A Theory
That eternal want of pence,
Which vexes public men.
—Alfred, Lord Tennyson
This chapter presents the main elements and outcomes of our model. The exposition here contains enough to reveal the features that we watched in history. A complete account of the model appears in part V.
Our theory allows us to interpret a pervasive and persistent depreciation of small denomination coins, exhibited for example in the data shown in figure 2.1. The six panels record estimates of the (inverse of the) silver content of small denomination coins from 1200 to 1800 for six countries. Increases in exchange rates of large for small coins and recurrent shortages of small coins accompanied these persistent depreciations in the silver content of small coins. Our theory identifies the source of the upward drifts in figure 2.1 and explains how they related to the concurrent shortages.1
How can something in short supply have its price fall over time? The demand side of our model gives our answer.2 The appropriate market signal for agents to economize on small denomination coins is a reduction in the rate of return on those coins relative to rates of return on other coins. A lower rate of return on small denomination coins occurs when those coins depreciate relative to large denomination ones. Depreciating exchange rates for small denomination coins are thus symptomatic of times when small coins render especially high “liquidity services.”
Figure 2.1 Indices of the mint equivalent of small coinage (number of small coins produced from a given weight of silver) in various countries, 1200 to 1800. All indices are set to 1 in 1400. Sources: Bernocchi (1976) and Galeotti ([1930] 1971) for Florence, Papadopoli (1893–1909) for Venice, Wailly (1857) for France, Challis (1992a) for England, Munro (1988), Van Gelder and Hoc (1960) for Flanders, García de Paso (2000b) for Castile.
In a commodity money system, coins might exchange “by weight” or “by tale.” In the former case, the exchange value of a collection of similar coins (say, pennies) would be determined by their aggregate weight; in the latter, by their aggregate number, one coin counting the same as another. For multiple denominations, circulation “by weight” refers by analogy to exchanges where the intrinsic content of a coin determines its value relative to coins of other denominations; when it doesn’t, the coin is said to circulate “by tale.”3
Figure 2.2 Portrait of Sir William Camden (1551–1623) by Marcus Gheeraerts (detail). Camden was a historian and antiquarian, headmaster of Westminster School, and Clarenceux king of arms. He endowed a chair in History at Oxford. His Latin motto, shown here beneath his coat of arms, was “pondere, non numero”: by weight, not by tale. (Bodleian Library, Oxford University).
A fiat money ipso facto exchanges by tale. Circulation by tale was common for commodity money systems too, despite the preference of Sir William Camden, a seventeenth-century gentleman, shown in figure 2.2, whose personal motto was “pondere, non numero” (“by weight, not by tale”).4 We shall focus on observations that seem explicable only if coins are at least sometimes valued by tale.5 Adam Smith ([1776] 1937, book I, ch. 5, 44; book IV, ch. 6, 517) and many other theorists of commodity money systems also noted that coins often circulate by tale. 6
A basic one-denomination theory
We use and extend a theory of commodity money that describes the demand and supply for coins made of a precious metal that we shall call silver. Ultimately, we shall use a multiple-coin version of the theory, but it is helpful to begin with a more standard version of the theory cast in terms of a single denomination, which we take to be the penny.
The basic one-denomination theory has the following features: (a) coins are made of valuable metal; (b) coins circulate by tale, not by weight; (c) the metal content of coins puts an upper bound on the price level (expressed in number of coins per consumption good). There are two methods for setting the quantity of new coins: (d) the government can instruct the mint to make coins on government account; alternatively, (e) the government can set up a system of unlimited minting in which citizens are free to purchase coins for silver at the mint at a set price of coins per unit of metal. An unlimited coinage regime puts a lower bound on the price level. The ideal single-commodity money system of the nineteenth century puts the upper and lower bounds close together, thereby tying the price level to the relative price of the metal in the coinage. We briefly describe each of these important features in turn.
