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Product Rule, marginal rates, ACA, college cost formulas etc
Old 11-03-2017, 04:58 AM   #1
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Product Rule, marginal rates, ACA, college cost formulas etc

I saw a post 2017 year end tax planning
where the poster was puzzled that ACA subsidy phaseouts of 10% or less could effectively lead to "marginal rates" like 15% or 17% as mentioned by other posters.
2017 year end tax planning
2017 year end tax planning

Here's an explanation of the phenomenon (with made up numbers, analogous to ACA calculations).
Suppose you would have to pay
10% tax on $50,000, that is, $5,000
but you would have to pay
10.1% tax on $51,000, that is, $5,151
so the extra $1,000 costs an extra $151 in tax, so a "marginal tax rate" of 15.1%
What happened? There is the extra 10.1% * $1000 = $101 which contributes an unsurprising 10.1% to the "marginal tax rate".
But there is also an extra 0.1% * $50,000 = $50 which contributes another (possibly surprising) 5% to the "marginal tax rate".

Think of the change in area when a rectangle grows from 50X10 to 51X10.1. There's an extra L-shaped region, and you'll miscalculate if you miss one of the legs of the L. See the rectangular picture in this.
World Web Math: The Product Rule


Some college cost formulas have something similar, e.g. a college may say that for income in the range $50k to $150k, you'll pay a percentage of income ranging from 0% to 20% increasing (linearly, 1% per $5k income above $50k) as a function of income, so
0% * $50k = $0
1% * $55k = $550
2% * $60k = $1,200
............
............
19% * $145k = $27,550
20% * $150k = $30,000

But even though the "rate" increases from 0% to 20%, the "marginal rate" increases from 10% to 50% !
Consider,
the first $5k income increase from $50k to $55k costs an extra $550 which is 11% of $5k.
The last $5k income increase from $145k to $150k costs an extra $30,000-$27,550=$2,450 which is 49% of $5k.

In formulas: for X between $50k and $150k, the percentage you pay is
R(X)=(X-$50k)/$500k which goes from 0% to 20% as X goes from $50k to $150k,
so the college cost is
C(X)=X*R(X)=X(X-$50k)/$500k which goes from $0 to $30k as X goes from $50k to $150k,
but the marginal cost (derivative of C(X) wrt X) is
C'(X)=(2X-$50k)/$500k which goes from 10% to 50% as X goes from $50k to $150k.

So this is one way marginal rates can be possibly surprisingly higher than expected.
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Old 11-03-2017, 06:47 PM   #2
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Looks like the message (first post) got out of moderation limbo.
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Old 11-18-2017, 03:38 PM   #3
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I've never thought of it quite that way, but that rectangle visualization works.

For me, I've given up on trying to understand the "why" of the numbers. Although it's possible, with a lot of noodling on it, it comes down the fact that the tax code is a patchwork of a bunch of cr*p that interacts in weird ways. There are credits and things that I have taken advantage of that phase out and different places...too many levers moving to understand why.

So my "solution" is to plug things into tax software and do "sensitivity analysis", which basically means move things that I have control over, and see what happens. That results in a bunch of "knob twiddling" until I find a reasonable sweet spot.

Most of my earning time there was nothing at all that I could change. Maybe choose to do tax deferred saving versus plain savings, and things like that. But now that I'm no longer a W2 wage earner, there's quite a few options to play with.
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Old 11-18-2017, 03:51 PM   #4
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This doesn't explain the ACA marginal rates at all. There is no higher % that is applied to all income in the tax code. I don't know about college cost formulas. I get the math of this post, but it doesn't apply.
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Old 11-18-2017, 08:26 PM   #5
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Quote:
Originally Posted by sengsational View Post
I've never thought of it quite that way, but that rectangle visualization works.

For me, I've given up on trying to understand the "why" of the numbers. Although it's possible, with a lot of noodling on it, it comes down the fact that the tax code is a patchwork of a bunch of cr*p that interacts in weird ways. There are credits and things that I have taken advantage of that phase out and different places...too many levers moving to understand why.

So my "solution" is to plug things into tax software and do "sensitivity analysis", which basically means move things that I have control over, and see what happens. That results in a bunch of "knob twiddling" until I find a reasonable sweet spot.

