The law of one price: an interpretation of the literature and some new evidence.

Author:Pippenger, John
 
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  1. INTRODUCTION

    The law of one price (LOP) is one of the fundamental ideas in economics, particularly international economics. Unfortunately the empirical evidence regarding the law appears at best mixed. Previous work has not provided a satisfactory explanation for this unsatisfactory situation. [For a review of some of the relevant literature, see (Goldberg and Knetter, 1997).] We believe that we can provide an explanation. In the strict version of the LOP found in dictionaries and encyclopedias, arbitrage is the mechanism behind the law. To effectively evaluate this strong version, prices must come from markets where arbitrage is possible.

    The empirical literature however often appears to use what we call a weak version of the law. This weak version does not depend on arbitrage. The weak version implicitly assumes that similar price indexes and prices should converge.

    This paper has two major objectives: (1) Use the distinction between weak and strong versions of the law of one price to explain why the evidence appears to be so mixed. [This distinction was suggested by Charles Engel.] (2) Use new data to provide half lives for price differentials that can, for the first time, be used to evaluate the strong version of the law. Section 2 discuses the meaning of the term "law of one price" and illustrates the importance of arbitrage for the strong version of that law. Section 3 reviews the literature from the perspective of Section 2 and uses the distinction between strong and weak versions of the LOP to explain why the empirical evidence appears to be so mixed. Section 4 uses Section 2 to develop our model of the LOP. Section 5 describes our data from international grain markets. Unlike prices used earlier to estimate half lives, our prices are from markets where arbitrage is possible. Section 6 reports our estimates of the half lives for spot price differentials in international grain markets. Unlike earlier estimates, we correct for the potential bias in OLS estimates where practical and also provide confidence intervals where practical. Section 7 summarizes our results and presents our conclusions. Our two primary conclusions are straightforward. (1) The failure to distinguish between strong and weak versions of the law of one price is a primary cause of the confusion about the evidence concerning the law of one price. (2) The strong version of law of one price works well in our grain markets with half lives between 3 and 8 weeks.

  2. ARBITRAGE AND THE LOP

    The role of arbitrage in the law of one price is crucial for our explanation for why the evidence appears to be so mixed regarding the LOP. Dictionaries and encyclopedias for economics make it clear that arbitrage is the mechanism behind the LOP. For example, The Penguin Dictionary of Economics defines the LOP as follows:

    The law, articulated by Jevons, stating that 'In the same open market, at any moment, there cannot be two prices for the same kind of article.' The reason is that, if they did exist, arbitrage should occur until the prices converge.

    Other dictionaries and encyclopedias that we have perused that include a reference to the LOP make similar references to arbitrage. By mentioning or appealing to arbitrage, most articles trying to test the LOP in commodity markets implicitly attribute the LOP to arbitrage. Many articles make the link between the LOP and arbitrage explicit. For example, in the first paragraph of his seminal article, (Isard, 1997) says the following:

    In the assumed absence of transport costs and trade restrictions, perfect commodity arbitrage insures that each good is uniformly priced (in common currency units) throughout the world--the "law of one price" prevails.

    Of course arbitrage is not the only mechanism that causes prices to converge. Even without arbitrage, competition normally limits the divergence between prices of similar goods. We would not expect differentials between sub-indexes of consumer price indexes in different countries for products such as tires or furniture to drift off to infinity. This expectation is apparently what is behind what we call the weak version of the law of one price. But the persistence in such differentials and the low power of tests for cointegration often combine to make it impossible to reject the null of no cointegration with normal sized data sets when arbitrage is not impossible. As pointed out below in more detail, the apparently mixed evidence regarding the LOP appears to be primarily the result of interpreting rejections of, or the failure to support, the weak version as rejections of, or the failure to support, the strong version. To illustrate the role of arbitrage in the strong version of law of one price, consider the following mental experiment: Following David Hume, suppose four-fifths of all the wheat in Europe disappeared over night. Spot prices in Rotterdam for wheat would rise far above the spot price in Gulf ports plus conventional transportation and transaction costs. This differential in spot prices would not create an opportunity for arbitrage because it would be impossible to move wheat from Gulf ports to Rotterdam within a couple of days. With arbitrage impossible, this potentially very large price differential would not reject effective arbitrage and the LOP.

