ON DISTANCE METRICS IN LOCATION PROBLEMS

Abstract

Product space metrics reflect the complementarity of product attributes and lead to different geometries in multidimensional spaces. This matters for optimal product positioning. I consider how a planner should design two substitute public goods with multiple attributes to best serve different needs. Attribute complementarity affects whether a given menu dominates another (using an extension of the Pareto criterion). Distancing the goods proportionately in every attribute guarantees improvement when tastes are maximally diverse. The intuition is that the planner must minimize overlap in the populations the goods serve, since benefits are wasted in appealing to the same individual with two offerings. (JEL C65, D78, H41)

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. It does not extend to non-Euclidean spaces. The Euclidean metric reflects a particular type of attribute complementarity, one of many intermediate cases between perfect substitute and perfect complement attributes, which are associated with non-Euclidean metrics.

Alternative metrics have been explored axiomatically in bargaining and social choice theory. Lehrer and Nitzan (1985) asked when a given choice rule could be “rationalized” by a metric on preference profiles.³ The class of admissible choice rules may be further restricted by imposing axioms on the metric. Pfingsten and Wagener (2003) derived axioms that induce the p-metrics; Conley, McLean, and Wilkie (2008) and Voorneveld, von den Nouweland, and McLean (2008) characterized the Euclidean metric (p = 2).⁴ This paper offers a different and nonnormative interpretation of alternate p-metrics. Here they relate to complementarity between decision criteria (attributes) that jointly determine preference.

For a fully diverse population that includes preferences for every possible mix of attributes, Pareto efficiency is not an informative welfare criterion, since it is impossible to relocate goods such that no one is made worse off. Therefore, I allow that improvements make everyone better off “by proxy.” It is enough to find a pairing that matches every loser from a change with a beneficiary whose gains at least offset the other's loss. While a “proxy improvement” implies an increase in total surplus, it is a stricter and more equitable standard. One difference is that proxy improvements would always win in majority voting.

Section I describes demand for public goods in multi-attribute space. Section II explains the connection between distance metrics and attribute complementarity. The geometry of the perfect substitutes case is given special attention, as it is often cited later on. Proxy efficiency is defined and motivated in Section III. Relocation of an isolated variety (which is welfare-neutral) is considered in Section IV. Relocation of non-isolated varieties is discussed in Section V, leading to the result that proportionate menu diversification is a welfare improvement by proxy. Section VI concludes with a brief discussion. All proofs are collected in the Appendix.

II. METRICS AND ATTRIBUTE COMPLEMENTARITY

When individuals choose among goods with multiple attributes, they will weigh a close fit with their tastes in one respect against disagreement in another. In comparing tax systems, voters are concerned with the total they must pay; they should be willing to trade a higher income tax against a lower property tax dollar-for-dollar. Hence the components of a tax system are perfect substitutes. When it comes to government-spending proposals, many voters have ethical priors that result in a perfect complements mentality. Suppose the issue is how to allocate funds between health care and unemployment benefits. Often the policies are ranked by comparing the least acceptable element of each.

Hence preference is affected by how the distance from a good is constructed from distances in single attributes: it depends on the metric. One can appreciate visually that different metrics reflect different degrees of complementarity between attributes. The spaces associated with the p-norms

• 

are the L^p spaces. I often refer to the extremes L¹ and L^∞ (as we shall see these are, respectively, the “perfect substitutes” and “perfect complements” cases) and to the familiar Euclidean space L². When p = ∞, the max norm ||x||_∞ = max _i≤n|x_i| arises, which induces the max metric d_∞(x,y) ≡ max _i≤n|x_i−y_i|.

The ball B (i,r_i) = {x : ||x−i||< r_i} about i is the set of individuals who find their preferred bundle of attributes is no farther than a distance r_i from i. Figure 1 plots this set in two dimensions for L¹, L², L^∞, together with an intermediate case (broken line) between p = 2 and p = ∞.

Figure 1. Ball with Respect to p-Norms

Since distance from i is constant along the boundary of B (i,r_i), |x₂−i₂| must decrease one-for-one with increases in |x₁−i₁| when the norm is ||·||₁, or stay fixed while |x₁−i₁| increases when the norm is ||·||_∞ and |x₁−i₁| < |x₂−i₂|. Hence deviations from the preferred attributes are treated as perfect substitutes in L¹ and as perfect complements in L^∞ (where only a reduction of the worst deviation makes a difference).

