Journal
of Economic
Dynamics
and Control
17 (1993) 7233758.
North-Holland
A dynamic equilibrium model of search,
production, and exchange*
Michele
Northwestern
Nobuhiro
Boldrin
University.
Evanston. IL 60208,
USA
Kiyotaki
University of’ Minnesota. Minneapolis,
Federal Reserve Bank of Minneapnlb.
MN 55455, USA
Minneapolis. MN
55480.
USA
University of’ Pennsylvania. Philadelphia, PA 19104. USA
Federal Resercc~ Bank of Minneapolis, Minneapoli.s. MN 55480,
USA
Randall
Received
Wright
May 1991, final version
received June 1992
We study a general equilibrium model where agents search for production
and trading opportunities, that generalizes the existing literature by considering a large number of differentiated commodities and agents with idiosyncratic
tastes. Thus, agents must choose nontrivial exchange as well as
production
strategies. We consider decreasing,
constant,
and increasing
returns to scale in the
matching technology,
and characterize
the circumstances
under which there exist multiple steady
state equilibria, or multiple dynamic equilibria for given initial conditions. We also characterize
the
existence of dynamic equilibria that are limit cycles. Equilibria are not generally optimal, and when
multiple equilibria coexist they may be ranked. We analyze comparative
statics and find that certain
intuitive results do not necessarily hold without restrictions on the stochastic structure.
1. Introduction
This paper analyzes a general equilibrium
model of search, production,
and
exchange, extending the standard search equilibrium
framework by assuming
Correspondence to: Michele
Boldrin, KGSM/MEDS,
Northwestern
University,
Evanston,
IL 60208,
USA.
*The authors are grateful to the National Science Foundation,
the Santa Fe Institute, the lnstitut de
Analisis Economica at the Universitat Autonoma de Barcelona, and the University of Pennsylvania
Research Foundation
for financial support. They would also like to thank David Levine, Dale
Mortensen, Manuel Santos, Michael Woodford, and seminar participants at the Federal Reserve Bank
of Minneapolis, Northwestern
University, University of Pennsylvania,
Keio University. Kyoto University, and the 1992 Summer Meetings of the Econometric Society for useful comments and suggestions.
0165-1889/93/SO6.00
(Q l993-Elsevier
Science Publishers
B.V. All rights reserved
724
M. Boldrin ef al.. Equilibrium
model fir search. production. and r.uchange
a large number of differentiated
commodities
and agents with idiosyncratic
tastes.’ In the usual framework, agents search for production
opportunities,
or
projects, which always yield one unit of homogeneous output, at some projectspecific cost. In our model, each project yields one unit of a particular
type of
output. As in the standard model, there is an exchange sector where agents with
output for sale search for potential trading partners, and when two traders meet
they exchange if and only if it is mutually agreeable. Due to the large number of
differentiated
commodities,
in our model, agents face nontrivial
decisions concerning trading strategies, as well as the decisions concerning production
strategies analyzed in the standard model.
We analyze both steady state and dynamic equilibria. We consider constant
(CRS), decreasing (DRS), and increasing (IRS) returns to scale in the meeting
technology. We characterize the cases in which there exist multiple steady state
equilibria,
the cases in which there exist multiple dynamic equilibrium
paths
(sometimes a continuum)
starting from given initial conditions, and the cases in
which equilibria display cycles. From the point of view of qualitative dynamics,
our model generates behavior similar to that of the one-good model analyzed in
Diamond and Fudenberg
(1989). However, we provide a complete analysis of
the global dynamics, as well as a characterization
of the various instances in
which indeterminacies
may or may not arise. Furthermore,
our use of the
Poincare-Bendixson
theorem allows us to make sure that asymptotically
stable
and ‘global’ limit cycles exists, something that cannot be verified by means of the
Hopf bifurcation
technique used by Diamond and Fudenberg.
There are a variety of reasons for being interested in our generalization
of the
standard equilibrium
search model. With many goods, whether or not a meeting
between two traders results in an exchange is endogenous;
with one homogeneous good, al/ meetings result in an exchange.2 The focus in our model is
therefore on search in the trading sector, rather than in the production sector; in
fact, most of the results go through even if we shut down search in the
production
sector (say, by assuming that all production
projects have zero cost,
or by assuming that traders are simply endowed with new output immediately
after they consume). Some of the economic implications
of the model change
when many goods are introduced
and exchange therefore becomes nontrivial.
