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Working Paper Series
The Big Problem with Small Change
Thomas J. Sargent and François R. Velde
Working Papers Series
Research Department
(WP-97-8)
Federal Reserve Bank of Chicago
The Big Problem of Small Change∗
Thomas J. Sargent
University of Chicago and Hoover Institution
François R. Velde
Federal Reserve Bank of Chicago
ABSTRACT
Western Europe was plagued with currency shortages
from the 14th to the 19th century, at which time a ‘standard
formula’ had been devised to cure the problem. We use a cashin-advance model of commodity money to define a currency
shortage, show that they could develop and persist under a
commodity money regime, and analyze the role played by each
ingredient in the standard formula. A companion paper documents the evolution of monetary theory, monetary experiments
and minting technology over the course of six hundred years.
v1.0
∗
October 1997
The views expressed herein are those of the authors and not necessarily those of the Federal Reserve
Bank of Chicago or the Federal Reserve System.
1. Introduction
The title of this paper paraphrases one by Carlo Cipolla.1 Like Cipolla, our subject is the
process through which western monetary authorities learned how to supply small change.
Along with many other writers, Cipolla described how Western Europeans long struggled
to sustain a proper mix of large and small denomination coins, and to break away from
the idea that a commodity coinage requires coins of all denominations to be full-bodied.2
The Carolingian monetary system, born about A.D. 800, had only one coin, the
penny. At the end of the twelfth century, various states began also to create larger denomination coins. From the thirteenth to the nineteenth century, there were recurrent
‘shortages’ of the smaller coins. Cipolla (1956, 31) states that: ‘Mediterranean Europe
failed to discover a good and automatic device to control the quantity of petty coins to be
left in circulation,’ a failure which extended across all Europe.3
By the middle of the nineteenth century, the mechanics of a sound system were well
understood, thoroughly accepted,4 and widely implemented. According to Cipolla (1956,
27):
‘Every elementary textbook of economics gives the standard formula
for maintaining a sound system of fractional money: to issue on government
account small coins having a commodity value lower than their monetary
value; to limit the quantity of these small coins in circulation; to provide
convertibility with unit money. . . . Simple as this formula may seem, it took
centuries to work it out. In England it was not applied until 1816, and in
the United States it was not accepted before 1853.’
Before the triumph of the ‘standard formula’, fractional coins were more or less
full bodied, and contained valuable metal roughly in proportion to their nominal values,
1
2
‘The Big Problem of the Petty Coins’, in Cipolla (1956).
The mix matters because small coins can make the same transactions as large coins, but not vice
versa. This is true so long as there is no ‘limited legal tender’ clause of the kind mentioned by Cipolla as
a possible fourth element of his ‘standard formula’.
3 These shortages are documented in Sargent and Velde (1997b).
4 For example, see John Stuart Mill (1857, chapter X, Section 2).
1
contradicting one element of the ‘standard formula.’ Supplies were determined by private
citizens who decided if and when to use metal to purchase new coins from the mint at
prices set by the government, contradicting another element of the standard formula. That
system produced chronic shortages of small coins, but also occasional gluts. Gradually
over the centuries, theorists proposed components of the ‘standard formula’; occasionally
policy makers even implemented some of them. Full implementation waited until 1816, in
Britain; and, over the following 60 years, in France, Germany, the United States and other
countries, culminating in the establishment of the ‘Classical Gold Standard’ with silver
and bronze or copper coinage as subsidiary money.
Our goal in this paper is to understand what made the medieval monetary system
defective, and why it took so long to implement the standard formula. We present a model
of supply and demand for large and small metal coins designed to simulate the medieval
and early modern monetary system, and to show how its supply mechanism lay vulnerable
to alternating shortages and surpluses of small coins. We extend Sargent and Smith’s
(1997) model to incorporate demands and supplies of two coins differing in denomination
and possibly in metal content. We specify cash-in-advance constraints to let small coins
make purchases that large coins cannot. Like the Sargent-Smith model, for each type of
coin, the supply side of the model determines a range of price levels whose lower and upper
boundaries trigger coin minting and melting, respectively. These ranges let coins circulate
above their intrinsic values. The ranges must coincide if both coins are to continue to
circulate. The demand side of the model delivers a sharp characterization of ‘shortages’
of small coins. ‘Shortages’ make binding our additional cash-in-advance constraint—the
‘penny-in-advance’ constraint that requires that small purchases be made with small coins.
This means that small coins must depreciate in value relative to large coins during shortages
of small coins. Thus in our model, ‘shortages’ of small coins have two symptoms: (1) the
quantity theory of money splits in two, one for large coins, another for small; and (2) small
coins must depreciate relative to large ones in order to render binding the ‘small change
in advance constraint’ and thereby provide a motive for money holders to economize on
small change. In conjunction with the supply mechanism, the second response actually
aggravates the shortage.5
5
The ‘standard formula’ mentioned by Cipolla solves the exchange rate indeterminacy problem
2
We use these features of the model to account for various historical outcomes,
and, among other things, to account for why debasements of small coins were a common
policy response to shortages of small change. Then we modify the original arrangement by
including one or more of the elements in the ‘formula’ recounted by Cipolla, and show the
consequences. We compare these consequences with the predictions and prescriptions of
contemporary monetary theorists, and with episodes in monetary history, in particular the
spectacular failure of a fiduciary coinage in 17th century Spain, and the lessons drawn by
contemporary observers. By way of explaining why it took so long to come to the ‘standard
formula’, we enumerate the theoretical, technological and institutional prerequisites for its
implementation.6 In a companion paper (Sargent and Velde 1997), we document how
‘theory’ was ahead of technology and institutions until 1816.
In her fascinating account of Britain’s adoption of the gold standard, Angela Redish
(1990) confronts many of the issues studied in this paper, and traces England’s inability to
adopt the gold standard before the 19th century to the problem of small change. She finds
that “technological difficulties (the threat of counterfeiting) and institutional immaturity
(no guarantor of convertibility)” were the main obstacles. According to Redish, Matthew
Boulton’s steam-driven minting press of 1786 overcame the first obstacle by finally giving
the government a sufficient cost advantage over counterfeiters. Redish identifies no such
watershed with respect to convertibility. Though she traces the Bank of England’s protracted negotiations with the Treasury over responsibility for the silver coinage, it is not
clear what the obstacle was, and how it was overcome. In her conclusion, Redish suggests
the earlier history of small change as a possible testing field for her theory. In some ways,
we are pursuing this project here.
This paper has a companion (Sargent and Velde 1997) that traces in detail the
histories of thoughts, technologies, and policies bearing on issues of coinage. That parallel
study motivated the questions and modeling decisions in the present paper.
inherent in any system with an inconvertible and less than full bodied fractional currency. Remember that
Russell Boyer’s (1971) original paper on exchange rate paper was titled ‘Nickels and Dimes’. See Kareken
and Wallace (1981) and Helpman (1981) for versions of exchange rate indeterminacy in models of multiple
fiat currencies. A ‘one-sided’ exchange rate indeterminacy emerges from the cash-in-advance restrictions
in our model, and determines salient predictions of the model.
