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Stationarity without Degeneracy in a Model of
Commodity Money
R. de O. Cavalcanti1 and D. Puzzello2
1

EPGE/FGV, Praia de Botafogo 190, Rio de Janeiro, RJ 22250, Brazil
ricardo.cavalcanti@fgv.br
2
Department of Economics, University of Illinois, Champaign, IL 61820, USA
dpuzzell@uiuc.edu

October 17, 2008

Abstract
We develop a model of macroeconomic heterogeneity inspired by
the Kiyotaki-Wright (1989) formulation of commodity money, with
the addition of linear utility and idiosyncratic shocks to savings. We
consider two environments. In the benchmark case, the consumer in
a meeting is chosen randomly. In the auctions case, the individual
holding more money can be selected to be the consumer. We show
that in both environments socially optimal trading decisions (that are
individually acceptable) are stationary and solve a tractable static optimization problem. Savings decisions in the benchmark case are remarkably invariant to mean-preserving changes in the distribution of
shocks. This result is overturned in the auctions case.
Keywords and Phrases: Macroeconomics with heterogeneous
savings; commodity money with linear adjustments; mechanism design;
auctions
JEL Classi…cation: E00, C00

1

Introduction

One attractive but commonly overlooked feature of monetary models is their
implications about heterogeneity. Contrary to an old tradition of aggregaWe thank Narayana Kocherlakota, Robert Molzon, Neil Wallace, Randy Wright, and
seminar participants at the SED meetings in Prague and at the Money, Bank, Payments
and Finance workshop at the Federal Reserve of Cleveland for useful comments and discussions.

1

tion, macroeconomists are increasingly studying uneven allocations of risk
across the population. Uninsured risk is no news to monetary theory, provided that the essentiality of money is properly taken into account. Since the
invisible hand fails or does not operate in a simplistic way in monetary models, consumption ‡
ows are instead organized by incomplete insurance and
dispersion of individuals across asset holdings. Monetary theory is therefore
in position to o¤er macroeconomists a coherent description of savings disparities resulting from a variety of patterns in monetary trades. This paper
is concerned with tractable descriptions of monetary frictions and the consequent heterogeneities. We are particularly interested in how that description
is facilitated by the property that savings decisions are heterogeneous but
stationary, and how such an outcome can be derived on e¢ ciency grounds.
In monetary theory, incomplete markets cannot be taken as a serious
primitive in the …eld because di¤erent formulations of incompleteness have
remarkably di¤erent implications for how money is used. Consensus is now
building that it is necessary to start with physical-environment assumptions, like imperfect monitoring and commitment, in order to o¤er predictions about savings paths. But because models of money are well known
for displaying multiplicity and nonstationarities, the curse of dimensionality associated to heterogeneity has become a serious obstacle for blending
monetary exchange and macroeconomic questions. In this paper, we formulate a model of monetary exchange and use constrained e¢ ciency to predict
a rich but tractable heterogeneity. Our model owns its tractability to the
extreme assumption that the only asset is commodity money, and that additions or subtractions of money holdings can be done at a linear utility
cost, conditional on the realization of idiosyncratic shocks. It is essentially
a version of the Kiyotaki-Wright (1989) model, with shocks to preferences
and unbounded holdings. As in their model, no markets open in our model.
Unlike their analysis, however, we do not need to impose stationarity, and
are able to o¤er predictions about the distribution of money that maximizes
ex-ante welfare.
The aim in this paper is to pursue a mechanism-design formulation of
e¢ cient allocations and to show that the optimum is constant, a result consistent with linear dynamics. In other …elds, the idea of using linearity in
order to facilitate the description of heterogeneity is not new.1 Most papers
in monetary theory make however an e¤ort to restore the aggregative struc1

Cavalcanti and Erosa (2007), for instance, use linearity in a version of the Lucas’ tree
s
economy, in which the productivity of a business depends on ownership shocks, in order
to predict business turnover rates.

2

ture of traditional macro models, and leave potential descriptions of heterogeneity unexplored. In a notable exception, Galenianos and Kircher (2006)
model a sequence of auctions that follow market interactions with possibly
heterogeneous outcomes in savings of money. Although they sidestep questions about stationarity and optimality to some extent, their formulation
of heterogeneity is similar in spirit to the one adopted here. In order to
make that comparison easier for the reader, we have included a section explaining how an auctions setup in the context of our environment could help
allocating resources in our model (more on this below). Lagos and Wright
(2005) is a standard reference for a di¤erent motivation: to impose stationarity and to appeal to quasi-linearity and markets in order to eliminate the
distribution of money and to evaluate in‡
ationary policies. Shi (1997) had
already pursued a model of degeneracy and policy evaluation, but did so
with a coordination of individuals according to ‘
families’ making it di¢ cult
,
to assess optimality in his model.2 These models remain attractive because
they can easily address policies related to the supply of …at money, as they
are usually stated, an issue that in the case of our model would certainly
require future research.
We succeed in providing a description of optima that is remarkably simple. Instead of setting up individual choices in sequence spaces or constructing value functions, we propose the analysis of a simple static problem. We
then show that the problem de…nes an upper bound on average welfare of
implementable allocations, and that its solution is actually implementable
as a constant sequence of consumption and savings decisions for the whole
economy. Because the optimum is shown to have low dimensionality, we can
ask what gives shape to the distribution of money, and pursue an answer in
the context of both the benchmark Kiyotaki-Wright formulation, and what
we call an auctions variation of the environment. In the benchmark environment, the consumer in a meeting is chosen as the realization of a random
variable, as usual, while in the auction environment we let the planner pick
who is the consumer between two candidates in a meeting (the other must
be the producer). We …nd that the optimum has the individual with largest
holdings of money being the consumer because such a choice implies the
weakest constrain on average utility. The tractability of this auction environment is evidence that the upper-bound argument can be generalized to
2

One concern in models in which individuals can commit to family (or another entity
such as government) plans is that gift-giving would render money inessential. Even when
individuals are anonymous (i.e., their identities are not observable) but meet in large
groups, special assumptions are needed to rule out certain trigger strategies. See Aliprantis
et al. (2007).

