What you actually keep
75 min
Two hosts talk the lesson through. The voices are synthetic; the script was written from this lesson and checked against it, and asserts nothing the lesson does not.
- Calculate the federal income tax on a given salary using marginal brackets, and explain why a raise into a higher bracket never lowers take-home pay
- Distinguish a marginal rate from an effective rate and say which practical question each one answers
- Compare what a deduction and a credit of the same size are worth, and explain who each is worth nothing to
- Explain what withholding is, and weigh the cost of over-withholding against the cost of under-withholding
Lesson 1 started with take-home pay and treated it as a given. You looked at what landed in your account and built everything on that. It was the right place to start, because the money you can spend is the money you have. But something happened between what your employer agreed to pay you and what showed up.
This lesson is about that gap. For Nadia, the $60,000 earner we're going to follow through it, the federal government took just over $9,600 last year, and she almost certainly couldn't tell you that number, name either of the two taxes that make it up, or say what happens to her next dollar.
So: not how to file, and not how to pay less. How it works, so you can read your own paystub, judge a claim about tax when you hear one, and answer the two questions that actually change decisions.
A note on scope first. The machinery here is United States federal income tax, and the numbers are for tax year 2025, from the IRS.12 That is the return filed in early 2026, so if you are earning in a later year a newer table already exists: the IRS keeps the current year's brackets and standard deduction on the same page, and everything below works the same way once you swap the numbers in.1 If you're somewhere else, the specific rates won't transfer but the structure almost certainly will, because nearly every income tax in the world is built from slices in the same way, and the last practice exercise tells you the three things to go and find out about your own system. If you're in the US, this lesson also ignores state income tax, which runs from nothing at all in eight states to a top marginal rate of 13.3% in California, the highest in the country.10 And the standing note for this course: this is education, not personalised advice. For anything with real money on it, and certainly anything with a filing deadline, check the current IRS page or ask someone who does this for a living.
The idea that fixes most of the confusion
Almost every wrong belief about income tax comes from one picture: that your income has a rate, and that rate is decided by which bracket you land in. If that picture were right, crossing into a higher bracket really could cost you money, and a lot of everyday advice about turning down overtime would make sense.
The picture's wrong, and the right one isn't much harder.
Think of your income as water poured into a set of containers stacked on top of each other. The bottom container fills first. When it's full, the water spills into the next one up, and so on. Each container has its own tax rate written on the side, and each is taxed only on what it holds. Your top container is almost never full.
So the rate on the container you're currently filling is the rate on your next dollar. It isn't the rate on the dollars sitting in the containers below, which were taxed at their own lower rates on the way up and aren't revisited. That's the model, and the rest of the lesson is what follows from it.
Before any numbers: if the containers work like that, what has to be true about the relationship between the rate on your top container and the share of your whole income that went to tax?
Show the answer
The share of your whole income has to be lower. Some of your money sat in containers with lower rates, and some of it (as you'll see in a moment) sat in a container with no rate at all. An average that includes 0% and 10% and 12% can't come out at 22%. That isn't a quirk or a loophole, it's a consequence of the shape, and it means the two numbers can never be equal for anyone who has crossed at least one bracket line.
Nadia's $60,000, sliced
Here's a real one, using the actual 2025 federal brackets for a single filer and the 2025 standard deduction of $15,750.12
Nadia earns $60,000. Before any bracket applies, the standard deduction comes off, which is a flat amount every filer can subtract without proving anything. That leaves $44,250 of taxable income. Now pour it in:
- The first $11,925 is taxed at 10%, giving $1,192.50.
- The remaining $32,325 is taxed at 12%, giving $3,879.00.
Total federal income tax: $5,071.50. That's 8.45% of her salary.
Look at the second bar. Her salary went up by $5,000, she's now, in the usual phrase, "in the 22% bracket", and the 22% bracket is that thin strip at the right-hand end. Everything to its left is taxed exactly as before. The raise cost $677.50 in extra federal income tax: $4,225 of it at 12% and $775 of it at 22%.