6 We will also discuss a sense in which the formula has redundant elements.
3
The remainder of the paper is organized as follows. Section 2 describes the model
environment, the money supply arrangement, and the equilibrium concept. Section 3 uses
‘back-solving’ to indicate possible co-movements of the price level, money supplies, and
national income; to illustrate perverse aspects of the medieval supply arrangements; and
to interpret various historical outcomes. Section 4 uses the model to study how aspects of
Cipolla’s ‘standard formula’ remedy the perverse supply responses by making small change
into tokens. Section 5 concludes with remarks about how, by learning to implement a token
small change, the West rehearsed a complete fiat monetary system.
2. The Model
In a ‘small country’ there lives an immortal representative household that gets utility from
two nonstorable consumption goods. The household faces cash-in-advance constraints.
‘Cash’ consists of a large and a small denomination coin, each produced by a governmentregulated mint that stands ready to coin any silver brought to it by household-owned
firms. The government specifies the amounts of silver in large and small coins, and also
collects a flat-rate seigniorage tax on the volume of newly minted coins; it rebates the
revenues in a lump sum. Coins are the only storable good available to the household. The
firm can transform either of two consumption goods into the other one-for-one and can
trade either consumption good for silver at a fixed international price. After describing
these components of the economy in greater detail, we shall define an equilibrium. For
any variable we let x denote the infinite sequence {xt }∞
t=0 . Table 1 compiles the main
parameters of the model and their units.
4
Variable Meaning
Units
φ
world price of silver
oz silver/cons good
b1
intrinsic content of dollar
oz silver/dollar
b2
intrinsic content of penny
oz silver/penny
γ1
melting point of dollar
dollars/cons good
γ2
melting point of penny
pennies/cons good
σi
seigniorage rate
(none)
b1 −1
mint equivalent of dollar
dollars/oz silver
b2 −1
mint equivalent of penny
pennies/oz silver
m1
stock of pennies
pennies
m2
stock of dollars
dollars
e
exchange rate
dollars/penny
p
price of cons goods
dollars/cons good
Table 1: Dramatis Personae.
The Household
The representative household maximizes
∞
X
β t u (c1,t , c2,t )
(1)
t=0
where we assume the one-period utility function u(c1,t , c2,t ) = v(c1,t ) + v(c2,t + α) where
α > 0 and v(·) is strictly increasing, twice continuously differentiable, strictly concave, and
satisfies the Inada conditions lim→0 v 0 (x) = +∞. We use consumption of good 1, c1 , to
represent ‘large’ purchases, and c2 to stand for small purchases. There are two kinds of
5
cash: dollars, whose stock is m1 , and pennies, whose stock is m2 . Each stock is measured
in number of coins, dollars or pennies. Both pennies and dollars can be used for large
purchases, but only pennies can be used for small purchases.7 A penny exchanges for et
dollars. Thus, the cash-in-advance constraints are:
pt (c1,t + c2,t ) ≤ m1,t−1 + et m2,t−1
(2)
pt c2,t ≤ et m2,t−1 ,
(3)
where pt is the dollar price of good i. The household’s budget constraint is
pt (c1,t + c2,t ) + m1,t + et m2,t ≤ Πt + m1,t−1 + et m2,t−1 + Tt ,
(4)
where Πt denotes the firm’s profits measured in dollars, and Tt denotes lump sum transfers
from the government. The household faces given sequences (p, e, T ), begins life with initial
conditions m1,−1 , m2,−1 , and chooses sequences c1 , c2 , m1 , m2 to maximize (1) subject to
(2), (3), and (4).
Feasibility
A household-owned firm itself owns an exogenous sequence of an endowment {ξt }∞
t=0 .
In the international market, one unit of either consumption good can be traded for φ > 0
units of silver,8 leading to the following restrictions on feasible allocations:
c1,t + c2,t ≤ ξt + φ−1 St ,
t≥0
(5)
where St stands for the net exports of silver from the country.
Coin Production Technology and the Mint
7 The assumption that pennies can be used for the same purchases as dollars is motivated by
several episodes in history during which small denominations overtook the monetary functions of large
denominations with great ease.
8 Alternatively, there is a reversible linear technology for converting consumption goods into silver.
6
Stocks of coins evolve according to
mi,t = mi,t−1 + ni,t − µi,t
(6)
where ni,t ≥ 0, µi,t ≥ 0 are rates of minting of dollars, i = 1, and pennies, i = 2. The
government sets b1 , the number of ounces of silver in a dollar, and b2 , the number of ounces
of silver in a penny. The government levies a seigniorage tax on minting: for every new
coin of type i minted, the government charges a flat tax at rate σi > 0.
The country melts or mints coins to finance net exports of silver in the amount
St = b1 (µ1,t − n1,t ) + b2 (µ2,t − n2,t ) .
(7)
Net exports of silver St correspond to net imports of φ−1 St of consumption goods.
The quantities 1/b1 and et /b2 (measured in number of dollars per minted ounce of
silver) are called by Redish (1990) ‘mint equivalents’. The quantities (1 − σ1 )/b1 , et (1 −
σ2 )/b2 are called ‘mint prices’, and equal the number of dollars paid out by the mint per
ounce of silver.
Government
The government sets the parameters b1 , b2 , σ1 , σ2 , and, depending on citizens’ minting decisions, collects revenues Tt in the amount
Tt = σ1 n1,t + σ2 et n2,t .
(8)
Below we shall describe other interpretations of σi partly in terms of the mint’s costs of
production. The only modification that these alternative interpretations require would be
to (8).
The Firm
7
The firm receives the endowment, sells it, mints and melts, pays seigniorage, and
pays all earnings to the household at the end of each period. The firm’s profit measured
in dollars is
Πt =pt ξt + (n1,t + et n2,t ) − (σ1 n1,t + σ2 et n2,t )
b2
b2
b1
b1
n1,t + n2,t + pt
µ1,t + µ2,t − (µ1,t + et µ2,t ) .
− pt
φ
φ
φ
φ
(9)
The first term measures revenues from the sale of the endowment; the next term is revenues
of minting, followed by seigniorage payments, minting costs, melting revenues and melting
costs.
The firm takes the price system (p, e) as given and chooses minting and melting
sequences (n1 , n2 , µ1 , µ2 ) to maximize (9) subject to (6) period-by-period.
Equilibrium
A feasible allocation is a triple of sequences (c1 , c2 , S) satisfying (5). A price system
is a pair of sequences (p, e). A money supply is a pair of sequences (m1 , m2 ) satisfying the
initial conditions (m1,−1 , m2,−1 ). An equilibrium is a price system, a feasible allocation,
and a money supply such that given the price system, the allocation and the money supply
solve the household’s problem and the firm’s problem.
