“money and banking in search equilibrium”
TRANSCRIPT
Money and Banking in Search Equilibrium¤
Ping HeUniversity of Pennsylvania
Lixin HuangCity University of Hong Kong
Randall WrightUniversity of Pennsylvania
August 11, 2003
Abstract
This paper develops a new theory of money and banking based on anold story about money and banking. The story is that bankers originallyaccepted deposits for safe keeping ; then their liabilities began to circulateas means of payment (bank notes or checks). We develop models wheremoney is a medium of exchange, but subject to theft, where theft may beexogenous or endogenous. We analyze how the equilibrium circulation ofcash and demand deposits varies with the cost of banking and the outsidemoney supply. We study cases with 100% reserves and with fractionalreserves.
¤We thank Ken Burdett, Ed Nosal, Peter Rupert, Joseph Haubrich, Warren Weber, Fran-cois Velde, Steve Quinn and Larry Neal for suggestions or comments. We thank the NSF andthe Federal Reserve Bank of Cleveland for ¯nancial support. The usual disclaimer applies.
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The theory of banking relates primarily to the operation of commer-cial banking. More especially it is chie°y concerned with the activ-ities of banks as holders of deposit accounts against which chequesare drawn for the payment of goods and services. In Anglo-Saxoncountries, and in other countries where economic life is highly devel-oped, these cheques constitute the major part of circulating medium.EB (1954, vol.3, p.49).
1 Introduction
This paper develops a new theory of money and banking based on an old story
about money and banking. This story is so well known that it is described nicely
in standard reference books like the Encyclopedia Britannica: \the direct ances-
tors of moderns banks were, however, neither the merchants nor the scrivenors
but the goldsmiths. At ¯rst the goldsmiths accepted deposits merely for safe
keeping; but early in the 17th century their deposit receipts were circulating in
place of money and so became the ¯rst English bank notes." (EB 1954, vol.3,
p.41). \The cheque came in at an early date, the ¯rst known to the Institute of
Bankers being drawn in 1670, or so." (EB 1941, vol.3, p.68).1
More specialized sources echo this view. As Quinn (1997, p.411-12) puts it,
\By the restoration of Charles II in 1660, London's goldsmiths had emerged as
a network of bankers. ... Some were little more than pawn-brokers while others1To go into more detail: \To secure safety, owners of money began to deposit it with the
London goldsmiths. Against these sums the depositor would receive a note, which originallywas nothing more than a receipt, and entitled the depositor to withdraw his cash on presen-tation. Two developments quickly followed, which were the foundation of `issue' and `deposit'banking, respectively. Firstly, these notes became payable to bearer, and so were transformedfrom a receipt to a bank-note. Secondly, inasmuch as the cash in question was deposited fora ¯xed period, the goldsmith rapidly found that it was safe to make loans out of his cashresources, provided such loans were repaid within the ¯xed period.
\The ¯rst result was that in place of charging a fee for their services in guarding theirclient's gold, they were able to allow him interest. Secondly, business grew to such a pitchthat it soon became clear that a goldsmith could always have a certain proportion of his cashout on loan, regardless of the dates at which his notes fell due. It equally became safe forhim to make his notes payable at any time, for so long as his credit remained good, he couldcalculate on the law of averages the exact amount of gold he needed to retain to meet thedaily claims of his note-holders and depositors." (EB 1941, vol.3, p.68). See Neal (1994) fora discussion of various other ¯nancial institutions and intermediaries at the time, includingscrivenors, merchant banks, country banks, etc.
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were full service bankers. The story of their system, however, builds on the ¯-
nancial services goldsmiths o®ered as fractional reserve, note-issuing bankers. In
the 17th century, notes, orders, and bills (collectively called demandable debt)
acted as media of exchange that spared the costs of moving, protecting and
assaying specie." Similarly, Joslin (1954, p.168) writes \the crucial innovations
in English banking history seem to have been mainly the work of the goldsmith
bankers in the middle decades of the seventeenth century. They accepted de-
posits both on current and time accounts from merchants and landowners; they
made loans and discounted bills; above all they learnt to issue promissory notes
and made their deposits transferrable by `drawn note' or cheque; so credit might
be created either by note issue or by the creation of deposits, against which only
a proportionate cash reserve was held."2
A story, even a very good story, is not a theory. What kind of a model could
we use to study banks as institutions whose liabilities may substitute for money
as a means of payment? Clearly we need a model where there is a serious role
for a medium of exchange in the ¯rst place. One such framework is provided
by recent work in monetary economics based on search theory. Of course, we
would need to relax the severe assumptions made in most of these models,
such as having all interaction take place in an anonymous bilateral matching
process, to even contemplate something like a bank. To this end we adopt certain
aspects of the framework in Lagos and Wright (2002), where agents interact in
decentralized markets at some points in time and in centralized markets at other
points in time. At the same time, there would have to be some friction over and
above those in the typical search-based model to generate a role for a second2According to Quinn (2002), e.g., \To avoid coin for local payments, Renaissance mon-
eychangers had earlier developed deposit banking in Italy, so two merchants could go to abanker and transfer funds from one account to another." It seems however that these earlierdeposit banks did not really provide checking services: transferring funds from one account toanother \generally required the presence at the bank of both payer and payee." (Kohn 1999b).
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means of payment, in addition to cash.
The historical literature gives us guidance here by telling us that money was
actually less than the ideal means of payment. Among other things, coins were
in short supply, were hard to transport, got clipped or worn, and could be lost or
stolen. With the expansion of commerce the need for a better means of payment
led to ¯nancial innovation, and in particular, the emergence of deposit banks
and bills of exchanges that reduced the need for actual cash (Kohn 1999a,b,c).
The idea presented here is exactly this: cash (outside money) is imperfect and
hence there arises a role for an alternative means of payment, which will be
¯lled by the demandable debt of banks (inside money). The sense in which
cash is imperfect in our model is that it is subject to theft while bank deposits
are relatively safe. More generally our focus is on banks' provision of payment
services, which today is important as ever, given recent moves towards hopefully
safer or more convenient services, including debit cards, electronic money, etc.
We hope this will provide not only a fresh perspective on banking, but will
also help to extend the search models to include some interesting institutions
other than simply money.3 We like to think of this alternative institution {
these deposits in ¯nalcial intermediaries { as representing in a stylized but rea-
sonable way modern checking accounts, or perhaps traveler's checks, because
in addition to their relatively high degree of acceptability a key feature is their
safety. This seems consistent with the historical view of the services provided
by banks or more generally by bills of exchange which were not payable to the
bearer. Although in principle other advantages to alternative assets (portability,
divisibility, recognizability, etc.) could be easily considered in related models,
3Previous work that tries to incorporate some notion of banks into search-theoretic modelsincludes Cavalcanti and Wallace (1999a,b), Cavalcanti et al. (2000), and Williamson (1999).Models that study the interaction among alternative media of exchange, inaddition to theabove, include Matsuyama at al. (1993), Burdett et al. (2001), and Peterson (2002).
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and from this one could get a theory of bank notes, we emphasize safety as
the key advantage of bank deposits in the present model and hence think of it
mainly as a theory of checking accounts.4
The rest of the paper is organized as follows. Section 2 presents the basic
assumptions of the model. Section 3 analyzes the simplest case, where theft
is exogenous, money and goods are indivisible, and we impose 100% reserves.
Section 4 endogenizes theft. Section 5 considers divisible goods. Section 6
relaxes the 100% reserves requirement. Section 7 concludes. We cannot in the
interests of space attempt a review of the large literature on banking, almost
all of which is (surprisingly?) pretty far removed from the microfoundations of
monetary economics into which the current paper ¯ts; the interested reader is
referred to the extensive survey by Gorton and Winton (2002).
2 Basic Assumptions
The economy is populated by a [0; 1] continuum of in¯nitely-lived agents. Ini-
tially, a fraction M 2 [0; 1] of the population are each endowed with one unit
4It may be useful to de¯ne some terms at this point. A bill of exchange is an orderin writing, signed by the person giving it (the drawer), requiring the person to whom it isaddressed (the drawee) to pay on demand or at some ¯xed time a given sum of money toa named person (the payee), or to the bearer. A cheque is a particular, form of bill, wherea bank is the drawee and it must be payable on demand. Quinn (2002) also refers to billsas \similar to a modern traveler's check." It is clear that safety was and is a key featureof checks. For one thing, the payee needs to endorse the check, so no one else can cash itwithout committing forgery; other features include the option to \stop" a check or \cross" it(make it payable only on presentation by a banker). Indeed, the word \check" or \cheque"originally signi¯ed the counterfoil or indent of an exchequer bill on which was registered thedetails of the principal part to reduce the risk of alteration or forgery; the check or counterfoilparts remained in the hands of the banker, the portion given to the customer being termed a\drawn note" or \draft."