She keeps $4,322.50 of the $5,000 before payroll tax, and about $3,940 after it. There's no arrangement of the brackets, at any income, where taking the raise leaves her with less. The higher rate can only ever apply to the dollars above the line, and those dollars didn't exist before.
Look at the short red bar underneath the first one too. That's Nadia's entire federal income tax, drawn on the same scale as her salary. People carry a picture of tax that looks like the long bar; it looks like the short one.
Someone earns $200,000 and says "I lose 32% of everything to federal income tax." Without doing the full sum, what's wrong with that sentence, and roughly which direction is the truth in?
Show the answer
Two things are wrong. First, 32% isn't even their top rate: at $200,000 with the standard deduction, taxable income is $184,250, which sits in the 24% bracket, so 32% is a bracket they haven't reached. Second, even their real top rate applies only to the top slice. The actual bill is $37,067, which is 18.5% of the salary. The truth is always well below the bracket number, and here it's below by nearly a factor of two.
Two rates, two questions
You now have both of Nadia's numbers, and they're worth naming properly, because they answer different questions and people mix them up constantly.
Her marginal rate is 12%. That's the rate on the next dollar she earns. It's the number for any decision about more: a raise, an overtime shift, a side job, a pre-tax retirement contribution. When lesson 7 asks whether to put a dollar into a traditional account or a Roth account, the marginal rate is the input to that decision, which is why this lesson comes first.
Her effective rate is 8.45%. That's the share of everything she earned that went to this tax. It's the number for any decision about totals: what share of a household's income the tax system takes, or how your burden compares with someone else's. Neither one is the "real" rate; they're two measurements of the same bill.
The red staircase is what people mean when they say "the 22% bracket". The navy curve is what they actually paid. The curve never catches the staircase, at any income on this chart or beyond it. And look at how it behaves in the range most people live in: doubling a salary from $40,000 to $80,000 moves the effective rate from 6.7% to 11.3%.
That gap is also the first thing to check on any claim you hear. When someone says the middle class pays a third of its income in tax, the useful question isn't whether it sounds right, it's which taxes are being counted. Federal income tax alone will never get you to a third at a middle income. Federal income tax plus both halves of payroll tax plus state income tax plus sales and property tax might. Neither version is dishonest; they're answers to different questions, and the arguing usually happens because nobody said which question.
Sal Khan works the same arithmetic on a whiteboard here, and if the container model hasn't quite landed, watching someone build the slices by hand is worth four minutes:
Deductions and credits are not the same size
Two words get used as though they were synonyms in almost every news story about tax, and they're not close.
A deduction comes off your income before the tax is worked out. A credit comes off the tax you owe, after. So a deduction is worth your marginal rate, and a credit is worth its face value. For Nadia:
- A $1,000 deduction removes $1,000 from the top of her taxable income, which was sitting in the 12% bracket. She saves $120.
- A $1,000 credit comes off the $5,071.50 she owes. She saves $1,000.
More than eight times as much, for the same headline number.
Both halves of that sentence need a qualification, and both qualifications matter most to the people with the least money.
Not every deduction reaches you. There are two kinds. Above-the-line deductions come off your income whatever else you do: a traditional 401(k) or IRA contribution, an HSA contribution. Itemised deductions (mortgage interest, state and local taxes, large charitable gifts) only help if they add up to more than the standard deduction, because you take one or the other, never both. Since the 2017 tax act roughly nine in ten filers take the standard deduction, up from about seven in ten before it: 89% of returns in tax year 2023, the most recent year the IRS has published, against 68% in 2017.11 So for most people, another $1,000 of itemised deduction is worth exactly nothing, which is why a homeowner's mortgage interest is often worth far less than they assume. This is the distinction to check before you believe any sentence beginning "you can deduct".
Not every credit pays out. A refundable credit pays you the balance if it's bigger than your tax bill; the Earned Income Tax Credit is the largest of those, worth $66 billion across 24 million returns in tax year 2023. The Child Tax Credit is larger in total, but only part of it is refundable, up to $1,700 per child, and that refundable part came to about half the EITC.12 A non-refundable credit can only take your bill to zero. So a family owing $900 in federal income tax gets $900 from a $2,000 non-refundable credit, not $2,000. Whether a given credit is refundable is one of the most argued-over choices in US tax design, precisely because it decides whether the credit is worth anything to households with little or no tax liability.
