Direct answer
Read what the rule is applied to, not what it is called. A rule applied to the amount above the line changes the price of each further unit and nothing else, so the line bends and more is always more. A rule applied to the whole amount reprices what you had already, so the line breaks and there is a range above the boundary where you keep less than you kept below it. Both can make the next unit a loss, and they do it for different reasons.
- The condition that matters most
- Which of the two a rule is cannot be inferred from the words threshold, band, limit or allowance. Those words are used for both shapes, often in the same document, so the only reliable test is what quantity the charge or the rate is multiplied by.
- What this does not tell you
- This explains a shape and names no rule. It contains no scheme, no jurisdiction, no threshold anybody set and no rate anybody publishes, and it cannot tell you which shape your own rule has. It says nothing about whether crossing a boundary is worth it, and nothing about tax anywhere.
- Last reviewed
The sentence that decides everything
Somewhere in every rule that starts at a boundary there is a sentence saying what the rule is applied to. It is usually short, it is rarely emphasised, and it is the whole of the mechanism:
- the amount above the line, or
- the whole amount.
Almost every wrong answer people reach about boundaries comes from assuming the first when the rule says the second. The assumption is reasonable, because the most familiar boundary anybody meets is an income tax band, and an income tax band really does work on the excess. Meeting a rule that works on the whole amount for the first time, with that habit already formed, produces an estimate that is wrong by roughly the size of everything below the line.
Reading the sentence takes a few seconds. Nothing else on this page is harder than that.
Two shapes follow from it
Draw what you keep against what you have, and the two readings are not two versions of one picture. They are different kinds of line.
Applied to the excess, the line bends. Below the boundary each unit is worth one unit; above it each unit is worth less than one. The rate at which you are climbing changes at the boundary and the height does not, because nothing has been charged on what you had before you got there. There is no point on that line where you would rather be standing further back.
Applied to the whole amount, the line breaks. The unit that carries you over does not cost the rate on itself. It triggers the rate on everything already behind it, so the line drops at the boundary and then resumes climbing from lower down. It has to climb back up before it reaches the height it had one unit before the line, and until it gets there you are keeping less than you kept while you were smaller.
The second shape has a name. A boundary where the value drops rather than the slope changing is a notch, and it is worth knowing the word because the documents that create notches almost never use it.
That range has a width, and the width is the part almost nobody estimates. If a rule applies a charge of twenty per cent to the whole amount, then crossing the line costs twenty per cent of the boundary itself, and it takes twenty-five per cent more than the boundary to get back to where you were. The rule looks like a rate of twenty per cent and behaves, at that one point, like a bill.
Applied to the excess: the line bends
Above the boundary each further unit is worth less than one. Nothing is taken from what was already there, so the height at the boundary is unchanged and no part of the line is below a point to its left.
Applied to the whole amount: the line breaks
The unit that crosses triggers the charge on everything already earned, so the line drops at the boundary. Across the shaded range you are keeping less than you kept one unit before the line.
The two pictures differ in one way that a description can carry as well as a drawing. In the first there is no vertical gap anywhere, and no point on the line sits below a point to its left. In the second there is a gap at the boundary, and every point in the shaded range sits below the height reached one unit before it.
Why the marginal answer is wrong, and by how much
Ask what one more unit costs and the honest answer, at a boundary of the second kind, is that the question has no arithmetic answer at that point. Away from the boundary each unit is worth whatever the rate leaves of it. At the boundary one unit carries a charge on everything behind it, and no rate expresses that, because a rate is a statement about a slope and the line has no slope where it breaks.
This is why the two facts have to be published separately, and why a page that gave you only the rate would be reproducing the mistake. The size of the drop and the price of a marginal unit are different quantities, and only one of them is a percentage.
The width of the range matters as much as the depth of the drop. A boundary described as expensive tells you nothing about how far you have to travel before it stops mattering, and on a rule applied to the whole amount that distance is proportional to the boundary rather than to the charge. Doubling the size of the business doubles the width of the range in which growing is a loss.
A continuous rule is not automatically a safe one
Here is the correction that matters most, because the tidy version of this subject gets it wrong.
It is tempting to finish the argument by saying that a rule applied to the excess is harmless: the line bends, nothing is repriced, more is always more. The first two are true and the third does not follow from them.
A rule applied to the excess can take more than the whole of the next unit. It happens when something is withdrawn at a percentage of the amount above a boundary, and the something being withdrawn is large relative to how wide the band is. The withdrawal adds a marginal rate equal to the size of what is being withdrawn divided by the width of the band. If a benefit worth sixty per cent of the band’s width is withdrawn evenly across it, the effective rate inside that band is sixty percentage points higher than outside it, and any baseline rate above forty points is enough to carry the total past one hundred per cent.