a. The government specifies that each penny contains b ounces of silver.7
b. As in modern fiat money systems (in which b equals zero), the theory assumes that coins exchange for goods by tale, not by weight. This means that the prices of goods are posted in number of coins per good, rather than ounces of silver per good. Whenever a coin buys more consumption goods than would the silver within the coin, the value by tale exceeds the value by weight, so that there is a fiat component to the tale value of the coin.
c. The metal content of coins puts an upper bound on the price level because, although circulation by tale lets coins be worth more than the intrinsic value of the silver they contain, they cannot be worth less, provided that people can without cost melt the coins to retrieve the silver. Let φ be the relative price of consumption goods in terms of silver, measured in ounces of silver per good. The price level pt must obey
where pt has the units of pennies per good. If the price level were ever to exceed 
, people would have the incentive to melt coins. That would drive the price level down, by diminishing the quantity of coins. The upper bound (2.1) is attained when the value by tale equals the value by weight.
One component of the theory of commodity money is the quantity theory of money, as encapsulated in a demand function for money or coins. Assume a demand function for coins of the simple form
where mt is the stock of coins measured in pennies, and kt is the demand for real balances at time t, measured in consumption goods. Through equation (2.2), the stock of coins mt determines the price level, subject to the upper bound expressed in the inequality (2.1).
Equation (2.2) and inequality (2.1) determine a maximum stock of coins:
People have the incentive to melt into silver stocks of coins exceeding the bound in (2.3). The upper bounds (2.1) and (2.3) embody the discipline on the issuers of coins provided by the commodity content of the coins.
In practice, there were two ways of determining the stock of coins mt in commodity money systems.
d. The government could set mt directly by making the mint a monopoly and by using government-owned silver to create new coins on government account. The government could set mt to any quantity less than 
, provided it had the required amount of silver bmt. In this system, the price level varied with mt as predicted by the quantity theory, up to 
.8
e. The government could surrender direct control over the quantity of coins. In this system, the government set a percentage σ ∈ (0, 1) that covered production costs and profits for the mint, as well as any seigniorage tax on minting. It instructed the mint to coin unlimited quantities of pennies for citizens who brought silver to the mint, in other words, to buy unlimited amounts of silver for a set price paid in pennies. For each ounce of silver, the mint offered 
pennies. The government shared net proceeds of 
per penny with the mint. A system of unlimited or “free” minting gives rise to a lower bound on the price level:
If the price level were to fall below p, people would have the incentive to bring silver to the mint to purchase coins, which would increase the stock of coins. The upper and lower limits on the price level in (2.1) and (2.4) are the “silver points” associated with a system of commodity money in which there is unlimited minting. These are the counterparts of the “gold points” for the nineteenth-century gold standard.
Feavearyear (1963, 2) summarized the conditions underlying a pure commodity money system:9
The efficiency of a metallic standard for controlling the value of money, given reasonable stability of value in the metal itself, depends upon the monetary regulations in force. To secure the maximum efficiency there must be complete freedom to exchange metal for money and money for metal at a fixed rate. There must be freedom of trade in the metal, with liberty to export and import it. If coins circulate, they must be issued by the Mint of accurate weight and fineness, in exchange for bullion in unlimited quantities and without charge; and they must be protected from clipping and from counterfeiting. Steps must be taken to replace regularly worn pieces, and there must be liberty to melt the coins if it pays to do so.
Multiple denominations
The basic theory is cast in terms of a single coin, the “penny.” Because this book is about repeated depreciations and shortages of small denomination coins, we must somehow put multiple denominations of coins into the theory. To do this, we modify both the demand and supply sides of the theory. Each side contributes to our explanation. Under a regime of unlimited coinage, the supply side of our model implies a multiple-coin version of the silver points: for each denomination of coin there are price levels that determine the minting and melting points, respectively, for that coin. A big part of our story will be about society’s long process of learning how to align the intervals for coins of different denominations to prevent shortages or surpluses of small denomination coins. We modify the demand side of the model to permit occasions when coins of different denominations are not perfect substitutes for one another, so that there can be shortages of small denomination coins. To capture the notion that it is difficult to make change, the demand side of our theory assigns small coins a special role in some transactions. Our theory of demand predicts that shortages of small coins will coincide with depreciations in the rate at which they exchange for large coins, contributing the key to understanding the trends observed in figure 2.1. Subsequent sections of this chapter describe the multiple denomination version of our theory.