There's a second consequence of all this, and it's the one the politics runs on. Because a deduction is worth your marginal rate, the same deduction is worth more to a higher earner: $1,000 of deduction saves $120 in the 12% bracket and $350 in the 35% bracket. A credit is worth the same to everyone who can use it.
Critics of particular deductions, the mortgage interest deduction most often, call this an upside-down subsidy: the higher your rate, the more the government pays toward the expense. Defenders answer that a deduction is a statement about what counts as income in the first place, so the larger saving is just the mirror of the larger rate that dollar would otherwise have faced, and the progressivity lives in the rate schedule rather than in the deduction. Both are descriptions of the same arithmetic, and both sides would add facts the other leaves out: that the distributional point applies mostly to itemised deductions, which nine in ten filers don't take, and that many deductions and credits phase out at higher incomes, which cuts the other way. Where the line should be drawn is a question about what a country ought to do, and that one's yours, not ours.
You hear that a proposed change gives families "a $2,000 tax break for childcare". What are the two questions to ask, and roughly how much does the answer change the number?
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Is it a deduction or a credit, and if it's a credit, is it refundable? If it's a refundable credit, it's worth $2,000 to a family. If it's non-refundable, it's worth $2,000 only to families whose tax bill is at least that big, and less or nothing below. If it's an itemised deduction, it's worth $2,000 multiplied by the family's marginal rate, and only if they itemise at all, which nine in ten don't. The same headline covers answers that run from zero to $2,000.
The tax that isn't in the brackets
Look at a real paystub and the income tax line isn't the only deduction, and for a lot of workers it isn't the largest.
In the US, payroll taxes fund Social Security and Medicare. For an employee they're 6.2% for Social Security on wages up to a cap that's reset every year ($176,100 in 2025, $184,500 in 2026) plus 1.45% for Medicare on all wages with no cap, so 7.65% off the top (SSA's contribution and benefit base page carries the rates and the whole history of the cap).3 Above $200,000 for a single filer an Additional Medicare Tax of 0.9% applies, taking the Medicare share to 2.35% on the excess. That $200,000 is written into the statute with no inflation adjustment attached to it, so unlike the Social Security cap sitting beside it, the threshold has not moved since the tax took effect in 2013.13 Your employer pays the same amount again on your behalf, and it never appears on your stub. The Congressional Budget Office allocates that half to you when it measures who bears federal taxes, on the reasoning that employers pass it on by paying lower wages than they otherwise would. CBO also says the incidence could differ from its allocation, and its own working paper on the question estimates that 58% of the burden of a broad payroll tax rise would fall on employees in the short run, so treat it as the standard working assumption rather than a measured fact.14 If you're self-employed you pay both halves yourself, 15.3%, and no employer withholds anything for you.3
Back to Nadia. Her federal income tax was $5,071.50. Her employee payroll tax is $4,590. Add them and her federal tax burden is $9,661.50, which is 16.1% of her salary, with payroll tax making up nearly half of it.
Her marginal rate on the next dollar is 12% counting income tax alone, 19.65% counting the employee payroll tax too, and higher still if you count her employer's share as hers. Anyone planning around a marginal rate has to say which taxes they're counting.
The shape of the payroll tax is also the reason the bracket table alone tells you so little. The Social Security part is flat within the cap and stops above it, so it takes a larger share of a modest income than of a large one, running opposite to the income tax brackets. Supporters of the design point out that the cap on the tax is paired with a cap on the benefit, and that the benefit formula replaces a much larger share of a low earner's wages than a high earner's, so the two halves have to be judged together. Critics answer that the tax on wages stops while income above the cap doesn't. People argue this in good faith from both directions; the arithmetic above is the part everyone agrees on.