Nothing bounds that sum at one. Where it passes one, the next unit leaves you worse off than not having it, and there is no break in the line anywhere. The amount withdrawn moves smoothly across both edges of the band. Only the rate steps, and a rate is allowed to step without anything jumping, because an accumulation of a step has no step in it.
So the two shapes fail in different ways, and the failure is not the thing that distinguishes them:
- a rule applied to the whole amount makes more leave you with less over a bounded range, because the line has to climb back to a height it already reached;
- a rule applied to the excess can make more leave you with less across the whole of a band, and there is no distance at which it stops.
Continuity tells you the line does not break. It tells you nothing about whether the line is going up.
Two boundaries that look like these and are not
Not everything that changes at a boundary is either of the above, and reading one of these as a break is its own error.
A cap or a floor binding. A quantity falls until it reaches a floor and then stops falling, or rises until it reaches a ceiling and then stops rising. The formula in force genuinely changes at that point, so a description written from the formulas alone will report a boundary. Nothing jumps. The value is the same on both sides of it and only the direction of travel changes, which is why a floor is a place a line flattens out rather than a place it falls off.
A break-even, a crossover, or a level of adequacy. These are the most common boundaries in personal finance and none of them is a rule at all. They are places where two continuous quantities happen to be equal: where a cost equals a saving, where one option overtakes another, where a reserve reaches the length of time it needs to cover. Nothing is applied to anything at that point. Crossing it changes which answer is larger, not what anything is multiplied by. A break-even is produced by arithmetic; a threshold is imposed by a rule, and only the second can reprice what you already had.
The practical difference is what you do with them. A break-even is a target and being just past it is fine. A boundary of the breaking kind is not a target, and being just past it is the worst place on the line.
How to read the rule in front of you
Four questions, in order. The first is the one that settles it.
- What is the charge or the rate multiplied by? If the answer is the part above the line, the line bends. If it is the whole of it, the line breaks. If the document does not say, this is the thing to find out, and it is worth more effort than any other sentence in it.
- Does anything appear or disappear at the boundary rather than change rate? An entitlement that stops existing, a fixed charge that starts, a status that switches on. These are level changes, and they are the same shape as a charge on the whole amount even when no percentage is involved.
- How wide is the band, compared with what is being withdrawn across it? This is the question that catches the continuous case that still takes more than the whole of the next unit. A narrow band and a large withdrawal produce a high rate for the same reason a steep hill is short and steep rather than long and steep.
- What is the boundary measured against? A share of one quantity is not the same line as a share of another, and the quantity a rule names is frequently not the one that comes to mind first. This does not change the shape, but it moves the boundary, and a correctly shaped answer at the wrong place is still wrong.
Where this shape turns up
Four UBWHY calculators price rules of these kinds. Each one models a different arrangement, each has its own locked model behind it, and none of them is a worked example of this page. What they have in common is the shape, which is the only thing this page is about.
The registration threshold calculator is the clearest case of a rule applied to the whole amount: a charge that arrives on turnover already earned, with a range above the boundary in which growing costs money. It computes the size of the drop and the width of that range for a threshold and a charge you supply, along with the two shares that decide how much of the charge you actually bear, which is a judgement no general page can make for you.
The benefit withdrawal band calculator is the continuous case. It computes the effective marginal rate inside a band from a benefit, a band and a count, and it is the one that shows how far past one hundred per cent an ordinary arrangement can go without any break in the line at all. It also carries a question this page does not touch, which is that the same household income split two different ways produces two different answers.
The tapered allowance calculator carries both shapes in one rule. The taper itself is continuous, and a second measure switches the withdrawal on rather than fading it in, so capacity jumps rather than sloping at that point. Its own mechanism is neither of those, though. It is that the contribution you are deciding on lands inside the measure that decides how much you may contribute, which makes the limit depend on a quantity the limit constrains, and that is a problem this page has nothing to say about.
The deposit band step calculator prices a boundary in the other direction. The rate is a step function of the ratio, so the whole balance is repriced at a boundary rather than only the part above it, and the practical consequence is that a unit of deposit is worth nothing across most of a band and worth a whole band of rate at the edge of one. What that page adds, and what this one cannot, is that reaching a boundary does two things at once and only one of them is the boundary’s doing.
What this does not tell you
It does not tell you which shape your own rule has. That is a fact about a document you are holding and not about arithmetic, and the words used for both shapes are the same words.
It publishes no threshold, no rate, no charge and no scheme, in any jurisdiction. The figures used above are round numbers chosen to make an identity visible, and they describe no real rule anywhere.
It says nothing about whether crossing a boundary is worth it. A range in which more leaves you with less is an arithmetic fact about one window and one quantity. Whether to grow into it, stop below it, or carry the cost and go through is a decision with far more in it than this shape, and none of the rest is here.