Arbitrage In Bimetallic Money Supplies: Evidence From The Exchange Rate

Stefan E. Oppers
University of Michigan

As a theoretical proposition, Gresham's Law is straightforward. In its most general form, it postulates that a money that has an intrinsic value higher than its face value will be removed from ordinary circulation, to be sold where it brings its highest price. Gresham's Law was the driving force behind disappearing pennies during the high copper prices of the late 1970s, the scarcity of small silver U.S. coinage after 1834, and the disappearance of gold coins from France before 1785, to name but a few examples. In each case, it paid to melt down the undervalued coins - those with a face value lower than their intrinsic value - and sell the metals in the commodities markets.

Despite its intuitive appeal and many compelling anecdotal examples of its effects, the practical importance of the arbitrage associated with Gresham's Law has been difficult to establish. Recently, the speed and efficacy of the arbitrage mechanism has been of interest in the revitalized literature on bimetallism. Friedman (1990), in an effort to explain the remarkable stability of the relative price of gold and silver in the 19th century, revived the old argument that bimetallic arbitrage took place more or less continuously in France during this time, never completely removing the undervalued coins from circulation. In a recent paper, however, Oppers (1993) cast doubt on his argument by showing that the bimetallic system has in fact two stabilizing effects on bullion prices. He showed that in addition to bimetallic arbitrage, the relative price of gold and silver will be stabilized through the effects of forward-looking rational expectations of bullion market traders. Observing a stable market ratio cannot be used as an argument in favor of the view that arbitrage took place continuously.

The recent literature on bimetallism therefore recasts a set of old questions. First, to what extent did bimetallic arbitrage contribute to the stability of the bimetallic system? Second, how effective was Gresham's Law in removing the undervalued coins from circulation? Or, equivalently, to what extent did both metals tend to circulate side by side in bimetallic money supplies? Both questions are closely linked. If Gresham's Law operated swiftly, a small premium on one of the metals could quickly move the bimetallic country to an effective monometallic standard. Further arbitrage would then be impossible, eliminating any stabilizing effect on the relative price of gold and silver. If, on the other hand, the mechanism of Gresham's Law operated only slowly, bimetallic arbitrage could provide stability to the market ratio for long periods of time.

The focus of the debate on bimetallic arbitrage has been the metal composition of bimetallic money supplies, in particular that of France during the 19th century. During this time, France and the countries of the Latin Monetary Union were the pivot of the international bimetallic monetary system. They had a mint ratio of 15.5 grams of silver for one gram of gold, and throughout the period, the market ratio remained very close to this value.

Recent research has offered important information on the metal composition of the available stock of French coins. Sicsic (1989) and Flandreau (1992) used data from a 1878 survey of the metallic money supply to construct time series for the supply of coins. Their research indicates that even though their relative prevalence fluctuated quite substantially with movements in the market ratio, both gold and silver coins were in existence throughout France's bimetallic period. This has led Flandreau, among others, to suggest that bimetallic arbitrage could operate continuously throughout France's bimetallic period.

This conclusion is not necessarily correct. In fact, evidence on the metal composition of the available stock of French coins does not offer much insight into the extent to which arbitrage could take place. Even if they were around, coins of the dearer metal might not have been available at face value. They could have been hoarded, and some (Rolnick and Weber, 1986) have claimed that undervalued coins might have traded at a premium in some cases. Without availability of undervalued coins at face value, arbitrage between the money supply and the bullion markets could not take place. To determine the possibility of bimetallic arbitrage, we would have to combine evidence on the stock of coins, the velocity of gold and silver coins, and the price at which they traded.

In this paper I show how we can use data on the exchange rate of the bimetallic currency to determine a measure of the metal composition of the French money supply that does give us an accurate indication of the extent to which arbitrage was possible. In the model I develop, the value of an asset denominated in a bimetallic currency depends on the expected value of the bullion received upon redemption of the asset. The expected value of the bullion received in turn depends not only on the relative bullion composition of the stock of coins, but also on the price at which gold and silver coins trade, and on the ease with which coins of each metal can be obtained.

