The Law of One Price

Karl Gunnar Persson, University of Copenhagen

Definitions and Explanation of the Law of One Price

The concept "Law of One Price" relates to the impact of market arbitrage and trade on the prices of identical commodities that are exchanged in two or more markets. In an efficient market there must be, in effect, only one price of such commodities regardless of where they are traded. The "law" can also be applied to factor markets, as is briefly noted in the concluding section.

The intellectual history of the concept can be traced back to economists active in France in the 1760-70's, which applied the "law" to markets involved in international trade. Most of the modern literature also tends to discuss the "law" in that context.

However, since transport and transaction costs are positive the law of one price must be re-formulated when applied to spatial trade. Let us first look at a case with two markets which are trading, say, wheat but with wheat going in one direction only, from Chicago to Liverpool, as has been the case since the 1850's.

In this case the price difference between Liverpool and Chicago markets of wheat of a particular quality, say, Red Winter no. 2, should be equal to the transport and transaction cost of shipping grain from Chicago to Liverpool. This is to say that the ratio of the Liverpool price to the price in Chicago plus transport and transaction costs should be equal to one. Tariffs are not explicitly discussed in the next paragraphs but can easily be introduced as a specific transaction cost at par with commissions and other trading costs.

If the price differential exceeds the transport and transaction costs, this means that the price ratio is greater than one, then self-interested and well-informed traders take the opportunity to make a profit by shipping wheat from Chicago to Liverpool. Such arbitrage closes the price gap because it increases supply and hence decreases price in Liverpool, while it increases demand, and hence price in Chicago. To be sure the operation of the law of one price is not only based on trade flows but inventory adjustments as well. In the example above traders in Liverpool might choose to release wheat from warehouses in Liverpool immediately since they anticipate shipments to Liverpool. This inventory release works to depress prices immediately. So the expectation of future shipments will have an impact on price immediately because of inventory adjustments.

If the price differential does not exceed the transport and transaction cost, this means that the price ratio is less than one, then self-interested and well informed traders take the opportunity to restrict the release of wheat from the warehouses in Liverpool and decrease the demand for shipments of wheat from Chicago. These reactions will trigger off an immediate price increase in Liverpool since supply falls in Liverpool and a price decrease in Chicago because demand falls.

Formal Presentation of the Law of One Price

Let P^L and P^C denote the prices in Liverpool and Chicago respectively. Furthermore, we also observe the transport and transactions costs, linked to shipping the commodity from Chicago to Liverpool, P^Tc. All prices are measured in the same currency and units, say, shillings per imperial quarter. What has been explained above verbally can be expressed formally. The law of one price adjusted for transport and transaction costs implies the following equilibrium, which henceforward will be referred to as the Fundamental Law of One Price Identity or FLOPI:

In case the two markets both produce and can trade a commodity in either direction the law of one price states that the price difference should be smaller or equal to transport and transaction costs. FLOPI then is smaller or equal to one. If the price difference is larger than transport and transaction costs, trade will close the gap as suggested above. Occasionally domestic demand and supply conditions in two producing economies can be such that price differences are smaller than transport and transaction costs and there will not be any need for trade. In this particular case the two economies are both self-sufficient in wheat.

A case with many markets will necessitate a third elaboration of the concept of the law of one price. Let us look at it in a world of three markets, say Chicago, Liverpool and Copenhagen. Assume furthermore that both Chicago and Copenhagen supply Liverpool with the same commodity, say wheat. If so, the Liverpool-Copenhagen price differential must be equal to the transport and transaction costs between Copenhagen and Liverpool and the Chicago-London price differential will be equal to the transport and transaction costs between Chicago and Liverpool. But what about the price difference between Chicago and Copenhagen? It turns out that it will be determined by the difference between transport and transactions costs from Chicago to Liverpool and from Copenhagen to Liverpool. If it costs 7 cents to ship a bushel of grain from Chicago to Liverpool and 5 cents from Copenhagen to Liverpool, the law of price difference between Copenhagen and Chicago will be 2 cents that is 7 – 5 = 2. If price is 100 cents per bushel in Chicago it will be 107 in Liverpool and 102 in Copenhagen. So although the distance and transport cost between Chicago and Copenhagen is larger than between Chicago and Liverpool, the equilibrium price differential is smaller! This argument can be extended to many markets in the following sense: the price difference between two markets which do not trade with each other will be determined by the minimum difference in transport and transaction costs between these two markets to a market with which they both trade.

