Abstract
We explore the price-setting protocol used by the money-changers guild, the Arte del Cambio, to set the exchange rate between domestic gold and silver coins in Renaissance Florence, a precursor in purpose and set-up to the contemporary system of posting interest rates. We show that the protocol is evolutionarily stable and designed to converge to the efficient price, which had important implications to the political economy of Florence and the stability of its monetary system. We also show that the predicted dynamics of convergence are consistent with surviving exchange rates from 1389 to 1432, thereby linking the empirical properties of the historical data to the institutional features of the guild that generated them.
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Notes
Our paper thus builds on recent literature on the microstructure of historical financial markets, which includes Neal and Davis (2006), Riva and White (2011), and Hautcoeur and Riva (2012), as well as the literature on using modern techniques for examining historical financial data, which includes Nogues-Marco and Vam Malle-Sabouret (2007), Fohlin and Reinhold (2010), Földvári and van Leeuwen (2011), Ho et al. (2013), and Annaert and Mensah (2014).
Referring to accounting and book transactions, de Roover (1944, p. 381) writes of a “French satirist, in the fifteenth century, [who] marveled at the ability of the Italians to do business without money. In dealing with them, he said, one never sees or touches any money; all they need to do business is paper, pen, and ink.”
Around 1430, in a proposal to deal with Florence's ballooning public debt, the silk merchant Andrea di Francesco Arnoldi recommended that shares in the public debt fund (Monte Comune) held by citizens could be transformed into current accounts on which they could make transfers amongst themselves for the settlement of private debts. By in effect monetizing the public debt, the government would thus be able to avoid repayment of its obligations. Such a proposal illustrates the Florentines' conception of money: "As Arnoldi explained, ‘Money and bank credit are both the same thing as cash.' He knew very well that most of the money circulating in Florence at the time was in fact bank money, not actual coin, for, as he went on to ask, how else could a city with no more than 150,000 florins in coin on hand at any one time ever have paid out the 4 million florins that the recent war with Lucca alone had cost? (Goldthwaite 2009; p. 427)."
While not entirely applicable to the case of Florence, it should be noted that there is a large literature examining discrepancies between the international market ratio and the fixed or legal ratio imposed by individual governments. For large discrepancies, Gresham's Law most likely would take effect, with the overvalued coin "driving out" the undervalued one. See Redish (2000), Weber (2009), and Nogues-Marco (2013) for very interesting discussions and examples.
The violent Ciompi revolt of 1378, a watershed moment deeply ingrained in Florentine memory, was precipitated by the distributional consequences of economic and business policy, and was precisely the type of upheaval that future generations of Florentine rulers had in mind and tried to avoid throughout the fifteenth century.
Created by the British Bankers Association in consultation with the Bank of England and others in 1986, LIBOR is currently managed by ICE. Under its aegis, interest rates for five major currencies and seven maturities (ranging from overnight to 12 months) are gathered from 11–18 of the world's major banks (the number depends on the currency) daily (except for holidays and so forth) at 11:00 a.m. London time. The submitted rates must be based on actual transactions associated with the cost of borrowing unsecured funds from other banks operating in the London money market, assuming a reasonable loan size. For each of the 35 currency-maturity interest rates combinations, the trimmed mean (with the upper and lower quartiles ignored) is calculated and announced to the market at approximately 11:45 a.m. as the current official benchmark value.
As for formal business education, commercial mathematics was taught in “reckoning” schools under the vigilance of a “reckoning master,” and Florence was the home to more than a dozen of these schools (Swetz 1987). According to Van Egmond (1976, p. 229) their curricula typically covered the topics found in modern secondary education. Instruction in the Florentine monetary system and its relationships to its foreign counterparts was also covered. Acquired skills included not only straightforward arithmetic and algebra but also more arcane (by modern standards) operations such as vigesimal and duodecimal divisions (Van Egmond 1976, p. 128).
During our sample period the weight of newly minted florins increased in 1390 from approximately 3.33 grams to 3.52 grams and remained at this weight until 1402 (Spufford 1986; pp. 6–23). The weight then reverted to 3.33 grams for the next 19 years. In 1421 the weight once again increased to 3.54 grams. The florin was increased again to 3.55 grams in 1422 to make it, according to Grierson (1981) comparable to the Venetian ducat in order to assist the Florentine merchants with business interests in the Levant. At this time, two different florins were minted – the stretto (19.5 mm in diameter) and the largo (21.5 millimeters). Toward the end of our sample period, 1432 or 1433, the weight was lowered to 3.54 g.
