Abstract
Many articles have been written about Harold Hotelling’s model of two competitors on a fixed line segment, competing by choosing mill price and location. Most of them have focused on the paradoxical case of crowding in the middle when demand is totally inelastic. Yet Hotelling conjectured that this would not happen if the consumers not only chose the least expensive supplier, but their demand were dependent on the price charged. However, surprisingly little has been written about the case with elastic demand. Even less has been attempted to put the problem in a dynamic format.
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- 1.
No doubt the paradox created its popularity. Scientists replaced distance by just “similarity” in some vague sense for competing products, or even for political opinions, all with doubtful measurability. Such vague analogies deprive such a good scientific model of its qualification as science. Further, to escape some consequences of the paradox, the unrealistic and most contrived idea of quadratic transportation costs was launched and surprisingly gained popularity. d’Aspremont et al. (1979).
- 2.
However, this analysis (without motivation) assumed that the competitors shared the market as a duopoly in a common boundary point, and that at the other ends each market extended to the boundary points of Hotelling’s fixed interval.
In a communication to the present author Dr. Helge Sanner pointed out that the last may not be true—the competitors might also end the markets where local demand dropped to zero. Unfortunately the present author has not been able to locate any publication by Dr. Sanner to cite on this.
Further thought, however, indicates that this case would never happen. If one endpoint only extends to the point where demand vanishes, the competitor in question would always profit from moving its location until an endpoint of the fixed interval is reached. This is because greater spatial symmetry of the market resulting from this always increases profit.
- 3.
Mill pricing, where consumers pay for full transportation costs is our case. However, it is by no means the only possibility. As the competitors are monopolists in their market areas, they can also themselves provide for delivery and apply price discrimination, provided they do not charge more for transport than its actual cost. It is well known that with linear demand perfect discrimination implies charging for exactly half the transportation cost.
- 4.
A more general formula for demand is \(q_{i}=\max \left( a-b\left( p_{i}+k\left| x-x_{i}\right| \right) ,0\right) \), but experience from work with linear models shows that parameter b has no independent influence, so we simplify by putting \(b=1\).
- 5.
- 6.
Actually they represent even more cases as a generic firm i could locate to left or right. Further a monopoly case can represent disjoint monopolies, or cutting out the competitor.
- 7.
We chose to denote the firm to the left 1, and the one to the right 2. This works in each single step, but it is fully possible, and shows up in numerical experiment, that each firm can move to the other side of its competitor, and so we must keep track of the numbering.
- 8.
Of course, a change of location will be followed by a change of mill price. But this is a further issue, location can be chosen so as to maximize sales alone.
- 9.
Notably, this time we did not to optimize with respect to location to get the formula. Demand, and therefore profit as well, is independent of location, and location itself indeterminate. Therefore, we have the disadvantage of not getting a definite location choice for the map we want to formulate. To solve this problem, consider that we deal with coexistent disjoint monopolies, whose maximum profits do not depend on location. We can choose either \(-1\) or 1. This actually means merging the case with cases 2 or 6.
- 10.
Note that this case can only occur when transportation cost is very low. As the entire interval \([-1,1]\) must be covered, market radius \(\frac{a-p}{k}\) must exceed unity, i.e. half the interval.
- 11.
Note that though price according to (30) is in the right interval between c and a, it is by no means certain that the location according to these formulas is reasonable—depending on parameters an “optimal” location may even fall outside the admissible interval \(\left[ -1,1 \right] \). We will therefore have to run a check of relevance with respect to region for each alternative once we proceed to formulating the map. Note further that in the monopoly cases (except for cutting out) the location like optimal price is independent of the competitor’s moves, whereas in the shared market cases 4 and 6 the best reply depends on the expected move by the competitor.
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Puu, T. (2016). Hotelling Duopoly Revisited. In: Matsumoto, A., Szidarovszky, F., Asada, T. (eds) Essays in Economic Dynamics. Springer, Singapore. https://doi.org/10.1007/978-981-10-1521-2_3
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