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FEDERAL RESERVE BANK OF CLEVELAND
pd
papers
NUMBER 14
By Ed Nosal and Guillaume Rocheteau
POLICY DISCUSSION PAPER
The Economics of Payments
FEBRUARY 2006
POLICY DISCUSSION PAPERS
FEDERAL RESERVE BANK OF CLEVELAND
The Economics of Payments
Ed Nosal is a senior
research advisor at the
Federal Reserve Bank of
Cleveland, and Guillaume
Rocheteau is an
economist at the Bank.
By Ed Nosal and Guillaume Rocheteau
In this paper we provide a survey of the payment literature in a unified framework. The
environment is a variant of the Lagos and Wright (2005) model of monetary exchange,
where some trades occur in bilateral meetings while others occur in more or less
decentralized markets. We use this basic environment to introduce alternative sets
of trading frictions that give rise to different payments instruments and/or payments
institutions. We investigate credit economies, monetary economies, and economies
in which money and credit coexist. We also study alternative assets, such as foreign
exchange, capital (equity), and government liabilities, which can be used as payment
instruments in conjunction with money. We introduce banks as deposit-taking institutions
whose liabilities circulate in the economy. We also provide an extension in which the
process of the settlement of debt for money is modeled and the potential social costs of
settlement are characterized. Finally, we investigate government policy responses to the
social costs introduced by various trading frictions.
Materials may be
reprinted, provided that
the source is credited.
Please send copies of
reprinted materials to the
editor.
POLICY DISCUSSION PAPERS
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ISSN 1528-4344
FEDERAL RESERVE BANK OF CLEVELAND
Table of Contents
Introduction ............................................................................................................... 1
The Basic Environment .............................................................................................. 2
Pure Credit Economies .............................................................................................. 3
Credit with Commitment ...................................................... 4
Credit with Public Recordkeeping ........................................ 6
Credit with Reputation ........................................................ 10
Related Literature ................................................................. 14
Pure Monetary Economies....................................................................................... 14
Money Is Memory ................................................................ 15
Indivisible Money and Currency Shortage .......................... 17
Indivisible Money and Lotteries .......................................... 20
Divisible Money ................................................................... 21
Related Literature ................................................................. 22
Coexistence of Money and Credit ........................................................................... 23
Value Functions.................................................................... 24
Terms of Trade...................................................................... 25
Equilibrium .......................................................................... 26
Related Literature ................................................................. 27
Alternative Media of Exchange ................................................................................ 28
Money and Capital ............................................................... 28
Dual-Currency Payment Systems ......................................... 31
Government Liabilities as Means of Payment ...................... 32
Related Literature ................................................................. 35
Banking .................................................................................................................... 36
Banks and Safekeeping Services .......................................... 36
Private Money ...................................................................... 38
Related Literature ................................................................. 41
Settlement ................................................................................................................ 41
The Environment ................................................................. 42
Frictionless Settlement......................................................... 43
Settlement and Liquidity ...................................................... 45
Settlement and Default Risk................................................. 48
Related Literature ................................................................. 50
Policy and Payments ................................................................................................ 50
Optimality of the Friedman Rule ......................................... 50
Trading Frictions and the Friedman Rule ............................ 52
Distributional Effects of Monetary Policy ............................ 55
Settlements and Monetary Policy ........................................ 56
Related Literature ................................................................. 58
References................................................................................................................ 59
FEDERAL RESERVE BANK OF CLEVELAND
Introduction
Economics is all about exchange, but exchange need not be seamless. How else can one
explain the existence of the myriad assets and institutions—domestic currency, bank deposits, bonds, capital, large value payments systems, such as Fedwire and CHIPS, foreign
exchange and the foreign exchange market, to name but a few—whose main purpose
is to facilitate trade? In the real world, trade between agents is not conducted in a frictionless environment, as, for example, it is in an Arrow–Debreu economy. Instead, real resources must be used in order for exchange to take place at all and, as a result, people
will attempt to design instruments of exchange that will economize on resource use.The
precise instrument or institution that one might use will depend upon the obstacles or
frictions that agents face in a particular trade. So, in our view, the existence of trading frictions implies that agents will use some sort of instrument, asset, or institution to facilitate
trade, and the instrument, asset, or institution that is actually used will depend upon the
nature of the trading friction(s) that agents face.
There does not really exist a well-defined literature on the economics of payments.
There are, of course, comprehensive literatures on credit, money, foreign currencies,
banking, and so on. But, by and large, these literatures have evolved independently of one
another and may have quite different focuses. For example, in the banking literature, there
are large bodies of works on bank runs and optimal lending contracts, but little time has
been spent on banks’ liabilities as a medium of exchange.An implication of this independent development is that the economic environments in these various literatures are not
necessarily comparable. Indeed, even within a literature there is considerable variation
in the specification of economic environments.This lack of comparability or a common
model environment is problematic if one is interested in understanding, for example, why
one set of payment instruments might emerge in one situation and not another. In this
Policy Discussion Paper, a common economic environment is used, one that models exchange between agents. Within this common economic environment, alternative sets of
trading frictions are introduced, where different sets of frictions may give rise to different
payments instruments and/or payments institutions. The benefit of our approach is that
one will be able to associate the particular trading frictions with payment instruments.
Government policy can be thought of in terms of attempting to counteract these trading
frictions, either directly or indirectly.
The paper is organized as follows. We begin by describing the common model environment that will be used throughout the paper; it is a variation of the Lagos–Wright
model of exchange.We then proceed to consider alternative sets of trading frictions that
give rise to different payments instruments. First we investigate the use of only credit
in exchange, then the use of only money. In reality, both money and credit are used to
facilitate exchange, so we next combine the trading environments of the previous sec1
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
tions to explain the coexistence of money and credit in trade.Although money seems to
be central to exchange, in practice money as well as other assets are used in trade. We
next study alternative assets, such as foreign exchange, capital (equity) and government
liabilities that can be used as payments instruments in conjunction with money. Banks
are intermediaries and have long been considered facilitators of trade, so we proceed
to show how a deposit-taking institution, whose liabilities circulate in the economy, can
be a useful institution of trade. In practice, virtually all debt is settled with some form of
money. But the very act of settlement can introduce additional frictions into a trading
environment. So we model the process of settlement of debt for money and characterize the potential social costs of settlement. Finally, we investigate government policy
responses to the social costs introduced by the various trading frictions that have been
discussed in previous sections.
The Basic Environment
We consider a simple model to describe different payments methods to carry out trades.
The benchmark model is as follows:Time is discrete and continues forever. Each period
is divided into two subperiods, day and night. During the day, trades occur in decentralized markets according to a bilateral matching process. Each agent meets a trading partner with probability σ . During the day, some agents can produce but do not want to
consume, while other agents want to consume but cannot produce. We call the former
agents buyers and the latter sellers, and the measures of buyers and sellers are normalized
to one.This generates a simple double-coincidence-of-wants problem in the decentralized
market. The double-coincidence-of-wants problem can be exacerbated by, for example,
having sellers produce different kinds of goods and having buyers wanting to consume
only certain types of goods. All this can be captured by the parameter σ . Below, we will
be explicit in terms of the lack of double coincidence problem.We will call the good that
is produced and traded during the day the search good, since buyers and sellers are randomly matched and trade is decentralized.
Exactly how production and trade are organized at night will depend upon the issue
that is under investigation. What can be said about the night market is that, in general, it
will be characterized by fewer frictions than those that plague the morning market. At
night, all agents can produce and consume.The good that is produced and consumed at
night will be called the general good.
Search goods can only be produced during the day and general goods can only be
produced at night.All goods, whether produced in the day or at night, are nonstorable, so
a search good cannot be carried over to the night and a general good cannot be carried
over to the next day.
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FEDERAL RESERVE BANK OF CLEVELAND
The period utility functions for buyers and sellers are given by
(1)
U b ( q, x ) = u( q ) + x ,
(2)
U s ( q , x ) = −c ( q ) + x ,
where q is the quantity of the search good consumed and produced during the day, and
x is the net consumption—the difference between what is consumed and produced—
of the general good at night. We assume u′( q ) > 0, u′′( q ) < 0, u( 0 ) = c ( 0 ) = c ′( 0 ) = 0,
c ′( q ) > 0, c ′′( q ) > 0, and c ( q ) = u( q ) for some q > 0. All agents discount between the
night and the next day at rate r = β −1 − 1, where β ∈ ( 0,1). Lifetime utility for agent
i, i ∈ {buyer,seller }, at date j is given by Σt∞= j β t − jU i ( q, x ). Let q * denote the efficient
(static) level of production and consumption of the search good, where q * is the solution to u′( q *) = c ′( q *). Note that the linear specification goods produced and consumed
at night implies that there is no benefit associated with producing the general good for
one’s own consumption.
We define a trade match to be a match between a buyer and seller during the day,
where the buyer wants the good that the seller can produce, the seller is actually able to
produce the good, and the buyer has the resources to pay for the good. We assume that
agents always have the option to “exit” or not participate in any particular market or markets, but can always return.
Generally speaking, we adopt the benchmark model specification in all of the discussions that follow.Without exception, the day market, with its bilateral matching of agents
and its lack of a double-coincidence-of-wants problem, will be common to all of the environments discussed below. At times, however, we may depart slightly from the specification of our benchmark model.When this does happen, we will be very clear in explaining
both how and why we are modifying the benchmark model.
Pure Credit Economies
In this section, we consider environments with the following characteristics: First, a
match between a buyer and a seller that is formed during the day is maintained at night.
The fact that agents are matched for the entire period allows them to make promises or
“write” debt contracts during the day, which can be settled at night. Second, there are
no frictions or costs associated with settling debt at night:An agent can settle his debt at
night by simply producing the general good and transferring it to his creditor.Third, there
are no assets, e.g., money, that agents can use for trade purposes. So agents can only trade
by using credit arrangements. An environment that satisfies these characteristics will be
referred to as a credit economy.We are interested in characterizing the set of allocations
that can be implemented as an equilibrium in a credit economy.
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POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
The extent to which a particular allocation can be implemented in a credit economy
depends on the degree of commitment that agents possess, as well as methods available
for punishing a debtor who reneges on his obligations. We will consider three related
environments in which credit arrangements can be sustained.We first assume that agents
have the ability to commit to repay their debts.We then consider an environment where
agents cannot commit but there exists a public recordkeeping device that can monitor
agents’ production levels. Finally, we assume that agents are able to form long-term partnerships, where trading relationships can be sustained by reputations.
Credit with Commitment
We first consider an economy where buyers can commit to repay their debts. Since commitment may be interpreted as a rather strong assumption, we will limit the extent to
which agents (and, in particular, buyers) are able to commit. We assume that in the day
market, buyers can promise to undertake future actions, but only for the subsequent
night market.This assumption implies, among other things, that private debt will not circulate across periods.
We will describe the set of allocations that are feasible in the sense that they are in accordance with agents’ willingness to trade.We restrict the set of allocations to those that
are symmetric across matches and that are constant across time.To find these allocations,
we assume the following simple trading mechanism:When a match is formed during the
day, the buyer and seller must consider implementing the allocation (q,y), where q is the
quantity of the search good produced by the seller for the buyer in the day and y is the
amount of the general good that the buyer promises to produce and deliver to the seller
at night. When a buyer and seller are in a trade match, they must decide simultaneously
whether to accept or reject allocation (q,y).This allocation is implemented only if both
agents accept it.
The sequence of events within a typical period is as follows: At the very beginning of
the period, all agents are unmatched. During the day, each agent finds a trading partner
with probability σ .The buyer and seller in a trade match decide simultaneously whether
to accept or reject the proposed allocation (q,y). If either player rejects the proposed
allocation, the match is dissolved; otherwise, the seller produces q units of the search
good for the buyer during the day and the buyer produces y units of the general good
for the seller at night. At the end of the period, all matches are terminated. Without loss,
unmatched agents and matched agents who rejected the proposed allocation simply exit
the night market and re-enter the day market in the next period.
The value function for a buyer evaluated at the beginning of the day market is
V b = σ [u( q ) − y ] + β V b ,
(3)
assuming, of course, that both the buyer and seller accept allocation (q,y). According to
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FEDERAL RESERVE BANK OF CLEVELAND
(3), the buyer meets a seller with probability σ and, in the event that he meets a seller,
he consumes q units of the search good and produces y units of the general good. Note
that there is no state variable that characterizes the lifetime expected utility of a buyer
since agents hold no assets at the beginning of the period, and the agents’ trading histories are irrelevant. In addition, since we focus on stationary allocations, time indexes are
suppressed on variables and functions.The value function of a seller evaluated at the beginning of the day market is
(4)
V s = σ [ −c ( q ) + y ] + β V s .
Equation (4) has an interpretation similar to (3), except for the fact that during the day
sellers produce (and buyers consume) the search good and at night sellers consume (and
buyers produce) the general good.
Since agents are able to commit, the only relevant constraints are buyers’ and sellers’
participation constraints, which are evaluated at the time that a match is formed. The
participation constraints indicate whether agents are willing to participate in a given
mechanism or to go along with a given allocation.These constraints are
(5)
u( q ) − y + β V b ≥ β V b ,
(6)
−c ( q ) + y + β V s ≥ β V s .
According to (5), a buyer will accept allocation (q,y) if the lifetime utility associated with
acceptance, the left-hand side of (5), exceeds the lifetime utility associated with rejection, the right-hand side of (5), or if his surplus from the trade, u( q ) − y, is non-negative.
Condition (6) has a similar interpretation for the seller. From (5) and (6), the set of incentive-feasible allocations, A C , is given by
(7)
{
}
A C = ( q, y ) ∈ R2+ : c ( q ) ≤ y ≤ u( q ) .
From (7) it is easy to check that {q *} × [c ( q *), u( q *)] ⊆ A C . This implies that the efficient
level of production and consumption of the search good, q*, is incentive-feasible for any
values of β and σ . In presence of commitment, the intertemporal nature of the trades
and any associated moral hazard considerations are irrelevant.
Hence, the gains from trade will be maximized if agents produce and consume q *
units of the search good in the day. And the level of output for the general good, y, will
determine how the gains from trade are split between the buyer and seller. But since any
allocation in A C is incentive-feasible, which of these allocations will be implemented?
Can agents somehow agree to produce q * units of the search good? If so, how is y
determined? One way to address these questions is to impose a bargaining procedure
for bilateral matches and to characterize the outcome of the bargaining procedure. For
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POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
example, we can assume that the allocation (q,y) is determined in accordance with the
generalized Nash bargaining solution, where the buyer’s bargaining power is θ ∈ ( 0,1).
The generalized Nash bargaining solution maximizes a weighted geometric mean of the
buyer’s surplus, u( q ) − y, and the seller’s surplus, −c ( q ) + y, from trade, where the
weights are given by the agents’ bargaining power.The generalized Nash bargaining solution is given by the solution to
θ
max [u( q ) − y ]
(8)
q, y
[ y − c ( q )]1−θ .
The solution to (8) is q = q * and y = (1 − θ )u( q *) + θ c ( q *). Note that the allocation is
efficient for any buyer’s bargaining power θ in the sense that the level of the search production is always at the efficient level q * . Furthermore, as one varies θ over [0,1], the
set of generalized Nash bargaining solutions is given by {q *} × [c ( q *), u( q *)]; as θ increases, the buyer’s share of the surplus increases and the seller’s share decreases.
Credit with Public Recordkeeping
In this section, we relax the assumption of commitment. In order to sustain trade in a
credit economy when agents cannot commit, they must experience some sort of negative consequence if they do not deliver on their promises. The punishment that we impose is that an agent can no longer use credit if he fails to pay back his debt obligation.
Furthermore, we will consider global punishments, in the sense that the entire economy
reverts to autarky if at least one agent behaves in an opportunistic manner.
When there is a large number of agents in the economy, there must be some sort of
public recordkeeping of agents’ trades if punishments are to be feasible and effective.We
assume that there exists a public-record device that provides all agents in the economy
with the list of quantities of the search and general goods that were produced and traded
during the period. In particular, the pair (q,y) is recorded for all agents in a trade match,
and this information is made available at the end of each period, i.e., at the end of each
night. Note that the public record lists only quantities, not the names of the agents who
produced the quantities. It is for this reason that any deviation from proposed play will
result in a global punishment. If names were associated with quantities, then nonglobal
(personalized) punishments would be possible. It turns out that very little is changed if
nonglobal punishments are possible, and we discuss the implications of nonglobal punishments at the end of this section.
The chronology of events is as follows:At the beginning of the day market, buyers and
sellers are randomly matched and meet a trading partner with probability σ . In each
trade match, an allocation (q,y) is proposed and agents simultaneously accept or reject
trade at those terms. If the allocation is accepted, then the seller produces q units of the
search good for the buyer. At night, the buyer chooses to either produce y units of the
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FEDERAL RESERVE BANK OF CLEVELAND
general good for the seller or to renege on his promise and produce nothing. At the end
of the night, a list of all the pairs of the day and night production levels for agents who
were in a trade match is publicly observed. Based on this list, agents decide whether or
not to enter the morning search market or exit the day and night markets forever to live
in autarky.
We restrict our attention to symmetric stationary allocations (q,y) that are incentivefeasible. Incentive-feasibility now implies that not only do the buyer and the seller agree
to allocation (q,y), as before, but that the buyer is willing to repay his debt when it is
his turn to produce.We assume that agents choose autarky at the end of the night whenever an allocation from any trade match is different from the proposed allocation (q,y).