This interpretation of metrics has economic content when a ball in L^p is related to consumption. Let the benefit for type x from consuming good i have the form

• (1)

(with α and β positive constants). The set of consumer types x with u_x ( i) > 0 is bounded by such that . Letting r_i = α/β, the types with positive valuations for i form the ball B ( i,r_i).

If goods a and b are on offer, and an individual x consumes at most one of them, then x chooses a if ||x−a|| < ||x−b|| and u_x ( a) > 0. Illustrations in Euclidean and L¹ space are given in Figures 2 and 3. Because these cases reappear throughout the paper, I explain the peculiarities of L¹ with some care.

Figure 2. Demand for Good a in L²

Figure 3. Demand for Good a in L¹

In Euclidean space (L²), a's demand area is the (shaded) intersection of B ( a,r_a) and the halfspace of points that are closer to a. The boundary of this halfspace passes through the intersection of the boundaries of B ( a,r_a) and B ( b,r_b). A shared boundary point satisfies , hence , which implies indifference. This property of the indifferent set is general (and independent of the metric) as long as the underlying preferences are transitive. If x's opportunity cost of consumption is c, then if x is in the boundary of B ( a,r_a) (i.e., u_x ( a) = c), and x is also in the boundary of B ( b,r_b) (i.e., u_x ( b) = c), x must be indifferent between a and b (since u_x ( a) = u_x ( b)).

While balls in Euclidean space are rotation invariant, the same is not true in spaces endowed with other metrics. Figure 3 depicts four cases arising in L¹.

Although the indifferent sets look idiosyncratic, they arise from the same three principles.

(1) At points x in the indifferent set such that min(a₁, b₁) ≤x₁≤ max(a₁, b₁) and x₂≤ min(a₂, b₂) or x₂≥ max(a₂, b₂), a vertical move away from a is also a move away from b, so vertical moves away from center preserve indifference. Analogously, at points x in the indifferent set such that x₁≤ min(a₁, b₁) or x₁≥ max(a₁, b₁) and min(a₂, b₂) ≤x₂≤ max(a₂, b₂), horizontal moves away from center preserve indifference.

(2) At points x in the set such that min(a₁, b₁) ≤x₁≤ max(a₁, b₁) and min(a₂, b₂) ≤x₂≤ max(a₂, b₂), moving away from a parallel to an axis means moving toward b. To stay indifferent, one must compensate for a step away from a in one dimension with a step away from b in the other. Hence the graph of the indifferent set has slope 1 (is parallel to the edges of the r-balls).⁵

(3) Points such that x₁ < min(a₁, b₁) and x₂ < min(a₂, b₂), or x₁ > min(a₁, b₁) and x₂ > min(a₂, b₂), all rank a and b identically. Suppose x in this region is indifferent between a and b. Any other point in the region can be reached by a series of steps parallel to an axis, where every step is away from a and b or toward both. Such moves cannot change the ranking of a and b; they preserve indifference. Hence if one point in this region is indifferent, all of them are. This causes the thick indifferent set in the lower right panel of Figure 3.

Only the lower left panel of Figure 3, where a and b differ in a single dimension, resembles the Euclidean case. At the other extreme, the L^∞ geometry is a 45° rotated version of that in L¹. The p-metrics with p∈ ( 1,2) and p > 2 generate curved indifferent sets that are topological analogs of those in L¹ and L^∞, respectively. The reader will at this point appreciate that the Euclidean geometry is rather special. The metric reflects complementarity between attributes. Since there is no compelling reason why Euclidean complementarity would be likelier than others, one might ask what difference the choice of metric will make in an economic problem. I consider in the remainder of the paper a particular issue in the provision of public goods under alternative preferences that are represented by different metrics.

III. PUBLIC GOOD MENUS

Although public goods are nominally free, demand (for most) is limited in practice. Selective participation reflects non-price costs (time and travel requirements) and the diversity of tastes. Many public goods are offered in varieties to appeal to different patrons. For example, library branches and parks are established in several neighborhoods to meet the location preferences of dwellers throughout the city. But physical place is often not the only dimension on which public good varieties are distinct. Libraries may have a unique genre focus and a quaint or modern atmosphere; parks offer unique wildlife and plant life. These attributes can be thought of as locations in an abstract multidimensional space.