Supply
Our supply theory gives gold or silver “points” for coins of each denomination and lets in the possibility that some denominations may disappear. For reference, table 2.1 catalogs symbols. To illustrate the supply mechanism, let all coins be silver. Let the relative price of consumption goods in terms of silver be φ, measured in ounces of silver per units of consumption good. An ideal commodity money makes the price level proportional to φ, where the factor of proportionality bi–1 is the number of coins of type i per ounce of silver, a parameter set by the government. The value by weight of coin i is γi ≡ φ/bi measured in coins per unit of the consumption good. A coin is valued by weight when the price level denominated in that coin is γi. When coins are valued by tale, γi serves as an upper bound on the price level.
For most of the remainder of this chapter, we assume two denominations, so that i = 1,2. Let coin 1 be the unit of account (the penny) and let ei be the market exchange rate of coin i, in units of coin 1 per coin i (e.g., pence per coin i), with e1 = 1. To facilitate the historical comparisons to follow, we choose pence to be the unit of account.10 The penny was the first and only coin for a long time in medieval Europe, and for much longer it served as the unit of account. Let p be the price level, in pence per unit consumption good. If all coins are full-weight, then p = eiγi for all i.
Table 2.1 Symbols.
| Variable | Meaning | Units |
| φ | world price of silver | oz silver / cons good |
| b1 | intrinsic content of penny | oz silver / penny |
| b2 | intrinsic content of dollar | oz silver / dollar |
| γ1 | melting point of penny | pence / cons good |
| γ2 | melting point of dollar | dollars / cons good |
| σi | seigniorage rate | (none) |
| b1−1 | mint equivalent of penny | pence / oz silver |
| b2−1 | mint equivalent of dollar | dollars / oz silver |
| m1 | stock of pennies | pence |
| m2 | stock of dollars | dollars |
| e | exchange rate | pence / dollar |
| P | price of cons goods | pence / cons good |
The government charters a mint and instructs it to sell newly minted coins to anyone offering to pay silver. The mint is not required to buy coins for silver. The government sets two parameters bi, σi for each coin i. For each ounce of silver brought to it, the mint must pay ei(1 − σi)/bi pence worth of coin i. Here σi ≥ 0 is a parameter to capture all costs that the mint incurs manufacturing coin i, including a seigniorage tax and a brassage fee. Angela Redish calls ei(1 − σi)/bi the mint price and ei/bi the mint equivalent for coin i.
If the price level were ever to fall below ei(1 − σi)γi, arbitrage profits would accrue to anyone taking silver to the mint and asking for coin i. A surge of minting would extinguish those profits by pushing the price level up. If the price level were ever to rise above eiγi, arbitrage profits would accrue to people melting coins of type i, and such coins would all disappear. Therefore, as long as coin i is in circulation, the absence of arbitrage profits on that coin requires that the price level must satisfy11
The left side of this inequality gives the rate at which consumption goods can be exchanged for silver and then taken to the mint to get coins of denomination i. The right side gives the rate at which coins of denomination i can be converted to consumption goods by melting the coins and then exchanging the silver for consumption goods. Notice that the transformations from consumption goods to pennies (on the left, by minting) and from pennies to consumption goods (on the right, by melting) are irreversible.
Coin by coin, (2.5) identifies silver points for melting (the right endpoint) or coining (the left endpoint) a commodity money. By making the range narrow, a commodity money system links the price level to φ, the relative price of consumption goods in terms of the metal.
