On the internal contradictions of the law of one price.

McChesney, Fred S. ; Shughart, William F. II ; Haddock, David D. 等

As stated originally, the venerable law of one price succinctly describes long-run equilibrium in a perfectly competitive market. The law was later amended, defining a market as the geographic area within which the same thing sells for the same price at the same time, allowance being made for transportation costs. Modified in that way, the law has two plausible interpretations. By one interpretation, every production site is a market. By the other, prices in fact do not differ by transportation costs. The transportation-cost amendment thus introduces internal contradictions that render the revised law of one price either useless or wrong. (JEL D40, L10)

 A market, according to the masters, is the
 area within which the price of a commodity tends
 to uniformity, allowance being made for
 transportation costs. That is, two places are in
 the same market for a good if the prices at the two
 places differ by transportation costs. (Stigler,
 1987, 77)

 Economic terms seem to pass in their historical
 development through a series of stages which,
 without pretension to rigidness, may be described
 as follows: first, no definition is given, but it is
 assumed that every one has a sufficiently clear idea
 of the subject to make a formal definition
 unnecessary; second, a definition is attempted
 and a number of exceptional forms are noted;
 third, with the further increase of data, the relative
 importance of the various forms changes, confusion
 in discussion is introduced, logomachy takes
 the place of constructive investigation; fourth, a
 complete classification of the forms embraced
 under the original term is made, and problems are
 investigated with reference to these classes. The
 bewildering vagueness of economic theory is
 largely due to the fact that the terms used are
 in all of these stages of development. (Moore
 1906, 211)

I. INTRODUCTION

Delineating the boundaries of markets engaged economists' attention early on. Many product attributes are important to consumers, but price is an objectively measurable margin along which rivalry is sharply engaged. Market prices transmit information about changing demand and supply conditions to buyers and sellers, and widening price--cost margins signal profit opportunities to which alert entrepreneurs respond. Price consequently has held center stage in economists' attempts to define markets.

Augustin Cournot (1927, 51), for instance, wrote that a market is "the entire territory of which the parts are so united by the relations of unrestricted commerce that prices take the same level throughout with ease and rapidity." Alfred Marshall (1930, 325) supplied a similar definition in his celebrated declaration that "the more nearly perfect a market is, the stronger is the tendency for the same price to be paid for the same thing at the same time in all parts of the market." More recently, Joseph Stiglitz (1993, 19) summarized what he called the law of the single price, stating that, "under this law, there is a uniform price in the market, and price differences are quickly eliminated by arbitrage." (1)

Marshall (1930, 325) introduced an important qualification, however, when he observed that, "but of course, if the market is large, allowance must be made for the expense of delivering the goods to different purchasers; each of whom must be supposed to pay in addition to the market price a special charge on account of delivery." That qualification subsequently found its way into Stigler's price theory textbook, quoted in the epigraph. Intentionally or not, the law of one price was thereby converted into a law of different prices. Other economists subsequently added transaction costs and information costs to the list of caveats, culminating in a multiplicity of prices nevertheless claimed to be consistent with the law of one price.

Economics has few laws. Perhaps the most famous is the (first) law of demand. It truly is a law, because price and quantity demanded are inversely related, ceteris paribus, at all times and in all places. But the amended law of one price is different. It holds--except (more often) when it doesn't.

The modern empirical literature abounds with apparent violations of the law. Elzinga and Hogarty (1978), for example, report that in 1975 the f.o.b. (free on board) price of bituminous coal was $27.03 per ton in eastern Kentucky, but $13.75 per ton in western Kentucky. That same year, the nationwide average charge for shipping coal via rail was $5 per ton. Inclusive of freight charges a ton of comparable coal sold two years later for $18 in New York, $38 in New Jersey, $23 in Illinois, and $34 in Wisconsin. Shrives (1978) was able to explain less than two-thirds of the observed variation in bituminous coal prices with a model containing 25 independent variables. Baye and Morgan (2001) document sizable and persistent differences in the prices quoted over the Internet for a seemingly homogeneous product (a 0-point, 30-year, conventional fixed-rate mortgage for a hypothetical New Jersey homebuyer). Asplund and Friberg (2001) likewise find that a given good in a given duty-free shop often sells at different prices quoted in different currencies, contrary to what the law of one price would have predicted.

This article argues that the numerous pricing anomalies interpreted by some economists as evidence that the law of one price does not hold--and by others as reason for adding new ceteris paribus conditions to save the law from accumulating empirical contradiction--reveal two basic problems: Modified to include transportation costs, the law is internally contradictory, and, furthermore, economists have tended to apply it overbroadly in attempts to force actual markets to conform to abstract models of pure competition. Stated in its received Marshall-Stigler form wherein transportation cost is the only complication allowed in the market for an otherwise homogeneous good, the very concept of a law of one price is fatally flawed. If the amended law is incoherent even in this simplest case, it remains incoherent in more complicated situations where additional market frictions are taken into account. Introducing other complications merely contributes auxiliary reasons why the law fails to hold in the original, transportation-cost-only case.