An appealing intuition is that diverse tastes are served best by “diverse” menus of distant goods. In practice, adjustments may have to be gradual, due to cost and time constraints. Thus, it is of interest whether diversifying an existing menu, by increasing the distance between goods, is welfare-improving. I examine this issue for the simplest case of two public goods and uniformly distributed tastes, on the basis of a suitable notion of welfare improvement. Whether the public prefers one menu over another will depend on the complementarity between attributes of the public good, unless the direction of diversification is restricted. I continue to emphasize the geometric aspect of the problem.

A menu M = ( a,b) contains public goods a and b, for each of which a planner chooses a mix of attributes from a feasible set D. Let this set be , where n≥ 2. Although a public good and its attribute mix are separate objects, no confusion should arise from denoting them by the same letter. A menu is then a pair of points in . I adopt the strict convention that a and a′ represent different attribute mixes for the same good, whereas a and b refer to different goods.

ASSUMPTION 1 Menu M∈D×D, where, n≥ 2.

Every individual prefers to consume a particular attribute mix in D, and every point in D is preferred by a nonempty subset of the population with the same density. I denote the group that favors attributes by x. Again, a group and its preferred attributes, or type, are not the same object, but they are referenced by the same letter.

ASSUMPTION 2 Types are continuously and uniformly distributed on every unit n-cube in D.

It is not important whether population mass is interpreted as an actual count or as a probability. What matters is that no one attribute mix is inherently more valuable to society than another. Thus, the social benefit of a menu depends on its breadth of appeal to people with different tastes.

An individual participates in at most one of the goods the menu offers: for example, one cannot visit two parks simultaneously and may always prefer the same one when facing the choice repeatedly over a period of time. Let the respective benefits to x from consuming a and b be u_x ( a) and u_x(b); x chooses whichever is preferable, provided it dominates not consuming at all. I assign a benefit of zero to abstinence.

ASSUMPTION 3 The value of menu M tois U_x ( M) = max{0, u_x(a), u_x ( b)}.

People prefer goods that are closer to their types. Let the distance from x to good i∈{a,b} be measured by ||x−i||, where ||·|| is a norm. The class of metrics covered includes the p-metrics

• 

which are induced by the p-norms.

Unavailability of the preferred design is one source of loss to individuals. In addition, there may be levies or costs specific to a good (but not the consumer's type). If the good is excludable, a price could be charged to break even at given costs, or target a specified loss (funded activities) or gain (fund-raising activities). Furthermore, consumers may bear search costs, or there may be a quality difference between the offered goods, such as between museums of international reputation and minor collections.⁶

To cover such cases, it is useful to require only that u_x ( i) decreases in the distance of type x from a given good i, but not across goods.

ASSUMPTION 4 For all types x,y∈D and goods i∈D, u_x ( i) ≥u_y ( i) if and only if||x−i||≤||y−i||, and u_x(i) ≤u_y(i) if and only if||x−i||≥||y−i||.

Assumption 4 effectively makes utility cardinal and thus allows for interpersonal comparisons. Note that it does not imply u_x ( i) ≥u_y ( j) when ||x−i||≤||y−j|| (where i and j are different goods), so it relaxes (1). It does imply u_x ( i) = u_y ( i) if and only if ||x−i|| = ||y−i||. Say x and y have a free-to-enter park at the same driving distance from their homes. Then x and y are equally content, regardless of where they live exactly.

I define the set of consumers of good a as

• 

Those who consume one of the goods, but are indifferent between them, are included in P_a. This convention does not affect the welfare analysis and simplifies notation. Accordingly,

• 

Consider the closed halfspace

• 

and the open halfspace

• 

Since x∈P_a only if , . Similarly P_b⊆H_ba, since y∈P_b only if y∈H_ba.

Assumptions 1–4 generate demands that have a particular geometry if a mild nontriviality requirement are added (and u_x ( i) is continuous in ||x−i||).

ASSUMPTION 5 Within a finite distance from any attribute mix, there existswith u_x ( i) ≤ 0.

Let B ( i,r_i) = {xs.t. ||x−i||< r_i} denote the open ball about i with radius r_i.

LEMMA 1. P_iis bounded for i∈{a,b}: there exists r_i < ∞such that P_i⊆B ( i,r_i).

PROPOSITION 1. P_i = B ( i,r_i) ∩H_ijfor i∈{a,b}, where r_i < ∞.

Proposition 1 says that those who consume good i constitute a capped n-sphere of types who are close enough to i to derive a nonnegative benefit from it, and who prefer it to the other variety.