3

more applications in macroeconomics.
The contrast between the two environments sheds light on what the
distribution of money represents in this linear context. We learn quite a
sharp lesson about the benchmark case. Because money is costly to acquire,
the optimal allocation has individuals economizing on money holdings in
a way that aligns private and social returns. In particular, trade takes
place in pairwise meetings with all surplus going to the consumer, and all
money holdings going to the producer. Due to linearity, after-trade holdings
are valued according to an average marginal utility, which is the mean of
the shock distribution. At the stage of making savings decisions (before
trade), individuals know that holdings of producers are irrelevant and need
not predict the distribution of money (it su¢ ces to know what the average
marginal utility is in case they are called to produce). The conclusion is
that individual savings decisions are invariant to changes in the distribution
of shocks that preserve the mean. Consequently, the distribution of money
can be computed residually in the benchmark case.
The picture is di¤erent in the auctions environment, although previous
…ndings about stationarity apply. Society economizes resources by having
individuals with large holdings being the consumers. Therefore, the return
of money is a¤ected by the way holdings are distributed. Savings decisions
now equilibrate two forces. It equalizes intertemporal costs and returns
and, at the same time, generates the distribution of money used to compute
money returns. The analysis is still easily accessible. We are able to avoid
a demanding …xed-point treatment in the space of distributions. We tackle
instead a …rst-order di¤erential equation that is necessary and su¢ cient to
characterize optimal savings in some class. We also o¤er, in our conclusion,
a discussion of how the …ndings square with a recent literature on money and
auctions. In that literature, the distribution of money re‡
ects indi¤ erence
among directed-search opportunities, not optimality as in our setup.

2

The benchmark environment

Our model is a version of Kiyotaky-Wright [8] in which only one good is
durable. This good, interpreted as commodity money or capital, can be
consumed and produced by all individuals in the population. We refer to
commodity money as just ‘
money’for simplicity.
Time is discrete and the horizon is in…nite. The economy is populated
by a continuum of individuals symmetrically divided according to an integer
number of types N , where N
3. There are N specialized goods per date

4

and one common good called money. Specialized goods are assumed divisible
and perishable. Money is assumed divisible and storable. Each period is
divided into two stages, and money can only be produced or consumed
in the second stage. In the …rst stage, people meet according to random,
pairwise meetings, and consumption of (specialized) goods by one person
can only be provided by the meeting partner, and only if a coincidence of
wants occurs. More speci…cally, type j enjoys good j but can only produce
good j + 1, modulo N . Since N > 2, no double-coincidence in which two
individuals consume can occur. In the second stage, each individual in
isolation consumes or produces money, according to an individual-speci…c
production function, to be de…ned below. Money can also be used in future
dates to pay for consumption of specialized goods.
Utility is separable across stages. If …rst-stage x 2 R+ is produced by
type j 1 and consumed by type j, in what we call a single-coincidence meeting, then the period utility for the consumer is u(x), and the period utility
for the producer is v(x), independently of j. In the second stage, each
individual is hit by an idiosyncratic preference/productivity shock s distributed according to the measure . The shock is distributed independently
and identically across individuals and over time. Production of money at the
second stage must be nonnegative, and if an individual in state s produces
m 2 R+ and consumes c 2 R+ units of money then a corresponding utility
‡ s(c m) takes place. Individuals discount future utilities according to
ow
the common discount factor , where 2 (0; 1), but there is no discounting
across stages of the same date. The functions u and v are de…ned on R+ and
assumed increasing, continuous and di¤erentiable. In addition, u is strictly
concave, v is convex, u v is bounded from above, and u0 (0) = +1. In
order to have aRcompact savings problem we impose a lower bound to s of
the form s >
s0 d (s0 ). For ease of exposition, we also …nd convenient to
R
choose normalizing units so that u(0) = v(0) = 0 and sd = 1. Hence is
assumed to have support in the Borel subsets of S ( ; +1).
People cannot commit to future actions and their personal histories are
private. The only assets are holdings of money. We assume that money
holdings and types are observable by participants in a meeting.
We also considered another economy in a second part of this paper. In
that economy, there are meetings in which both individuals can produce, yet
only one can be the producer, and the other one must be the consumer. We
shall investigate whether an allocation resembling an auction over holdings
of money can implement the optimum. Because the description of allocations in this second environment is a straightforward extension of the benchmark environment, we proceed with a discussion of implementability for the
5

benchmark environment only, and then make the necessary quali…cations in
the second part of this paper.