During this time period, the unit of account monies were based on the "£/s/d" system, where one lira (pound, lb.) equals 20 soldi (schillings, s.), which in turn equals 240 denari (pence, d.). This system was modeled after the rubric developed by Charlemagne in the early ninth century. Evans (1931) and Shaw (1896, pp. 301–309), among many others, provide detailed descriptions of the Florentine monetary system including the role of unit of account monies and their relation to the physical coins of Florence.
According to Sargent and Velde’s (2002, p. 37) model, for the florin and denaro or its companion coins to circulate freely after accounting for costs of minting, the exchange rate between the two coins could not be greater (less) than the exchange rate value that aligned the melting (minting) point for the florin with the minting (melting) point of the denaro.
Najemy (2008, p. 42) points out from the Statuti dell'Arte del Cambio di Firenze (1299–1316) that: "The consuls of the money-changers' guild swore an oath to ‘govern [regere] the guild and all the money-changers of the city and its territory who are bound to us by oath, for the common utility of the whole guild, in good faith and without fraud' or favoritisms. They also promised ‘to gather to render justice [ad ius reddendum] every Monday and Friday and any other days they wish.'" For instance, according to Rule 70, a money-changer found guilty of being untruthful concerning his monetary dealings could be subjected to the “rack or other corrective instruments [such as the strappado]” at the Guild’s headquarters.
The environment in which these banks operated required that they had to be in constant contact and implicitly trust one another. This was true even in the face of the frequent political intrigues among the merchant-banking families of the ruling elite. As Goldthwaite (1987, p. 23) points out, although the banks were organizationally independent entities, “… they had to work through the far-flung web of relations that they all depended on. Their sensibilities about the system were such that relations within the business community were not often disturbed by the city’s vicious factional politics. Even political exiles were not excluded.” To emphasize his point, Goldthwaite mentions that when Cosimo de’ Medici went into exile from Florence in 1433 after narrowly escaping a death sentence he simply moved to Venice and continued to conduct business with Florence from there, and that when he returned to power in Florence less than a year later, he facilitated the exile of his opponents who then continued their business as usual from afar.
They note further that success in business often was a stepping stone toward other elite activities, like becoming a city councilor, an ambassador, a rentier, or even an art patron or humanist scholar.
As an international example of surmountable barriers to entry and thus low concentration, the Medici Bank had a presence in almost all the major banking centers—London, Bruges, Avignon, Geneva, Lyon, Venice, Naples—but it did not dominate business in any one these places. Moreover, in the last quarter of the fourteenth century, more than 140 Florentine firms were operating at one point or another, and surviving records indicate that many merchant-bankers operated abroad without doing business with the Medici (Goldthwaite 1991).
To help avoid these problems fiorini di suggello were sometimes used. These were florins that had been certified by assayers at the Florentine Mint, who then sealed them in a leather bag, thereby guaranteeing these coins were uncirculated and genuine.
We thank the anonymous reviewers for pointing out that such an arrangement is consistent with centuries-old European traditions relying on pure brokers to certify "public" (official) prices. On the one hand, this puts pure brokers under the regulatory purview of Florence (as elsewhere in Europe); on the other, the rationale of this legal "tradition" is to avoid the conflicts of interest that arise when public or official prices come from the private activities of bankers. Such conflicts of interest can result in a huge temptation to engage in the manipulation of prices and rates, which was at the heart of the recent LIBOR scandal.
Obtained from the Archivio di Stato di Firenze (ASF), the original document is a handwritten entry contained in the Deliberazioni of the Zecca (mint), years 1491–1494: Ufficiali della moneta poi maestri di Zecca, n.67, cc.17v-19: "lex quod quattrini albi non possunt recusari pro eorum valuta". Bernocchi (1978, p. 9–10) provides a direct transcription of this archival document. An English translation of this document was provided by Cynthia C. Craig and Lenisa V. Chang. Crucial interpretations of the document’s contents were supplied in an e-mail communication dated May 5, 2015 from Giampiero Nigro (Prof., Ordinario di Storia Economica, Dipartimento di Scienze per l'Economia e l'Impresa, Università degli Studi di Firenze and Direttore scientifico della Fondazione Istituto Internazionale di Storia Economica "F. Datini") to the authors.
"Sotto pena a c(i)aschuno sensale, che non rechasse la vera valuta, dì per dì, di quello et quanto harà facto chome di sopra, di fiorini larghi 25 per c(i)aschuno et per c(i)aschuna volta…". Half of the penalty is assessed to the Guild, a quarter to the individual reporting the violation, and the remaining quarter to the commune of Florence. To put the fine in perspective, the typical scale of annual salaries in Florentine banks ranged from about 20 florins for beginners to 100 florins for well trained branch supervisors (de Roover 1999; p. 45).