The next section supplies analytical background by discussing the assumptions under which there is a strong "tendency for the same price to be paid for the same thing at the same time in all parts of the market." Section III then notes that the amended law of one price actually has two plausible interpretations. Does the law mean that a market can be defined endogenously as the territory wherein prices differ only by transportation costs? Or does it mean that a market, once exogenously defined by some other criterion, will exhibit prices that differ only by transportation costs?

Under the first interpretation of the law every production site is a market. Construing the law of one price that way adds more confusion than clarity to the definition of economic markets. Under the second interpretation, the amended law is simply wrong: Prices in fact do not differ by transportation costs. Wherever located, all sellers in a market must charge the same price for the same product. The owners of favorably positioned firms may enjoy Ricardian rents in this case, but they are not thereby able to charge different prices.

It is useful to emphasize at the outset what this article is not about. The connection to the practice of defining markets for antitrust purposes is tangential at best. It is true, of course, that the antitrust laws require enforcers and courts to determine the set of products--and the extent of the geographic area in which the sellers of those products compete--that will be deemed relevant when assessing the impact of mergers and other business practices on economic performance. Depending on how narrowly or broadly a market is defined, a firm does or does not have market power, or a proposed business consolidation does or does not go beyond the market-share thresholds that trigger antitrust concerns. Market definition gradually has become an art (if not a science) because of its role in the enforcement of competition law. But the law of one price evolved separately from--indeed, was stated well before--the rise of modern antitrust. Nor is the article primarily an exercise in the history of economic thought. Although the law of one price surely originated in the work of the "masters," including Cournot, Jevons, and Marshall, it remains a standard part of the canon of price theory today. (2)

In the end, the law of one price has become for modern economists an analog of the Ptolemaic system of the universe for medieval astronomers. Rather than abandon the notion that the sun moved around Earth, astronomers such as Tycho Brahe invented new and increasingly fabulous theories to account for inconveniently inconsistent observations, in that way hoping to shore up the model of a geocentric universe. All else equal, however, science prefers simpler explanations to complex ones, and in time Ptolemaic notions were abandoned altogether in favor of the Copernican heliocentric model. Increasingly encumbered with addenda and asterisks to make it seem to work, the law of one price, meant originally as a description of the tendency for prices in a market to equilibrate, likewise has been gutted of analytical content.

Stripped of interpretative errors and irrelevant complications, the law of one price, properly understood, is seen to be neither more nor less useful than the abstract competitive model on which it was founded. Being a direct implication of that model, the law of one price states that, once a market has been defined, prices will tend to converge so that, in the long run, the same price will prevail in all parts of the market. Plainly, however, at any point in time a tendency toward price equilibration and a single price are not the same thing.

II. THE EXTENT OF THE MARKET

In the textbook model of pure competition, with all its simplifying assumptions (including product homogeneity, perfect information, and perfect resource mobility) all sellers necessarily charge the same price. All production and consumption decisions implicitly take place at a single point in economic space-time. There are no locational advantages or any other differences across products or firms that matter to buyers and that would cause the demand curves perceived by sellers to slope downward. Under these conditions, any price differences that emerge are quickly eliminated by arbitrage as the market moves toward long-run equilibrium.

Clearly, a useful economic model does not conclude that the products making up a naturally occurring market must sell at the same price. The goods and services that consumers treat as substitutes frequently are different physically. Even if this is not the case, sellers differ as to location, promptness of delivery, their willingness to extend credit, offering and honoring warranties, repairing or replacing defective items, and so on. Buyers likewise differ as to location, willingness to accept delivery, promptness of payment, creditworthiness, penchant for returning items, desire for pre-and postsale services, and so on. Prices will adjust to reflect differences in the many nonprice attributes of transactions and, indeed, as emphasized by Stigler (1968), variations in these nonprice attributes are competitive substitutes for variations in price. (3)

However, as modified to incorporate transportation costs, the law of one price is a simpler proposition. The theory received from those masters to whom Stigler refers abstracts from the myriad strategies sellers employ to distinguish their products from those offered by rivals. (4) In eschewing realism for tractability, and by downplaying the dynamic forces that in Marshall's (1930, 333) colorful language cause prices to be "tossed hither and thither like a shuttlecock, as one side or the other gets the better in the 'higgling and bargaining' of the market," the amended law of one price is a statement about the boundaries of markets wherein products are differentiated along a single dimension, namely, the locations of sellers. All other complicating factors possibly influencing prices implicitly are held constant.