Analytical Strategy
We proceed sequentially to extract restrictions that our model places on co-movements
of the price level and the money supply. The firm’s problem puts restrictions on these comovements, and the household’s problem adds more.
Arbitrage Pricing Conditions
8
The firm’s problem puts the price level inside two intervals.9 Define γi = φ/bi and
rearrange (9):
−1
Πt =pt ξt + 1 − σ1 − pt γ1−1 n1,t + et 1 − σ2 − pt (et γ2 )
n2,t
−1
+ pt γ1−1 − 1 µ1,t + et pt (et γ2 ) − 1 µ2,t .
(10)
Each period, the firm chooses ni,t , µi,t to maximize Πt subject to non-negativity constraints
ni,t ≥ 0, µi,t ≥ 0 and to the upper bound on melting: mi,t−1 ≥ µi,t , for i = 1, 2. The form
of (10) immediately implies the following no-arbitrage conditions:10
n1,t ≥ 0;
= if pt > γ1 (1 − σ1 )
(11a)
n2,t ≥ 0;
= if pt > et γ2 (1 − σ2 )
(11b)
µ1,t ≥ 0;
= if pt < γ1
(11c)
µ2,t ≥ 0;
= if pt < et γ2
(11d)
µ1,t ≤ m1,t−1 ;
= if pt > γ1
(11e)
µ2,t ≤ m2,t−1 ;
= if pt > et γ2 .
(11f )
Implications of the Arbitrage Conditions for Monetary Policy
These no-arbitrage conditions constrain the mint’s policy if both coins are to exist.11
The constraints put pt within both of two intervals, [γ1 (1 − σ1 ), γ1 ] (corresponding to
dollars) and [et γ2 (1 − σ2 ), et γ2 ] (corresponding to pennies). See Figure 1. Only when the
price level pt is at the lower end of either interval might the associated coin be minted.
Only when the price is at the upper end might that coin be melted. Therefore, if the lower
ends of the intervals do not coincide, (i.e., if γ1 (1 − σ1 ) 6= et γ2 (1 − σ2 )), then only one type
9
See Usher (1943) for a discussion of these constraints on the price level.
These restrictions must hold if the right side of (10) is to be bounded (which it must be in any
equilibrium); their violation would imply that the firm could earn unbounded profits.
11 See the related discussion in Usher (1943, 1997–201).
10
9
of coin can ever be minted. Equating the lower ends of the intervals (by the government’s
choice of (bi , σi )) makes the mint stand ready to buy silver for the same price, whether
it pays in pennies or dollars. Equating the upper ends of the intervals makes the ratio of
metal contents in the two coins equal the exchange rate. If the upper ends of the intervals
don’t coincide, then one type of coin will be melted before the other. Pennies are said to be
full-bodied if 1/b1 = e/b2 , that is, if the melting points (upper limits) of the two intervals
coincide. Household preferences and equations (2) and (3) imply that et γ2 ≥ γ1 . If this
inequality is strict, the intrinsic content of pennies is less than proportionate to their value
in dollars, in which case pennies are ‘light’.12
minting
melting
dollars
(1−σ1)γ1
γ1
minting
melting
pennies
e(1−σ2)γ2
pt
eγ2
Figure 1: Constraints on the price level imposed by the arbitrage conditions.
Thus, if pennies are not full-bodied, a sufficient rise in the price level will make large
coins disappear. If the mint prices differ, a sufficient fall in prices will prompt minting of
12
By substituting Πt = pt ξt , (8), and (6) into (4) we obtain
pt (c1,t + c2,t ) ≤ pt ξt + µ1,t + et µ2,t − (1 − σ1 ) n1,t − et (1 − σ2 ) n2,t .
Using the no-arbitrage conditions (11) in this expression and rearranging leads to
h
i
c1,t + c2,t ≤ ξt + γ1−1 (µ1,t − n1,t ) + γ2−1 (µ2,t − n2,t ) .
The term in square braces equals net imports of consumption goods. Thus, as usual, manipulation of
budget sets at equilibrium prices recovers a feasibility condition.
10
only one of the two coins. The perpetual coexistence of both coins in the face of price
fluctuations requires that pennies be full-bodied and that equal mint prices prevail for
both coins; that is, the intervals must coincide, and therefore the seigniorage rates must
be equal. This is not possible if we reinterpret the σi ’s in terms of the production costs
for the two types of coins.
Interpretations of σi
In using (8), we have interpreted σi as a flat tax rate on minting of coins of type i.
But so far as concerns the firm’s problem and the arbitrage pricing restrictions, σi can be
regarded as measuring all costs of production borne by the mint, including the ‘seigniorage’
it must pay to the government. On this interpretation, in setting σi , the government is
naming the sum of the seigniorage tax rate and the mint’s costs of production. If a
government was unwilling to subsidize production of coins, then the costs of production
served as a lower bound on σi .
The government could decide to set gross seigniorage σi to 0, by subsidizing the
mint. In this circumstance, our two coins could coexist only if pennies were full-weight
(their intrinsic content being proportional to their face values), and if the price level never
deviated from γ1 .
We discuss in the companion paper the attitudes of medieval writers as well as
the actual policies followed by governments. A tradition of thought advocated setting
σi = 0 but it was not followed in practice until the 17th century; other jurists thought
that σi should remain close to production costs, except in cases of clearly established fiscal
emergency.
On the other side, restraints were placed on the government’s freedom to set σi by
potential ‘competitors’ to the mint, such as counterfeiters and foreign mints, and by the
government’s ability to enforce laws against counterfeiting and the circulation of foreign
11
coins. Let σ̃i be the production costs for counterfeiters, or for arbitrageurs taking metal to
foreign mints and bringing back coins (inclusive of transport costs). A wide gap between
σi and σ̃i was difficult to maintain unless government’s enforcement powers were strong.
If they were not, σ̃i placed an upper bound on σi .13
A government could maintain positive seigniorage if the costs of production of licensed mints were smaller than those of competitors. For example, Montanari ([1683]
1804, 114) argues that the death penalty for counterfeiting, while impossible to enforce
strictly, adds a risk premium to counterfeiters’ wage bill, thereby increasing σ̃i when the
same technology is used by all. Furthermore, if a government was able to restrict access to
the mint’s technology, or if it could secure the exclusive use of a better technology, it could
set seigniorage above the mint’s production costs, up to the level of competitors’ costs.14
Per coin production costs differed between small and large denomination coins. The
Medieval technology made it significantly more expensive to produce smaller denomination
coins.15 In situations that tied the σi ’s to the costs of production, different production
costs implied different widths of our no-arbitrage intervals. This meant either that pennies
had to be less than full-bodied or that the mint prices differed. In either case, price level
fluctuations could arrest production or, by stimulating melting, cause the disappearance
of one coin.
