Why Tax Brackets Don't Punish Raises

People worry a raise could push them into a higher bracket and cut their take-home pay. It can't: the higher rate applies only to the dollars above the threshold. Tax owed is a continuous function of income, with corners but no jumps.

How it works

Continuity

In one sentence

$f$ is continuous at $a$ if $\lim_{x\to a}f(x) = f(a)$ , which requires both one-sided limits to exist and equal $f(a)$ . A piecewise function is continuous at a break point when its pieces agree there.

General formula

{\htmlData{sym=0}{T(I)}} = \begin{cases} {\htmlData{sym=3}{r_1}} {\htmlData{sym=1}{I}} & {\htmlData{sym=1}{I}} \le {\htmlData{sym=2}{b}} \\ {\htmlData{sym=3}{r_1}} {\htmlData{sym=2}{b}} + {\htmlData{sym=4}{r_2}} ({\htmlData{sym=1}{I}} - {\htmlData{sym=2}{b}}) & {\htmlData{sym=1}{I}} > {\htmlData{sym=2}{b}} \end{cases}, \qquad \lim_{{\htmlData{sym=1}{I}}\to {\htmlData{sym=2}{b}}^-}T = \lim_{{\htmlData{sym=1}{I}}\to {\htmlData{sym=2}{b}}^+}T = T({\htmlData{sym=2}{b}})

Continuity means small changes in input give small changes in output. Taxing only the dollars above the threshold makes that true at the threshold too.

What the symbols mean

$T(I)$
Tax owed
Thousands of dollars of tax on income I.
$I$
Income
Taxable income in thousands.
$b$
Threshold
Where the higher rate starts.
$r_1$
Lower rate
Tax rate on income up to b.
$r_2$
Higher rate
Tax rate on income above b.

Up to the threshold $b$ income is taxed at $r_1$ ; only the part above $b$ is taxed at the higher $r_2$ .
At $I = b$ both formulas give $r_1b$ : the left and right limits agree with $T(b)$ , so $T$ is continuous.
The graph has a corner at $b$ (the slope jumps from $r_1$ to $r_2$ ) but no jump in value.
If instead the higher rate applied to all income past $b$ , $T$ would jump by $(r_2 - r_1)b$ : a raise could lower take-home pay.

Learn the math

ContinuityCalc I · Lesson 12, Limits