The specific language of the statute regarding the calculation of the official rate is as follows: If only one money-changer reports the rate he applied, then this rate will be taken from then on to be the official exchange rate of the following day (Et se uno solo sensale sarà quello che rechera la valuta per lui quel dì fatta, quella sola s'intenda essere et da ora sia la vera valuta per lo dì sequente). If on the same day, two money-changers disagree on the aformentioned rate, in this case the Guild manager must combine the two values and take the mean, which will be the official rate for the following day (E se saranno, in uno medesimo dì, più sensali discordi nel rechare la valuta predetta, in tal chaso detto proveditore…debba chong(i)ungnere dette due valute insieme et di poi pigliarne la metà, et quella gli resterà sia la vera valuta per lo dì sequente). In addition, if there are three money-changers who disagree, all three rates will be added and divided by three, and that will be the official rate for the following day (Et essendo tre e' sensali discordi, chong(i)ungha tutte e tre dette valute et riduchale a una somma, di poi ne pigli di tutta la somma et terço, et quello terço gli resta sia la valuta per lo dì sequente). As so on for four or more (Et chosi essendo quattro o più).
Strictly speaking, the statute governs the exchange rate for quattrini albi, a new type of silver coin introduced in the late fifteenth century. However, the statute itself states that the procedure is in fact based on and identical to the protocol for determining the exchange rate for gold coins, oro a grossi (…in quel modo et forma che loro fanno anch'alle valute de oro a grossi…Et in effecto facci detto proveditore di dette valute in quel modo che lui fa di quelle da oro a grossi).
Bernocchi (1978, p. 10) refers to the provision of October 22, 1471: “where large florins must be paid, pay lire in grossi or quattrini or soldini, according to the value for large florins and such coins…the value will be that posted by the Guild, where the official rate is placed (….et stiesene alla valuta che sarà posta all'arte del cambio, dove si ponga la vera valuta…).”
A possible explanation for the scarcity of records on manual exchange is that, despite the well known Florentine penchant for detailed record keeping and having veritably invented modern accounting practices, there is a surprising peculiarity in the books of Florentine bankers: the lack of a cash account. Goldthwaite (1985) suggests that such a practice was meant to conceal usurious activities arising from money-lending. When it was periodically necessary to draw a balance (e.g., for tax reporting purposes), the banker would take inventory of his cash box. Changes or discrepancies over time could then be attributed to profits from activity in the cash box that did not get recorded, such as manual exchange. From this perspective, more detailed records on manual exchange would have unraveled the camouflage it provided to earnings from usurious activities. Thus, it would seem that the scarcity of detailed records, at least for manual exchange, was by design. For a further discussion of the conception of usury during this period, see fn. 26.
On January 25, 1318: “Madonna Giemmina and her son Lapo…gave…seven gold florins: of which we have cash 6 gold florins and s. 37 d. 6 piccioli, and s. 25 was paid for the tax on rent, based on lb.3 s. 2 and a half to the florin, as the table agent gave this day (…come diede il tavoliere questo dì).” On May 10, 1322: “Monna Fia…gave…one gold florin, of which we have lb. 3 s. 5 and a half, as was stated in the table (…come ne dava la tavola).” In a contract pertaining to the sale of a farm in San Quilico il Vecchio on May 7, 1281: “We made a pact with them to sell to them in four years and they will give us lb. 400 piccioli in gold florins for what they will be worth at the tables (…e deranoci lb. quatrocento di picioli in fiorini d'oro per quello ke varranno comunemente a le tagole).” See also fn. 9.
MSS Florence, Archivio di Stato, Miscellanea Republica, Box 33.
Goldthwaite (2009, p. 412) notes that with a continual influx of various foreign coins that lacked uniform standards (a town of any size could have its own mint) and that were frequently and often irresponsibly debased, “monetary instability…and uncertainty about coins abounded, and speculators were always ready to exploit the situation.” The record books of Lippo di Fede del Sega (ca. 1285 - ca. 1363) provide insight into the activities of such a professional speculator and arbitrageur. “Taking advantage of variance in the value of coins from one place to another, he bought foreign coins to have them reminted at the local mint as Florentine coins, at times receiving back not coins but blanks suitable for minting elsewhere.” Del Sega traveled extensively throughout Tuscany, as well as to Perugia, Bologna, and Venice, to buy and sell coins in this manner.
