Paradox & Proof Entry #0157 Classified Declassified

Why a growing company can have shrinking margins on every sale

Total profit and profit per unit are separate quantities that can move in opposite directions. Growth is reported in one of them and health is assumed in both.

No visual record attached The written record below is complete.
Plate 125 — two arrows from one ledger, pointing apart.

Intuition test — answer before you read on

Revenue doubles, total profit hits a record, and per-unit margin falls by a third. Which is wrong?

A company doubles revenue and reports record profit. In the same period the margin on each individual sale falls by a third. Both statements come from the same accounts, and a reader given only one of them will form a confident and incomplete view of the business.

What everyone sees

Growth is read as a single direction of travel, so a record profit implies that everything underneath it improved. Announcements encourage this by leading with totals, and totals are the figures that travel into headlines and summaries.

What is actually happening

A total is a rate multiplied by a volume, and the two factors are independent. Discounting to win volume, entering cheaper segments or adding lower-margin lines can raise the total while lowering the rate. Neither figure is more real than the other; they answer different questions, one about scale achieved and one about the economics of the next sale. Confusing them is a category error, not an accounting dispute.

Why it stays hidden

The divergence hides in the choice of headline. Rates require a denominator and denominators require explanation, so summaries default to totals. Nothing in the total signals the direction of the rate, and a reader would need the ratio explicitly to see it. The pattern is legal, ordinary and entirely disclosed to whoever reads far enough.

A total is a rate times a volume. Growth in the product says nothing about the direction of either factor.

A total is a rate times a volume. Growth in the product says nothing about the direction of either factor.

The hidden part — entry #0157

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A total is a rate times a volume. Growth in the product says nothing about the direction of either factor.

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Sources & further reading 3
  1. Brealey, Myers & Allen, "Principles of Corporate Finance", 2020
  2. Simpson, "The Interpretation of Interaction in Contingency Tables", Journal of the Royal Statistical Society, 1951
  3. Huff, "How to Lie with Statistics", 1954

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