(Indeed, having all agents revert to autarky is an equilibrium outcome in this situation.)
During the day, matched sellers and buyers agree to a trade (q,y) if
(9)
−c ( q ) + y + β V s ≥ 0,
(10)
u( q ) − y + β V b ≥ 0.
Condition (9), which is the seller’s participation constraint, says that a seller prefers allocation (q,y) plus the continuation value of participating in future day and night markets,
β V s , to autarky at the time when the match is formed. The seller compares the payoff
associated with acceptance to that of autarky because if the seller rejects the proposal,
a (0,0) trade will be recorded and such a trade will trigger global autarky.The condition
(10) has an interpretation similar to (9) but for the buyer; i.e., the buyer prefers to go
along with the suggested trade (q,y) rather than go to autarky. Note that the participation constraints (9) and (10) differ from the participation constraints when agents could
commit, (5) and (6), in that now agents go to autarky if they do not accept the proposed
allocation (q,y).
Since the buyer produces the general good after he consumes the search good, we
now need to check that the buyer is, in fact, willing to produce. The buyer will have an
incentive to produce the general good for the seller if
(11)
− y + β V b ≥ 0.
The left-hand side of inequality (11) is the buyer’s payoff if he repays his debt by producing y units of output for the seller and the right-hand side is his continuation payoff of
zero if he defaults (since the economy reverts to autarky). Clearly, if the buyer’s incentive
constraint (11) is satisfied, then so is his participation constraint (10).
Equations (3) and (4) still represent the beginning-of-period value functions for the
buyer and seller, respectively. Using these Bellman equations, the seller’s participation
constraint (9) and the buyer’s incentive constraint (11) can be re-expressed, respectively, as
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POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
(12)
−c ( q ) + y ≥ 0
(13)
σ [u( q ) − y]
≥ y,
r
where r = β −1 − 1. Condition (12) simply says that the seller has to get some of the surplus from the trade match to be willing to participate in the trade. Note that this participation condition does not depend on discount factors or matching probabilities. Condition (13) is the incentive constraint for the buyer to repay his debt.The left-hand side of
(13) is the buyer’s expected payoff beginning next period if he pays back his debt; it is
the discount sum of the expected surplus from trade in future periods.This term depends
on the frequency of trades, σ , and on the discount rate, r. The right-hand side of (13) is
the buyer’s (lifetime) gain if he does not produce the general good for the seller. Not surprising, because a necessary—but not sufficient—condition for inequality (13) to hold is
that the buyer’s surplus from the trade is positive, i.e., u( q ) − y ≥ 0.
The set of incentive-feasible allocations, A PR , when agents cannot commit but when
public recordkeeping is available, is obtained by combining inequalities (12) and (13), i.e.,
σ
⎧
⎫
A PR = ⎨( q, y ) ∈ R2+ : c ( q ) ≤ y ≤
u( q ) ⎬ .
r +σ
⎩
⎭
(14)
The set of incentive-feasible allocations, A PR , is smaller than the set of incentive-feasible
allocations when agents can commit, A C .This is a consequence of the additional buyer
incentive constraint, (11), which must be imposed when buyers are unable to commit to
repay their debts. Note that the set A PR expands as the frequency of trades, σ , increases
or as agents become more patient, i.e., when r decreases and when r → 0, A PR → A C .
The efficient production and consumption level of the search good, q*, is incentivefeasible if
c ( q *) ≤
(15)
σ
u( q *).
r +σ
The right-hand side of (15) is increasing in σ and decreasing in r. Suppose that inequality (15) holds for a particular σ and r. Then if σ decreases, the probability of finding a
future match decreases and, hence, the buyer has a greater incentive not to produce y
since the benefit of avoiding autarky has now been reduced. If σ falls sufficiently, then
the buyer will not, for sure, produce any general good for the seller and, therefore, the efficient level of production and consumption of the search good is not incentive-feasible.
Similarly, if buyers discount the future more heavily, i.e., if β decreases or if r increases,
the buyer will have a greater incentive not to produce y since he cares more about his
current payoff than future payoffs. Once again, if r increases sufficiently, the efficient level
of production and consumption of the search good is not incentive-feasible.
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FEDERAL RESERVE BANK OF CLEVELAND
Another way to think about inequality (15) is as follows. For each level of search friction in the day market, σ ∈ ( 0,1], there exists a threshold for the discount factor, β (σ ),
such that if β ≥ β (σ ), then the efficient allocation ( q *, y*) is incentive-feasible. This
threshold β (σ ) is a decreasing function of σ , which means that the efficient level of
production and consumption of the search good, q*, is easier to sustain when there are
few frictions in the day market. If, however, β (σ ) ≤ β , incentive-feasible allocations are
such that the output level of the search good, q, is inefficiently low, i.e., q < q * .
Instead of characterizing the set of all incentive-feasible allocations one can, alternatively, focus on the allocation that would be suggested by a bargaining solution; for instance, the generalized Nash bargaining solution. The bargaining outcome, (q,y), in a
trade match now needs to be restricted to take into account the condition under which
the buyer has an incentive to repay his debt.When agents are unable to commit, but there
is a public recordkeeping device, the generalized Nash bargaining solution is given by
(16)
θ
max [u( q ) − y ]
q, y
(17)
s.t.
[ y − c ( q )]1−θ ,
− y + β V b ≥ 0.
The solution to this problem is given by q = q * and y = (1 − θ )u( q *) + θ c ( q *) ≡ y * if
β V b ≥ y*; and θ u′( q )[ y − c ( q )] = c ′( q )(1 − θ )[u( q ) − y] and y = β V b otherwise.When
β V b ≥ y*, the generalized Nash bargaining solution here corresponds exactly to the generalized Nash bargaining solution when agents are able to commit. Otherwise, the bargaining solutions will differ. When the buyer lacks commitment, he has an “outside option” of not producing the general good for the seller. If the proposed bargaining outcome
( q *, y*) does not provide the buyer with sufficient surplus, then the buyer will choose
not to produce the general good for the seller, and this happens when β V b < y * . In such
a situation the unconstrained generalized Nash bargaining solution ( q *, y*) will not be
incentive-feasible and the constrained generalized Nash bargaining solution (q,y) will
be characterized by q < q * and y < y * . When agents are unable to commit, it can be
shown that ( q *, y*) is the generalized Nash bargaining solution if
(18)
⎡θ (σ + r ) − r ⎤
c ( q *) ≤ ⎢
⎥ u( q *).
⎣ θ (r + σ ) ⎦
Inequality (18) implies that a necessary condition for ( q *, y*) to be the generalized Nash
bargaining solution is that the term in the square brackets on the left-hand side of the inequality is greater than zero, or that θ > r /( r + σ ). Hence, buyers must have sufficiently
high bargaining power in order to obtain the efficient consumption and production of
the search good. Note also that conditions (15) and (18) coincide whenever θ = 1, i.e.,
when buyers have all the bargaining power.
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POLICY DISCUSSION PAPERS
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We conclude this section with a brief discussion of punishments. First, we have assumed that if agents in a trade match do not accept the proposed offer, then the economy
will forever revert to autarky in the next period. Suppose instead that we assume that
only outputs produced from accepted proposals become part of the public record—i.e.,
agents could, without impunity, reject the proposed offer. Now, the agents’ participation
constraints are given by (5) and (6), instead of (9) and (10); for this case, it can be shown
that all of the above results still go through. This, perhaps, should not be so surprising
because as long as agents receive some surplus from trade, they will always accept the
proposed offer. Second, we have assumed that if an agent defects from proposed play,
then the economy will revert to global autarky forever.This is because an agent defecting
from equilibrium play is not identified by other agents in the economy, except his trading
partner. Since the trading partner is of measure zero, the defector could escape individual
punishment even if his trading partner and subsequent trading partners of that trading
partner could credibly pass on the message that a particular agent defected from equilibrium play. If, however, in addition to the list of outputs that are observed at the end of
the night market, the name associated with each output is observed, then it is possible to
support credit arrangements through individual punishments; that is, the above credit
arrangements can be sustained without having to revert to global autarky in the event of
a defection from a proposed allocation.
Credit with Reputation
Any credit arrangement necessitates some degree of cooperation between a buyer and a
seller, or a debtor and a creditor. As is well known, cooperation is more easily attainable
when agents repeatedly interact with one another. Intuitively, with repeated interactions,
agents are able to develop reputations for behaving appropriately. In this section, we rule
out the existence of both commitment and public recordkeeping but introduce the notion of reputation by allowing pairs of agents to repeatedly interact with one another via
a long-term partnership. We do this by assuming that agents who are in a trade match
during the day can form a partnership that can be maintained beyond the current period.
That is, agents can continue their trade match or partnership into the next period (day)
if they so desire.
We allow for both the creation and destruction of a partnership. At the end of each
period, an existing partnership is destroyed with some probability λ ∈ ( 0,1). One can
justify the exogenous destruction of a partnership by supposing that sellers produce different types of goods and buyers only value a subset of these goods.A match destruction
can be interpreted as an event where the buyer receives a preference shock (at night)
with the result that he no longer wants to consume the good that the seller produces. In
this case, the partnership is no longer viable and agents split apart. More generally, agents
can choose to terminate a partnership at the end of any period. For example, the seller
10
FEDERAL RESERVE BANK OF CLEVELAND
may choose to dissolve the partnership at the end of the period if the buyer does not
deliver on his promise to produce the general good.This sort of termination is important
because it provides the seller with a punishment vehicle—namely, the destruction of the
value asset of an enduring match or partnership—which is required in order to make a
partnership viable in the first place.
The chronology of events is as follows. At the beginning of the day, unmatched buyers and sellers participate in a random matching process. Each unmatched agent finds
a partner with probability σ ; that is, with probability σ the buyer is matched with a
seller whose search good he desires to consume. In each match, an allocation (q,y) is
proposed, which agents can either accept or reject. If both agents accept the offer, the
seller produces q units of the search good for the buyer in the day. At night, the buyer
chooses whether or not to produce y units of the general good for the seller.At the end of
the night, agents decide simultaneously to stay together or to split apart. If either or both
agents decide to destroy the partnership, then (in equilibrium) their best response will be
to enter the random matching process at the beginning of the day in order to find a new
trading partner.A partnership, which is basically a continuation of a trade match, can only
be formed during the random matching process at the beginning of the day.
We will characterize the set of symmetric stationary equilibrium allocations for this
economy. Let et denote the measure of new trade matches and existing partnerships
during the day of period t.Assuming that buyers do not renege on their promises, the law
of motion for et is
(19)
et +1 = (1 − λ )et + σ [1 − (1 − λ )et ].
According to (19), if there are et partnerships in period t, a fraction (1 − λ ) of them will
be maintained in period t + 1. Among the 1 − (1 − λ )et agents who are unmatched at the
beginning of t + 1, a fraction σ find new partners. In the steady state, et +1 = et = e which,
from (19), implies that
(20)
e=
σ
.
σ + λ (1 − σ )
Let Vmb be the value function of a buyer who is in a partnership at the beginning of a
period and Vub the value function of a buyer who is not. Then, assuming that the buyer
does not defect,
(21)
Vmb = u( q ) − y + λβ Vub + (1 − λ ) β Vmb ,
(22)
Vub = σ Vmb + (1 − σ ) β Vub .
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POLICY DISCUSSION PAPERS
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According to (21), the buyer receives q units of search goods in the day and produces y
units of general goods at night.The partnership is exogenously destroyed with probability λ , in which case both the buyer and the seller go to the random matching process at
the beginning of the day of the next period to find a new partner. According to (22), an
unmatched buyer finds a seller who produces a search good that he desires to consume
with probability σ .
Let Vms be the value function of a seller who is in a partnership at the beginning of the
period and Vus the value function of a seller who is not.Then,
(23)
Vms = −c ( q ) + y + λβ Vus + (1 − λ ) β Vms ,
(24)
Vus = σ Vms + (1 − σ ) β Vus .
According to (23), the seller produces q units of the search good during the day and
consumes y units of the general good at night.With probability λ the partnership is dissolved, in which case the seller enters the random matching process at the beginning the
next period. According to (24), the seller is matched with a buyer who likes the search
good that the seller produces with probability σ .
For (q,y) to be an equilibrium allocation, three sets of conditions have to be satisfied.
First, agents who enter the day search market unmatched, and subsequently become
matched, will accept the proposed allocation (q,y) if the following (participation) constraints hold,
(25)
−c ( q ) + y + λβ Vus + (1 − λ ) β Vms ≥ β Vus ,
(26)
u( q ) − y + λβ Vub + (1 − λ ) β Vmb ≥ β Vub ,
If the seller and buyer accept the allocation (q,y), then their expected payoffs are
given by the left-hand sides of (25) and (26), respectively. If, however, the allocation is
rejected, the continuation payoffs are given by the right-hand sides of (25) and (26),
respectively. Second, if the buyer does not receive a preference shock, then a matched
buyer and the seller will agree to continue their partnership if
(27)
−c ( q ) + y + λβ Vus + (1 − λ ) β Vms ≥ Vus ,
(28)
u( q ) − y + λβ Vub + (1 − λ ) β Vmb ≥ Vub .
If the seller and the buyer choose to continue the partnership, their payoffs at the beginning of the subsequent period are given by the left-hand sides of (27) and (28), respectively. If the seller and buyer choose to dissolve the partnership, the expected payoffs at
the beginning of the subsequent period are given by the right-hand sides of (27) and (28),
12
FEDERAL RESERVE BANK OF CLEVELAND
respectively. Clearly, if inequalities (27) and (28) hold, then inequalities (25) and (26), respectively, hold as well. Finally, a buyer in a partnership must be willing to produce the
general good for the seller at night.This requires that
(29)
− y + λβ Vub + (1 − λ ) β Vmb ≥ β Vub .
If the buyer produces y units of the general good, then his expected payoff is given by the
left-hand side of (29). If, however, he deviates and does not produce, then the partnership
will be dissolved at the end of the period and the buyer starts the next day search market
seeking a new match; the utility associated with this outcome is given by the right-hand
side of (29).
The set of incentive-feasible allocations that can be sustained by reputations, which
can be inferred from the value functions (21)–(24) and constraints (27)–(29), is given by
(30)
A R = {( q, y ) : c ( q ) ≤ y ≤ β (1 − λ )(1 − σ )u( q )}.
Note that inequalities (27) and (28) require that c ( q ) ≤ y ≤ u( q ) in order for both
the seller and the buyer to continue with the partnership. The buyer’s incentive-compatibility condition, (29), however, generates the endogenous borrowing constraint
y ≤ β (1 − λ )(1 − σ )u( q ). This endogenous borrowing constraint indicates that the maximum amount the buyer can promise to repay at night depends on the buyer’s tastes,
β , the stability of the match, λ , and market frictions, σ .The buyer can promise to repay
more: (i) the more patient he is, i.e., the higher is β ; (ii) the more stable are his preferences, i.e., the lower is λ ; (iii) the greater is the matching friction, i.e., the lower is σ ; and
(iv) the higher is his consumption, q, the next day.
Note that from (30), when q > 0, the set of incentive-feasible allocations, A R , is empty
when all matches are destroyed at the end of a period, i.e., when λ = 1, or when an agent
can find a partner in the day market with certainty, i.e., when σ = 1. The existence of
credit relationships relies on the threat of termination, but such a threat has bite only if
matches, for exogenous reasons, are not destroyed with high probability or if it is difficult
to create a new trade match.
From (30), the efficient production and consumption level of the search good, q*, is
implementable if and only if
(31)
c ( q *) ≤ β (1 − λ )(1 − σ )u( q *).
Agents are able to trade the quantity q * through long-term partnerships if the average
duration of a long-term partnership is high, i.e., if λ is low, and if the matching frictions
are severe, i.e., if σ is low.
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POLICY DISCUSSION PAPERS
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Related Literature
Pairwise credit in a search-theoretic model was first introduced by Diamond (1987a,b,
1990).The environment described by Diamond is similar to that in one of his earlier paper (Diamond, 1982), where agents are matched bilaterally and trade indivisible goods.
The number of trades is given by a matching function that exhibits increasing returns to
scale. Diamond allows agents to use lotteries in order to endogenize terms of trade.As in
our setup credit is repaid with goods. The punishment for not repaying a loan is permanent autarky.
Kocherlakota (1998a,b) describes credit arrangements in different environments, including a search-matching model based on a public record of individual transactions. He
uses mechanism design to characterize the set of symmetric, stationary, incentive-feasible
allocations. Kocherlakota and Wallace (1998) extends the model to consider the case
where the public record of individual transactions is updated after a probabilistic lag.
They establish that society’s welfare increases as the frequency with which the public
record is updated increases. As pointed out by Wallace (2000), this is the first model that
formalizes the idea according to which technological advances in the payment system improve welfare.The model by Kocherlakota and Wallace has been extended by Shi (2001)
to discuss how the degree of advancement of the credit system affects specialization.
Most search-theoretic models of the labor market assume long-term partnerships. A
canonical model is provided by Pissarides (2000). However, in these economies, trades do
not involve credit and are free of moral hazard considerations. Corbae and Ritter (2004)
consider an economy with pairwise meetings where agents can form long-term partnerships to sustain credit arrangements. A related model of reciprocal exchange is also presented by Kranton (1996).
Aiyagari and Williamson (1999) consider a random-matching model in which agents
receive random endowments that are private information. Exchange is motivated by risksharing.The social planner designs the optimal dynamic contract. It acts as a financial intermediary that opens accounts for the different agents. Optimal allocations have several
features similar to those of real-world credit arrangements, including credit balances and
credit limits.