The argument in the preceding paragraph has important implications for the relationship between distance and price differences. It is often argued that the difference between prices of a commodity in two markets increases monotonically with distance. But this is true only if the two markets actually trade directly with each other. However, the likelihood that markets cease to trade directly with each other increases as the distance increases and long distance markets will therefore typically be only indirectly linked through a third common market. Hence the paradox illustrated above that the law of one price difference between Chicago and Copenhagen is smaller despite the larger geographical distance than that between Copenhagen and Liverpool or Chicago and Liverpool. In fact it is quite easy to imagine two markets at a distance of two units both exporting to a third market in between them at a distance of one unit from each of them and enjoying the same price despite the large distance.

Efficient Markets and the Law of One Price

In what follows we typically discuss the "law" in a context with trade of a particular commodity going in one direction only, that is FLOPI = 1.

In a market with arbitrage and trade, violations of the law of one price must be transitory. However, price differentials often differ from the law of one price equilibrium, that is FLOPI is larger or smaller than 1, so it is convenient to understand the law of one price as an "attractor equilibrium" rather than a permanent state in which prices and the ratio of prices rest. The concept "attractor equilibrium" can be understood with reference to the forces described in the preceding section. That is, there are forces which act to restore FLOPI when it has been subject to a shock.

A perfectly efficient set of markets will allow only very short violations of the law of one price. But this is too strong a condition to be of practical significance. There are always local shocks which will take time to get diffused to other markets and distortions of information will make global shocks affect local markets differently. How long violations can persist depends on the state of information technology, whether markets operate with inventories and how competitive markets are. Commodity markets with telegraphic or electronic information transmission, inventories and no barriers to entry for traders can be expected to tolerate only short and transitory violations of the law of one price. News about a price change in one major market will have immediate effects on prices elsewhere due to inventory adjustments.

A convenient econometric way of analyzing the nature of the law of one price as an "attractor equilibrium" is a so-called error correction model. In such a model an equilibrium law of one price is estimated. If markets are not well integrated one cannot establish or estimate FLOPI. Given the existence of a long-run or equilibrium price relationship between markets, a violation is a so called "innovation" or shock, which will be corrected for so that the equilibrium price difference is restored. Here is the intuition of the model described below: Assume first that Liverpool and Chicago prices are in a law of one price equilibrium. Then, for example, the price in Chicago is subject to a local shock or "innovation" so that price in Chicago plus transport and transaction costs now exceeds the price in Liverpool. That happens in period t-1, and then the price in Liverpool will increase in the next period, t, while the price in Chicago will fall. Prices will fall in Chicago because demand for shipments will fall and it will increase in Liverpool because of a fall in supply when traders in Liverpool stop releasing grain from the warehouses in expectation of higher prices in the future. Eventually the FLOPI = 1 condition will be restored but at higher prices in both Liverpool and Chicago.

To summarize, the logic behind the error correction model is that prices in Liverpool and Chicago will react if there is a dis-equilibrium, that is when the price differential is larger or smaller than transport and transaction costs. In this case the prices will adjust such that the deviation from equilibrium is decreasing. The error correction model is usually expressed in differences of log prices. Let. The error correction model in this version is given by:

whereare statistical error terms with are assumed to be normally distributed with mean zero and constant variances. Please, note that errors are not the "error" that figures in the term "error correction model." A better name for the latter would be "shock correction model" or "innovation correction model" to evade misunderstanding.

and are so-called adjustment parameters which indicate the power of FLOPI as an "attractor equilibrium." The expected sign of the parameter is negative and it is positive for. To see this, imagine a case where the expression in the parenthesis above is larger than one. Then price in Liverpool should fall and increase in Chicago.