We also note here that, while declining, checks are still the most common means of paymentin the US. According to the Philadelphia Inquirer (Feb.14, 2003, p.A1), \There has been a20% drop in personal and commercial check-writing since the mid 1990s, as credit cards,check cards, debit cards, and online banking services have reduced the need to pay withwritten checks. [But] Checks are still king. The latest annual ¯gures from the Federal Reserveshow 30 billion electronic transactions and 40 billion checks processed in the United States."Moreover, \While credit cards reduce the number of checks that need to be written in retailstores, the credit-card balance still has to be paid every month. And, at least for now, that isusually done by mail { with a paper check."
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of money, which is an object that is consumed or produced by anyone but has
potential use as a means of payment (for simplicity if not historical accurracy,
money here is most easily interpretted as ¯at, although it would not be espe-
cially hard to redo things in terms of commodity money, say following Kiyotaki
and Wright [1989] or Velde et al. [1999]). To begin, in this section we follow the
early literature in the area and assume that goods as well as money are indivisi-
ble and that agents can store at most one unit of money at a time. While there
have been many recent advances in monetary theory that do away with these
restrictive assumptions, there is no doubt that even such simplistic models yield
insights into monetary economics, and they have some advantages in terms of
tractability; hence, we want to study this case ¯rst.
Agents want money not for its own sake but to use as a medium of exchange
in a random matching process where specialized nonstorable goods are traded.
As mentioned above, these goods are indivisible here, but this is relaxed below.
As they are nonstaorable, the goods are produced for immediate trade and
consumption. Trade is di±culy because in the random trading process there
is a standard double coincidence problem: only a fraction x 2 (0; 1) of the
population can produce a specialized good that you want. A meeting where
someone can produce what you want is called a single coincidence meeting;
for simplicity, there are no double coincidence meetings, so we can ignore pure
barter, but it would not be hard to relax this. Consuming a specialized good
that you want conveys utility u. Producing a specialized good for someone else
conveys disutility c < u.
Although money is potentially useful as a means of payment, it is unsafe;
it can be stolen. We assume for simplicity that goods cannot be stolen (you
cannot force anyone to produce). Also, because each individual can store at
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most one unit of money, only individuals without money steal { so it is truly
money that is the root of all evil here. In each meeting with an agent holding
cash in the exchange process you attempt to steal it with probability ¸, which
is exogenous for now, but this will be endogenized below. Given that you try
to steal it, with probability ° you are successful. Theft has a cost z < u. We
allow for z ¡ c and ° ¡ x to be positive or negative. Thus, crime can be more
or less costly than honest trade, and could be more or less likely to succeed. Of
course, if z · c and ° ¸ x then crime would dominate honest work, since it is
easier and more likely to pay o®, while if c · z and x ¸ ° then honest work
dominates. If z < c and x > ° , for instance, there is a trade o® { crime is less
costly by has a lower probability of success.
In general we need to introduce notation to distinguish between di®erent
types of money. The ¯rst type is outside money, the cash that can be stolen.
We assume that the M units of money with which agents are initially endowed
is this unsafe outside money. Each one can, however, potentially deposit his
cash in a bank in exchange for a checking account that is perfectly safe. As
we said above, one way to think of these assets is that they are like traveller's
checks: they are signed by the agent when he gives up his cash, and must be
signed a second time when exchanged for goods; if not endorsed in this way,
the bank is under no obligation to redeem the check for cash. Even if agents
could potentially steal checks, they would be of no value as long as we rule out
forgery. Hence, checks are safe in equilibrium.
Following Lagos and Wright (2002), each period before the random trading
process begins, agents are all together in a centralized location. In this ¯rst
subperiod, say in the morning, agents cannot produce the specialized goods that
will be traded in the random matching market that convenes that afternoon.
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However, they can produce and consume a so-called general good, which for
simplicity conveys linear utility (this can be relaxed without di±culty for our
purposes). Thus, consuming Q units of general good conveys utility Q and
producing Q units conveys disutility ¡Q. General goods are perfectly divisible,
but they are produced only in the morning and are nonstorable, so that they
cannot be used to trade for special goods in the random matching market that
convenes in the afternoon. Moreover, since afternoon meetings are anonymous
agents cannot trade bilateral promises to exchange specialized goods for general
goods to be delivered next morning. Any interest or fees at the bank will be
settled in terms of general goods.
Later, agents (but not banks) meet in the afternoon market's anonymous
bilateral matching market. Anonymity implies that agents will never surrender
goods without getting something tangible in return (Kocherlakota 1998; Wallace
2001). Hence in this market we must have quid pro quo. In principle, this can
be accomplished with cash or a check, since in practice a check is as good as
cash because banks credibly commit to redeeming the former as long as it has
been signed by the depositor. We simply assume these commitments by banks
are enforced, say by legal authorities, but it is not hard to imagine reputation
playing such a role. In any case, as there may be both inside and outside money
circulating, we let M0 be the total amount of cash and M1 the total amount of
cash plus checking deposits.
3 Exogenous Theft
In this section we study the model where stealing is exogenous: if agent i with
money meets agent j without money, j will simply try to rob i with some ¯xed
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probability ¸.5 When j tries to steal i's money, with probability ° he succeeds,
while with probability 1¡° he fails and must leave empty handed. With proba-
bility 1¡¸, j does not try to rob i but rather acts as a \legitimate businessman"
{ which means that, if he can produce a specialized good that i likes, they can
trade. We ¯rst study the case where the only asset is money, which yields a
simple extension of the most basic search model of monetary exchange (e.g.
Kiyotaki and Wright 1993). After studying this case, we introduce banking.
3.1 Money
Throughout the paper we use V1 to represent the value function of an agent
with 1 unit of money and V0 the value function of an agent with no money. We
have the °ow Bellman equations
rV1 = (1 ¡ M)(1 ¡ ¸)x(u + V0 ¡ V1) + (1 ¡ M)¸°(V0 ¡ V1)
rV0 = M (1 ¡ ¸)x(V1 ¡ V0 ¡ c) + M¸°(V1 ¡ V0 ¡ z);
where r is the rate of time preference. For example, the ¯rst equation says that
the °ow value to holding money rV1 is the sum of two terms: the ¯rst is the
probability of meeting someone without money who does not try to rob you
and can produce a good you like, (1 ¡ M )(1 ¡ ¸)x, times the gain from trade
u + V0 ¡ V1; the second is the probability of meeting someone without money
who robs you, (1 ¡ M)¸° , times the loss V0 ¡ V1.
We are interested in monetary equilibria (there are always nonmonetary
equilibria that are ignored from now on). The incentive condition for agents to
produce in order to acquire money is V1¡V0¡c ¸ 0. Note that we do not impose
the symmetric condition for stealing, V1 ¡ V0 ¡ z ¸ 0 since, as we said, theft is5One could also assume that a certain fraction of the population always attempt to rob
the people they meet with money and the rest never do, but it is slightly easier to assumeeveryone does so with some probability.
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exogenous for now: with probability ¸ agents cannot help themselves and must
act like criminals. However, do impose participation constraints V0 ¸ 0 and
V1 ¸ 0, since we allow agents to drop out of the economy and live in autarky with
a payo® normalized to 0. It is clear from the incentive condition V1¡ V0 ¡ c ¸ 0
that the binding participation constraint will be V0 ¸ 0. Hence, a monetary
equilibrium here simply requires that when agents participate and produce in
exchange for money { i.e. that the conditions V0 ¸ 0 and V1 ¡ V0 ¡ c ¸ 0 both
hold.
In order to describe the regions of parameter space where these conditions
hold, and hence where a monetary equilibrium exists, de¯ne
CM =(1 ¡ M)(1 ¡ ¸)xu + M¸°zr + (1 ¡ M)(1 ¡ ¸)x + ¸°
CA =(1 ¡ M )[¸° + (1 ¡ ¸)x]u
r + (1 ¡ M)[¸° + (1 ¡ ¸)x]¡ ¸°z
(1 ¡ ¸)x:
Figure 1 depicts CM and CA in (x; c) space using the properties in the following
Lemma (we omit the easy proof).
Lemma 1 (a) x = 0 ) CM = M¸°zr+¸° , CA = ¡1. (b) C0
M > 0, C0A > 0. (c)
CM = CA i® (x; c) = (x¤; z), where x¤ = [r+(1¡M)¸°]z(1¡M)(1¡¸)(u¡z) .
We can now verify the following.
Proposition 1 Monetary quilibrium exists i® c · minfCM ; CAg.