Pure Monetary Economies
In the previous section, credit arrangements could be sustained because agents could
commit to repay their debt, observe a record of other agents’ trades, or were able to form
enduring relationships. In this section, we assume that none of these options is possible.
As a result, trade cannot be mediated by credit, and some alternative payments instrument must emerge if trade is ever to occur. The payments instrument that we consider
here is fiat money, which is a durable but intrinsically useless object. Fiat money is essen14
FEDERAL RESERVE BANK OF CLEVELAND
tial to the economy in the sense that its existence allows buyers and sellers to trade with
one another; the introduction of fiat money generates (desirable) outcomes that would
not be possible in its absence.
The economic environment is the benchmark model. During the day, buyers and sellers are randomly matched; in a trade match, the seller produces the search good for the
buyer. We will consider two versions of the night market. The first version enables us to
compare money economies with credit economies. In this version, buyers and sellers
who are matched in the day can continue their relationship at night; at the end of the period, all matches are dissolved.This setup is similar to that of the credit economies in the
previous section. The second version of the night market is a more “standard” setup for
models of money. In this version, matches are destroyed at the end of the day, and at night
there exists a competitive market where all agents can trade money for the general good.
Money Is Memory
How does a monetary economy compare with a credit economy in terms of the set of
allocations that can be implemented? To address this question, we structure a monetary
economy to mirror a credit economy when public recordkeeping is possible.
At the beginning of time, each buyer is endowed with one unit of an indivisible and
durable object, fiat money, that is intrinsically useless.The sequence of events in a typical
period is as follows: At the beginning of the day, buyers and sellers are matched pairwise
and at random. In each trade match, an allocation (q,d am) is proposed where q is the
amount of search good that the seller produces, and d am is the transfer of one unit of
money from the buyer to the seller. The allocation can depend on the money balances
of the buyer and seller in the match.The buyer and the seller in a trade match simultaneously accept or reject the offer. If one of the agents rejects the offer, then the match is
dissolved; otherwise, the trade takes place and agents remain matched into the night. At
night, an allocation (y,d pm) is proposed where y is the amount of the general good that is
produced by the buyer for the seller and d pm is the money transfer from the seller to the
buyer.The matched agents decide simultaneously on whether to accept or reject this allocation. If it is accepted by both agents, then it is implemented.At the end of the period,
all matches are dissolved.
We will consider allocations such that all buyers at the beginning of each period hold
one unit of money. This guarantees that whenever a buyer is matched with a seller, he
is able to trade. If matched buyers and sellers follow their equilibrium strategies of accepting the proposed (incentive-feasible) allocations and producing output for one unit
of money, then buyers will, in fact, always begin each period with one unit of money in
hand and sellers with no units of money. The mechanism that we consider is such that
if a matched buyer does not hold one unit of money at the beginning of the period, or,
if the seller holds more than zero units of money, then the proposed allocation will be
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POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
(q,d am) = (0,0); and if the seller does not hold exactly one unit of money at night, then
the proposed allocation is (y,d pm) = (0,0). If any one of these circumstances arises, it
necessarily implies that in the past, an agent departed from equilibrium play. Hence, an
agent’s money holdings reveal if he behaved according to the proposed allocation in the
past.
The value function of a buyer holding one unit of money at the beginning of the day is
V b (1) = σ [u( q ) − y ] + β V b (1).
(32)
According to (32), the buyer meets a seller with probability σ . If this event occurs, then
the buyer will consume q units of the search good in exchange for his unit of money.
At night the buyer gets his unit of money back in exchange for y units of the general
good. If the buyer does not hold money, the mechanism proposes no trade and, therefore,
V b ( 0 ) = 0. (Alternatively, the mechanism could allow the buyer to get his money back if
it proposes an allocation that has y = β V b (1), which implies that V b ( 0 ) = 0 ).
The value function of a seller holding zero units of money at the beginning of a
period is
V s ( 0 ) = σ [ −c ( q ) + y ] + β V s ( 0 ).
(33)
According to (33), a seller meets a buyer with probability σ . In the event a trade match
occurs, the seller produces q units of the search good for the buyer in exchange for one
unit of money. At night the seller gives the unit of money back to the buyer in exchange
for y units of the general good. If the seller holds a different amount than zero at the beginning of a period, he cannot trade and, therefore, V s ( m ) = 0 for m > 0, i.e., in this situation, the mechanism will propose the offer (0,0).
For the allocations {(q,1),(y,1)} to be incentive-feasible, agents must be willing to
participate in a trade.This requires
(34)
u( q ) − y + β V b (1) ≥ β V b (1),
(35)
−c ( q ) + y + β V s ( 0 ) ≥ β V s ( 0 ),
(36)
− y + β V b (1) ≥ β V b ( 0 ),
(37)
y + β V s ( 0 ) ≥ β V s (1).
Conditions (34) and (35) require the buyer and the seller, respectively, to accept the trade
during the day, while conditions (36) and (37) require the buyer and the seller, respectively, to accept the trade at night. Conditions (34) and (35) imply that c ( q ) ≤ y ≤ u( q ), i.e.,
the buyer and seller surpluses from trade must be non-negative. Clearly, condition (37) is
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FEDERAL RESERVE BANK OF CLEVELAND
satisfied whenever (35) holds. Finally, condition (36), in conjunction with the value function (32), implies that
(38)
y≤
σ
u( q ).
r +σ
Hence, the set of implementable incentive-feasible allocations in this monetary economy,
A M , is
(39)
σ
⎧
⎫
A M = ⎨( q, y ) ∈ R2+ : c ( q ) ≤ y ≤
u( q ) ⎬ .
σ
r
+
⎩
⎭
The set of incentive-feasible allocations in the monetary economy is identical to the set of
incentive-feasible allocations in a credit economy with a public record, i.e., A M = A PR ,
where the set A PR is described in (14). In this sense, money is equivalent to a public recordkeeping mechanism, i.e., money is memory. The explanation for this result is as follows.The money balances of an agent are a state variable that conveys some information
about his past trading behavior.The money balances of an agent indicate whether he has
defected from a given allocation by not producing for his trading partner when it was his
turn to produce. Money here can also be interpreted as a license to consume or as collateral.The buyer who transfers his unit of money to the seller knows that he won’t be able
to consume in future periods if he does not get his unit of money back at night.
Indivisible Money and Currency Shortage
We now assume that matches are dissolved at the end of the day. As well, we will also
impose a specific pricing mechanism for both search goods and general goods. During
the day, the terms of trade in bilateral matches are determined according to a bargaining
process.We assume that the buyer has all of the bargaining power and makes a take-it-orleave-it offer to the seller. At night, we assume that the market for general goods is competitive, where price-taking agents can buy and sell units of money in exchange for general goods at the market-clearing price, φ , where φ represents how many general goods
can be purchased at night for one unit of money. As is typically the case for Walrasian markets, the way in which agents meet or trades actually take place is not made explicit. Since
agents trade with “the market,” they are anonymous to one another; i.e., agents who purchase goods do not know which agents produced them, and agents who receive money
do not know which agent previously held the money.This anonymity precludes the use
of credit arrangements in the general goods market.
Regarding the quantity of money in the economy, we assume that M < 1.This implies
that not all buyers can be endowed with money:There is a currency shortage.The reason
that we investigate this case is twofold. First, currency shortages were common until the
17
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
mid-ninteenth century and they can be captured by this assumption. Second, this assumption was used in the early search-theoretic literature of money for tractability purposes.
The value function for a buyer holding m units of money at the beginning of the period satisfies
V b ( m) = σ
(40)
max
( q ,d )∈O ( m )
{u(q ) + W
}
( m − d ) + (1 − σ )W b ( m ),
b
where O(m) is the set of offers that are acceptable by sellers and that are feasible given
the money balances of the buyer, and W b is the value function of a buyer at the beginning
of the night.According to (40), a buyer with money at the beginning of the day is randomly matched with a seller with probability σ . In a trade match, he consumes q units of the
search good and delivers d units of money to the seller, where both q and d are chosen
optimally in the set of acceptable offers. The value function of a buyer in the Walrasian
market at the beginning of the night satisfies
{
}
ˆ)
W b ( m ) = max x − y + β V b ( m
(41)
m
ˆ ,x , y
ˆ = y + φ m.
s.t. x + φ m
(42)
The budget constraint (42) simply says that the buyer finances his end-of-period money
balances, m̂ , and consumption, x, with production of the general good, y, and with money balances brought into the night, m. Substituting x – y from the budget identity (42)
into the maximand of (41) we obtain
ˆ + βV b (m
ˆ )}.
W b ( m ) = φ m + max{−φ m
(43)
m
ˆ
Note that equation (43) tells us that the buyer’s choice of end-of-period money balances, m̂ , is independent of the money balances brought in from the day, m. This comes
from the linearity of the utility function that eliminates wealth effects. Furthermore, it is
straightforward to see that W b ( m ) = φ m + W b ( 0 ).
Sellers spend all of their money balances at night in the general goods market since
they do not require money in order to trade in the morning search market, and it is costly
to hold money balances in the morning search market. Hence, the value function for a
seller at the beginning of the period is given by
∫
V s = σ [−c ( q ) + φ d ]dF ( q, d ) + β V s ,
(44)
where F(q,d) is the distribution of offers made by buyers, which depends on the distribution of money balances across buyers. According to the value function (44), the seller
18
FEDERAL RESERVE BANK OF CLEVELAND
is matched to a buyer with probability σ , and the offer (q,d) is a random draw from the
distribution F(q,d).
s
s
A seller in a trade match will accept the offer (q,d) if it satisfies −c ( q ) + φ d + β V ≥ β V .
Consequently, the set of offers the buyer with m units of money can make, O(m), is
given by
O ( m ) = {( q, d ) : −c ( q ) + φ d ≥ 0, d ≤ m}.
The buyer will extract all of the surplus in the trade match, which implies that the buyer’s offer (q,d) will satisfy
(45)
−c ( q ) + φ d = 0.
Equations (44) and (45) imply that V s = 0, and equations (40) and (45) may be
rearranged to read
V b ( m ) = σ max ⎡⎣u c −1 (φ d ) − φ d ⎤⎦ + W b ( m ).
d∈{ 0,…,m}
Substituting this expression for V b (m) into the buyer’s value function at the beginning of
the night, equation (43), the buyer’s problem, can be re-expressed more compactly as
(46)
{
}
ˆ + σ max ⎡u c −1 (φ d ) − φ d ⎤ ,
max −rφ m
⎦
m
d∈{ 0,…,m
ˆ} ⎣
ˆ
where r = (1 − β ) / β . Equation (46) has a simple interpretation.The buyer faces a tradeoff when determining his money holdings: There is a cost associated with holding real
balances, which is equal to agents’ rate of time preference; the cost of bringing m̂ balˆ . But there is also a benefit associated with holding real
ances into the next period is rφ m
balances, which is equal to the expected surplus that is obtained in the search market;
the expected surplus is given by σ [u( q ) − φ d ]. Since r > 0, it is easy to check from (46)
that buyers will not hold more money than they expect to spend if they are matched in
ˆ.
ˆ and c ( q ) = φ m
the search market; this implies that d = m
If money were perfectly divisible, the buyer’s maximization problem (46) would have
a unique solution. However, because money is indivisible, this solution may not be attainable. It can be checked that (46) has, at most, two solutions which are two consecutive
integers. Furthermore, since M, the quantity of money per buyer, is less than one, marketclearing implies that buyers must be indifferent between holding 0 or 1 unit of money.
Therefore, m̂ = 1 and, from problem (46),
(47)
rφ = σ {u( q ) − φ }.
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POLICY DISCUSSION PAPERS
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The seller’s participation constraint (45) implies that c ( q ) = φ and, therefore,
c (q ) =
(48)
σ
u( q ).
r +σ
An equilibrium in this environment is the quantity of search goods produced, q, that satisfies (48). Given our assumptions about c and u, it is easy to check that there exists a
unique q > 0 that satisfies (48).
Let us turn to a comparative static analysis of the purchasing power of money as fundamentals change. From (48), ∂q / ∂σ > 0 and ∂q / ∂r < 0. As the matching probability σ
increases, a buyer has a higher chance to trade during the day, which makes money more
valuable.As a consequence, the quantities traded increase.As the rate of time preference,
r, increases, agents become more impatient, and the cost of holding money increases. As
a consequence, the value of money falls, and agents trade less. Note that the purchasing
power of money is independent of M, the quantity of money in the economy.
If we measure social welfare by the sum of utilities of buyers and sellers, then
W = σ M[u( q ) − c ( q )]. Here money is not neutral and an increase in M, for M ∈ ( 0,1),
raises welfare by increasing the number of trades in the economy. Hence, a change in M
has no effect on the intensive margin, which is the quantity produced in a particular
trade match, but affects the extensive margin, which is the number of trade matches.We
will see later how this extensive margin result disappears when money is assumed to be
perfectly divisible.
Depending on the precise functional form of u and c and the values of β and σ , the
equilibrium value of the amount of search goods produced, q, can be greater than, less
than, or equal to the efficient level, q * . Hence, a problem associated with the indivisibility of money is that sometimes the amount of search goods produced, q, is greater than
what is socially efficient, q * . This can easily be seen from (48), if we assume that r ≈ 0.
Indivisible Money and Lotteries
When money is indivisible, there are circumstances under which the buyer could be
made better off if he could somehow give up only a fraction of his money balances to
the seller; this happens when q > q * . Before turning to the case of perfectly divisible
money, we show that there is a way to achieve this outcome when money is indivisible
by introducing lotteries over the outcome (q,d), d ∈ {0,1}. Since there is no (social) benefit associated with having a lottery over output, the take-it-or-leave-it offer by the buyer of the search good can be described by ( q,τ ), where q is the amount of the search
good produced by the seller, and τ is the probability that the buyer surrenders his unit
of money to the seller.
Consider a trade match between a buyer and a seller.As above, equilibrium in the general goods market will require that buyers hold either zero or one unit of money at the
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FEDERAL RESERVE BANK OF CLEVELAND
end of the period. In a trade match, the take-it-or-leave-it offer that the buyer makes to the
seller, ( q,τ ), solves the problem
(49)
max[u( q ) − τφ ]
q ,τ
s.t. − c ( q ) + τφ ≥ 0, and
0 ≤ τ ≤ 1.
The solution is q = q * and τ = c ( q *) / φ if τ ≤ 1; otherwise τ = 1 and q < q * satisfies
(48). The solution to the buyer’s choice of money balances to bring into the next period—where the problem is now described by (46), except that φ d is replaced by τφ d—
is given by
(50)
rφ = σ [u( q ) − τφ ],
which is a modified version of equation (47). An expression for τ can be obtained by
substituting the seller’s participation constraint, −c ( q ) + τφ = 0, into (50) and rearranging, i.e.,
(51)
⎧
⎫
rc ( q *)
τ = min ⎨
,1⎬ .
⎩σ [u( q *) − c ( q *)] ⎭
Hence, q = q * iff τ ≤ 1 and, from equation (51), τ ≤ 1 iff c ( q *) ≤ σ u( q *) /( r + σ ). So if
allocation ( q *, c ( q *)) is incentive-feasible in the “money is memory” environment (see
the section on Money is Memory and the definition of A M in [39]) or in a credit environment with public recordkeeping (see the section on Credit with Public Recordkeeping and the definition of A PR in [14]), then it can be implemented as an equilibrium in a
monetary economy with indivisible money by a take-it-or-leave-it offer when buyers can
use lotteries, i.e., if ( q *, c ( q *)) ∈ A M = A PR , then τ ≤ 1. If τ = 1, then the level of search
good production, q, is given by equation (48), and q ≤ q * .
Divisible Money
We now turn to the case where money is perfectly divisible when at night the market
for general goods is competitive. Since there is no technological constraint that prevents
the stock of money from being divided evenly across buyers, there can be no shortage
of currency. In this environment, the buyer’s value functions satisfy (40) and (41), and his
choice of money balances is still given by (46), but now d ∈[0, m
ˆ ]. Since money is costˆ . Recognizing that the seller’s participation conly to hold, it will be the case that d = m
straint will bind, i.e., c ( q ) = φ m, it will be convenient to rewrite the buyer’s choice of
money balances problem (46) as
(52)
ˆ + σ {u[q ( m
ˆ )] − c[q ( m
ˆ )]}}.
max{−rφ m
m
ˆ
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POLICY DISCUSSION PAPERS
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The solution to problem (52) when money is perfectly divisible is given by
u′( q )
r
=1+ ,
c ′( q )
σ
(53)
since, from the seller’s participation constraint, dq / dm = φ / c ′( q ). It can be checked
that there is a unique q satisfying (53) and it is such that ∂q / ∂σ > 0 and ∂q / ∂r < 0. So,
the comparative static results are similar to those obtained in the model with indivisible money. The term r / σ is a measure of the cost of holding real balances: this is the
product of the rate at which agents depreciate future utility, times the average number
of periods for a match to occur. As this cost increases, buyers reduce their real balances
and output falls.As the rate of time preference approaches 0, q tends to q * . Finally, M is
neutral; it affects neither the quantity of search goods produced, q, nor the frequency of
trades, σ .