Nadia's federal income tax was $5,071.50 on $60,000, which we called 8.45% effective and 12% marginal. Add payroll tax and her total federal burden is $9,661.50. Before reading on: what does that do to the pair, and which of the two numbers moves more?
Show the answer
Both move, and the marginal moves more in proportion. Her effective rate goes from 8.45% to 16.1%, roughly double. Her marginal rate goes from 12% to 19.65%, because the next dollar faces 6.2% Social Security and 1.45% Medicare on top of the 12% income tax. The point isn't the exact figures, it's that "my rate" means nothing until you say which taxes you're counting, and the bracket table counts only one of them.
One more thing about that paystub, since exercise 3 sends you to look at yours. Beyond the three federal lines you'll usually find state and sometimes local income tax, and above them a set of pre-tax deductions: a 401(k) contribution, health insurance premiums, an HSA. Those come out before the federal income tax is calculated, which means they shrink the taxable income you just learned to slice. That's lesson 7's subject, and it's the mechanical reason a retirement contribution costs you less than its face value.
Where the myth is true
Brackets can't punish extra income. Something else can, and the reason the myth survives is that people have genuinely been burned and were then told the wrong thing did it.
You've now seen that brackets can never make a raise cost you money. And yet plenty of people will tell you, sincerely, that extra income left them worse off, and some of them are right. What could be doing it, if not the brackets?
Show the answer
A benefit that shrinks or stops. Brackets slice, so the most they can ever take is part of the dollar that crossed the line. But a payment, subsidy or discount that phases out as your income rises isn't a slice, it's a step, and a big enough step can withdraw more than the dollar that triggered it. Everything that follows is about those steps.
Two shapes, and they behave differently. A phase-out withdraws a benefit gradually as income rises. It never takes more than the extra dollar, but it can take a startling share of it. A cliff switches a benefit off entirely at a threshold, and a cliff really can take thousands from one extra dollar.
Phase-outs are the common case, and they fall hardest on low and moderate earners, because that's where most income-tested programmes operate. The Congressional Budget Office measured the combined effect of income tax, payroll tax and the phase-out of benefits like SNAP and health insurance subsidies, and found that low- and moderate-income workers faced an effective marginal rate of about 31% on average, with individual cases running far higher.5 That is a higher marginal rate than the bracket table shows for most of the people it describes, and it's the reason the sentence "work more and you'll keep more" is truer for some households than others. Medicaid eligibility, childcare subsidies and income-driven student loan repayment are all built the same way, on a threshold or a taper, though the 31% figure above does not measure them: that study covered SNAP and health insurance subsidies.
Cliffs are rarer and sharper. The live example, as this is written in September 2026: premium tax credits for health insurance bought on the US marketplace. The American Rescue Plan capped what a household paid for a benchmark silver plan at 8.5% of income from 2021, with no upper income limit, and the Inflation Reduction Act extended that through 2025. The enhancement expired at the end of 2025, and the original rule returned: no premium tax credit at all above 400% of the federal poverty level.15 For the 2026 coverage year, a single person in most states hits that line at 400% of the 2025 guideline of $15,650, which is $62,600, since marketplace eligibility for a year uses the previous year's guidelines. The 2026 guideline is $15,960, so for 2027 coverage the same line sits at $63,840.6
Here's what a cliff does, in numbers. On KFF's 2026 estimate, a 60-year-old earning $62,000 pays about $6,175 a year for the benchmark plan, because their premium is capped at a share of their income. The same person in the same city earning $64,000 pays about $14,931, the full price, because they're over the line.7 Two thousand dollars of extra income costs about $8,750. As KFF puts it, that person now pays the same premium as someone earning $160,000.
Whether that cliff should exist is a live political argument and this lesson takes no side in it. What matters for you is the shape. Brackets are safe. Thresholds aren't. If you receive an income-tested benefit of any kind, find out where its threshold sits and how it behaves there, because that's the only place where earning more can leave you with less.
A friend earning $95,000 is offered a bonus that would take her to $110,000. She receives no income-tested benefits of any kind, and she's heard that $110,000 crosses into the 24% bracket. Do the arithmetic: what does the bonus actually cost her in federal income tax?