IV. PROXY EFFICIENCY

Menus cannot usually be ranked by the Pareto criterion. Every attribute mix is ideal for some member of the population. Two menus will differ in at least one good's specifications, so one expects that preference is not unanimous. I introduce the following relaxation of the Pareto criterion; it retains much of the flavor.

DEFINITION. Menu M′ is a proxy improvement over M if there exists a bijective map h on such that U_h(x)(M′) ≥U_x ( M) for all , and U_h(x)(M′) > U_x ( M) for at least one x. If no such improvement on M exists, then M is proxy efficient.

Thus, M′ improves on M if one can pair everyone who loses by the replacement of M with M′ with someone who has an offsetting gain, and no one is paired twice. Suppose a planner is able to arrange marriages for all members of the population before changing the menu (and married individuals regard their partner's happiness as a perfect substitute for their own). There must be a way to match people so that all couples become weakly better off (and one couple is strictly happier).

While this notion of efficiency has utilitarian content (it tolerates individual losses for the overall good), it also preserves some of the Pareto criterion's egalitarian spirit. Unlike surplus measures, it never allows large benefits for a minority to supersede small losses for the majority. It is “democratic”: M′ is an improvement only if it would win against M in majority voting. The same is not true if menus are ranked by surplus. A proxy improvement implies a surplus increase, but not conversely.

To motivate proxy efficiency rigorously, I show in the remainder of this section that all menus are Pareto efficient under Assumptions 1–5, and with linear utility, so that this criterion is not informative. The claim is not quite as obvious as might appear. Any alternative menu must differ in at least one good's attributes, and although there is a type who favored the original attributes, this type is not necessarily harmed. Suppose u is nonlinear in distance, and the goods have nonzero prices π_a and π_b. Consider Figure 4, set in Euclidean space with u_x ( i) = α−β(||x−i||² + π_i) and π_a > π_b. (Equivalently, u_x ( i) ≡α_i−β||x−i||² where α_a < α_b.)

Figure 4. Pareto Domination in L²

The move from a to a′, that is, from menu M = (a,b) to M′= (a′, b), is Pareto-dominated. Every type who demands b before and after a's location change is neutral (in the striped area). The set of indifferent types (the solid black line) shifts to the left (the broken black line) after the move. No one switches from b. Everyone who consumes one of the goods and is closer to a′ than to a (to the right of the dotted gray line) previously preferred and still prefers b. (This set includes a's target a′.) Therefore, the move has no beneficiaries. But every consumer in the white region is strictly worse off.⁷

The uninformativeness of the Pareto criterion with linear u can be stated as a formal result.

PROPOSITION 2. Any menu M = ( a,b) ∈D×D with P_a, P_b≠Ø is Pareto efficient if u is linear in distance:

• 

for i∈{a,b}. That is, for any alternative menu M′ = (a′, b′) ∈D×D, there exists x∈P_a∪P_bsuch that U_x(M′) < U_x ( M).

What happens in Euclidean space with linear u? The indifferent set is then nonlinear, as in the left panel of Figure 5, and a′ is always in . This creates the shaded area of types benefiting from the move to a′, hence it is not Pareto-dominated. A corresponding example for L¹ is displayed in the right panel of Figure 5. The reader may check that the indifferent sets are drawn in accordance with the principles given earlier.⁸ Again , so the menu (a′, b) cannot be dominated.

Figure 5. No Pareto Domination with u Linear in ||·||, L² (Left) and L¹ (Right)

Two additional scenarios in L¹ are drawn in Figure 6.

Figure 6. Additional Examples of Nondomination in L¹

In this setting, it would be futile to compare menus by the Pareto criterion. Proxy efficiency is as discriminating a welfare measure as surplus for our purposes, yet more appealing from a normative standpoint.

V. EFFICIENCY OF ISOLATED GOODS

Before discussing the two-goods location problem, it is useful to consider isolated goods. Graphically, the demand areas are nonintersecting balls in attribute space. Changing the attributes of an isolated good turns out to be welfare-neutral. This will not come as a surprise, since the uniformly populated space has no “special” points, and proxy efficiency places equal weight on all members of the population.

When an isolated good is moved from a to a′, there exist bijective mappings between B ( a,r_a) and B (a′, r_a) and between and such that the paired types are at an equal distance from the goods they consume, and thus attain equal benefits.