In this stylized world of static, long-run competitive equilibrium, two sellers are said to be in the same market if the prices they charge differ only by the cost of transportation between their separate locations. In that world, sellers quote prices f.o.b.: The price paid by any buyer is equal to the price at the seller's plant (the "mill price") plus the cost of shipping the product to him or her. Buyers are free to make their own transportation arrangements (and pay for them). Alternatively, if the seller offers the option of delivering the product to the buyer's location, the cost of shipping is invoiced separately. Under these assumptions, Stigler (1987, 77) writes, "the price of a commodity 'tends to uniformity' for one reason: the buyers at point B refuse to pay more than the price at point A plus the cost of transportation, and the buyers at point A act similarly. Or the sellers act in this manner."

In other words, according to Stigler, the market price is p + t, the mill price plus the unit cost of transportation, and locations A and B supposedly are both "in" the same market because the prices at the two points differ only by the cost of transportation between them. Marshall (1930, 325) supplies a particularly apt example of the logic underlying the law of one price:

 The whole of the Western World may, in a
 sense, be regarded as one market for many kinds
 of stock exchange securities, for the more valuable
 metals, and to a less[er] extent for wool and cotton
 and even wheat; proper allowance being made for
 the expenses of transport, in which may be
 included taxes levied by any customs houses
 through which the goods have to pass. (emphasis
 added)

The amended law of one price, although usually expressed in terms of sellers' transportation costs, does have one other embellishment: "mobility of customers [is] ... sufficient to ensure the tendency to uniformity in price, allowance being made for transportation costs of consumers." Buyers, as Stigler (1987, 78) observed, sometimes face lower transportation costs than sellers: "A cotton farmer will have a relatively small area in which he will sell his crop; the buyers may deal in every cotton-picking state." Fundamentally, however, recognizing that buyers are sometimes cheaper transporters than sellers are does not alter the thrust of the analysis. Two places are in the same market if the prices at the two places differ only by the cost of transportation between them, regardless of the identity of the low-cost shipper.

But is that proposition useful or even true?

III. THE FUNDAMENTAL ECONOMICS OF THE LAW OF ONE PRICE

An initial problem attends analysis of the amended law of one price: As normally stated, the law has at least two interpretations. Does the law mean that a market is definable as a location in which price is set by the marginal costs of production, differing only by transportation costs, ceteris paribus? Or does the law mean that once a market has been identified, prices therein predictably will be uniform, allowing for transportation costs? In other words, does equality of prices (but for transportation costs) define a market endogenously? Or does the delineation of the market, however it is defined exogenously, imply uniform prices (but for transportation costs)?

As will be discussed in this section, those propositions are not logically the same. Nor is the economic evaluation of the two plausible interpretations the same. Our analysis confines attention to the ceteris paribus conditions contemplated by the Marshall-Stigler statement of the law of one price: Reference to prices differing only by transportation costs means that all else (e.g., product quality) is held constant.

Equality of Price Defines a Market

Let us interpret the modified law of one price, first, as stating that one can define places as belonging in a single market by their having the same price but for transportation cost. All firms by assumption have equal production costs. However, transportation costs to particular customers differ, being a function of distance from one of a firm's plants.

Consider the situation portrayed in Figure 1. Let two firms, 1 and 2, operate plants at points of local resource supplies, points A and B respectively. Spatially, A and B are located at distances OA and OB, measured from arbitrary point O. (For simplicity, there is no firm closer to point O than firm A, nor are there any customers to the left of point A or to the right of point B.) Each firm charges the f.o.b. price that covers its marginal production costs (OP), to which is added unit transportation cost ([t.sub.1] for firm 1 and [t.sub.2] for firm 2). (5) Thus, purchasers at points A and B will pay prices OP. Purchasers located away from points A and B would have to pay prices increasing with distance as shown along OP + [t.sub.1] or OP + [t.sub.2].

To turn to the question of interest, are points A and B in the same market? Under the first interpretation of the amended law of one price, the answer is no. Out to point M, all buyers will purchase from firm 1, the one producing at point A. But beyond OM, prices paid will no longer be firm 1's f.o.b. price plus transportation costs. Buyers at those distances will prefer to buy from firm 2. Point M is the boundary between two markets. Firm 1 serves all of the customers in the space up to M; firm 2 serves all of the customers beyond M.