2.1

Allocations

An allocation is a description of what happens in all stages and dates. We
build on [5] and de…ne allocations as trade and savings plans for all contingencies without imposing stationarity, but our application turns out to
be much simpler, and the proof for the existence of an optimum is done by
construction. As a result, we do not need to de…ne the planner’ dynamic
s
programming problem and can avoid much the questions raised in [5] about
compactness and admissibility of the state space.
We assume that the economy starts at the second stage of date zero, and
with zero money endowments. Part of the planner’ problem is to choose a
s
sequence of Borelean measures f t g1 describing the distribution of money
t=1
holdings in R+ at the start of all dates.
In order to …x ideas, let us take date t and assume that the initial distribution of money, t , has already been chosen. An allocation plan (for
date t) is a trade plan for the …rst stage, (g; h), and a savings plan for the
second stage, f . More formally, the function g : R2 ! R+ determines the
+
quantity of output produced by producers for each kind of single-coincidence
meeting (m; n), where the consumer is type j and has m units of money,
and the producer is type j 1 (modulo N ) and has n units of money. The
function h : R2 ! R+ determines the corresponding after-trade holdings of
+
the consumer in these single-coincidence meetings. As a result, in meeting
(m; n), the after-trade holdings of the producer is n + m h(m; n). The
function f : S R+ ! R+ describes …nal holdings for an individual at the
end of the second stage, and is de…ned over pairs (s; m), where s 2 S is the
productivity state at the second stage and m is the holdings of money at
the beginning of the second stage.
It should be clear that an allocation plan fully describes what happens
in a date. In the …rst stage, trade takes place in single-coincidence meetings
according to (g; h). An individual starting the second stage with state (s; m)
and leaving with holdings m0 = f (s; m) has consumed c = maxfm m0 ; 0g
and enjoyed utility s(m m0 ). A plan is said feasible if h(m; n) m for all
(m; n) 2 R2 .
+
We restrict attention to allocation plans that are continuous and thus
measurable functions. Similar arguments to those applied by [5] would then
show that an allocation plan maps a Borelean measure t into another Borelean measure t+1 in the obvious way. In particular, at t = 0, the economy
6

starts at the second stage without money endowments, and a date-0 savings
plan f0 , together with , generates 1 . More generally, we say that the sequence fft 1 ; ht g1 generates f t g1 . An allocation, and thus a complete
t=1
t=1
description of what happens in all meetings, is a sequence of feasible plans
fft 1 ; gt ; ht g1 . We also include f t g1 among the objects associated to
t=1
t=1
an allocation whenever it is clear from the context that f t g1 is implied
t=1
by a particular sequence fft 1 ; gt ; ht g1 .
t=1

2.2

Implementability

When an allocation is …xed, a game is formed with individuals in each stage
choosing an agree/disagree strategy at each contingency in response to the
plan prescribed by the allocation. Hence we only allow individual defections.
There are also two implications of sequential individual-rationality: individuals who disagree at the …rst stage are allowed to preserve any holdings
brought to a meeting; and individuals who disagree at the second stage, are
free to deviate from the plan (f ) and choose di¤erent holdings m0 2 R+ as
the state for the next date. An allocation is implementable if it is feasible
and if agree in all contingencies is a subgame perfect equilibrium.

2.3

The welfare criteria

The planner’ problem is to …nd an implementable allocation that maxis
mizes average discounted utility. The goal is to …nd a savings and goods
exchange path that achieves the highest average discounted utility under
implementability.
We express average utility by parts with the help of some simplifying notation. Let us use u g and v g to denote utility ‡
ows, for each (m; n), when
the output is g(m; n). With respect to the law of motion of holdings, f h in
general, and (f h)(s; m; n) in particular (which is short for f (s; h(m; n))),
describe …nal holdings of an individual after being the consumer in meeting
~
(m; n) and drawing productivity s. Using also h(m; n) n + m h(m; n),
~ it is understood that the individual was the prowith the composition f h
ducer. For completeness, we let the function i denote the projection with
respect to the …rst coordinate of pairs (m; n), so that i(m; n) = m and,
hence, (f i)(s; m; n) describes the law of motion for those in no-coincidence
meetings. Finally, in our application, double integration with respect to
m and n can be written in a compact form as integration with respect to
the product measure , where (A B) is de…ned as (A) (B) for cartesian
products of Borel sets A and B in R+ .
7

Let us consider the consequences of allocation plan (f; g; h) for average
utility at some date in which ; and thus , are …xed. In single-coincidence
meetings, the sum of the ‡ of utilities accruing to the consumer and proow
ducer in meeting (m; n) is u(g(m; n)) v(g(m; n)). Hence, using a compact
notation, we write the average over m and n as
Z
(u g v g)d .
The average utility, in the second stage, for those who were consumers in
those meetings is
Z Z
s(h f h)d d
That average for those who were producers is
Z Z
~
~
s(h f h)d d ,

while the average utility for those experiencing no-coincidence meetings is
Z Z
s(i f i)d d :
An individual …nds a single coincidence meeting as consumer with prob1
1
ability N , as producer with probability N , and none of the above with
R
R
2
~
probability 1 N . Since (h + h)d = 2 id then
Z Z
Z Z
1
1 ~
2
sh + sh + (1
)si d d =
sid d .
N
N
N
Thus, second-stage average utility is
Z Z
1
~
s i
(f h + f h) + (1
N

2
)f
N

i

d d .