LEMMA 2. For all a,a′∈D and, there exists a bijective mapping h : B ( a,r_a) →B (a′, r_a) (with restriction) such that h( x) = y only if u_y ( a) = u_x(a′) and u_x ( a) = u_y(a′).

Because there is for every type x a type y with u_y(a) = u_x(a′) and u_y(a′) = u_x(a), relocation of the isolated good conserves benefits. Figure 7 illustrates the proof for ||·||₁ and ||·||₂. Lemma 2 reflects the translation invariance of norms: the ball B (a′, r_a) can be constructed from B ( a,r_a) by subtracting a+ a′ from every x∈B ( a,r_a) (and multiplying by −1). Equivalently, observe that any (the dotted area in Figure 7) is associated with a unique vector v such that x = a + v. A point at the same distance from the preferred good can be constructed as y = a′−v. Such a mapping is bijective and can be extended to the whole ball B ( a,r_a). While the result is quite intuitive in Euclidean space, it holds in all normed vector spaces (even if the indifferent set is thick somewhere).

Figure 7. Value Conservation with p-Norms, p = 2 (Left) and p = 1 (Right)

In are the types who are disadvantaged when a is replaced by a′; the set contains the beneficiaries. Lemma 2 implies that losses and gains balance and, more strongly, balance in pairs. Relocating an isolated good is welfare-neutral with repect to proxy efficiency.

VI. NEIGHBORING GOODS

With consumption restricted to one good, any menu where someone prefers both goods to abstinence is wasteful. In Figure 8, moving one good from a toward b, say to a′, generates three regions: (1) neutral types who neither gain nor lose because of the move. This is the striped area. Note that anyone who originally consumed b and still does after the move must be neutral. (2) Types who are better off. This includes anyone who switches from b to a′ (they would not do so unless they could now do better than b) and anyone now in the demand area of a′ who is closer to a′ than to a (i.e., to the right of the dotted equidistant line). This is the gray area. (3) Types who are worse off: anyone who originally consumed a and now drops out or is more distant from a′ than from a. This is the solid white area.

Figure 8. Closer Goods in Euclidean Space

Lemma 2 states that the portion of B (a,r_a) that lies in the halfspace (to the left of the dotted equidistant line) creates the same benefits as the portion of B (a′, r_a) that lies in the complementary halfspace (to the right of the equidistant line). But some of is absorbed into b's demand area (the blackened part of the striped area). In other words, some types who benefit from the move because they are close to a′ do not realize the benefit by consuming a′. Instead they continue to prefer b, and the benefit is lost. Because gains for these types no longer offset losses to their associates under h, the move toward b is not welfare-neutral as in isolation. It is welfare-reducing by the proxy criterion.

These observations suggest that it may never be desirable for goods to be closer together. Consider relocations from a that maintain the distance to b. Figure 9 depicts the Euclidean and L¹ cases.

Figure 9. Distance-Preserving Relocation of a, L² (Top) and L¹ (Bottom)

In the Euclidean case, benefit and loss areas correspond.⁹ But in L¹ the move to a′ causes an increase in the gain area (equivalent to the black striped region), while the distance from b has not changed. According to Lemma 2, the light gray area and the union of the white and black striped areas yield the same benefits. Since only the white area favors the good at a, and it can be mapped to an identical (strict) subset of the gray area, which prefers a′, the relocation is strictly welfare-improving. There are nearby moves such that ||a′−b|| < ||a−b|| that are still beneficial.

Thus it is possible for closer goods to be strictly preferred by the population. While welfare is not directly linked to menu diversity as measured by distance, we have a result which, combined with the special symmetry of Euclidean spaces, does imply that welfare is distance-monotonic in the case of L².

PROPOSITION 3. Let M = ( a,b) and M′ = (a′, b′), where a′and b′are strict convex combinations of a and b. Then M is a proxy improvement on M′.

Proposition 3 is true when the utility from consumption derives from any norm-induced metric. The idea of the proof is that B (a,r_a) ∩B (b,r_b) increases as a approaches b along a given line. Therefore the efficiency loss from the overlap increases when a′ is a convex combination of a and b (and b′ = b, or b′ is itself a convex combination of a and b). In Figure 10, the striped area is the added overlap from the move to a′.