Proof: Subtracting the Bellman equations and rearranging implies
Vm ¡ V0 =(1 ¡ ¸)x[(1 ¡ M )u + Mc] + ¸Mz
r + (1 ¡ ¸)x + ¸:
Algebra implies V1 ¡ V0 ¡ c ¸ 0 i® c · CM and V0 ¸ 0 i® c · CA . ¥
Naturally, monetary equilibrium is more likely to exist when c is lower or x
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Figure 1:
bigger.6 Also notice that either of the two constraints c · CM and c · CA may
bind. It is easy to see that welfare, as measured by average utility
W = MV1 + (1 ¡ M)V0 = M(1 ¡ M ) [(1 ¡ ¸)x(u ¡ c) ¡ ¸°z] ;
is decreasing in ¸ and ° . However, the welfare cost of crime here is due to
either the pure resource cost z or the opportunity cost of thieves not producing;
stealing per se is a transfer not an ine±ciency. In any case, when ¸ = 0, so that
CM = CA = (1¡M)xur+(1¡M)x , we essentially have the model in Kiyotaki and Wright
(1993). Allowing money to be subject to theft provides an simple but not
unreasonable extension of the basic search-based model of monetary exchange.
We now show it can be used to build a theory of banks.
6One might also expect @CM=@¸ < 0, but actually this is true i® z < ~z = (1¡M)(r+°)xuM [r+(1¡M)x)]° ;
thus, for large z, when ¸ increases agents are more willing to acept money. This is because¸ measures not only the probability of being robbed but also the probability of trying to robsomeone else; when z is very big, if this probability goes up agents are more willing to workfor money to keep themselves from crime. This e®ect goes away when ¸ is endogenized.
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3.2 Banking
We now introduce institutions where agents can make deposits into accounts
on which they can write checks. Checks are safe by assumption: thieves cannot
steal them, or, if they could, checks are redeemable unless signed.7 In this sec-
tion we assume 100% reserve requirements: a bank simply keeps your money
in its vault and earns revenue by charging a fee Á for this service, paid each
morning in terms of general goods. We assume the banking industry is compet-
itive and banks incur constant cost a to manage each depositor's account. As
a result, Á = a and bank pro¯ts are 0 in equilibrium. Let µ be the probability
an agent with money decides each morning to put his money in the bank (or,
if he already has an account, to not withdraw it). Let M0 = M(1 ¡ µ) and
M1 = M0 + Mµ = M . Let Vm be the value function of an agent trying to buy
with cash, and Vd the value function of an agent with money in his checking
account, exclusive of the fee a. Hence, for an agent with 1 dollar in the morning,
V1 = maxfVm; Vd ¡ ag.
Although checks will be perfectly safe, it facilitates the discussion for now
to proceed more generally and let °m and °d be the probabilities that one can
successfully steal from someone with money and from someone with a checking
account. Bellman's equation for an agent with asset j 2 fm; dg is then8
rVj = (1 ¡ M )(1 ¡ ¸)x(u + V0 ¡ V1) + (1 ¡ M )¸°j (V0 ¡ V1) + V1 ¡ Vj:
7A model could have equilibria where stolen checks, even if not redeemable, still circulateas a kind of bizzare ¯at money. We ignore this.
8The value of entering the decentralized market with asset j is
Vj =1
1+ r[(1¡M)(1¡ )x(u+ V0) + (1 ¡M)¸°jV0 + ³V1]
where ³ = 1 ¡ (1 ¡M)(1 ¡ )x¡ (1 ¡M)¸°j. The ¯rst term is the expected payo® froma trade, the second is the expected payo® from a theft, and the third is the expected payo®from leaving the market with an asset worth 1 dollar that can be used as cash or depositednext period. Multiplying by 1+ r and subtracting Vj from both sides yields the equation inthe text.
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Notice the last term disappears if V1 = Vj ; e.g., if there is no banking this
reduces to the equation for V1 = Vm in the previous section. For an agent
without money we have
rV0 = (1 ¡ ¸)Mx(V1 ¡ V0 ¡ c) + ¸ [M0°m + (M ¡ M0)°d ] (V1 ¡ V0 ¡ z):
In principle we could allow a thief to steal from a money holder or depositor with
any probability °m or °d, but to reduce notation and little loss in generality we
set °m = ° and °d = 0.
Bellman's equations can be rewritten
rVm = (1 ¡ M)(1 ¡ ¸)x(u + V0 ¡ V1) + (1 ¡ M)¸°(V0 ¡ V1) + V1 ¡ Vm
rVd = (1 ¡ M)(1 ¡ ¸)x(u + V0 ¡ V1) + V1 ¡ Vd
rV0 = (1 ¡ ¸)M x(V1 ¡ V0 ¡ c) + ¸M0°(V1 ¡ V0 ¡ z):
We also have V1 = maxfVm; Vd ¡ ag = µ(Vd ¡ a) + (1 ¡ µ)Vm, from which it is
clear that
µ = 1 ) Vd ¡ a ¸ Vm; µ = 0 ) Vd ¡ a · Vm; and µ 2 (0; 1) ) Vd ¡ a = Vm:
Equilibrium satis¯es this condition, plus the incentive condition for money to
be accepted, V1 ¡ V0 ¡ c ¸ 0, and the participation condition, V0 ¸ 0.
To characterize the parameters for which di®erent types of equilibria exist,
de¯ne
C1 = ¡ (1 ¡ M)uM
¡ ¸°z(1 ¡ ¸)x
+[r + (1 ¡ ¸)x + ¸° ]a
M (1 ¡ M)¸(1 ¡ ¸)°x
C2 =(1 ¡ M )(1 ¡ ¸)xu ¡ ar + (1 ¡ M)(1 ¡ ¸)x
C3 = ¡ (1 ¡ M)uM
+[r + (1 ¡ ¸)x + (1 ¡ M )¸°]a
M(1 ¡ M)¸(1 ¡ ¸)°x
where a = (1 + r)a. Figure 2 show the situation in (x; c) space for two cases,
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z < C4 and z > C4, where
C4 =a
(1 ¡ M)¸°;
and we assume C4 < u, or a < (1 ¡ M)¸°u. The following Lemma establishes
that the Figures are drawn correctly by describing the relevant properties of Cj ,
and relating them to CM and CA from the case with no banks; again the easy
proof is omitted.
Figure 2:
Lemma 2 (a) x = 0 ) C1 = 1 or ¡1, C2 = ¡a=r < 0, C3 = 1. (b) C02 > 0,
C 03 < 0, and C1 is monotone but can be increasing or decreasing. (c) C1 = CM
i® (x;c) = (¹x; C4), C2 = C3 i® (x; c) = (~x; C4), and C1 = CA i® (x; c) = (~x; ~C),
where
¹x =a(r + ¸°) ¡ M(1 ¡ M)¸2°2z
(1 ¡ M)(1 ¡ ¸)[(1 ¡ M )¸°u ¡ a]
~x =a[r + (1 ¡ M )¸°]
(1 ¡ M)(1 ¡ ¸)[(1 ¡ M )¸°u ¡ a]:
(d) ¹x > ~x i® z < C4 < ~C and ¹x < ~x i® z > C4 > ~C .
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We can now prove the following:
Proposition 2 (a) µ = 0 is an equilibrium i® c · min(CM ; CA; C1). (b) µ = 1
is an equilibrium i® C3 · c · C2. (c) µ 2 (0; 1) is an equilibria i® either:
z < C4, c 2 [C3; C1] and c · C4; or z > C4, c 2 [C1; C3] and x ¸ ~x.
Proof: Consider µ = 0, which implies M1 = M0 = M and V1 = Vm. For this
to be an equilibrium we require Vm ¡V0 ¡ c ¸ 0 and V0 ¸ 0, which is true under
exactly the same conditions as in the model with no banks, c · min(CM ; CA).
However, now we also need to check Vm ¸ Vd ¡ a so that not going to the bank
is an equilibrium strategy. This holds i® c · C1.
Now consider µ = 1, which implies M0 = 0 and V1 = Vd ¡ a. In this case,
V1 ¡ V0 ¡ c ¸ 0 holds i® c · C2. We also need V0 ¸ 0, but this never binds
since µ = 1 implies rV0 = M(1 ¡¸)x(V1¡ V0¡ c) ¸ 0 whenever V1 ¡V0 ¡ c ¸ 0.
Finally, we need to check Vd ¡ a ¸ Vm, which is true i® c ¸ C3.