Show the answer
$3,300, and the interesting part is why. At $110,000 her taxable income is $94,250, and the 22% bracket runs all the way to $103,350 of taxable income, so she doesn't cross into 24% at all. She isn't where she thinks she is on the table. Every dollar of the bonus faces 22%, her tax goes from $12,349 to $15,649, and she keeps $11,700 of the $15,000 before payroll tax. Take it. Most people who think they're near a bracket edge are further from it than they imagine, because they forget the standard deduction sits underneath everything.
What the top of the table does and doesn't tell you
One more thing about brackets, because the 37% figure gets quoted a lot and it's the most misunderstood number in the table.
Nobody pays their top rate on their whole income; that's just the model again. Beyond that, the top of the schedule isn't the only thing operating up there. Long-term capital gains and qualified dividends, which is most of what large investment portfolios produce, are taxed on a separate and lower schedule of 0%, 15% and 20%, not on the ordinary income brackets at all.16 Long-term there means held for more than a year. That's established fact and it's worth knowing before lesson 7, because it's part of why where you hold an investment changes what it costs you.
What the highest earners actually pay is measured very differently by different researchers, and the disagreement is worth understanding as a disagreement. IRS Statistics of Income, which counts realised income and federal income tax and nothing else, puts the top 1% at an average effective rate of 26.3% in 2023, earning 20.6% of income and paying 38.4% of all federal income tax.8 Saez and Zucman count unrealised gains and assign corporate tax to shareholders, and get top-end rates far lower, with the system close to flat overall.9 Auten and Splinter, using the same tax data with different assumptions about unreported income and transfers, get rates rising steeply to about 50% at the very top.9 They aren't arguing about arithmetic. They're arguing about what counts as income and who bears a tax, which is partly an empirical question and partly a definitional one, and that's why the answers differ by so much. Anyone who quotes one of those numbers without saying which method produced it is giving you half a fact.
What people get wrong
- "A raise pushed me into a higher bracket, so I take home less." Never from brackets. Sometimes from a phase-out or a cliff, which is worth checking.
- "My tax rate is 22%." That's the rate on the next dollar. The rate on all the dollars is much lower, and the gap grows with income.
- "A deduction and a credit are both a tax break." They differ by a factor of your marginal rate, and each has a group it's worth nothing to: itemised deductions for the nine in ten who take the standard deduction, non-refundable credits for households with no tax bill.
- "A big refund means I did well." It means you lent money at 0% for the best part of a year.
- "Checking the bracket table tells me what I pay." It leaves out the standard deduction underneath and the payroll tax alongside, and those two move the answer in opposite directions.
- "Rich people pay 37%." Nobody pays their top rate on their whole income, and the honest answer to what they do pay depends on a measurement dispute that's worth knowing about rather than picking a side in.
Withholding, and the refund that isn't a bonus
The last piece is withholding, which is where most people's actual feelings about tax live.
You don't pay income tax in April. You pay it all year: your employer takes an estimated amount out of every paycheque and sends it in on your behalf. The estimate comes from a formula applied to the form you filled in when you were hired (Form W-4), and like every estimate it's usually a bit wrong.4
Too high and you get a refund. Too low and you get a bill. A refund is your own money coming back, having been held for you at no interest. One exception matters: if your refund is larger than the extra you had withheld, the difference is a refundable credit, and that really is money arriving rather than money returning. For most filers, though, the refund is the loan being repaid.
Take Priya, who over-withholds by $200 a month and gets $2,400 back each spring, and who also carries $4,000 on a card at 22%. Had she put that $200 a month against the card instead, month by month, she'd have avoided about $260 of interest over the year and still owed nothing in April. The refund cost her $260 to receive.
Two honest qualifications. If you don't carry high-rate debt, the cost is much smaller: on these numbers, roughly what $2,400 would earn sitting in a savings account for an average of half a year, so somewhere near $50. And some people save nothing they can reach, so for them a refund is a savings account with a lock on it, which is the same pre-commitment idea lesson 2 built the automatic transfer on. If that's you, keep doing it and know the price.
