Abstract

Advances in the theoretical literature have extended the Hotelling model of spatial competition from a uniform distribution of consumers to the family of log-concave distributions. While a closed form has been found for the equilibrium locations for symmetric log-concave distributions, the literature contains no closed form solution for the socially optimal locations. We provide a closed form solution for the socially optimal locations: one mean-deviation away from the median. We also derive a formula for the excess differentiation ratio which complements the bounds previously derived in the literature, and establish the invariance of this ratio to a form of mean preserving spread. The equilibrium duopoly locations of several types of commonly used distributions were discussed in [1] . This paper provides the closed form solutions for the socially optimal locations to the same set of distributions. We calculate welfare improvements arising from regulation of firm location and show how these vary with the distribution of consumers. While regulating firm locations is sufficient to optimize welfare for symmetric distributions, additional price regulation is required to ensure social optimality for asymmetric distributions. These results are significant for urban policy over firm/store locations.

Keywords

Socially Optimal Firm Locations, Distributions, Spatial Competition, Duopoly

1. Introduction

Transport costs are a primary driver of the location decisions of firms in models of spatial economics. Transport costs can occur in an enormous range of contexts: inputs costs for intermediate products or in commute times for workers; or in output markets as search costs, shopping costs or delivery costs for consumers. A large, but by no means exhaustive, survey is provided by [2] . From a firm’s perspective the essential issue of spatial economics is that it cannot be close to everything¹. Identifying how firms handle this trade-off in choosing locations, and how these choices impact on society, gives rise to a central research theme of the literature, and it is to this area that this paper contributes.

Most theoretical firm location models assume a uniform consumer distribution; this is seldom observed in real life. In their seminal contributions [3] and [4] established the existence of price equilibrium for log-concave densities. [5] extended this line of research by providing sufficient conditions for the existence of pure strategy location-price equilibrium for log-concave consumer densities, and for the symmetric case, a closed form equation for the equilibrium locations and prices.

The analysis of socially optimal locations in the literature has not established a closed form for the socially optimal locations and as a result the comparison of equilibrium and optimal locations—in terms of the ratio of competitive product differentiation to the optimal differentiation—has produced bounds but not an exact result. We fill this gap by providing explicit formulas for both the socially optimal locations and the ratio of equilibrium to optimal differentiation.

The research questions of this paper are: 1) what are the socially optimal locations of firms within the canonical, quadratic Hotelling model; and 2) how do the equilibrium, firm level location decisions compare with the socially optimal locations? Our contribution is to extend the analysis of social optimality beyond the typical assumption of uniformly distributed consumers. In most real world settings, non-uniform consumer distributions would be more realistic—our results apply to important specific densities like the Normal and Logistic. One of the few theoretical works, which utilizes a non-uniform consumer distribution, is that of [6] where a symmetric triangular density was used. This is extended in unpublished work by [7] , where the consumer distribution can take several forms ranging from uniform (rectangular) to triangular (densities in-between the two assume a trapezoidal shape). In [8] , a state space approach where different states give rise to different consumer distributions was used to model uncertainty in spatial duopolies.

[1] identified a list of distributions where unique pure strategy equilibrium exists and also determines their locations. This paper follows [1] and examines the socially optimal locations of these distributions. Although [9] showed that there is excessive differentiation in the uniform case of the two-stage location-price game, and [5] supports the existence of excessive differentiation by deriving bounds results about the social-optimum locations; we are the first to examine precisely excess differentiation for a wider range of specific distributions, including asymmetric distributions.

The amount of differentiation and the total transport cost² savings from government intervention are discussed in this paper. We also consider the effect of regulating the firms to the social-optimum locations and allowing them to set competitive prices. This decomposes the welfare gains from regulation into spatial and pricing decisions. This separation is important in practice because location in many industries is much cheaper to regulate/monitor than prices.