Finally consider µ 2 (0; 1), which implies M0 = M(1 ¡ µ) 2 (0; M) is en-
dogenous and V1 = Vd ¡ a = Vm. The Bellman equations imply
(1 + r)(Vd ¡ Vm) = (1 ¡ M)¸°(V1 ¡ V0):
Inserting Vd ¡ Vm = a and
V1 ¡ V0 =(1 ¡ ¸)x[(1 ¡ M )u + Mc] + M0¸°z
r + (1 ¡ ¸)x + (1 ¡ M + M0)¸°;
we can solve for
M0 = (1¡M)¸(1¡¸)°x[(1¡M)u+Mc]¡[r+(1¡¸)x+(1¡M)¸°]a(C4¡z)(1¡M)¸2°2 :
We need to check M0 2 (0; M ), which is equivalent to µ 2 (0; 1). There are two
cases, depending on the sign of the denominator: if z < C4 then M0 2 (0; M)
i® c 2 (C3; C1), and if z > C4 then M0 2 (0; M) i® c 2 (C1; C3). We also need
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Figure 3:
to check V1 ¡ V0 ¡ c ¸ 0, which holds i® c · C4, and V0 ¸ 0, which holds i®
x ¸ ~x. When ¹x > ~x the binding constraint is c · C4, and when ¹x < ~x the
binding constraint is x ¸ ~x. ¥
Based on the above results, Figure 3 shows the situation when ¹x and ~x are
in (0;1).9 In the case z > C4, shown in the left panel, we always have a unique
equilibrium, which may entail µ = 0, µ = 1, or µ = © where we use the notation
© for any number in (0; 1). In the case z < C4, shown in the right panel, we
may have a unique equilibrium but we may also have multiple equilibria with
µ = 1 and µ = ©, or with all three equilibria. Recall that without banks the9The assumption a < (1¡M)¸°u mentioned above guarantees ~x > 0, and it is easy to see
that ¹x > 0 i®M(1 ¡M)¸2°2z < (r+ ¸°)a. We also have
¹x < 1 i® a < ¹a =(1¡M)2 (1¡¸)°u +M(1¡M)¸2°2z
r + ¸° +(1¡¸)(1¡M)
~x < 1 i® a < ~a =(1¡M)2¸(1¡ )°ur +(1¡M)(1¡¸+ ¸°) :
We do not need ¹x and ~x in (0; 1), but if ~x > 1, e.g., equilibria with µ > 0 disappear.
16
monetary equilibrium exists i® c · minfCM ; CAg. Hence, if it exists without
banks monetary equilibrium still exists with banks, and it may or may not
entail µ > 0. However, there are parameters such that there are no monetary
equilibria without banks while there are with banks. In these equilibria we must
have µ > 0, although not necessarily µ = 1. The important economic point is
that for some parameters, and in particular for relatively large x and c, money
cannot work without banking but it can work with it.
Figure 4 shows how the set of equilibria evolves as a falls. For very large a
banking is not viable, so the only possible equilibrium is µ = 0. As we reduce
a two things happen: in the region where there was a monetary equilibrium
without banks, the nature of the equilibrium changes as some or all of the
agents start using checks; and in some regions where there were no monetary
equilibria without banks a monetary equilibrium emerges. As a falls further
the region where µ = 0 shrinks. As a falls still further we switch from the case
z < C4 to the case z > C4; thus, for relatively small a equilibrium must be
unique, but it could entail µ = 1, µ = 0, or µ = ©. As a falls even further,
eventually there can be no equilibrium with µ = 0, and the only possibility is
that checks will completely drive currency from circulation. Note however that
this is not especially robust: en the next section when we endogenize theft we
will see that banking can never completely drive out money.
An very similar picture of the equilibrium set would emerge if we let M fall,
since reducing M raises the demand for checking services just like reducing a.
One reason for this result is the fact that less cash implies a higher demand for
checking because when M is small the number of criminals (1 ¡ M)¸ is big.
However, we can circumvent this by considering what happens when we vary
M and adjust ¸ to keep (1 ¡ M)¸ constant. The results are shown in Figure 5;
17
Figure 4:
the top row has z > C4 and the bottom z < C4, and in both rows the left panel
has M smaller than the right panel. Lower M makes µ > 0 more likely, but the
reason now is not that lower M implies more criminals; rather, lower M simply
makes money more valuable, which makes you more willing to pay to keep it
safe.10
We close this section with a few details. First, there are other slices of10In any case, the results are consistent with the historical evidence that people are more
likely to use demandable debt as means of payment when money is in short supply (a classicreference is Ashton [1945] while a more recent study is Cuadras-Morato and Ros¶es [1998]).
18
Figure 5:
parameter space that one could use to illustrate the results; for example, the
following ¯gure shows the equilibrium set in (z; a) space for progressively larger
values of c, which will be useful for comparison with one of the generalizations
below where c is endogenous. Second, we want to emphasize that equilibria
with µ 2 (0;1) may be considered particularly interesting because they yield
the concurrent circulation of cash and checks. Obviously, the reason cash and
checks may be both used even though they have di®erent rates of return { i.e.,
19
one has a fee Á = a > 0 and the other does not { is the fact one is safer. In other
words, the model does not display rate of return dominance. Still, we think it
captures something simple but still interesting and historically relevant about
banking.
Figure 6:
20
4 Endogenous Theft
In this section we endogenize the decision to be a thief.11 One reason is purely
generality, but beyond that the model with ¸ exogenous does have some features
that one may wish to avoid for certain issues (e.g. when M goes down the crime
rate mechanically increases). Moreover, one interesting implication of a model
with ¸ endogenous is that checks can never completely drive currency from
circulation: if no one uses cash, then no one will choose crime, but if there is
no crime then cash is perfectly safe and no one would pay for checks. Hence
it is more likely to have concurrent circulation. So, while many points can be
made with ¸ exogenous, we also want to work things out with ¸ endogenous.
Paralleling the previous section, we start with the case where money is the only
medium of exchange and then introduce banks.
4.1 Money
Let ¸ now be the probability an individual without money chooses to be a
thief at the beginning of each period, given beliefs about the choices of others.
In a symmetric equilibrium, Bellman's equation for V1 is the same as before,
repeated here for convenience as
rV1 = (1 ¡ ¸)(1 ¡ M)x(u + V0 ¡ V1) + ¸(1 ¡ M)°(V0 ¡ V1):
Bellman's equations for producers and thieves are now
rVp = M x(V1 ¡ Vp ¡ c) + (1 ¡ Mx)(V0 ¡ Vp)
rVt = M °(V1 ¡ Vt ¡ z) + (1 ¡ M °)(V0 ¡ Vt);
where in this model V0 = maxfVt ; Vpg = (1 ¡ ¸)Vp + ¸Vt.
11The basic strategy for modeling crime follows Burdett et al. (2003).
21
The equilibrium conditions are
¸ = 1 ) Vt ¸ Vp , ¸ = 0 ) Vt · Vp, and ¸ 2 (0; 1) ) Vt = Vp ;
plus the incentive condition V1 ¡ V0 ¡ c ¸ 0. Notice that in a model with ¸
endogenous, we do not have to check the participation constraint V0 ¸ 0: since
an agent can always set ¸ = 0, we have rV0 ¸ Mx(V1¡ V0 ¡ c), which has to be
non-negative as long as the incentive condition holds. The next observation is
that we can never have an equilibrium with ¸ = 1, since we cannot have agents
accepting money when everyone is a thief: if V1 ¡ V0 ¡ c ¸ 0 then
¸ = 1 ) rV1 = (1 ¡ M)°(V0 ¡ V1) · ¡(1 ¡ M)¹°c < 0;
and therefore V1 ¡ V0 ¡ c < 0. Hence, we must have ¸ < 1 in any equilibrium.
De¯ne the following thresholds:
c0 =(1 ¡ M)xu
r + (1 ¡ M)x
c1 =(x ¡ °)(1 ¡ M)xu + °(r + x)z
x[r + (1 ¡ M )x + M° ]
c2 =[r + M x + (1 ¡ M )° ]°z
(r + °)x:
The next Lemma establishes properties of these thresholds illustrated in Figure
4, shown for the case ° > x¤ = rz(1¡M)(u¡z) (in the other case the region labeled
¸ = © disappears). Again we omit the simple proof.
Lemma 3 (a) x = 0 ) c0 = 0, c1 = c2 = 1. (b) c00 > 0, c0
2 < 0. (c) c0 = c1
i® (x; c) = (x¤; z) and c1 = c2 i® (x; c) = (°; z). (d) c0, c1 ! u as x ! 1.
We can now establish the following:
Proposition 3 (a) ¸ = 0 is an equilibrium i® either: x < x¤ and c 2 [0; c0]; or
x > x¤ and c 2 [0; c1]. (b) ¸ 2 (0; 1) is an equilibrium i® x > ° and c 2 [z; c1),
or x < ° and c 2 (c1; z].
22
Figure 7:
Proof: Equilibrium with ¸ = 0 requires Vt ¸ Vp and V1 ¡ Vp ¡ c ¸ 0.
