Abstract
In this chapter, we illustrate some classic spatial models. We provide a collection of the principal classic spatial models, by illustrating their characteristics and their main results. In particular, we discuss the Hotelling linear market and the Salop circular market, both under uniform pricing and under price discrimination. The location vertical differentiation model is also discussed.
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Notes
- 1.
Obviously, the importance of the spatial dimension has been well recognized even before Hotelling. For example, Thunen (1826), Launhardt (1885), Marshall (1890), and Weber (1909) developed relevant frameworks to understand the implications of space for consumers and firms’ behavior. However, none of these models has been used for plenty of applications as the Hotelling one and its epigones.
- 2.
It should be observed that the main purpose of spatial models is to derive endogenously the “locations” of firms. However, it might be useful to start with the case of exogenous locations.
- 3.
v is assumed to be sufficiently high so that the market is always covered in equilibrium.
- 4.
This is not the only possible timing. For example, one might consider a simultaneous choice of location and price. However, the sequential timing is more reasonable when one considers that it is often more difficult to modify the location/product characteristic rather than the price.
- 5.
That is, the number of firms must be an integer.
- 6.
Note that there is no contradiction with the equilibrium prices in Thisse and Vives (1988) in the DD subgame. In that case the transportation costs were sustained by the consumers. Here the transportation costs are sustained by the firm. Therefore, the profit margin is the same.
- 7.
Indeed,
$$ {\displaystyle \begin{array}{c}{\pi}_i\left({z}_A,{z}_B,p\ast \right)=\int {\int}_S\left[{f}_j\left({z}_j,z\right)+{c}_j-{f}_i\left({z}_i,z\right)-{c}_i\right]\rho (z) dz\\ {}=\int {\int}_S\left[{f}_j\left({z}_j,z\right)+{c}_j\right]\rho (z) dz-\int {\int}_S\min \left[{f}_j\left({z}_j,z\right)+{c}_j,{f}_i\left({z}_i,z\right)+{c}_i\right]\rho (z) dz\\ {}=\int {\int}_S\left[{f}_j\left({z}_j,z\right)+{c}_j\right]\rho (z) dz-K\left({z}_A,{z}_B\right)\end{array}}. $$ - 8.
The continuity of function K on S also guarantees that the location equilibrium exists.
- 9.
Colombo (2011) extends to the case of endogenous locations.
- 10.
However, it does not emerge in the case of hyperbolic demand function (Colombo 2016).
- 11.
In other words, the lower price is not sufficient to compensate for the higher sensitivity to the price of consumers.
- 12.
It can be shown that ϑ is the inverse of the marginal rate of substitution between income and quality. That is, consumers have different incomes, and wealthier consumers have a lower marginal utility of income and a higher ϑ.
- 13.
Under some appropriate restrictions on the parameters, this conjecture is correct in equilibrium.
- 14.
However, this conclusion is not always true: if the lowest level of quality is particularly low so that the market is uncovered, the low-quality firm would end up with zero demand. In this case, there is less than maximal differentiation in equilibrium.
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Colombo, S. (2020). Classic Spatial Models. In: Colombo, S. (eds) Spatial Economics Volume I. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-40098-9_1
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