Tax bands: marginal and effective rates

Set an income and watch two rates at once. One is charged on the next unit you earn; the other is what you paid across the whole. They are never the same number, and nearly every argument about tax is really an argument about which one is meant.

Worked example — Tax bands: marginal and effective rates

Marginal rate	40.0%
Effective rate	17.8%
Kept, out of the next 1,000	600 of 1,000

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Two rates, and only one of them is the one people quote 🖖

A band charges its rate on a slice, never on the whole. Take an income of 60,000 on the two-band preset. The first 10,000 is the allowance and costs nothing. The stretch from 10,000 to 40,000 is charged 20%, which is 6,000, and that 6,000 does not change however much is earned above it. Only the last 20,000 meets the 40% rate, for 8,000. Total 14,000, take-home 46,000. So the marginal rate is 40% and the effective rate is 23.3%, and both are correct because they answer different questions: what the next unit costs, and what the whole cost on average. Drag the income upward and the two lines never meet. At 300,000 the effective rate has reached 36.7%; at a million, 39.0%. It approaches 40% and never arrives, because the untaxed slice and the 20% slice are still down there being cheap, and they stay the same size forever.

Why a raise cannot make you poorer, when the system is bands 🖖

The worry is specific and very common: take the promotion, cross into the next band, end up with less. Under bands alone it cannot happen, and the reason has nothing to do with where the thresholds sit. Take-home is income minus tax. Each extra unit earned adds its marginal rate to the tax, so it adds one minus that rate to the take-home. While no rate exceeds 100%, that quantity is positive, and a line whose slope is always positive cannot come back down. Drag the income through every threshold on the two-band preset and watch: the take-home line bends, twice, and never turns. What the bend costs is real enough — 800 of the next 1,000 is kept below the higher threshold, 600 above it — but 600 is more than nothing, and that is the entire argument. It is worth noticing what this argument does not cover. It is a claim about bands, and it holds for every rate in the table precisely because it is about the rates in the table.

The withdrawal is the thing worth watching 🖖

Switch on the withdrawn allowance and drag the income to 110,000. An extra unit is charged 40% like its neighbours, but it also destroys half a unit of allowance, and that half-unit is then charged 40% as well. Forty plus twenty is sixty: the rate on extra income is 60%, and it appears nowhere in the table. Now push past 125,140 and watch the marginal rate fall to 45%. A rate that drops as income rises looks like a drawing error and is not — it is the withdrawal running out. The place it runs out is not a chosen number either. The allowance is 12,570 and it goes at 1 for every 2 earned, so it lasts 25,140 beyond 100,000. A threshold that looks arbitrary is usually the end of a taper, and this is how to recognise one. Then switch the cliff on instead, for the version with no subtlety: withdraw a lump at a threshold, and one extra unit of income costs 1,999.40.

Problems solved in full

Income of 60,000 offered a raise to 61,000 on the two-band preset 7 steps
On the two-band preset, an income of 60,000 is offered a raise to 61,000. Work out what is actually kept — then find the income where that reasoning stops being safe.
1. Take the income apart before touching any rate. The first 10,000 is the allowance and is charged nothing.
  
  $10, 000 \times 0 = 0$
2. The stretch from 10,000 to 40,000 is charged 20%: 6,000. This is the step that gets skipped, and skipping it is where the fear of a raise comes from — that 6,000 stays 6,000 no matter what is earned above it.
  
  $(40, 000 - 10, 000) \times 0.20 = 6, 000$
3. What is left, the 20,000 from 40,000 up to 60,000, is charged 40%: 8,000.
  
  $(60, 000 - 40, 000) \times 0.40 = 8, 000$
4. Tax is 14,000, take-home 46,000, effective rate 23.3%. The marginal rate is 40%. Neither figure is wrong and they are not the same fact.
  
  $T = 14, 000, N = 46, 000, \frac{14 , 000}{60 , 000} = 23.3%$
5. Now the raise. The extra 1,000 sits entirely in the top band, so 400 is charged and 600 is kept. Take-home goes from 46,000 to 46,600.
  
  $1, 000 \times (1 - 0.40) = 600 \Rightarrow 46, 600$
6. That 600 is the general case rather than a coincidence of these numbers: take-home rises by one minus the marginal rate. No rate here exceeds 100%, so it always rises, and no raise on this system can leave anyone worse off.
  
  $\frac{d N}{d I} = 1 - r > 0 (r < 1)$
7. Now switch on the withdrawn allowance and drag the income to 110,000. The same 1,000 returns 400. The argument above has not broken — it was always a statement about the rates in the table, and 60% is not one of them.
  
