## Abstract

The Hotelling model was a very clever way to deal with Bertrand oligopoly. The commodities were considered as perfectly homogenous, but the suppliers were separated in geographical space and thus had local monopoly areas. Local price, mill price accrued with transportation costs was what mattered for the consumers and this made it possible to charge different mill prices for the same commodity. If the transportation cost was very high then the monopoly areas could be disjoint, and there would be no competition at all, otherwise they shared a market boundary which would provide for competition. A lowered mill price would decrease the proceeds per unit of product, but it would also increase the market area through pushing away the boundary from the mill’s location. We are intentionally vague on space and boundary. One would naturally want the issue set in true 2-dimensional geographical space, though economists almost never managed. So, Hotelling’s setting is a line interval in which the competitors locate—he refers to an intercontinental railway with stops at the locations, later vulgarized to ice-cream vendors on a shore. Well 2D is difficult, so we should not blame him too much. Worth blame is, however, that he assumed demand to be inelastic. Consumers would choose the cheapest supplier—mill price plus accrued transportation charge—but once this was done they would buy just one unit no matter what the price was. This seems contradictory. As location choice is free, Hotelling concludes that the competitors both crowd in the centre of the interval, and then we are back at the mess of Bertrand’s original case. The cleverly introduced space goes out again. In a short passage Hotelling notes that if demand were elastic then this extreme crowding would not occur, the competitors would locate apart, not at the quartiles as would be socially optimal, but somewhat closer due to the contested middle area. It is a mystery that Hotelling did not make this his main case. Everybody was thrilled at the paradox of crowding in the middle, and the extremely popular model was interpreted in different ways by analogy—similarity of competing goods, convergence to the centre of political ideologies, and whatever in terms of “variables” not defined and not even measurable. Only a few years after Hotelling’s article Lerner and Singer in an ingenious graphical analysis showed that the crowding did not occur, even when the consumers just had a reservation price. This means they would buy one unit if local price did not exceed the reservation price, otherwise they would buy nothing. Around 1940 Smithies set up the entire model with linear demand functions, but he claimed that the revenue integrals were too difficult to evaluate. They were not, rather quite simple, but it took 60 years before the present author did it in 2002, and it seems nobody touched the issue in between. This chapter presents the analysis and deals with some minor details, also devising a dynamic process which shows fast convergence—a bit disappointing if one wants complex dynamics. In the stubs we also suggest putting the case in the two dimensions of geographical space, though here the integrals indeed become problematic, at least if one assumes an Euclidean distance metric. All becomes much simpler with a Manhattan metric, so this is what we suggest. The model is defined, but its analysis is left to the reader.

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