Fiat objects, be they divisible or indivisible, can be valued in an environment where
there is a double-coincidence-of-wants problem and agents cannot use credit. The value
of money depends on the fundamentals of the economy, including agents’ rate of time
preference, r, and the extent of the search-matching frictions, σ . When money is perfectly divisible, the number of trades is maximized but the quantities traded are too low
when r > 0.When money is indivisible and scarce, the number of trades is too low since
not all buyers can be endowed with money. However, the quantities traded can be too
high if agents are unable to use lotteries.
Related Literature
Using a mechanism design approach, Kocherlakota (1998a,b) has established that the
technological role of money is to act as a societal memory that gives agents access to
certain aspects of the trading histories of their trading partners. As a corollary, imperfect
knowledge of individual histories is necessary for money to play an essential role in the
economy (Wallace, 2000). The recordkeeping role of money was emphasized by Ostroy
(1973), Ostroy and Starr (1974, 1990) and Townsend (1987, 1989), among others.
Diamond (1984) was the first to introduce fiat money into a search-theoretic model
of bilateral trade. However, money was not essential in Diamond’s environment since all
matches were double-coincidence-of-wants matches. Kiyotaki and Wright (1989, 1991,
1993) have introduced a double-coincidence-of-wants problem into a search-theoretic
environment to explain the emergence of a medium of exchange and the essentiality of
fiat money. These models were based on important restrictions: money and goods were
indivisible and agents could hold at most one unit of an object. Shi (1995) and Trejos and
Wright (1995) have relaxed the assumption of indivisible goods to endogenize prices.
Wallace and Zhou (1997) have used a related framework to explain currency shortages.
Berentsen, Molico, and Wright (2002) have introduced lotteries. The assumption of indi22
FEDERAL RESERVE BANK OF CLEVELAND
visible money and its implications for the efficiency of monetary exchange are discussed
in Berentsen and Rocheteau (2002). Search models with divisible money include Shi
(1997), Green and Zhou (1998), and Lagos and Wright (2004).The formalization adopted
in this section follows the one in Lagos and Wright.
Alternative models of monetary exchange are surveyed in Wallace (1980) and
Townsend (1980).
Coexistence of Money and Credit
Actual economies differ from the pure credit and the pure monetary economies described in the previous sections in that many modes of payments coexist. For example,
some trades are conducted through credit arrangements, while other trades are based on
monetary exchange.Why do different means of payment coexist? How does the presence
of monetary exchange affect the use and the availability of credit? How does the availability of credit affect the value of money? We address these questions below.
One way to explain the coexistence of monetary exchange and credit arrangements
is to introduce some heterogeneity among agents and/or trading matches. For example,
some agents may have the ability to commit to repay their debt or to have their trading
histories publicly observable, while others don’t. The former set of agents will be able
to trade using credit arrangements, while the latter set of agents will need to use money.
In this section, we explain the coexistence of money and credit by introducing heterogenous matches: Some matches will be short-lived and last only one period, while other
matches will be longer-lived and can be productive for many periods. The use of credit
will not be incentive-feasible in short-lived matches, since the buyer will always default
on repaying his obligation at night. In contrast, the buyer’s behavior in a long-lived match
is disciplined by reputation considerations that will trigger the dissolution of a valuable
relationship following a default.
In this section, we extend the long-term partnership environment described in the
section on Credit with Reputation by allowing the possibility of short-term partnerships
to arise.When unmatched agents arrive at the beginning of a period, with probability σ
they find a long-term trade match and, with probability σ s they can enter into a shortterm trade match. We assume that 0 < σ = σ + σ s < 1. A short-term match is destroyed
with a probability of one at the end of the day, and a long-term match will be destroyed
exogenously, with probability λ < 1 during the night.
The timing of events is as follows: Buyers enter the day market either attached, i.e., in
a long-term trade match (or partnership), or unattached. Unattached buyers and sellers
participate in a random matching process. Note that since there is the same number of
buyers and sellers, there is also the same number of unattached buyers and unattached
sellers.After the matching process is completed, all matched sellers, i.e., those in either a
long-term or short-term relationship, produce the search good for buyers. All short-term
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POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
matches are destroyed at the end of the day.The night begins with buyers who are in a
long-term partnership producing the general good for sellers. Buyers then realize a preference shock, which is followed by the opening of a competitive general goods market,
where the general good is exchanged for money. As a result of the preference shock, a
fraction, λ , of long-term trade matches are dissolved.
We assume a specific pricing mechanism for traded goods. For search goods, buyers make take-it-or-leave-it offers to sellers; for general goods, the market is competitive,
where one unit of money trades for φ units of the general good.
The aim of this section is to present an environment where money and credit can
coexist and to see how these two payments systems may affect one another.To this end,
we will focus on a particular class of equilibria that exhibit two features. First, money is
valued but is only used in short-term trade matches. Second, the buyer’s incentive-compatibility constraint in long-term matches—that the buyer is willing to produce the general good for the seller—is not binding. This latter assumption implies that the buyer in
a long-term trade match can obtain the efficient quantity of the search good, q*, without
having to use any money. At the end of this section, we will briefly discuss the implications of relaxing these assumptions.
Value Functions
The Bellman equations for the value functions for a buyer assume that the equilibrium
has money being used in only short-term trade matches. Consider first a buyer who enters the day market unattached, with m units of money. The value function for such a
buyer is
Vub ( m ) = σ V b ( m ) + σ sVsb ( m ) + (1 − σ − σ s )Wub ( m ),
(54)
The buyer finds a long-term trade match that has value V b ( m ) with probability σ ; with
probability σ s , he finds a short-term match whose value is Vsb ( m ). The buyer remains
unattached with probability 1 − σ − σ s and enters the night market with m units of
money with value Wub ( m ).
Consider first a buyer holding m units of money who is in a short-term trade match.
The buyer makes a take-it-or-leave-it offer (q s ,d s ) to the seller, where q s is the amount
of the search good that the seller produces and d s is the amount of money transferred
from the buyer to the seller. Both the seller’s output and the money transfer will depend
on the money holdings of the buyer.The utility to the buyer in a short-term trade match,
evaluated in the day, is
Vsb ( m ) = u[qs ( m )] + Wub [m − ds ( m )].
(55)
The buyer will consume q s units of the search good in the day and will enter the com-
24
FEDERAL RESERVE BANK OF CLEVELAND
petitive general goods market with m – d s units of money. The value of the (now unmatched) buyer at night satisfies
(56)
ˆ ) + β Vub ( m
ˆ )}.
Wub ( m ) = max{φ ( m − m
m
ˆ
The buyer acquires m̂ − m units of money at the price φ in terms of general goods at the
competitive market, in order to readjust his balances to the desired level m̂ . Recall that
ˆ ) and that Wub ( m ) is linear in m, i.e., Wub ( m ) = φ m + Wub ( 0 ).
x − y = φ (m − m
Consider now a buyer who enters the period in a long-term partnership. The buyer
consumes q units of the search good, which is assumed to be independent of money
balances, m, that he might hold (since we focus on equilibria where the buyer’s incentivecompatibility constraint in long-term matches is nonbinding).The value function for such
a buyer at the beginning of the period is
(57)
V b ( m ) = u( q ) + W b ( m, y ),
where W b ( m, y ) is the value of the matched buyer at night holding m units of money
and a promise to produce y units of the general good for his partner in the trade match.
The value function of the matched buyer at night satisfies
(58)
ˆ ) + β Vub ( m
ˆ )}.
Wb ( m, y ) = − y + (1 − λ ){φ m + β Vb ( 0 )} + λ max{φ ( m − m
m
ˆ
At the beginning of the night, the buyer fulfills his promise and produces y units of the
general good for the seller. If the trade match is not exogenously destroyed, the buyer consumes his real balances, φ m, since money is not needed in a long-term relationship.Alternatively, if the partnership breaks up at night, an event that occurs with probability λ , the
buyer has the opportunity to readjust his money balances in the competitive general goods
market before he proceeds to the next period in search of a new trading partner. In this
case, the buyer will choose to bring m̂ money balances into the next period. Note from
b
b
(58) that W b ( m, y ) is linear in both m and y , i.e., W ( m, y ) = − y + φ m + W ( 0, 0 ).
Terms of Trade
We first examine the amount traded in the case where buyers and sellers are in a shortterm partnership.The buyer makes a take-it-or-leave-it offer, (q s ,d s ), such that ds ≤ m and
the seller will accept as long as −c ( qs ) + φ ds ≥ 0. As described in the section on Divisible Money, the offer will be characterized by qs ( m ) = q * and φ d = c ( q *) if φ m ≥ c ( q *);
otherwise qs = c −1 (φ m ).
The optimal choice of money balances for a buyer who is not in a long-term relationship satisfies
(59)
ˆ + σ s {u[qs ( m
ˆ )] − c[qs ( m
ˆ )]}},
max{−rφ m
m
ˆ
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POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
which is identical to problem (52) in the section on Divisible Money and leads to the familiar first-order condition,
u′( qs )
r
=1+ .
′
c ( qs )
σs
(60)
For buyers and sellers who are in a long-term partnership, we focus on equilibria
where the incentive-compatibility constraint, W b ( m, y ) ≥ Wub ( m ), does not bind.When
this constraint does not bind, a buyer in a long-term relationship proposes (q,y) such
that
max[u( q ) − y]
q, y
s.t.
− c ( q ) − y ≥ 0,
which gives q = q * and y = c ( q *).
Equilibrium
We consider a steady-state equilibrium where money is only used in short-term matches.
For an allocation ( qs , q , ys , y ) to be an equilibrium, we need to check that the buyer in
a long-term partnership is willing to repay y at night. The incentive-compatibility constraint requires W b ( m, y ) ≥ Wub ( m ), or, from the linearity of the value functions,
W b ( 0, y ) ≥ Wub ( 0 ).
(61)
With the help of equations (54)–(58), and after some rearranging, inequality (61) can be
rewritten as
c ( q *) ≤ (1 − λ ) β {(1 − σ )u( q *) − σ s [u( qs ) − c ( qs )]},
(62)
where q s satisfies equation (60). If inequality (62) holds, then there exists an equilibrium where buyers and sellers who are in a long-term relationship consume and produce
q = q * units of the search good and y = c ( q *) units of the general good and use credit
arrangements to implement these trades. Buyers and sellers in short-term partnerships
trade q s units of the search good for y s = c(q s ) units of real balances.
Perhaps not surprisingly, when σ s = 0, the incentive condition (62) is identical to
the one obtained in a model where money was absent and trade in long-term relationships was supported by reputation (see the definition of A R given by [30] in the section on Credit with Reputation). So when allocation ( q *, c ( q *)) ∈ A R , this allocation
can be implemented in the long-term relationship—in a world where credit and money
coexist—as long as σ s is sufficiently small. If the frequency of short-term matches, σ s ,
increases, then, from (60), agents will increase their real balance holdings; as a result the
incentive-constraint (62) becomes more difficult to satisfy. Hence, the availability of mon-
26
FEDERAL RESERVE BANK OF CLEVELAND
etary exchange in the presence of a long-term partnership increases the attractiveness of
defaulting on promised performance.
In this section, we have only described one type of equilibrium, where money is not
needed in long-term trade matches.There exist other kinds of equilibria when the incentive-constraint (61) binds. In those equilibria, agents will use money in both short-term
and long-term trade matches, but fewer money balances will be needed in long-term
relationships. In these kinds of equilibria, money can be used to weaken the buyer’s
incentive-compatibility constraint. Hence, payment in long-term trade matches be will a
combination of money and a promise to repay output in the future.
Finally, an interesting extension of this model would consist of introducing money
growth in order to investigate how inflation affects the buyer’s incentive-compatibility
constraint, that is, in a long-term relationship. We conjecture the following: As inflation
increases, the cost of holding real balances is higher and, as a consequence, the amount
of search good traded in a short-term trade match, q s , decreases. Hence, buyers obtain a
smaller surplus in short-term trade matches which raises the cost of defaulting in longterm trade matches. If the incentive-compatibility constraint (61) is binding, inflation
would relax this constraint, although it would also reduce the quantities traded in shortterm matches. Depending on the relative magnitudes, it is possible that a mild inflation
can deliver a better outcome than a zero inflation. If, however, one allows for negative
money growth rates, i.e., deflation, then the cost of holding real balances can be driven to
zero, which implies that all trades can be conducted with money and this generates the
efficient allocation.
Related Literature
Shi (1996) has constructed a search-theoretic environment where fiat money and credit can coexist, even though money is dominated by credit in the rate of return. A credit
trade occurs when two agents are matched and the buyer in the match does not have
money. Collateral is used to make the repayment incentive-compatible, and debt is repaid
with money. In this approach, monetary exchange is superior to credit in the sense that
monetary exchange allows agents to trade faster. Li (2001) extended Shi’s model to allow
private debt to circulate and to investigate various government policies including openmarket operations and public-debt policies.
Townsend (1989) investigates the optimal trading mechanism in an economy with
different locations, where some agents stay in the same location while other agents move
from one location to another. The optimal arrangement implies the coexistence of currency and credit. Currency is used between strangers, i.e., agents whose histories are
not known to one another, and credit is used among agents who know their histories.
Kocherlakota and Wallace (1998) consider a random-matching economy with a public
record of all past transactions that is updated only infrequently. They show that in this
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POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
economy there are roles for both monetary transactions and some form of credit. Jin and
Temzelides (2004) consider a search-theoretic model with local and faraway trades.There
is recordkeeping at the local level so that agents in local meetings can trade with credit.
In contrast, agents from different neighborhoods need to trade with money.
Corbae and Ritter (2004) consider a model of long-term and short-term partnerships
similar to the one presented in this section. It is shown that the presence of money
weakens incentive-compatibility conditions. Aiygari and Williamson (2000) construct a
dynamic risk-sharing model where agents can enter into a long-term relationship with a
financial intermediary.They introduce a transaction role for money by assuming random
limited participation in the financial market. In each period, agents can defect from their
long-term contracts and trade in a competitive money market in each succeeding period.
Aiyagari and Williamson show that the value of this outside option depends on monetary
policy.
Alternative Media of Exchange
We have examined economies where credit and fiat money are used as means of payment. Even though, in practice, fiat money plays a singular role in facilitating transactions,
a large variety of assets and commodities can be and are used as means of payment. For
example, real commodities, such as gold and silver, capital or claims on capital, government securities, and foreign currencies, to name but a few, have been known to be used
to purchase goods. A number of interesting questions naturally arise. Is money useful
when capital can be used as a medium of exchange? Is it useful to have several currencies and how are exchange rates determined? Can money and interest-bearing bonds coexist?
We will now study economies where agents can choose among different media of
exchange: money and something else.We will draw conclusions regarding the possibility
and the need of having different payment instruments simultaneously circulating.
Money and Capital
A direct way for a buyer in our model to obtain the search good would be to give the
seller what he values: the general good.This direct method of payment, however, is technically infeasible in the benchmark model since it is assumed that goods fully depreciate
at the end of the subperiod in which they are produced. The general good is produced
at night and cannot carried over to the next period to pay for a search good. In what follows, we relax the assumption that all goods are perishable.
The economic environment is that of the benchmark model, except that the general
good can be stored.Trade matches are destroyed at the end of the day, and the market for
the general good is competitive.We consider an economy where agents have access to a
storage technology that enables them to carry over the general good from one period to
28
FEDERAL RESERVE BANK OF CLEVELAND
the next. The storage technology is linear, which means that each unit of stored general
good generates A ≥ 0 units of the general good in the following period. More specifically,
if a unit of the general good is stored at night, then it turns into A units of the general
good the next morning; the A units can be held over the day and consumed (or stored) in
the subsequent night. The case where A = 0 corresponds to the assumption that goods
are perishable, while the case where A = 1 corresponds to a pure storage technology.
If A > 1, then the storage technology is productive, whereas if A < 1, the storage technology is characterized by depreciation. For convenience, we will refer to a good that is
stored as capital.
Consider a buyer holding a portfolio (m,k) of money and capital at the beginning of
the day.This portfolio implies that the buyer stored k/A units of the general good from the
previous night.We denote (q,d m ,d k ) as the terms of trade in a bilateral match where q is
the amount of the search good that the buyer receives from the seller, d m is the transfer
of money from the buyer to the seller, and d k is the transfer of capital from the buyer to
the seller.The buyer’s lifetime expected utility satisfies
(63)
V b ( m, k ) = σ {u[q ( m, k )] + W b ( m − dm ( m, k ), k − dk ( m, k ))} + (1 − σ )W b ( m, k ).
According to (63), a buyer who meets a seller consumes q units of the search good
and transfers d m units of money and d k units of capital to the seller. The terms of trade
(q,d m ,d k ) depend on the buyer’s initial portfolio.The utility of the buyer at night obeys
(64)
{
}
ˆ − kˆ + β V b ( m
ˆ , Akˆ) ,
W b ( m, k ) = φ m + k + max −φ m
m
ˆ ,kˆ
ˆ + k − kˆ. Note that the valsince, from the buyer’s budget constraint, x − y = −φ m − φ m
ue function for the buyer at the beginning of the next period depends upon Ak units of
goods.
The terms of trade in bilateral matches are determined by take-it-or-leave-it offers by
buyers.An optimal offer for the buyer solves
(65)
max [u( q ) − dk − φ dm ] s.t.
q ,dm ,dk
dm ≤ m,
− c ( q ) + dk + φ dm ≥ 0,
dk ≤ k.
