Methodology

Progressive Slice Taxation

Why crossing a stamp duty threshold does not reprice the whole purchase, and where a genuine cliff does exist.

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%.

T = \sum_{i = 1}^{n} t_{i} max (0, min (P, u_{i}) - u_{i - 1})

The tax equals the sum, over each band, of that band's rate times the part of the price lying inside it.

P: price
t: the rate for band i
u: the upper bound of band i

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Reliefs, unlike rates, can be withdrawn rather than tapered — and that produces a genuine cliff.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.

Calculators that use this method

UK Stamp Duty (SDLT) CalculatorStamp Duty Land Tax on a residential purchase in England or Northern Ireland, with first-time buyer relief and the additional-property surcharge.
Ontario Land Transfer Tax CalculatorProvincial land transfer tax on an Ontario property purchase, with the first-time buyer refund.

Basis

HMRC SDLT residential rates for England and Northern Ireland, effective 1 April 2025.
Ontario Land Transfer Tax Act rate schedule.