▲% · Progressive Tax
A progressive tax doesn't apply the higher rate to your whole income — only to the part above the bracket line.
Your raise pushes you into a higher tax bracket. Are you actually worse off?
What are we trying to understand?
'A raise pushed me into a higher bracket, so I actually lose money' — the most widespread math myth among working people. If it were true, nobody would welcome a raise.
The truth: income tax slices your income into bracket-sized pieces and applies each bracket's rate only to its own piece. The higher rate touches only the part above the line.
Once you've seen this cake-slicing with your own eyes, it becomes clear why the marginal rate (the last slice's rate) differs from the effective rate (your overall average), and what news about 'raising the top rate' actually means for you. (We use 2024 US federal brackets, single filer.)
Change one assumption at a time
The result follows the inputs and assumptions. Check the units and model before treating it as an actual outcome.
Predict first — taxable income moves from $46,000 to $48,000, entering the 22% bracket. Can your after-tax income actually fall?
Run the trial yourself — just before vs just after the line
Read the calculation another way
Cross a bracket and the higher rate applies to your entire income — a net loss.
The higher rate applies only to the excess above the line. If taxable income rises by $1, tax rises by at most $0.37 — after-tax income always increases. Losing money by crossing a bracket is structurally impossible.
'A 24% tax rate' means you pay 24% of your income in tax.
24% is the marginal rate (the last slice's rate). Your actual burden — the effective rate — is much lower. At $120,000 of taxable income, the marginal rate is 24% but the effective rate is about 18%.
Why the calculation works, and when
Write the experiment's 'slice by bracket' as formulas and you have all of progressive taxation. The last formula is the bridge between this lab and the Derivative lab.
Slice, tax, and add
Slice the income into bracket pieces, multiply each piece by its bracket's rate, and add. No piece ever affects another piece's rate.
Effective rate — what you actually pay
Total tax over total income. Because the cheap lower slices are mixed in, it's always below the marginal rate.
Marginal rate — it's secretly a derivative
'If income rises by $1, how much does tax rise?' — the instantaneous slope of the tax curve, i.e., a derivative. 'Marginal' in economics news almost always means a derivative.
Try the idea in another setting
Raises and bonuses
'The bonus isn't worth it because of taxes' is wrong. The raise may be taxed at a higher rate, but after-tax income always goes up.
Filing with side income
To predict tax on freelance or side income, you need this bracket structure. The key question: as incomes stack, which bracket do they fill up to?
Capital gains on property
Capital gains taxes are progressive too. Selling for a bit more isn't a 'tax bomb' — the higher rate touches only the excess.
Why tax deductions favor high earners
Retirement-account deductions are 'worth more' to high earners because a deduction carves off your most expensive final slice — the marginal rate.
Now explain it in your own words
Follow the mathematics behind the calculation
Related lab
Derivatives
The marginal rate is the derivative of the tax function. Through the derivative's lens of 'the rate of change right now', marginal cost and marginal utility all show the same structure.
Go to the Derivatives lab →Related lab
Functions
The tax curve is a piecewise-linear function whose slope changes at each bracket. See the staircase (marginal rate) and the bent line (tax) with function eyes.
Go to the Functions lab →Previous lab
Take-home Pay
How progressive tax actually shows up in your paycheck — see the previous lab.
← Take-home Pay labRecommended next
Expected Value
Taxes done, next comes the world of uncertain choices. The single number for comparing lotteries, insurance, and investments — the Expected Value lab is in the works.
Go to the Expected Value lab →