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.

2

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."

5

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 .

14

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

15

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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