Inserting the value functions and simplifying, the former reduces to c · c1 and
the latter to c · c0. Hence, ¸ = 0 is an equilibrium i® c · minfc0; c1g, and the
binding constraint will depend on whether x is below or above x¤. Equilibrium
with ¸ 2 (0; 1) requires Vt = Vp and V1 ¡ V0 ¡ c ¸ 0. We can solve Vt = Vp for
¸ = ¸¤, where
¸¤ =(° ¡ x)x[(1 ¡ M)u + M c] ¡ (r + x)(°z ¡ xc)
(° ¡ x)(1 ¡ M)(xu ¡ xc + °z):
It can be checked that ¸¤ 2 (0; 1) i® c 2 (c1; c2) when x < ° and ¸¤ 2 (0; 1)
i® c 2 (c2; c1) when x > °. The condition V1 ¡ V0 ¡ c ¸ 0 can be seen to hold
i® c · z when x < ° and i® c ¸ z when x > ° . Hence, ¸ = ¸¤ 2 (0; 1) is
an equilibrium i® c 2 (c1; z] when x < °, and i® c 2 [z; c1) when x > ° . This
completes the proof. ¥
As seen in the ¯gure, a monetary equilibrium is again more likely to exist
23
when c is low or x is high. Given a monetary equilibrium exists, it is more
likely that ¸ = 0 when c is low or x is high, since both of these make honest
production relatively attractive. Given x, it is more likely that ¸ 2 (0; 1) when
c is bigger. When x¤ < ° (e.g., when z is small, as shown in the ¯gure), there
is a region of (x; c) space with x < ° where an equilibrium with ¸ 2 (0; 1) exists
and is unique. Regardless of x¤, as long as c0 > z at x = 1, there exists a
region where equilibrium with ¸ 2 (0; 1) and ¸ = 0 coexist. In the case where
¸ 2 (0; 1) exists uniquely it has the natural comparative static properties, such
as @¸=@c > 0; in other case things are reversed.
An interesting economic aspect of the results is the following. Notice that
in constructing equilibrium with ¸ 2 (0; 1) we need to guarantee that ¸ < 1,
but this condition is actually never binding: for any x, as ¸ increases we hit
the incentive constraint for money to be accepted before we reach ¸ = 1 (when
x < °, we have ¸ < 1 i® c < c2 while V1 ¡ V0 ¡ c ¸ 0 i® c · z; and when x > ° ,
¸ < 1 i® c > c2 while V1¡V0 ¡c ¸ 0 i® c ¸ z). For example, in the region where
¸ 2 (0; 1) exists uniquely, as c increases we get more thieves, but long before
the entire population resorts to crime people stop producing in exchange for
cash and the monetary equilibrium breaks down. In any case, as in the simpler
model with ¸ ¯xed, again the possibility of crime hinders the use of money, and
therefore provides an incentive for agents to ¯nd a safer alternative means of
payment, as we analyze next.
4.2 Banking
With banking, Bellman's equations for Vm and Vd are
rVm = (1 ¡ M)(1 ¡ ¸)x(u + V0 ¡ V1) + (1 ¡ M)¸°(V0 ¡ V1) + V1 ¡ Vm
rVd = (1 ¡ M)(1 ¡ ¸)x(u + V0 ¡ V1) + V1 ¡ Vd ;
24
and V1 = maxfVm; Vd ¡ ag. Again V0 = maxfVp ;Vtg, where
rVp = Mx(V1 ¡ Vp ¡ c) + (1 ¡ M x)(V0 ¡ Vp)
rVt = M0°(V1 ¡ Vt ¡ z) + (1 ¡ M0°)(V0 ¡ Vt):
Equilibrium requires the choice of crime to satisfy
¸ = 1 ) Vt ¸ Vp ; ¸ = 0 ) Vt · Vp; and ¸ 2 (0; 1) ) Vt = Vp ;
and the choice of going to the bank satisfy
µ = 1 ) Vd ¡ a ¸ Vm; µ = 0 ) Vd ¡ a · Vm; and µ 2 (0; 1) ) Vd ¡ a = Vm:
In this section, since things are more complicated, we analyze possible equi-
libria one at a time. In principle, there are nine qualitatively di®erent types of
equilibria, since each endogenous variable ¸ and µ can be 0, 1, or © 2 (0; 1), but
we can quickly rule several combinations.
Lemma 4 The only possible equilibria are: µ = 0 and ¸ = 0; µ = 0 and
¸ 2 (0; 1); and µ 2 (0; 1) and ¸ 2 (0; 1).
Proof: Clearly ¸ = 1 cannot be an equilibrium, as then no one accepts
money. If ¸ = 0 then there are no thieves, so money is perfectly safe and we
cannot have µ > 0. Finally, if µ = 1 then Vt = 0 and so we cannot have ¸ > 0
(for generic parameters). This leaves the possibilities listed above. ¥
Proposition 4 µ = 0 and ¸ = 0 is an equilibrium i® x < x¤ and c 2 [0; c0], or
x > x¤ and c 2 [0; c1].
Proof: Given ¸ = 0, it is clear that µ = 0 is a best response. Hence the only
conditions we need are V1 ¡ V0 ¡ c ¸ 0 and Vp ¸ Vt. With µ = 0 these are
equivalent to the conditions from the model with no banks. ¥
25
Figure 8:
To proceed with equilibria where ¸ > 0, de¯ne
c11 =¡B0 §
pB2
0 ¡ 4A0C0
2A0
where
A0 = °x2[r + (1 ¡ M)x + M° ]
B0 = ¡[(1 ¡ M )°xu + (x ¡ °)a](x ¡ °)x ¡ [2(r + x) ¡ M(x ¡ °)]°2xz
C0 = (x ¡ °)2xua + [(1 ¡ M)°xu + (x ¡ °)a](x ¡ °)°z + (r + x)°3z2:
The function c11 is shown in Figure 8 for both positive and negative values of x,
even thoug x < 1 makes no economic sense. As the solution to a quadratic, c11
generically takes on 2 or 0 values. Notice that c+11 ! 1 as x ! ¡r+M°
1¡M from
below, and c¡11 ! ¡1 as x ! ¡ r+M°
1¡M from above. Also, (c+11; c
¡11) ! (¡1; ¡1)
as x ! 0 from below, and (c+11; c
¡11) ! (1; 1) as x ! 0 from above. Also,
(c+11;c¡11) !
³a
° (1¡M) ; u´
as x ! §1, where the ¯gure is drawn assuming
a < u°(1 ¡ M). Finally, notice in the ¯gure that there is a region of (0; 1)
where there is no solution; this is true for big a, but for smaller a this region
disappears and there are two solutions for all x 2 (0; 1).
26
Figure 9:
This is seen more clearly in Figure 9, which shows c11 as well as c0 and c1
for x 2 [0; 1] and c 2 [0; u]. In the left panel, drawn for large a, c11 does not
exist in the neighborhood of x = ° (for still bigger a, c11 would not even appear
in the ¯gure). As a shrinks, c11 exists for more values of x, and at some point
it does exist in a neighborhood of x = ° as in the middle panel. As a shrinks
further, c11 exists for all x > 0, as in the right panel. One can also show the
following, again assuming u > limx!1 c¡11 = a
°(1¡M ) (the other case is similar).
Lemma 5 (a) for x > ° , if c11 exists then c+11 < c1; for x < ° , if c11 exists
then c1 < c¡11; if c11 exists in the neighborhood of x = ° then c11 = c1 at
(x; c) = (°; z). (b) for x > ° , c+11 ! c1 as a ! 0; for x < ° , c¡11 ! c1 as a ! 0.
Based on these results, we have the following.
Proposition 5 µ = 0 and ¸ 2 (0; 1) is an equilibrium i® either: x > ° , c 2
[z; c1), and c =2 (c¡11; c
+11); or x < °, c 2 (c1; z], and c =2 (c¡
11; c+11).
Proof: In this case the conditions for ¸ 2 (0; 1) are exactly the same as in the
model without banks, but we now have to additionally check Vm ¡Vd +a ¸ 0 to
guarantee µ = 0. Algebra implies this condition holds i® A0c2 + B0c + C0 ¸ 0,
which is equivalent to c =2 (c¡11; c
+11). ¥
27
The region where equilibrium with µ = 0 and ¸ 2 (0; 1) exists is the shaded
area in Figure 9. The economics is simple: in addition to the conditions for
¸ 2 (0; 1) from the model with no banks, we also have to be sure now that people
are happy carrying cash instead of checks, which reduces to c =2 (c¡11; c
+11). For
large a this is not much of a constraint. As a gets smaller we eliminate more of
the region where ¸ 2 (0; 1) is an equilibrium. When a ! 0 the relevant branch
of c11 converges to c1, and this equilibrium vanishes.
We now consider equilibria with µ 2 (0;1) and ¸ 2 (0; 1). To begin, we have
the following lemma.
