# Locational Product Differentiation

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- Additional Contributors: None
Categories: CategoryModel CategoryMicro CategoryIndustrialOrganisation CategoryImperfectCompetition

Related Models: Cournot_Model Monopolistic_Competition

Created: 2008-07-15

Suggested Citation: See license page.

Contents

## Introduction

In locational models we represent product differentiation via location in a notional space (this could be actual physical space or a conceptual product space defined by the product characteristics). Users and firms/products are located at different points in this space. Users (or symmetrically) firms must incur a 'travel' cost. User location and demand is usually given and the focus is on firms location and pricing choices. In what follows we present two well-known, and analytically tractable cases.

## Linear City

### Model

- A unit mass of consumers uniformly distributed over a city of length 1. Thus we identify consumers with location on $$[0,1]$$
Two firms/stores $$A,B$$ located at $$a,b$$ respectively (WLOG set $$b > a$$). Initially we ignore location choice and set $$a = 0, b= 1$$ respectively. Competition is of Bertrand form, i.e. in price.

- Transportation costs are of form $$c(d)$$ where $$d = d(\textrm{location of consumer},\textrm{location of product})$$ is distance between consumer and the firm. We will restrict the function $$c$$ to two particular cases:
- Linear: $$c(d) = k d$$
- Quadratic: $$c(d) = k d^{2}$$

Thus a consumer living at $$t$$ incurs cost $$-k t$$ of going to $$A$$ and $$- k (1-t)$$ of going to $$B$$ under linear costs.

- Consumer utility given by: $$u(t, p_{X},X) = \phi - c(t, X) - p_{X}$$ for consumer of index $$t$$, purchasing from firm $$X$$ (if no purchase is made utility is 0).
- Denote demand for firm $$X$$ by $$D_{X}(p_{A},p_{B})$$

### Solving

Demand $$D_{A}$$ is equal to the index, $$t$$, of the consumer who is indifferent between the two products. I.e. $$t$$ defined by:

$$ \[ u(t,p_{A},A) = u(t,p_{B},B) \] $$

In the special case where firms are located at the two extremes ($$a = 0, b = 1$$) both quadratic and linear costs yield:

$$ \[ D_{A}(p_{A},p_{B}) = \frac{(p_{B} - p_{A} + k)}{2k} \] $$

Taking a firm with marginal cost $$c$$ the firm solves standard monopolists problem:

$$ \[ \textrm{max} \Pi^{X} = (p_{X} - c) D_{X} \] $$

Solving yields $$p_{X} = c + k$$ with $$\Pi^{X} = k/2$$

Moving away from the special case and allowing for arbitrary $$a, b$$ the linear cost model presents certain analytic problems as the demand function is discountinuous in price. This can result in the non-existence of any pure-strategy price equilibrium (see aspremont_ea_1979 for details. A mixed strategy equilibrium will still exist, see maskin_ea_1986).Thus attention is restricted to the case of quadratic costs.

Using the condition for the marginal consumer we obtain:

$$ \[ D_{A}(p_{A},p_{B},a,b) = a + \frac{b-a}{2} + \frac{ p_{B} - p_{A} }{ 2k(b-a) } \] $$

With (Nash) equilibrium prices given by:

$$ \[ p_{A}(a,b) = c + k(b-a)\frac{2 + b + a}{3} \] $$

$$ \[ p_{B}(a,b) = c + k(b-a)\frac{4 - b - a}{3} \] $$

Using the envelope theorem $$(\frac{\partial \Pi^{A}}{\partial p_{A}} = 0)$$ we have:

$$ \[ \frac{d \Pi^{A}}{da} = (p_{A} - c) ( \frac{\partial D_{A}}{\partial a} + \frac{\partial D_{A}}{\partial p_{B}}\frac{\partial p_{B}}{\partial a} ) \] $$

Calculating and substituting and using previous results for prices we obtain:

$$ \[ 0 > \frac{d \Pi^{A}}{da} \] $$

Which implies that firm A will locate at the extremity, i.e. $$a = 0$$. This is an interesting result demonstrating that the strategic effects (competitors lower price response) outweighs any business stealing effect of locating closer together.

## Circular City

Here we replace the linear city of the first example with a circular one. We assume that firms are distributed uniformly spaced around the circle. This then results in an analogous situation to the linear city for each stretch between two firms is a replication of the linear city above but of length 1/N when there are N firms. The main benefit is to endogenize the number of firms by imposing a fixed cost of $$f$$ and allowing free-entry (a zero profit condition).

Remark: economides_1984 demonstrates that the symmetric distribution where firms locate equidistantly still holds when entry location is endogenized -- at least in the case of quadratic costs. He considers a three stage game in which firms first choose whether to enter, then choose locations and then compete on prices. He shows that there exists a free-entry symmetric equilibrium.