We have extracted the preceding restrictions from the requirement that equilibrium
13 See Usher (1943, 201): “Seigniorage presented no special problem unless the amount exceeded the
average rate of profit attractive to gold and silversmiths, or to mints in neighboring jurisdictions. Beyond
this limit, the effective monopoly of coinage might be impaired by illegal coinage of essentially sound coins,
or by the more extensive use of foreign coin.”
14 Philip II, king of Spain, referred explicitly to this advantage when he put a new technology to use
in making small copper coinage in 1596. Mentioning the new mechanized mint of Segovia, he declared that
“if we could mint the billon coinage in it, we would have the assurance that it could not be counterfeited,
because only a small quantity could be imitated and not without great cost if not by the use of a similar
engine, of which there are none other in this kingdom or the neighboring ones.”
15 This was true of competitors’ costs as well: Montanari notes that the risk premium induced by
the death penalty is the same across denominations. For arbitrageurs taking metal to foreign mints,
transportation costs made the operation worthwhile only for the larger coins; the near-uniformity of
Medieval seigniorage rates on gold coins, contrasted with much greater variation on smaller coinage, bears
this out.
12
prices should not leave the firm arbitrage opportunities. We now turn to additional restrictions that the household’s optimum problem imposes on equilibrium prices and quantities.
The Household’s Problem
The household chooses sequences c1 , c2 , m1 , m2 to maximize (1) subject to (4), (2)
and (3), as well as the constraints mi,t ≥ 0 for i = 1, 2. Attach Lagrange multipliers λt ,
ηt , θt and νi,t , respectively, to these constraints. The first-order conditions are:
u1,t
= λt + ηt
pt
u2,t
= λt + ηt + θt
pt
−ν1,t = −λt + β (λt+1 + ηt+1 )
−ν2,t = −et λt + βet+1 (λt+1 + ηt+1 + θt+1 )
(12a)
(12b)
(12c)
(12d)
with corresponding relaxation conditions. Conditions (12a), (12b) and (12c) lead to the
following:
u1,t ≤ u2,t ;
u1,t
u1,t+1
≤
;
β
pt+1
pt
= if et m2,t−1 > pt c2,t
(13a)
= if m1,t > 0 and m1,t−1 + et m2,t−1 > pt (c1,t + c2,t )
(13b)
Subtracting (12c) from (12d), rearranging, and imposing that ν2,t = 0 (because of the
Inada conditions on u), we find:
λt
et
et+1
−1
= βθt+1 − ν1,t .
(14)
Suppose θt+1 > 0 and ν1,t = 0: this requires et > et+1 . In words, if the ‘penny-in-advance
constraint’ is binding and positive holdings of dollars are carried over from t to t + 1,
pennies must depreciate in terms of dollars from t to t + 1. This makes sense, because if
the household holds money from t to t + 1 and also wishes at t + 1 that it had held a higher
proportion of pennies (i.e., if (3) is binding at t + 1), it is because it chose not to because
et > et+1 , indicating that dollars dominate pennies in rate of return. Thus ‘shortages’ of
13
pennies occur only after dollars dominate pennies in rate of return. To describe the rate
of return on pennies further, write (12c) and (12d) at t, and (12a) and (12b) at t + 1, and
make the necessary substitutions to obtain:
et
et+1
=
u2,t+1
≥ 1.
u1,t+1
(15)
In models with only one cash-in-advance constraint but two currencies, this equation holds
with equality. Inequality (15) embodies a form of ‘one-sided’ exchange rate indeterminacy,
and leaves open the possibility of a class of equilibrium exchange rate paths that for which
the small change is not appreciating relative to large coins.
3. Equilibrium Computation via ‘Back-solving’
We utilize the same ‘back-solving’ strategy employed by Sargent and Smith (1997) to describe possible equilibrium outcomes. Back-solving takes a symmetrical view of ‘endogenous’ and ‘exogenous’ variables.16 It views the first-order and other market equilibrium
conditions as a set of difference inequalities putting restrictions across the endowment, allocation, price, and money supply sequences, to which there exist many solutions. We use
back-solving to display aspects of various equilibria. For example, we shall posit an equilibrium in which neither melting nor minting occurs, then solve for an associated money
supply, price level, endowment, and allocation. We shall construct two examples of such
equilibria, one where the penny-in-advance constraint (3) never binds, another where it
occasionally does. We choose our sample equilibria to display particular adverse operating
characteristics of our money supply mechanism.
Equilibria with neither melting nor minting
When neither melting nor minting occurs, c1,t + c2,t = ξt , and mit = mi,−1 ≡
mi , i = 1, 2. In this case, the equilibrium conditions of the model consist of one or the
16
See Sims (1989, 1990) and Diaz-Giménez et al. (1992).
14
other of the two following sets of equations, depending on whether the penny-in-advance
restriction binds:
when (3) binds at t:
m1 = pt c1,t
et m2 = pt c2,t
when (3) does not bind:
m1 + et m2 = pt ξt
et m2 ≥ pt c2,t
c1,t + c2,t = ξt
et−1
u2,t
=
=1
u1,t
et
u1,t+1
u1,t
≥β
pt
pt+1
or
c1,t + c2,t = ξt
et−1
u2,t
=
≥1
u1,t
et
u1,t+1
u1,t
≥β
pt
pt+1
Notice how when (2) does not bind, there is one quantity theory equation in terms of the
total stock of coins, but that when (3) does bind, there are two quantity theory equations,
one for large purchases cast in terms of dollars, the other for small purchases in terms of
the stock of pennies.
In what follows, we begin by examining equilibria where the first set of conditions
applies at all times. We will show how to construct stationary equilibria. Then we will
study a variation in the endowment that brings into play the second set of equations,
when the penny-in-advance constraint binds, all the while maintaining the requirement
that neither melting nor minting occurs. Throughout, we take the mint policy (σi and γi )
as fixed.
Stationary Equilibria with no minting or melting
We describe a stationary equilibrium with constant monies, endowment, consumption rates, price level, and exchange rate.
Proposition 1. (A stationary, ‘no-shortage’ equilibrium)
15
Assume a stationary endowment ξ, and initial money stocks (m1 , m2 ). Let (c1 , c2 ) solve
u 1 = u2
(16)
c1 + c2 = ξ,
(17)
Then there exists a stationary equilibrium without minting or melting if the exchange rate
e satisfies the following:
/
A = (γ1 (1 − σ1 ) , γ1 ) ∩ (eγ2 (1 − σ2 ) , eγ2 ) 6= O
m1 + em2
∈A
ξ
ξem2
≥ c2 .
em2 + m1
(18)
(19)
(20)
Proof: Let p = (m1 + em2 )/ξ. By (19), p will satisfy (11) in such a way that coins are
neither melted nor minted. Condition (2) is then satisfied with equality and (3) is satisfied
with inequality by (20). Since (3) does not bind, θt = 0 and conditions (12) are satisfied
with a constant e.
