I cracked up when I got to the marathon example. When I ran a half marathon I realized about 80% of the way through that I was on track to finish under 2:30:00 and pushed myself to make it happen. I should have guessed that sort of behavior would show up in the statistics!
After this year's Paris marathon, I ran the same per-minute graphs, and they match perfectly the "overall study" graphs with more than 9 million finishes in the article. I also added graphing by age category and gender. I don't want to deduce too much but I think it showed that young men are the most "competitive" (what I mean by that is targeting a specific time) since there are the clearest "goal time" peaks in the graphs.
IIRC There is also a discontinuity when it comes to the histogram of ages of marathon runners, because runners are binned into age groups and there is a more runners at the youngest ages of each group, I guess because it’s younger runners in each group that are more likely to run if they feel more likely to place well in their group
I think there's also the opposite – there are a number of places in large marathons allocated for "good for age" entrants, which I would guess are more likely to be taken up by people at the younger end of those buckets.
The UK tax system also has a bunch of unfortunate cliffs, and tapers that create >60% marginal tax rates and worse. There's a calculator here that illustrates it well https://tax-cliffs.britishprogress.org/calculator
The childcare cliff edge is probably the worst, but the personal allowance taper isn't ideal either as it's compressed over a relatively short income range
And of course all the thresholds remain frozen, creating plenty of fiscal drag on top.
The Economist has this crazy graph of UK VAT thresholds and how they are causing companies to stay small, to keep their turnover under the threshold which avoids a lot of paperwork. It's the most infuriating thing, we are literally throttling our companies, the engines of growth, because of some accounting stupidity.
Hey, at least they can exist. Many businesses in the US can no longer exist at small scale, because the barriers to entry are too high because the entrenched interests achieved total regulatory capture.
I remember when my grandfather died, and my grandmother got part of his pension.
It put her pension above a max threshold for old age / low pension (cheaper phone bill, no tv taxes, ...) by, ... hold your hats... 2 Euro.
Father called the service ... "sorry, we can not do anything about that".
She lost over 100 Euro in benefits because technically... she was not low pension anymore but was financially WAY more poor then when my grandfather was alive.
So when my grandfather was alive, their combined pension was higher, they qualitied as having a low pension. When only *part* of my grandfather pension got combined into my grandmothers. Well, now your not poor...
I do not remember the exact pensions but it was a massive haircut, with the insulting on top, losing those benefits. The lost benefits made it from threading water, to drowning.
Without my dad financial help, she will have lost the house. Ironically, if she did not get my grandfathers pension part, she will have qualitied for more social assistance, and will have been better off.
> Sometimes, those lines really screw people over.
Another angle is how they affect the living. Plenty of young adults delay moving out from their parents, or starting a career, because changes in combined household income would cross one of those magic threshold, and ending up screwing the parents and younger siblings for life.
Such delaying may not be as immediately critical to the person, but it alters the course of their lives in significant ways.
> The UK tax system also has a bunch of unfortunate cliffs, and tapers that create >60% marginal tax rates and worse.
Oh I don't doubt it. In my native country of Belgium the tax rate can go to 69% and more: 50% income tax (used to be as high as 52.5% and maybe even 55%, don't remember) and then, on top of that, an addition 19% social wellfare tax. That's not mentioning stuff like, if you manage to buy stocks with what you have left, 30% state tax (+ crazy bank fees) on any dividends you receive from those stocks.
And then the brand new (1st jan 2026) 10% tax on any capital gain, once again: in addition to all the already existing taxes.
Add some arbitrary decisions on top of that, where if you happen to maybe have bought/sell a bit too many stocks or if you happen to have managed to put a too big part of your wealth into stocks, they may consider you're a "professional investor" and then your capital gains are taxed at your income rate.
Some are going to say: "Oh but create a company, get back the VAT, do this/do that".
I don't want to basically become a fiscal lawyer just to not be assraped by ever more taxes.
Instead of trying to "optimize", I just voted --before they'll introduce an exit tax (I'm sure that's coming)-- with my feet, my money and the future wealth I was going to create and left the country.
For what was I getting in return? According to a flemish university an estimated 55 000+ undocumented illegal migrants are now living in the capital city of Belgium, Brussels, alone. That's 5% of the population of the city that are undocumented migrants. They don't live in proper conditions. They're a drag on the economy. They're not happy, they're not thriving. Locals aren't happy either.
I didn't vote for this.
To me once politicians go too far, that's what they deserve: the middle class living the city in drones and being replaced by poor people. As I tell of my friends and family who also left the city: I just went a bit further and left the country altogether.
Following the example of my friend and ex-neighbor who did so decades ago: he left for SoCal (btw I didn't go to the US) and created two startups. Second one was a big hit. Belgium (and the EU) saw exactly zero return of his incredible drive and talent.
Another recent statistics: 40% of the people of Brussels are living close to the poverty line.
But at least socialists got the utopia they dreamed of: they're sure to never lose an election again. The middle class is dying in Brussels. The poor won't vote against socialism.
I just packed and left and advice anyone who can flee a sinking ship to do so. Many countries in Europe have decided to commit suicide and to turn themselves into slums.
I think this is just the beginning for may EU cities/countries: there's much more pain ahead --and although AI is going to be used as a scapegoat-- this has absolutely nothing to do with AI and all to do with political decisions.
It pains to see how my native city has been dragged into poverty... But good riddance.
As for the next gen: from day one I raised my kid in english. At 11 y/o she's totally fluent and our thinking was simple: english is at least understood mostly everywhere in the world. She'll have to move because I fear it's not just my native city that is toast.
It is highly likely the 50k undocumented immigrants aren't "a drag on the economy" but "people increasing margins of business by being exploited in almost slave-like fashion, and doing the work you don't want your kids to do".
The marathon one has a simple - and fun - explanation: it’s great to run with people!
Many (most?) marathons have pace runners who run the course, hewing to each 30 minute and 15 minute finish. So there’ll be a 3:30:00 pacer, a 3:45:00 pacer and so on. Your local pacer might even have their pace on their shirt and may even have a flag so they’re super hard to miss as you’re running.
One friend of mine runs with a speaker, to play music and keep everyone’s spirits up.
By the end of the race they’d acquired a small army of marathoners! You could see their smiles for miles, and when they all finished together they had a huge party in the recovery area with the speaker :)
I’m surprised the marathon time discontinuity isn’t bigger :)
> One of the most common comments I've seen online about this graph and/or this paper is that this is due to pace runners provided by the marathon. Section 4.4 of the paper gives multiple explanations for why this cannot be the case, once again indicating that people tend to comment without reading the paper.
Note that this is about comments on the paper itself. Somebody reading this article could not reasonably be expected to have read the paper before commenting on it, not least of which because the link to it is dead (but also because reading a research paper before making a casual remark about a single graph in an article with many, wherein it is not remotely related to the main thesis of the article, is not a particularly reasonable expectation).
The quote is from the page in OP, its not necessary to circumvent the dead link to see the remark (not that finding the original article is very hard in any case).
Yes, I understand that the quote is from the article in the OP. The point is the quote is about people commenting on the paper. A comment on the article is not a comment on the paper, and it's entirely reasonable to leave a comment on the article without reading an auxiliary paper that is both not directly available and not relevant to the larger point of the article.