The solution to this bargaining problem is q ( m, k ) = q * and φ dm + dk = c ( q *) if
φ m + k ≥ c ( q *); otherwise q = c −1 (φ m + k ). Notice that it is the real value of the entire
portfolio, φ m + k, that is relevant to determining the terms of trade and not the composition of the portfolio.
If we substitute the buyer’s beginning-of-period lifetime utility, V b , given by (63), into
buyer’s beginning-of-night lifetime utility, W b , given by (64), recognizing that the buyer
29
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
extracts all of the surplus from a trade match, the buyer’s choice for the optimal portfolio
can be expressed as
⎧ ⎛1− β
max ⎨− ⎜
ˆ
m
,
k
ˆ ⎩ ⎝ β
(66)
⎫
⎞
⎛1− β A ⎞ ˆ
ˆ −⎜
ˆ , Akˆ) ⎤ − c ⎡q ( m
ˆ , Akˆ) ⎤⎦ ⎬ .
⎟φ m
⎟ Ak + σ u ⎡⎣q ( m
⎣
⎦
β
A ⎠
⎠
⎝
⎭
{
}
The expression in the brackets should look familiar: The first two terms represent the
cost of carrying money and capital, respectively, into the subsequent period, and the third
term represents the expected surplus that is obtained in a trade match. Since the quantity
of search goods proposed in the bargain, q(m,k), depends only on the real value of the
buyer’s portfolio, φ m + k, buyers will be willing to hold both money and capital if and
only if they both offer the same return, i.e., if and only if A = 1. If A > 1, capital dominates money in its rate of return and agents will hold only capital goods to make transactions. In this case, the quantity traded satisfies
u′( q )
1− β A
.
=1+
c ′( q )
σβ A
(67)
Note that the quantity of search goods traded in bilateral matches increases with the rate
of return of capital. As the rate of return of capital approaches the discount rate, i.e., as
Aβ approaches one, the quantity traded, q, approaches its efficient value, q * . If A < 1,
then capital generates a lower rate of return than money and, as a result, buyers will hold
only money for transaction purchases in equilibrium, i.e., buyers will not store any of the
general good.
Let us return to the case of a pure storage technology, i.e., A = 1, so that money and
capital can coexist as a means of payments.The quantities of goods traded will be identical to the quantities traded in the monetary economy studied in the previous section.
However, the presence of money is welfare-improving because it frees up real resources
for alternative uses: Capital that was previously used as a medium of exchange can now
be consumed. This is a standard argument in favor of a fiat money regime rather than a
commodity standard.
As it now stands, the model cannot explain how capital and money can coexist if capital dominates money in its rate of return.A crude way to obtain coexistence is to impose
costs on the use of capital as a medium of exchange. For example, productive capital
may be costly to use as a medium of exchange because it is not as easy to transport as
money, or because claims on capital may be easier to forge than fiat money. Hence, if the
net return to capital—net of transportation or expected forgery costs—equals one, then
money and capital can coexist. An alternative approach to obtain coexistence is based
on the existence of legal restrictions. Suppose that there exists a government that wants
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FEDERAL RESERVE BANK OF CLEVELAND
to promote the use of fiat money. The government can promote the use of fiat money
by refusing to accept capital for any transactions with private agents. To the extent that
agents want to or have to transact with the government, they will then want to hold some
of the (lower rate of return) money. Here, the lower return on money is compensated by
its higher “liquidity.”
Dual-Currency Payment Systems
Real economies are endowed with many different currencies. We now examine the coexistence of two monies to see whether two currencies can be valued and used in payments. We also want to determine whether there are gains in having more than one currency. Our investigation here is slightly different from that of the previous section. The
previous section examined the coexistence of a fiat money and a real object; this section
examines the coexistence of two fiat monies.
We now turn to an economy where two fiat monies—called money 1 and
money 2—can be used as mediums of exchange. We assume that the stocks of both
monies, M1 and M2 , are constant and that agents are free to use any currency in a trade.
The market for general goods is competitive, where one unit of money 1 buys φ1 units of
the general good and one unit of money 2 buys φ2 units of the general good.
Consider a buyer holding m1 units of currency 1 and m2 units of currency 2. His beginning-of-period value function, V b (m 1 ,m 2 ), satisfies
(68)
V b ( m1 , m2 ) = σ {u[q ( m1 , m2 )] + W b [m1 − d1 ( m1 , m2 ), m2 − d2 ( m1 , m2 )]}
+(1 − σ )W b ( m1 , m2 ).
The interpretation of the value function (68) is similar to that of the value function
V b (m,k) given in (63).The value function of the buyer at night is given by
(69)
{
}
ˆ1 − φ2 m
ˆ1, m
ˆ 2 + βV b (m
ˆ2) ,
W b ( m1 , m2 ) = φ1m1 + φ2 m2 + max −φ1m
m
ˆ 1m
ˆ2
and the interpretation of this value function is similar to that of W b (m,k) given in (64).
The terms of trade are determined by take-it-or-leave-it offers by buyers. It is straightforward to show (and should come as no surprise) that the solution to the buyer’s bargaining problem is given by q = q * and φ1d1 + φ2 d2 = c ( q *) if φ1m1 + φ2 m2 ≥ c ( q *); otherwise q = c −1 (φ1m1 + φ2 m2 ). As in the previous section, the terms of trade depend only on
the real value of the whole portfolio of the buyer. Substituting V b , given by its expression
in (68), into (69), and using the solution for the buyer’s bargaining problem, the buyer’s
optimal portfolio satisfies
(70)
max {−r (φ1m1 + φ2 m2 ) + σ {u[q ( m1m2 )] − c[q ( m1m2 )]}}.
m1 ,m2
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POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
Since q(m 1 ,m 2 ) depends only on the real value of the buyer’s portfolio, φ1m1 + φ2 m2 ,
(70) does not pin down the composition of the portfolio. The buyer is indifferent between holding one currency or another.The buyers-take-all assumption implies that
c ( q ) = φ1m1 + φ2 m2 ;
(71)
from the first-order condition (70), q satisfies
u′( q )
r
=1+ .
c ′( q )
σ
(72)
While equation (72) uniquely determines the value of q, equation (71) is effectively left to
determine both φ1 and φ2 . Equivalently, for any exchange rate ε , there exists a price for
money 2, φ2 = c ( q ) /[ε M1 + M2 ], that solves (71). Consequently, the nominal exchange
rate ε = φ1 / φ2 is indeterminate.
The indeterminacy of the exchange rate can be resolved if the government simply
imposes a certain exchange rate when trading with private agents.The government can
implicitly impose an exchange rate by refusing to accept one of the monies in trades
with private agents. Assume, for example, that the government will only accept the first
currency.Then buyers will only want to hold the first currency, and the second currency
will lose its value, φ2 = 0. Finally, it turns out that multiple currencies are not useful in
this economy since the equilibrium allocation is the same as the one in the single currency economy.That is, agents trade the same quantities q in all matches, irrespective of
the number of currencies.
Government Liabilities as Means of Payment
In real economies, financial institutions (e.g., banks) whose liabilities are used by private
agents as mediums of exchange (deposit and saving accounts) hold government securities. This phenomenon reflects an intermediation activity on the part of the financial
institution consisting of transforming illiquid assets into more liquid ones. We will now
describe economies where agents hold government bonds and fiat money.We will investigate whether government bonds that pay interest can be used as a means of payment.
This discussion will require that we formalize the notion of illiquid bonds. We will conclude with a discussion on whether illiquid assets have a role to play in the economy.
Consider an economy where agents can use both money and government bonds as
mediums of exchange.A one-period government bond is issued at night and is redeemed
for one unit of money in the night market of the subsequent period. Government bonds
are of the pure discount variety and are perfectly divisible, payable to the bearer, and
default-free. (These assumptions make money and bonds close substitutes.) The flow of
bonds sold by the government each period is equal to B. The government finances the
interest payments on bonds, if any, by lump-sum taxation at the end of each period.
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FEDERAL RESERVE BANK OF CLEVELAND
Since matured bonds are exchanged for money one to one, the price of matured bonds,
in terms of night goods, is φ . Let ω be the price of newly-issued bonds in terms of night
commodities. If ω < φ , newly-issued bonds are sold at a discount for money.The one-period rate of return on newly issued bonds rb = φ / ω − 1.
Can newly issued bonds ever be sold at a discount? If bonds were sold at a discount
for money, then agents would prefer to sell all their money for bonds, since bonds are as
liquid as money but provide a rate of return. (A standard backward-induction argument
would generalize this result to the case where the length of the maturity period is more
than one period.) So, in equilibrium, money and newly issued bonds are perfect substitutes, i.e., ω = φ . This result is then similar to the dual-currency economy of the section
on Terms of Trade, where the exchange rate between the two currencies is unity and the
two nominal assets are traded at par. Hence, interest-bearing government bonds cannot
coexist with fiat money unless one assumes some form of illiquidity associated with government bonds.
We introduce an arbitrary restriction on the use of bonds in bilateral meetings during
the day in order to generate a form of illiquidity for bonds. A buyer holding a portfolio
of b units of bonds can use only a fraction g ∈[0,1] of his bonds to make a payment in
a bilateral match during the day. If g = 0, bonds are fully illiquid, whereas if g = 1, they
are perfectly liquid. One can interpret this illiquidity of bonds as stemming from the fact
that bonds are not as easily transportable as money, are not perfectly divisible, or involve
costs to be recognized. One can also view this constraint as an arbitrary form of a cash-inadvance requirement. Since terms of trade are determined by take-it-or-leave-it offers by
buyers, the quantities traded in bilateral matches during the day satisfy
c ( q ) = φ ( m + gb ),
which is the seller’s participation constraint.
The value function for buyers, V b (m,b), which is similar to (68), can be expressed as
V b ( m, b ) = σ {u[q ( m, b )] + W b ( m − dm , b − db )} + (1 − σ )W b ( m, b ),
where
{
(
)}
ˆ − ω bˆ + β V b m
ˆ , bˆ ,
W b ( m, b ) = φ ( m + b ) + max −φ m
m
ˆ ,bˆ
and d m (d g ) represents the transfer of money (bonds) from the buyer to the seller in the
morning.The buyer’s portfolio problem is given by the solution to
{
}
ˆ + {σ u[q ( m, b ) − c[q ( m, b )] .
max −ω bˆ( r − rb ) − φ mr
m
ˆ ,bˆ
33
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
The first-order conditions of this problem with respect to b̂ and m̂ , respectively, are
⎛ u′( q ) ⎞ ω ( r − rb )
− 1⎟ =
⎜ ′
σ gφ
⎝ c (q )
⎠
(73)
and
⎛ u′( q ) ⎞ r
− 1⎟ = .
⎜ ′
⎝ c (q )
⎠ σ
(74)
Equating the right-hand sides of (73) and (74), so that buyers are indifferent between
holding money and bonds, we obtain
rb =
r (1 − g )
.
1 + gr
The one-period rate of return on government bonds depends on the degree of liquidity
of bonds. If bonds are perfectly liquid, i.e., g = 1, then r b = 0 and ω = φ . If bonds are illiquid, i.e., g = 0, then r b = r.
An alternative way to formalize legal restrictions is to describe the government as
a subset of sellers in the economy whose trading behaviors are specified exogenously.
Government agents refuse to accept government bonds in payment and can also influence terms of trade by choosing the price at which it will sell its output. In general, the
interest rate on government bonds would depend on the size of the government, i.e., the
fraction of sellers who are government agents, as well as the government’s trading policy,
i.e., the price of its output. We can rationalize the coexistence of interest-bearing bonds
and money if bonds are illiquid. But so far we have not explained why bonds should be
illiquid:The presence of illiquid bonds does not generate better allocations.
In order for government bonds to pay interest, they must be illiquid. And, up to this
point, we have arbitrarily imposed a form of illiquidity on bonds. But why should bonds
be illiquid? An interesting way to justify the illiquidity of bonds is to show that their
presence can, in fact, raise society’s welfare.To make this point, the environment can be
modified as follows: Suppose that each period is divided into three subperiods; the early
morning, the day, and the night.The day and night are as before. During the early morning,
however, buyers receive a preference shock: with some probability they want to consume, and with the complement probability they do not want to consume. The buyer’s
portfolio of money and bonds is chosen at night, before the buyer knows whether he will
want to consume the next period, and buyers are allowed to readjust their portfolios in
the early morning after they receive their preference shock. If bonds are perfectly liquid,
there is no reason to trade money for bonds in the early morning, since they are perfect
substitutes. If bonds are made illiquid, however, buyers will have an incentive to reallocate their portfolios in the early morning. Buyers with a positive preference shock will
34
FEDERAL RESERVE BANK OF CLEVELAND
sell bonds to be able to consume more during the day, while buyers with a negative preference shock will buy bonds that pay interest. An alternative interpretation of this result
is as follows. Buyers with a positive preference shock would like to borrow from buyers
with a negative shock. Since they cannot commit to repay their debts, the (private) loan
market is inactive. In contrast to private agents, the government can commit to pay off in
the future.Therefore, instead of selling their own debt, buyers with a positive preference
shock sell the government’s debt and buyers with a negative shock buy it.
Related Literature
Kiyotaki and Wright (1989) constructed an environment where all commodities can
serve as means of payment and where agents can choose which one to use. Models of
commodity monies include Sargent and Wallace (1983), Burdett and Wright (2001), Velde, Weber, and Wright (1999), and Li (2003). The existence of a monetary equilibrium
when agents have access to a linear storage technology was studied by Wallace (1980)
in the context of an overlapping-generations model. Lagos and Rocheteau (2004) studied how money and capital can compete as means of payment in a search environment.
Shi (1999), Aruoaba and Wright (2003), and Aruoba, Waller, and Wright (2004) described
search economies where agents can accumulate capital but capital is illiquid, in the sense
that it cannot be used as a means of payment in bilateral matches.
The first search-theoretic environment with two currencies was proposed by Matsuyama, Kiyotaki, and Matsui (1993) and was extended by Zhou (1997) to allow for currency exchange. They considered a two-country economy where the two countries are
imperfectly integrated and establish conditions on parameters under which one currency
is used as an international currency.They also showed that a uniform currency dominates
in terms of welfare. Head and Shi (2003) extended the previous analysis to propose a
dual-currency economy where terms of trade are endogenous and monies are perfectly
divisible. Legal restrictions were introduced by Li and Wright (1998). Trejos and Wright
(1996) and Craig and Waller (2000) survey the search literature on dual-currency payment systems.The proposition of the indeterminacy of the exchange rate was established
by Karakeen and Wallace (1981) in an overlapping-generations economy.
The coexistence of money and bonds has been discussed by Bryant and Wallace (1979),
Aiyagari,Wallace, and Wright (1996), Kocherlakota (2001), Shi (2004a,b), and Wallace and
Zhu (2004). According to Bryant and Wallace (1979), interest-bearing government bonds
are socially inefficient because of intermediation costs to transform large-denomination
bonds into perfectly divisible intermediary liabilities.Aiyagari,Wallace, and Wright (1996)
introduced government agents to explain why government bonds are sold at a discount.
The effects of the government’s trading behavior on the equilibrium outcome have been
studied more generally in Li and Wright (1998). Kocherlakota (2001) established the
proposition according to which illiquid bonds can raise society’s welfare when agents
are subject to idiosyncratic shocks.
35
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
Banking
Payment systems involve financial intermediaries—in particular, banks—that not only
supply liabilities that can circulate as mediums of exchange but also provide credit to finance productive investments. What kind of economic environments give rise to intermediaries that can issue debt that can be used as a mediums of exchange? What is the relationship between inside (bank) money and outside (fiat) money? Although the theory
of banking is still in a very early stage of development, we present here two models that
attempt to address these questions.
Banks and Safekeeping Services
Historically, banks have played a role in providing safekeeping services by storing gold
and silver specie in their vaults. In exchange for their assets, agents receive bank notes
that are much safer to hold. Since bank notes can serve as means of payment—and, as
a result, circulate in the economy—banks are able to loan out some the assets that they
hold for safekeeping.
We now describe a simple model that can account for a demand for safe bank notes.
Buyers can hold two types of assets: commodity money and bank notes. Commodity
money can be “minted” from general goods, according to a linear technology. In particular,
each unit of night good can be transformed into one unit of commodity money, and this
process is fully reversible (e.g., “coins” can be minted or melted at no cost, and melted
coins can be consumed). Commodity money is exactly like the capital described in the
section on Money and Capital, when A = 1.
At night, banks exchange bank notes for commodity money. Let φ represent the value
of a bank note in terms of general goods. For convenience, we assume that each bank
note is a claim to one unit of commodity money.Therefore, φ = 1. Banks charge agents a
per period fee equal to γ , measured in terms of the general good, for each unit of commodity money deposited in their vaults.The fee is payable to the bank at the end of the
following period.
Buyers and sellers are matched at the beginning of each period. A buyer is matched
with a seller with probability one. Sellers are divided into two types: honest sellers and
thieves. The fraction of sellers who are thieves is equal to p. Honest sellers can produce
the search good during the day market (in exchange for money) and consume at night.
Thieves cannot produce anything during the day, but have the ability to steal a buyer’s
commodity money.We assume that whenever a thief meets a buyer, he can steal a fraction
λ of the buyer’s commodity money:Thieves are unable to steal bank notes.Thieves, like
honest sellers, consume at night.An agent can sell some or all of his commodity money to
a bank, receiving one bank note for each unit of commodity money deposited.