For analogy with the linear model denote two of adjacent firms by $$A,B$$. Then (for firm A) demand is given by ($$p$$ is the price of other firms (which is assumed the same by symmetry):

$$ \[ D_{A}(p_{A},p) = \frac{p - p_{A} + k/N}{k} \] $$

And the firm seeks to maximize:

$$ \[ \Pi_{A} = (p_{A} - c) \frac{p - p_{A} + k/N}{k} - f \] $$

Solving by differentiating wrt $$p_{A}$$ and setting $$p_{A} = p$$ yields:

$$ \[ p = c + \frac{k}{N} \] $$

Solving for $$N$$:

$$ \[ \frac{p - c}{N} - f = 0 \Rightarrow \frac{k}{N^{2}} - f = 0 \Rightarrow N^{e} = \sqrt{k/f} \] $$

This in turn gives:

$$ \[ p^{e} = c + sqrt{kf} \] $$

NB: thus an increase in the number of firms has a dual benefit. On the one hand it reduces the average 'distance' between a consumer and a product $$= n/4$$. On the other it reduces prices.

## Generalized Location Model

- Start from pure product space
- Factor this out into a defined set of product characteristics that can be approximated as a linear continuum or sphere (e.g. ease of use, power, speed ...). cf. hedonic price indices and regressions.
- Break product space down into a set of principal components with, one hopes, a relatively small dimension.
- Consider distributions of consumers over this space.

Simple case two compenents and we can take the sphere: $$S^{2}$$

### Existing Work

Format:

- Paper
- City Type
- Consumer Distbn
- Results
hanson_2004 _Location Discrimination in Circular City, Torus Town, and Beyond_. However although this generalizes to torii of arbitrary dimension it has a set of fairly strong assumptions (e.g. location of shops is forced to a be a grid) that make it rather limited.

- Tabuchi and Thisse (1995)
- Hotelling
- Triangle (with peak at centre of city)
- Explicitly solve the price-location problem for two firms in the presence of a symmetric triangular consumers distribution. In this case any symmetric location cannot be an equilibrium, due to a discontinuity of the reactions functions generated by the non-differentiability of the consumers' density at its modal value. Instead the model exhibits two subgame perfect asymmetric equilibria characterised by strong product differentiation.

Symmetric and Asymmetric Equilibria in a Spatial Duopoly, Marcella Scrimitore, 2003, http://ideas.repec.org/p/wiw/wiwrsa/ersa03p194.html

- Hotelling
- Trapezoid
- Abstract:

"In this paper, we assume that consumers are distributed according to a trapezoid distribution. This allows a simple parametrization of the degree of consumers' concentration, which includes the uniform and the triangular distribution as limit cases, and makes possible to solve the price-location problem as a function of the concentration index. Therefore we are able to find a more general explicit solution which covers those previously discussed in the literature. The basic results of the paper are the following. A symmetric equilibrium exists for all values of the concentration parameter, provided that the density is differentiable at the centre of its support. A higher degree of the consumers' concentration around the middle induces firms to move inwards, in order to locate closer to the growing share of consumers: competition in the highlyopulated central area of the market reduces differentiation and strengthens price competition. The overall equilibrium shows clearly that the demand effect outweighs the strategic effect. However the symmetric equilibrium may be not unique. When concentration becomes sufficiently high, two asymmetric specular equilibria coexist with the symmetric one. They arise for a degree of concentration lower than that implied by a triangular distribution, with price-location choices collapsing in the limit to those identified by Tabuchi and Thisse. At these equilibria one firm locates in the central area of the market, while the other locates outside the market space. These results are consistent with the idea that a higher concentration of consumers around the centre induces firms to reduce the optimal product differentiation and offer theoretical support to the intuition that homogeneity of consumers might have important implications in terms of reducing the firms' market power. However, our findings suggest that in models of spatial competition realistic representations of the demand side may generate a "strange" interplay between the strategic effect and the demand effect which may cause a failure of the uniqueness property and weakens the economic interpretation of equilibria."

## Bibliography and References

- **Gabszewicz, C.J. and Thisse, J.-F.d'Aspremont, C.; ** On Hotelling's Stability in Competition, Econometrica pp. 1415-1151, 1979.
- **Bonano, G. and Zeeman, C.** Limited Knowledge of Demand and Oligopoly Equilibria, Journal of Economic Theory pp. 276-283, 1985.
- **Dixit, Avinash and Stiglitz, Joseph E.** Monopolistic Competition and Optimum Product Diversity, American Economic Review, 1977.
- **Economides, N.** Symmetric Equilibrium Existence and Optimality in Differentiated Product Markets, 1984.
- **Ethier, Wilfred** National and International Returns to Scale in the Modern Theory of International Trade, American Economic Review pp. 389-405, 1982-06.
**Hanson, R. and Zeeman, C.** Location Discrimination in Circular City, Torus Town, and Beyond, 2004?. <hanson.gmu.edu/torustwn.pdf>

- **Hart, Oliver** Monopolistic Competition in the Spirit of Chamberlin: A General Model, Review of Economic Studies, 1985.
- **Hart, Oliver** Monopolistic Competition in the Spirit of Chamberlin: Special Results, Economic Journal, 1985.
- **Tirole, Jean** The Theory of Industrial Organization, 1988.