Price, demand and pricing tactics

In depth

Point and midpoint formulas

The point formula measures each change against the starting values: $E = \dfrac{(Q_2 - Q_1)/Q_1}{(P_2 - P_1)/P_1}$. It is quick, but the answer depends on which point you start from. In Lantern’s hypothetical price test, supermarkets that sold the can at $3.49 averaged 400 cans a month each, and comparable stores that sold it at $3.14 averaged 455. Starting from $3.49, quantity rises by $55 \div 400 = 13.75\%$ and price falls by $0.35 \div 3.49 \approx 10.03\%$, so $E \approx -1.37$. Starting from $3.14 and going the other way, quantity falls by $55 \div 455 \approx 12.09\%$ and price rises by $0.35 \div 3.14 \approx 11.15\%$, so $E \approx -1.08$. Same two points, two answers.

The midpoint formula, also called the arc elasticity, avoids the problem by measuring each change against the average of the two values:

For the test, quantity changes by $55 \div 427.5 \approx 12.87\%$ and price by $-0.35 \div 3.315 \approx -10.56\%$, so $E \approx -1.22$ in either direction. Use the midpoint formula whenever you have two observed price–quantity pairs. Use the point formula when you are given an elasticity and want to predict the effect of a change from today’s price; the prediction is approximate, and less reliable the larger the change. Analysts with many observations estimate elasticity by regression on the logarithms of price and quantity, a method covered in the analytics courses.

Elasticity is not the slope. Along a straight-line demand curve the slope is the same everywhere, but the same change in units is a smaller percentage of a large quantity, and the same change in price a larger percentage of a low price. So a straight-line demand curve is elastic at high prices and inelastic at low ones. One consequence matters in practice: no product has “an” elasticity. It has an elasticity at a price, over a range of prices, in a market and over a time horizon.

Profit is not revenue

Lantern’s test shows elastic demand, so cutting the price to $3.14 raises revenue. Whether it raises profit is a different question. Every extra can carries its own $0.74 cost of goods, and the lower price reduces the contribution margin on every can, including the cans that would have sold anyway. A price cut pays only if the extra cans bring in more contribution than the lower margin gives away.

Let $CM$ be the contribution per unit at the current price, $Q_1$ the current volume and $\Delta P$ the change in the price the seller receives, negative for a cut. If variable cost per unit is unchanged, each unit contributes $CM + \Delta P$ after the change. Total contribution is unchanged when $(CM + \Delta P)\,Q_2 = CM \times Q_1$, so $Q_2 \div Q_1 = CM \div (CM + \Delta P)$. Subtracting 1 from both sides gives the volume change that keeps contribution where it was:

Measure $\Delta P$ and $CM$ in the same units: both in dollars per unit, or both as percentages of the current price, in which case $CM$ is the contribution margin ratio. Dividing the top and bottom by the price shows that the two give the same answer. For a price cut, the result is the volume increase needed just to stand still. For a price rise, it is negative: the largest fall in volume the business can absorb before contribution drops. The formula assumes that variable cost per unit and fixed costs stay the same; if a cut needs extra capacity or more promotion, the extra units must cover those costs too.

Price changes are percentages of the price the seller receives. Read across a row: the thinner the margin, the more volume a cut must win and the more volume a rise can afford to lose.

Two patterns stand out. Cuts are expensive: at a 30% contribution margin ratio, a 10% price cut must raise volume by half just to keep contribution where it was, while a 10% rise can lose a quarter of the volume and still break even. And the thinner the margin, the stronger that asymmetry, which is why businesses with thin margins have the most to lose from a price war.

Apply it to Lantern’s test. Because the trade keeps a fixed 32% of the shelf price, cutting the shelf price from $3.49 to $3.14 cuts Lantern’s net price from $2.3732 to $3.14 \times 0.68 = 2.1352$, a change of $\Delta P = -0.238$, and its contribution per can from $1.6332 to $1.3952. The volume increase needed is $0.238 \div (1.6332 - 0.238) = 0.238 \div 1.3952 \approx 17.1\%$, from 400 to about 468 cans per store per month. The test found 455, an increase of 13.75%. Revenue rose by about 2.3% while contribution fell by about 2.8%, as the challenging example works through step by step.

New products: skimming or penetration

A new product’s first price has to be set before there is any sales history to measure elasticity from, and it is hard to raise once buyers have seen it. Two classic strategies frame the choice. Price skimming launches at a high price to sell to the buyers who value the product most, then lowers the price in steps to reach the next group. Penetration pricing launches at a low price to win as many buyers as possible early in the product life cycle, accepting a low margin per unit in exchange for volume.

Evidence on what firms actually do is a useful corrective. In a study of 663 digital cameras launched under 79 brand names, about 20% followed a skimming price path and 20% a penetration path, while the other 60% were launched at the market price; the authors found that market pricing dominates in practice (Spann, Fischer and Tellis, 2015). Skimming and penetration are the two ends of a range, and many launches sit near the going rate.