> Runners who were on pace to finish between 3:45 and 3:55 or between 4:05 and 4:15 ran approximately 6%–8% slower in the last 2.195 kilometers. In contrast, runners who were on pace to finish close to the 4-hour mark (3:55–4:02) ran only 4%–6% slower in the last 2.195 kilometers. [...] Panel (b) of Figure 7 zooms out to show this normalized pace for runners across a wider range of 40-kilometer-pace times. The same qualitative pattern around 4 hours is observed at other round numbers in the distribution. Thus, there is clear evidence that runners finish the last 2.195 kilometers relatively faster when they are close to a round number than when they are farther away.
I sincerely believe that a better idea in my opinion is to eliminate all means testing in government benefits altogether. No, higher taxes is not means testing. They are not the same thing at all. Removing means testing means we want the service to be used by all, which means the service should be at least usable by people who have the option to use something else. On a similar note, medicare clawback rules are cruel and unusual punishment. To punish the stupid and ill prepared while rewarding those who did tax planning five years or more before retirement is cruel.
There is a hilarious one in the Indian direct tax (Income tax) code right now:
If your (taxable) income is less than 7L (inr), no tax is assessed.
If your income is greater than 7L, but upto 12L, tax is assessed, but a tax _rebate_ (a discount on payable tax) is granted for the exact amount of tax owed. This means net tax paid till 12L is zero.
But if your income is 12L + 1, this rebate no longer applies, so the entire tax assessed (on the 7L+ income) is now due. This means that a person who earns 12L+1 INR makes ~70K inr less, post-tax, than the person who earns 12L INR.
After complaints, to correct this, they added another layer: If your taxable income is >12L but less than 12L+~70K (the tax thats due), then the maximum tax due is just the amount your income exceeds 12L bye. So you stagnate at the same post-tax salary from 12L to 12L+70K.
Gosh that Polish language scores graph is good. Clean bell curve, slight skew, classic little bump at 100 because the measure is truncated, and then this giant mess at 30 that looks more like something out of a heart rate monitor than a normal distribution.
> this giant mess at 30 that looks more like something out of a heart rate monitor than a normal distribution.
Right, it's as if someone shot the heart with a defibrillator - because that's a good analogy of what is actually happening, given the "arbitrary negative effects for people" failing the exam are basically determining the course of one's future (or at least feel like this to the people and their parents at the time).
In particular, having passed "matura" is the lowest level kind of threshold used (both formally and informally) by employers considering job applicants; much like some higher-skill jobs still require a specific university degree, not having "matura" will cut you out of a large chunk of low-skill and entry-level jobs.
Couple failed "matura" with any kind of criminal record - and the two, I imagine, are probably correlated - and you can be screwed for life, economically speaking. It is a big deal, and teachers know it.
IDK what the current system is, but for my cohort, it was possible to repeat the exam in subsequent years - but like with any kind of degree, chances of someone doing that drop sharply with temporal distance; at some point, people just find a path for their lives and are more interested in present-day challenges than going back to finish some degree.
> A simple fix for the problems mentioned above would be to have slow phase-outs instead of sharp thresholds.
How about the even simpler fix of not having phase-outs? This generally works: the cost of a subsidy generally does not increase as the recipient pays more taxes, so the higher total tax paid by a higher-income person will easily pay for the cost of a subsidy. And giving higher-income/wealthier people the same subsidies as poor people may help them appreciate the ways in which the subsidies are helpful and the ways in which they suck, which can help the whole system improve.
Beautiful! I noticed this on the chess ratings distribution on Lichess: players like to cross a nice multiple of 100 and make an extra effort to avoid dropping below if they can: https://imgur.com/a/Db7fQdX
Indian laws have the same too. For taxation 'surcharge' applied to high income groups, a patchwork called "marginal relief" fixes it [1], however, leaves ranges in income where 100% of incremental income goes away in taxes.
Another place where thresholding is misplaced is minors (below the age of 18 years) being subject to different laws (this is for India at least), e.g., seriously different punishments for crimes like homicide or rape. I have heard that crime groups then purposefully involve minors, push the blames on them if caught, and effectively get away for much less.
> Another place where thresholding is misplaced is minors (below the age of 18 years) being subject to different laws (this is for India at least), e.g., seriously different punishments for crimes like homicide or rape. I have heard that crime groups then purposefully involve minors, push the blames on them if caught, and effectively get away for much less.
This also used to be common in the US, but nowadays for (violent?) crimes minors can be charged as adults, so this kind of thing is only done for drugs and organized stealing rather than murders and violent crimes.
I think it more or less still is. (Memorialized in The Offspring's Come Out and Play: "If you're under 18, you won't be doing any time.")
> but nowadays for (violent?) crimes minors can be charged as adults, so this kind of thing is only done for drugs and organized stealing rather than murders and violent crimes.
Minors are very occasionally charged as adults, but still mostly get lesser sentences for all kinds of crimes. Gangs prefer to recruit minors, probably in part because minors are stupid/immature enough to join gang life.
I am not sure. What I have heard is that the minor pleads guilty for the entire crime (possibly in exchange for some money behind the scenes).
The broader point is that being a couple days or months younger or older than eighteen years should not result in a drastically different outcomes to begin with.
> The broader point is that being a couple days or months younger or older than eighteen years should not result in a drastically different outcomes to begin with.
The problem is that anytime you create a division there are always some cases just on the border, and if the outcome difference is sized to the difference between the centers of the classes the jump from on size of the border to another gets large.
But on the other hand if the jump isn't large you end up saying that 13 year olds and 30 year olds should be treated the same, which is clearly just worse.
You can try smoothing out the jump at the edges by adding more cuts and doing less changes from one step to the next, but I don't think you gain much in terms of recruitment of youths compared to just pushing the breakpoint fore lenient sentences to larger ages than 18.
I was on a jury for a case like that. The charges included committing the other crimes in the presence of a minor. The minor in reference was a few days away from his 18th birthday, clearly acting on his own volition, and my understanding is the minor bad already been given essentially a slap on the wrist and his record cleared when he turned 18. The judge explained after the trial was over that his sentencing has significant flexibility to take those kinds of factors into account.
Jim Roskind had a really nice talk [https://www.youtube.com/watch?v=_uaaCiyJCFA] at AWS Reinvent 2022 which highlighted the fence post problem. Basically, AWS was targeting P50 and P90 latencies in many services, and sure enough there were big clumps of latencies right under those target fence posts as engineers were basically gaming the metrics and nerfing the >P90 latencies to make it happen.
I don't think that counts as gaming the metric. It's an example of a metric that even when targeted in good faith results in consequences management didn't intend.
Good faith starts with understanding that "90" in P90 is an arbitrary number, and that the exact sampling threshold is not important to why anyone cares about that metric.
The more interesting part of the talk is how using what he calls "fenceposts" naturally results in bunching up against the fenceposts because even small regressions that cross the fenceposts are blocked but regressions of any size that don't cross a fencepost go unnoticed.
It's just a plain bad metric. Useful due to brevity though.
A decent metric for testing against might consist of a target mean, the sigma specifying a gaussian about that mean, a minimum sample size, and a hard upper cutoff.
> There are a number of ways someone could do this; one commonly suggested scheme was to buy put options that were expected to expire worthless, allowing the buyer to (probably) take a loss.
> That means if an individual buying ACA insurance was going to earn $55k, they'd be better off reducing their income by $6440 and getting under the $48,560 subsidy ceiling than they are earning $55k.