Condition (18), a consequence of the no-arbitrage conditions, requires that the
exchange rate be set so that there exists a price level compatible with neither minting nor
melting of either coin, of the type described in proposition 1. Condition (19) means that
the total nominal quantity of money (which depends on e) is sufficient for the cash-inadvance constraint. Condition (20) puts a lower bound on e: the share of pennies in the
nominal stock must be more than enough for the penny-in-advance constraint not to bind.
Conditions (19) and (20) together imply that pc1 ≥ m1 , that is, dollars are insufficient for
large purchases.
Figure 2 depicts the determination of c1 , c2 , p, and satisfaction of the penny in
advance constraint (3) with room to spare. There is one quantity theory equation cast
in terms of the total money supply. That there is room to spare in satisfying (3) reveals
16
c2
expansion path
ξ
slope: (et-1m2/m1)
F
etm2/p
C
U curve
ξ=c1+c2
m1/p
c1
Figure 2: Stationary equilibrium, non-binding constraint.
an exchange rate indeterminacy in this setting: a range of e’s can be chosen to leave the
qualitative structure of this figure intact. Notice the ray drawn with slope
et−1 m2
.
m1
So long
as the endowment remains in the region where the expansion path associated with a unit
relative price lies to the southeast of this ray, the penny-in-advance constraint (3) remains
satisfied with inequality. But when movements in the endowment or in preferences put the
system out of that region, it triggers a penny shortage whose character we now study.
Small coin shortages
Consistent with the back-solving philosophy, we display a few possible patterns of
endowment shifts that generate small coin shortages.
First, note that another way to interpret (19) and (20) is to take m1 , m2 , e as fixed
17
and to formulate these conditions as bounds on ξ:
m1 + em2
m1 + em2
(21)
<ξ<
γ1
γ1 (1 − σ1 )
m1 + em2
m1 + em2
(22)
<ξ<
eγ2
eγ2 (1 − σ2 )
m1 + em2
ξ≤α
(23)
m1 − em2
(where (23) is written for the logarithmic utility case). These equations reveal a variety of
ways of generating small coin shortages with a change in ξ, starting from a given stationary
equilibrium. The simplest, which we explore first, is a one-time change in ξ within the
intervals defined in (21) and (22) but that violate (23): no minting or melting occurs, but
a shortage ensues. Another way is as follows: a shortage of small change arises when the
relative stock of pennies is insufficient relative to dollars (violation of (23)). Suppose the
lower bound in (22) is lower than that in (21). Then a shift in the endowment can lower
the price level to the minting point for dollars without triggering any minting of pennies,
resulting in a (relative) decrease in the penny supply; for some values of the parameters,
this can generate a shortage in the period following the minting of dollars.
Small coin shortage, no minting or melting
We begin by studying the situation that arises when, following an epoch where
constant money supplies and endowment were compatible with a stationary equilibrium,
there occurs at time t a shift in the endowment. In Figure 2, for the utility function
v(c1,t ) + v(c2,t + α) we drew the expansion path traced out by points where indifference
curves are tangent to feasibility lines associated with different endowment levels. The
expansion path is c2 = max(0, c1 − α), and so has slope one or zero. If
em2
m2
> 1, i.e., if
pennies compose a large enough fraction of the money stock, then the ray c2 /c1 equaling
this ratio never threatens to wander into the southeastern region threatening to render (3)
binding, described above. However, when
em2
m1
< 1, growth in the endowment ξ can push
the economy into the southeastern region, which makes (3) bind and triggers a depreciation
of pennies.
18
Thus, suppose that ξt is high enough that the intersection of the expansion path
with feasibility (c1 +c2 = ξt ) is above the ray et−1 m2 /m1 . This means that at t, our second
subset of equations determines prices and the allocation. If we assume that ξt is such that
neither minting nor melting occurs (an assumption that must in the end be verified), then
equilibrium values of c1 , c2 , e, p can be computed recursively. Given et−1 , the following
three equations can be solved for c1 , c2 , e at t:
c1,t + c2,t = ξt
et−1
u2,t
=
u1,t
et
c2,t
et m2
=
.
m1
c1,t
(24)
(25)
(26)
These can be combined into a single equation in c2 :
m1
u1,t (ξt − c2,t , c2,t )
c2,t
=
u2,t (ξt − c2,t , c2,t )
et−1 m2 ξt − c2,t
(27)
c2
m1
Define f (c2 ) = u2,t ξ−c
− u1,t . The fact that the expansion path intersects the
2 et−1 m2
feasibility line below the ray et−1 m2 /m1 implies that f (c2 ) > 0 for (ξ − c2 , c2 ) on the
expansion path. As c2 → 0, lim f (c2 ) < 0 (note that u2 (ξ, 0) = u0 (α) < +∞). Therefore,
there exists a solution c2 to (27). Then c1 is determined from (24), and et (which satisfies
et−1 > et by construction) is given by (25). It remains to check that pt = m1 /c1 ∈ A and
that the Euler inequality
βpt−1
pt
βu1,t
pt
≤
u1,t−1
pt−1
holds. There is room to satisfy the inequality if
is small enough.
Figure 3 depicts the situation when constraint (3) is binding but no minting or
melting occurs. We use the intersections of various lines to represent the conditions (24),
(25) and (26), and to determine the position of the consumption allocation at point C.
The feasibility line F represents (24). Condition (26) means that the ray passing through
point C has slope et m2 /m1 : this defines et . Finally, condition (25) can be transformed
(using (26)) into the following:
c1 +
u2,t
et−1 m2
c2 = c1 +
c1 .
u1,t
m1
19
(28)
c2
ξ
A
C
U curve
R
slope: (et-1m2/m1)
slope: -u2/u1
F
ξ=c1+c2
D
c1
c1+(u2/u1)c2 = c1+(et-1m2/m1)c1
Figure 3: Effect of a shift in endowments.
Define point A to be the vertical projection of C onto the ray et−1 m2 /m1 . The lefthand side of (28) is the point at which a line, parallel to the feasibility line and drawn
through point A, intersects the x-axis. The right-hand side is the point where the tangent
to the indifference curve at C intersects the x-axis. Condition (25) requires that these
intersections coincide. Note that, when the constraint does not bind, points A and C
coincide, and the tangent to the indifference curve coincides with the feasibility line.
Logarithmic example
Suppose in v(·) = ln(·) in equation (1). For this specification we can compute
pt , et , c1,t , c2,t by hand for the no-minting, no-melting case in which pt ∈ A, defined in
(18). When
et−1 m2
m1
>
ξt −α
,
ξt +α
the constraint (3) does not bind; when the inequality is
reversed, it does bind, requiring a decrease in et relative to et−1 . When (3) does not bind,
the time t equilibrium objects are c1,t =
ξt +α
2 , c2,t
20
=
ξt −α
2 ,
pt =
m1 +et−1
.