Equation (2.5) imposes silver points throughout the denomination structure. The medieval monetary authority intended that coins of all denominations should be full-bodied. Cipolla’s standard formula suspends (2.5) for all but one standard coin, and makes all other coins tokens that the government promises to convert into the standard coin on demand.
We complete our model of supply with identities that track stocks of coins:
where mit is the stock of coin i carried from time t to time t +1, nit ≥ 0 is the quantity of coins of type i newly minted at time t, and μit is the quantity of coins of denomination i melted at time t, which must satisfy mit-1 ≥ μit ≥ 0. The no-arbitrage requirements imply nit = μit = 0 if ei(1 − σi)γi < p < eiγi; ei(1 − σi)γi = p if nit > 0; and eiγi = p if μit > 0. That is, there is neither melting nor minting of coin i if the price level is strictly between its silver points. There is melting only if the price level is at or above the upper limit, the melting point. There is minting only if the price level is at or below the lower limit, the minting point. Later we shall modify (2.6) to incorporate open market operations.
Demand
The quantity theory of money rests on a demand function for an aggregate M = Σi eimi, where mi is the number of coins or notes of denomination i. Aggregation assumes that different denominations are perfect substitutes. The perfect substitutes assumption is justified when (1) different denominations bear identical rates of return, presenting no advantage to holding one denomination rather than another, and (2) it is costless to make change. But shortages of some denominations are symptomatic of distinct demand functions for different coins because somehow feature (1) has broken down.
Because coins and notes do not pay nominal interest, denominations have identical rates of return if and only if the exchange rates ei are constant over time. Consider the case of two coins, with coin 1 the penny and coin 2 the dollar. When pennies depreciate relative to the dollar, dollars have a rate of return greater than pennies by a factor of 
1. This signals a shortage of pennies and makes people economize on them.
We require a theory of demand that lets different denominations be perfect substitutes, but that occasionally permits shortages of some denominations. Our theory identifies one class of transactions to be made by any and all denominations and another class that requires specific small denominations.12 In particular, we allow small denomination coins to make all types of transactions but let large denomination coins make only some.
Standard models have one demand function for money and one quantity theory equation. This can be true of our theory too, but only sometimes. At other times, our model has two or more demand functions for money and two or more quantity theory equations. During shortages, one quantity theory equation holds for real balances aggregated over all of the nonscarce currencies, and others hold for each of the scarce coins.
We illustrate the demand side of our model with the special case of two denominations. Let m1 be the supply of the small denomination and m2 be the supply of the large one; et is the exchange rate of small for large denominations (pence per dollar). Parameterizing the demand for money in terms of the “Cambridge k” gives the following theory. During times without currency shortages,
During shortages of small denominations,
Here the kit’s sum to the Cambridge k and measure purchases of different sizes. The kit’s express demands for stocks of coins mit−1 carried over from time t − 1 to t. System (2.7)−(2.8) expresses that small denomination coins can be used for all transactions and that only they can be used for some transactions. The variable k1t measures transactions that require small change. We refer to constraint (2.7b) as the penny-in-advance constraint. It is slack in regime (2.7) but binding in regime (2.8). Equations (2.7) prevail during normal times when all denominations are perfect substitutes and equations (2.8) prevail during shortages. When et = et−1, large and small denomination coins are perfect substitutes. But strict inequality (2.8c) induces holders of small coins to economize and makes constraint (2.7b) hold with equality.
Movements in the variables k1t, k2t reflect changes in preferences and growth or contraction in income. The formal presentation in part V links these parameters explicitly to the endowment and preference structure of the model.
Equations (2.7c)–(2.8c) make et a nondecreasing sequence. This prediction lets the model match the upward drift indicated, for example, in figure 2.1. The model predicts that shortages will coincide with periods when rates of return are less on small denomination coins than on large ones, dissolving people’s indifference about denominations. The price that signals shortages is the rate of return et/et−1, not the level et. The market manages a shortage by temporarily giving a low return to small coins: (2.7c)–(2.8c) translate this into a permanent effect on e. This feature of demand creates problems in conjunction with the supply mechanism.