I am told by a CPA: In the US tax system, one is limited to deduct $3,000 of net capital loss against their otherwise taxable income. These put options would generate a capital loss. Therefore, this strategy would not accomplish the goal stated in the article.
> someone who was desperately trying to lose money before the end of the year
Isn't donating to charity a lot easier than gambling on options? It's both legal and extremely easy to donate 5k to a tax-deductible charity of your choice...
I also appreciate discontinuities and while I won't comment on the data in the paper itself without cross-referencing, I will say that a couple of these examples hold true for me from applied observation over the years. When I was old enough to start caring about insurance for health and property, or became a parent and had to begin forecasting costs for college and what loans really represented, I began looking at observable data much differently. Working in the software industry, you begin to see the complicated systems at work at the C-Level, and the seemingly odd relationships with unrelated organizations start to become clear. Being an educated voter, a discerning consumer of products, and turning a critical eye on world news all require the ability to see processes, their patterns and the discontinuities within them. While there may not always be a useful explanation behind all of them, seeing them in the first place is essential to navigating so-called reality successfully.
The reddit explanation in the post addresses your question I believe. If someone is at a 28 or 29 a few "charity" points can be found in subjectively-graded tests.
I think that really just reflects the fact that on subjectively graded tests the score really doesn’t have that many significant figures of accuracy. That a regrading can find 3 to 5 points by being more generous - or presumably take 3 to 5 off by being harsher - says that really you could save a lot of effort by treating the final grades as bucketing into 10 point bands and treating 25-35 as the actual cutoff.
Unfortunately this is the final exam from pre-university education. You can think of it as one's first educational degree, a step below bachelor's. It's expected - particularly by employers - that everyone should achieve it.
it depends what the purpose of grading is. if the people is to purely assess, grading question by question is better. if the purpose is to provide feedback, knowing how the student did in general can substantially help with identifying what to emphasize
I think because the gradient is simply too confusing for laypeople to understand.
Even for a simple system like US social security that has a gradient. For every $2 you make over the limit, you lose $1 in benefits. I've heard countless times misconceptions of people thinking they'd be losing money (as in literally having less money net) by working.
Assuming that normies cannot understand gradients or feedback loops is IMO the source of a large amount of "tactical-level" problems in society, particularly around healthcare, taxation, and politics.
The better alternative would be to assume that normies are, in fact, capable of understanding these tools, and in the process forcing them to understand them by setting an expectation. I mean, this is what people should be told as answer when they ask, "what will learning math be useful for me in my life?". No, multiplying polynomials will not be useful to 99% of the people. But having a feel for basic linear algebra and feedback loops would be, because they describe the behavior of the simple and interesting control systems in any and all areas of life - which includes the Social Security examples of yours, too.
The worst thing is, letting majority of people off the hook here doesn't just impact them individually; in democracies, it prevents systems require certain level of understanding from being created in the first place. As shown in plenty of examples brought up in the discussion thread here.
So I guess one answer to "how is any of that math going to be useful for me in the future?" is, "if you all learn it, it will allow you to stop keeping yourselves in poverty in stupid and entirely unintentional ways".
Given that normies don't geneally understand tax brackets, I can understand not wanting to do more complex stuff either.
I don't know how many people I've heard express that they didn't want to move into the next tax bracket because they didn't want to pay that rate on their whole income, but it's a lot.
A very small minority are worried about hitting a ceiling that removes eligibility for important services, and some are worried about the weird phaseouts where marginal rates are very high, but most just don't understand.
Maybe that means having a more complicated system is fine because it's already incomprehensible to most, but I dunno.
Tax brackets are very easy to explain, it doesn't even require math beyond basic arithmetics. You can draw a bunch of lines on paper and show how different slices are taken from each.
The problem is that the intuitive understanding doesn't line up with that, so that's what people have by default - and then we don't teach how it actually works as a basic skill! Sure, there's no shortage of explainers, but you need to be actually looking for them in the first place.
And the best way to teach that in school? Give students a small allowance that grows with each year, but they have to "pay taxes" on it to continue receiving it, complete with different brackets etc. I bet you that all the "normie" kids will figure it out in no time.
FWIW, I didn't understand tax brackets when I was at university because nobody had bothered to explain them. I absolutely started off with the common misconception that on the margin going from e.g. the 20% bracket to the 40% bracket would double my tax bill.
The UK didn't have civics classes in school; I think this would have been much more useful than my mandatory RE lessons with an exam centred on one of the gospels.
A single benefit usually has an appropriate incentive structure, but a lot of people get multiple benefits -- even from different levels of government (local, state, federal) -- and adding up phase-outs in different systems can result in marginal phase-outs rates above 100%. It's hard to avoid that entirely given that we want to have a lot of transfers to the bottom of the income distribution while phasing those out by roughly the median. It would be easier to avoid phase-outs above (say) 80% of marginal income is we only had federal and state aid as predictable money transfers, but for various reasons we provide a lot of transfers in-kind or with limited authorized uses. Those limitations aren't necessarily wrong, but they do mean that transfers aren't fungible, so there's an incentive to provide transfers for other "good" uses, and that diversity is what makes it hard to bound the marginal phase-outs for everyone.
This does happen in Finnish tax system. Your tax rate (percent with one decimal) is calculated based on your annual gross income. Rates are supposed to be calculated smoothly, and they are certainly calculated for each individual separately.
In reality they are step functions. It is surprisingly common to have people refuse promotions because if would put them above an income tax threshold, bump up their rate, and end up with less money after taxes in the end.
The UK tax system is far from fair but at least it has clear brackets: income above threshold X is taxed at rate Y.
Because that is a marginal system, (and unless they've messed up the calculations, which they haven't in this case) you should never end up with less from earning more. Can you give an example of two income amounts where the lower income ends up with more money after-taxes than the higher income?
Or is it the additional municipal, church, or health levies mentioned on that page which have the discontinuities?
I've encountered people talking about turning down a promotion to not "get above a tax bracket". They have invariably been wrong and just not understood the basic math, or done any research.
Lacking those basic skills might be a reason not to promote someone though, so perhaps it all works out for the best
This does NOT happen with Finnish tax rate. Your marginal tax rate can jump but that only for the euros after that bump. Your total tax rate goes up in more slowly. You’re always better off with the extra euro of income.
The UK system doesn't either though. The rate for 100-125k is higher than for 125k+ due to the phasing out of the personal allowance. It gets worse if you have kids and can even result in a >100% marginal rate.
I've heard the same thing -- if they take a job they will lose money. What they really mean is that if they take a job (trade time for money), they will lose some of the pension they have already earned. This is a real economic loss (even if they might have a few more bucks at the end of the week) to say nothing of their lost time.
I have a different take. I think the state wants the system to be this rigid because, paradoxically, that makes it more flexible for them.
For example, they can keep certain tax exemption thresholds low while overall incomes rise. That means more people gradually fall into higher tax brackets or lose exemptions, even though the money they earn is worth less over time.
The rigidity itself is not a bug but a feature. It is a sneaky way to raise the taxes without openly raising taxes. But it's just a theory.
I agree, but almost everyone today can use a computer or smartphone. They can type in their income, and the computer can calculate it, providing them an average number of what their percentage of actual taxable income was- I think Turbo Tax and other software might do something like this.
They don't have to understand how it works to do their own taxes.
They can, but what would make them think their initial misunderstanding is even wrong? While some might think "surely the system isn't like that..." many would default to "of course they want to steal my money" etc.
