Tax brackets explained: can a raise make you take home less?

Anyone looking into their income eventually hits the same question: how much is left after taxes? Many income tax systems are progressive, charged in brackets. And around them floats a stubborn myth: “if I get a raise, I’ll jump into a higher bracket and take home less.” The math behind it is a piecewise function, and once you see how it works, the myth falls apart in a few lines.

The key idea: each rate applies only to the slice of income that falls inside its bracket. Your whole income is not taxed at the highest rate it reaches. It gets sliced, and each slice pays its own bracket’s rate.

To study this without depending on any real tax table (they change every year), we’ll use a made-up schedule built just for this example. Amounts are monthly, in US dollars:

Someone earning $7,000 does not pay 20$7,000. They pay 0% on the first $2,000, 10$3,000, and 20% only on the $2,000 above$5,000.

Let $x$ be income and $T(x)$ the tax owed. Within each bracket, the tax is a linear function: the bracket’s rate times $x$, minus a constant that accounts for the lower slices having paid less.

Where do 200, 700 and 1,700 come from? From requiring the graph to have no jumps. At $x = 5000$, bracket 2’s formula gives 0.10 · 5,000 − 200 = 300, and bracket 3’s gives 0.20 · 5,000 − 700 = 300. At $x = 10000$, both neighboring formulas give 1,300. The function is continuous: it changes slope at each threshold but never jumps. Its graph is a polyline that gets steeper, and the slope of each piece is exactly that bracket’s rate.

Two different rates describe the same tax:

• Marginal rate: the rate on your next dollar earned, the slope of the piece you’re on. At $7,000, it’s 20%.
• Effective rate: the share of your total income that goes to tax, i.e. tax divided by income.

Check with the formula. $7,000 is in bracket 3, so$T(7000)$ = 0.20 · 7,000 − 700 = 1,400 − 700 = 700. Matches.

Now suppose a raise to $10,500. By slices: 0 + 300 + 1,000 (the$5,000 in bracket 3 at 20%) + 150 (the $500 above$10,000 at 30%) = $1,450. By bracket 4's formula: 0.30 · 10,500 − 1,700 = 3,150 − 1,700 = 1,450. Take-home pay goes from$6,300 to $9,050, and the effective rate rises from 10% to about 13.8%, still far below the 30% marginal rate.

Picture someone earning exactly $10,000 who gets a$100 raise, crossing into the 30% bracket. The fear is that 30% now applies to everything. It doesn’t.

• At $10,000: tax of$1,300, take-home $8,700.
• At $10,100: tax of$1,330, take-home $8,770.

Of the extra $100,$30 goes to tax and $70 stays with you. Because$T$is continuous and its slope never reaches 100$x - T(x)$ is strictly increasing: its slope is 1 minus the marginal rate, always positive. More gross pay always means more net pay.

The myth comes from mixing up two models. If tax were “bracket rate times your whole income,” the function would jump: at $10,000 the tax would be 20$2,000, and at $10,100 it would be 30$3,030, and take-home would indeed drop. Bracket schedules are designed precisely to avoid that cliff. Income-tested benefits, which sit outside the tax table, can create real cliffs in specific cases; that’s a separate calculation.

1. Applying the bracket rate to your entire income. This is what feeds the myth. Each rate covers only its slice.
2. Confusing marginal and effective rates. Saying “I pay 30% in taxes” when only the top slice does. At $10,500 the effective rate is about 13.8%.
3. Dropping the subtraction constant. Computing 0.20 · 7,000 = 1,400 and stopping there doubles the tax.
4. Getting the slice width wrong. The $5,000 to$10,000 bracket is $5,000 wide, not$10,000.
5. Treating the example table as real. These numbers are fictional; they teach the structure, not your actual tax bill.