Interactions of supply and demand
If the operation of (2.5) coin by coin is to be consistent with positive amounts of each type of coin, intervals must be aligned to let the price level satisfy
If the price level exceeded the right limit of (2.9), coins of all denominations for which p > eiγi would have been melted.
Demand specification (2.7)–(2.8) makes the price level satisfy
The right inequality allows eiγi < p for i = 2,… and consequently allows melting of all but the small coins. The aspect of our theory that opens this possibility is the asymmetry in the uses of large and small denomination coins incorporated in (2.7)–(2.8) (small denomination coins can be used for all transactions, but large coins cannot). In the two-coin case, when eγ2 < p and m2t = 0, the quantity theory prevails in terms of small denomination coins:
Chapter 14 tells how this possibility was to be realized in a seventeenth-century Castilian experiment that aligned intervals so that
and that augmented the sources of supply in (2.6) to allow government purchases of coin 2 with coin 1, an open market operation. The Castilian open market purchases of silver coins were originally intended to keep m2t > 0, implying that p ≤ eγ2 and thereby retaining the upper bound on the price level associated with the melting point for silver large denomination coins. Such operations would raise silver revenues for the government without causing inflation. But there is a bound on the amount of silver that can be raised with open market operations: the initial total stock of silver coins. Beyond the bound, more revenues can be raised only if the government engages in inflationary finance. To accommodate that, (2.6) must be modified to include a term for deficit finance, as was discovered in Castile in the late 1620s. Chapter 14 studies the Castilian episode in detail.
Economics of interval alignments
Various alignments across coins of the minting and melting intervals are possible, depending on the settings for σi, the per coin sums of production costs and seigniorage taxes. It is difficult to avoid misalignments. Consider the following cases.
1. If σ1 = σ2, the intervals can be aligned by setting the bi’s so that 
.
2. Per coin cost differences create problems in aligning the intervals, and present a choice about whether to equalize the mint prices (ei (1 – σi)/bi) or the mint equivalents (ei/bi). Evidently, if σ1 > σ2, the intervals cannot coincide. The case σ1 > σ2 is realistic because it embodies the notion that per unit value, small coins are more expensive to manufacture than large coins. (It takes as much labor to strike a small coin as a large one.) When σ1 > σ2, we can align the right limits by setting eγ2 = γ1, which implies eγ2(1 – σ2) >γ1 (1 – σ1). This inequality implies that the small denomination coin 1 won’t be minted. Alternatively, when σ1 > σ2, we can align the left limits, but then eγ2 <γ1, which sets the melting point for large coins below that for small, allowing price levels that cause large coins to disappear.
3. The σi’s determine the widths of the bands. Small σi’s produce small bands, but require small or possibly negative seigniorage taxes, negative if the government subsidizes the minting of a coin to offset part or all of the production costs.
4. Consider a period in which et drifts upward, causing the interval for dollars to drift rightward relative to the interval for pennies. Suppose that the price level expressed in pence also drifts up, always remaining within the interval for dollars. Eventually, the price level can reach the melting point for pennies, threatening to eradicate the stock of pennies. This situation can be remedied by debasing pennies, that is, reducing b1, thereby raising γ1, and shifting the interval for pennies to the right.
5. A situation when the large coin has disappeared because eγ2 < p can be remedied by shifting the interval for coin 2 (the dollar) to the right by increasing e or by increasing γ2 through a decrease in b2, a debasement of the large coins. In seventeenth-century Castile, the market and the monetary authorities repeatedly devalued small coins, in effect resetting e.13
6. The government can push σi above production costs if it can monopolize the best technologies for producing coins and can detect and punish counterfeiters.14
Perverse dynamics
The dynamics in case 4 are perverse because they are self-aggravating. The demand specification makes et a monotone nondecreasing sequence, et rising during periods of shortages. The response to shortages that arise even while the price level is in the interior of all intervals is to shift the interval for coins not in short supply to the right relative to intervals for coins experiencing shortages. The rise in et thus hastens the day when the price level will give incentives to traders eventually to melt the coin in short supply. Unless a policy intervention changes some of the parameters, say through a debasement or an exogenous resetting of the exchange rate, the mechanism cannot give the signal for minting. This perversity was one of the main problems that was to be cured by Cipolla’s standard formula.
