Loss leading, or pricing some goods below cost, has a long and controversial history; for instance, most US states still today prohibit below-cost pricing of gasoline. The standard worries are two-fold. First, stores will take advantage of uninformed consumers by luring them in with a discount on one item and charging high prices on the rest. Second, stores will keep the prices low until their competitors are forced out of business. Neither of these stories seem to explain the very common practice of loss leading on grocery staples or other everyday purchases, where surely consumers are aware of the full cost of the bundles they regularly buy, and exit or entry is less salient.
Chen and Rey show an additional intuitive reason for loss leading: screening. Imagine there are two goods, A and B. Large stores sell both, while specialty or discount retailers sell only B, with unit costs cLA, cLB and cSB; the specialty retailer has a cost (or quality) advantage in B. Let consumers dislike shopping, with a heterogeneous cost of shopping for shopper i of s(i) for each store they patronize. Let consumers have homogeneous unit demand (vA>cLA and vB>cLB) for both A and B. If only the large store exists, it can’t screen by shopping cost, so it just sets a uniform price for the bundle of goods A and B to maximize profit; this means that those with low shopping costs will earn some rents since I keep the price low enough that even high shopping cost folks buy. If, on the other hand, the specialty retailers exist, the large store can sell B at below cost, keeping the combined price of A+B the same as before. This ensures that the large store continues to extract full rent from the high shopping cost buyers, and allows full extraction of willingness to pay for good A from low shopping cost buyers (who now visit both stores).
The authors prove that whenever the large retailer finds it worthwhile to price such that at least some shoppers buy both A and B at the large store, then that store will loss lead with B. As long as the distribution of shopping costs is sufficiently high, the large store earns higher profits when they face small store competition than under monopoly, since the small store can be used to screen for shopping costs, and hence for willingness to pay. This flavor of result is general to having only one competitor rather than a competitive fringe of small firms, as well as other loosened assumptions. Banning loss leading increases total social welfare as well as consumer surplus; those who shop at both venues are made better off, as are those who have shopping costs just too high to make shopping at both venues worthwhile, while every other consumer and the large firm earn the same surplus.
Note that the exact same model applies to platform competition. If Microsoft sells an OS and a media player, and small firms also sell media players, then the small firms allow Microsoft to screen for people who have different costs for learning a new technology, and hence earn higher profits than under uniform pricing. Likewise, the authors mention a large airplane manufacturer in Europe who sells more types of airplanes than their rivals. Rivals allow the large manufacturer to subsidize the planes which have competition and increase the price of planes that do not have competition by providing a way to screen for willingness to pay for a single manufacturer of parts should the planes break.
Also, as a matter of theory, it is interesting that banning loss leading, and hence banning a form of price discrimination based on shopping cost, raises social welfare; the usual result is that price discrimination improves total welfare vis-a-vis the monopoly uniform price. What is happening here comes from competition: efficiency is gained when everyone with shopping cost lower than the difference in costs for producing B, cLB-cSB, buys from the small store. The ability to subsidize B causes some of those shoppers to buy both items in the large store, inefficiently.