Psychological pricing

Psychological pricing sets prices with an eye to how buyers perceive numbers, not only to what the product is worth to them. The best-known form is the charm price, a price just below a round number: $2.99 rather than $3.00. Thomas and Morwitz (2005) explain why it can work. People read prices from left to right and give the leftmost digit most weight, so $2.99 can feel noticeably cheaper than $3.00. They found that this left-digit effect occurs only when the leftmost digits differ, and that it is more likely the closer together the two prices being compared are.

The evidence is not uniform. Sokolova, Seenivasan and Thomas (2020) find the bias is stronger when prices are judged side by side, as on a shelf, than when one of them is recalled from memory, and they note that the size of the effect varies substantially across published studies. Treat charm prices as a real but context-dependent effect, worth testing rather than assuming. For Lantern the research has a direct implication: $3.49 and $3.50 share a left digit, so the 49-cent ending is unlikely to do much on its own; the effect would matter in a choice between, say, $2.99 and $3.09. Round prices can work the other way: OpenStax notes that luxury goods are often given even prices, which suggest quality rather than a bargain.

Reference prices, discounts and promotions

Buyers judge a price against a reference price, the price they expect or regard as normal for the product. Reviewing the research, Kalyanaram and Winer (1995) set out three well-supported generalisations: buyers use reference prices when they choose between brands; they form them largely from the prices they have seen in the past; and they react more strongly to a price above their reference price, which feels like a loss, than to one the same distance below it, which feels like a gain.

Those findings explain the hidden cost of discounting. Each promotion adds a low price to shoppers’ memories and pulls their reference price down. A brand that is on deal every other week teaches its buyers that the deal price is normal, so the regular price starts to feel like a loss and full-price sales suffer. Frequent deals can also teach buyers to wait for the next one or to stock up during it, so part of the extra volume is borrowed from later weeks rather than created. Lantern’s default plan spends $20,000 on trade promotions. The case for them is trial by new buyers; the risk is a lower reference price among the buyers Lantern already has.

Sellers also supply external reference prices: “was $4.29”, “recommended retail price”, “compare at”. These must be genuine. OpenStax describes the US Federal Trade Commission’s view that comparing a price with a “suggested” price at which the product isn’t actually sold, or raising a price just before a “buy one, get one free” offer, is deceptive. Other jurisdictions have their own consumer-protection rules on price comparisons, so check the local law before advertising one.

Bundling

Bundling sells two or more products together for one price, usually less than the sum of their separate prices: a fast-food value meal, a hotel room with breakfast, a suite of software modules. Pure bundling sells only the bundle; mixed bundling offers the bundle and the separate items. Bundles can raise the amount each customer spends, and they can capture more of the value customers place on products when their tastes differ.

Priced separately, the best the seller can do with the reporting module is $600 (both customers buy at $300; at $500 only X buys) and with billing $400 (both buy at $200, or Y alone at $400): $1,000 in all. Offered as a bundle at $700, both customers buy, for $1,400. The bundle works because the two customers value the modules in opposite ways, so their values for the pair are closer together than their values for either module. For software an extra copy costs almost nothing, so the extra revenue is nearly all contribution. For physical goods, check that the bundle’s price still covers the variable costs of everything in it.

Price discrimination

Price discrimination, often called segmented pricing in marketing, charges different customers different prices for the same product when the difference is not explained by a difference in cost. Cinemas and museums charge students less; hotels and airlines charge different rates for the same room or seat depending on when you book, whether you can cancel and which day you stay or travel. It works when three conditions hold: the segments differ in price sensitivity; the seller can tell them apart, by a student card or by letting customers sort themselves through conditions such as advance purchase; and buyers paying the low price cannot easily resell to those paying the high one.

The conditions attached to cheaper rates are called fences. A non-refundable rate booked a month ahead suits a leisure traveller but not a business traveller whose plans may change, so each group chooses the rate meant for it. A fence that leaks, letting price-insensitive buyers take the cheap rate, gives revenue away. The applied example works through the arithmetic for a hotel.

Legal and ethical limits

Pricing freedom has limits that differ from country to country, so treat what follows as general principles, not legal advice. Agreements between competitors to set prices (price fixing) are prohibited by competition law in many jurisdictions. Advertised comparisons with former or recommended prices must be genuine. Selling below cost to drive a rival out of a market and then raising prices (predatory pricing) can break antitrust law. Charging competing business customers different prices for the same goods is restricted in some jurisdictions, and price differences based on characteristics protected by anti-discrimination law raise serious legal problems. OpenStax notes that many US states prohibit price gouging during declared emergencies.

The law is the minimum. Customers also judge prices for fairness, and OpenStax notes that customers who feel they are not getting a fair deal from price discrimination may stop being loyal. Two practical tests help: could you explain the price difference openly to the customers who pay more, and would you be comfortable seeing it described in the press?