ξt
When (3) binds,
pt c2,t
et−1 m2
m1 −et−1 m2 α, c1,t = ξt − c2,t , pt = m1 /c1,t , et = m2 , where the
< ξξtt −α
+α implies that et−1 > et . It must be checked that pt ∈ A in
the solutions are c2,t =
condition that
et−1 m2
m1
each case.
Permanent and transitory increases in ξ
Having determined the new exchange rate et after a shift in the endowment ξt , we
can determine what happens if the shift is permanent or transitory. If it is permanent,
then the constraint (3) will continue to bind. The reason is that, since et < et−1 , the
ray et m2 /m1 is in fact even lower than et−1 m2 /m1 , which means that the expansion path
remains above the ray, and the penny constraint continues to bind. This situation cannot
continue without minting or melting indefinitely, however. Thus, a permanent upward shift
in the endowment from the situation depicted in Figure 2 to that in Figure 3 would impel
a sequence of reductions in the exchange rate until eventually the price level is pushed
outside the interval A.
As for a temporary (one-time) increase in ξt , it might prompt further depreciations
in the exchange rate even if the endowment immediately subsides to its original level. The
reason is that the reduction in et−1 induces a permanent downward shift in the
em2
m1
ray
that enlarges the (3)-is-binding southeastern region.
By shifting the interval for (e(1 − σ2 )γ2 , eγ2 ) to the left, the reduction of et hastens
the day when pennies will be melted and postpones the day when they might be minted,
without a government adjustment of γ2 . Not until the advent of the ‘standard formula’
described in the introduction was this perverse mechanism to be set aside.
Aggravated small coin shortage through minting of dollars
A shortage of small coins can also occur as a consequence of minting, independently
of the “income effect” we have described. Assume that all coins are full-bodied, so that the
21
bounds of the intervals coincide to the right (γ1 = eγ2 ), but that production costs require
that a higher seigniorage be levied on small coins (σ2 > σ1 ), so that the left boundaries
do not coincide (γ1 (1 − σ1 ) > eγ2 (1 − σ2 )).
In the previous section, we considered “small” increases in the endowment ξ; that
is, increases that led to movements in the price level pt within the intervals dictated by
the arbitrage conditions. We now consider “large” increases that will induce such a fall
in the price level that it reaches the minting point for large coins (pt = γ1 (1 − σ1 )). The
structure of coin specifications and minting charges means that small coins will not be
minted. As a result, m1 increases while m2 remains unchanged, and the ratio em2 /m1
falls. The intuition garnered from Figure 3 suggests that, for large enough increases in m1 ,
trouble may occur; a shortage of small change results, because the share of pennies in the
total money stock falls too far.
We begin again from a stationary equilibrium with no minting or melting. At time
t, the endowment increases from ξ0 to ξt , with ξt > (m1 + em2 )/γ1 (1 − σ1 ). Then minting
occurs, that is, pt = γ1 (1 − σ1 ). From the binding constraint (2),
c1,t + c2,t =
m1 + et m2
.
γ1 (1 − σ1 )
As before, two situations can arise, depending on whether (3) is binding or not.
We will look for equilibria where it is not binding at t, so that u2 = u1 ; combined with
the binding cash-in-advance constraint, one can solve for c1,t and c2,t . By assumption,
ξt − c1,t − c2,t > 0: that amount is minted, and n1,t = γ1 (ξt − c1,t − c2,t ).
At t + 1, it can be shown that
γ1 (1 − σ1 ) <
m1,t + et−1 m2
< γ1
ξt
or, in other words, that no more minting occurs if (3) does not bind. But (3) will bind, if
the following holds:
ξt − α
et−1 m2
.
<
m1,t
ξt + α
22
This turns out to be a second-degree polynomial in ξt , which will be positive for large
enough values of ξt .
Thus, if enough dollars are minted, pennies become relatively short of supply.
Small coin shortages, secularly declining e
We have thus shown two ways that endowment growth can induce small coin shortages, with or without minting. We now discuss how such episodes affect the monetary
system over time, in particular the relation between small and large coins.
A shortage of small coins manifests itself in a binding penny-in-advance constraint
(3), and is associated with two kinds of price adjustments, one ‘static’, the other ‘dynamic’.
First, the quantity theory breaks in two to become two separate quantity theory equations,
one for small, another for large coins taking the forms pt = m1,t /c1,t and et =
m2,t
pt c2,t .
The
mitosis of the quantity theory is the time-t consequence of a shortage of small coins.
A second response is dynamic, and requires that et < et−1 , so that pennies depreciate in
terms of dollars. This response equilibrates the ‘demand side’ but has perverse implications
because of its eventual effects on supply. For fixed b2 , a reduction in e shifts the interval
[e(1 − σ2 )γ2 , γ2 ] to the left. This hastens the occasion when pennies will be melted, and
reduces the chances that pennies will again be minted.
This penny-impoverishing implication of a shortage-induced fall in e isolates a force
for the government to adopt an offsetting reduction in b2 eventually to resupply the system
with a new, lighter penny. Sargent and Smith’s (1997) analysis of a non-inflationary
debasement can be adapted to simulate such a debasement.17
An alternative way to realign the two intervals in response to a shortage-induced
17 This experiment stresses the ‘circulation by tale’ axiom embedded in our framework: for the
purposes of the cash-in-advance constraint (3), a penny is a penny. To analyze a debasement as described
in the text, one has to keep track of two kinds of pennies, old and new. See Sargent and Smith (1997) for
an analysis in a one-coin system.
23
reduction in e would be to raise b1 , thereby shifting the interval for large coins also to
the left. Such a small-coin-shortage-induced ‘reinforcement’ of large coins would evidently
diminish the price level.
2
lire / Florentine pound
10
picciolo (4d)
grosso (60−80d)
10
1
1250
1300
1350
1400
1450
1500
Figure 4: Plot of 1/b for two Florentine silver coins (1250–1530). The dots plot an index
of the price (in pennies) of the gold florin, or 1/et in our model.
While the model thus identifies either a debasement of small coins or a reinforcement
of large coins as a workable policy for keeping both large and small coins in existence,
history records more debasements of small coins. Cipolla reports secular declines in e
and b2 , in the face of long periods of stable b1 . For example, in Florence the gold florin
retained a constant metal content for centuries, while petty coins were recurrently debased.