Lemma 6 If there exists an equilibrium with ¸ 2 (0; 1) and µ 2 (0;1), then
¸ =¡B §
pB2 ¡ 4AC
2A(1 ¡ M )and µ = 1 ¡ x[a ¡ (1 ¡ M)¸°c]
° [a ¡ (1 ¡ M)¸°z]
where A = °xu, B = ¡[(1 ¡ M)°xu + M °xc + (x ¡ °)a], and C = (r + x)a.
Proof: In this equilibrium we have V1 = Vm = Vd ¡ a and V0 = Vp =
Vt . Solving Bellman's equations and inserting the value functions into these
conditions gives us two equations in ¸ and µ. One is a quadratic that can be
solved for ¸ = ¡B§p
B2¡4AC2A(1¡M) . The other gives us µ as a function of the solution
for . ¥
For now, in order to reduce the number of possibilities, we concentrate on the
smaller root ¸ = ¡B¡pB2¡4AC
2A(1¡M) ; it is not impossible to have equilibrium where
¸ = ¡B+p
B2¡4AC2A(1¡M ) , but it only happens for a very small set of parameters. In
order to see when these equilibria exist, de¯ne:
c3 =¡k +
p4(r + x)°xuaM°x
c4 =¡(1 ¡ M)2°xu + [2(r + x) ¡ (x ¡ °)(1 ¡ M )]a
(1 ¡ M)M °x
c5 =[r + Mx + (1 ¡ M)° ]a
(1 ¡ M)M°x
28
c6 =k
[2(r + x) ¡ Mx]°
c7 =k +
¡p
k2 ¡ 4[r + (1 ¡ M)x]°xua2[r + (1 ¡ M )x]°
c8 =¡k + 2(r + x)°z
M x°
c9 =xua ¡ kz + (r + x)°z2
M°xz
c10 =(x ¡ °)k + 2(r + x)°2z
[2(r + x) ¡ M(x ¡ °)]x°
where we let k = (1¡M )°xu+(x¡°)a to reduce notation. We now prove some
properties of the cj's and relate them to c0, c1 and c11, continuing to assume
a < °(1 ¡ M)u.
Lemma 7 (a) x = 0 ) c3 = c4 = c5 = c8 = c9 = c10 = 1 and c6 = ¡ a2r < 0.
(b) c03 < 0, c0
4 < 0, c05 < 0, c0
6 > 0, c08 < 0, c0
9 < 0. (c) c7 ¸ 0 exists i®
x ¸ x7 2 (0; 1); if c7 exists then c¡07 < 0, c+0
7 > 0 and (c+7 ; c¡
7 ) !³
a° (1¡M) ; u
´
as x ! 1; c7 2 (c3; c0) and (c+7 ; c¡
7 ) ! (c0; 0) as a ! 0. (d) c10 = c1 at
(x; c) = (°; z); c6, c8 and c10 all cross at c = z, and c7, c9 and c11 all cross
at c = z, although the values of x at which these crossing occur need not be in
(0; 1). (e) c3 = c6 = c7 at a point where c7 is tangent to c3; c3 = c10 = c11 at a
point where c11 is tangent to c3.
The cj are shown in (x;c) space below for various values of a progressively
decreasing to 0. We use these curves to characterize the existence region for
equilibrium with µ 2 (0; 1) and ¸ 2 (0; 1). The shaded area is the region
where equilibrium with µ 2 (0; 1) and ¸ 2 (0; 1) exist, as proved in the next
Proposition.
Proposition 6 µ 2 (0; 1) and ¸ 2 (0;1) is an equilibrium i® all of the following
conditions are satis¯ed: (i) c > c¸ = maxfc3; minfc4; c5gg; (ii) c > maxfc8; c9g
29
Figure 10:
and either c < c6 or c 2 (c¡7 ; c+
7 ); and either (iii-a) x > ° , c > maxfz; c10g, and
c =2 (c¡11;c
+11), or (iii-b) x < ° and either c > c10 or c 2 (c¡
11; c+11).
Proof: The previous lemma gives us ¸ as a solution to a quadratic equation
and µ as a funciton of , assuming that an equilibrium with µ 2 (0; 1) and
¸ 2 (0; 1) exists. We now check the following conditions: when does a solution
to this quadratic in ¸ exist; when is that solution in (0; 1); when is the implied
µ in (0; 1); and when is V1 ¡ V0 ¸ c?
First, a real solution for ¸ exists i® B2 ¡ 4AC ¸ 0, which holds i® either of
the followng hold:
c ¸ ¡[(1 ¡ M )°xu + (x ¡ °)(1 + r)a] +p
4(r + x)(1 + r)°xuaM°x
= c3
30
c · ¡[(1 ¡ M )°xu + (x ¡ °)(1 + r)a] ¡p
4(r + x)(1 + r)a°xuaM °x
= c03
Second, given that it exists, ¸ > 0 i® B < 0 i®
c >¡(1 ¡ M)°xu ¡ (x ¡ °)(1 + r)a
M °x= c00
3 .
It is easy to check that c3 > c003 > c0
3, and so ¸ > 0 exists i® c ¸ c3.
It will be convenient to let W = V1 ¡ V0. Subtracting the ¯rst two Bellman
equations, we have Vm ¡ Vd = (1¡M)¸°1+r W , and hence by the equilibrium condi-
tion for µ 2 (0; 1), Vm ¡ Vd = ¡a, we have W = a¸(1¡M)° = ¡B+
pB2¡4AC
2(r+x)° after
inserting . We can now see that ¸ < 1 is equivalent to W > a(1¡M)° . Straight-
forward analysis shows this holds i® c > minfc4; c5g. Hence we conclude that
there exists a ¸ 2 (0; 1) satisfying the equilibrium conditions i®
c > c¸ = maxfc3; minfc4; c5gg:
We now proceed to check µ 2 (0; 1) and W ¸ c. Rearranging Bellman's
equations gives us
1 ¡ µ =M0
M=
x(W ¡ c)°(W ¡ z)
.
Hence, we conclude the incentive condition W ¸ c and µ < 1 both hold i®
W > max(c; z). Algebra shows that W > c holds i® c < c6 or c¡7 < c < c+
7 , and
that W > z holds i® c > min(c8;c9).
For the last part, µ > 0 holds i® (x ¡ °)W < xc ¡ °z. When x > ° , µ > 0
is therefore equivalent to W < xc¡°zx¡° , which holds i® c > c10 and c =2 (c¡
11; c+11).
Moreover, notice that when x > °, c < z implies W < xc¡°zx¡° < c, and therefore,
we need c > z as an extra constraint. When x < ° , µ > 0 is equivalent to
W > xc¡°zx¡° , which is equivalent to c > c10 or c¡
11 < c < c+11. This completes the
proof. ¥
31
The following ¯gure puts together everything we have learned in this section
and shows the equilibrium set for decreasing values of a. For very big a the
equilibrium set is just like the model with no banks. As a decreases, equilibria
with µ > 0 emerge, in addition to those with µ = 0 (and either ¸ = 0 or
0 < ¸ < 1), and we also expand the set of parameters for which there exists a
monetary equilibrium. That is, for relatively high values of x and c there cannot
be a monetary equilibrium without banks, because too many people would be
thieves and cash would be too risky; once banks are introduced, for intermediate
a values, agents will deposit their money into safe checking accounts. The fall
in a actually has two e®ects on the value of money: the direct e®ect is that it
makes it cheaper for agents to keep their money safe; the indirect e®ect is that
as more agents opt to do so the number of thieves changes.
The next Figure shows what happens as M increases. Again, the lower is the
supply of M the more likely it is that we observe equilibrium with µ > 0. Clearly
this is not due to the number of theives (1¡M)¸ increasing mechanically, as was
true in the previous section, since ¸ is now endogenous. Finally, we emphasize
that as long as a > 0, no matter how small, we can never have µ = 1 in
equilibrium. Thus, even if banking is almost, free some cash will still circulate
because as more and more people opt for checking the number of criminals goes
down and cash actually becomes very safe. The more general point is that as
the cost of cash substitutes falls there can be general equilibrium e®ects that
make the demand for these substitutes fall relative to the demand for money,
and the net e®ect may be that cash is driven entirely out of circulation.
32
Figure 11:
5 Prices
So far we have been dealing with indivisible goods and money in the decentral-
ized market, which means that every trade is a one-for-one swap. Here we follow
the approach in Shi (1995) and Trejos and Wright (1995) and endogenize prices
by making the specialized goods divisible. We do this here only in the case
where ¸ is ¯xed, although in principle one could also do it with ¸ endogenous.