See Houkes (2004) for numerous references in this regard. Goldthwaite (1985, pp. 31–32) provides an excellent perspective on the conception of usury during this period: “By the fifteenth century the simplistic usury prohibition of an earlier era was being rapidly eroded. Entrepreneurs ever in need of capital…were not to be held back from moral imperatives, however much they may have regretted some of their methods…on the death bed. Subterfuges for the charging of interest on loans were highly developed, the most notable instrument adapted to this purpose being the bill-of-exchange, the legitimacy of which in its primary function the theologians could not deny…and even some churchmen [recognized] the need of the economy for capital and the legitimate demands men with money to spare could make for others' use of it. Nevertheless, doubts lingered on, and bankers were particularly vulnerable to them.”
From Bernocchi (1978; p. 7), “La colonna della quotazione del Fiorino d'oro al netto di provvigioni…dalla indicazione ‘Dassene',” whereas “…la colonna della quotazione ufficiale dalla indicazione ‘Vale’”.
We thank an anonymous reviewer for pointing out this distinction.
According to Goldthwaite (2009, p. 223): “The gathering of so many merchants at the [Geneva] fairs facilitated clearance of exchange operations among them, while the seasonal timing of the fairs, each lasting about a week, at four equal intervals throughout the year made it possible for them to do this on four dates—Epiphany (6 January), Easter, at the beginning of August, and All Saints' Day (1 November)—since the fairs, not the travel time of the bills, now determined the dates for exchange transactions.”
Padgett and McLean (2002) note that there was some fluidity between the activities of domestic and international banks in the period from 1350 to 1434, with many of the former also involved in lucrative international business. For our purposes, such local banks could be considered informed since they would have had access to relevant international news. However, we surmise that the quality, speed, and breadth of their information was likely inferior to that obtained through the geographically extensive and well established branch networks of large international banks. In any case, to the extent that most small local banks had a local rather than international focus, de Roover's rough dichotomy between banco grossi and banco a minuti is still useful for broadly delineating which types of money-changers were, respectively, informed or uninformed.
A third type of bank, in addition to the banchi grossi and banchi a minuti, was the banchi di pegno, or low level pawnshop. Pawnbrokers were considered to be manifest usurers and, as such, were rigorously excluded from the Arte del Cambio. However, upon paying an annual fine for committing the “detestable sin of usury”, they were then "to be free and absolved from further censure, penalty, or exaction" (de Roover 1999; p. 14). Thus, these pawnbrokers were in effect licensed to operate by the city. From 1437 onward licenses were granted solely to Jews, and in 1481 the commune granted Jewish pawnbrokers the right to engage in manual exchange alongside their interest generating lending and deposit operations with Christians, thus contributing to a gray, or unofficial, parallel market. Staley (1906, p. 176) notes that this gray market relieved some pressure on the official market and thus was tolerated by the Guild. However, because they were barred from Guild membership, these non-sanctioned money-changers, while allowed tables in the marketplace, were not allowed to cover their tables with a green cloth, nor were they allowed to have a chair on which to sit as Guild members did. Thus, it appears that manual exchange was an ancillary activity for such pawnbrokers and it is therefore unlikely their quoted rates differed much from the official Guild rate. In any case, their rates did not affect the calculation of the official rate, and any “black market” effects or feedback into the official rate would have been neutralized because Guild money-changers changed their rates freely throughout the day in response to market conditions.
It is possible that a knowledgeable money-changer may attempt to profit either by withholding their private information and trading on it the next day or by purposefully applying higher or lower rates than the rate that he expects in order to profit via speculation. This is unlikely for three interrelated reasons. First, any personal gains to the money-changer would come at the expense of mispricing in the official Guild rate, in turn negatively impacting a large portion of the money-changer's own business in the form of book transactions. Second, an individual money-changer's rate is but one of many and to have a meaningful effect it would have to be noticeably different from the rates applied by the other money-changers and, thereby, likely to be noted with suspicion by the Guild manager, who by statute could impose heavy fines and punishments for misdealing. Finally, the communality of the guilds, based on the spirit of fiducia, cannot be discounted. Although it would be naive to assume that this ethos ensured that all money-changers always acted in accordance, it likely served to limit such behavior. In any case, it seems unlikely that one could consistently profit at the expense of their colleagues without eventually being reported (recall that a quarter of the fine is rewarded to the notifier) and reprimanded accordingly.
