Incentives and elasticity

The ratio. Percent change in units over percent change in price. It is negative for almost everything, because people buy less of what got dearer, and the minus sign is usually said out loud rather than dropped: "minus one point eight". The size is what matters. Below 1 in absolute value is inelastic: buyers barely move, because they need the thing, have no substitute, or spend so little on it that they do not notice. Above 1 is elastic: buyers walk away, because a substitute is one click over, or the thing is a luxury, or the price is a big part of their budget.

The revenue rule. Revenue is price times units. Raise the price and one goes up while the other goes down; which one wins is the whole question. On an inelastic product, units fall by less than the price rose, so revenue rises. On an elastic product, units fall by more than the price rose, so revenue falls. At exactly -1 the two cancel and revenue does not move. The figure draws it: the amber strip is what the higher price gains on the units you kept, the hatched strip is what you lost on the units that left, and elasticity is the ratio of their widths.

With the real number. On the 203 Olist products, each compared with itself month by month, the default answer is about -1.8. Under a constant-elasticity assumption, a 10 percent price rise goes with 15.7 percent fewer units and 7.2 percent less revenue. Had the answer been the pooled -0.108, the same rise would go with 1.0 percent fewer units and 8.9 percent more revenue. Same rows, opposite sign on the revenue line, and the difference is only whether a product was compared with itself or with other products. Deep-dive session 2 shows exactly how that happens; tonight the point is that the revenue rule flips at -1, and the honest number sits well past it.

The same product can be elastic in one setting and inelastic in another, and the settings are the interesting part. Substitutes: one brand of bed linen on a marketplace with fifty brands is elastic; bed linen as a category is not. Time: people cannot change their commute the week petrol rises, but they can change their car over three years, so the long-run response is bigger than the short-run one. The gasoline meta-analysis by Havranek, Irsova and Janda 2012 in Energy Economics, as reported, puts the short-run figure at about -0.09 and the long-run at about -0.31 after correcting for publication bias: both inelastic, and the long run three times the short. Share of budget: a 10 percent rise on a R$ 3 item is not noticed; on a R$ 3,000 item it is a family conversation. The scope of the price change: raising your price alone is elastic; the whole market raising together is not, because buyers have nowhere to go.

The data habit: never quote an elasticity without saying what it is an elasticity of. "Demand for our product is elastic" means nothing until you add "against the other sellers on the platform, over a month, for the 203 products that moved". The number is always attached to a scope, and the scope is often the part a leader most needs to hear.

Put a tax of a fixed amount per unit between buyer and seller. Whatever the buyer pays, the seller now receives that minus the tax. On the picture that is a vertical wedge inserted between the demand curve and the supply curve: the top of the wedge is what buyers pay, the bottom is what sellers keep, and the wedge slides left along the two curves until its height equals the tax. Fewer units are sold than before, and the old price sits somewhere inside the wedge.

Where it sits is the whole story. If demand is steep, meaning buyers cannot easily switch, the wedge pokes mostly upward: buyers pay most of the tax in a higher price and sellers keep nearly what they kept. If demand is flat, meaning buyers can switch, the wedge pokes mostly downward: the seller cannot pass much on without losing the sale, so the seller keeps less and bears most of it. The same logic runs on the supply side: a seller who can easily stop supplying bears little, a seller who cannot bears a lot.

And it does not matter which side the law names. A tax "on sellers" and a tax "on buyers" of the same size produce the same wedge, the same units, the same buyer price and the same seller take. The law decides who fills in the form. The elasticities decide who is poorer. That is why a "tax on platforms" ends up in seller fees, why a "fee on sellers" ends up in shelf prices, and why the person who can walk away is always the person who pays least.

The units that stop being sold because of the wedge are not a transfer to anyone. The buyer who would have paid a bit more than the seller needed, but not the seller's price plus the tax, simply does not buy, and both sides lose the surplus that trade would have created. Economists call that the deadweight loss of the tax, and it is larger when both curves are flat, because more trades sit close to the line. A tax on something people cannot avoid raises money with little deadweight; a tax on something people can easily skip raises less and destroys more.

The same picture describes any wedge, not only a government's. A marketplace commission is a wedge between what a buyer pays and what a seller keeps. A payment fee is a wedge. A delivery surcharge is a wedge. In each case the question "who really pays the commission" has the same answer as the tax: the side that can walk away pays less. A seller who can list elsewhere passes the commission on; a seller who cannot absorbs it. Whether the platform sets a "seller fee" or a "buyer fee" is a labelling choice, and the market re-labels it.

Fit the line five reasonable ways and you get five numbers. Compare each product with itself and the slope is -1.788. Add month effects so that the marketplace's growth is removed too: -1.854. Weight each month by how many units it sold, so big months count more: -2.216. Drop the months where only a single unit sold: -1.480. Include freight in the price: -1.881. None of these is wrong. Each is a defensible choice about which rows to trust and how much, and a person who reports only one of them is hiding the other four.

So the statement that survives is in two scopes. The range: on these rows, the within-product elasticity is between -1.5 and -2.2 depending on the weighting and the row filter. The default: about -1.8. And the two things it is not: it is not -0.1, which is what you get by pooling all the products together without comparing each with itself, and it is not zero, which is what a shuffled placebo gives (mean -0.002 across 200 draws, standard deviation 0.159). The real slope sits more than 11 placebo standard deviations from zero. The range is wide; the sign and the size are not in doubt.

Publishing a range is not weakness. It is the only shape that lets a reader see how much the answer depends on choices you made, and it is the shape that stops the next person from quoting "-1.788" as if the third decimal meant something.

Two cousins of the price elasticity earn their keep. Cross-price elasticity is the percent change in units of one thing when the price of another thing changes. Positive means substitutes (dearer coffee, more tea); negative means complements (dearer printers, fewer cartridges). It is the number behind every "if we raise the price of A, what happens to B" question, and on a marketplace it is the number that says whether two sellers are really competing. Income elasticity is the percent change in units when income changes: above 1 for luxuries, between 0 and 1 for necessities, below 0 for the things people buy less of as they get richer. It is why a recession hits some categories and leaves others alone.

The word itself was formalised by Alfred Marshall in Principles of Economics, 1890, Book III, chapter IV, where he sets out "elasticity of demand" as the ratio of proportional changes and works through how it varies with price and with the wealth of the buyer. Earlier writers had the idea in pieces; Marshall gave it the name and the ratio. It is a short chapter, and reading it is a good way to notice that the questions have not changed since 1890, only the data available to answer them.