33
Figure 12:
As before we start with money and later add banking. Without banks, we have
rV1 = (1 ¡ M)(1 ¡ ¸)x[u(q) + V0 ¡ V1] + (1 ¡ M)¸°(V0 ¡ V1)
rV0 = M(1 ¡ ¸)x[V1 ¡ V0 ¡ c(q)] + M ¸°(V1 ¡ V0 ¡ z);
where u(q) is the utility from consuming and c(q) the disutility from producing
q units. We assume u(0) = 0, u(¹q) = ¹q for some ¹q > 0, u0 > 0, u00 < 0 and,
without loss in generality, normalize c(q) = q .
While any bargaining solution could be used, for simplicity we assume the
34
agent with money gets to make a take-it-or-leave it o®er.12 Given c(q) = q, the
take-it-or-leave it o®er will be to trade the cash for q = V1 ¡V0 units of output,
and so
rV0 = M ¸°(V1 ¡ V0 ¡ z):
Rearranging Bellman's equations, we have q = CM (q) where
CM (q) =(1 ¡ M)(1 ¡ ¸)xu(q) + M ¸°z
r + (1 ¡ M)(1 ¡ ¸)x + ¸°
is the same as the threshold CM de¯ned in the model with indivisible goods,
except that u(q) replaces u. We also need to check the participation condition
V0 ¸ 0, which holds i® q · CA(q) where
CA(q) =(1 ¡ M)[ ° + (1 ¡ ¸)x]u(q)r + (1 ¡ M)[¸° + (1 ¡ ¸)x]
¡ ¸°z(1 ¡ ¸)x
is the same as the threshold CA, except that u(q) replaces u. This can be reduced
to q ¸ z, which is actually obvious from the expression rV0 = M¸°(V1 ¡V0¡ z).
Hence, a monetary equilibrium exists i® the solution to q = CM (q) satis¯es
q · CA(q), or equivalently q ¸ z. A particularly simple special case is the one
with z = 0, since the participation condition automatically holds automatically
and there always exists a unique monetary equilibrium q 2 (0; ¹q), as well as
a nonmonetary equilibrium q = 0. Interestingly, for z > 0 a nonmonetary
equilibrium does not exist. The reason is as follows: even if you believe others
give q = 0 for a dollar, you would still be wiling to produce some q > 0 in
exchange for money since this keeps you from stealing, which has a cost if z > 0
and no bene¯t given q = 0; hence 0 is not a ¯xed point. This e®ect would go
away if ¸ were endogenous, however. In any case the equilibrium price level is
p = 1=q. As long as z is not too big, we have @q=@¸ < 0 and @q=@° < 0 { i.e.
more crime means money will be less valuable and prices will be higher.12When ¸ is endogenous, take-it-or-leave o®ers by buyers would be a bad assumption be-
cause it implies sellers gets no gains from trade, and so crime becomes a dominant strategy.
35
With banks, we have
rVm = (1 ¡ M)(1 ¡ ¸)x[u(q) + V0 ¡ V1] + (1 ¡ M)¸°(V0 ¡ V1) + V1 ¡ Vm
rVd = (1 ¡ M)(1 ¡ ¸)x[u(q) + V0 ¡ V1] + V1 ¡ Vm
rV0 = M(1 ¡ µ)¸°(V1 ¡ V0 ¡ z);
where V1 = µ(Vd ¡ a) + (1 ¡ µ)Vm. Equilibrium again requires q = V1 ¡ V0 and
V0 ¸ 0, and now also requires
µ = 1 ) Vd ¸ Vm; µ = 0 ) Vd · Vm; and µ 2 (0; 1) ) Vd = Vm:
Consider ¯rst µ = 0. We need the same conditions as in the model with no
banks, q = CM (q) and q ¸ z, but now we additionally need Vm ¸ Vd ¡ a.
This latter condition holds i® q · C1(q), where C1 is the same as in the model
with indivisible goods except that u(q) replaces u, which can be reduced to
q · C4 = a=(1 ¡ M )¸°. In what follows we will write q0 for the value of q in
equilibrium with µ = 0. Then the previous condition has a natural interpretation
as saying that for µ = 0 we need the cost of banking a to exceed the beni t,
which is the probability of meeting someone who steals your money (1 ¡ M)¸°
times the value of money in this equilibrium q0.
Now consider equilibium with µ = 1. In this case the bargaining solution
q = V1 ¡ V0 implies q = C2(q), where C2 is the same as above except u(q)
replaces u,
C2 =(1 ¡ M )(1 ¡ ¸)xu(q) ¡ a
r + (1 ¡ M)(1 ¡ ¸)x:
For small values of a there are two solutions to q = C2(q) and for large a there
are none. Hence, we require a below some threshold, say a2, in order for there
to exist a q = C2(q) consistent with this equilibrium. Since there is no crime
when µ = 1 the participation condition V0 ¸ 0 holds automatically, but we still
36
need to check Vm · Vd ¡ a. This holds i® q ¸ C3(q), where C3 is the same as
in the model with indivisible goods except that u(q) replaces u. The condition
q ¸ C3(q) here can be reduced to q ¸ C4 = a=(1 ¡ M)¸° . If q1 denotes an
equilibrium value of q when µ = 1, this condition says that we need the cost
of banking a to be less than the beni¯t, which is the probability of meeting
someone who steals your money (1 ¡ M)¸° times the value of money q1. This
requires a to be below some threshold, say a1.
Hence, whenever a · minf a1; a1g, q = C2(q) exists and satis¯es q ¸ C4,
which implies Vm · Vd ¡ a. There can potentially be two di®erent values of
q in the equilibrium with µ = 1, although it is also possible that one solution
to q = C2(q) exceeds C4 while the other does not, in which case there is only
one equilibrium q . Also note that the conditions for µ in the two equilibria
considered so far are not mutually exclusive: µ = 0 requires C4 ¸ q0 and µ = 1
requires C4 · q1, but the equilibrium values q0 and q1 are generally not the
same.
Now consider equilibrium with µ 2 (0; 1). This requires Vm = Vd ¡ a, which
can be solved for q = C4 = a=(1 ¡ M )¸° . As in the model with q ¯xed, we
can now solve for M0 and check M0 2 (0;M ). Recall that with q ¯xed there
were two possibilities that admit M0 2 (0; M ): either z < C4, c 2 [C3;C1] and
c · C4; or z > C4, c 2 [C1; C3] and x ¸ ~x. It is easy to check that when q is
divisible and we use take-it-or-leave-it o®ers the latter possibility will violate the
participation condition V0 ¸ 0. Hence this leaves the other possibility, which is
now summarized as q = C4 > z and
C3(C4) · C4 · C1(C4):
37
6 Fractional Reserves
We now allow banks to lend money. Since we saw above how the model can be
complicated with endogenous prices theft, in this application we revert to the
case where ¸ and q are exogenous. If someone without money borrows from
the bank, he can choose to hold the loan as cash or make a deposit. There is
an up-front payment of ½ when borrowing from the bank; equivalently, we can
reinterpret this as a coupon payment of r½ each period. There is a required
reserve ratio ® set by the government. While there is a cost a for each unit of
deposit service, there is no cost for the loan service but this is easily relaxed.
Banks charge a fee of Á for deposit services, but since they can lend money it
is not neccessarily the case that Á = a; indeed Á can be negative (interest on
checking accounts). It is easy to check that in a competitive market, banks
will lend out as much as possible, therefore the reserve constraint is binding. As
above, M0 is the measure of agents with cash and M1 the measure of agents with
cash or deposits. Now L is the measure of agents with loans, and D the measure
of agents with deposits. We have the following relationships: L + M = M1 and
D + M0 = M1.
Bellman's equations become
rVm = (1 ¡ ¸)(1 ¡ M1)x(u + V0 ¡ V1) + ¸(1 ¡ M1)°(V0 ¡ V1) + V1 ¡ Vm
rVd = (1 ¡ ¸)(1 ¡ M1)x(u + V0 ¡ V1) + V1 ¡ Vd
rV0 = (1 ¡ ¸)M1x(V1 ¡ V0 ¡ c) + ¸M0°(V1 ¡ V0 ¡ z);
where V1 = maxfVm; Vd ¡Ág, and we require V1 ¡V0 ¸ c and V0 ¸ 0, exactly as
above. We now also have the following equilibrium conditions: ¯rst, the interest
rate that clears the loan market is ½ ¸ V1 ¡ V0 with equality if L > 0; and
38
second, zero pro¯t by banks requires a = (1 ¡ ®)r½ + Á, since the left hand side
is per period cost per dollar deposited and the right hand side is per period
revenue (this reduces to the zero pro¯t condition used above, a = Á, when
® = 1). Moreover, the identity L = (1 ¡ ®)D implies, using L + M = M1 and
D + M0 = M1, we have M1 ¡ M = (1 ¡ ®)(M1 ¡ M0) or
M0 =M ¡ ®M1
1 ¡ ®:
Suppose no one makes deposits, M1 = M0 = M . We still require the
incentive and participation conditions, V1¡V0 ¸ c and V0 ¸ 0, which hold under
the same conditions as above c · minfCM ;CAg, and we have to be sure that
the loan market clears at L = D = 0. The demand for loans is 0 if ½ ¸ V1 ¡ V0,
and as in the previous sections deposits are 0 if Vd ¡ Vm · Á = a ¡ (1 ¡ ®)r½
after inserting the zero pro¯t condition. Hence, any ½ between V1 ¡ V0 and
(a + Vm ¡ Vd )=(1 ¡ ®)r will clear the market at L = D = 0, and so we simply
need to check that this interval is non-empty. This is equivalent to checking
Vm ¡ Vd + a ¸ (V1 ¡ V0)(1 ¡ ®)r, which generalizes the equilibrium condition
that no one go to the bank from the model with ® = 1. Algebra implies that
this holds i® c · C1 where
C1 =[r + (1 ¡ ¸)x + ¸° ]a
[(1 ¡ ®)r(1 + r) + (1 ¡ M)¸° ]M (1 ¡ ¸)x¡ (1 ¡ M)u
M¡ ¸°z
(1 ¡ ¸)x
Notice that this reduces to the expression for C1 in the model with no loans
when ® = 1.