Also, as discussed above, sufficiently large discrepancies between the denaro-florin exchange rate and the silver-gold market ratio determined in international bullion markets can lead to a form of arbitrage at the expense of the Guild. For example, if florins are overvalued in terms of denari relative to the international market ratio, then an arbitrageur in Florence can sell florins for an excess amount of denari, convert these denari into silver (by melting the equivalent amount of silver coins or buying the equivalent weight in silver bullion from a Cambio bank), and then sell this amount of silver for gold in the international bullion market. The arbitrageur will end up with an excess amount in weight of gold (bullion) over the original amount of gold (florins) he started with. Conversely, if florins are undervalued in terms of denari relative to the market ratio, then the arbitrageur can arrange to end up with an excess amount of silver over what he started with, again profiting at the expense of the Guild. Note, however, minting and transportation costs impose limits to arbitrage, creating a band around the exchange rate protecting money-changers from arbitrage losses. Also, see fn. 10.
The assumption that agents know their information type was necessary in the previous section to characterize the static Nash equilibria, but it is no longer required in the dynamic setting.
In a similar but modern context, Seru et al. (2010) empirically show that investors learn from their trading experience and find that some become better with experience while others stop trading once they realize that their ability to make profitable trades is poor.
In addition to population distributions, one can also view p and q as probabilities for mixed strategies.
Evolutionary economic theory suggests that that the social mechanisms of learning and imitation are sufficient to deliver the replicator dynamic. For instance, according to Malaith (1998, p. 1348), “[i]t is important to note that successful behavior becomes more prevalent not just because market forces select against unsuccessful behavior, but also because agents imitate successful behavior.” Moreover, Friedman (1953), in is support of flexible versus fixed exchange rates, argues that market participants with rational beliefs will dominate the market.
Note, Eq. (9) can be rearranged to get a better sense of the deviations from the efficient price that are caused by mispricing: \(V_{t + 1} = V_{t}^{ * } + \lambda [qu\varepsilon_{t} + (1 - q)(V_{t} - V_{t}^{ * } )]\), where the mispricing is a weighted average of signal noise and the deviation of today's official rate from the true intrinsic rate, with the weights determined by the proportion of uninformed money-changers choosing strategy S or O, respectively.
Equation (12) is derived using the definition of covariance along with the fact that increments of the random walk are independent from each other.
The emphasis of the model is on the role of information in the Guild rather than inventory management concerns related to liquidity demand (i.e., non-arbitrage transactions arising from ordinary customers' liquidity needs). It is unlikely that serial dependence is induced by liquidity. As mentioned above, both money-changer types, through sensali, were highly cognizant of local liquidity conditions and thus were likely able to forecast future liquidity demand correctly on average. If so, then pricing errors due to liquidity trading would by definition be uncorrelated.
Andrei and Cujean (2010) show that in centralized markets with rational agents, rumors circulated by word-of-mouth (which are analogous to the noisy signals received by uninformed money-changers), once identified as such, may cause price reversals. Koudijs (2016) demonstrates that while news-related trades result in positively correlated price changes, non-news-related trades are associated with negatively correlated price changes.
The data are also now available on the Medieval and Early Modern Data Bank (MEMDB) maintained by Rutgers University.
The original data are comprised of 10,514 trading days. Booth and Gurun (2008) have identified 43 potential data errors that they maintain are not economically feasible (e.g., a negative commission) and likely the result of transcription errors. Since it is not possible for us to correct these data discrepancies, we remove them from our data, resulting is a working sample of 10,471 daily exchange rate observations.
The silver unit in these data is based on the quattrini (which is equivalent to four denari). Beginning in 1418, archival records indicate that a parallel exchange rate based on the silver grossi (the equivalent of 66 denari) was also used. The grossi based denaro-florin exchange rate was typically somewhat smaller than the quattrino based denaro-florin rate. For example, on March 1, 1418, the quattrino gross rate was 3.6 denari higher than the grosso rate. This difference led Spufford (1986, p. 34) to suggest that money-changers preferred handling grossi rather than quattrini. The numeraire gold coin in these data is the stretto (weighing 3.5 grams and measuring 19.5 millimeters in diameter; see fn. 8). Weight changes (also discussed in fn. 8) do not appear to materially affect our statistical results for two reasons. First, old and newly minted coins circulated at the same time and, as Grierson (1981) points out, not many of the 3.33 gram florins were minted. Second, we conduct the statistical analysis on exchange rate returns versus levels. Since there are only five weight changes during our sample period, of which only two are meaningful, their impact, if any, will be swamped by the remaining observations.