The objective to be maximized in the planner’ problem is the date-0 diss
counted sum of average utility, or W , de…ned as

W (fft 1 ; gt ; ht g1 )
t=1

=

Z

sf0 (s; 0)d +

1
X
t=1

t

Z Z

At d t d ,

where the term At inside the integral is
si +

1
(u gt
N

v gt

sft ht
8

~
sft ht )

(1

2
)sft i.
N

In the right-hand side of the equation de…ning W , the …rst term is the social
cost at date t = 0 of savings decisions leading to the initial distribution of
money, and the second is the present value of stage-1 and stage-2 utilities
for t 1.
Remark 1 If, for all t, ft is constant in its second argument (varies only
with s) then consumption at t R 1 of ft (s; ) units of commodity at marginal
+
utility s0 gives average utility s0 ft (s; )d (s0 ) = ft (s; ) since, as assumed,
R 0
s d (s0 ) = 1. Integrating now ft (s; ) over s, then average utility at t + 1
from consumption of time-t investment can be written in simple terms as
R
sft (s; )d , so that the expression for W becomes
Z

(s

)f0 (s; 0)d +

1
X
t=1

t

1
N

Z

(u

v) gt d

t

Z

(s

)ft (s; )d

.

Notice also that by treating u gt v gt as identically equal to zero at t = 0,
this expression for W can be further simpli…ed to
1
X
t=0

t

Z

(s

)ft (s; )d +

N

Z

(u

v) gt+1 d

t+1

.

When ft is constant in its second argument, t+1 is uniquely determined by
and ft for all t. Thus, in this case, W corresponds to a discounted sum
of independent terms.

3

Optimality

We shall construct a constant allocation (f ; g ; h ) and show that it is
optimal. All three functions in the constructed allocation are constant in
the second coordinate, so that output and after-trade holdings of consumers
do not depend on holdings of producers, and savings do not depend on
stage-1 outcomes. We shall …rst present (f ; g ; h ) as continuous solutions
of particular problems, assuming that such solutions exist, and then show,
in a lemma below, that the allocation is actually well de…ned. We then …nish
the section with the statement about optimality.
The construction proceeds in two steps. First, given some money holdings m of the buyer, the optimal exchange of goods in a single-coincidence
meeting involves the maximization of the joint surplus under the constraint
that there is enough money for the producer to be compensated for his
disutility (see P1). Second, given this optimal trading decision, the optimal
9

savings decision m0 given some shock s is determined. This choice is pinned
down by trading o¤ the bene…t with the cost of money (see P2). More
details are provided next.
Consider the sum of stage-1 utility ‡
ows in a single coincidence meeting in which the consumer has m and the producer has n units of money.
Consider now the problem of maximizing this over choices of output g(m; n)
restricted to v(g(m; n)) m. Let now g (m; n) be a continuous, constantin-n, solution to
max fu(y) v(y) : v(y) mg .
(P1)
y

We construct h as the slack in the constraint of the problem de…ning g ,
that is, h (m; n) = m v(g (m; n)) for all m and n. We next consider a
one-period savings problem, for a given realization s > 0, in order to de…ne
f (s; m) as a continuous solution to
max
0
m

(s

)m0 +

N

u(g (m0 ; n))

v(g (m0 ; n))

:

(P2)

In P2 the values of m and n are immaterial. Notice that the interpretation of
P2 as a ‘
savings’problem applies in the sense that a choice m0 implies forgone
present utility sm0 , a possible future expected ‡ N (u g
ow 1
v g ) , and a
R 0
0 s d (s0 ). Applying then
residual expected future utility m
to discount
R
the future and using the normalization s0 d (s0 ) = 1 yields the objective in
P2. Clearly, P2 suggests that savings decisions are di¤erent depending on
the costs of savings s that agents face.
Lemma 2 There exists a continuous and bounded solution g to P1 so that
(f ; g ; h ) are well de…ned and constant with respect to their second coordinates.
Proof. The objective in P1 is a continuous and concave function bounded
from above. Existence follows because the constraint set is compact for
s 2 S, that is, for s > . Uniqueness and continuity of g for m in [0; v(y)),
where y = arg maxy fu(y) v(y)g, follows from the strict concavity of u v
and a straightforward application of the theorem of the maximum. For m
v(y), continuity requires that g (m; n) = y. Hence h is also continuous.
That g (m; n) and h (m; n) are bounded and constant in n is trivial. Since
g is bounded and continuous then f is continuous and well de…ned.3
3
The optimization problem P1 is an optimization problem in parametric form, where
the parameter is given by the buyer’ money holdings. Depending on the values of the
s
parameter m, the set of maximizers is a correspondence. In this paper, without loss of
generality we focus on a continuous selection.

10

Proposition 3 (i) The constant allocation (f ; g ; h ) is implementable,
(ii) W is bounded from above by W (f ; g ; h ) on the set of implementable
allocations and, therefore, (f ; g ; h ) is optimal.
Proof. We start discussing implications of second-stage deviations for implementability. Let an arbitrary implementable allocation fft 1 ; gt ; ht g1
t=1
be …xed, and consider a given date t. Since individuals can freely deviate
from ft at the second stage, and such deviations produce a linear payo¤, the
continuation utilities associated to after-trading in the …rst-stage must be
linear in money holdings. As a result, producers in a single-coincidence meetR
R 0
~
ing (m; n) say agree if and only if v(gt (m; n)) + ht (m; n) s0 d
Rn 0 s d ,
and consumers say agree if and only if u(gt (m; n)) + ht (m; n) s d
R
R
m s0 d . Since s0 d = 1, the participation constraints implied by linearity and individual rationality are equivalent to
u(gt (m; n))

m

ht (m; n)

v(gt (m; n)).