Using location as a form of regulation is of interest to governments. For example, the Australian Competition and Consumer Commission found in an inquiry into grocery stores in Australia that incumbent supermarkets have used planning objection processes to deter new entry thus lessening competition. Proceedings in the UK Competition Commission inquiry into Groceries [12] resulted in legal orders regulating the location related behavior of large grocery chains. Governments may use location regulation as a means to regulate competition.

Pricing regulation is hard to administer when stores sell a variety of products. The Australian government attempted to regulate prices of grocery chains by publishing the prices charged by each store for a basket of goods; this proved to be too costly and the website was shut down [13] . This may indicate that location regulation may be a more feasible way to ensure social welfare.

The question of price regulation is only binding under asymmetry, and hence prior to our analysis of asymmetric densities, price regulation could not be considered in any meaningful way³. There is of course more to the location of a firm than the stylized Hotelling model, but it is the workhorse of spatial competition and as we show the welfare implications are sensitive to the choice of consumer distribution.

2. Social-Optimum Location in the Quadratic Transport Cost Model

We use a quadratic transport cost model as specified in [9] , but generalize beyond uniform consumer density. There are two firms on the real line and each produces a homogeneous good at a constant marginal cost, henceforth normalized to zero. Firm i is located at x_i, where I = 1, 2. Without loss of generality, the firm on the left will be denoted as firm 1 and the firm on the right will be firm 2, x₁ ≤ x₂. There is a continuum of consumers, whose mass is normalized to unity without loss of generality. The transport cost incurred by the consumer located at x from consuming the good at x_i is:

(1)

where t is the amount it costs to travel to purchase the good. For simplification, we let t = 1.

The consumer density is, ≥for all x, and its cumulative distribution function is denoted by. We assume is twice differentiable on its support [a, b], where a may be −∞ and/or b can be ∞.

The social-optimum problem is simply to minimize the aggregate transport costs, T, in serving consumers:

(2)

where µ is the boundary between the group of consumers who go to firm 1 and those who go to firm 2. Since demand is inelastic and production costs are constant and identical across firms, the socially optimal value of µ, denoted µ_s, is

. (3)

since it is clearly optimal to serve each consumer from the closer firm.

Remark 1. As shown in [5] , if is logconcave then the first order conditions yields the unique socially optimal locations, i.e. the unique solution to the programme

(4)

2.1. Logconcave Consumer Densities

Given Remark 1, we naturally restrict attention in this paper to logconcave densities. Although logconcavity is sufficient for the existence of a unique pair of socially optimal location, additional conditions are required to guarantee existence of a unique competitive equilibrium, see [5] .

Building on these results, we provide an explicit characterization of the socially optimal locations for symmetric distributions.

Proposition 1. For any symmetric, log-concave distribution with a mean/median of M and mean-deviation, the social-optimum duopoly locations are one mean-deviation away from the median:

(8)

Proof. Let be a symmetric, log-concave probability density function with supports (−b, b) and median M = 0. [5] have shown that the socially optimal locations for duopoly firms are symmetric around the median for distribution. The distance between either firms and the median is

(9)

To prove the proposition, we need to show that this distance equals one mean deviation or

(10)

Since X is symmetrically distributed, on the left-hand-side,

(11)

Integrating by parts gives

(12)

and on the right-hand-side,

(13)

which establishes the result for M = 0. Extension to M ≠ 0 is trivial. ■

For log-concave densities, symmetry of the density is sufficient to make the socially optimal locations of the two firms symmetric about the median/mean of the consumer density. One might expect that as consumer locations become more dispersed, the locations of the two firms should also become further apart. We show in Proposition 3 on spread that this loose intuition is precisely true to the degree that dispersion of consumer preferences is measured by the mean deviation. With a formula for the socially optimal locations in hand, it is now possible to analyze the excess differentiation of the competitive locations compared to the socially optimal locations. Using results from Proposition 1, we solve the system of equations⁴ for each of the density functions which satisfy [5] ’s three conditions (i.e. the distributions listed in Table 1)—the results are given in Proposition 2.

Proposition 2. The socially optimal locations and transport costs for five of the logconcave distributions given in Table 2, where’s are the social-optimum location for firm i, and T^S’s are the social-optimum transportation costs.