Figure 4 plots the evolution of 1/b2 for two Florentine silver coins, the picciolo and the
grosso, during the Middle Ages (for the gold Florin, b1 remained constant). A pattern
of recurrent debasements, as describe by Cipolla, is apparent. The graph also displays
another piece of information, namely the price of the gold florin in terms of silver pennies:
24
this corresponds to 1/et in our model.18
The Affair of the Quattrini
Figure 5 displays fourteenth century time series for the mint equivalent eγi and
the mint price e(1 − σ)γi for two petty coins of Florence, the grosso and the quattrino.19
The figure thus traces the two intervals shown in Figure 1. For most of the century the
intervals overlapped, with the interval for the smaller denomination quattrino containing
the interval for the grosso. A glaring exception occurred from 1371 to 1403, when the
interval for the quattrino lay above that for the grosso. Our model predicts that this spells
trouble, and Cipolla’s account confirms it.
During this exceptional period there occurred the ‘Affair of the Quattrini,’ another
of Cipolla’s beautiful chapter titles (Cipolla 1982), sparked by a devaluation of the quattrino in 1371 designed to stimulate its coining in order to displace ‘foreign’ small coins
flowing into Florence from Pisa. Our theory would imply during this period that (a) the
price level would move upward into the interval dictated by the interval for the quattrino,
that (b) quattrini would be coined, and that (c) grossi would disappear. Cipolla’s account
confirms implications (a) and (b), and is silent but not inconsistent with implication (c).
To protect it, the florin was permitted to appreciate in terms of petty coins. The authorities faced public pressure to reverse this ‘inflation’. Cipolla describes a 1381 ‘anti-inflation’
government policy to acquire and melt quattrini and so reduce the stock of money. Within
our model, this policy makes sense, though not too much could have been expected of it: a
‘quantity theory’ policy could move the price level downward within the band determined
by bi , σi , but could not drive it outside that band.
18 Strictly speaking, our model assumes that the technological rate of transformation between the
metals used in large and small coins is constant, for example, silver is used in both. In the case of the
florin, the gold/silver ratio fluctuated over time; in particular, in the first half of the 14th century gold
prices increased; adjusting for that variation would lower the dots in Figure 4 in that period.
19 Between 1345 and 1365, a Florentine lira exchanged at par for 4 grossi and 60 quattrini.
25
55
50
45
lira/pound silver
40
quattrino (4d)
35
30
25
20
15
1300
grosso (30−80d)
1320
1340
1360
1380
1400
1420
1440
1460
1480
Figure 5: Evolution of the upper and lower bounds on quattrini and grossi, Florence.
4. Arrangements to eliminate coin shortages
This section describes two money supply mechanisms that, within the context of
our model, eliminate shortages of small coins. After briefly describing these mechanisms,
we will scrutinize them in terms of how they incorporate some or all of the ingredients
in Cipolla’s recipe, and will determine to what extent some ingredients of the recipe are
redundant.
A ‘standard formula’ regime
We change the supply mechanism to implement a version of Cipolla’s ‘standard
formula’, retaining the demand side of the model. It is as if the government tells the mint
to set up a ‘pennies department’ that operates like a ‘currency board’ for pennies. The
26
rules for supplying dollars are not changed from those described above. But now the mint
is required to convert pennies into dollars and dollars into pennies, upon demand and at
a fixed exchange rate e, named by the government. Assume that pennies are produced
costlessly by the mint, and so are truly tokens (b2 = 0). The government requires the
mint to carry a non-negative inventory of dollars Rt from each date t to t + 1. The mint
increases its inventory of dollars only when it buys dollars, and decreases it only when it
buys pennies.
This regime imposes the following laws of motion for stocks of dollars and pennies:
m1,t = m1,t−1 + n1,t − µ1,t − (Rt − Rt−1 )
m2,t = m2,t−1 + e−1 (Rt − Rt−1 )
Rt ≥ 0,
where Rt is the stock of dollars held by the mint from t to t + 1. The law of motion for
dollars can be rewritten as
(m1,t + Rt ) = (m1,t−1 + Rt−1 ) + n1,t − µ1,t .
For the firm, it does not matter whether it melts dollars or whether pennies are exchanged
for dollars that are then melted; only the total stock of dollars m1,t + Rt counts.
Under this money supply mechanism, the firm’s profits become:
Πt = pt ξt + n1,t − σ1 n1,t − pt
b1
b1
n1,t + pt µ1,t − µ1,t ,
φ
φ
(29)
subject to the constraints n1,t ≥ 0 and m1,t−1 + Rt−1 ≥ µ1,t ≥ 0.
The absence of Rt from its profits signifies the firm’s indifference to the choice
of m2,t (as long as Rt ≥ 0), because the firm always breaks even when it buys or sells
pennies.20 The firm’s indifference lets the demand side of the model determine the stock
20
The condition Rt ≥ 0 can be ensured by choosing initial conditions such that em2,0 ≤ R0 .
27
of m2,t appropriately. More precisely, one can find paths for the money stocks that are
consistent with the mint’s profit maximization and that satisfy the household’s first-order
conditions with (3) not binding.
A subset of the no-arbitrage conditions now obtains:
n1,t ≥ 0;
= if pt > γ1 (1 − σ1 )
(30a)
µ1,t ≥ 0;
= if pt < γ1
(30b)
m1,t−1 + Rt−1 ≥ µ1,t ;
= if pt > γ1
(30c)
This regime forces pt into the interval [(1 − σ)γ1 , γ1 ]. In particular, pt can never rise above
γ1 because that would mean melting down all dollars, including those backing the pennies,
and the economy would have no money stock. This regime also solves the exchange rate
indeterminacy problem by administering a peg.
The positive stocks of Rt carried by the mint are socially wasteful, as indeed are
the stocks of silver being used in large coins.
Variants of the standard formula
The preceding version of a standard formula regime omits Cipolla’s stipulation to
‘limit the quantity of the small coins’ in circulation. Though our model renders that stipulation redundant, various writers included them, and insisted on limiting the legal tender
of small coins and strictly limiting the quantity issued, often as supplements – though occasionally apparently as alternatives to pegging the exchange rate et by converting either
coin into the other at par.
The effect of limited legal tender is evidently to modify the cash-in-advance constraint (2), because pennies are no longer accepted in payment of large purchases. Instead,
two separate cash-in-advance constraints are imposed.
28
The ‘standard formula’ without convertibility: the Castilian experience
One striking feature of the ‘standard formula’ and its reserve requirement is the
apparent wastefulness of Rt (in fact, of m1,t + Rt ). We now consider a regime in which the
convertibility requirement is simply removed. As before, the rules for supplying dollars
are identical to the earlier ones. Pennies are token, that is, costless to produce. The
government does not require an inventory of dollars to back the pennies, but sets the
seigniorage rate on pennies σ2 = 1. The firm’s profit is then (29), to be maximized subject
to n1,t ≥ 0 and m1,t−1 ≥ µ1,t ≥ 0. Again, the firm is indifferent to the values taken by
µ2,t and n2,t . Those values are set exogenously by the government, and m2,t follows the
law of motion (6). The government’s budget constraint (8) is
Tt = σ1 n1,t + et n2,t .