Spontaneous debasements: invasions of foreign coins
The monetary authorities could remedy a shortage by debasing the smaller coin, that is, raising b1 to match the rightward shift in the large coin’s interval. Even without such measures by the policy authorities, the market sometimes produced another remedy with the “spontaneous debasements” that occurred with “invasions” of foreign currencies (for example, see chapter 9). Those invasions happened when a nearby sovereign minted coins roughly similar to the domestic small coins, but at a mint price sufficiently higher than the domestic mint price to make minting worthwhile. Instead of exporting goods and importing silver to be coined by the domestic mint, residents chose to export goods and import foreign coins.15
Having access to foreign mints thus offered traders choices among mints. When domestic pennies were scarce, traders could sell domestic dollars to the foreign mint and receive foreign pennies to augment the domestic pennies.
The choice among mints and systems of coinage opens possible indeterminacies sparked by indifference among alternative types of pennies. Laws forbidding the use of foreign currencies appear frequently in the historical record. In situations of indifference, the laws can be regarded as a “tie-breaking” device. So long as there were enough domestic pennies, the laws could be obeyed costlessly. During shortages, the laws were often disobeyed or resisted. In 1339, the king of England prohibited certain foreign coins in Ireland, but rescinded his order shortly after protests by the people suffering from a shortage of small denominations (Ruding 1840, 1:212).
Cipolla (1956, 64), Lane and Mueller (1985, 267), Glassman and Redish (1988, 82) and others have interpreted these invasions of foreign coins as reflecting the operation of Gresham’s law. In our model, there is no need to appeal to that ambiguous law.16 Indeed, it was not a case of one coin chasing another, but one coin supplementing the other. An invasion of foreign pennies was a consequence, not a cause, of a shortage of domestic pennies.
We have identified inherent weaknesses in the way multiple denominations were supplied in many early commodity money systems. The standard formula remedies these weaknesses by prescribing particular open market operations. In doing so, it draws attention to the opportunity cost of commodity money and puts before governments the temptation to overissue token coins or currencies. Learning to use the standard formula involved learning to to resist that temptation.
Opportunity cost
In the special case of two denominations, the stock of silver in the money supply is m1b1 + m2b2. The silver money stock is worth
units of consumption good, where the inequality follows from (2.5).17 The left side of (2.12) measures the capital cost of maintaining a commodity money system. This cost can be saved under a well-managed fiat money system. If m1 + em2 can be limited without the automatic supply mechanism, silver can be conserved by driving both bi’s toward 0. Setting bi’s equal to 0 for all coins would implement a pure fiat standard.
An open market operation from Castile
In the 1590s, economists in Castile recommended that the supply mechanism be altered to permit the government to exchange one coin for another at the prevailing exchange rate. When there are no shortages of particular denominations, the perfect substitutes assumption makes private parties willing to accept such open market operations at the prevailing p and e.
Suppose that γ1 > eγ2, so that the intervals are aligned as in the Castilian case (2.11). This inequality implies that b2 > eb1. Consider a government purchase of dollars for pennies. This open market operation is constrained by
Through this operation the government acquires – (m2t – m2t–1) b2 units of silver at the cost of (m1t – m1t-1) b1 units of silver required to produce the pennies. Taking the difference and using (2.13) shows that the government’s net proceeds in silver are
These are positive whenever γ1 > eγ2. They are larger when b1 is smaller (i.e., when γ1 is higher).