  $0.40 + \frac{1}{2} (0.40) = 0.60$
Answer

46,600, so 600 of the 1,000 is kept. A marginal rate of 40% against an effective rate of 23.3%. The argument from the slope protects you from bands and not from withdrawals: turn one on and the same raise returns 400, charged at a rate of 60% that no line in the table mentions.
A raise of 2,000 at an income of 59,000, on the preset with a cliff in it 7 steps
Switch to the cliff-edge preset. You earn 59,000 and someone offers you 2,000 more. Work out what you would actually keep, and decide whether to take it.
1. Start with the tax at 59,000, sliced the same way as any band system. The first 10,000 is the allowance and costs nothing, the next 30,000 meets 20%, and the 19,000 left over meets 40%. Take-home is 45,400 and the effective rate card reads 23.1%.
  
  $T = 30, 000 (0.20) + 19, 000 (0.40) = 13, 600 ⟹ N = 45, 400, \frac{13 , 600}{59 , 000} = 23.1%$
2. The next unit of income is charged 40% exactly like the one before it, so 60 of every 100 is kept. The marginal rate card reads 40.0%, which is the least alarming number on the page.
  
  $\frac{d N}{d I}_{59, 000} = 1 - 0.40 = 0.60$
3. At 60,000 something else happens. The cliff withdraws a lump of 2,000 the moment income reaches the threshold, all of it at once, with no fraction and no taper. Tax goes from 13,600 to 16,000 across a single unit of income.
  
  $T (60, 000) = 6, 000 + 8, 000 + 2, 000 = 16, 000 ⟹ N = 44, 000$
4. The card for the next 1,000 measures a step of a thousand rather than a step of one, and it reads −1,400. Two cards, one income, and they disagree about whether the next thousand is worth having. Both are right: a step of one cannot see a threshold standing a thousand away.
  
  $44, 000 - 45, 400 = - 1, 400$
5. The raise on offer is 2,000, which lands at 61,000. Take-home 44,600, against 45,400 before it. You would be 800 a year behind for doing more work.
  
  $N (61, 000) = 44, 600 < 45, 400 (- 800)$
6. So find the raise that helps. Above the threshold take-home is six tenths of income plus 8,000, and that clears 45,400 at 62,333.33. The first whole number that works is 62,334, a raise of 3,334.
  
  $N (I) = 0.6 I + 8, 000 \geq 45, 400 ⟹ I \geq 62, 333.33$
7. The two branches say why. Below the threshold take-home is 0.6I + 10,000 and above it 0.6I + 8,000: parallel lines, 2,000 apart. At a slope of 0.6 it takes 3,333.33 of income to climb 2,000, and that is the width of the hole.
  
  $N (I) = 0.6 I + 10, 000 (I < 60, 000), N (I) = 0.6 I + 8, 000 (I \geq 60, 000)$
Answer

Refuse the 2,000 and ask for 3,334. The raise on offer costs 800 a year; the smallest one that leaves you level takes you to 62,334.

The width of the hole is C/(1−r), which is worth carrying around. A lump of 2,000 withdrawn where the rate is 40% swallows 3,333 of income; the same lump where the rate is 20% swallows only 2,500, because more of each extra unit survives to fill the gap. On this system everyone earning between 60,000 and 63,333 is behind someone who stopped just short of the threshold, and none of them can see it from the marginal rate. That card measures the next unit, and a cliff is not made of units.

$Δ I_{m i n} = \frac{C}{1 - r} = \frac{2 , 000}{0.60} = 3, 333.33$

References (2)

The band table the "withdrawn allowance" preset reproduces, including the rates and thresholds: HM Revenue & Customs, "Income Tax rates and Personal Allowances: Current rates and allowances." GOV.UK.
The withdrawal itself, which is what pushes the rate on extra income above every rate in the table: HM Revenue & Customs, "Income Tax rates and Personal Allowances: Income over £100,000." GOV.UK.

By the 42Math team · Updated 2026-08-26

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Two bands - An allowance of 10,000, then 20%, then 40%. At an income of 45,000 the marginal rate is 40% and the effective rate is 17.8% — the two figures that get swapped for one another in almost every conversation about tax.
A withdrawn allowance - A real published system, with the allowance taken away at 1 for every 2 earned above 100,000. From 100,000 to 125,140 an extra unit of income is charged 60%, while the highest rate printed in the table is 45%.
A cliff edge - The same two bands, with 2,000 withdrawn outright once income reaches 60,000. Earn one unit more than 59,999 and take-home drops by 1,999.40 — the case where a raise really does leave you worse off.