
20.36%
19.03%
-1.33%
$178,650
$43,483
$41,913
-$1,570
24.34%
23.46%
-0.88%
$388,350
$112,684
$113,633
$950
29.02%
29.26%
0.24%
$500,000
$151,761
$157,208
$5,447
30.35%
31.44%
1.09%
$29,188,238
$10,192,644
$29,188,238
$18,995,594
34.92%
100.00%
65.08%

As you can see, there is one major problem with my proposed formula: it's an exponential function, so tax as a percentage of income keeps rising indefinitely. At the $29,188,238 income level, the Chaffee tax would be 100% of income. Above that, people would owe more than they earned (i.e. the income tax rate would be more than 100%). This is clearly absurd.

How can I alter my "Chaffee tax" formula so that it maintains or improves its fit to current rates, but then levels off and asymptotically approaches an effective tax rate of 35% (or any other arbitrary maximum tax rate)?
Click to expand...

Don't have the time right now to see if this will work, but it sounds like you need something that is more like a logistic equation. There are many other folks on this forum way smarter than me that can give you their thougths as well, and in way more depth.
 
conradchaffee,

As you described, the tax rates are broken into brackets. So, apparently, the function is not continuous. You have sudden discontinuities. Hence, you can use a step function or a heavy side function to represent the phenomenon by combining all cases into a single equation.

Step function is usually defined as: step(x)=1 if x>0 and step(x)=0 if x<0

For instance, for the 3 brackets,

"10% on taxable income from $0 to $8,700, plus
15% on taxable income over $8,700 to $35,350, plus
25% on taxable income over $35,350 to $85,650, plus


you can write,

Tax(I)=(10/100)*step(8700-I) + (15/100)*step(35350-I)*step(I-8700) + (25/100)*step(85650-I)*step(I-35350)

Of course, you can expand the equation to handle as many brackets as possible.


Thus, you get no error here. Step function step(x), the way I defined above may be defined differently at x=0 depending on the application. At x=0, you can adjust the equation suitably.

Cheers,
Sai.
 
Hello Sai,

Thank you for your response. You are, of course, correct: as it stands, the US income tax can indeed be exactly described by a single continuous function. I should have been more exact in my request.

What I'm looking for is a smooth curve, rather than a series of straight lines of differing slopes, which is what the income tax is currently. My hope is to come up with a simple idea that would be easier to use than the current system.

My current formula:

Chaffee tax = (Income^1.2845)/133

isn't perfect in this regard, because some people might not understand the idea of raising a number to a fractional exponent. But nevertheless, it's still relatively simple, and it produces a smooth curve.

Any thoughts in this regard?

Cheers,
Conrad
 
Conrad,

I apologize if I misread your question.

Approximating a step function with a continuous function doesn't seem like a scientific approach to me. However, if you are trying to fit the best fit for the function, then you can do that using trial and error. It would be easier to use either Matlab (or octave; octave is free). To start, you need a sample set of points that represent the exact step function. In your case, may be you can come up with 15-20 points. You can use these to interpolate and find your function.

Method 1. You can use generalized equations and find the R^2 value. The best fit will have an R^2 closest to 1.0. You just have to use trial and error. Obviously linear functions wont work. You can try quadratic and some combinations of linear and non-linear functions like: y=(Ax+B)*exp(Cx+D) ; (Ax^2+Bx+C)*exp(Dx^2+Ex+F) or some combination with a trigonometric function. You can even try generalized higher order polynomial equations if you like. Using Matlab, its easy to find the unknown coefficients of these equations. Try many combinations and look for your R^2 value. You will never be able to get a good match because you are approximating a step function with a curve, unless you want to use the complicated bezier curves or splines -which I am sure you don't want.

Method 2. I think this is what you may want although the equation may be fussy. Simply plot the exact step function in Microsoft excel. Then you can right click and plot the best fit curve. You can click on the best fit curve to change the parameters of the curve. You can change the order of the curve and play with it to get the desired fit. It will also show you the corresponding equation.

However, I still think you should just use the step function. When you can exactly represent the conditions given in the form of an equation, why do you want to fit a curve, especially when you have a complicated set of conditions? May be you should plot the data and then think about it.

Cheers,
Sai.

conradchaffee said:
Hello Sai,

Thank you for your response. You are, of course, correct: as it stands, the US income tax can indeed be exactly described by a single continuous function. I should have been more exact in my request.

What I'm looking for is a smooth curve, rather than a series of straight lines of differing slopes, which is what the income tax is currently. My hope is to come up with a simple idea that would be easier to use than the current system.

My current formula:

Chaffee tax = (Income^1.2845)/133

isn't perfect in this regard, because some people might not understand the idea of raising a number to a fractional exponent. But nevertheless, it's still relatively simple, and it produces a smooth curve.

Any thoughts in this regard?

Cheers,
Conrad
Click to expand...
 
Last edited:
Conrad

I get it; you are not trying to duplicate the current tax structure but to propose an alternative.

Sir michael had a good idea, namely to start by thinking about the logistic function. If you believe that as a matter of policy no citizen should be entirely exempt from the income tax, it captures that idea. It also preserve the idea that those who have higher incomes pay a higher tax rate. It does, however, have policy implications that you may not like or that are politically impossible. One is that the rate of increase in the tax rate is highest in the middle of the income range. That is, the poor would pay a very low but positive rate; the rich would pay a high rate that would approach but never reach a maximum rate, and the middle class would pay a moderate rate that increased rapidly as income increased, which might have undesriable effects on incentives.

I am sure that there are ways to tweak the function to do almost anything, but I'd start there.
 
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