Not to mention the amount of misinformation that can get passed around
I get that, but let me give you a typical example. Your tax software as a box where it asks you to enter your income. It automatically estimates how much your tax refund as soon as you type in one field, even though it is just the first page. It updates automatically, and changes often whenever you add more deductible things (and non-deductible things).
If they are curious, they can type in three different incomes, even if they aren't planning on submitting the tax form with the wrong amounts. So if their income is $59,000 and their tax bracket is say, 15% for that, instead of say, 24% for 60,000-the next end of that bracket- e.g. $75k or $95k) they might get a much smaller tax return if they type $60,000. But then if they earned a little more, like $61,000, they could type in that and see what their return is, along with the rate.
If it is a gradient with smooth progression for each dollar increase, then they can immediately see how much more they are paying in taxes from 15% to 20%. But in reality, in a smooth/gradient-based progressive tax increase, those amounts would be centered in the middle of those brackets (like 15% would right in the the center between $35,000 and $59,999, or $48,500, and the $59,000 income would be closer to 19.9% and the $61,000 income might be $20.1%.
It wouldn't be considered fraud if they are just determining what they would pay if they earned a little bit more. In fact, if they just discovered an invoice check they received from self-employment, and forgot to report it, they might have to type in different amounts, if they didn't report the right amount the first time. Everyone would pay the same tax rates at each income level (with $1000 intervals, or even $1 intervals.
The system could still be transparent because it can show a chart of the tax rates for each $1,000 increase, even if the tax brackets only provides an estimate for each bracket. I do agree some people can't read charts very well, but if they included 100 thin bars for taxable rates under $100,000, people would be able to see the estimated percentage that they would be paying. It might be 15%, 15.5%, 16%, 16.5%, or 17%, and so on for $48,500. $49,000, $49500, $50,000, and so on, respectively.
The idea with this is that it would disincentivize people from misreporting because if they report earning $500 less, they aren't saving much more money (not to mention illegally)- their tax rate might be 0.1% less, and on a $49500 income that might only be $250 less in taxes over reporting $50000. The risk of getting caught is not to going worth it if this method is in place. (Actually not in any case, but at least they can see it's not going to be a huge decrease in taxable income by reporting a little less, so the only way they can attempt to pay less is by reporting a lot less, and the penalties for that would be larger)
It's especially hard when you have a combination of many government programs at all levels (federal, state, local, many different kinds of taxes, many different kinds of welfare). Even if every individual program uses a gradient, it's still possible that summing all the programs together leads to a >100% effective marginal tax rate.
Why do we have legislated cliffs instead of gradients? Because approximately nobody understands lerp. And linear interpolation is the simplest (nontrivial) gradient scheme. Consequently, we get cliffs or, if we're lucky, lookup tables that approximate gradients with stair-step successions of small cliffs.
This isn’t a very satisfying explanation to me. I mean, the idea of a cliff causing bad incentives is extremely widely understood (to the point where people under progressive taxation policies avoid taking overtime because they misunderstand the policy and believe the cliff to be there).
I wonder why we don’t define these things in terms of logistic functions or something. That’s high school or at least first semester college math, right?
I was surprised to see the "Big Beautiful Bill" tax incentive for financing a new US-made car had a gradual phase out above $100,000, although it's implemented as a step function instead of a linear one.
It's very well established internationally that income taxes are defined by gradients. I have no idea why politicians want to reinvent them so often in other kinds of taxes.
I hear this explanation a lot but I think it's a convenient fiction. Lawmakers, at least in the US, don't seem to write their own legal texts very often. Sometimes they don't even read them [1]! Congress has no problem producing monstrously complicated laws, in any case.
I think it's likely that politicians and their funding sources have found ways to profit off of these discontinuities. The infamous Medicare "donut hole" [2] was arguably a "benefit discontinuity" of the sort mentioned by the author and pharmaceutical companies profited off of it (more than a hypothetical Medicare structure without a donut hole, not relative to the spending cap that replaced it—which profits them even more).
One way to reduce income may be to buy a bunch of stocks with high variance - for any that is in the red during the year, sell and rebuy them. Any that has a gain, leave them be.
[ EDIT: Since the undated article is actually from Feb 2020, predating the expansion of federal health insurance subsidies, I withdraw everything I said below. ]
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The opening story is fabricated and/or bullshit.
Last year (2025) there was no limit on income for health insurance subsidies. That ended for this year, but last year there would have been no reason for anyone who knew what they were doing to try to lose money to drop their income (especially in the cited range of $48-55k/year).
That is the case this year, in most states (thankfully not where I live), but that's not what TFA is talking about.
Suspicious? It certainly makes me skeptical that the author has got the details of the other examples correct.
It's always a good idea, and it's rather an obvious point - not to mention, something schools try to drill into kids since they're single-digit years old.
At this point I treat lack of dating articles as being done intentionally, and as evidence of malicious intent - i.e. that the entity publishing it has a reason they don't want readers to know the creation date, and there's really no reason other than trying to get away with lies or otherwise screw the reader over.
(Above is a heuristic I apply by default; this post is an example of a case-by-case exception, because of the type of site, my impression of articles read in the past, and of the author themselves.)
This might vary state-by-state, CA MediCal for instance did limit the subsidies based on income last year. I don't think it was an all-or-nothing cutoff, but I do think there were points around the 50k mark where the delta between your subsidy and the one for the lower bracket was higher than the loss you'd take by a few thousand dollars.
The subsidies were from the federal government, managed as part of your federal income tax return. They had nothing to do with additional state subsidies.
The basic story was that until the end of 2025, nobody in the USA had any reason to pay more than (roughly) 8.3% of their AGI for health insurance.
I cracked up when I got to the marathon example. When I ran a half marathon I realized about 80% of the way through that I was on track to finish under 2:30:00 and pushed myself to make it happen. I should have guessed that sort of behavior would show up in the statistics!
After this year's Paris marathon, I ran the same per-minute graphs, and they match perfectly the "overall study" graphs with more than 9 million finishes in the article. I also added graphing by age category and gender. I don't want to deduce too much but I think it showed that young men are the most "competitive" (what I mean by that is targeting a specific time) since there are the clearest "goal time" peaks in the graphs.
IIRC There is also a discontinuity when it comes to the histogram of ages of marathon runners, because runners are binned into age groups and there is a more runners at the youngest ages of each group, I guess because it’s younger runners in each group that are more likely to run if they feel more likely to place well in their group
I think there's also the opposite – there are a number of places in large marathons allocated for "good for age" entrants, which I would guess are more likely to be taken up by people at the younger end of those buckets.
The UK tax system also has a bunch of unfortunate cliffs, and tapers that create >60% marginal tax rates and worse. There's a calculator here that illustrates it well https://tax-cliffs.britishprogress.org/calculator
The childcare cliff edge is probably the worst, but the personal allowance taper isn't ideal either as it's compressed over a relatively short income range
And of course all the thresholds remain frozen, creating plenty of fiscal drag on top.
The Economist has this crazy graph of UK VAT thresholds and how they are causing companies to stay small, to keep their turnover under the threshold which avoids a lot of paperwork. It's the most infuriating thing, we are literally throttling our companies, the engines of growth, because of some accounting stupidity.
Graph: https://www.economist.com/cdn-cgi/image/width=600,quality=10... from this article: https://www.economist.com/britain/2024/04/22/how-to-fix-brit...
Hey, at least they can exist. Many businesses in the US can no longer exist at small scale, because the barriers to entry are too high because the entrenched interests achieved total regulatory capture.