(1)

Moreover, since continuation utilities are linear in after-trade holdings, then
linearity of the payo¤ on savings decisions imply that only one-period deviations needed to be considered at a time. In particular, the payo¤ Rt 1 at
t 1 from a stage-2 deviation to m0 , when the shock is s and the distribution
at t is t , can be written as
Rt
N

1 (m

Z h

0

; s;

t)

=

2
) m0 +
N

sm0 + (1

u(gt (m0 ; n)) + ht (m0 ; n)

(2)

i
~
v(gt (n; m0 )) + ht (n; m0 ) d t (n).

Hence, fft 1 ; gt ; ht g1 is implementable if and only if (1) holds and ft (s; )
t=1
maximizes Rt (m; s; t+1 ) in m at all dates.
We now argue that (f ; g ; h ) is implementable. It is straightforward
to verify that. Because g solves problem P1, then y = g (m; n) and x =
h (m; n) solve
maxfu(y)
(x;y)

v(y) : u(y)

m

x

v(y)g.

Hence g and h satisfy the participation constraints (1), and individuals say
agree to (f ; g ; h ) in all …rst-stage meetings. We now claim that individuals
agree with f at the second stage of all dates. This follows because f is
constructed as the maximizer of problem P2, and because (2) coincides with
the objective in P2 on [0; y] (de…ned in the proof of Lemma 1) when gt+1 = g
11

and ht+1 = h (because n drops out in this case from (2) the value of t+1
becomes irrelevant): Hence individuals also say agree to (f ; g ; h ) at the
second stage in all dates and the proof to the …rst part of the proposition is
complete.
We now show that W (f ; g ; h ) is an upper bound on welfare of implementable allocations. Let us again …x an alternative, implementable allocation fft 1 ; gt ; ht g1 and assume that f t g1 is its corresponding sequence
t=1
t=1
of meetings distributions. Let t 0 be arbitrary. Because (g ; h ) was constructed so as to maximize u v subject to participation constraints in any
meeting (m; n), it follows that
(u

v) g

(u

v) gt+1 :

Integrating the objective in problem P2 with respect to implies that f is
such that, for any m and n,
Z
(s
)f (s; ) + (u v) g (f (s; ); ) d (s)
N
Z
(s
)ft (s; m) + (u v) g (ft (s; m); ) d (s)
N
Z
(s
)ft (s; m) + (u v) gt+1 (ft (s; m); n) d (s):
N
Changing now variables in both the …rst and the third integrals yields
Z
Z
(s
)f (s; )d +
u(g (m0 ; )) v(g (m0 ; )) d f 1 (m0 )
N
Z Z
Z
(s
)ft (s; m)d d! t (m) +
(u gt+1 v gt+1 )d t+1 ,
N
where ! t is the distribution of after-trade holdings associated with
fft 1 ; gt ; ht g1 , and f 1 is the measure of end-of-stage-2 holdings induced
t=1
by f [that is, if A is Borel subset of R+ then f 1 (A) = (f 1 (A))]. For
this last step, we use the fact that ! t and ft must generate t+1 . It is now
trivial to verify that this inequality implies
W (f ; g ; h )

W (fft

1
1 ; gt ; ht gt=1 ):

The proof of the proposition is thus complete.
Remark 4 This result shows that the simple problem P2 gives an upper
bound on average discounted welfare for any allocation that is individually
12

rational. Since it can be implemented in an individually rational way, the
solution to P2 provides an optimal plan.4
The constraint on the …rst part of the problem P1 relies on the expected
value of money, thus the optimum is invariant to mean preserving changes
in the distribution of shocks. This is no longer the case in the environment
with auctions considered next.

4

The environment with auctions

We change the environment in order to allow for a selection of consumers in
some meetings.

4.1

Mixing standard and auction meetings

For each meeting (m; n) there is now a realization of a Bernoulli shock. Its
distribution is iid over meetings and dates. With probability 1
consumption and production must take place as in the benchmark environment; that
is, there is a single coincidence only if the consumer is type j and the producer is type j 1, for some j. With probability , however, both individuals
can produce the good that the meeting partner likes, but only one of the
traders can be the producer.
Our goal is to show that a modi…cation of the upper-bound argument
used to construct the optimum in the benchmark case can also be used to
describe the optimum in this new environment. As we shall see, an interesting aspect of the optimum is that it is desirable to choose the individual
holding the largest quantity of money to be the consumer, as if that choice
is the result of a …rst-price auction (i.e., the probability of being a buyer or
a seller is endogenous). The reader can easily verify that the essence of this
optimum would be preserved in extensions where the number of people in
a meeting is larger than two, provided that only one individual can be the
consumer, and only one of the remaining participants in the meeting can be
the producer.
Let us call this new kind of meeting, an event that happens with probability , an auction meeting, and call the other realization, that happens with
4

Another implication of this result is that optimal trading exchanges involve the buyer
take-it-or-leave-it o¤er. In other words, we show that the buyer take-it-or-leave-it trading
rule is optimal among all the incentive compatible trading rules (including also the ones
arising under Nash bargaining).

