Pricing with the number

The rule. A price rise raises revenue when demand is inelastic (absolute elasticity below one) and lowers it when demand is elastic (above one). Under a constant-elasticity curve the arithmetic is exact: revenue after a rise of p percent is (1 + p/100) to the power (1 + elasticity), minus one. At -0.108 the exponent is 0.892, so a 10 percent rise gives 1.10 to the 0.892, which is +8.9 percent. At -1.788 the exponent is -0.788, and 1.10 to the -0.788 is -7.2 percent. The sign flipped because the elasticity crossed one.

Why the range and not the point. Deep-dive session 2 printed -1.788 at the default, -2.216 weighted by units, -1.881 with freight in the price, -1.480 without single-unit months, and -1.854 with month effects. The sentence for a leader is: on these rows, the within-product elasticity is between -1.5 and -2.2 depending on the weighting and the row filter, about -1.8 at the default; it is not -0.1 and it is not zero. Under every number in that range a 10 percent rise loses revenue, between 4.7 and 10.8 percent. The decision does not depend on which decimal you believe, which is exactly what a range is for.

The assumption, said out loud. Constant elasticity means the same percentage response at every price, which is what the log-log line assumes. It is fine for a 5 or 10 percent move near the prices the panel saw; it is a guess for a 50 percent move, and no panel can tell you what happens at a price nobody was charged.

Published elasticities are almost all below one in absolute value, because most of what economists have measured is a necessity bought by everyone. Gasoline: about -0.09 in the short run and about -0.31 in the long run, the publication-bias-corrected figures reported by Havranek, Irsova and Janda 2012, Energy Economics 34(1) 201-207. Food away from home 0.81 and eggs 0.27, in absolute value as the paper reports them, from Andreyeva, Long and Brownell 2010, American Journal of Public Health 100(2). Nobody stops driving to work because petrol rose 10 percent; people do skip a restaurant meal.

A marketplace product with a slope of about -1.8 is a different animal: a bath towel, a phone case or a garden tool with a dozen substitutes one search away. That is why the within-product Olist number is elastic where petrol is not, and it is the reason the sign of a price rise on a marketplace is the opposite of the sign at a petrol station. It is not a general e-commerce elasticity; the panel is 203 products on one marketplace in 2017 and 2018, and no number on this page is a claim about any other market.

First degree charges each buyer their own ceiling. It needs the seller to know every buyer's willingness to pay, and to stop the R$ 60 buyer reselling to the R$ 90 one. Pure first degree is rare; haggling in a bazaar and a car dealer's negotiated price are the closest real cases, and personalised online prices are the modern attempt. It is the one people mean when they say "they charge what you will pay", and it needs information no seller fully has.

Second degree posts one menu and lets buyers pick. Small, medium, large; economy and business; a free tier and a paid one. The seller does not know who is who, and does not need to: the menu is designed so that the high-ceiling buyer chooses the dear line on their own and the low-ceiling buyer chooses the cheap one. What it needs is that the lines differ in something the high-ceiling buyer cares about and the low-ceiling one can live without. Part 3 is entirely about this one.

Third degree charges by group, using a marker the buyer cannot easily fake: a student card, an age, a postcode, a country, a business email domain. Cinema tickets, regional pricing and educational licences. It needs the marker to correlate with willingness to pay, and it needs the wall to hold: a student cannot buy for a non-student without it becoming first-degree resale.

The test of which degree a price is: who did the sorting? If the seller did it with a marker, third. If the buyer did it by choosing, second. If the seller did it one buyer at a time, first.

Foundations session 1 introduced surplus: the gap between what a buyer would have paid and what they did pay, summed over buyers. At one price of R$ 70, the buyer whose ceiling was R$ 90 keeps R$ 20 of surplus; the buyer whose ceiling was R$ 60 keeps nothing and buys nothing, and the seller made no sale that would have been worth R$ 60 minus cost. Discrimination moves both: it takes some of the first buyer's R$ 20, and it makes the second sale at R$ 60 happen. Whether that is good or bad for buyers as a group depends on which effect is larger, which is why the same mechanism is called "personalised pricing" by one side and "price gouging" by the other.

The condition all three degrees share is that resale must be hard. If the R$ 60 buyer can sell to the R$ 90 buyer, every degree collapses back to one price. Digital goods, tickets with a name on them and services consumed in person are the cases where resale is hard, which is why they are where discrimination is most visible.

The menu works when nobody wants to move down. The high-ceiling customer must prefer the dear version at its price to the cheap version at its price. That is the whole design constraint: the gap in features between two lines has to be worth at least the gap in price to the customer you want on the upper line. If it is not, they drop down and the upper line earns nothing. This is why the difference between a free tier and a paid one is so often a limit that a heavy user hits and a light user never does: seats, storage, history, an export button.

The cheap version is sometimes the dear one with something removed. Shapiro and Varian's examples, as reported, include software shipped with features switched off and a printer sold in a slower version at a lower price. It looks perverse to spend money making a product worse. It is not: the worse version exists to make the low-ceiling sale without giving the high-ceiling customer a reason to take it. The cost of the damage is the price of the sorting.

Freemium, said plainly. A free tier is a version with a price of zero. It is second-degree discrimination whose bottom line is set where the marginal cost of one more user is close to nothing, which is true of information goods and not of towels. Its job is to catch every customer whose ceiling is above zero and below the first paid price, and its design question is the same as any other line's: what does the paid line have that a customer with a real ceiling will not do without?

Every mechanism on this page is a way of working around one fact: a customer's ceiling is inside their head and nothing in a transaction table records it. A sale at R$ 70 says the ceiling was at least 70; a non-sale says nothing, because the non-buyer is not in the table at all. Surveys ask, and people answer strategically or wrongly. The elasticity from deep-dive session 2 is the best a panel can do: the average response of the customers who were there to the prices that were charged. It cannot say what the customers who were not there would have paid, and it cannot say what happens at a price nobody was charged.

What settles a pricing decision is a price test: the same product shown at two prices to two comparable groups, with the assignment decided by a random draw rather than by a seller looking at demand. That removes the thing the panel could not, the seller's choice, and the difference in units between the groups is a causal elasticity at those two prices. How to size, run and read that test is the subject of learn-experimentation-with-phoebe. And where a customer's value over time rather than in one purchase comes into a pricing decision, the lifetime value that a free tier is betting on, that arithmetic lives in learn-customer-retention-with-phoebe, which owns customer acquisition cost and lifetime value; a menu is designed on the elasticity, and paid for on the lifetime.