Same in slovenia for eg. kindergarten prices, 1eur over the line, 90eur more per month in fees.
I remember when my grandfather died, and my grandmother got part of his pension.
It put her pension above a max threshold for old age / low pension (cheaper phone bill, no tv taxes, ...) by, ... hold your hats... 2 Euro.
Father called the service ... "sorry, we can not do anything about that".
She lost over 100 Euro in benefits because technically... she was not low pension anymore but was financially WAY more poor then when my grandfather was alive.
So when my grandfather was alive, their combined pension was higher, they qualitied as having a low pension. When only *part* of my grandfather pension got combined into my grandmothers. Well, now your not poor...
I do not remember the exact pensions but it was a massive haircut, with the insulting on top, losing those benefits. The lost benefits made it from threading water, to drowning.
Without my dad financial help, she will have lost the house. Ironically, if she did not get my grandfathers pension part, she will have qualitied for more social assistance, and will have been better off.
Sometimes, those lines really screw people over.
> Sometimes, those lines really screw people over.
Another angle is how they affect the living. Plenty of young adults delay moving out from their parents, or starting a career, because changes in combined household income would cross one of those magic threshold, and ending up screwing the parents and younger siblings for life.
Such delaying may not be as immediately critical to the person, but it alters the course of their lives in significant ways.
What's most infuriating it's only the job income, work, that's taxed like this.
Passive income of any sort? Bribes from the lobbying groups? Absolutely not.
This is how you tell genuinely left-wing politicians from "free market progressives": look at whom they are trying to tax.
Exactly.
> The UK tax system also has a bunch of unfortunate cliffs, and tapers that create >60% marginal tax rates and worse.
Oh I don't doubt it. In my native country of Belgium the tax rate can go to 69% and more: 50% income tax (used to be as high as 52.5% and maybe even 55%, don't remember) and then, on top of that, an addition 19% social wellfare tax. That's not mentioning stuff like, if you manage to buy stocks with what you have left, 30% state tax (+ crazy bank fees) on any dividends you receive from those stocks.
And then the brand new (1st jan 2026) 10% tax on any capital gain, once again: in addition to all the already existing taxes.
Add some arbitrary decisions on top of that, where if you happen to maybe have bought/sell a bit too many stocks or if you happen to have managed to put a too big part of your wealth into stocks, they may consider you're a "professional investor" and then your capital gains are taxed at your income rate.
Some are going to say: "Oh but create a company, get back the VAT, do this/do that".
I don't want to basically become a fiscal lawyer just to not be assraped by ever more taxes.
Instead of trying to "optimize", I just voted --before they'll introduce an exit tax (I'm sure that's coming)-- with my feet, my money and the future wealth I was going to create and left the country.
For what was I getting in return? According to a flemish university an estimated 55 000+ undocumented illegal migrants are now living in the capital city of Belgium, Brussels, alone. That's 5% of the population of the city that are undocumented migrants. They don't live in proper conditions. They're a drag on the economy. They're not happy, they're not thriving. Locals aren't happy either.
I didn't vote for this.
To me once politicians go too far, that's what they deserve: the middle class living the city in drones and being replaced by poor people. As I tell of my friends and family who also left the city: I just went a bit further and left the country altogether.
Following the example of my friend and ex-neighbor who did so decades ago: he left for SoCal (btw I didn't go to the US) and created two startups. Second one was a big hit. Belgium (and the EU) saw exactly zero return of his incredible drive and talent.
Another recent statistics: 40% of the people of Brussels are living close to the poverty line.
But at least socialists got the utopia they dreamed of: they're sure to never lose an election again. The middle class is dying in Brussels. The poor won't vote against socialism.
I just packed and left and advice anyone who can flee a sinking ship to do so. Many countries in Europe have decided to commit suicide and to turn themselves into slums.
I think this is just the beginning for may EU cities/countries: there's much more pain ahead --and although AI is going to be used as a scapegoat-- this has absolutely nothing to do with AI and all to do with political decisions.
It pains to see how my native city has been dragged into poverty... But good riddance.
As for the next gen: from day one I raised my kid in english. At 11 y/o she's totally fluent and our thinking was simple: english is at least understood mostly everywhere in the world. She'll have to move because I fear it's not just my native city that is toast.
It is highly likely the 50k undocumented immigrants aren't "a drag on the economy" but "people increasing margins of business by being exploited in almost slave-like fashion, and doing the work you don't want your kids to do".
The marathon one has a simple - and fun - explanation: it’s great to run with people!
Many (most?) marathons have pace runners who run the course, hewing to each 30 minute and 15 minute finish. So there’ll be a 3:30:00 pacer, a 3:45:00 pacer and so on. Your local pacer might even have their pace on their shirt and may even have a flag so they’re super hard to miss as you’re running.
One friend of mine runs with a speaker, to play music and keep everyone’s spirits up.
By the end of the race they’d acquired a small army of marathoners! You could see their smiles for miles, and when they all finished together they had a huge party in the recovery area with the speaker :)
I’m surprised the marathon time discontinuity isn’t bigger :)
> One of the most common comments I've seen online about this graph and/or this paper is that this is due to pace runners provided by the marathon. Section 4.4 of the paper gives multiple explanations for why this cannot be the case, once again indicating that people tend to comment without reading the paper.
Note that this is about comments on the paper itself. Somebody reading this article could not reasonably be expected to have read the paper before commenting on it, not least of which because the link to it is dead (but also because reading a research paper before making a casual remark about a single graph in an article with many, wherein it is not remotely related to the main thesis of the article, is not a particularly reasonable expectation).
The quote is from the page in OP, its not necessary to circumvent the dead link to see the remark (not that finding the original article is very hard in any case).
Yes, I understand that the quote is from the article in the OP. The point is the quote is about people commenting on the paper. A comment on the article is not a comment on the paper, and it's entirely reasonable to leave a comment on the article without reading an auxiliary paper that is both not directly available and not relevant to the larger point of the article.
> https://faculty.chicagobooth.edu/devin.pope/research/pdf/Web...
404 Not Found
Had you tried to read the paper, you’d have noticed the dead link.
How’s the wind up on your high horse?
> Runners who were on pace to finish between 3:45 and 3:55 or between 4:05 and 4:15 ran approximately 6%–8% slower in the last 2.195 kilometers. In contrast, runners who were on pace to finish close to the 4-hour mark (3:55–4:02) ran only 4%–6% slower in the last 2.195 kilometers. [...] Panel (b) of Figure 7 zooms out to show this normalized pace for runners across a wider range of 40-kilometer-pace times. The same qualitative pattern around 4 hours is observed at other round numbers in the distribution. Thus, there is clear evidence that runners finish the last 2.195 kilometers relatively faster when they are close to a round number than when they are farther away.
From "Reference-Dependent Preferences: Evidence from Marathon Runners" https://devingpope.com/assets/files/Website_Marathons.pdf
Not sure why you think not making any effort to find the article beyond clicking a link is a flex, but ok.
I'm from Boston and I'd never heard of this, but reading a bit it sounds like Boston is unusual in not having pace runners.
Given how hilly the course is, this makes sense: running an even pace the whole distance is not a good plan.
I sincerely believe that a better idea in my opinion is to eliminate all means testing in government benefits altogether. No, higher taxes is not means testing. They are not the same thing at all. Removing means testing means we want the service to be used by all, which means the service should be at least usable by people who have the option to use something else. On a similar note, medicare clawback rules are cruel and unusual punishment. To punish the stupid and ill prepared while rewarding those who did tax planning five years or more before retirement is cruel.