36
FEDERAL RESERVE BANK OF CLEVELAND
Consider a buyer with a portfolio of z units of commodity money and b bank notes at
the beginning of a period.The buyer’s lifetime expected utility satisfies
(75)
V b ( z , b ) = pW b [(1 − λ ) z , b] + (1 − p ){u[q ( z , b )] + W b [ z − dz ( z , b ), b − db ( z , b )]}.
Equation (75) has the following interpretation: A buyer meets a thief with probability p.
The thief steals a fraction λ of the buyer’s commodity money, so that the buyer enters
the night market with (1 − λ ) z units of commodity money and b units of bank notes.
With probability 1 – p, the buyer meets an honest seller, in which case he trades d z units
of commodity money and d b bank notes for q units of the search good. Terms of trade
(q,d z ,d b ) depend on the buyer’s portfolio (z,b).
The utility of the buyer at night satisfies
(76)
{
}
W b ( z , b ) = z + b + max − zˆ − bˆ(1 + βγ ) + β V b ( zˆ, bˆ ) .
ˆz .bˆ
According to (76), at night the buyer chooses his portfolio ( zˆ, b̂ ) of commodity money
and bank notes for the following period. Recall that one unit of commodity money can be
melted into one unit of general goods at no cost, one unit of general goods can be minted
into one unit of commodity money, and, for each bank note, the buyer must pay a fee of
γ to the bank in the following period.
The terms of trade in bilateral matches are determined by take-it-or-leave-it offers
by buyers. Therefore, q = q * if z + b > c ( q *); otherwise q = c −1 ( z + b ). Substituting
V b (z,b) by its expression given by (75) into (76), the buyer’s portfolio choice problem
at night can be reformulated as
(77)
{
{
}}
max − zˆ( r + pλ ) − bˆ( r + γ ) + (1 − p ) u ⎡⎣q ( zˆ, bˆ) ⎤⎦ − c ⎡⎣q ( zˆ, bˆ ) ⎤⎦ .
ˆz ,bˆ
The terms r + pλ and r + γ in (77) represent the costs of holding commodity money
and bank notes, respectively. Since commodity money and bank notes are perfect substitutes in the day market, buyers choose to hold all their wealth in bank notes whenever pλ > γ and they choose to hold all their wealth in commodity money whenever
pλ < γ . In the knife-edge case where pλ = γ , buyers are indifferent between holding
bank notes or money.These conditions are quite intuitive.The cost of depositing one unit
of money at the bank is γ , and the benefit of holding a bank note is to avoid the loss of
a fraction λ of one’s monetary wealth when meeting with a thief, an event that occurs
with probability p. So, for example, if pλ < γ , the cost associated with holding bank notes
exceeds the benefit; therefore, agents will hold all of their wealth in commodity money. In
the case where pλ > γ , only bank notes are used in the day to implement trades.Those
bank notes are fully backed by commodity money in banks’ vaults.
37
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
Private Money
We now consider an environment where banks provide two types of services:They help
to finance productive investments and they issue notes that can serve as mediums of exchange.The benchmark model will be slightly modified in order to accommodate banking. Instead of having marketplaces that open sequentially within a period, the “day–
night” structure, we will assume that within a period two sectors are open simultaneously.
There is a search sector, which mirrors the search market in the benchmark model, where buyers and sellers are bilaterally matched and sellers can produce the search
good. There is also a banking sector, where agents can trade with a bank. Trading with a
bank entails either selling an investment project to the bank in exchange for a bank note,
where the input for an investment project is the general good, or selling a bank note to
the bank in exchange for the general good, which is obtained by liquidating an investment project from the bank’s portfolio.The bank sector here mirrors the night subperiod
in the benchmark model.
At the beginning of each period, agents are allocated randomly between the two sectors. With probability π , an agent visits the search sector; with probability 1 − π , the
agent visits the banking sector. If an agent enters the search sector, then with probability
1
2
he is a buyer and with probability
1
2
he is a seller. (An agent’s preferences over search
goods are the same as in the benchmark model and are given by u(q) – c(q). In the
search sector, buyers and sellers are matched with probability one. Whether or not a
buyer and a seller can trade depends on the each agent’s portfolio of assets at the time
they are matched.
In the banking sector, an agent always has the opportunity to fund an investment
project. The investment project is indivisible and costs y units of the general good to
be initiated; that is, to initiate the project, the agent must produce y units of the general
good at a utility cost of y.The project pays off Ay in terms of utility when it is liquidated,
where A > 1. The investment project, which is not portable, is “deposited” at the bank,
in exchange for an indivisible bank note.The investment project can be liquidated in any
period after the project is initiated; an investment project will be liquidated (and consumed) if an agent presents the bank with a bank note. We assume that an agent cannot
hold more than one bank note at a time; denote ρ as the proportion of agents holding a
bank note at the end of a period.
We focus on steady-state equilibria where bank notes circulate in the search sector.
Let Vi denote the value of an agent in the search sector holding i ∈ {0,1} bank notes at
the beginning of a period, and Wi the value of an agent in the banking sector. Consider
first an agent in the banking sector.The value functions W0 and W1 satisfy the following
Bellman equations:
38
FEDERAL RESERVE BANK OF CLEVELAND
(78)
W0 = max{− y + πβ V1 + (1 − π ) β W1 ,πβ V0 + (1 − π ) β W0 },
(79)
W1 = Ay + max{− y + πβ V1 + (1 − π ) β W1 ,πβ V0 + (1 − π ) β W0 }.
According to (78), an agent without money in the banking sector can choose either to
fund an investment project or not; if a project is funded, then the agent receives a bank
note. If the agent funds an investment project, he produces y units of the general good
and starts the next period with one bank note. In the subsequent period, the agent goes
either to the search sector with probability π or, with probability 1 − π , remains in the
banking sector. If the agent chooses not to fund an investment project, then he starts the
next period with no bank note.According to (79) an agent with a bank note in the banking sector redeems this bank note for Ay units of consumption goods.The agent can then
decide whether or not to fund a new investment project. Equations (78) and (79) imply
that
(80)
W1 = W0 + Ay.
We are interested in steady-state equilibria where bank notes circulate in the search
sector. For such an equilibrium to occur, some agents who enter the banking sector must
be willing to fund an investment project; otherwise, the stock of bank notes would fall
to zero, as bank notes are redeemed over time. It must also be the case that not all agents
who enter the banking sector want to fund an investment project; otherwise, all agents in
the economy will eventually end up with a bank note and no trades would take place in
the search sector. In equilibrium, agents must be indifferent between funding an investment and not funding an investment, which implies that
(81)
− y + πβ V1 + (1 − π ) β W1 = πβ V0 + (1 − π ) β W0 .
Now let’s turn to the value functions of an agent in the search sector.The value functions for an agent in the search market with and without a bank note, V1 and V0 , respectively, satisfy the following two Bellman equations:
(
V0 = ρ2 [−c ( q ) + πβ V1 + (1 − π ) β W1 ] + 1 −
(83)
V1 = 1−2ρ [u( q ) + πβ V0 + (1 − π ) β W0 ] + 1 − 1−2ρ [πβ V1 + (1 − π ) β W1 ].
(
ρ
2
)[πβ V
(82)
0
+ (1 − π ) β W0 ],
)
According to (82), an agent with no bank note becomes a seller with probability
1
2
and,
with probability ρ , is matched with a buyer who has a bank note. In this case, the agent
produces q units of the search good and starts the following period with one bank note,
after which he will go either to the banking sector, with probability π or to the search
39
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
sector, with probability 1 − π . With probability 1 − ρ / 2, an agent with no bank note
does not trade in the search market and starts the next period as an agent with no bank
notes. Equation (83) has a similar interpretation.An agent holding a bank note becomes a
buyer with probability
1
2
and, with probability 1 − ρ , is matched with a seller who does
not have a bank note. In this case the agent gets q units of the search good and starts the
subsequent period without a bank note. With probability 1 − (1 − ρ / 2), an agent with a
bank note does not trade in the search market and starts the next period once again as
an agent with a bank note.
The terms of trades in bilateral matches in the trading sector are determined by takeit-or-leave-it offers by buyers.A seller is indifferent between accepting a trade or rejecting
it, if
−c ( q ) + πβ V1 + (1 − π ) β W1 = πβ V0 + (1 − π ) β W0 .
(84)
According to (84), the seller is indifferent between producing q units of the search good
for the buyer, starting the next period with a bank note or producing nothing, starting the
next period without a bank note.
From equations (78), (81), (82), and (84), we obtain that
W0 = V0 = πβ V0 + (1 − π ) β W0 .
It is easy to see that V 0 = W 0 = 0. Hence, from (81) and (84) we can deduce that c(q) = y.
We will assume that y ≤ c ( q *), so that in the search sector there is no need to introduce
lotteries over money transfers. Hence, the purchasing power of a bank note in the search
sector is determined by the cost of funding an investment project in the banking sector,
i.e., one bank note buys q = c −1 ( y ) units of the search good.
Finally, we determine the measure ρ of agents with bank notes in the steady state.
Equation (81) represents the indifference between investing and not investing in the
project. Since V0 = W 0 = 0, and, from (80), W 1 = Ay, the value of holding a bank note in
the search sector, as expressed by equation (81), can be rearranged to read
⎛ 1 − (1 − π ) β A ⎞
V1 = ⎜
⎟ y.
πβ
⎝
⎠
(85)
Equation (85) implies that, as the return of an investment project increases, the value of
holding a bank note in the trading sector must fall in order to keep agents indifferent between funding a project or not.The value of holding a bank note in the search sector, as
expressed by (83), can be simplified to read
V1 =
(86)
40
1− ρ
⎛ 1− ρ ⎞
u( q ) + ⎜1 −
⎟ y.
2
2 ⎠
⎝
FEDERAL RESERVE BANK OF CLEVELAND
One can obtain an expression for ρ by equating the right-hand sides of equations (85)
and (86); after some rearranging we obtain
(87)
⎛1− β A
⎞ 2y
.
1− ρ = ⎜
+ A − 1⎟
⎝ πβ
⎠ u( q ) − y
So ρ ∈ ( 0,1) if
(88)
1 − πβ {1 + [u( q ) − y]/ 2 y}
1 − πβ
< βA<
.
1−π
1−π
According to (88), the return of an investment cannot be too high or too low for bank
notes to circulate. If the rate of return on an investment is too low, agents do not find it
worthwhile to fund projects; if it is too large, all agents want to fund investment projects,
so that no trades take place in the trading sector.
This model generates an equilibrium where banks play two roles. They “finance” investment projects by purchasing an illiquid asset with bank notes and, as a result, they
provide the economy with a liquid asset: bank notes, which facilitate trades between
buyers and sellers.
Related Literature
The model of banking based on crime was proposed by He, Huang, and Wright (2005).
The model has been extended to endogenize the rate of crime (i.e., the number of thieves
in the economy) and to allow for a money multiplier.The model of private money is similar to the one in Williamson (1999, 2002). In addition, Williamson shows that even if private monies can be subject to lemon problems and counterfeiting, the introduction of fiat
money can decrease welfare. Cavalcanti and Wallace (1999a,b) and Wallace (2004) proposed a model of private money where a subset of agents, called banks, are monitored.
Those agents can issue notes that can be used as a medium of exchange by nonbank
agents.They show that the presence of inside money enlarges the set of allocations that
are incentive-feasible.A model of private money with reserve management has been provided by Cavalcanti, Erosa, and Temzelides (1999).
Settlement
In actual economies, fiat money plays a dual role: It serves as a medium of exchange to
facilitate trade and it is used to settle debt. In this section, we will consider economies
where monetary exchange and credit coexist, and where debt must be settled with money.The fact that money is required to settle debt can generate liquidity problems in credit
markets. These liquidity problems will affect the relative price of money, which in turn
can distort the allocation of resources. Hence, liquidity problems in credit markets can
spill over into product markets. This line of reasoning has been used to justify the need
41
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
for an elastic supply of currency, which is one of the founding principles of the establishment of the Federal Reserve System. In this section we examine a model where debt obligations must be settled with money (that is, a debt obligation cannot be settled by simply
producing output). We introduce realistic frictions into the settlement process which, in
turn, generate liquidity problems in credit markets.The specific nature of the settlement
friction is a mismatch between the time a debtor can repay his debt and the time a creditor needs to be repaid.
The Environment
We consider an environment in which credit and money coexist and where money is
used to settle debt obligations. In order to present the ideas in the most economical way,
we make a slight departure from the benchmark model.We now divide a period into four
subperiods: morning, day, night, and late night.As in the benchmark model, the day subperiod is characterized by bilateral matching with the production and consumption of the
search good and the night subperiod is characterized by the production and consumption of the general good. In terms of the two new subperiods, the morning subperiod
mirrors the night subperiod in that agents produce and consume the general good; the
late-night subperiod is the time where agents settle their debts. In the late night, debtors
and creditors ultimately go to a central meeting place in order to settle, with money, any
outstanding debt that was issued during the previous subperiods.
In contrast to the benchmark model, where agents are infinitely lived, now agents live
for only four subperiods. Buyers are born at the beginning of a period—in the morning—and die after the settlement phase in the late night of the same period. Sellers are
born in the day subperiod and die at the end of the morning of the subsequent period. So
in any particular morning, the economy will be populated with “young” buyers and “old”
sellers; in all other subperiods, the economy is populated with buyers and sellers who are
born in the same period. We assume that debts can only be issued in bilateral meetings
and agents can commit to repay their debts.
During the day, buyers and sellers are matched, where buyers consume the search
good while sellers produce it. In both the morning and night subperiods, the market for
general goods is competitive. In contrast to the benchmark model, we assume that buyers
can produce the general goods but obtain no utility from consuming them, while sellers
consume market goods but cannot produce them. Buyers’ preferences are described by
the instantaneous utility function
U b ( q, y ) = u( q ) − y,
where y is the buyer’s total production of the general good and q is the consumption of
the search good. Similarly, the preferences for the seller are given by
42
FEDERAL RESERVE BANK OF CLEVELAND
U s ( q , x ) = −c ( q ) + x ,
where x is the seller’s total consumption of the general good and q is the amount of the
search good that is produced. Note that agents do not discount utility across subperiods.
In order to capture the coexistence of money and credit, we assume that buyers are
heterogenous in terms of when they can produce. Half of the buyers can only produce
in the morning market, and the other half can only produce in the night market. We call
the former early producers and the latter late producers. Early-producer buyers can use
money to trade in the day. In contrast, late-producer buyers are unable to obtain any
money in the morning. But they are able to repay any debt that is issued in the day by
producing for money at night. For simplicity, we eliminate any search-matching frictions
by setting the matching probability σ to one.
To summarize, the timing of events and the pattern of trade will be as follows: At the
beginning of a period, a measure one of buyers is born. Half of these buyers, the early
producers, can produce in the morning. In a competitive market, buyers produce general
goods in exchange for money and “old” sellers exchange money for the general good. Old
sellers die at the end of the morning and are replaced by a measure one of new-born
sellers at the beginning of the day. In the (day) search market, each buyer is matched
with a seller. Half of the buyers—the early producers—trade with money and the other
half—the late producers—trade with credit. In order to settle their debts at the end of the
night period, buyers who traded with credit will produce general goods in exchange for
money in a competitive market; sellers exchange money for the general good. In the late
night, buyers and sellers arrive in a meeting place for the purpose of settling debts. Sellers
who receive money in the late-night settlement subperiod will spend it in the morning of
the following period before they die.
We focus on stationary equilibria. Money is traded for general goods in competitive
markets but in the two different subperiods, so we distinguish two prices for money. Let
φ am be the price of money in terms of general goods in the morning and φ pm the price
of money at night.
Frictionless Settlement
We consider first the case where there are no frictions in the late-night settlement phase:
All debtors and creditors arrive simultaneously at a central meeting place and all debts
are settled instantaneously.
Consider first a buyer who is an early producer.This buyer produces general goods in
the morning to get m units of money, which he spends in a bilateral match in the day for
q m units of the search good. The quantity q m is determined by a take-it-or-leave-it offer
m
am
pm
by the buyer.The seller’s participation constraint is −c ( q ) + max(φ ,φ )m ≥ 0, since
a seller can spend the money he receives either at night or in the following morning. If
43
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
φ pm < φ am , sellers will spend their money in the following morning; but this outcome
is inconsistent with the clearing of the general goods market at night. Therefore,
max(φ am ,φ pm ) = φ pm . (Note that the buyer has no incentive to use debt because debt
would have to be repaid with money at the end of the period. But the value of money
received in the settlement phase is φ am ≤ φ pm . ) The early-producer buyer’s choice of
money balances solves the problem
(89)
max[−φ am m + u( q m )],
(90)
s.t. c ( q m ) = φ pm m.
m ,q m
Substituting m from (90) into (89) and taking the first-order condition for q m , we obtain
u′( q m ) φ am
=
.
c ′( q m ) φ pm
(91)
From (91) q m = q * if and only if φ am = φ pm ; if φ pm > φ am , then q m > q * .
At the end of the morning, all of the money in the economy is held by half of the buyers, i.e., the early-producer buyers. Hence, equilibrium in the money market (or in the
general goods market) in the morning implies M = m/2 and, from the seller’s participation constraint (90), q m satisfies
c ( q m ) = 2 Mφ pm .