If this first interpretation is what the Marshall-Stigler version of the law of one price means, it is economically problematic for two reasons. First, more obviously, the law of one price effectively means that every plant comprises its own spatial (geographic) market. Area AM is one market; area MB defines a separate market. Within each of these markets, there is not one price but a multitude of prices tailored to individual customers' locations. Customers in AM pay firm 1's f.o.b. price plus the cost of transportation from point A. Customers in MB pay firm 2's f.o.b. price plus its transportation costs. Only at location M are both firms in the same market in the sense that they charge the same price, OP + [t.sub.1] = OP + [t.sub.2]. (6)

This leads to the paradoxical result that apart from the market boundary at M, firms are in the same market only if they are located atop one another. The moment plants diverge spatially, under the modified law of one price they each define a separate market if they charge f.o.b. prices. In making every firm a market, the transportation-cost amendment surely creates a definition of market that will seem of little operational use to most economists and likely confuse most noneconomists. Why use the term market if its meaning is identical to that of firm or plant? Or if market refers to multiple firms at the same point, why make special note of transportation costs?

There is a second problem with this first interpretation of the amended law of one price. In Figure 1, would firms in fact charge prices equal to marginal production cost (OP) plus transportation cost, as the law of one price says they would? Seemingly not, absent further assumptions. Out to point M, firm 1 can charge all of its customers up to some maximum price just [alpha] below firm 2's production and transportation costs. Likewise, beyond point M, firm 2 can charge all of its customers a maximum price just A less than firm 1's production and transportation costs. (7)

Again to quote Stigler (1987, 77), the law of one price works "for one reason: the buyers at point B refuse to pay more than the price at point A plus transportation, and the buyers at A act similarly." But this Stiglerian notion of price formation assumes that only buyers matter in the setting of price within a market. If every firm is a market, then sellers by definition have some market power that must also be taken into account. In Figure 1, the sellers at A and B have no incentive to offer prices calibrated solely to their own production and transportation costs. (8) Only at M, where both firms are in the same market, need the price that one firm charges be tightly constrained by the other firm. At that single point the price might actually equal the sum of marginal production and transportation costs. (9)

In other words, if under the amended law of one price each spatially separated plant defines a market, that plant has a locational monopoly. It therefore will not charge a price equal to its production and transportation costs, but rather a higher price limited by the other firm's production and transportation costs. Ironically, if there truly was a law by which price varies within a market only by transportation cost, firm 1's price would be a negative function of distance between points A and M. The highest prices would be charged at sites most isolated from competition from firm 2's plant at point B and the lowest prices at the market boundary. (10)

In short, if the amended law of one price is interpreted to mean that a market is defined by the area surrounding a firm (plant) where prices are the same but for transportation cost, two implications follow. First, every firm (plant) is a market, essentially stripping the term market of any useful meaning. Second, although price in the market is indeed a function of distance under this interpretation, the limit price does not vary positively with own-firm distance, as the transportation-cost-modified law of one price suggests. Rather, it varies positively with distance from other firms (i.e., price is inversely related to own-firm distance)--the very opposite of what the amended law predicts. (11) For firm 1, prices are higher at A (near the plant) than at M (further away). The same relationship between limit price and distance would hold for firm 2.

A Market Implies Price Equality

The key implication of the foregoing section is that for a firm to charge a price at a point dictated by its own marginal production cost plus transportation, there must be other firms competing with it at that point, holding price equal to marginal cost (including both production and transportation costs). So, if one observes different firms selling in the same place, each constraining other firms' ability to raise price above marginal production and transportation costs, that place would seem to define a market as the term is popularly used.

But in this locale (point M) where multiple firms compete, if there is one price, is that single price equal to firms' production plus transportation cost, as the law of one price claims? Consider the situation depicted in Figure 2. A set of firms located at point A is willing to supply any quantity of output up to [q.sub.A] units at price p. Another firm (or set of firms) located at point B has a higher supply price of p + t for sales in A.

Given multiple firms at A, if the demand for the product were D, the market price would be p and firms located at B would not be in the market. (12) The law of one price would apply, but only because all firms selling in A were located in A, as discussed in the previous section. But by hypothesis, firms located at both A and B have been observed selling in A. So, the demand for the product must be D'.

For firms at B to sell in A, the market price must be p + t. (13) Under the law of one price, all of the firms in the market charge the same price. But if so, some of them earn inframarginal rents owing to their more favorable location, because they charge prices exceeding their production and transportation costs. Locational rents are earned on sales to customers at A by local firms who would have been willing to supply them at price p, but actually can get p + t. (14) Once again, price is determined not by a firm's own transportation costs but as limited by the transportation costs of the more remote firm(s), just the opposite of what the amended law of one price posits.