The error runs the other way too, and it has a price the lesson owes you. Withhold far enough below what you actually owe and the IRS charges an underpayment penalty. The safe harbours are the thing to remember: broadly, if you've paid in 90% of this year's tax or 100% of last year's (110% at higher incomes) through withholding and estimated payments, no penalty applies however large the April bill is.4 So the target isn't "as little withheld as possible", it's "land near zero, on the safe side of the harbour". This matters most for people with self-employment or variable income, who have no employer withholding anything at all and make quarterly estimated payments instead.
If your withholding is badly off in either direction, the fix is a new W-4 and the IRS's own withholding estimator, which takes about ten minutes with a paystub in front of you.4
Practice
These take about half an hour between them, on paper, and they're worth more than rereading the lesson.
- Find your own effective rate. Take last year's tax return or a recent paystub. Write down your gross pay for the year and the total federal income tax for the year, and divide the second by the first. Keep both numbers, the rate and the dollar amount. Then write your marginal rate beside them from the bracket table. Most people are surprised how far apart they are.
- Slice three incomes without a calculator app. Using the 2025 single-filer brackets and the $15,750 standard deduction, work out the federal income tax on $35,000, $75,000, and $150,000, slice by slice. (Answers, to check: $2,071.50, $7,949.00, and $25,067.00.)
- Read your paystub properly. Find the federal income tax line, the Social Security line and the Medicare line. Check that Social Security is 6.2% and Medicare 1.45% of your gross for that period, then multiply each by the number of pay periods in a year. Now find the other lines: state and local tax, and any pre-tax deductions. If the federal income tax line, annualised, is far from the dollar figure you wrote in exercise 1, your W-4 is worth revisiting.
- Find your cliffs. Write down every income-tested thing your household receives: marketplace premium credits, student aid, an income-driven loan repayment plan, SNAP, childcare help, a state or local programme. For each, find the threshold and whether it's a phase-out or a cliff. If you receive none, write "none", and know your raises are safe.
- If you're not in the US. Find your own country's bracket table and its standard deduction or personal allowance, and do exercises 1 and 2 with those. Then find your equivalent of payroll tax, the contribution that funds the pension and the health system, and work out whether it's flat, capped, or bracketed. Those three facts are most of what this lesson is, in your system.
Before the quiz, close the page and answer from memory: why can't a bracket ever make a raise cost you money? What's the difference between a marginal rate and an effective rate, and which one do you use for a raise? What's a $1,000 deduction worth compared with a $1,000 credit, and who is each one worth nothing to? What genuinely can make extra income leave you worse off? Then check what you missed.
Connections
This closes a gap lesson 1 left open. When you audited your spending against take-home pay, the difference between that and your salary was tax, and you can now account for all of it: income tax in slices, payroll tax flat off the top, both delivered monthly by withholding.
It also sets up lesson 7. The choice between a traditional retirement account and a Roth account is a bet about one number: your marginal rate now compared with your marginal rate when you take the money out. That sentence was meaningless before this lesson and should be clear now.
Three habits from earlier lessons got another workout. Lesson 1's fixed-versus-variable distinction has a new extreme case, since payroll tax is the most fixed cost there is: it's a fixed percentage, it's taken before you see the money, and no amount of budgeting touches it. Lesson 1's discipline of asking what a statistic measures is exactly what separates a marginal rate from an effective one, and what separates the three answers about what top earners pay. And lesson 4's insistence on real versus nominal is why bracket thresholds move every year, since they're indexed to inflation so that a raise which only keeps pace with prices doesn't quietly push you into a higher slice.
Go deeper
- Kapoor, Dlabay and Hughes, Personal Finance, part 1. The standard US college textbook's own treatment of taxes inside a personal finance course, including the filing mechanics and tax planning strategies this lesson leaves out. Widely available second-hand and in libraries.
- Tax Policy Center Briefing Book, especially "How do federal income tax rates work?" and "What is the standard deduction?". The best free analytical treatment, and it states its own assumptions, which is what makes it usable when the subject is contested.