13

probability 1 , a standard meeting. We restrict the set of allocations as follows. An allocation is now fet ; ft 1 ; gt ; ht g1 , where the last three sequences
t=1
can be given the same interpretation as before, while et : R2 ! f0; 1g de+
…nes a selection of who is the consumer in auction meetings. If type j holds
m, type k holds n, and the meeting is an auction, then et (m; n) = 1 means
the consumer is the one holding the largest amount (if the holdings are not
equal) and et (m; n) = 0 means the consumer is the one holding the least
amount. (For completeness, one can assume that the consumer is chosen
according to a randomization device with probability 1 when m = n).
2
If m _ n denotes maxfm; ng and m ^ n denotes minfm; ng then, after
the consumer is selected in an auction meeting, output is
qt (m; n) = et (m; n)gt (m _ n; m ^ n) + [1

et (m; n)]gt (m ^ n; m _ n)

while the after-trade holdings of the consumer is
pt (m; n) = et (m; n)ht (m _ n; m ^ n) + [1

et (m; n)]ht (m ^ n; m _ n):

The set of allocations is restricted because we are not allowing (g; h) to vary
across auctions and standard meetings, but this restriction imposes no loss
of generality as it follows that the optimum features et = 1 in all auction
meetings. Before we present the argument, we conclude the presentation
of the environment with the de…nition of implementability and the welfare
criteria.
If fft 1 ; gt ; ht g1 is implementable in the benchmark environment, then
t=1
fet ; ft 1 ; gt ; ht g1 is implementable in the new environment because saying
t=1
agree or disagree to et presents no new participation constraints.
Hence fet ; ft 1 ; gt ; ht g1 is implementable if and only if fft 1 ; gt ; ht g1
t=1
t=1
is implementable in the benchmark environment. The expression for average
utility W (fet ; ft 1 ; gt ; ht g1 ) is the same as that for W (fft 1 ; gt ; ht g1 ),
t=1
t=1
with the exception that the term At inside the integral is now extended to
(1

)At + (u qt

v qt + 2si

sft pt

sft pt ),
~

with the understanding that pt (m; n) = n + m pt (m; n).
~
We assume that has a density bounded away from zero on the interval
S ( ; s), for some s > .

4.2

The upper bound

As before, we …rst construct a particular candidate, and show later that
it is in fact an optimal allocation. In our construction, the candidate is
14

stationary (constant in t). On the one hand, in all auction meetings, e = 1.
On the other hand, in both standard and auction meetings, trade takes
place according to g and h , as in the exchange scheme for the benchmark
allocation (but now, in auction meetings g (m; n) and h (m; n) are applied
to m n). The novelty is that the optimum savings function, now denoted
f a , is chosen among possibly many solutions to a …xed-point problem.
We …nd it convenient to limit ourselves to smooth and monotone solutions as follows. A function f : S R+ ! R+ is a candidate optima if it
is di¤erentiable, strictly decreasing in the …rst argument, constant in the
second, and solves moreover
max

(s

m

where, for all m

)m + (1

)

1
R(m) + F (m)R(m)
N

(P3)

0,
R(m) =

max fu(y)
y

v(y) : v(y)

mg

and
F (m) = (fs 2 S : f (s; )

mg).

(FP)

In P3, the function R is a short representation for (u g
v g ).
It follows that when
= 0, the unique candidate is f = f , and thus
f a = f . For > 0, however, problem P3 di¤ers from P2 because now the
distribution of money is relevant. For each candidate f , in allocations of the
kind (e; f; g; h) = (1; f; g ; h ) a savings level m is payo¤-relevant in auction
meetings only when the producer has n
m, an event taking place with
probability F (m).

4.3

Existence and uniqueness

In this subsection we present su¢ cient conditions for the existence of a
unique solution to P3. We examine optimality in the next subsection.
Lemma 5 Let D denote the cdf of and let f 1 denote the inverse of f
(with respect to s). Then f is a solution to P3 if and only if D f 1 = x
for x : I ! [0; 1] solving
x0 =

1
N

+1

x

R0
R

1
D
R

1

(x)

,

(DE)

together with the auxiliary condition
x(m0 ) = 1 ,
where I = [m0 ; m ) is de…ned by R0 (m0 ) = (s
15

(IC)
)N=(1

) and R0 (m ) = 0:

Proof. Recall that R0 is strictly decreasing, with R0 (0) = +1 and R0 (m ) =
0. Moreover, if f is continuous then F is di¤erentiable and the solution m =
f (s; ) for each s must be interior and satisfy the tangency condition
s

= (1

)

1 0
R (m) + [R0 (m)F (m) + R(m)F 0 (m)].
N

Since f is decreasing, F (f (s; )) = 0. Thus, taking into account the de…nition
of m0 , if f (s; ) 6= m0 then
(s

)m0 + (1

)

1
R(m0 ) >
N

(s

)f (s; ) + (1

)