Concerning the debasement policy with respect to the market ratio, Goldthwaite (2009, p. 488) notes: "[Florence] faced a serious monetary problem when, in the second quarter of the fourteenth century, the gold-silver ratio fluctuated radically, but from midcentury on, the government followed a strictly defensive debasement policy with respect to silver coins so as not to disrupt the local market. The debasement of coins was slow, almost imperceptible…New coins therefore did not disrupt the market, where older coins, which had lost some of their intrinsic value as a result of natural wear, circulated. Florence thereby avoided some of the shocks [experienced] in England and the Low Countries, where anger of workers over wages paid in [debased] coin…could lead to labor unrest and even violence."
This calculation assumes that returns are continuously compound and is a standard approach in the economics and finance literature because the first difference in this case results not only in a stationary variable but also in an economically meaningful one.
Spikes may be the result of significant news that will likely impact the gold-silver ratio. Cipolla (1983), in his study of monetary policy in fourteenth century Florence, describes how coin shortages can be caused by differences in their commodity and face values and uses what he poetically calls “The Affair of the ‘Quattrini'” as an example.
We would like to thank an anonymous referee for pointing out the existence of strings of several consecutive days in the data where the official rate does not change, i.e., strings of zero returns. These in fact occur quite frequently in the data, for varying lengths of time. However, this phenomenon is entirely consistent with our model. First, in an efficient-markets, random-walk context, if there is no new (i.e., unexpected) news, then prices do not change. According to our model, if information diffusion is low, then the uninformed will choose to simply use the current official rate, while the informed will also use the official rate (if there is no “new” news). Thus, the official rate can remain constant for consecutive days at a time. Also, when there is full information diffusion, and all agents know there is no “new” news, again everyone will use the official rate, thus resulting in no change for tomorrow's official rate, possibly for several days in a row. Inasmuch as there does not seem to be a systematic pattern that relates when such strings occur in the data to the data itself, and to the extent that there is an economic basis for their occurrence (which our theoretical model helps to explain), our empirical analysis is valid.
Changing technology plays a crucial role in both the speed and availability of information. By way of illustration, Koudijs (2016) reports that stock market news was sent by boat from London to Amsterdam twice a week in the late 1700s but that delivery would often be delayed several days by inclement weather. Kielbowicz (1985) examines the U.S. Post Office, which started express mail services in 1825 and encouraged newspapers to use this service. Policy makers believed that the speedy delivery of news would neutralize the advantages enjoyed by commodity speculators with private channels of information. Garbade and Silber (1978) document that the establishment of the transatlantic cable in 1866 permitted individuals in New York to know foreign prices in London in a day rather than in weeks, thereby enabling individuals to better estimate future prices. Booth et al. (2014a, b) point out that modern computer technology permits information travel at almost the speed of light and transactions to be recorded in milliseconds. In the U.S. market designers are questioning whether markets have become so fast that barriers may be needed to slow down algorithmic trading.
The mechanism design perspective of the florin protocol is mentioned in Sect. 3 above and is discussed further in the next section.
To be clear, we do not intend to imply here that the Geneva fairs became the sole source information. However, the periodic timing of the fairs can be usefully employed as an informal benchmark for our estimated time frame. With the fairs taking place every three months, and therefore a roundtrip usance of six months, and then adding (conservatively) a couple of months for complete information diffusion (i.e., \(u \to 0\)), gives a range of three to eight months. Thus, our finding that convergence to efficiency occurs, on average, in six months appears to fall sensibly within this range.
The similarity of the florin fix and the London Interbank Offer Rate (LIBOR) was noted in the Introduction. Prior to its being administered by ICE, which operates under the regulatory oversight of the U.K. (Financial Services Act of 2012), LIBOR was calculated using the supplier banks' best estimates. These estimates may or may not have been based on actual transactions. Thus, the estimates were subject to possible manipulation in order for the bank to profit by betting on the direction of interest rates. In 2012 traders from UBS and several other banks were accused of manipulation of LIBOR. After much discussion of alternative remedies, including a change in the administrating body, it was decided, among other things, that (1) the determination of LIBOR would be regulated by the government; (2) banks' submitted rates must be based on actual transactions that can be documented and audited; and (3) rate manipulation was officially defined to be a crime and was associated with specific sanctions. These three conditions were met by the florin fix. However, LIBOR is still determined by the top 20 largest banks rather than the entirety of the membership (roughly 250 banks), which is a point of departure from the florin fix. However, requiring all member banks to submit rates was a main recommendation for reform after the scandal (The Economist 2012).
