On Distance Metrics in Location Problems

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I. INTRODUCTION

The positioning of brands, placement of stores, or design of public goods is usually treated as an optimal location problem in a product space. Firms and planners have an incentive to create value by endowing goods with attributes close to what many consumers prefer. If more than one attribute matters to consumers, closeness is an ambiguous notion; various nonequivalent distance measures are available. These metrics have an economic interpretation in terms of the substitutability/complementarity of attributes. While the Euclidean metric is often used by default, I argue that many others represent plausible consumer preferences.

The Euclidean metric invokes a special demand geometry that is rotation invariant and thus preserves aspects of one-dimensional analysis, where there is no role for direction independently of distance. In non-Euclidean demand environments, location problems have different solutions that are sensitive to both distance and direction. (1)

As a case in point, I consider the question whether offering a more differentiated pair of public goods increases welfare for a population with fully diverse (uniformly distributed) tastes. This statement is always true when preferences are "Euclidean," but false otherwise. Welfare may strictly deteriorate if the goods are differentiated (moved farther apart in attribute space) in the wrong directions.

I show that, for all p-metrics, differentiating proportionately in all attributes guarantees a welfare improvement. Proportionate differentiation is a movement along the line through the initial good locations, hence in a fixed direction. The result is consistent with the notion that solutions in Euclidean environments only generalize if the direction of differentiation is restricted.

Related (single-attribute) problems have been studied in different contexts. Social choice theorists considered ways to arrange two goods on a line which satisfy efficiency and consistency criteria. Ehlers (2002, 2003) found that Pareto optimality and fairness requirements select the "extreme-peaks rule," which places the goods at the smallest and largest locations someone in the population prefers. This is also the only admissible rule if Nash's and Arrow's independence axioms are imposed instead of fairness (Ehlers 2001). (2) Under the extreme-peaks rule, goods will be farther apart if the population's tastes are more diverse.

In ranking fiscal policies, all individuals prefer more of a public good. Yet they are actually offered a bundle of service and taxes. Because valuations of the services vary, there is disagreement about the ideal level of taxation. In theory, individuals move to the jurisdiction where the most acceptable fiscal policy is in force or achievable through voting. This is formally equivalent to consuming the preferred good. Perroni and Scharf (2001) examine this problem with individually preferred fiscal policies distributed uniformly on the real line. In equilibrium, jurisdictions are equally sized intervals that select the median policy by majority voting. Thus, policies are evenly spaced along the line, which is efficient for a given number of jurisdictions.

Even spacing is the natural extension of the "maximal-distance principle" to more than two goods. It also occurs in multi-store monopoly. Under typical assumptions, one location arrangement is no more costly to the monopolist than another, for a fixed number of plants. If consumers bear the transport cost and the monopolist can partially appropriate the benefits of reducing it, the monopolist places the plants as a planner would. With consumer types uniformly distributed, as in the study of Katz (1980), Pal and Sarkar (2002), or Matsumura (2003), even spacing of stores occurs in equilibrium.

The thrust of these one-dimensional examples, where optimal locations are in some sense distance-maximizing, carries over to higher dimensions as long as the metric is Euclidean. …