The parameters and indicate the speed at which "innovations" are corrected, the larger the parameters are for a given magnitude of the "innovation," the more transitory are the violations of the law of one price – in other words, the faster is the equilibrium restored. The magnitudes of the parameters are an indicator of the efficiency of the markets. The higher they are, the faster will the equilibrium law of one price (FLOPI) be restored and the more efficient markets are. (The absolute values of the sum of the parameters should not exceed one.) The magnitude of "innovations" also tends to fall as markets get more efficient as defined above.

It is convenient to express the parameters in terms of the half life of shocks. Half life of a shock measures the time it takes for an original deviation from the equilibrium law of one price (FLOPI) to be reduced to half. The half life of shocks has been reduced dramatically in the long-distance trade of bulky commodities like grain – that is distances above 1500 km. From the seventeenth to the late nineteenth centuries, the half life was reduced from up to two years to only two weeks in international wheat markets, as revealed by the increase in the adjustment parameters. The major reason for this dramatic change is the improvement in information transmission.

The adjustment parameters can also be illustrated graphically and Figure 1 displays the stylized characteristics of adjustment speed in long-distance wheat trade and indicates a spectacular increase in grain market efficiency, specifically in the nineteenth century.

Read Figure 1 in the following way. At time 0 the two markets are in a law of one price equilibrium (FLOPI), that is prices in the two markets are exactly equal (set here arbitrarily at 100), and the ratio of prices is one. In this particular graphical example we abstract from transport and transactions costs. Now imagine a shock to the price in one market by 10 percent to 110. That will be followed by a process of mutual adjustment to the law of one price equilibrium (FLOPI) but at higher prices in both markets compared to the situation before the shock. The new price level will not necessarily be halfway between the initial level and the level attained in the economy which was subject to a shock. Adjustments can be strong in some markets and weak in others. As can be seen in Figure 1, the adjustment is very slow in the case of the Pisa (Italy) to Ruremonde (Netherlands). In fact, a new law on price equilibrium is not attained within the time period, 24 months, allowed by the Figure. This indicates very low, but still significant, adjustment parameters. It is also worth noting the difference in adjustments speed between pre-telegraph Chicago-Liverpool trade in the 1850's and post-telegraph trade in the 1880's.

Figure 1

Adjustment Speed in Markets after a Local Shock in Long-distance Wheat Markets
Cases from 1700-1900.

Note: The data underlying the construction are from Persson (1988) and Ejrnæs and Persson (2006).

It is worth noting that the fast speed of adjustment back to the law of one price recorded for single goods in the nineteenth century contrasts strongly with the sluggish adjustment in price indices (prices for bundles of goods) across economies (Giovanini 1998). However, some of these surprising results may depend on misspecifications of the tests (Taylor 2001).

Law of One Price and Convergence

The relationship between the convergence of prices on identical goods and the law of one price is not as straightforward as often believed. As was highlighted above, the law of one price can exist as an "equilibrium attractor," despite large price differentials between markets, as long as the price differential reflects transport and transaction costs and if they are not prohibitively high. So in principle the adjustment parameters can be high, despite large price differentials. For example, the Chicago to Liverpool trade in the nineteenth century was based on highly efficient markets, but transport and transaction costs remained at about 20-25 percent of the Chicago price of wheat. However, historically the convergence in price levels in the nineteenth century was associated with an improvement in market efficiency as revealed by higher adjustment parameters. Convergence seems to be a nineteenth-century phenomenon. Figure 2 below indicates that there is not a long-run convergence in wheat markets. Convergence is here expressed as the UK price relative to the U.S. price. Falling transport costs, falling tariffs and increased market efficiency, which reduced risk premiums for traders, compressed price levels in the nineteenth century. Falling transport costs were particularly important for the landlocked producers when they penetrated foreign long-distance markets, as displayed by the dramatic convergence of Chicago to UK price levels. When the U.S. Midwest started to export grain to UK, the UK price level was 2.5 times the Chicago price. However, the figure exaggerates the true convergence significantly because the prices used do not refer to identical quality goods. As much as a third of the convergence shown in the graph has to do with improved quality of Chicago wheat relative to UK wheat, a factor often neglected in the convergence literature.