Figure 10. B ( a,r_a) ∩B ( b,r_a) Increases if ||a−b|| Is Reduced along

The perfect symmetry of a ball in Euclidean space implies that “direction doesn't matter”: for an arbitrary a′, every ã such that ||ã−b|| = ||a′−b|| generates identical intersections and distance relationships. So if the distance between a and b is reduced in any manner, there is an equivalent move that satisfies Proposition 3, hence is inefficient.¹⁰ However, other spaces do not exhibit such symmetry; the same argument does not apply.

VII. CONCLUSION

A product space metric reflects the extent to which consumers will tolerate a mismatch in one attribute of a good if they are well matched in another. Thus, it expresses the complementarity of product attributes, a property of demand with implications for optimal choices. I demonstrated that a planner's options for improving a menu of substitute public goods are sensitive to the metric. There is, however, a policy that always increases welfare when tastes are maximally diverse, namely to distance the goods proportionately in every attribute.

The finding that anticonvex diversification is welfare-improving is ultimately consistent with the intuition that maximally distant goods are a solution to the menu design problem. But the optimal direction from a given arrangement is of practical interest when the planner is budget-constrained and adjustments are costly. One wants to achieve the greatest welfare improvement with a small outlay on a particular variety and needs to know in which direction it should be relocated.

In practice, attributes are not equally malleable. A park's physical location can only be adjusted at prohibitive cost, but adding a playground may be possible. (This is not a quality change. Some patrons may want quietude, so the playground is not universally preferred.) One has to caution against the hypothesis that moving away from other varieties (neighboring parks that have no playgrounds) in one attribute is welfare-improving. Say the park in question is also unique in offering a rose bed. If quietude and the scent of roses are complements, the playground could deter more visits than it attracts.

In the case of single-attribute goods, distance metrics are equivalent and every move “away” is anticonvex. Hence increasing distance between goods is indeed welfare-improving; attribute complementarity is not an issue. But the one-dimensional setting is not a valid representation of richer preferences that care about multiple attributes. The paper also demonstrates the special nature of multidimensional Euclidean spaces that reflect an intermediate complementarity between attributes.

The analysis applies beyond the public goods context to situations where pricing is independent of variety locations. In this sense, it is about pure location effects. Even some competitive situations satisfy this criterion. Zhang (1995) argues that Bertrand duopolists have an incentive to announce price-matching policies. It is conceivable that firms would never find it optimal to adjust their prices if the product location changes, given the anticipated response from competitors.

But even in fully competitive situations, firms typically want to appeal to different customer bases, that is, minimize their market overlap, to encourage soft pricing. Differentiation may not be effective unless it is done proportionately in all relevant features. Selective changes in only some of the features could have adverse effects, making the distinction sharper in the design, but not necessarily in the value to consumers. Such product differentiation could actually cause fiercer competition.

Footnotes

• 1

  In multidimensional extensions of the Hotelling model (Irmen and Thisse 1998; Ansari, Economides, and Steckel 1998; Tabuchi 1994) duopolists locate at maximum distance in one axis direction and at minimum distance in all others. This complexity arises from restricting the consumer-preferred attributes to a cube (the higher-dimensional analog of the line interval), not from an intrinsic importance of direction. If tastes were sufficiently spread out, so that the shape of their boundary had no effect, optimal locations would simply maximize the distance between goods, as in the one-dimensional case.

• 2

  Miyagawa (2001) previously argued that the only solutions that satisfy both Pareto optimality and a different fairness restriction are the “left-peaks” and “right-peaks” rules. They place the goods at the two smallest (distinct) locations, respectively the two largest (distinct) locations, that are most preferred by someone in the population.

• 3

  Consider the set of consensus profiles that unanimously prefer some allocation. An allocation is rationalized by a metric if it is elected by a consensus profile that is closest to the actual profile in terms of the metric. Metric rationalizability of a choice rule is equivalent to the Pareto property (if an allocation is preferred by all, then it is implemented).

• 4

  Related to metric rationalizability are research programs that exploit the duality between social choice problems and multi-criteria optimization. Conley et al. and Voorneveld et al. belong to this literature The problem is to balance multiple objectives when it is not possible to maximize with respect to each simultaneously. The metric aggregates shortfalls in component objectives, and the least costly solution in this sense is chosen.

• 5

  This assumes a≯b and a≮b (that is, a is located southeast or northwest from b), as in the diagrams. Else the slope is −1.

• 6

  Prices are taken to be fixed by exogenous motives; they are not a choice variable in menu design. Not within the scope of the discussion are private goods, where pricing and locations are interdependent.