Such calculations lay beneath a seventeenth-century Castilian experiment to be studied in chapter 14. The calculations identify an open market operation that is not inflationary and that substitutes a cheap money for an expensive one, leaving the total quantity of money unaltered. The silver that the government can obtain by such an operation is bounded by (b2 – b1e) m2t−1, the amount raised by setting m2t = 0 in (2.13), thereby making the entire stock of money into pennies.
An open market operation for the standard formula
The standard formula eliminated possible misalignments of the intervals (2.5). It altered the supply arrangement by doing away with all of the intervals except for one dominant coin. It made subsidiary coins into tokens. Tokens are created not by coining precious metals but by promising to conduct open market purchases and sales of intrinsically valueless tokens for a coin of precious metal.
In the two-coin case, form the exchange rate weighted sum of (2.6) across coins to get
The standard formula suspends the supply arrangement (2.6) for all coins except the last (i = n, where we have taken n = 2 in most of this chapter). Other coins i < n are to be tokens with no valuable metal (bi = 0 for all i < n). The mint is told to supply only the nth coin in exchange for silver. The mint is also told to stand ready to buy and sell token coins for the dollar at a fixed exchange rate e.18 In (2.15) we set μ1t = n1t = 0, because token coins are not created or destroyed by minting or melting precious metal (bi = 0 for a token coin) but by being exchanged for dollars. Token coins are warehouse certificates for metal coins, printed on cheap metal.19
The supply of token coins satisfies
where Rt is the stock of reserves in dollars that the mint holds to “back” pennies. Equation (2.16) implies
This states that token coins m1t are created only through open market operations. These specifications let us write (2.15) as
Now (2.5) applies only for coin n = 2 with e being the penny-dollar exchange rate. This is a Gordian knot solution to the enduring problem of aligning the intervals: the standard formula aligns the intervals by eliminating all intervals but one.
Figure 2.3 Indices of prices in terms of the unit of account in England, the United States, France, Spain. Sources: United Kingdom, Schumpeter-Gilboy index from 1661 to 1799 and Rousseaux index from 1800 to 1913, both in Mitchell (1988), Twigger (1999) after 1913. France: price index computed by David Weir from 1726 to 1860, Singer-Kerel cost-of-living index for Paris from 1861 to 1913, Insee consumer price index after 1913. United States: index of wholesale prices from 1749 to 1913 in Warren and Pearson (1933, 10), consumer price index after 1913. Spain, García de Paso (2000b) from 1600 to 1700, Reher and Ballesteros (1993) after 1701.
Transparency of opportunity cost
The standard formula regime makes part of the opportunity cost of the commodity money regime more evident. The reserves Rt backing the tokens lie in vaults. It would be profitable to lend the reserves at interest and thereby convert the backing for the tokens from metal to credit. Lending those reserves is one step toward a fiat money system.
Trusting the government with bi = 0
The waste from keeping those reserves may have been one of the factors that prompted Keynes to call a commodity standard a “barbarous relic.” A fiat money system disposes of all reserves behind a currency. In doing so, it also eliminates the automatic supply mechanism constraining the price level. A low-inflation fiat money system replaces that automatic mechanism with an enlightened government that commits itself to limit the quantity of a pure token, no-cost currency.
Because most nineteenth-century proponents of a commodity money system did not trust governments properly to manage a fiat money system, they were willing to pay the resource costs associated with setting up and maintaining a commodity money system. In light of the high inflation episodes that many countries experienced in the twentieth century after they abandoned commoditymonies,20 it is difficult to criticize them for that. See figure 2.3, which records price levels in Castile, France, England, and the United States. The inflationary experience of the twentieth century, the century of paper money, is unprecedented.
1 In chapter 16, we describe how William Lowndes collected the English data in figure 2.1 for the purposes of establishing a precedent behind the debasement that he was proposing to cure a shortage of small change in 1695.
2 We must remember that we are in the domain of monetary and financial theory: lower rates of return on an asset are what make people want to diminish their holdings of it.