However, after the convergence forces had been exploited, trade policy was reversed. European farmers had little land relative to farmers in the New World economies, such as Argentina, Canada and U.S. and the former faced strong competition from imported grain. A protectionist backlash in continental Europe emerged in the 1880's, continued during the Great Depression and after 1960, which contributed to price divergence. The trends discussed above are applicable to agricultural commodities but not necessarily to other commodities because protectionism is commodity specific. However, it is important to note that long-distance ocean shipping costs have not been subject to a long-run declining trend despite the widespread belief that this has been the case and therefore the convergence/divergence outcome is mostly a matter of trade policy.

Figure 2
Price Convergence, United States to United Kingdom, 1800-2000

(UK price relative to Chicago or New York price of wheat)

Source: Federico and Persson (2006).

Note: Kernel regression is a convenient way of smoothing a time series.

The Law of One Price, Trade Restrictions and Barriers to Factor Mobility

Tariffs affect the equilibrium price differential very much like transport and transaction costs, but will tariffs also affect adjustment speed and market efficiency as defined above? The answer to that question depends on the level of tariffs. If tariffs are prohibitively high, then the domestic market will be cut off from the world market and the law of one price as an "equilibrium attractor" will cease to operate.

The law of one price can also, of course, be applied to factor markets – that is markets for capital and labor. For capital markets the law of one price would be such that interest rate or return differentials on identical assets traded in different locations or nations converge to zero or close to zero – that is the ratio of interest rates should converge to 1. If there are significant differences in interest rates between economies, capital will flow into the economy with high yields and contribute to leveling the differentials. It is clear that international capital market restrictions affect interest rate spreads. Periods of open capital markets, such as the Gold Standard period from 1870 to 1914, were periods of small and falling interest rate differentials. But the disintegration of the international capital markets and the introduction of capital market controls in the aftermath of the Great Depression in the 1930s witnessed an increase in interest rate spreads which remained substantial also under the Bretton Woods System c.1945 to 1971(73), in which capital mobility was restricted. It was not until the capital market liberalization of the 1980s and 1990s that interest rate differences again reached levels as low as a century earlier. Periods of war, when capital markets cease to function, are also periods when interest rates spreads increase.

The labor market is, however, the market that displays the most persistent violations of the law of price. We need to be careful, however, in spotting violations, in that we need to compare wages of identically skilled laborers and take differences in costs of living into consideration. Even so, huge real wage differences persist. A major reason for that is that labor markets in high income nations are shielded from international migration by a multitude of barriers.

The law of one price does not thrive under restrictions to trade or factor mobility.

References:

Ejrnæs, Mette, and Karl Gunnar Persson. "The Gains from Improved Market Efficiency: Trade before and after the Transatlantic Telegraph," Working paper, Department of Economics, University of Copenhagen, 2006.

Federico. Giovanni and Karl Gunnar Persson. "Market Integration and Convergence in the World Wheat Market, 1800-2000." In New Comparative Economic History, Essays in Honor of Jeffrey G. Williamson, edited by Timothy Hatton, Kevin O'Rourke and Alan Taylor. Cambridge, MA.:MIT Press, 2006.

Giovanini, Alberto. "Exchange Rates and Traded Goods Prices." Journal of International Economics 24 (1988): 45-68.

Persson. Karl Gunnar. Grain Markets in Europe, 1500-1900: Integration and Deregulation. Cambridge: Cambridge University Press, 1998.

Taylor, Alan M. "Potential Pitfalls for the Purchasing Power Parity Puzzle? Sampling and Specification Biases in Mean-Reversion Tests of the Law of One Price," Econometrica 69, no. 2 (2001): 473-98.