The most straightforward example of the value of a bimetallic asset is the exchange rate. The model in this paper shows how we can use data on the exchange rate to easily estimate a parameter I call "arbitrage potential." This parameter is a composite of the relative stock of gold and silver coins, their relative availability (velocity) and the price at which they traded, and thus indicates the extent to which gold and silver coins were available for trade at face value, or the extent to which bimetallic arbitrage was possible.

To derive the measure of arbitrage potential, I assume that the value of the bimetallic asset is determined by the expected value of the bullion received upon redemption. Take for example France in the 19th century. With a bimetallic monetary system, redemption of franc banknotes could in principle occur in gold or silver. How did this affect the exchange rate of the franc? The value of a franc note should depend on the expected amount of silver and gold that could be obtained upon redemption. One would therefore expect arbitrage between the exchange market and the monetary systems of Britain and France to ensure that the value of the exchange rate remained close to that determined by the relative bullion values of the currencies. If, for example, the exchange rate of the franc was lower than the bullion values warrant, the following arbitrage opportunity would arise. With the pound price of francs lower than the corresponding value of bullion, the arbitrageur would buy franc notes in London, transport them to Paris to redeem them for bullion. Once back in London, the silver could then be sold for a price higher than that paid for the franc notes initially. Therefore, the exchange rate between the franc and, say, the British pound - a gold currency - could not diverge substantially from the "bullion value" of the franc. In particular, it was a function of the proportion of the face value the franc note that was expected to be exchanged for silver, [[lambda]]

To see this, realize the value of a franc note in the exchange market depended on the expected amount of silver and gold that could be obtained upon redemption. Using silver as a numeraire, we get:

value of Fr1 note (in grams of silver) = [[lambda]]([[Delta]]SFr) + (1-[[lambda]])([[Delta]]GFr * x)

where [[lambda]] is equal to the proportion of the face value of the note that is redeemed in silver coin, [[Delta]]SFr is the fixed silver content of the franc, and [[Delta]]GFr is the fixed gold content of the franc. x is the market ratio, in ounces of silver that rade for one ounce of gold.


value of [[sterling]]1 note (in gold) = [[Delta]]G[[sterling]],

where [[Delta]]G[[sterling]] is the fixed gold content of the pound.

The exchange rate is the relative value of these two assets:

where the error E is introduced by the cost of arbitrage. It is assumed to be in the range -ø < E < ø. We can rewrite (1) as


and [[sterling]]/S is the market price of silver in London in pounds per ounce. The relationship between the franc exchange rate and the value of silver in pounds depends on the proportion of face value the franc note that is expected to be exchanged for silver, [[lambda]]. Equation (2) can easily be estimated.

What was the relationship between [[lambda]] and the relative circulation of gold and silver in France? The choice of metal in which the face value of the banknote will be paid is up to the payer, and it is assumed the payee cannot observe the preference of the payer regarding the payout metal. However, he does observe the proportion of the money supply consisting of silver coins. This corresponds to the average propensity to use silver as the medium of exchange, and is therefore the best estimate of [[lambda]], the mix of gold and silver coins the payee can expect to receive in exchange for the note. Therefore, [[lambda]] is equal to the relative proportion of silver in the French money supply, weighed by the velocity of silver coins. The paper describes how [[lambda]] can be easily interpreted as a measure of arbitrage potential.

Our measure of arbitrage potential does two things. First, it accurately indicates the extent to which arbitrage was possible in the bimetallic money supply. High arbitrage potential indicates the undervalued coins are readily available at face value, so that bimetallic arbitrage can easily take place. Low arbitrage potential indicates that the undervalued coins have either mostly disappeared from circulation, or now trade at a premium, making bimetallic arbitrage difficult. Second, the changes in arbitrage potential over time, together with evidence on coinage and changes in the total money supply, will enable us to determine the extent of arbitrage activity actually taking place. This will shed light on the speed with which the mechanism of Gresham's Law operated to remove the undervalued coins from ordinary circulation.