Extra taxes on the stupid should be illegal.
Yes, exactly. Means-testing just makes a service more expensive by inserting needless middlemen.
If we are going to tax things that we want less of; would not taxing stupidity seem like good public policy?
There is a hilarious one in the Indian direct tax (Income tax) code right now:
If your (taxable) income is less than 7L (inr), no tax is assessed. If your income is greater than 7L, but upto 12L, tax is assessed, but a tax _rebate_ (a discount on payable tax) is granted for the exact amount of tax owed. This means net tax paid till 12L is zero. But if your income is 12L + 1, this rebate no longer applies, so the entire tax assessed (on the 7L+ income) is now due. This means that a person who earns 12L+1 INR makes ~70K inr less, post-tax, than the person who earns 12L INR.
After complaints, to correct this, they added another layer: If your taxable income is >12L but less than 12L+~70K (the tax thats due), then the maximum tax due is just the amount your income exceeds 12L bye. So you stagnate at the same post-tax salary from 12L to 12L+70K.
How much relative to 70K is this 12L? Sorry, I am not Indian.
12L is 1.2 million
Gosh that Polish language scores graph is good. Clean bell curve, slight skew, classic little bump at 100 because the measure is truncated, and then this giant mess at 30 that looks more like something out of a heart rate monitor than a normal distribution.
> this giant mess at 30 that looks more like something out of a heart rate monitor than a normal distribution.
Right, it's as if someone shot the heart with a defibrillator - because that's a good analogy of what is actually happening, given the "arbitrary negative effects for people" failing the exam are basically determining the course of one's future (or at least feel like this to the people and their parents at the time).
In particular, having passed "matura" is the lowest level kind of threshold used (both formally and informally) by employers considering job applicants; much like some higher-skill jobs still require a specific university degree, not having "matura" will cut you out of a large chunk of low-skill and entry-level jobs.
Couple failed "matura" with any kind of criminal record - and the two, I imagine, are probably correlated - and you can be screwed for life, economically speaking. It is a big deal, and teachers know it.
IDK what the current system is, but for my cohort, it was possible to repeat the exam in subsequent years - but like with any kind of degree, chances of someone doing that drop sharply with temporal distance; at some point, people just find a path for their lives and are more interested in present-day challenges than going back to finish some degree.
> A simple fix for the problems mentioned above would be to have slow phase-outs instead of sharp thresholds.
How about the even simpler fix of not having phase-outs? This generally works: the cost of a subsidy generally does not increase as the recipient pays more taxes, so the higher total tax paid by a higher-income person will easily pay for the cost of a subsidy. And giving higher-income/wealthier people the same subsidies as poor people may help them appreciate the ways in which the subsidies are helpful and the ways in which they suck, which can help the whole system improve.
That would require a tax hike to fund though, which is not very palatable.
politics, can’t have the government give a dime to the rich directly.
Beautiful! I noticed this on the chess ratings distribution on Lichess: players like to cross a nice multiple of 100 and make an extra effort to avoid dropping below if they can: https://imgur.com/a/Db7fQdX
Indian laws have the same too. For taxation 'surcharge' applied to high income groups, a patchwork called "marginal relief" fixes it [1], however, leaves ranges in income where 100% of incremental income goes away in taxes.
[1] https://cleartax.in/s/marginal-relief-surcharge
Another place where thresholding is misplaced is minors (below the age of 18 years) being subject to different laws (this is for India at least), e.g., seriously different punishments for crimes like homicide or rape. I have heard that crime groups then purposefully involve minors, push the blames on them if caught, and effectively get away for much less.
> Another place where thresholding is misplaced is minors (below the age of 18 years) being subject to different laws (this is for India at least), e.g., seriously different punishments for crimes like homicide or rape. I have heard that crime groups then purposefully involve minors, push the blames on them if caught, and effectively get away for much less.
This also used to be common in the US, but nowadays for (violent?) crimes minors can be charged as adults, so this kind of thing is only done for drugs and organized stealing rather than murders and violent crimes.
> This also used to be common in the US
I think it more or less still is. (Memorialized in The Offspring's Come Out and Play: "If you're under 18, you won't be doing any time.")
> but nowadays for (violent?) crimes minors can be charged as adults, so this kind of thing is only done for drugs and organized stealing rather than murders and violent crimes.
Minors are very occasionally charged as adults, but still mostly get lesser sentences for all kinds of crimes. Gangs prefer to recruit minors, probably in part because minors are stupid/immature enough to join gang life.
Why cant they just scan the minor's texts/calls, send them to therapy and rehabillitation and RICO the fuck out of their "superiors"?
I am not sure. What I have heard is that the minor pleads guilty for the entire crime (possibly in exchange for some money behind the scenes).
The broader point is that being a couple days or months younger or older than eighteen years should not result in a drastically different outcomes to begin with.
> The broader point is that being a couple days or months younger or older than eighteen years should not result in a drastically different outcomes to begin with.
The problem is that anytime you create a division there are always some cases just on the border, and if the outcome difference is sized to the difference between the centers of the classes the jump from on size of the border to another gets large.
But on the other hand if the jump isn't large you end up saying that 13 year olds and 30 year olds should be treated the same, which is clearly just worse.
You can try smoothing out the jump at the edges by adding more cuts and doing less changes from one step to the next, but I don't think you gain much in terms of recruitment of youths compared to just pushing the breakpoint fore lenient sentences to larger ages than 18.
I was on a jury for a case like that. The charges included committing the other crimes in the presence of a minor. The minor in reference was a few days away from his 18th birthday, clearly acting on his own volition, and my understanding is the minor bad already been given essentially a slap on the wrist and his record cleared when he turned 18. The judge explained after the trial was over that his sentencing has significant flexibility to take those kinds of factors into account.
Generally organised crime groups has some skill at avoiding being provably responsible for crimes, it's a precondition for remaining active.
Jim Roskind had a really nice talk [https://www.youtube.com/watch?v=_uaaCiyJCFA] at AWS Reinvent 2022 which highlighted the fence post problem. Basically, AWS was targeting P50 and P90 latencies in many services, and sure enough there were big clumps of latencies right under those target fence posts as engineers were basically gaming the metrics and nerfing the >P90 latencies to make it happen.
I don't think that counts as gaming the metric. It's an example of a metric that even when targeted in good faith results in consequences management didn't intend.
Good faith starts with understanding that "90" in P90 is an arbitrary number, and that the exact sampling threshold is not important to why anyone cares about that metric.
The 99 in his TM99 is also an arbitrary number.
The more interesting part of the talk is how using what he calls "fenceposts" naturally results in bunching up against the fenceposts because even small regressions that cross the fenceposts are blocked but regressions of any size that don't cross a fencepost go unnoticed.
It's just a plain bad metric. Useful due to brevity though.
A decent metric for testing against might consist of a target mean, the sigma specifying a gaussian about that mean, a minimum sample size, and a hard upper cutoff.
[dead]
> There are a number of ways someone could do this; one commonly suggested scheme was to buy put options that were expected to expire worthless, allowing the buyer to (probably) take a loss.
> That means if an individual buying ACA insurance was going to earn $55k, they'd be better off reducing their income by $6440 and getting under the $48,560 subsidy ceiling than they are earning $55k.