The point is that in either set of circumstances, the price is not one that "tends to uniformity, allowance being made for transportation costs." Two places are not in the same market "if the prices at the two places differ by transportation costs." For two places to be in the same market, it is necessary that the prices at the two places be the same, period. (15) If the sellers of an otherwise homogeneous product differ only as to location, they must charge the same price if they compete in one market. If they do not charge the same price, they are in the same market only if the boundary line between two other markets is defined as "the market." (16)

Reevaluating the Law of One Price

Recognizing that (unlike the law of demand, for example) the law of one price does not necessarily hold always and everywhere, economists have for a long time been ambivalent about it. As noted, the law abstracts from any number of nonlocational factors, owing to which a buyer might not patronize a seller who is closest to him. Promptness of delivery, reputation, willingness to extend credit, warranty policies, and willingness to repair or replace defective items all may differ among firms. Other things are not always equal.

More recently, though, economists have offered more Coasean reasons why, ceteris paribus, the law of one price might not hold. Failures of the law have been ascribed to information or transaction-cost factors, for example. On the demand side, echoing Stigler (1961), Stiglitz (1993, 19) discusses imperfect information and buyer search costs as reasons why, "in fact, many markets are marked by noticeable differences in prices." (17) Building on the work of Varian (1980), in which consumers are either informed or uninformed about the distribution of prices and sellers randomly conduct sales, Baye and Morgan (2001, 2002) propose a model of Internet pricing wherein a gatekeeper charges fees to firms wanting to advertise prices and to consumers wanting to access that information. Prices for homogeneous goods are shown to be dispersed in equilibrium under those assumptions. Asplund and Friberg (2001) rely on "costs of arbitrage" to explain why the prices of the same goods quoted in different currencies at the same duty-free shops fail to converge in apparent violation of the law of one price. Similarly, Goldberg and Knetter (1997, 1270) conclude that "deviations in the law of one price" observed in otherwise integrated international markets "appear to be largely a result of third-degree price discrimination."

On the supply side, Carlton and Perloff (2000, 365) write, "Because prices reflect costs in competition, economists expect purchasers to pay for FOB pricing and for actual freight under competition." But in fact, Carlton and Perloff continue, firms often do not charge for shipping, using uniform delivered pricing (a single freight charge, regardless of distance) instead "because it is simple and saves on administrative costs.... It appears that uniform delivered pricing is often followed as long as the variation in freight charges among customers is 10 percent or less." (18) Or, as Hay (1999, 196) writes, the law may not hold at any given point because temporally the system is in disequilibrium: "While it is true that in a 'perfect' market prices for identical products tend toward equality, there is generally enough friction in the system that the process does not work instantly."

In short, some modern economists who have considered the law of one price do not seem to regard it as falsifiable. When price does not equal marginal production plus transportation costs, dei ex machinis, such as asymmetric or costly information, price discrimination, transaction costs or disequilibrium, are invoked to save the law. Being consistent with any observed pattern of prices, the law of one price predicts nothing.

Recognizing that as Varian (1980, 651) notes, "most retail markets are ... characterized by a rather large degree of price dispersion," instead of attempting to rescue the law of one price from empirical contradiction, others have been prompted to recommend jettisoning it altogether: Again quoting Varian (1980), "economists have belatedly come to recognize that the 'law of one price' is no law at all." Routinely observed phenomena, such as f.o.b. pricing, the willingness to absorb freight to sell to distant customers and the "high degree of overlap and cross-shipping found among spatially dispersed producers," all "obviate the usefulness" of the law, according to Elzinga and Hogarty (1973, 51).

The analysis here illustrates that neither of these reactions is justified. As stated originally, the law of one price was never meant to accommodate frictions like transaction and information costs. Nor was it expected to hold in situations where underlying demand and cost conditions are in flux, except to point toward the equilibrium to which prices would tend to converge. Indeed, internal contradictions are introduced even in the starkest case where transportation cost is the only complication allowed. In that simplest case, the most straightforward interpretation of the amended law of one price implies that every firm or plant is a separate market, that prices within that market vary by customers' locations and that prices in one market may well be a negative function of transportation costs from adjacent markets, not a positive function of transportation costs for the particular market itself. Interpreted correctly, the law of one price is a law of one price, not a law of one price adjusted for differences in transportation costs. In other words, under this alternative interpretation transportation costs generate Ricardian rents for favorably located sellers but do not result in price differences within a market.

There is, in short, a law of one price, but it is not the one stated by Alfred Marshall or George Stigler. The transportation-cost qualification they appended erroneously added an element of spatial product differentiation to a proposition that was meant to apply only to perfectly competitive markets, wherein knowledge is perfect and goods are costlessly mobile. The original meaning can be restored by defining a market as Cournot did: "The entire territory of which the parts are so united by the relations of unrestricted commerce that prices take the same level throughout with ease and rapidity." Jevons's law of indifference, as quoted in Edgeworth (1987, 786) is equally apt: "In the same open market, at any one moment, there cannot be two prices for the same kind of article." Alternatively, one can define a market in amended Stiglerian terms as the area within which the price of a commodity tends to uniformity. That is, two places are in the same market for a good if the prices at the two places are the same.

