A hosted invoice extractor charges per document — how do you find the monthly volume where running your own becomes cheaper?

## Compare shapes, not prices A build-or-buy comparison for a prediction service is a comparison of **cost shapes**. A hosted extractor quoted per scanned document is almost entirely **variable**: process nothing this month and the bill is near zero. Running your own extractor is mostly **fixed**: the serving capacity has to exist before the first document of the month arrives, the retraining run happens on its own schedule, and the people who keep the pipeline alive are paid whether volume doubles or halves. Write both totals for a month of volume `V`: - buying: `p * V`, where `p` is the quoted price per document - running your own: `F + m * V`, where `F` is the monthly floor and `m` is what one more document genuinely costs you Setting them equal gives the **volume crossover**: ``` V* = F / (p - m) ``` The denominator carries the argument. The floor is paid down by the *gap* between the quoted price and your own marginal cost — not by the price alone, and not by your marginal cost alone. ## A worked crossover With `F` = 9,000 a month, `m` = 0.01 per document and `p` = 0.05 per document: | documents / month | buy (`0.05 * V`) | own (`9,000 + 0.01 * V`) | cheaper | |---|---|---|---| | 90,000 | 4,500 | 9,900 | buy | | 225,000 | 11,250 | 11,250 | equal | | 400,000 | 20,000 | 13,000 | own | `9,000 / (0.05 - 0.01) = 225,000`. Two arithmetic slips are common on the way there. Dividing by `p` alone gives 180,000 and quietly assumes your own documents are free to process. Dividing by `m` gives 900,000 and uses a number that is not a saving at all. ## Draw the line the same way on both sides The comparison is only honest if `F` and `m` are drawn with the same pen on both options. - **The buy side is not purely variable.** Integration, the connector that submits documents, and the exception queue that handles low-confidence fields are yours on either option. - **Costs common to both options cancel**, so they may be left out — but only after you have checked they really are common. Reviewing a wrongly read total is work whoever extracted it. - **The floor is what must exist at zero volume**: idle serving capacity, amortised retraining, the standing share of a person's time, and any capacity held for the month-end burst but unused the rest of the month. - **The marginal figure is only what scales with a document.** If a cost does not move when one more invoice arrives, it is not in `m`. - **A quoted price with a committed minimum is not variable either** — a floor on their side is a floor on yours. ## The volume crossover is not the time to payback The two questions are routinely confused, and they give different verdicts. 1. The **volume crossover** compares two *recurring* monthly costs and answers "at what volume does running our own become the cheaper bill?" 2. **Time to payback** compares the *one-off* engineering spend to reach a first version against the monthly saving earned after the crossover, and answers "how long until that spend is recovered?" 3. A team can be past the crossover and still lose: two quarters of build effort, then a volume forecast that flattens in the third, and the saving never accumulates far enough. Because of that, the one-off build effort does not belong in `F`. Keep it in the payback calculation where it answers a question about time. ## Growth makes it a dated decision This month's volume is the wrong input, because the decision lives as long as the commitment does. At 90,000 documents a month growing 8% month on month, `1.08^11` is about 2.33 (roughly 210,000) and `1.08^12` about 2.52 (roughly 227,000), so the crossover arrives in **month 12**. That does not say "build now". It says: buy, write the crossover and the growth assumption down beside the decision, and revisit before month twelve. The crossover itself moves while you wait — a renegotiated price lowers it, a falling marginal cost raises it, a flattening forecast delays it. And past the crossover the gap is still not the whole argument: exit cost, where customer documents are processed, and the value of keeping the decision reversible all sit outside this arithmetic.