• 7

  The restriction that individuals consume only one good is critical, for else a′ must benefit from having the ideal attribute mix provided.

• 8

  In particular, the dark gray triangles connected by a dotted line mark the types who are equidistant from a and a′. The solid black line is the indifferent set between a and b; the partially coinciding broken black line that between a′ and b.

• 9

  Although P4 does not state that there exists a bijection from to , this is true (only) in Euclidean space if ||b−a|| = ||b−a′||. Then b belongs to the equidistant line between a and a′, which is an axis of symmetry between and .

• 10

  More precisely, in Euclidean space every arrangement M′ = (a′, b′) such that can be obtained from by a linear mapping and rotation about . These are isomorphisms in a vector space. In particular, if ||ã−b|| < ||a−b||, we can choose a convex combination M′ = (a′, b′) of M = (a,b), so Proposition 3 implies is less efficient.

• 11

  Although x could have chosen a in menu M while being indifferent between a and b, this would imply y was indifferent between a and b also, in which case y∉P_b, contrary to what was assumed.

• 12

  The expression λy′ + (1 −λ)z = λ(y−κγ) + (1 −λ)(y−K(y)γ) reduces to y− (λκ + (1 −λ) K (y)) γ.

Appendix

Proof of Lemma 1

Assumption 5 guarantees that there is an x such that ||x−i|| < ∞ and u_x ( i) ≤ 0. If y∈P_i, then it must be that ||y−i|| < ||x−i||; else Assumption 4 implies u_y ( i) ≤u_x ( i) ≤ 0. Hence, for all y∈P_i, ||y−i|| < ||x−i|| < ∞. Letting r_i = ||x−i||, we have P_i⊆B ( i,r_i) with r_i < ∞.

Proof of Proposition 1

Since P_i is bounded by Lemma 1, r_i = sup {||x−i||s.t. u_x(i) > 0} exists, and r_i < ∞. Because r_i is an upper bound, x∈P_i( ⇒ u_x ( i) > 0) only if ||x−i||< r_i ( ⇔ x∈B ( i,r_i)). Hence P_i⊆B ( i,r_i). Since P_i⊆H_ij, we have P_i⊆B ( i,r_i) ∩H_ij.

If x∈B ( i,r_i) ∩H_ij, then ||x−i||< r_i and x∈H_ij. Suppose x∉P_i, that is, u_x ( i) ≤ 0. There exists y∈H_ij such that ||x−i|| < ||y−i||< r_i, so u_y ( i) = 0. Now, for all z∈H_ij with ||z−i||≥||y−i||, Assumption 4 implies u_z ( i) ≤u_y ( i) ≤ 0. But then ||y−i|| is a bound of {||x−i|| s.t. u_x ( i) > 0}, and r_i is not the least upper bound, a contradiction. Thus x∈P_i, that is, B ( i,r_i) ∩H_ij⊆P_i.

Proof of Proposition 2

Consider menus M = {a,b} and M′ = {a′, b′}. I show first that M′ cannot improve on M if a∈P_a and b∈P_b, then that a∈P_a and b∈P_b if u is linear and P_a, P_b≠Ø. Suppose a∈P_a and b∈P_b, and consider type x who favors design a and type y who favors design b. If x chooses a′ in M′, then x is worse off unless a = a′, since ||x−a′||≥ 0 = ||x−a||. If y chooses b′ in M′, then y is worse off unless b = b′.

Thus, M′ cannot be a Pareto improvement on M if x chooses a′ and y chooses b′ in M′, since a = a′ and b = b′ imply no one is strictly better off. On the other hand, if x chooses b′ in M′, then x is worse off unless α_b > α_a, or α_b = α_a and b′ = a. If y chooses a′ in M′, then y is worse off unless α_a > α_b, or α_a = α_b and a′ = b. Hence we can rule out a Pareto improvement if x chooses b′ and y chooses a′: one of x and y would be worse off, except if a′ = b and b′ = a, in which case no one in the population can be strictly better off.

The only remaining possibilities are (1) a = a′ and b ≠ b′ or (2) a ≠ a′ and b = b′. In (1), y must be worse off, since y chose b over a in M (hence would suffer from switching to a′), and u_y ( b′) ≤u_y ( b) since ||y−b′|| > 0 = ||y−b||. In (2), x must be worse off, since x chose a over b in M (hence would suffer from switching to b′), and u_x ( a′) ≤u_x ( a).¹¹ It follows that a Pareto improvement is not possible unless a∉P_a or b∉P_b or both.