- IRS: federal income tax rates and brackets, the page to check each year rather than trusting any figure printed here, and credits and deductions for individuals, which lists what actually exists rather than what a headline claims.
- Khan Academy's Financial Literacy course has a taxes and tax forms unit covering filing, W-2s and 1099s, which this lesson deliberately leaves alone.
- CBO, Effective Marginal Tax Rates for Low- and Moderate-Income Workers, 12 pages, if you want to see how the phase-out arithmetic is actually done.
Sources
[1] Internal Revenue Service, "Federal income tax rates and brackets", tax year 2025. https://www.irs.gov/filing/federal-income-tax-rates-and-brackets
[2] Internal Revenue Service, Publication 501, "Dependents, Standard Deduction, and Filing Information", 2025 edition; standard deduction $15,750 for a single filer under 65. https://www.irs.gov/publications/p501
[3] Social Security Administration, "Contribution and benefit base"; 6.2% and 1.45% employee shares, 15.3% combined for the self-employed, wage cap $176,100 in 2025 and $184,500 in 2026. https://www.ssa.gov/oact/cola/cbb.html . The Additional Medicare Tax of 0.9% above $200,000 is at https://www.irs.gov/taxtopics/tc751
[4] Internal Revenue Service, "Tax withholding", the Tax Withholding Estimator, and Topic 306 on the underpayment penalty and its safe harbours. https://www.irs.gov/individuals/employees/tax-withholding , https://www.irs.gov/individuals/tax-withholding-estimator , https://www.irs.gov/taxtopics/tc306
[5] Congressional Budget Office, "Effective Marginal Tax Rates for Low- and Moderate-Income Workers in 2016", November 2015; 31% average effective marginal rate. https://www.cbo.gov/publication/50923
[6] US Department of Health and Human Services, ASPE poverty guidelines; the 2025 guideline of $15,650 for a household of one in the 48 contiguous states, of which 400% is $62,600. https://aspe.hhs.gov/topics/poverty-economic-mobility/poverty-guidelines
[7] KFF, "A steep subsidy cliff looms for older middle-income enrollees if ACA enhanced tax credits expire", 8 October 2025, checked 6 September 2026; the $6,175 and $14,931 benchmark premium estimates for a 60-year-old at $62,000 and $64,000, and the comparison with a $160,000 earner. https://www.kff.org/quick-insights/a-steep-subsidy-cliff-looms-for-older-middle-income-enrollees-if-aca-enhanced-tax-credits-expire/
[8] Tax Foundation, "Who pays federal income taxes?", summarising IRS Statistics of Income for tax year 2023; top 1% average effective rate 26.3%, 20.6% of income, 38.4% of federal income tax paid. https://taxfoundation.org/data/all/federal/who-pays-federal-income-taxes-tax-year-2023/
[9] Emmanuel Saez and Gabriel Zucman, "The rise of income and wealth inequality in America", Journal of Economic Perspectives 34(4), 2020. https://gabriel-zucman.eu/files/SaezZucman2020JEP.pdf . Gerald Auten and David Splinter, "Income inequality in the United States: using tax data to measure long-term trends", Journal of Political Economy, 2024. https://davidsplinter.com/AutenSplinter-Tax_Data_and_Inequality.pdf
[10] Tax Foundation, "State Individual Income Tax Rates and Brackets, 2026", 17 February 2026; "Eight states, including New Hampshire, which repealed its interest and dividends tax as of 2025, levy no individual income tax at all", and "Top marginal rates span from 2.5 percent in Arizona and North Dakota to 13.3 percent in California." Washington is counted separately because it taxes capital gains income only. https://taxfoundation.org/data/all/state/state-income-tax-rates-2026/
[11] Internal Revenue Service, Statistics of Income, Table 1.2, "All Returns: Adjusted Gross Income, Deductions, and Tax Items, by Size of Adjusted Gross Income and by Filing Status". Tax year 2023: 143,309,624 returns claimed the standard deduction out of 160,602,107 total, or 89.2%, and 15,106,257 itemised. Tax year 2017: 104,013,115 out of 152,903,231, or 68.0%, with 46,852,675 itemising. Tax year 2023 is the most recent year published. https://www.irs.gov/statistics/soi-tax-stats-individual-statistical-tables-by-size-of-adjusted-gross-income (files 23in12ms.xls and 17in12ms.xls)