Most of my earning time there was nothing at all that I could change. Maybe choose to do tax deferred saving versus plain savings, and things like that. But now that I'm no longer a W2 wage earner, there's quite a few options to play with.
I definitely agree that it is good to plug a range of numbers into tax software, or online calculators, or a spreadsheet. I like to try to understand what's going on, but even if I (sometimes) can, the number crunching can confirm it.
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Old 11-18-2017, 08:28 PM   #6
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Quote:
Originally Posted by RunningBum View Post
This doesn't explain the ACA marginal rates at all. There is no higher % that is applied to all income in the tax code. I don't know about college cost formulas. I get the math of this post, but it doesn't apply.
Yes it definitely applies. I'll explain further. (Next post.)
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Old 11-18-2017, 08:31 PM   #7
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The formula is (implicitly) here.
https://www.law.cornell.edu/uscode/text/26/36B
Look at the table in (b)(3)(A)(i) and consider the words "within an income tier ... on a sliding scale in a linear manner".

If you go through the analysis as in the OP, you'll find that in the range 250% up to 300% of FPL, the applicable percentage goes (linearly) from 8.05% up to 9.5% but the marginal rate of change (with respect to household income) of household income times applicable percentage goes from 15.3% up to 18.2%.

This 18.2% comes purely from ACA and is on top of any other tax effects. Also 18.2% is the highest marginal rate in this situation (except for the "cliffs" at 1.33*FPL and 4*FPL for course).

This 18.2% is much higher than the percentages in the linked table, and that's the point I've been trying to make.



Also you can consider bigger jumps, e.g. going from 2.5*FPL to 3*FPL

8.05% * 2.5 * FPL = 0.20125 * FPL
9.5% * 3 * FPL = 0.285 * FPL

The average rate of change in this range is (0.285-0.20125)/(3-2.5) = 16.75%
which again is a much higher percentage than the ones explicitly in the table.

To intuitively see what's going on consider "extra L-shaped region" in the OP.
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Old 11-18-2017, 09:31 PM   #8
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OK, you are right. That's another stupid thing about the way they do the ACA subsidies along with the 400% cliff. If you are $1 under 300% FPL, adding that extra $1 to take you to 300% costs you $10. That's for a single, probably more for a couple or family. I don't think it applies to the 2 yr old Roth conversion thread you woke up, but whatever.
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Old 11-18-2017, 10:37 PM   #9
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^ It definitely applies to that thread The 29% Bracket in Roth Conversions since the OP of that thread was calculating the cost of Roth conversions due to various tax stuff including ACA.

If people are trying to understand how their take-home income (or tax or credit or subsidy etc) depends on a whole bunch of parameters, the stuff I described applies to parts where rates/phaseins/phaseouts vary in ways other than piecewise constant, and ACA is a case of this.

Also I don't know what you mean by " If you are $1 under 300% FPL, adding that extra $1 to take you to 300% costs you $10." That sounds like a totally different phenomenon, perhaps rounding effects.
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Old 11-19-2017, 07:18 AM   #10
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I was using actual numbers from 2016 for form 8962. If a single person has a MAGI of $35,309 (299% FPL), their part of the premium would be 0.0963 of that (table 2 of the 8962 instructions), or $3400 annually. If they had a MAGI of $35,310 (300%FPL), their part would be 0.0966 of that, or $3410. $10 more toward the premium on just $1 more income. Rounding is part of it, but the other part is the rising application factor (line 7 of 8962) up to 300% FPL, which was your point. I think this is the worst case example. $34,310 (291% FPL) uses a factor of 0.0939 for $3221, so $189 less for $1000 less income, for example, so an 18.9% "tax rate" on that $1000. $36310 uses the 0.0966 factor as everything 300-400% FPL uses, so the contribution is $3507--$97 more for the extra $1000 over 35310, a 9.7% (actually 9.66) rate on that $1000.


So the "marginal" rate on anything above 300% FPL up to the cliff is a flat 9.66% rate. Below 300% it is a lot higher. I had not realized that.


Using a real case like this is a lot more clear than a vague, made up generic formula. If you had just said in the first post that the ACA subsidy uses a rising % factor, and pointed to the form 8962 instructions, I would've looked for myself and not argued.
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Old 11-19-2017, 12:05 PM   #11
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I assumed people knew how it was set up, with the sliding rates, since they were crunching numbers and observing marginal rates much higher than the numbers you see in the linked table
https://www.law.cornell.edu/uscode/text/26/36B

I was explaining this phenomenon.
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Old 12-07-2017, 01:02 AM   #12
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This Kitces blog post goes into some detail about this.
https://www.kitces.com/blog/how-the-...inal-tax-rate/
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