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Acknowledgments
We thank Cynthia C. Craig and Lenisa V. Chang for their Italian to English translation of excerpts from Mario Bernocchi’s definitive work on Florentine coinage. We also thank Mercedes Carrara and Alessandra Bernocchi for introducing us to Giampiero Nigro who graciously explained to us the nuances of the Florentine domestic exchange market. We cannot thank him enough for his thoughtful contributions. We appreciate the help of Francesca Klein of the Archivio di Stato di Firenze in obtaining the original fifteenth century document describing exchange rate statutes and protocols. We also appreciate the helpful comments of Kenneth Frank, Charles Hadlock, and participants in the 2012 American Economics Association Annual Meeting in Chicago, IL (Insights from Financial History session) on earlier versions of this paper. Finally, we thank Claude Diebolt, the editor of this journal, for his encouragement and two anonymous reviewers for their important and insightful comments. All errors are our own. In Memoriam: Dr. James L.C. Chang.
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Appendices
Appendix A. The dynamic game and asymptotic stability
A.1. Setup of the dynamic model
Assume that the present state of the current round of the florin fix is some \((p,q)\), where p is the proportion of Informed (I) money-changers choosing to apply their Signal (S), so that \((1 - p)\) represents the proportion of the informed choosing to use the Official (O) rate; and q is the proportion of Uninformed (U) money-changers choosing to apply their Signal (S), so that \((1 - q)\) represents the proportion of the uninformed choosing to use the Official (O) rate. With the distribution of actions of U fixed at q, I's expected payoff from choosing S is given by \(F_{I}^{S} = q\alpha \varPi^{S,S} + (1 - q)\alpha \varPi^{S,O}\), while the expected payoff from choosing O is \(F_{I}^{O} = q\alpha \varPi^{O,S} + (1 - q)\alpha \varPi^{O,O}\). Thus, the average expected payoff for the population of Informed money-changers is then given by \(\bar{F}_{I} = pF_{I}^{S} + (1 - p)F_{I}^{O}\). Similarly, fixing the distribution of I's actions at p, U have expected payoffs of \(F_{U}^{S} = p(1 - \alpha )\varPi^{S,S} + (1 - p)(1 - \alpha )\varPi^{O,S}\) and \(F_{U}^{O} = p(1 - \alpha )\varPi^{S,O} + (1 - p)(1 - \alpha )\varPi^{O,O}\), with an average expected payoff for the population of Uninformed money-changers of \(\bar{F}_{U} = qF_{U}^{S} + (1 - q)F_{U}^{O}\).
Next, it is necessary to specify the dynamics by which the population states evolve. A hallmark of evolutionary games is the notion that actions that have higher payoffs given the current distribution of behaviors tend over time to displace those that have lower payoffs (Friedman 1991). This is the so-called replicator dynamic (see fn. 38). Intuitively, the growth rate of a particular choice of action is assumed to be proportional to its relative payoff, or \(\frac{dp}{dt} = p(F_{I}^{S} - \bar{F}_{I} )\) and \(\frac{dq}{dt} = q(F_{U}^{S} - \bar{F}_{U} )\). After substituting and rearranging terms, we obtain the system of ordinary differential equations (ODE) in Eqs. (7) and (8).
A.2. Asymptotic stability
Friedman (1998) formally defines an evolutionary equilibrium as follows: State s ∈ S is a fixed point of the dynamic F if F(s) = 0 and is locally asymptotically stable if every open neighborhood N ⊂ S of s has the property that every path starting sufficiently close to s remains in N and converges asymptotically to s. Such asymptotically stable fixed points are referred to as evolutionary equilibria.
Since this definition specifies convergence for points sufficiently close to s, it governs local stability. With respect to global convergence properties, from inspection of Table 1 it is clear that, regardless of whether \(u > u^{ * }\) or \(u < u^{ * }\), the price-fixing game is dominance solvable (i.e., iteratively removing strategies leaves only one strategy for each money-changer population), and that the dominance solution corresponds to the game’s unique Nash equilibrium in each respective case. Sandholm (2010, p. 260) shows that for dominance solvable games, all interior solution trajectories of any imitative dynamic (including the replicator dynamic) converge to the dominance solution (Theorem 7.4.5).
