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 This is extended in unpublished work by [7] , where the consumer distribution can take several forms ranging from uniform (rectangular) to triangular (densities in-between the two assume a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.

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:

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:

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

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

[1] derive the bounds for the ratio between social-optimum and equilibrium differentiation for symmetric distributions. Specifically they find

where

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

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:

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

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

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

The first derivative is

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

of divergence, R, increases with c.

As c increases for the Power distribution, it can be observed that the optimum amount of product differentiation decreases while the equilibrium amount of differentiation increases. The divergence between the two gets larger as c increases. The need for government intervention may increase as the distribution gets more skewed (Figure 2).

In [5] , it was determined that variance affects the equilibrium outcome of the location game. Subsequently, [1] compared the Uniform, Normal and Logistic for the sensitivity of location to changes in variance, as they are most commonly used to describe populations. Figure 3 shows the effect of variance on the amount of differentiation in the social optimum outcomes.

We find that the socially optimum amount of differentiation is the most sensitive to variance under the Uniform distribution; distance between firms increase at a faster rate compared to the Logistic and the Normal distributions. The Logistic is the least sensitive to changes in variance. This result is consistent with the equilibrium analysis in [1] .

We calculate R to test the effects of changes in variance on the amount of divergence of equilibrium from optimum, Table 8 contains these values. As derived in Proposition 3, variance does not have any effect on the amount of divergence of the equilibrium from the optimum (R). The amount of divergence from the equilibrium is largest for the logistic distribution and smallest for the uniform distribution.

Note: the values of D^* are found in [1] .

In [5] , an implicit system of equations was provided which we build upon to give an explicit closed form solu-

tion for the social-optimum locations for symmetric consumer densities—each firm should be located one mean- deviation away from the median. With this result we are also able to provide an explicit formula for the ratio of excess differentiation R (socially optimal differentiation relative to equilibrium differentiation). We show that the excess differentiation ratio is invariant to a multiplicative increase in the dispersion of consumers around their mean/median.

We derive the social-optimum locations and prices for densities with unique pure strategy equilibria in a two- stage location-price game and examine the degree of excess product differentiation. The results show that firms under the Uniform and Rayleigh densities have the greatest excess differentiation at their equilibrium locations.

This paper also looked at the potential saving from the regulation of firm decisions via the calculation of the savings ratio (S) in Section 3.1. We found that the Laplace has the lowest S and thus give governments the least incentive to intervene. The Power, Uniform and Rayleigh distributions exhibit high savings ratios and would benefit the most from government intervention. Policy makers and urban planners should consider consumer distributions when deciding to regulate firm location decisions.

The implicit social-optimum locations provided by [5] apply to both symmetric and asymmetric densities as they are transport cost minimizing. But the solution does not take into account the redistributive effect of prices in the pricing stage. The social-optimal locations for symmetric densities are chosen such that firms will naturally choose equal prices at the pricing stage. As long as prices are equal, the indifferent consumer remains at the median and transportation costs are minimized. On the other hand, under asymmetric densities (such as the Rayleigh distribution) firms will not choose equal prices (at the pricing stage) due to the asymmetric nature of the distribution. When prices are unequal, more consumers go to the cheaper firm (than socially optimal); this will cause transportation costs to deviate from the minimum. Thus, the sole regulation of location will not work for asymmetric densities as firms will not choose symmetric prices. This problem can be tackled by price regulation or one can work out an alternative location solution which will minimize transport costs where firms can set their own prices. The latter is a question open for future research.