Progressive Slice Taxation

Each rate applies to its own slice

Transaction taxes such as UK Stamp Duty Land Tax and Ontario Land Transfer Tax are slice taxes. Each band's rate applies only to the portion of the price that falls inside that band, not to the whole price. The total is the sum over bands of the width of the price inside that band times the band's rate.

This is the source of the most persistent misconception about these taxes. People expect a cliff at each threshold — that a purchase one pound over a boundary is taxed entirely at the higher rate — and there is none. A three hundred thousand pound purchase pays nothing on the first hundred and twenty-five thousand, two percent on the next hundred and twenty-five, and five percent on the last fifty: five thousand pounds, an effective rate of 1.7%.

Where a real cliff does exist

Reliefs, unlike rates, can be withdrawn rather than tapered — and that produces a genuine cliff. UK first-time buyer relief applies in full up to five hundred thousand pounds and not at all above it. A first-time buyer paying five hundred thousand owes ten thousand; one paying five hundred thousand and one pound owes fifteen thousand.

That is a five thousand pound step for one pound of price, and it is worth knowing about before agreeing a figure a few thousand above the line. It is the one place in this system where the intuition about thresholds is correct.

The same shape appears elsewhere

Once you recognise the pattern, it turns up throughout the site. Utility tariffs with tiered rates, progressive income tax, and any pricing schedule with volume breaks all compute the same way. The implementation here is one shared function over a table of bands, which is why the UK and Ontario calculators produce structurally identical breakdowns from entirely different tables.

Marginal and average are different numbers, and neither is the other

In a slice system the MARGINAL rate applies only to the next unit earned, while the AVERAGE rate is the total bill divided by the whole amount. The average is always lower, because the lower slices were taxed at lower rates.

This is why describing somebody as a forty per cent taxpayer is a statement about their top slice and not about their bill. Someone whose income just crosses into a forty per cent band pays close to nothing at that rate and an overall proportion far below it, and the gap between the two figures narrows only as income rises well past the threshold.

The distinction decides different questions. The average rate answers what was paid. The marginal rate answers what an extra hour, an extra job or a bonus is worth after tax — and for any decision about doing more work, the marginal rate is the only relevant one.

Earning more cannot leave you worse off — in the rates themselves

The persistent belief that a raise can push someone into a band and reduce their take-home pay is, for a pure slice system, arithmetically impossible. Only the amount ABOVE the threshold is taxed at the higher rate, so crossing it changes what happens to the next pound and nothing about the previous ones.

The belief survives because something like it is true elsewhere. Real cliffs exist, and they are created by things that are WITHDRAWN rather than by rates that rise: a benefit that stops at a threshold, an allowance that disappears, an exemption that no longer applies, a childcare entitlement lost outright.

Those are genuine discontinuities, and at a genuine discontinuity a pound of extra income really can cost more than a pound. The useful mental separation is that rate bands are continuous and eligibility rules are not, so when someone reports being worse off after a raise, the thing to look for is an entitlement rather than a tax table.

Tapers create marginal rates that appear on no rate table

Between a clean band and a clean cliff sits the taper: an allowance withdrawn gradually over a range, commonly at a fixed rate per unit of income above a threshold.

A taper that removes one unit of allowance for every two units of income means that, within the taper band, each extra unit is taxed at its band rate AND exposes half a unit that was previously untaxed. The effective marginal rate inside such a band can be half again as high as the headline rate around it, and it falls back down afterwards.

So the effective marginal rate is not monotonic. It rises through the taper, drops when the allowance is exhausted, and rises again at the next band — a shape no published rate table shows, because the table describes the rates and the taper lives in the allowance rules. Anyone planning around a threshold needs the effective curve rather than the bands.

Slice and slab are different systems and only one is safe

Some taxes are not sliced at all. A SLAB system applies a single rate, determined by the total, to the WHOLE amount — and it therefore has a cliff at every threshold, where crossing by one unit re-rates everything below it.

The consequence is that transactions bunch just below slab thresholds, and prices distort around them: sellers and buyers both have an interest in landing under the line, and values in the range just above a threshold become difficult to achieve. Several transaction taxes were historically slab and were reformed to slice for exactly this reason.

So identifying which system applies is the first question about any banded charge, and it is not something the rate table makes obvious. A slice calculation applied to a slab tax under-states the bill, sometimes by a large amount, at every point above a threshold — and the pages here say which they are computing.