Contribution Limit When the Allowance Tapers

How this is worked out

Why this is a circle rather than a subtraction

An allowance is withdrawn against a measure of income. That alone is a subtraction and needs no page. What makes this different is that the contribution you are deciding onlands in that measure, so the more you put in, the smaller the allowance that permitted you to put it in.The amount you may contribute depends on your income including your contribution, so the limit depends on a quantity the limit constrains. You cannot work it out without first assuming an answer. That is a fixed point, and it is the only one on this site.

There are two income measures, and they do different jobs

Both have to be over their own thresholds before anything is withdrawn, and they are measures of different things with different definitions. A reader who checks only one of them can be wrong in either direction: far above the first threshold and not tapered at all, or comfortably below the second and tapered the moment they contribute.

The arithmetic

Write c for a contribution, A for the allowance before anything is withdrawn,F for the floor, T for the level of the first measure at which withdrawal starts, r for the withdrawal ratio, G for the second measure's threshold,K for the carry-forward, and k for each measure's share of the contribution:

first(c)      = first base  + k_first  × c
second(c)     = second base + k_second × c

allowance(c)  = A                                              second(c) ≤ G
              = clamp( A − r × max(0, first(c) − T), F, A )    second(c) > G

capacity(c)   = allowance(c) + K

the answer    = the largest c ≥ 0 with c ≤ capacity(c)

The last line is the whole of it, and note what it says: the answer is the largest contribution that fits, which is not quite the same as the contribution that equals the capacity it produces. Usually they are the same figure. Sometimes there is no such contribution at all, and the two sections below are those two facts.

Why the answer is always unique

Contributing more can only lower the allowance or leave it alone — it can never raise it — so capacity minus the contribution is strictly falling. A strictly falling quantity changes sign at most once, so the set of contributions that fit is a single unbroken range starting at nothing.There is exactly one largest one, always. Nothing in that argument needs capacity to be continuous, which matters, because it is not.

Why guessing and re-guessing is not the method

The natural approach is the one the circle suggests: guess, work out the allowance that guess produces, use it as the next guess. It is what a reader with a spreadsheet reaches for, and its convergence is not a property of the mechanism.It fails in two unrelated ways, and one of them at a slope where it ought to work.The verification behind this page runs it rather than describing it, on every canonical arrangement and from several starting points.

So this page solves the circle rather than approaching it. Capacity is straight in pieces — at most four of them — and on a straight piece the crossing has an exact answer that a division finds in one step. There isno tolerance, no iteration count and no convergence criterion anywhere in the calculation, and the divisor is never near zero because contributing can never raise the allowance.

When the equation has no solution at all

The second measure switches the withdrawal on rather than fading it in, so capacity does not slope down at that point — it jumps. Where a contribution is large enough to carry that measure over its threshold, the allowance that permitted the contribution stops existing because of it.Below the crossing every contribution is permitted, above it none is, and no contribution equals the capacity it produces. The largest permitted contribution is still exact and still unique, and it is still the answer to the question — what it is not is a solution.

Where that happens this page says so in the headline rather than in a footnote, and prints the capacity that waits one unit past the answer. Publishing the un-collapsed capacity as though it were reachable above the boundary, or the collapsed one as though it were the limit, would each be wrong in a way no reader could see — and either would invite the inference that one more unit costs one more unit, when it can cost the whole allowance.

Why your own figure is published beside the answer

The figure you reach by looking at your income as it stands is what an unaided reader arrives at, and it is never too small — it is the right answer to a question about a contribution you have not made. This page publishes it, publishes the answer, and publishes the difference, because a page that showed only the answer would leave you with no way to see that your own arithmetic was wrong or by how much. The difference is the figure this calculation exists to produce.

The two facts that look like one and are not

Whether the second measure is above its threshold, and what became of your allowance, are different facts and the result reports both.A reader whose withdrawal ratio is zero is squarely inside the taper and has lost nothing. A page that read the first as the second would tell that reader they had lost an allowance they still have, so the two are separate rows, both in words, and neither is a tick or a colour.

The limits this calculation imposes on itself

• Every figure is yours. This page publishes no allowance, no floor, no threshold, no withdrawal ratio and no charge rate, and there is no field that could carry one. That is not caution — it is what makes one surface correct in every jurisdiction that has this mechanism rather than correct in one.
• Neither income measure is defined here, and the definitions are the hard part. Both have different rules, both have to be over their thresholds before anything is withdrawn, and a document showing your income shows neither. This page computes correctly on whatever two figures it is given, and a reader who supplies the wrong two gets a correct answer to a different question.
• The charge is at a rate you state. The mechanism says the charge falls at your own marginal rate, and this page neither knows that rate nor supplies one. Enter 0 if you do not want the charge priced.
• This is one period and one allowance. There is no carry-forward schedule, no ordering rule between periods, and no unused-allowance history beyond the single figure you state.
• Nothing here says whether contributing is worth doing. A limit is a ceiling, and what a contribution is worth depends on relief, on what else the money would do and on when you need it — three quantities that are not in this arithmetic and have no field on this page.

What is not modelled

• any country, jurisdiction, tax authority, scheme, pension, employer or statute
• any allowance, floor, threshold, withdrawal ratio or charge rate of its own — all twelve inputs are yours
• either income measure’s definition, or any rule for computing one from a payslip
• any carry-forward schedule, ordering rule between periods, or unused-allowance history beyond the single figure you state
• any relief, refund or tax consequence other than the charge on the excess at the rate you state
• whether contributing is worth doing, at any amount
• more than one period, and more than one allowance

The class of mistake this page corrects — a proportion worked out against a base your own action moved, and a figure you had in mind that was never available rather than lost — is taught in the explainer: why losses require disproportionately larger gains. That explainer is about a balance and this page is about an allowance; what they share is the moving base, and it is the part that makes unaided arithmetic wrong in both.