Suppose a∉P_a. Since and a∈B ( a,r_a), this means . Then u_a ( b) > u_a ( a) and, with linear u, α_b−β||a−b|| > α_a−β||a−a|| = α_a, which leads to

• (1)

Let . Because x chooses a over b in M, u_x ( b) ≤u_x ( a), hence:

• (2)

Together with (1), (2) implies

• ((A.1))

Since norms have the triangle property ||x−b||≤||x−a|| + ||a−b||, inequality (A.1) is not possible. Thus a∈P_a.

Suppose therefore that b∉P_b, that is, b∉H_ba. Then u_b ( a) ≥u_b ( b), giving

• (3)

Let y∈B ( b,r_b) ∩H_ba. Because x chooses b over a in M, u_y ( b) > u_y ( a), hence

• (4)

Together with (3), (4) implies

• ((A.2))

The triangle inequality ||y−a||≤||a−b|| + ||y−b|| invalidates inequality A.2. Thus b∈P_b.

The premise that M′ is a Pareto improvement over M leads to contradictions in all cases.

Proof of Lemma 2

The argument is by construction. Let

• ((A.3))

I claim that h is a bijective map on (1) B (a,r_a) and (2) , into B (a′, r_a) and respectively. In addition, I claim that (3) h( x) = y only if u_y ( a′) = u_x ( a) and u_y ( a) = u_x(a′).

Using Equation (A.3) to substitute for x and h( x),

• ((A.4))

Now ||x−a|| = ||y−a′|| and ||y−a|| = ||x−a′|| follow from the axiomatic property of norms that ||αx|| = |α|||x|| (let α = −1). Since u_x ( a) = u_y(a′) if and only if ||x−a|| = ||y−a′||, and u_x(a′) = u_y ( a) if and only if ||x−a′|| = ||y−a||, the mapping satisfies (3).

If h( x) belongs to the specified co-domains, then it is one-to-one. On domain (1) B ( a,r_a), it is immediate from ||x−a|| = ||h(x) −a′|| that ||x−a||≤r_a only if ||h(x) −a′||≤r_a, so h( x) ∈B (a′, r_a).

Moreover,

• 

follows from Equation (A.4) and implies ||x−a||≤||x−a′|| if and only if ||y−a′||≤||y−a||. Hence u_x ( a) ≥u_x(a′) if and only if u_y(a′) ≥u_y ( a), and we have if and only if . Therefore entails , so that h is also one-to-one on domain (2) . Because the inverse of h is one-to-one by analogous reasoning, h is onto and then bijective.

Proof of Proposition 3

Clearly, would imply that M improves on M′. Therefore it is sufficient to show: if a′ = λ₁a + ( 1 −λ₁) b and b′ = λ₂a + ( 1 −λ₂) b with λ₁,λ₂∈ ( 0,1), then . Since P_a∪P_b = B ( a,r_a) ∪B ( b,r_b) and , we have

• 

Given B ( a,r_a) and B ( b,r_b), denote the direction from a toward b by the vector γ≡ ( b−a) /||b−a||. Then every point x′∈B ( b,r_b) can be written as x′ = x + κγ for some x∈B ( a,r_a) and fixed κ = ||b−a||. In particular, y′ = y + κγ in the boundary of B ( b,r_b) is the image of y in the boundary of B ( a,r_a). For a boundary point y of B ( a,r_a), let z = y + K ( y) γ be the farthest boundary point of B ( a,r_a) in direction γ such that K ( y) ≥ 0. See Figure 11 for examples in L² and L¹.

Figure 11. Construction of y′ and z in L² and L¹ Space

Observe that B ( a,r_a)∩B ( b,r_b) is the union of convex combinations of y′ and z, for all y such that κ≤K. So¹²

• 

If κ is larger, fewer y satisfy κ≤K ( y); note that κ increases in the distance between a and b. If a′ and b′ are convex combinations of a and b, then ||b−a||≥||b′−a′||. Hence κ′ in M′ is smaller than κ in M and enlarges the intersection B ( a′, r_a) ∩B ( b′, r_b). Therefore B ( a,r_a) ∩B ( b,r_b) ⊆B ( a′, r_a) ∩B ( b,r_b), which is sufficient.