(92)
Now let’s turn to the problem of a late-producer buyer. In his bilateral match during
the day, the late-producer buyer must issue an IOU to pay for the search good, which will
be repaid in the late-night settlement subperiod.The buyer repays the debt by producing
output for money at night.The terms of trade in the day match are determined by a takeit-or-leave-it offer (q d ,d) by the buyer, where q d is the amount of search good produced
by the seller and d is the amount of dollars that the buyer commits to repay in the latenight settlement subperiod. (It might be convenient to think of the “m” in offer (q m ,m)
as referring to a buyer who uses money to purchase search goods and the “d” in offer
(q d ,d) as referring to a buyer who uses debt.) The offer (q d ,d) is given by the solution
to the buyer’s problem
(93)
max[u( q d ) − φ pm d ]
(94)
s.t.
q d ,d
− c ( q d ) + φ am d = 0.
The seller’s participation constraint has the price φ am , since the seller spends the
money obtained in the late-night settlement subperiod the next morning.The solution to
the buyer’s problem (93)–(94) is
44
FEDERAL RESERVE BANK OF CLEVELAND
(95)
u′( q d ) φ pm
=
.
c ′( q d ) φ am
From (95), q d = q * if and only if φ am = φ pm . If φ am < φ pm , then q d < q * .
If φ pm > φ am , then sellers holding money at the beginning of the night will spend all of
it so that at the end of the night all of the money is held by the late-producer buyers, i.e.,
d/2 = M. If φ pm = φ am , then sellers holding money are indifferent between spending it
at night or in the following morning. In this case, d / 2 ≤ M . From the seller’s participation constraint (94),
(96)
⎧=⎫
c (q d ) ⎨ ⎬ 2 Mφ am
⎩≤ ⎭
⎧>⎫
if φ pm ⎨ ⎬ φ am .
⎩=⎭
A steady-state equilibrium is a list ( q m , q d ,φ am ,φ pm ) that satisfies (91), (92), (95), and
(96). It is easy to check that q m = q d = q * and φ am = φ pm = c ( q *) / 2 M is an equilibrium. (And one can show that this is the unique equilibrium for some specifications for u
and c, e.g., c(q) = q and u( q ) = 2 q ). In this equilibrium, the price of money is the same
in the morning and night markets, and the efficient quantity of the search good q * is
traded in all matches. Note that early-producer buyers, who use money, are as well off as
late-producer buyers, who use credit. So, if buyers could choose using money or debt in
bilateral matches, they would be indifferent between the two means of payment.
Settlement and Liquidity
We now introduce settlement frictions. Settlement frictions are captured by assuming
that debtors and creditors arrive and leave the late-night settlement period at different
times. So a liquidity problem may arise if a creditor leaves the late-night settlement period before his debtor arrives. To be more specific, the timing during the late-night settlement period is as follows: All of the creditors and a fraction α of debtors arrive at the
beginning of the late-night settlement period. Then a fraction δ of the creditors depart,
after which the remaining (1 − α ) debtors arrive. Finally, the remaining (1 − δ ) creditors
and all of the debtors leave the late-night settlement period; debtors, who are buyers, all
die, and creditors move into the morning of the next period. We will sometimes refer to
creditors (debtors) as being early-leaving (-arriving) and late-leaving (-arriving), where the
meaning is obvious. These arrival and departure frictions will create a need for a resale
market for debt during the late-night settlement period.We denote ρ as the value of onedollar of debt in terms of money in this market.
There are agents in the economy who are neither creditors nor debtors; for example,
sellers who produce search goods for money during the day. These sellers may have an
incentive to forgo (some) consumption in the night market and instead provide liquidity
in the late-night settlement period. More specifically, sellers who have money balances at
the beginning of the night may, in the late-night settlement period, want to buy the IOUs
45
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
of early-leaving creditors that will be repaid by late-arriving debtors. For simplicity, and
with no loss in generality, we assume that sellers with money who choose not to spend
all of it at night always arrive at the beginning of the late-night settlement period and
always stay until the end.
The problem of an early-producer buyer must now take into account the possibility
that a seller who receives money for producing search goods during the day may want to
use some of it to purchase debt in the late-night settlement period.A seller who receives
pm
units
one unit of money in a bilateral match during the day can spend it at night for φ
of the general good, or he can buy 1/ ρ IOUs in the late-night settlement period and then
purchase φ am / ρ units of the general good the following morning. Since φ pm ≥ φ am / ρ
is required for the night money market to clear, the seller’s participation constraint is still
given by c ( q m ) = φ pm m. Hence, the early-producer buyer’s day-bargaining problem is
(still) given by (89)–(90). As well, the solution to this problem is characterized by (91),
and the quantity produced in this day match, q m , satisfies equation (92).
The late-producer buyer’s day-bargaining problem must also take into account the
frictions that affect settlement in the late-night period. More specifically, since creditor
sellers may have to sell their IOUs at a discount if they need to leave the settlement phase
before their debtors arrive, the participation constraint of a seller who trades output for
debt will be affected. Let θ denote the expected value to the seller of a one-dollar IOU
expressed in dollars.The buyer’s bargaining problem can be represented by
(97)
max[u( q d ) − φ pm d ]
(98)
s.t. c ( q d ) − θφ am d = 0,
q d ,d
where θ satisfies
θ = δα + δ (1 − α ) ρ + (1 − δ )
(99)
α
+ (1 − δ )(1 − α ).
ρ
Equation (99) has the following interpretation.With probability δ a seller holding a onedollar IOU must leave the late-night settlement place early. If his debtor has already arrived, an event which occurs with probability α , the IOU is redeemed for one dollar. Otherwise, the IOU is sold at the price ρ .With probability 1 − δ , the seller with a one-dollar
IOU does not need to leave early.Therefore, the IOU is redeemed for one dollar, irrespective of the arrival time of his debtor. However, if the debtor arrives early, an event which
occurs with probability α , the creditor can use the dollar he receives to buy 1/ ρ IOUs
that will be redeemed for 1/ ρ dollars at the end of the settlement phase.The solution to
the late-producer buyer’s bargaining problem (97)–(98) is given by
46
FEDERAL RESERVE BANK OF CLEVELAND
(100)
u′( q d ) φ pm
=
.
c ′( q d ) θφ am
If φ pm > φ am / ρ , then sellers who hold money at the beginning of the night prefer spending it at night rather than the following morning. As a consequence, the equilibrium of
the money market implies d/2 = M. If, however, φ pm = φ am / ρ , then sellers are indifferent between spending money at night or in the morning, so that d / 2 ≤ M . Since
c ( q d ) = θφ am d , we have
(101)
⎧=⎫
c ( q d ) ⎨ ⎬ 2 Mθφ am
⎩≤ ⎭
⎧>⎫φ
if φ pm ⎨ ⎬
.
⎩=⎭ ρ
am
Let us turn to the equilibrium of the second-hand debt market in the late-night settlement period. Denote Δ = M − d / 2 as the funds that sellers with money retain at night
so that they can purchase second-hand IOUs in the late-night settlement period. Note
that ρ , the price of IOUs in the late-night settlement period, cannot be greater than one;
otherwise, no one would buy second-hand IOUs.Therefore, ρ ≤ 1. The supply of “funds,”
i.e., money, by creditors who are repaid early and leave late is (1 − δ )α d / 2. (Recall that
half of the sellers in the day are paid with IOUs.) The supply of funds of sellers who transacted with money in the day is Δ .The demand of funds from creditors who leave early
is δ (1 − α ) d ρ / 2. From this, the market-clearing price ρ satisfies
(102)
⎡ (1 − δ )α d + 2Δ ⎤
ρ = min ⎢
,1⎥ .
⎣ δ (1 − α ) d
⎦
A steady-state equilibrium is a list (φ am ,φ pm , ρ , q m , q d ) that satisfies (91)–(92) and
(100)–(102). We distinguish between two types of equilibria: one where ρ = 1 and one
where ρ < 1. If ρ = 1, then there is no liquidity shortage in the late-night settlement
period: Second-hand IOUs are sold at par, ρ = 1 and θ = 1 . The equilibrium conditions
are then identical to those of the economy without any settlement frictions in the latenight settlement period, so that qm = qd = q * and Δ = 0. From (102), ρ = 1 requires
that (1 − δ )α / δ (1 − α ) ≥ 1, or equivalently, α ≥ δ . Intuitively, there is no liquidity shortage if the measure of debtors who arrive early in the settlement place, α , is larger than
the measure of creditors who leave early, δ . Creditors who are repaid by early-arriving
debtors can use this money to purchase the IOUs of creditors who need to sell them, the
earlier-leaving creditors.
Let us now turn to equilibria where second-hand debt is sold at a discount in the latenight settlement period, i.e., where ρ < 1. The equilibrium is liquidity-constrained in the
sense that the amount of money available at the late-night settlement period just prior to
the departure of the early-leaving creditors is insufficient to clear debts at their par value.
One can first show that if ρ < 1, then Δ > 0, which implies that sellers with money pro47
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
vide additional liquidity in the late-night settlement period by only spending a fraction of
their money balances at night.To see this, suppose to the contrary that Δ = 0. Then, from
(102), ρ = (1 − δ )α / δ (1 − α ) and, from (99), θ = 1. But then, the equations determining
( q m , q d ,φ am ,φ pm ) are exactly the same as those derived in the model with no settlepm
am
ment frictions implying that φ = φ , which contradicts the no-arbitrage condition
φ pm ≥ φ am / ρ . Therefore, Δ > 0, when ρ < 1.
When ρ < 1 and Δ > 0, condition (101) implies that φ pm = φ am / ρ , which means that
sellers with money are indifferent between spending money at night or the following
morning. Since φ pm > φ am and θ < 1, equations (95) and (100) imply that the quantities
traded in the day’s bilateral matches must satisfy q d < q * < q m : Buyers who trade with
money in the day receive more output than those who trade with credit.
The liquidity shortage during the late-night settlement period affects the allocation of
resources by making money more valuable at night than in the morning. Indeed, since
unsettled debts are sold at a discount during the late-night settlement period, there is an
additional demand for liquidity at night. The fact that money is more valuable at night
allows early-producer buyers to consume more, whereas the consumption of late-producer buyers falls.
Settlement and Default Risk
We now introduce an idiosyncratic risk of late-producing buyers defaulting on their debt.
We formalize the default risk by assuming that the debtor can produce at night with
probability γ and, with probability (1 − γ ), he is unable to produce and, hence, defaults
on his debt. A debtor does not know whether or not he will default before night.Therefore, in the day, buyers and sellers have symmetric information in their bilateral matches.
We assume that debtors who are unable to produce, and hence default on their debt, do
not show up at the late-night settlement period.
The problem of an early-producer buyer is still given by (89)–(90) since transactions
are conducted with money. The problem of a late-producer buyer, however, is now
given by
(103)
max[u( q d ) − γφ pm d ]
(104)
s.t. − c ( q d ) + θ ′φ am d = 0.
q d ,d
According to (103), the buyer receives q d from the seller and repays his debt with probability γ . According to (104), the seller who receives a promise of d dollars can expect
to have θ ′d dollars, at the end of the period, that can be spent the following morning,
where θ ′, the expected value of a one-dollar IOU, now reflects not only any settlement
frictions but also the possibility of default.The solution to (103)–(104) implies that
48
FEDERAL RESERVE BANK OF CLEVELAND
(105)
u′( q d ) γφ pm
=
.
c ′( q d ) θ ′φ am
In the absence of any settlement frictions, it will be the case that θ ′ = γ and, therefore,
q d = q * . The default risk is reflected in the amount of money that the buyer commits to
repay, but the quantity of output traded in bilateral matches remains efficient.
Consider a seller who has money at night and who contemplates buying a second-hand IOU in the late-night settlement period. The probability that a secondhand IOU will be repaid conditional on the fact that the debtor did not arrive early is
γ (1 − α ) /(1 − γα ) . (There are three possible events for an IOU: It is not repaid, which
occurs with probability 1 − γ ; it is repaid early, which occurs with probability γα ; it is
repaid late, which occurs with probability γ (1 − α ).) Therefore, the price ρ of a secondhand IOU cannot be greater than γ (1 − α ) /(1 − γα ) , in order for debt to clear in the resale
market. If liquidity in the central clearing market is not constrained, then ρ = ρ *, where
(106)
ρ* =
γ (1 − α )
.
1 − γα
From (106), in the absence of a liquidity constraint, the price of IOUs simply reflects the
probability of default (conditional on the fact that they have not been redeemed early).
The expected value of an IOU in the day satisfies
(107)
⎛
⎞
ρ*
+ (1 − δ )(1 − α ) ⎟ + δ (1 − γα ) ρ .
θ ′ = γ ⎜ δα + (1 − δ )α
ρ
⎝
⎠
Equation (107) has the following interpretation:The debtor arrives early with probability
γα . With probability δ the creditor leaves early, in which case he gets the par value of
the IOU.With probability 1 − δ , he can stay late and use his money to buy a second-hand
IOU at the price ρ .The probability that the second-hand IOU is repaid is ρ *. The debtor arrives late with probability γ (1 − α ). If the creditor can wait, with probability 1 − δ ,
he receives one dollar at the end of the settlement phase. Finally, if the debtor does not
arrive early (because he defaults or because he arrives late), an event that occurs with
probability 1 − γα , and if the creditor leaves early, with probability δ , then the creditor
sells his IOU at the price ρ . If liquidity is not scarce in the late-night settlement period,
ρ = ρ * and θ ′ = γ .
The equilibrium is not liquidity-constrained whenever γα (1 − δ ) ≥ δ (1 − αγ ) ρ * which,
from (106), is equivalent to α ≥ δ . This is precisely the condition we had in the absence
of default risk.The fact that the rate of repayment γ does not influence the condition for
a liquidity shortage can be explained as follows: Consider an increase in the repayment
49
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
rate. On the one hand, the number of creditors who are repaid early increases, so there
is more liquidity in the late-night settlement period. On the other hand, the price of second-hand debt increases in the late-night settlement period, so the demand for liquidity
is higher.The two effects just cancel each other.
Related Literature
The model of settlement presented in this section is closely related to the one by Freeman (1996a,b). Freeman considered an overlapping-generations economy with heterogenous agents. Some agents trade with debt, while others trade with money.As in our analysis, all debts are settled in a central clearing house. Freeman (1999) extends the model to
allow for aggregate default risk. Green (1999) shows that the role of the central bank as
a clearing house can be assumed by ordinary private agents. Zhou (2000) discusses the
literature. Temzelides and Williamson (2001) consider two related models, a model with
spatial separation and a random-matching model, and investigate different types of payment arrangements: monetary exchange, banking with settlement, and banking with interbank lending.They show that payment systems with net settlement generate efficiency gains, and interbank lending can support the Pareto-optimal allocation in the absence
of idiosyncratic shocks.
Policy and Payments
Optimality of the Friedman Rule
In this section, we determine the optimal growth rate of the money supply in the economy with divisible money that was examined in the section on Divisible Money. Let M t
represent the stock of money at date t, and π the constant rate of growth of the money
supply, i.e., Mt = Mt −1 (1 + π ). Money is injected or withdrawn in a lump-sum fashion in
the centralized market: If π > 0, then injections of money take place at the beginning of
the centralized market; if π < 0, then money is withdrawn at the end of the centralized
market.Without loss of generality, we assume that money transfers go only to the buyers.
We focus on steady-state equilibria where the real value of the money supply is constant
over time, i.e., φt Mt = φt +1 Mt +1 . In equilibrium, the price of money in terms of general
goods is falling at rate π .
To take into account that the price of money is not constant across time, we write the
value functions V b and W b as functions of the buyer’s real balances, φt mt . The Bellman
equations for V b and W b are given by
{
}
V b (φt mt ) = σ u( q (φt mt )) + W b (φt mt − φt dt ) + (1 − σ )W b (φt mt )
= σ {u( q (φt mt )) − φt dt } + W (φt mt )
b
50
FEDERAL RESERVE BANK OF CLEVELAND
and
W b (φt mt ) = max{φt mt − φt mt +1 + β V b (φt +1mt +1 )}.
mt +1
We assume that prices are determined by take-it-or-leave-it offers by buyers.This implies
that the quantity traded in a match satisfies c ( qt ) = φt mt whenever φt mt ≤ c ( q *). The
buyer’s problem at time t can be generalized to read
(108)
{
}
ˆ + φt +1m
ˆ + β ⎡σ ( u ( q (φt +1m
ˆ ) ) − c ( q (φt +1m
ˆ ) ) )⎤ .
max −φt m
⎣
⎦
m
ˆ
According to (108), the buyer at night who wishes to hold φt +1m
ˆ real balances the following day must produce φt m̂ of night goods. In the following day, he trades with probability
σ , in which case he extracts all the surplus of the match. Denote the buyer’s choice of
ˆ and the nominal interest rate as i, where 1 + i = (1 + r )(1 + π ).
real balances as z = φt +1m
Through the seller’s participation constraint, there is a one-to-one relationship between z
and q for all z ≤ c ( q *), i.e., z = c(q). Hence, the buyer’s problem (108) can be rewritten
more compactly as a choice for q,
(109)
max {−ic ( q ) + σ [u( q ) − c ( q )]},
q∈[ 0,q *]
The first-order condition to the buyer’s problem (109) is simply
(110)
u′( q )
i
=1+ .
c ′( q )
σ
This equation is similar to (53), except for the fact that the real interest rate, r, has been
replaced by the nominal interest rate, i. The cost of holding real balances, i, generates a
wedge that is proportional to the average length of time to complete a trade in the day
market, 1/ σ .
