The elasticity bench, and defending a number

Every elasticity anyone reports was fitted on some rows in some way, and the question that sorts the useful ones from the rest is: what was compared with what? Across products (rung 1) compares a bath towel with a phone case, and the slope of +0.008 says price tells you nothing about that comparison, which is true and useless. Pooled (rung 2) throws all product-months in one cloud, and the -0.108 is mostly rung 1 with a little within-product signal leaking through. Within product (rung 3) subtracts each product's own mean price and mean units before fitting, so the only variation left is a product moving against itself. That is the comparison a pricing decision is about, and it reads -1.788.

Rung 4 subtracts each month's mean too, which takes out the marketplace growing, and moves the slope to -1.854. That it moves so little is itself a finding: the growth every product shared was already mostly absorbed by comparing products with themselves. Rung 5 is not a fit. It shuffles the prices inside every product, so that each product keeps its own prices and its own units but the pairing between them is destroyed, and refits rung 3. Two hundred times over, the slope averages -0.002 with a standard deviation of 0.159. The real slope is more than eleven of those standard deviations away.

The standard errors on the bench are the naive kind, computed as if every row were independent, and the page says so. They are a floor on the uncertainty, not a ceiling. The placebo is the honest measure of how far the answer sits from noise, and it is why the break button exists.

Weight months by units makes a month with 121 sales count for more than a month with 2, and moves rung 3 to -2.216. Price includes freight swaps the seller's price for the buyer's cost, and moves it to -1.881. Drop single-unit months removes the 16.5 percent of rows where one sale happened, and moves it to -1.480. Category restricts the panel to one shelf, where the within-product slope runs from -1.283 (garden tools, 13 products) to -3.883 (bed, bath and table, 22 products) with standard errors wide enough that most of the categories cannot be told apart.

None of the four is the right setting. Together they are the range: -1.5 to -2.2, about -1.8 at the default. A number that stays in that band whatever you press is a number with a shape you can describe, and describing the shape is what defending it means.

Press it on rung 3. Inside every product the prices are shuffled among that product's own months, so the set of prices and the set of units are unchanged and only their pairing is destroyed. The slope collapses to somewhere within about 0.16 of zero. Press it again and the real pairing returns with the real slope. A detector that cannot be made to fail has not been tested; this one fails on demand, in the direction it should.

What it cannot prove is causation. The shuffle shows that the pairing of price and units inside a product carries the slope. It does not show that the price caused the units rather than the seller reading the units and setting the price. Product effects remove "cheap and popular", month effects remove "the marketplace grew", the placebo removes "chance", and none of the three removes "the seller chose". That last step needs a price that was changed for a reason unrelated to demand, which is what an experiment is, and the experimentation course owns it.

"So what is the number?" Between -1.5 and -2.2, about -1.8 at the default. A 10 percent rise across the board projects to lose 7 percent of revenue, not gain 9. The range is not hedging; it is the four filters on the bench, and each end of it is a setting somebody could reasonably choose.

"How do you know it is real?" Because it disappears when it should. Shuffle prices inside every product and the slope falls to zero within 0.16. The real slope is more than eleven of those standard deviations away. And because it holds when the rows are cut four ways.

"So we should raise prices, or not?" The number says a uniform rise loses revenue on these products. It does not say why buyers respond, it does not say which product to move, and it does not rule out that sellers were reading demand and setting price after it. The next step is a price test on a few products, which the experimentation course designs, not a spreadsheet extrapolation from this slope.

It does not show an elasticity for the marketplace: the 203 are the products that sold a lot and repriced, and the honest scope is "for the products that can be read". It does not show a per-product elasticity, because deep-dive session 1 found that a third of those are noise. It does not show a causal effect, for the reason in the break button card. And it does not show what happens at prices outside the range the sellers actually tried, which within a product was a median max-to-min ratio of 1.29; a 30 percent rise is a projection past the data.