I am told by a CPA: In the US tax system, one is limited to deduct $3,000 of net capital loss against their otherwise taxable income. These put options would generate a capital loss. Therefore, this strategy would not accomplish the goal stated in the article.
> someone who was desperately trying to lose money before the end of the year
Isn't donating to charity a lot easier than gambling on options? It's both legal and extremely easy to donate 5k to a tax-deductible charity of your choice...
I assume a part of the point is gambling - you either lose exactly as much as you planned, or you "win big" and still come out ahead.
donating would go against the American spirit of individualism where you only care about yourself. after all, who ever helped you for free, right?
Tax the bots for UBI.
I also appreciate discontinuities and while I won't comment on the data in the paper itself without cross-referencing, I will say that a couple of these examples hold true for me from applied observation over the years. When I was old enough to start caring about insurance for health and property, or became a parent and had to begin forecasting costs for college and what loans really represented, I began looking at observable data much differently. Working in the software industry, you begin to see the complicated systems at work at the C-Level, and the seemingly odd relationships with unrelated organizations start to become clear. Being an educated voter, a discerning consumer of products, and turning a critical eye on world news all require the ability to see processes, their patterns and the discontinuities within them. While there may not always be a useful explanation behind all of them, seeing them in the first place is essential to navigating so-called reality successfully.
Would test score problem be solved if teachers graded individual questions, not entire test?
The reddit explanation in the post addresses your question I believe. If someone is at a 28 or 29 a few "charity" points can be found in subjectively-graded tests.
I think that really just reflects the fact that on subjectively graded tests the score really doesn’t have that many significant figures of accuracy. That a regrading can find 3 to 5 points by being more generous - or presumably take 3 to 5 off by being harsher - says that really you could save a lot of effort by treating the final grades as bucketing into 10 point bands and treating 25-35 as the actual cutoff.
Isn't that equivalent to just setting the passing threshold to 25, with the same incentives?
… which is effectively what result this process seems to produce already.
If you grade individual questions, you don't know the total score.
The incentive to find the extra point would partially disappear.
Unfortunately this is the final exam from pre-university education. You can think of it as one's first educational degree, a step below bachelor's. It's expected - particularly by employers - that everyone should achieve it.
it depends what the purpose of grading is. if the people is to purely assess, grading question by question is better. if the purpose is to provide feedback, knowing how the student did in general can substantially help with identifying what to emphasize
I never understood why taxes or similiar absolute points aren't gradients instead.
I think because the gradient is simply too confusing for laypeople to understand.
Even for a simple system like US social security that has a gradient. For every $2 you make over the limit, you lose $1 in benefits. I've heard countless times misconceptions of people thinking they'd be losing money (as in literally having less money net) by working.
Assuming that normies cannot understand gradients or feedback loops is IMO the source of a large amount of "tactical-level" problems in society, particularly around healthcare, taxation, and politics.
The better alternative would be to assume that normies are, in fact, capable of understanding these tools, and in the process forcing them to understand them by setting an expectation. I mean, this is what people should be told as answer when they ask, "what will learning math be useful for me in my life?". No, multiplying polynomials will not be useful to 99% of the people. But having a feel for basic linear algebra and feedback loops would be, because they describe the behavior of the simple and interesting control systems in any and all areas of life - which includes the Social Security examples of yours, too.
The worst thing is, letting majority of people off the hook here doesn't just impact them individually; in democracies, it prevents systems require certain level of understanding from being created in the first place. As shown in plenty of examples brought up in the discussion thread here.
So I guess one answer to "how is any of that math going to be useful for me in the future?" is, "if you all learn it, it will allow you to stop keeping yourselves in poverty in stupid and entirely unintentional ways".
Given that normies don't geneally understand tax brackets, I can understand not wanting to do more complex stuff either.
I don't know how many people I've heard express that they didn't want to move into the next tax bracket because they didn't want to pay that rate on their whole income, but it's a lot.
A very small minority are worried about hitting a ceiling that removes eligibility for important services, and some are worried about the weird phaseouts where marginal rates are very high, but most just don't understand.
Maybe that means having a more complicated system is fine because it's already incomprehensible to most, but I dunno.
Tax brackets are very easy to explain, it doesn't even require math beyond basic arithmetics. You can draw a bunch of lines on paper and show how different slices are taken from each.
The problem is that the intuitive understanding doesn't line up with that, so that's what people have by default - and then we don't teach how it actually works as a basic skill! Sure, there's no shortage of explainers, but you need to be actually looking for them in the first place.
And the best way to teach that in school? Give students a small allowance that grows with each year, but they have to "pay taxes" on it to continue receiving it, complete with different brackets etc. I bet you that all the "normie" kids will figure it out in no time.
FWIW, I didn't understand tax brackets when I was at university because nobody had bothered to explain them. I absolutely started off with the common misconception that on the margin going from e.g. the 20% bracket to the 40% bracket would double my tax bill.
The UK didn't have civics classes in school; I think this would have been much more useful than my mandatory RE lessons with an exam centred on one of the gospels.
A single benefit usually has an appropriate incentive structure, but a lot of people get multiple benefits -- even from different levels of government (local, state, federal) -- and adding up phase-outs in different systems can result in marginal phase-outs rates above 100%. It's hard to avoid that entirely given that we want to have a lot of transfers to the bottom of the income distribution while phasing those out by roughly the median. It would be easier to avoid phase-outs above (say) 80% of marginal income is we only had federal and state aid as predictable money transfers, but for various reasons we provide a lot of transfers in-kind or with limited authorized uses. Those limitations aren't necessarily wrong, but they do mean that transfers aren't fungible, so there's an incentive to provide transfers for other "good" uses, and that diversity is what makes it hard to bound the marginal phase-outs for everyone.
This does happen in Finnish tax system. Your tax rate (percent with one decimal) is calculated based on your annual gross income. Rates are supposed to be calculated smoothly, and they are certainly calculated for each individual separately.
In reality they are step functions. It is surprisingly common to have people refuse promotions because if would put them above an income tax threshold, bump up their rate, and end up with less money after taxes in the end.
The UK tax system is far from fair but at least it has clear brackets: income above threshold X is taxed at rate Y.
Are you talking about this tax system? <https://nordisketax.net/pages/en-GB/taxation/?country=finlan...>
Because that is a marginal system, (and unless they've messed up the calculations, which they haven't in this case) you should never end up with less from earning more. Can you give an example of two income amounts where the lower income ends up with more money after-taxes than the higher income?
Or is it the additional municipal, church, or health levies mentioned on that page which have the discontinuities?
I've encountered people talking about turning down a promotion to not "get above a tax bracket". They have invariably been wrong and just not understood the basic math, or done any research.
Lacking those basic skills might be a reason not to promote someone though, so perhaps it all works out for the best
This does NOT happen with Finnish tax rate. Your marginal tax rate can jump but that only for the euros after that bump. Your total tax rate goes up in more slowly. You’re always better off with the extra euro of income.
See graph here: https://www.veronmaksajat.fi/tutkimus-ja-tilastot/tuloverot/...
The UK system doesn't either though. The rate for 100-125k is higher than for 125k+ due to the phasing out of the personal allowance. It gets worse if you have kids and can even result in a >100% marginal rate.
You might just be taking people too literally.
I've heard the same thing -- if they take a job they will lose money. What they really mean is that if they take a job (trade time for money), they will lose some of the pension they have already earned. This is a real economic loss (even if they might have a few more bucks at the end of the week) to say nothing of their lost time.