    Arbitrage would be possible using forward contracts. Forward prices in Rotterdam would also rise. Higher forward prices in Rotterdam than at Gulf ports would create an opportunity for wheat dealers to engage in profitable arbitrage. To eliminate risk, within as short a period of time as is possible, arbitrageurs would enter into at least three forward contracts: One contract would be for wheat to be delivered on board ship in a Gulf port in say two weeks. Another contract would secure the ship to load the wheat in two weeks and would fix the freight rate. A third contract would sell the wheat for dollars in Rotterdam when the ship is scheduled to arrive. All these contracts are forward contracts and all prices are forward prices. This combination of forward contracts meets all the conditions for arbitrage including identical products, resale, and no risk. If these forward contracts are omitted, the transaction is no longer arbitrage because it involves risk.

    Arbitrage raises forward prices in Gulf ports relative to what they would have been without arbitrage. Arbitrage lowers forward prices in Rotterdam relative to what they would have been. Arbitrage also bids up forward freight rates relative to what they would have been. Arbitrage does not directly reduce the differential in current spot prices, but it indirectly reduces that differential. Higher forward prices in Gulf ports encourage grain elevators there to hold over more wheat for future delivery. The reduced supply of wheat for sale in spot markets raises spot prices in Gulf ports. Lower forward prices in Rotterdam encourage grain elevators there to offer more wheat in spot markets, reducing current spot prices in Rotterdam.

    After accounting for transaction costs, effective arbitrage equalizes the appropriate forward prices for identical products in different locations. Arbitrage does not provide a direct link between spot prices. But arbitrage provides an important indirect link between spot prices. Competition limits the spread between spot and forward prices for identical products in each location. With the LOP holding for forward prices, the link between spot and forward prices in each individual location limits the divergence in spot prices between locations.

    With effective arbitrage, after accounting for the transaction costs, differentials between the relevant forward prices should be relatively small and not highly autocorrelated. Half lives should be very short. With no direct market mechanism to reduce differentials in spot prices, spot differentials will be larger and more persistent. The fact that we find half lives of just a few weeks for spot prices suggests that the half lives for the appropriate forward prices would be very short.

  3. THE LOP LITERATURE AND ITS INTERPRETATION

    In spite of a prodigious amount of empirical work, there is still no clear consensus about the validity of the law of one price. A major factor contributing to this lack of consensus is that the literature often uses two versions of the "law of one price", a weak and a strong version, without clearly distinguishing between them. The weak version assumes that just similar price indexes or similar prices should converge. The strong version found in dictionaries and encyclopedias depends on arbitrage. If arbitrage is not possible, the failure of prices to converge does not reject, or fail to support, the strong version. (Parsley and Wei, 1996) and (Engel and Rogers, 2001) do not use prices where arbitrage is possible, but they also do not appeal to arbitrage in discussing the LOP. As a result, their conclusions about the weak version of the LOP are less likely to be misinterpreted as conclusions about the strong version. Unfortunately much of the empirical literature is more easily misinterpreted. When articles appeal to arbitrage in their discussion of the LOP and then use prices for products that are not identical to reject the LOP, even careful readers can misinterpret the results. (Richardson, 1978) is one example of this potential misinterpretation. The title of the article, Some Empirical Evidence on Commodity Arbitrage and the Law of One Price, would lead readers to believe that he is testing and rejecting the strong form of the LOP. His rejection itself is unambiguous: "Second, it is notable that the 'law of one price' fails uniformly." (Richardson, 1978, p. 347) But Richardson can only reject the weak version because he uses sub-categories of consumer price indexes such as Bakery Products.

    Many articles that reject the LOP up until about 2001 do something similar. For example (Fraser, Taylor and Webster, 1991) give the impression that they are...

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