IV. CONCLUSIONS AND IMPLICATIONS

The foregoing analysis demonstrates that, encumbered with its modern transportation-cost proviso, the law of one price is incoherent. It cannot therefore be rescued by amending it further to control for other complications of the real world.

The law of one price restated by the masters, as Stigler refers to them, is a more subtle proposition than those writing about it have grasped. Errors of interpretation have been multiplied by failing to be clear about what the transportation-cost amendment implies. If it means that price equality defines a market, then the law is of no use, except in that small minority of instances in which transportation cost is essentially zero: Every production site is a market, robbing the latter term of any relevance. It is also wrong. Price does not necessarily vary positively with a producer's own transportation costs; the upper limit on price varies positively with other producers' transportation costs.

Alternatively, if the amended law means that a market (however defined) will exhibit prices that are equal but for spatially dispersed firms' own transportation costs, the law again is incorrect. It is not true, as Stigler (1987, 77) asserts, that "two places are in the same market for a good if the prices at the two places differ by transportation costs." Rather, any number of plants can compete in any given market place as long as each quotes the same price, though they may or may not compete in other marketplaces. Under the usual simplifying assumptions, all other things equal, as spatially separated firms face important transportation costs, the number of places in which they compete will diminish. Within a market, however, it is only the marginal producer's transportation costs that matter. Corn shipped to town from distant locations, as Adam Smith long ago recognized, sells at the same price as corn shipped from nearby farms, whose owners thereby benefit.

Stripped of interpretive errors, the law of one price must prevail. It identifies a tendency for equilibration that follows as a logical consequence of the assumptions of the model of pure competition and, being an implication of it, is therefore neither more nor less useful than that model. Qualifying the law of one price to make allowance for transportation costs turns out to have been a mistake, subsequently multiplied by trying to force it also to be consistent with other messy properties of the real world, such as transaction and information costs. That error has caused economists to treat the law of one price the same way that Montana's drivers treated the 55 mph legal speed limit. Most knew the law was on the books (or, more accurately, in the books), but few observed it or expected others to.

(1.) In the same passage, Stiglitz refers to the law of the single price as a "major principle of economics." Nearly a century ago, Henry Moore listed what William Stanley Jevons called the law of indifference--that "there is but one price for commodities of the same quality in the same market"--as one of the hallmarks of (pure) competition. Indeed,"--as quoted in Moore (1906, 214), Jevons thought they were the same thing: "This law of indifference, in fact, is but another name for the principle of competition which underlies the whole mechanism of society." That was Francis Edgeworth's interpretation as well. Edgeworth (1987, 786; emphasis added) wrote that Jevons's law of indifference rests on "certain ulterior grounds: namely, certain conditions of a perfect market," one being that "monopolies should not exist." He specifically ruled out the existence of market "power in virtue of which a proprietor of a theater, in Germany for instance, can make a different charge for the admission of soldiers and civilians, of men and women." Edgeworth went on to say in the next sentence that "the indivisibility of the articles dealt in appears to be another circumstance which may counteract the law of indifference in some kinds of market, where price is not regulated by the cost of production."

(2.) One of this journal's referees called the law of one price the "mother of all economic laws." Nowadays the proposition appears in a number of popular principles of economics texts under the heading of international trade and finance. According to Lipsey et al. (1999, 129), for example, "the law of one price states that when a product that can be cheaply transported is traded throughout the entire world, it will tend to have a single worldwide price--the world price." Similarly, O'Sullivan and Sheffrin (1998, G-6) state the law of one price as "the theory that goods easily tradable across countries, should sell at the same price, expressed in a common currency." Mankiw's (2004, 689) discussion of purchasing power parity likewise is based on the law of one price, which "asserts that a good must sell for the same price in all locations."

(3.) Economists were aware of the analytical consequences of product differentiation even before Jevons (1970, 137) wrote that price differences for even otherwise identical products can "arise from extraneous circumstances, such as the defective credit of the purchasers, their imperfect knowledge of the market, and so on." The multidimensional character of competition was in fact recognized as early as 1844 in the writings of the great French engineer-economist Jules Dupuit. See Ekelund (1970) for a summary of Dupuit's contributions to the analysis of price discrimination and product differentiation. Later, Edward Chamberlain (1962), Joan Robinson (1969), and Nicholas Kaldor (1934), among others, explored the significance of "gaps in the chain of substitutes" that might be used to identify distinct market boundaries in common situations where, as the last of these masters put it, "different producers are not selling either 'identical' or "different' products, but 'more or less different products'--the demand confronting them being neither completely sensitive nor completely unsensitive to the prices charged by other producers."