[12] Internal Revenue Service, Statistics of Income, Table 3.3, "All Returns: Tax Liability, Tax Credits, and Tax Payments", tax year 2023: earned income credit $66.3 billion on 24,439,936 returns, of which $55.9 billion was the refundable portion; refundable child tax credit or additional child tax credit $34.5 billion on 17,626,084 returns, of which $32.1 billion was refundable; non-refundable child and other dependent credit a further $84.4 billion. https://www.irs.gov/statistics/soi-tax-stats-individual-statistical-tables-by-size-of-adjusted-gross-income (file 23in33ar.xls). The $1,700 refundable cap per child for tax year 2025 is from IRS, "Child tax credit": "The Child Tax Credit is worth up to $2,200 per qualifying child" and the Additional Child Tax Credit is available "up to $1,700 per qualifying child depending on your income". https://www.irs.gov/credits-deductions/individuals/child-tax-credit
[13] 26 U.S.C. §3101(b)(2), which imposes the 0.9% tax on wages "in excess of" $250,000 on a joint return and $200,000 "in any other case" for taxable years beginning after 31 December 2012, and attaches no cost-of-living adjustment to either figure. https://www.law.cornell.edu/uscode/text/26/3101 . IRS, "Questions and answers for the Additional Medicare Tax", confirms the same thresholds still apply: "Additional Medicare Tax went into effect in 2013". https://www.irs.gov/businesses/small-businesses-self-employed/questions-and-answers-for-the-additional-medicare-tax
[14] Congressional Budget Office, The Distribution of Household Income in 2021, September 2024, Appendix A, "Incidence of Federal Taxes": "CBO also allocates the employer's share of payroll taxes to employees because employers appear to pass on their share of payroll taxes to employees by paying lower wages than they otherwise would. The incidence of those taxes could differ from CBO's allocation, though." https://www.cbo.gov/publication/60706 . The 58% short-run estimate is from the working paper CBO cites there: Dorian Carloni, Revisiting the Extent to Which Payroll Taxes Are Passed Through to Employees, Working Paper 2021-06, June 2021, which also states that "Empirical evidence on the incidence of payroll tax changes in the United States is limited". https://www.cbo.gov/publication/57089
[15] KFF, "What we know so far about 2026 ACA marketplace enrollment, premiums, and deductibles", Matt McGough, Jared Ortaliza, Justin Lo and Cynthia Cox, 19 May 2026; "The enhanced premium tax credits established by the American Rescue Plan in 2021 and extended through 2025 by the Inflation Reduction Act", their expiry "at the end of 2025", and "Under the enhanced premium subsidies, people with incomes above 400% of the poverty level had their premium payments for a benchmark silver plan capped at 8.5% of income", above which "eligibility for premium tax credits ends". https://www.kff.org/affordable-care-act/what-we-know-so-far-about-2026-aca-marketplace-enrollment-premiums-and-deductibles/
[16] Internal Revenue Service, Topic no. 409, "Capital gains and losses", for the 0%, 15% and 20% schedule on net capital gain and the one-year holding period: "Generally, if you hold the asset for more than one year before you dispose of it, your capital gain or loss is long-term." https://www.irs.gov/taxtopics/tc409 . For qualified dividends, IRS Publication 550 (2025), Investment Income and Expenses: "Qualified dividends are the ordinary dividends that are subject to the same 0%, 15%, or 20% maximum tax rate that applies to net capital gain." https://www.irs.gov/publications/p550
The bracket arithmetic in this lesson was computed from sources [1], [2] and [3] for tax year 2025; the marketplace premium figures are from [7]. All of it is federal only. State income tax, where it exists, is on top.
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