[ Note from the editor: I am pleased to introduce our first guest blog from long-time friend and collaborator, Graham Sack. The following draws out the lessons of a bygone economic theory for a discipline in crisis. Graham’s history of the “Hotelling line” is an apt and informative follow-up to my recent lengthy post on Paul Krugman and the dangerous but seductive simplicity of reductive economic models. Another example of how insufficiently self-critical knowledge can be led astray. ~ drferris ]
In March 2009, as the markets lay in shambles and prognosticators debated the merits of the recently enacted stimulus bill, there quietly passed the eightieth birthday of the ‘Hotelling line,’ an influential microeconomic model first proposed by Stanford professor Harold Hotelling in his article “Stability in Competition” (PDF). The paper launched two new sub‐disciplines within economics—the study of spatial competition and product differentiation—and is required reading for graduate students and aspiring microeconomists. Along with Edgeworth’s box and Nash’s equilibrium, it is one of the foundational metaphors of the discipline, arresting for its simplicity, clarity, and explanatory power. It is also fundamentally flawed.
Hotelling begins his article by imagining a line of fixed length (usually visualized as a boardwalk) along which consumers are evenly distributed. There are two sellers in the market and each selects a location and a price. Consumers then choose a preferred supplier by weighing price against the disutility of traveling long distances. The sellers reädjust their location and price, trying to maximize profits, until an equilibrium is achieved.
The popularity of the model derives from the fact that it can be interpreted literally or figuratively. The line can represent geographic space (such as a street or linear city) or can function as a metaphor for a spectrum of similar, substitutable products (such as ice cream flavors or automobile models). In the latter case, the firm’s location on the line represents its place in the product spectrum, while the distance to a consumer represents how far the firm’s product is from that consumer’s ideal. This flexibility of interpretation enables the model to describe two apparently unrelated phenomena: (1) where businesses locate (spatial competition) and (2) what products they produce (product differentiation).
Although the metaphor is elegant, Hotelling botched the implementation. To simplify calculations, he assumed that the disutility of travel (also interpreted as transportation cost) was a linear function and arrived at the conclusion that both sellers will locate in the middle of the line because any deviation to the right or left would give share to the competitor. Hotelling concluded that sellers tend towards “minimal differentiation,” producing near‐identical products with only nominal variations. He read the phenomenon into numerous facets of American life, from the homogeneity of shoes and apple cider to the similarity of Democratic and Republican party platforms.
For fifty years, Hotelling’s conclusion was accepted. Then, in 1979, a trio of European microeconomists led by Claude d’Aspremont (himself a Stanford grad) showed that Hotelling’s original result was incorrect. In fact, it was more than incorrect. It was the opposite of correct. The assumption of linear transportation costs in the original paper led to discontinuities in the consumer utility function and therefore no solution existed: it turned out there was no ‘stability’ in “Stability in Competition.” When d’Aspremont fixed the continuity problem by assuming quadratic transportation costs, he arrived at a result diametrically opposed to Hotelling’s: the two sellers will locate at opposite ends of the boardwalk rather than in the middle—there will be maximal differentiation rather than minimal differentiation. Since 1979, literature on Hotelling’s line has proliferated: authors have shown that whether one arrives at minimal or maximal differentiation depends on numerous factors, including the functional form of the transportation costs, the type of competition (price vs. quantity), and whether search and advertising costs are present.
The salient question, of course, is how fifty years could go by without anyone catching Hotelling’s mistake. The math is not difficult (for academic economists, anyway), requiring only algebra and basic calculus. Why did no one see the error? One answer is that the model’s power as a metaphor overwhelmed and obscured inconsistencies in its logic and predictions. Hotelling’s mapping of the abstract idea of “difference” onto concrete, easily visualizable space is, along with Marshall’s supply and demand curves, one of the most elegant acts of representation in all of economics, and economists, like poets, were persuaded by symmetry, beauty, and concision. The artistry of Hotelling’s model temporarily eclipsed the incongruities of minimal differentiation.
Hotelling’s article was published in Econometrica in March of 1929, a mere six months before the Great Stock Market Crash. Unlike macroeconomics, faulty microeconomics doesn’t blow up markets or create depressions—at least not directly—and Hotelling’s error is negligible compared to those made by central bankers and policymakers of the time. But the difference is one of degree rather than of kind. Hotelling made a basic conceptual error in his model of economic behavior and applied his results too broadly. His model was accepted uncritically by others and reproduced for half a century.
The relevance to contemporary economics should be apparent. In the aftermath of the financial crisis and a year of chastening public rebukes, economists are under pressure to re‐examine their methods of representation and argumentation and to consider the factors that persuade them to believe in one model over another. The story of the Hotelling line is an example of how powerful metaphors can illuminate, but also blind. A greater attention to metaphor may help contemporary economists to develop more realistic and descriptively powerful models and avoid multi‐decade errors such as the theory of “minimal differentiation.”
~ Graham Alexander Sack
Great article, Graham. Thanks for the contribution. You are so right to draw attention to the power of metaphor and its use and abuse. More to come on that…
With respect to product differentiation, check James Surowiecki’s recent “Financial Page” in the New Yorker, “Soft in the Middle,” on Apple’s iPad, General Motors, and the shrinking middle of the consumer market.
Surowiecki notes Apple’s fundamental gamble with the iPad (as with their other products) that people will pay for quality. With respect to the Hotelling line, their gamble is to locate at one end of the line — in this case, the high end of the consumer market — high quality, high price.
For Apple, which has enjoyed enormous success in recent years, “build it and they will pay” is business as usual. But it’s not a universal business truth. On the contrary, companies like Ikea, H. & M., and the makers of the Flip video camera are flourishing not by selling products or services that are “far better” than anyone else’s but by selling things that aren’t bad and cost a lot less. These products are much better than the cheap stuff you used to buy at Woolworth, and they tend to be appealingly styled, but, unlike Apple, the companies aren’t trying to build the best mousetrap out there. Instead, they’re engaged in what Wired recently christened the “good-enough revolution.” For them, the key to success isn’t excellence. It’s well-priced adequacy.
These two strategies may look completely different, but they have one crucial thing in common: they don’t target the amorphous blob of consumers who make up the middle of the market. Paradoxically, ignoring these people has turned out to be a great way of getting lots of customers, because, in many businesses, high- and low-end producers are taking more and more of the market….
While the high and low ends are thriving, the middle of the market is in trouble. Previously, successful companies tended to gravitate toward what historians of retail have called the Big Middle, because that’s where most of the customers were. These days, the Big Middle is looking more like “the mushy middle”… The products made by midrange companies are neither exceptional enough to justify premium prices nor cheap enough to win over value-conscious consumers….