3 Our model assumes circulation by tale in the first sense, as it applies to coins of a given denomination. The exchange rate between multiple denominations is determined by the market, so that coins may or may not circulate by tale in the latter sense.
4 John Locke agreed (see chapter 16).
5 Sargent and Smith (1997) let coins differ by weight and assume circulation by tale.
6 Also see Bernstein (2000, 83).
7 In chapter 16, we describe a version of our model that allows underweight pennies that contain b/δ units of silver, where δ ∈ (0,1). The existence of underweight coins widens the silver points described below.
8 Notice that so long as , the government collects seigniorage revenues when it creates new coins. The seigniorage revenues generated at time t equal mt−mt−1 pt 
minus the value in terms of consumption goods of the quantity of silver put in the new money, 
. These revenues are positive whenever
9 He proceeded to discuss deviations from these conditions in England in the form of mint charges, laws against export, and others.
10 We use the English word “penny” to translate the Latin denarius, Italian denaro, French denier, as coin and as unit of account. The plural of the coin is “pennies,” that of the unit is “pence.”
11 Our specification of supply is consistent with the reasoning of Adam Smith ([1776] 1937, book I, ch. 5, 44). Smith described how coins must be valued more than by weight if citizens are ever voluntarily to use metal to buy coins at a price that covers the mint’s costs of production and seigniorage fees. Smith pointed out that if there were a delay in delivering the coins after the metal was paid to the mint, an interest factor would contribute an additional wedge beyond the seigniorage fee and production costs. Smith presumed valuation by tale, and considered the interval of values above their values by weight that is created by seigniorage fees, production costs, and delivery delays.
12 A shortage indicates an additional component of value beyond the explicit rate of return. The Lagrange multiplier on our additional restriction on transactions contributes this additional component of value (see chapter 21).
13 In our discussion of the Castilian episode, we shall add a parameter x that the government can manipulate to alter the exchange rate.
14 Concern about counterfeiting is a continuing theme in our narrative. Feavearyear (1963, 171) writes the following about the temptation to counterfeit in Britain in the eighteenth century: “Here we have a picture of the difficulty which persisted almost down to modern times. So long as the coins contained much less than their face value of metal, there was a great temptation to copy them. The technique of coining was still not sufficiently good to give to the money a stamp which it was beyond the skill of the forger to imitate even upon a scale large enough to make it worth while in the case of halfpennies and farthings, and the machinery for the detection of crime was bad.”
15 In a similar vein, Feavearyear (1963, 169) wrote that in the eighteenth century,
“Considerable quantities of foreign and counterfeit pieces were imported and passed off as English. In point of fact, so long as the Government was unable to find a method of providing the country with a sound and adequate coinage, the importation and issue of counterfeit or light silver was a good thing. The coins in circulation were now definitely tokens, and the only disadvantage of the Mint’s neglect to produce them was that it created a shortage. The counterfeiter tended to fill up the void, and he could do no harm to the standard.”
16 The entry on Gresham’s law in the index of Friedman and Schwartz (1963) is “Gresham’s law, misapplication of.” Also see Rolnick and Weber (1986). Sargent and Smith (1997) describe various senses of Gresham’s law, and some situations where a version of the law applies and others where it does not.
17 See Friedman (1951) who estimates the right side of (2.12) for the United States as a measure of the waste in a commodity money system. The advisors of the king of Castile in 1596 may also have done such a calculation (see chapter 14). See Sargent and Wallace (1983) for an argument that Friedman’s calculation overstates the cost of maintaining a commodity money system. Friedman (1951) remarked that in terms of their values in consumption goods, the resources tied up as a commodity money don’t depend on the particular commodity used. This is true in our model.
18 Remember that the mint had earlier been told only to buy coins for silver, and not to sell. Now it is told both to buy and to sell at a fixed price.
19 For convenience, we assume that token coins are stamped on intrinsically valueless metal.
20 See Rolnick and Weber (1997).