(4.) Empirical applications of the law of one price have attempted to grapple with complications like product differentiation by, for example, testing whether the prices of two candidates for inclusion in the same market tend to converge over time, as in Horowitz (1981), or whether the prices of the candidates are sufficiently correlated to warrant inclusion in the same market, as in Stigler and Sherwin (1985). An alternative approach that avoids the problem of determining the price at which a heterogeneous good sells is the "shipments test" proposed by Elzinga and Hogarty (1973, 1978).

(5.) Firms 1 and 2 may have equal or different production and transportation costs without any important points of the analysis here being affected. Transportation costs need not be a linear function of distance.

(6.) F.o.b. pricing thus creates a natural territorial monopoly for each seller. Ironically, Stigler (1949) himself noted this point. Alternatively, because the same price is paid for the same thing at the same time only by the customers who are located a specific distance from a seller's plant, perhaps every point on OP+ [t.sub.1] and OP+ [t.sub.2] is a separate "market."

(7.) The hypothetical price schedules shown in Figure 1 are maxima, not necessarily the prices a profit-maximizing firm will charge. The appropriate model here is one of limit-entry pricing, as in Modigliani (1958). The limit price represents an upper bound, and the spatial set of prices actually chosen within that constraint will correspond to the quantities at which marginal revenue equals marginal cost. Depending on cost and demand conditions, Greenhut and Greenhut (1975) show that those prices might increase away from a plant at a rate faster than shipping charges do, decrease at the same rate as shipping charges from the competing plant fall, or more likely track some pattern in between. The set of prices charged might induce a new firm to enter and locate between points A and B, as in the standard Hotelling (1929) model. Entry would cause the two existing firms to exit parts of their markets, but would result in a firm's charging a price equal to its mill price plus its own transportation cost only if the two firms were located atop one another or at razor point M.

(8.) If the market is contestable, price will be held to potential entrants' marginal cost. But except in the unlikely event of building its plant atop that of an existing firm, the law of one price would still dictate that the existing firm could charge a price more than its own production plus transportation cost. The exercise of local market power might likewise be constrained by buyers, if they can ship more cheaply than sellers (see note 10).

(9.) No statement of the law of one price refers to the number of firms, nor requires that the "one price" be a competitive one. Strictly speaking, the law of one price is a positive proposition, but it has obvious normative implications. Prices that just cover marginal (production plus transportation) costs maximize output and wealth, ceteris paribus. Obviously, in the absence of price discrimination, the first interpretation of the law of one price depicted in Figure 1 implies just the opposite: higher prices, reduced output, and deadweight welfare losses.

(10.) As already noted, buyers may be lower-cost shippers than sellers. But that possibility does not change any of the analysis here. Buyers closer to point A would still purchase only from firm 1, ceteris paribus; buyers closer to B would still purchase only from firm 2. Each firm operates in an isolated market over the relevant spatial distance. Moreover, firm I would still charge a price based on the cost of transportation from firm 2's market (that cost now determined by the cost to buyers, not sellers, of transporting from point B); firm 2's prices would be determined according to buyers' cost of transporting from firm 1's market.

(11.) To the extent that firm 2 faces lower transportation costs than those shown in Figure 1, the size of its market relative to that of firm 1 increases, and the prices that the two firms charge will fall. To go one step further, to the extent that other firms face nonprohibitive transportation costs, entry by other firms becomes possible, and their production plus transportation costs will set an upper limit on prices any firm can charge. In the limit, if transportation costs are zero (e.g., securities or foreign exchange markets), then all firms wherever located are competitors, but then any interest in transportation costs as affecting price and defining markets disappears.

(12.) The only way firms at B can compete for sales at A would be to absorb freight charges, that is, to quote prices to customers in A that are less than the prices they charge in B plus transportation costs. That strategy is characteristic of a delivered pricing system where all sellers designate A as a basing point and which would, at least under some plausible circumstances, as shown by Haddock (1982) and Carlton (1983), be consistent with procompetitive behavior. But it would be inconsistent with the law of one price. The inclusion of transportation costs in determinations of what should constitute competitive pricing nevertheless is common in antitrust cases. Recently, for example, a well-known economist testified on behalf of plaintiffs that at tempts to orchestrate a collusive agreement should be inferred from evidence that defendant sellers charged lower prices to distant customers, "although the transportation costs of delivering to those customers were higher." See A. A. Poultry Farms, Inc. v. Rose Acre Farms, Inc., 881 F.2d 1396 (7th Cir. 1989) (Easterbrook, J.).