I apply the model to the case of France in the late 1840s and 1850s, a period when a large increase in world gold output created bimetallic arbitrage opportunities in France. The estimates of [[lambda]] show that the French money supply was effective monometallic in silver before 1850. With the market ratio above the mint ratio, arbitrage potential was low. In late 1850, however, a decline in the market ratio caused silver coins to become undervalued and arbitrage potential to shift up. Massive arbitrage started to take place, replacing silver coins with gold coins. In a matter of less than a decade, the French monetary system moved from an effective silver standard to an effective gold standard: already by 1859, this transition was completed. Silver coins no longer traded at face value, and arbitrage potential was near zero.

The development of the relative bullion composition of the money supply by itself, however, does not necessarily give a clear picture of the exact timing and extent of bimetallic arbitrage that took place in France. For example, even with a declining relative share of silver, the absolute size of the stock of silver could be constant, declining, or even rising, depending on what happens to the size of the overall money supply. Only when the absolute size of the silver stock is declining is arbitrage taking place. Examining changes in gold coinage at the mint does not provide an adequate answer either. It is true that bimetallic arbitrage requires the minting of gold coins, but an increase in gold coinage does not necessarily indicate that arbitrage is taking place. Instead of replacing silver coins, the increase in gold coins in circulation could in principle merely add to the original money supply. To determine when arbitrage was taking place, we need to combine the evidence on total gold and silver coinage with some measure of the changes in the size of the money supply; an increase in gold coinage indicates arbitrage only if it replaces silver coins and therefore does not increase the money supply.

Thus, using data on French income, prices, and coinage I construct the following picture of the changes in the French money supply after 1850. Between 1849 and 1851, the share of silver coins declined from close to one to below 50%. Nominal income was fairly constant, however, indicating that the disappearing silver coins were most likely being replaced by gold coins through arbitrage. This was supported by an increase in gold coinage, as well as the likely return to general circulation of gold coins previously hoarded or previously traded at a premium. During 1852, the market ratio was mostly close to the mint ratio, rendering arbitrage unprofitable. Indeed, gold coinage ceased almost completely, the money supply was largely constant, and the share of silver coins stopped declining.

In the next three years, the money supply increased by almost 50%. Looking at the big increase in gold coinage during this time, it is clear that a large part of this increase consisted of gold coins. The share of silver coins correspondingly declined in 1853, but seems to have recovered slightly in 1854 and 1855. In 1856 the money supply had started to decline, accelerating to a fall of more than 15% in 1858. Since minting of gold coins remained high, this means the amount of silver in circulation must have been declining. Indeed, the estimated value of [[lambda]] drops to zero by 1858.

The years of heaviest bimetallic arbitrage in France therefore seem to have been 1850, 1851, 1857, and 1858. Large gold coinage between 1853 and 1855 instead added to the money supply without removing silver coins.

Two conclusions can be drawn from this research. First, it seems that Gresham's Law was rather effective in removing undervalued coins from ordinary circulation. In less than a decade, in response to a silver premium of less than 2 3/4%, France's money supply - the largest of its time, containing around 10% of the cumulative production of silver since the end of the 15th century - was transformed from effectively monometallic in silver to effectively monometallic in gold, with most of the arbitrage occurring in just four years. Second, because bimetallic arbitrage could only take place during periods of effective bimetallism, only during relatively short periods of time did bimetallic arbitrage contribute to the stability of the market ratio. During most of the bimetallic period of the 19th century, other forces - such as the mean-reverting expectations of bullion-market traders pointed out by Oppers (1993) - played a major role in keeping the market ratio close to the French mint ratio.


Flandreau, Marc (1992), "Coin Memories: Estimates of the French Metallic Currency 1840-1878," Mimeo, University of California.

Friedman, Milton (1990), "Bimetallism Revisited," Journal of Economic Perspectives, 4, pp. 85-104.

Oppers, Stefan E. (1993), "A Model of the Bimetallic System." University of Michigan Research Forum on International Economics Discussion Paper No. 332, July.

Sicsic, Pierre (1989), "Estimation du Stock de Monnaie Metallique en France a la Fin du XIXeme Siecle," Revue Economique, pp. 709-735.