From (110), it is clear that the optimal monetary policy corresponds to the Friedman
rule, which requires the nominal interest rate i to be set equal to zero, or equivalently,
that the rate of growth of the money supply be negative and approximately equal to the
rate of time preference. Intuitively, by reducing the cost of holding real balances to zero,
the Friedman rule maximizes the demand for real balances and, therefore, the quantities
traded in individual matches.The allocation of the monetary equilibrium under the Friedman rule coincides with the socially efficient allocation, i.e., q = q * .
The result that the Friedman rule generates the first-best allocation is sensitive to the
choice of the bargaining solution. To see this, let’s assume that the terms of trade, (q,d),
are determined by the symmetric Nash solution, i.e., (q,d) is given by the solution to
51
POLICY DISCUSSION PAPERS
(111)
NUMBER 12, DECEMBER 2005
max[u( q ) − φ d ][−c ( q ) + φ d ] s.t. d ≤ m.
q ,d
If the constraint d ≤ m binds (and it will in equilibrium), then the relationship between
q and z = φ m is given by
(112)
z=
c ′( q )u( q ) + u′( q )c ( q )
.
u′( q ) + c ′( q )
Since there is a one-to-one relationship between z and q for all q ∈[0, q *], the buyer’s
choice of real balances can be rewritten as a choice of q, i.e.,
(113)
max {−iz ( q ) + σ [u( q ) − z ( q )]}.
q∈[ 0,q *]
At the Friedman rule, the buyer simply chooses q to maximize his surplus, u(q) – z(q).
Using (112), the buyer’s surplus can be re-expressed as
(114)
u( q ) − z ( q ) =
u′( q )
[u( q ) − c ( q )].
u′( q ) + c ′( q )
According to equation (114), the buyer receives a fraction u′( q ) /[u′( q ) + c ′( q )] of the
match surplus. Since u′( q ) /[u′( q ) + c ′( q )] is decreasing in q, it is easy to show that the
buyer’s surplus is decreasing in q when q is close to q * . Therefore, buyers choose an
inefficiently low value for q, even when the cost of holding real balances is zero. This
inefficiency is due to the nonmonotonicity of the Nash bargaining solution, according
to which the buyer’s surplus can fall even if the match surplus increases. However, despite the fact that real balances are “too low,” the optimal monetary policy is still the
Friedman rule.
Trading Frictions and the Friedman Rule
There are two dimensions associated with trading in a search environment: the quantities traded in individual matches, sometimes called the intensive margin, and the number
of matches, sometimes called the extensive margin. Monetary policy can affect both margins.As we have seen, monetary policy affects agents’ choices of real balances and, therefore, the intensive margin. But it can also affect agents’ costs of participating in the market
and, therefore, the extensive margin. In the previous section, we saw that the Friedman
rule takes care of the intensive margin because it maximizes buyers’ real balances and,
therefore, the quantities traded in bilateral matches. However, it is not at all clear whether
the Friedman rule generates an efficient extensive margin and, more generally, whether it
is the optimal monetary policy when the number of trades is endogenous.
In order to generate an extensive-margin effect, we have to slightly alter our benchmark model. Assume now that there is a unit measure of ex ante identical agents that
can choose to be buyers or sellers in the day market. Suppose, for instance, that there are
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FEDERAL RESERVE BANK OF CLEVELAND
two goods produced during the day: an intermediate good and a final consumption good,
which requires the intermediate good as an input.The final consumption good can only
be consumed by the agent who produces it, and agents have to specialize in one of the
production technologies.Agents who produce intermediate goods are sellers, while those
who produce final goods are buyers. Hence, the intermediate good will be produced and
traded in bilateral matches between buyers and sellers during the day.
The fraction of buyers is denoted by n, whereas the fraction of sellers is 1 – n. The
matching technology between buyers and sellers is as follows:A buyer meets a seller with
probability 1 – n, the fraction of sellers in the population. A seller meets a buyer with
probability n, the fraction of buyers in the population. As a consequence, the number
of matches in the day market is n(1 – n). The number of matches is maximized when
n = 12 .
Clearly, the way in which the terms of trade are determined will affect an agent’s
choice of which side of the market to participate in, i.e., whether to be a buyer or a seller.
Here we will assume that the terms of trade are determined by a simple proportional
bargaining solution, according to which the buyer’s surplus is a fraction θ ∈ ( 0,1) of the
total match surplus, i.e.,
(115)
u( q ) − φ d = θ [u( q ) − c ( q )],
where q is the level of intermediate goods produced in a match and d is the monetary
transfer from the buyer to seller. Furthermore, the trade (q,d) is pairwise Pareto-efficient
so that q = q * if φ m ≥ (1 − θ )u( q *) + θ c ( q *) and d = m otherwise. Assuming the constraint d ≤ m is binding, there is a simple relationship between q and z = φ m,
(116)
z = (1 − θ )u( q ) + θ c ( q ).
b
The buyer’s expected utility at night W (φ m ) satisfies a Bellman equation similar to
(113) except that σ is replaced by 1 – n. Hence, the buyer’s choice of real balances:
(117)
max {−iz + (1 − n )θ {u[q ( z )] − c[q ( z )}}.
z
The problem in (117) takes into account the fact that the buyer obtains a fraction θ of
the surplus of a match.The solution to the buyer’s problem (117) is given by
(118)
i
u′( q ) − c ′( q )
=
.
(1 − n )θ (1 − θ )u′( q ) + θ c ′( q )
An agent will be indifferent between a buyer and a seller if W b (φ m ) = W s (φ m ). Since
both value functions are linear, the choice of being a buyer or seller is independent of the
53
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
money balances of the agent when he enters the centralized market. After some calculation, the condition W b (0) = W s (0) yields
(119)
−i[(1 − θ )u( q ) + θ c ( q )] + (1 − n )θ [u( q ) − c ( q )] = n(1 − θ )[u( q ) − c ( q )].
Equation (119) can be explained as follows: The left-hand side is the buyer’s payoff. It is
the sum of two elements: the cost of carrying z = (1 − θ )u( q ) + θ c ( q ) real balances and
the expected surplus of a match.The right-hand side is the seller’s payoff, which is simply
the seller’s expected surplus of a match.
We first ask whether the Friedman rule generates the first-best allocation.As the nominal interest rate, i, tends to zero, equation (118) implies that q approaches q*, and equation (119) implies that n approaches θ , the buyer’s bargaining power. As before, the
Friedman rule generates the efficient intensive margin; this is true even though the buyer
does not have all the bargaining power. However, if θ is different from one-half, the
composition of buyers and sellers will be socially inefficient at the Friedman rule. The
requirement that θ = 12 for the composition of buyers and sellers to be efficient is related
to the Hosios (1990) condition for efficiency in search models, according to which efficiency is achieved when an agent’s bargaining power coincides with his contribution to
the matching process. A buyer’s bargaining power is θ , whereas his contribution to the
matching process is 1 – n, the fraction of sellers in the economy. From equation (119), it
is straightforward to see that the condition θ = n is satisfied at the Friedman rule.Therefore, if i = 0 and the buyer’s bargaining power coincides with his contribution to the matching process, then θ = 12 .
We now ask whether a deviation from the Friedman rule can be optimal.We measure
social welfare by W = n(1 − n )[u( q ) − c ( q )]. The effect of a change in the inflation rate
on the number of buyers is given by
(120)
dn
(1 − θ )u( q *) + θ c ( q *)
=−
< 0.
di i↓0
u( q *) − c ( q *)
As the cost of holding real balances increases, the number of buyers decreases.This effect
can be easily understood from equation (119). An increase in inflation has a direct negative effect on buyers, which is given by the right-hand side of (119). When a match occurs, this cost of holding the real balances is sunk and, hence, cannot be recovered by the
buyer.The effect on welfare of a deviation from the Friedman rule is given by
(121)
dW
di
i ↓0
⎛1
⎞
= −2 ⎜ − θ ⎟[(1 − θ )u( q *) + θ c ( q *)].
⎝2
⎠
So a deviation from the Friedman rule is optimal whenever θ > 12 . In this case, there are
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FEDERAL RESERVE BANK OF CLEVELAND
too many buyers and, therefore, too few trades.An increase in inflation reduces the number of buyers and increases the number of sellers, and, therefore, increases the number
of trades. Inflation also reduces the quantities traded in individual matches, but since this
has only a second-order effect close to the Friedman rule, overall welfare will increase.
Distributional Effects of Monetary Policy
Inflation can be beneficial when the number of trades is inefficient because it affects
agents’ decisions to participate in the market. Monetary policy can also have a positive effect on the extensive margin when the distribution of money balances is not degenerate.
To capture this distributional effect of monetary policy, we modify the benchmark model
as follows: Buyers and sellers live for only three subperiods—they are born at night and
die at the end of the following period.Agents can, therefore, potentially trade three times:
In the night when they are born, on the following day, and in the night just before they
die. Assume that agents do not discount across periods. This implies that the Friedman
rule corresponds to a zero inflation rate. In order to obtain a nondegenerate distribution
of money balances across agents, we assume that only a fraction p of newly born buyers
get access to the centralized general goods market, say, because they are productive.The
remaining 1 – p (unproductive) buyers are excluded and, therefore, are unable to acquire money to be able to trade in the next day’s market.
The problem of a newly born buyer who has access to the centralized night market,
which is similar to problem (108), is
(122)
ˆ
ˆ +φ m
ˆ + σ {u[q (φ m
ˆ
max {−φt m
t +1
t +1 )] − c[q (φt +1m )]}} .
m̂
Since the buyer has access to the centralized general goods market when he is born, he
can produce to accumulate the m̂ money balances he needs to trade in the next day’s
search market. If he doesn’t meet a seller during the day, he spends his money balances
in the night before he dies; if he does meet a seller, we assume that he captures the entire
ˆ as the buyer’s choice of real balances for the
surplus from the match. Denote z = φt +1m
next day’s search market.The buyer’s problem (122) can be simplified to read
(123)
max {−π z + σ {u[q ( z )] − c[q ( z )]}}.
z
The first-order condition for this problem is
u′( q )
π
=1+ .
c ′( q )
σ
Therefore, whenever the money supply is constant, i.e., when π = 0, newly born buyers who are not excluded from the centralized general goods market can trade q = q *
55
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
units of the search good the following day. However, if the money supply is constant, then
those (unproductive) buyers who are excluded from the night market when they are
young receive no money transfers and, therefore, cannot consume during the day.
Assume now the there is a steady state inflation and that money is injected into the
economy through lump-sum transfers to buyers. Let T t denote a transfer at night in period t.We have Tt = Mt +1 − Mt = π Mt . Let m t represent the money balances of buyers in
the morning of period t who had access to the centralized general goods market when
they were young. Hence, equilibrium in the money market requires that
(124)
pmt + (1 − p )Tt −1 = Mt .
Using the definition of T t and (124), we obtain
(125)
⎛ 1 + pπ ⎞
mt = Mt ⎜
⎟
⎝ p + pπ ⎠
and
(126) Tt −1 =
pπ
mt .
1 + pπ
Note from (126), that Tt −1 < mt so that unproductive buyers have less money balances
than buyers who had access to the night market when young. Let q denote the quantities traded by unproductive buyers. Since c ( qt ) = φt mt and c ( qt ) = φt Tt −1 from the
buyer-takes-all assumption we have
(127)
c ( qt ) =
pπ
c ( qt ).
1 + pπ
From (127) qt < qt
and limπ →∞ qt = qt . So the planner faces a trade-off between
smoothing consumption across buyers and preserving the purchasing power of real balances. Welfare is measured by W = σ p[u( q ) − c ( q )] + σ (1 − p )[u( q ) − c ( q )]. It can be
checked that a deviation from price stability has a beneficial effect on welfare, i.e.,
(128)
dW
dπ
π ↓0
⎡ u′( 0 ) ⎤
− 1⎥ pc ( q *) > 0.
= σ (1 − p ) ⎢
⎣ c ′( 0 ) ⎦
An increase in inflation from π = 0 is optimal because it allows unproductive buyers to
consume, while the negative effect on productive buyers’ welfare is only second-order.
Settlements and Monetary Policy
Assume for the time being that there is no default risk and that there is a liquidity shortage in the late-night settlement period. There will be a liquidity shortage when the fraction of creditors who depart early, δ , is greater than the fraction of debtors who arrive
56
FEDERAL RESERVE BANK OF CLEVELAND
early, α . In this situation, the market clearing price for debt in the late-night settlement
period, ρ , will be less than one. As well, sellers who produced search goods for money during the day will only spend a fraction of their balances in the competitive general
goods market at night and will use the remainder to purchase IOUs in the late-night settlement period.This implies that the levels of search good production will be inefficient;
in particular, q d < q * < q m , where q d is the amount of search goods produced for a buyer who purchases with an IOU and q m is the amount of search goods produced for a
buyer who purchases with money. Hence, the late-night liquidity problem results in inefficient levels of production of search goods during the day.
Suppose now that there exists a central bank that can provide “liquidity” to the latenight settlement period. More specifically, the central bank purchases Δ cb amount of
IOUs from early-leaving creditors in exchange for fiat money.When the late-arriving debtors come to the late-night settlement period, the central bank will exchange the IOUs for
fiat money. Recall that the supply of funds by creditors who are paid early and stay late is
(1 − δ )α d / 2 and that the face value of bonds of the creditors who leave early and whose
issuers arrive late is δ (1 − α ) d / 2. If Δ cb ≥ (δ − α ) d / 2, then the liquidity problem is
solved and it will be the case that ρ = 1. This temporary supply of liquidity by the monetary authority resembles either a discount window policy or an open market operation.
As an open market operation, the central bank purchases (δ − α ) d / 2 units of bonds
before the early-leaving creditors depart and “sells” the bonds back after the late arriving
debtors arrive. As a discount window policy, the central bank stands ready to purchase
second-hand IOUs at their par value, with the understanding that the IOUs have to be repurchased at their par value before the late-night settlement period ends. One can imagine one of the late-leaving creditors—call him the clearinghouse—gathers (δ − α ) d / 2
units of IOUs from early-leaving creditors, exchanging them with the central bank for
money, with the understanding that the IOUs will be repurchased before the late-night
settlement period ends. When the late-arriving debtors arrive, the clearinghouse can obtain money from the debtors whose creditors have already left, repurchase the debt from
the central bank, and return the IOUs to the original issuers. The increase in the money
supply that results from the open market operation or discount window policy is not
inflationary, since the IOUs purchased by the monetary authority are all redeemed within
the period so that the stock of currency remains constant across periods. This policy is
also in accordance with the real bills doctrine, which considers that the stock of money
should be allowed to fluctuate to meet the needs of trade.
Note that a central bank is not needed in order to overcome the liquidity problem. Suppose that a late-leaving creditor, say a clearinghouse, purchases the debt of early-leaving
creditors with his own IOUs, with the understanding that the IOU’s of the clearinghouse
can be exchanged for money at the beginning of the next period, before the early-morning general goods market opens.When the late-arriving debtors arrive, the clearinghouse
57
POLICY DISCUSSION PAPERS
NUMBER 12, DECEMBER 2005
will exchange the debt that it holds for money. At the beginning of the next period,
the clearinghouse can repurchase its debt with money before the general goods market
opens. Hence, as long as the clearinghouse is able to repurchase the debt it has issued,
the liquidity problem that arises due to the settlement frictions can be overcome by private agents: It is not necessary for a central bank to exist in order to deal with a liquidity
problem that may arise due to settlement frictions.
Related Literature
The result according to which the optimal monetary policy requires setting the nominal
interest rate to zero or, equivalently, deflating at the rate of time preference, comes from
Friedman (1969). Different definitions and interpretations of the Friedman rule are discussed in Woodford (1990). The optimal monetary policy in a search model with divisible money was first studied by Shi (1997), who showed that the Friedman rule is optimal when agents’ participation decisions are exogenous.The ability of the Friedman rule
to generate the first-best allocation when the terms of trade are determined according to
some bargaining solution are discussed in Rauch (2000), Lagos and Wright (2005), and
Rocheteau and Waller (2004).
The importance of trading frictions and search externalities for the design of monetary policy was first emphasized by Li (1995, 1997), who established that an inflation
tax could be welfare enhancing when agents’ search intensities are endogenous. However, her results are subject to the caveat that prices are exogenous. Shi (1997) found a
related result in a divisible-money model where prices are endogenous. In Shi’s model,
each household has a large number of members who can be divided between buyers
and sellers. When the composition of buyers and sellers is inefficient, a deviation from
the Friedman rule can be welfare improving. Rocheteau and Waller (2004) discuss Shi’s
finding under alternative bargaining solutions. Berentsen, Rocheteau, and Shi (2001) established the result according to which the efficient allocation is achieved when both the
Hosios rule and the Friedman rule are satisfied.A necessary condition for a deviation from
the Friedman rule to be optimal is that the Hosios condition is violated. Rocheteau and
Wright (2004) study the optimal monetary policy in a model with free entry of sellers
under alternative pricing mechanisms.
The welfare-improving role of a monetary expansion through distributional effects has
been studied by Levine (1991), and in a search-theoretic environment by Molico (1997),
Deviatov, and Wallace (2002), and Berentsen, Camera, and Waller (2004).
Freeman (1996a) and (1996b) provides a framework to study the settlement of debt
with money in a model with no aggregate risk. Freeman (1999) adds aggregate risk to the
analysis. Green (1999) provides a general overview of a settlements model and demonstrates that a central bank is not needed to provide liquidity to the economy if there is a
“shortage”; agents in the model are able to provide the requisite liquidity.
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FEDERAL RESERVE BANK OF CLEVELAND
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