From the firm’s no-arbitrage conditions (30), we can bound pt below, by the minting point
γ1 (1−σ1 ); but there is no upper bound on pt : if pt > γ1 , µ1,t = m1,t and dollars disappear,
but pennies remain in circulation, and the cash-in-advance constraint (or quantity theory
equation) determines the price level:
pt (c1,t + c2,t ) = et m2,t−1 .
We start from given initial stocks m1,0 + em2,0 and a constant endowment ξ, such
that γ1 (1 − σ1 ) < (m1,0 + em2,0 )/ξ < γ1 , and consider alternative policies for the path
of the penny stock. Suppose that a shift in endowment occurs as in section 3, so that no
minting or melting of dollars takes place. In the absence of any change in m2,0 a penny
shortage would develop. The shortage can be remedied, or prevented, by an appropriate
increase in m2 , one that raises the ray of slope em2 /m1 in Figure 2 so as to put the
intersection of the expansion path and the resource constraint in the southeastern region.
Issues of pennies can proceed in the absence of any further shifts in endowments.
Suppose ξ remains constant, but the government issues quantities of pennies every period.
29
When constraint (3) isn’t binding, the price level is determined by the binding constraint
(2), and the issue of pennies will lead to a (non-proportionately) rising price level, until pt =
γ1 . At that point, dollars begin to be melted, and successive increases in the penny stock
displace dollars at the constant rate e, maintaining m1,t + em2,t = γ1 ξ. Once all dollars
have been melted, the economy becomes a standard cash-in-advance fiat currency economy,
whose price level is governed by the quantity theory equation (2), namely et m2,t−1 = pt ξ.
Further increases in the penny stock result only in increases in the price level (and in
seigniorage revenues for the government).21
Should a government wish to bring about the return of dollars, it can only do so by
lowering the price level to γ1 (1 − σ1 ). It could do so by acting on m2,t : either by retiring a
portion of m2,t , a costly option envisaged by the Castilian government between 1626 and
1628 and ultimately rejected; or by changing m2,t−1 through a redenomination, the policy
followed in 1628.
The change in m2,t−1 and the consequent change in et have to occur “overnight,” in
an unanticipated fashion, or (14) would be violated. We look for an equilibrium in which
minting of dollars occurs, which requires pt = γ1 (1 − σ1 ). The household enters the period
holding only pennies, so that (2) is γ1 (1 − σ1 )(c1,t + c2,t ) = et m2,t−1 . (Note that, since the
household holds only pennies, (3) cannot be binding). The resource constraint becomes
c1,t + c2,t = ξ − γ1−1 n1,t . The two conditions combine into
γ1 ξ = n1,t + et m2,t−1 / (1 − σ1 ) ,
(31)
a joint condition on n1,t and et . The requirement that γ1 ξ > n1,t > 0 induces an interval
in which et must lie, namely
et ∈
γ1 (1 − σ1 )
0,
pt−1
where pt−1 was the price level prior to the monetary operation.
21
The ratio pt /γ1 can also be thought of as the premium on dollars in terms of pennies.
30
In the following period t1 , neither minting nor melting occurs, because (31) implies
γ1 (1 − σ1 ) ≤
n1,t + et m2,t−1
< γ1
ξ
However, to ensure that (3) does not bind at t + 1, the stock of new dollars n1,t = m1,t+1
must not be too large: m1,t+1 ≤ pt+1 ct+1 , or (c2,t+1 /ξ)n1,t ≤ et m2,t−1 . This places a
further restriction on et , namely:
γ1 (1 − σ1 ) γ1 (1 − σ1 )
c2
et ∈
,
c2 + (1 − σ1 ) ξ
pt−1
pt−1
(32)
where c2 lies on the expansion path and on the feasibility line in Figure 2.
Condition (32) requires the government to engineer a devaluation of pennies of the
“right” extent. If it is too small, no dollars are minted; if it is too large, too many dollars are
minted and another increase in m2 will be required to relieve the binding penny-in-advance
constraint.
We have assumed that the government carries out this operation in an unanticipated
manner. After 1628, the Castilian public surely viewed subsequent manipulations of m2
with some suspicion. Expectations of further reforms altered the demand for pennies, but
our simple model is not equipped to pursue the analysis in that direction.
Fiat Currency
To attain a version of Lucas’s (1982) model of fiat money, we would suspend the
original technology for producing both coins, and let them be produced costlessly. Alternatively, we can think of ‘widening the bands’ – i.e., driving the γi ’s to infinity. Like
Lucas, we would simply award the government a monopoly for issuing coins. We get a
pure quantity theory. The government fixes paths for mit , i = 1, 2, being careful to supply
enough pennies (i.e., to keep (3) from binding). Condition (2) at equality determines the
price level as a function of the total money supply. Condition (3) imposes a lower bound
on the quantity of pennies needed to sustain a fixed e equilibrium; equation (2) imposes
31
an upper bound on the amount of pennies that can be issued via ‘open-market operations’
for dollars.
5. Concluding Remarks
We designed our model to help us understand problems with the arrangements for minting
more or less full-bodied coins that prevailed for centuries throughout Western Europe. Our
model ascribes rules for operating the mint that copy historical ones, and focuses on the
difficulty those rules create for simultaneously maintaining two commodity currencies. The
model extends insights from single-currency commodity money models,22 where minting
and melting points impose bounds within which the price level must stay to arrest arbitrage
opportunities. In those one-commodity money models, when the price level falls enough
(i.e., when currency becomes scarce), new coins will be minted; and when the price level
rises enough, coins will be melted.23
With two currencies, there are distinct melting and minting points for each currency, and this causes trouble. We posit a particular model of demand for coins that, in
conjunction with the two sets of melting and minting points, makes shortages of the smaller
denomination coins arise when national income fluctuates. We analyze the perverse price
adjustments fostered by the historical supply arrangement, how they served to aggravate
shortages over time, and how they left debasement as the preferred relief.
The vulnerability to recurrent shortages of small coins that characterized the historical supply arrangement eventually prompted its repair in the form of a huge ‘onceand-for-all debasement’ of small coins, in the form of a permanent system of token small
change. In that system, the government acts as a monopolist for small coins. It can peg
the exchange rate of small for large coins, either by always choosing a proper quantity of
22
See Sargent and Smith (1997).
Sargent and Smith (1997) analyze two-commodity monies in the form of gold and silver coins, but
do not link these metals to denomination as we do here. In particular, they have no counterpart to our
constraint (3).
23
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small ones, or by maintaining convertibility.
Establishing a system of token small change was a fateful step on the road to
creating a fiat money system for all currency. But refining the idea of fiat money and
actually implementing it were destined to take centuries. Historical episodes, such as
one in Castile in the 17th century, highlight the difficulty of establishing even a token
system for small coins in the face of technological imperfections, and of the pressures that
governments frequently had to raise revenues.
33
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