Proof. See online working paper at: http://dx.doi.org/10.2139/ssrn.1615884. ■

The results for the Uniform case are already well known; the results for the other four distributions are new and allow for an analysis of spatial competition in the Hotelling framework which is no longer tied to the uniform distribution. The Rayleigh distribution is asymmetric and has socially optimal locations which are not symmetric about either its mean (≈0.886) or its median (≈0.833). Thus the symmetry result for the first four distributions in Proposition 1 does not hold in general for asymmetric densities. In the Rayleigh case the left firm is located in the dense left part of the distribution and the right firm is located well out into the long right tail which goes off to infinity. The locations are sufficiently to the right to place the boundary µ_S to the right of the median and the mean.

(15)

The above can only be solved explicitly for fixed levels of c. Hence the value of µ_S depends on c. To examine the relationship between µ_S and c, the following total differential equation is obtained.

(16)

The first part on the right-hand-side, , is positive. To determine the sign of the second part, we check the first order conditions and concavity of the numerator and denominator. The result shows that both functions are positive given the constraints c ≥ 1, and Equation (15) holds. Therefore, the value of µ_S increases with c, where c determines the curvature of Power distribution’s density function.

Knowing the direction of change of µ_S resulted from a change in c, we can now derive the change in and from Equations (5) and (6). The deviation of firms’ location from µ_S is

(17)

A check on the second order conditions for dµ_S/dc shows that it is a concave function with maximum value at µ_S = 0.5 and c = 1, i.e.. We then have

(18)

since the denominator increases at a faster rate than the numerator. From here we know that the deviation of firms’ locations from µ_S decreases as µ_S approaches 1. As such, product differentiation decreases as the curvature of the density function increases.

To illustrate, we plot the graph of µ_S against c and solve numerically for Power distributions with integer values 2 ≤ c ≤ 10. The graph and values are in Figure 1 and

(19)

where

(20)

R is inversely proportional to the amount of excess differentiation. These bounds show there is always substantial excess differentiation in the location-price game [5] , also show that increasing the dispersion of consumers (in the form below) increases the equilibrium differentiation⁵. In contrast, we now show that the excess differentiation ratio is invariant to this kind of increase in the dispersion of consumers.

Proposition 3. If consumers are distributed according to a continuous, log-concave, symmetric density f on which satisfies the technical restrictions of [5] 6 then

(21)

and the differentiation ratio, , is invariant to the mean preserving spreads of the form

, with s > 0, i.e..

Proof. From [5] , the conditions on f give. The expression for R then follows from direct substitution of the previous proposition:

(22)

Using this explicit expression we now derive the invariance result, without loss of generality assume M = 0. Let E_s denote the expectation with regard to the distribution f_s, then

(23)

where z = y/s. Simple substitution gives the result with M ≠ 0. Thus for the mean preserving spread given by f_s we have

(24)

For this type of mean preserving spread, the increase in the absolute deviation from the median/mean is linear in s while the decrease in the density of the mean is at the rate 1/s, so the two effects exactly balance. That is, in this case the tendency for socially optimal locations to increase because of greater dispersion is proportionally the same as the tendency for competitive locations to move apart due to decreased competition for the marginal consumer at M.

The exact amount of differentiation that each distribution results in is shown in Table 4 and Table 5. As [5] observe that amongst the symmetric distributions the Uniform and Laplace distributions exhibit the greatest and least divergence respectively. However, in extending the analysis to asymmetric distributions, we see that asymmetric distributions have greater divergence than any of the symmetric distributions in the group.

For the Power distribution, under a duopoly location-then-price game, the product differentiation⁷ is

(25)

The first derivative is

(26)

so the amount of product differentiation increases as c increases. Examples are given in Table 5. Since the product differentiation increases under equilibrium and decreases under social-optimum conditions, the amount

Table 4. Amount of differentiation (x₂ − x₁) for equilibrium and social-optimum locations and their ratios.