1
R(f (s; )),
N

contradicting that f is a solution to P3. Hence f (s; ) = m0 and a similar
argument shows that f ( + ; ) = m . Let now h : I ! ( ; s] be the inverse of
f with respect to the …rst coordinate. Since the FP condition is equivalent to
F (m) = 1 D[h(m)], where D is the cdf of , then F 0 (m) = d(h(m))h0 (m)
for m in the image of f , where d is the density of . Changing variables
once again with x : (m0 ; m ) ! (0; 1), x(m) = D(h(m)) then, because
s = D 1 (x(m)) and F 0 = x0 , this …rst order condition can be written as
the di¤erential equation (DE ), with the understanding that x; x0 , R; and R0
are functions of m. Because f (s; ) = m0 , then an initial condition for (DE )
is (IC ).
Lemma 6 The initial value problem (DE)-(IC) has at most one solution.
If the distribution of shocks is uniform, or s is su¢ ciently low, then it has
a solution.
Proof. As it is standard, in order to examine existence and uniqueness of
solutions to (DE-IC) on the interval I, we let
H(t; r) =

1
N

+1

r

R0 (t)
R(t)

D

1 (r)

R(t)

denote the function de…ned by the right-hand side of (DE) with r = x and
t = m, with the understanding that R and R0 are continuous and bounded
functions on I (since
< 1 implies m0 > 0 and R0 (m0 ) < 1). If is
uniform then D is linear, and so H is also linear in r. As a result, (DE)
de…nes a linear ordinary di¤erential equation with nonconstant coe¢ cients.
A basic result in the theory of linear ordinary di¤erential equations is that
initial value problems are uniquely solvable, and solutions are de…ned on all
of I.
16

Even if D is not linear, because H and @H are bounded (since the density
@r
of is assumed bounded away from zero), then another basic result on the
theory of …rst-order di¤erential equations is that the initial value problem
(DE-IC) has at most one solution (see Theorem 2 in Brauer and Nohel,
1986, p. 400). In this case, existence on some interval can be demonstrated
according to details of the bounds on H according to the Picard-Lindelöf
theorem (see Theorem 1 in Brauer and Nohel, 1986, p. 389). To do that,
we have to …nd bounds of the right-hand side of (DE) so that the interval
around m0 for which the theorem applies includes (m0 ; m ). We state below
the theorem for completeness.
Theorem. Assume that M is a bound for jHj, that a = m
m0 , and
that b = 1. Suppose that H and @H are continuous and bounded in the
@r
b
rectangle f(t; r) : t m0 < a; 1 r < bg with jHj M . Let = minfa; M g.
Then the successive approximations j given by
0 (t)

= 1;

j+1 (t) = 1 +

Z

t

H(m;

m0

j (m))dm

(j = 0; 1; 2; :::)

converge (uniformly) on the interval ft : t m0 < g to a solution of DE
that satis…es initial condition IC.
We now provide a bound M for H such that implies (m0 ; m )
ft : t m0 < g. On the one hand, since D 1
and r 0, then
1

H(t; r)

N

On the other hand, since r
H(t; r)

1

R0 (t)
R(t)

+1

1
N

+1

R0 (m0 )
.
R(m0 )

1,

R0 (t)
N R(t)

1 (r)

D

s
.
R(m0 )

R(t)

Hence
jHj

1
max
R(m0 )

and since, by construction, s
M=
Since
= minfa;

+ (1

)

)

1
N

b
g = minfm
M
17

R0 (m0 ); s

1
) N R0 (m0 ), then jHj

= (1
+ (1

1
N

R0 (m0 )
.
R(m0 )
m0 ;

1
g;
M

;
M , where

in order to have
large so that

m
+ (1

m0 , as desired, it su¢ ces to have m0 su¢ ciently
)

1
N

R0 (m0 )
R(m0 )

1
m

m0

:

Making now explicit the dependence of R on parameters, we …nd that
R0 (m0 )
1
=
R(m0 )
u(y0 ) v(y0 )
and
s

= (1

)

N

u0 (y0 )
v 0 (y0 )

u0 (y0 )
v 0 (y0 )

1

1 ,

where y0 = v 1 (m0 ): Hence, if s ! then m0 ! m . Thus
s
is su¢ ciently low. The proof is now complete.

4.4

=m

m0 if

Optimality with auctions

We compare solutions to P3 in terms of the average payo¤ !, de…ned for
each f as
Z
1
!=
(s
)f (s; ) +
R(f (s; )) + F (f (s; ))R(f (s; )) d (s).
N
Proposition 7 (i) If f solves P3 then (1; f; g ; h ) is implementable. (ii) If
fet ; ft 1 ; gt ; ht g1 is implementable and ! is an upper bound on the average
t=1
payo¤ in P3 then W (fet ; ft 1 ; gt ; ht g1 ) !=(1
).
t=1
Proof. The same reasoning of the proof of Proposition 1 applies to statements (i) and (ii). Let us suppose f solves P3 and let us …x allocation
(1; f; g ; h ). Since individuals take the distribution of money as given (pdf
F de…ned in P3), individual rationality is equivalent to having f maximize
the right-hand side of P3, as assumed. Thus (1; f; g ; h ) is implementable.
Regarding part (ii), it is straightforward to show the producer constrained is weakened by increases in holdings of the consumer. More formally, that (u v) g (m; n) is weakly increasing in m and constant in n.
Hence, the social payo¤ u v is bounded from above by the choice e = 1
and g = g . As in the proof of Proposition 1, integrating individuals’payo¤s
resulting with respect to yields again an aggregate payo¤ ! that coincides
with the average objective in P3. As a result, (1
)W (fet ; ft 1 ; gt ; ht g1 )
t=1
is bounded above by any ! bounding the average payo¤ of the set of solutions
to P3.
18

Remark 8 As in the benchmark case, the savings choice is determined by
trading o¤ bene…ts and costs. Now, there is an additional term in the bene…ts
re‡ecting the fact that the probability of being a buyer is endogenous and is
captured by the cdf of money holdings. Thus, the entire distribution of money
holdings matters and it arises as a solution of a di¤ erential equation.