(13.) For a mathematical demonstration of this same point, showing that in Nash equilibrium, Betrand duopolists selling products differentiated only by location both charge prices of p* = c + t, where c is the (constant) unit cost of production at the seller's plant and t is the unit cost of transportation to customers uniformly distributed along a line segment of length 1, see Mas-Colell et al. (1995, 396-98). Mas-Colell et al. do not address the implications of their result for market definition, however.

(14.) These locational rents may be quasi-rents, eventually capitalized in input prices. In that case, all firms charge the same price (p + t) and none earns above-normal returns. Such an equilibrium is insightfully described by Adam Smith (1976, 366-67), the most renowned of the masters:

 The greater the number and revenue of the
 inhabitants of a town, the more extensive is the
 market which it affords to those of the country;
 and the more extensive that market, it is always
 the more advantageous to a greater number. The
 corn which grows within a mile of the town, sells
 there for the same price with that which comes
 from twenty miles distance. But the price of the
 latter must generally, not only pay the expence of
 raising and bringing it to market, but afford too
 the ordinary profits of agriculture to the farmer.
 The proprietors and cultivators of the country,
 therefore, which lies in the neighborhood of the
 town, over and above the ordinary profits of
 agriculture, gain, in the price of what they sell, the
 whole value of the carriage of the like produce
 that is brought from more distant parts, and they
 save, besides, the whole value of the carriage in
 the price of what they buy. Compare the
 cultivation of the lands in the neighborhood of
 any considerable town, with that of those which
 lie at some distance from it, and you will easily
 satisfy yourself how much the country is benefited
 by the commerce of the town.

(15.) Price equality is a necessary but not sufficient condition for defining a market. Though eggs may sell for $13 per case in Moscow and $13 per case in Chicago, the two cities are probably not in the same market.

(16.) Although the same price must prevail in all parts of a properly defined economic market, one should not confuse the discussion here with legal (antitrust) definition of markets. According to [section] 1.1 of the Department of Justice/ Federal Trade Commission Horizontal Merger Guidelines, promulgated on 2 April 1992, antitrust market definition asks what alternatives are (or will become) available to consumers if, hypothetically, one firm (or set of firms) attempts to impose a "small but significant and nontransitory" increase in price on them. Hence, for purposes of antitrust analysis, places that are not currently "in" the market on our interpretation of the law of one price might well be included when delineating the boundaries of the area within which the forces of supply and demand interact to constrain any attempt to exercise monopoly power; see, for example, Landes and Posner (1981).

(17.) If the appropriate interpretation of the amended law is that depicted in Figure 1, imperfect information might also lead to more than one firm selling in a market where another firm, perfectly informed, could price so as to maintain a total monopoly. Firm 1 may not know perfectly what firm 2's costs are, and so choose a limit price that is high enough to allow firm 2 to compete with firm 1.

(18.) Transportation typically occurs from shipping hubs (often large cities) rather than from every single point on the spatial continuum. In Illinois, producers in Joliet are some 50 miles closer to Los Angeles than those in Waukegan. But these cities are more or less equidistant from Chicago, the point where shipments to Los Angeles are aggregated, and so face equivalent costs to ship to Los Angeles.

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FRED S. McCHESNY, WILLIAM F. SHUGHART II, and DAVID D. HADDOCK *

* We benefited from discussions with William Breit, Kenneth Elzinga, Barry Hirsch, David Laband, John Mayo, Russell Sobel, and Robert Tollison and from the comments of Michael Reksulak and Hilary Shughart. Helpful comments also were received in presentations at the meeting of the Southern Economic Association in Tampa, Florida, and at New York University's Colloquium on Market Institutions and Economic Processes. The thoughtful suggestions of the editor and two anonymous referees were of particular value in improving the article. Thanks also to Michael Reksulak and Birsel Tavukcu for help with the diagrams and to Lina Zhou for efficient research assistance. As is customary, however, we accept full responsibility for any remaining errors.

McChesney: Class of 1967/James B. Haddad Professor of Law and Professor, Kellogg School of Management, Northwestern University, 357 E. Chicago Avenue, Chicago, IL 60611. Phone 1-847-425-1134, Fax 1-847-328-4213, E-mail [email protected]

Shughart: F. A. P. Barnard Distinguished Professor of Economics and holder of the Robert M. Hearin Chair, Department of Economics, University of Mississippi, P.O. Box 1848, University, MS 38677-1848. Phone 1-662-915-7579, Fax 1-662-915-6943, E-mail [email protected]

Haddock: Professor of Law and Professor of Economics, Northwestern University, 3238 Arthur Andersen Hall, 2001 Sheridan Road, Evanston, IL 60208-2600. Phone 1-847-491-8225, Fax 1-847-491-7001, E-mail [email protected]