Proposition 7 Equilibrium with µ = 0 and M1 = M exists i® c · minfCM ; CA ; C1g.
Now suppose that µ > 0, so that M1 = M=[1 ¡ µ(1 ¡ ®)]. The way to
think about this \money multipler" is exactly as in the elementary textbooks:
of the intitial money supply M a fraction µ is deposited, which means banks
39
can make (1 ¡ ®)µM loans, of which a fraction µ is deposited, which means
banks can make (1 ¡ ®)2µ2M more loans, etc. The total money supply is
M1 = M +(1¡®)µM +(1 ¡®)2µ2M + ::: = M=[1 ¡ µ(1 ¡®)]. Notice that there
is always a demand for these loans because ½ is determined to clear the loan
market. Moreover, Á = a ¡ (1 ¡ ®)r½ is set to keep pro¯t 0. Also, ½ = V1 ¡ V0
must be positive because V1 ¡ V0 ¸ c, and hence Á must be less than a and
possibly even negative; as in footnote 1, \in place of charging a fee for their
services in guarding their client's gold, they were able to allow him interest."
Notice however, that when Á < 0 checks strictly dominate cash { they are safer
and bear interest { and so must drive cash from circulation (µ = 1).
Consider ¯rst the case where µ = 1, which implies Vd ¡Vm ¸ Á. Eliminating
Vd ¡ Vm, this can be rewritten
Á · ¸(1 ¡ M1)°1 + r
½(1 + r)Á ¡ (1 ¡ ¸)x [(1 ¡ M1)u + M1c] ¡ ¸°M0z
r + (1 ¡ ¸)x + ¸°M0
¾:
Similarly, loan market clearing now requires ½ = V1¡V0, and eliminating V1¡V0
implies
½ =(1 ¡ ¸)x [(1 ¡ M1)u + M1c] + ¸°M0z ¡ (1 + r)Á
r + (1 ¡ ¸)x + ¸°M0:
Inserting Á = a¡ (1 ¡ ®)r½ and, becuase we have µ = 1, M0 = 0 and M1 = M® ,
the previous equation implies the loan market equilibrium is
½ =(1 ¡ ¸)x[(1 ¡ M
® )u + M® c] ¡ a
r [®(1 + r) ¡ r] + (1 ¡ ¸)x:
Inserting ½ into the zero pro¯t condition now gives the equilibrium value of
Á, and we can insert these into the above conditoin for Vm · Vd ¡ Á to see that
it holds i®
c ¸ a[r + (1 ¡ ¸)x + ¸(1 ¡ M1)° ](1 ¡ ¸)[r(1 + r)(1 ¡ ®) + ¸(1 ¡ M1)° ]M1x
¡ (1 ¡ M1)uM1
= C3
40
Notice this reduces to the expression for C3 used above in the special case when
® = 1. We also need to check V0 ¸ 0, which as before cannot bind when µ = 1,
and V1 ¡ V0 ¸ c, or equivalently ½ ¸ c, which holds i®
c · (1 ¡ ¸)(1 ¡ M1)xu ¡ a¡r(1 + r)(1 ¡ ®) + r + (1 ¡ ¸)(1 ¡ M1)x
= C2:
This reduces to the expression for C2 used above when ® = 1. Summarizing,
Proposition 8 Equilibrium with µ = 1 and M1 = M=® exists i® c 2 [C3; C2].
Now suppose µ 2 (0; 1), which requires Vd ¡ Vm = Á. Eliminating Vd ¡ Vm
we now get
Á =¸(1 ¡ M1)°
1 + r
½(1 + r)Á ¡ (1 ¡ ¸)x [(1 ¡ M1)u + M1c] ¡ ¸°M0z
r + (1 ¡ ¸)x + ¸°M0
¾;
which is the same the condition for the case µ = 1 except with strict equality.
The loan market clearing conditon ½ = V1 ¡ V0 is the same as in the case µ = 1,
which we repeat for convenience:
½ =(1 ¡ ¸)x [(1 ¡ M1)u + M1c] + ¸°M0z ¡ (1 + r)Á
r + (1 ¡ ¸)x + ¸°M0:
Inserting Á = a ¡ (1 ¡ ®)r½ and, in this case, M0 = M¡®M11¡® , we can reduce
these to two equations in ½ and M1; one can interpret these as a deposit supply
function, say M1 = MS (½), which comes from comparing Vd and Vm, and a loan
demand function, say M1 = MD(½), which comes from comparing V1 and V0.
Eliminating ½ from these supply and demand functions gives us
f (M1) = AmM 21 + BmM1 + Cm = 0;
where
Am = (1 ¡ ®)¸(1 ¡ ¸)°x(u ¡ c) + ®¸2°2z
Bm = (1 ¡ ®)(1 ¡ ¸)f¡¸°xu ¡ [¸° + (1 ¡ ®)r(1 + r)]x(u ¡ c)g
41
¡¸2°2Mz ¡ ®¸2°2z + ¸°a
Cm = [(1 ¡ ®)(1 ¡ ¸)xu + ¸°Mz][¸° + (1 ¡ ®)r(1 + r)]
¡f(1 ¡ ®)[r + (1 ¡ ¸)x] + ¸°(1 ¡ ® + M)ga
We can see Am > 0, so f (M1) is convex. If ® = 1 then
Am = ¸2°2z
Bm = ¡¸2°2Mz ¡ ¸2°2z + ¸°a
Cm = ¸2°2Mz ¡ ¸°M a
We can see M1 = M is a solution, and the other solution is M1 = 1 ¡ a¸°z .
In general we need B2m ¡ 4AmCm ¸ 0 and ¡Bm§
pB2
m¡4AmCm
2Am2 [M; 1).
Moreover, the participation constraint requires V1¡V0 ¸ c, which implies ½ ¸ c.
We can solve for ½, which satis¯es the following quadractic equation:
0 = A½½2 + B½½ + C½
with A½ = ¸°f(1 ¡ ®)[(1 ¡ ¸)x ¡ r2] + ¸°(M ¡ ®)g
B½ = ¸°f(1 + r)a ¡ (1 ¡ ®)(1 ¡ ¸)xc + ¸°(® ¡ M )zg
+r(1 + r)(1 ¡ ®)[(1 ¡ ®)(1 ¡ ¸)x(u ¡ c) + ¸°®z]
C½ = ¡(1 + r)a(1 ¡ ®)(1 ¡ ¸)x(u ¡ c) + ¸°®z
We can see that ½ = ¡B½§p
B2½¡4A½C½
2A½.
From the montonicity of the money supply function, we know that in equi-
librium bigger value of ½ corresponds to bigger value of M1 if there are two
interior solution.
7 Conclusion
We have analyzed some models of money and banking based on explicit frictions
in the exchange process. Although simple, we think these models capture some-
42
thing interesting and historically accurate about banking, and they or at least
various extensions may have something to contribute to our undertanding of
¯nancial institutions much more generally. Certainly, it seems natural to want
to have models in which both money and banking arise in a natural and endoge-
nous way. Some possible extensions include allowing aggregate uncertainty and
hence the possibility of bank runs, something studied in nonmonetary banking
theory extensively since Diamond and Dibvyg (see the Gorton and Winton sur-
vey). Also, since modern banks actually perform a wide varierty of functions, it
may be interesting to try to study these in the context of the model (e.g. del-
egated lending, monitoring, etc.). A wide variety of applications seem feasible.
The goal here was to provide a ¯rst pass at a natural and fairly simpler class
of models where money and banking arise endogenously and in accord with the
historical record.
43
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