5

Remarks about …at money and directed search

We have constructed a model of macroeconomic heterogeneity, inspired by
Kiyotaki and Wright (1989) and driven by shocks to a linear-utility savings
decision to hold a commonly desired medium of exchange. We have applied mechanism design with individual defections, and shown that in the
benchmark case the optimum can be constructed by a static maximization
problem. In that case, the optimum is not only time-invariant, but individual savings are actually invariant to the distribution of money.
Our upper-bound argument is so simple that it is tempting to conjecture
that it may help demonstrating that the optimum in …at-money models in
the spirit of Lagos and Wright (2005) is also stationary. We concede that
a …xed rate of money growth would translate into a …xed s in our model,
and thus optimality could be stated relative to a …xed s. A full discussion
of optimality, however, would have to address whether levels of s su¢ ciently
close to are feasible, like in the Friedman rule, and it is not clear that this is
an interesting application of the upper-bound argument, at least because the
optimum with near zero-opportunity cost of holding money is well known.5
Although stationarity is a natural prediction of linear models, and that
is why our upper-bound argument is so …tting, the property that savings are
invariant to distributions is not robust to a simple and appealing change in
our model. In situations where a choice over consumer/producer status is
physically feasible, allocating consumption to wealthy individuals is in fact
socially optimal. We have shown that a di¤erential equation can be used
to describe the optimum in a class of smooth allocations. The shape of
the optimal distribution of money re‡
ects two forces. First, di¤erent people
face di¤erent savings opportunities. Since output in meetings is divisible,
there is an intensive margin to be explored, and a consequent dispersion in
money holdings. This is the benchmark explanation for heterogeneity. Sec5

The analysis of Lagos and Rocheteau (2008) also implies that the coexistence commodity and …at monies is not interesting in linear models like ours. Commodity money
would be driven out of the economy if the opportunity cost of …at money can be made
su¢ ciently low.

19

ond, because consumption opportunities are scarce, it is optimal to allocate
consumption according to wealth, so that savers explore the distribution of
holdings when making decisions. This is the auctions or extensive-margin
explanation for heterogeneity.
We have thus linked heterogeneity and optimality. We recognize however that optimality need not be the only explanation for heterogeneity (and
if frequently the least adopted in macroeconomics). An alternative can be
found in the ‘
directed search’ literature that builds on Burdett and Judd
(1983). Their work focus on price dispersion resulting from precisely a lack
of coordination by individuals that can sample prices from subsets of sellers. Julien et al. (2006) allow individuals to select sellers in a version of
the Kiyotaki-Wright model with indivisible money, and consider an auction
process to select the consumer according to bids in output quantities. This
indivisibility restriction has been removed to allow bids of money by Galenianos and Kircher (2006), who also appeal to linearity in savings costs, but
restrict consumption in meetings to be constant.
Two noticeable features of these monetary applications, in comparison to
Burdett and Judd (1983), is that a seller can only serve one buyer at a time
and, in equilibrium, buyers are allocated to sellers randomly. An econometrician observing pairs of sellers and (auction winner) buyers in their models,
or pairs of auction traders in our model, could not tell which model is which.
(If actual trades are observed, then the planner could ‘
disguise’our model
by choosing, if necessary, implementable allocations with constant output or
constant money transfers across meetings). Galenianos and Kircher (2006)
provide nevertheless a promising avenue for future research, that could incorporate elements of optimality from our analysis. They o¤er an explicit
discussion of asymmetric information in meetings (and a valuable literature
review), which we have abstracted from completely. We conjecture that the
upper-bound argument can be generalized in the presence of asymmetric
information and a core-deviation concept.

References
[1] C. D. Aliprantis, G. Camera and D. Puzzello, Contagion equilibria in
a monetary model, Econometrica 75 (2007), 277-282.
[2] F. Brauer and J. Nohel, Introduction to di¤ erential equations with applications, Harper & Row Publishers Inc., New York, 1986.

20

[3] K. Burdett and K. L. Judd, Equilibrium price dispersion, Econometrica
51 (1983), 955-969.
[4] R. Cavalcanti and A. Erosa, A theory of capital gains taxation and
business turnover, Economic Theory 32 (2007), 477-496.
[5] R. Cavalcanti and P. Monteiro, On e¢ cient distributions of money,
mimeo.
[6] M. Galenianos and P. Kircher, A model of money with multilateral
matching, Journal of Monetary Economics 55 (2008), 1054-1066.
[7] B. Julien, J. Kennes and I. King, Bidding for money, Journal of Economic Theory 142 (2008), 196-217.
[8] N. Kiyotaki and R. Wright, On money as a medium of exchange, Journal of Political Economy. 97 (1989), 927-954.
[9] R. Lagos and G. Rocheteau, Money and capital as competing media of
exchange, Journal of Economic Theory 142 (2008), 247-258.
[10] R. Lagos and R. Wright, A uni…ed framework for monetary theory and
policy analysis, Journal of Political Economy 113 (2005), 463-484.
[11] S. Shi, A divisible search model of …at money, Econometrica 65 (1997),
75-102.

21


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