However, to determine whether such dynamics result in asymptotically stable, evolutionary equilibria, we must examine the properties of the Jacobian matrix of the system of ODE in Eqs. (7) and (8) evaluated at its fixed-point steady states. The 2 × 2 Jacobian matrix is given by:
where\(\begin{aligned} J_{11} (p,q) & = (1 - 2p)[q(\varPi^{S,S} - \varPi^{O,S} ) + (1 - q)(\varPi^{S,O} - \varPi^{O,O} )]\alpha , \\ J_{12} (p,q) & = p(1 - p)[(\varPi^{S,S} - \varPi^{O,S} ) - (\varPi^{S,O} - \varPi^{O,O} )]\alpha , \\ J_{21} (p,q) & = q(1 - q)[(\varPi^{S,S} - \varPi^{S,O} ) - (\varPi^{O,S} - \varPi^{O,O} )](1 - \alpha ), \\ J_{22} (p,q) & = (1 - 2q)[p(\varPi^{S,S} - \varPi^{S,O} ) + (1 - p)(\varPi^{O,S} - \varPi^{O,O} )](1 - \alpha ). \\ \end{aligned}\)
Setting Eqs. (7) and (8) equal to zero and solving for fixed points, it is clear that steady-states occur at the corners of the unit square, \((p,q) = \{ (0,0),(0,1),(1,0),(1,1)\}\). To determine whether a fixed point is asymptotically stable, it is necessary to determine whether the eigenvalues of the Jacobian matrix (13) evaluated at that point are both real and negative. A necessary and sufficient condition is that the trace and determinant at that point are such that \(tr(J) < 0\) and \(Det(J) > 0\), respectively.
We thus see that asymptotic stability depends on the parameter u. When \(u > u^{ * }\), the only stable fixed point is at \((p,q) = (1,0)\). Specifically, the Jacobian evaluated at this point is given by:
with trace \(tr[J(1,0)] < 0\) and determinant \(Det[J(1,0)] > 0\), which allows us to conclude \((p,q) = (1,0)\) is an asymptotically stable, evolutionary equilibrium (the relationship between the parameter u and the relative sizes of the various expected profit outcomes is discussed in the setup of Eqs. (2) through (6) and summarized in the notes to Table 1). This point corresponds to the outcome (I choose S, U choose O), which is the Nash equilibrium of the static game when \(u > u^{ * }\).
In a similar manner, when \(u < u^{ * }\), the only stable fixed point is \((p,q) = (1,1)\), so that:
with \(tr[J(1,1)] < 0\) and \(Det[J(1,1)] > 0\). This point corresponds to the outcome (I choose S, U choose S), which is a Nash equilibrium of the static game when \(u < u^{ * }\). Thus, both \((p,q) = (1,0)\) and \((p,q) = (1,1)\) are Nash and evolutionary equilibria.
Appendix B. Subsampling procedure
If we conducted all our statistical tests in the standard way based solely on returns relative to the first exchange rate observation in the series, it is possible that these results may not hold when repeating the analysis at different starting points. Thus, to achieve maximal usage of the limited available historical denaro-florin exchange rate data, and hence the most robust statistical results possible, we perform repeated tests under the subsampling procedure of Politis and Romano (1994). Unlike standard bootstrap techniques, subsampling is valid for stationary serially dependent and heteroskedastic time series (Politis et al. 1997).
In general, the method involves taking overlapping subsamples of size b of the original data, computing statistics for each subsample, then looking at the empirical distribution of the statistics. Specifically, the procedure is carried out as follows. Consecutive overlapping subsamples of size b = 9472 exchange rates are formed (starting from the first observed exchange rate, then second, third, etc.), yielding a total of 1000 subsamples (there are 10,471 observations and thus 10,471 − 9472 + 1 = 1000 subsamples). For each subsample, n-day returns are obtained for n = 1, 25, 50, 75, 100,…, 300 days. Then, the autocorrelation function (ACF) and test statistics are computed, yielding an empirical distribution of 1000 Q-statistics for each n-day return interval and lag length k = 5, 10, and 15. Confidence intervals (\(Q_{2.5}^{ * }\),\(Q_{97.5}^{ * }\)) are then obtained by taking the 2.5th and 97.5th percentiles of the empirical distribution of the Q-statistics.
Note, the interval n = 300 days corresponds to a maximum return horizon of roughly 1 year (twelve 25-day months), and with a subsample size of 9472 trading days, this leaves 31 returns with which to estimate the ACF and compute Q-statistics in each subsample. Also, column (c) of Table 2 reports the number of observations (returns) used in the calculation of the ACF and Q-statistics for each subsamples. (Note, letting T denote the size of the full sample and B the size of the subsample, the standard error of returns for the full sample differs from the standard error of the subsample by a factor of \(\sqrt {B/T}\)).
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Booth, G.G., Chang, S.S. Domestic exchange rate determination in Renaissance Florence. Cliometrica 11, 405–445 (2017). https://doi.org/10.1007/s11698-016-0146-5
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DOI: https://doi.org/10.1007/s11698-016-0146-5