On Distance Metrics in Location Problems

By Roessler, Christian | Economic Inquiry, January 2013 | Go to article overview

On Distance Metrics in Location Problems


Roessler, Christian, Economic Inquiry


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. …

The rest of this article is only available to active members of Questia

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this article

This article has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this article

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited article

On Distance Metrics in Location Problems
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Help
Full screen

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

    Cited passage

    Thanks for trying Questia!

    Please continue trying out our research tools, but please note, full functionality is available only to our active members.

    Your work will be lost once you leave this Web page.

    Buy instant access to save your work.

    Already a member? Log in now.

    Oops!

    An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.