I have a different take. I think the state wants the system to be this rigid because, paradoxically, that makes it more flexible for them.
For example, they can keep certain tax exemption thresholds low while overall incomes rise. That means more people gradually fall into higher tax brackets or lose exemptions, even though the money they earn is worth less over time.
The rigidity itself is not a bug but a feature. It is a sneaky way to raise the taxes without openly raising taxes. But it's just a theory.
I agree, but almost everyone today can use a computer or smartphone. They can type in their income, and the computer can calculate it, providing them an average number of what their percentage of actual taxable income was- I think Turbo Tax and other software might do something like this.
They don't have to understand how it works to do their own taxes.
They may not trust it if they can't understand it.
They can, but what would make them think their initial misunderstanding is even wrong? While some might think "surely the system isn't like that..." many would default to "of course they want to steal my money" etc.
Not to mention the amount of misinformation that can get passed around
I get that, but let me give you a typical example. Your tax software as a box where it asks you to enter your income. It automatically estimates how much your tax refund as soon as you type in one field, even though it is just the first page. It updates automatically, and changes often whenever you add more deductible things (and non-deductible things).
If they are curious, they can type in three different incomes, even if they aren't planning on submitting the tax form with the wrong amounts. So if their income is $59,000 and their tax bracket is say, 15% for that, instead of say, 24% for 60,000-the next end of that bracket- e.g. $75k or $95k) they might get a much smaller tax return if they type $60,000. But then if they earned a little more, like $61,000, they could type in that and see what their return is, along with the rate.
If it is a gradient with smooth progression for each dollar increase, then they can immediately see how much more they are paying in taxes from 15% to 20%. But in reality, in a smooth/gradient-based progressive tax increase, those amounts would be centered in the middle of those brackets (like 15% would right in the the center between $35,000 and $59,999, or $48,500, and the $59,000 income would be closer to 19.9% and the $61,000 income might be $20.1%.
It wouldn't be considered fraud if they are just determining what they would pay if they earned a little bit more. In fact, if they just discovered an invoice check they received from self-employment, and forgot to report it, they might have to type in different amounts, if they didn't report the right amount the first time. Everyone would pay the same tax rates at each income level (with $1000 intervals, or even $1 intervals.
The system could still be transparent because it can show a chart of the tax rates for each $1,000 increase, even if the tax brackets only provides an estimate for each bracket. I do agree some people can't read charts very well, but if they included 100 thin bars for taxable rates under $100,000, people would be able to see the estimated percentage that they would be paying. It might be 15%, 15.5%, 16%, 16.5%, or 17%, and so on for $48,500. $49,000, $49500, $50,000, and so on, respectively.
The idea with this is that it would disincentivize people from misreporting because if they report earning $500 less, they aren't saving much more money (not to mention illegally)- their tax rate might be 0.1% less, and on a $49500 income that might only be $250 less in taxes over reporting $50000. The risk of getting caught is not to going worth it if this method is in place. (Actually not in any case, but at least they can see it's not going to be a huge decrease in taxable income by reporting a little less, so the only way they can attempt to pay less is by reporting a lot less, and the penalties for that would be larger)
It's especially hard when you have a combination of many government programs at all levels (federal, state, local, many different kinds of taxes, many different kinds of welfare). Even if every individual program uses a gradient, it's still possible that summing all the programs together leads to a >100% effective marginal tax rate.
Sometimes they are. The US Earned Income Tax Credit being probably the biggest example.
https://taxfoundation.org/research/all/federal/earned-income...
Why do we have legislated cliffs instead of gradients? Because approximately nobody understands lerp. And linear interpolation is the simplest (nontrivial) gradient scheme. Consequently, we get cliffs or, if we're lucky, lookup tables that approximate gradients with stair-step successions of small cliffs.
This isn’t a very satisfying explanation to me. I mean, the idea of a cliff causing bad incentives is extremely widely understood (to the point where people under progressive taxation policies avoid taking overtime because they misunderstand the policy and believe the cliff to be there).
I wonder why we don’t define these things in terms of logistic functions or something. That’s high school or at least first semester college math, right?
I was surprised to see the "Big Beautiful Bill" tax incentive for financing a new US-made car had a gradual phase out above $100,000, although it's implemented as a step function instead of a linear one.
It's very well established internationally that income taxes are defined by gradients. I have no idea why politicians want to reinvent them so often in other kinds of taxes.
Because that's harder to write the laws for
I hear this explanation a lot but I think it's a convenient fiction. Lawmakers, at least in the US, don't seem to write their own legal texts very often. Sometimes they don't even read them [1]! Congress has no problem producing monstrously complicated laws, in any case.
I think it's likely that politicians and their funding sources have found ways to profit off of these discontinuities. The infamous Medicare "donut hole" [2] was arguably a "benefit discontinuity" of the sort mentioned by the author and pharmaceutical companies profited off of it (more than a hypothetical Medicare structure without a donut hole, not relative to the spending cap that replaced it—which profits them even more).
[1] https://www.pennstatelawreview.org/penn-statim/dont-be-silly... (2013) ...which argues that this is a good thing!
[2] https://www.medicalnewstoday.com/articles/what-is-the-medica...
Not if you assume people could understand basic math, such ... oh, any continuously valued polynomial ....
Haven’t seen danluu stuff here for a while!
One way to reduce income may be to buy a bunch of stocks with high variance - for any that is in the red during the year, sell and rebuy them. Any that has a gain, leave them be.
The wash sale rule (in the US) makes it a lot harder to pull this off.
yeah, slow phase-outs seems ideal- I have often wondered about that with gross and and taxable income.
[ EDIT: Since the undated article is actually from Feb 2020, predating the expansion of federal health insurance subsidies, I withdraw everything I said below. ]
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The opening story is fabricated and/or bullshit.
Last year (2025) there was no limit on income for health insurance subsidies. That ended for this year, but last year there would have been no reason for anyone who knew what they were doing to try to lose money to drop their income (especially in the cited range of $48-55k/year).
That is the case this year, in most states (thankfully not where I live), but that's not what TFA is talking about.
Suspicious? It certainly makes me skeptical that the author has got the details of the other examples correct.
Goes to show that having a creation date placed somewhere near the top of an article or blog post is a good idea.
It's always a good idea, and it's rather an obvious point - not to mention, something schools try to drill into kids since they're single-digit years old.
At this point I treat lack of dating articles as being done intentionally, and as evidence of malicious intent - i.e. that the entity publishing it has a reason they don't want readers to know the creation date, and there's really no reason other than trying to get away with lies or otherwise screw the reader over.
(Above is a heuristic I apply by default; this post is an example of a case-by-case exception, because of the type of site, my impression of articles read in the past, and of the author themselves.)
The post is from February 2020.
I can't see a date on the post (on mobile) but the archives link has month and year.
Well then, I take it all back.
This might vary state-by-state, CA MediCal for instance did limit the subsidies based on income last year. I don't think it was an all-or-nothing cutoff, but I do think there were points around the 50k mark where the delta between your subsidy and the one for the lower bracket was higher than the loss you'd take by a few thousand dollars.
The subsidies were from the federal government, managed as part of your federal income tax return. They had nothing to do with additional state subsidies.
The basic story was that until the end of 2025, nobody in the USA had any reason to pay more than (roughly) 8.3% of their AGI for health insurance.
[flagged]
(2020)