For instance, in the simplest case of a CRS meeting technology, equilibrium
is
inefficient in our model while it is efficient in the one-good model. This is due to
the way in which trading strategies are determined
and, in particular, the fact
that an exchange requires that both agents want to trade.
‘References to the standard
equilibrium
search model include Diamond (1982, 1984a. b) and
Diamond and Fudenberg (1989). See also Mortensen (1989, 1991). Coles (1991), and the literature
cited in these papers.
‘In some versions of the one-good model, both traders in any given meeting experience a common
‘taste shock’ that determines whether an exchange is made. Our approach is quite different and is
based upon specialization
in production
and consumption.
M. Boldrin et al., Equilibrium
model,for search, production.
and e.uchange
725
It is precisely this difficulty of direct exchange resulting from the large number
of goods that leads to a role for fiat money as a medium of exchange in the
simple (steady state and CRS meeting technology)
versions of this model
analyzed in Kiyotaki and Wright (199la, b). Hence, from the perspective of
monetary theory, generalizing
the model to include many goods is essential. In
this paper, however, we concentrate
exclusively on the nonmonetary
economy,
since its analysis is already fairly complicated and interesting. Nevertheless, our
welfare results do provide insights into the welfare-improving
role of money,
and our techniques for analyzing general meeting technologies and dynamics in
the model could potentially be applied to monetary versions of the framework in
the future.
The rest of the paper is organized as follows. Section 2 describes the model
and defines equilibrium.
Section 3 introduces the dynamic analysis by studying
the CRS and DRS cases. Section 4 discusses in detail the dynamic behavior of
the IRS case. Section 5 derives some welfare implications
of the model and, in
particular,
demonstrates
the above-mentioned
result that the equilibrium
is
inefficient even under CRS. Section 6 describes the comparative
static properties
of the model and demonstrates
how some counterintuitive
results can occur,
although we also provide a simple restriction on the stochastic structure that
can be ‘used to rule them out. Section 7 concludes. Most of the proofs are
relegated to the appendix.
2. The basic model
Consider an economy with a continuum
of infinite-lived
agents, with total
measure normalized to one, indexed by points around a circle of circumference
two. There is also a continuum
of commodities
indexed by points on the same
circle. Goods are indivisible and come in units of size one. They are also perfectly
storable, but only a single unit at a time. Individuals have idiosyncratic
tastes for
these goods: the agent indexed by point i most prefers the good indexed by i, and
receives utility u(z) from consuming a unit of good j, where z is the length of the
shortest arc between i and j along the circle, and u: [0, I] + L%is twice continuously differentiable
with u’(z) < 0.
We do not require u(e) to be either concave or convex, in general, but for some
results below we do need to assume that u’(z) + zu”(z) I 0 (which holds automatically if u is concave). We assume that u(O) > 0 > u(l), where 0 is also the
utility from consuming
nothing. We note here for future reference that the
distance along the circle between an agent’s ideal commodity and a commodity
type drawn at random from the circle is uniformly distributed
in [0, 11.
To acquire commodities,
agents search in a production
sector at a cost in
terms of disutility of w0 per unit of continuous
time. They locate potential
production
projects stochastically
according to a Poisson process with constant
arrival rate c( > 0. Each project yields a unit of good i at a cost in terms of
instantaneous
disutility c, where i is drawn randomly from the circle and c is
drawn independently
from the cumulative
distribution
function (CDF), F(c).
Both commodity
type i and cost c are observed upon location of a project,
before a production
decision is made. Let the greatest lower bound of the
support of F be c; for some results, it is important whether c > 0 or c = 0. We
will also typically assume F(c) is differentiable and strictly increasing in order to
simplify the presentation.
Individuals
do not consume their own output; rather, once production
takes
place, agents proceed to an exchange sector where they look to trade their
output for something else that they can consume. In the exchange sector, there is
a search (or storage) cost u’, per unit of time, and potential trading partners are
located according to a Poisson process with arrival rate B 2 0. The number of
meetings in the exchange sector depends on the number of agents searching, N,
and we write m = m(N). This means the arrival rate for a representative
agent is
B = B(N) = m(N)/N. We always assume m(0) = 0. In the CRS case, m(N) = /IN,
and the arrival rate is constant, B(N) = fi. By analogy, B’ > 0 is referred to as
increasing returns (m convex) and B’ < 0 as decreasing returns (m concave).
In any case, meeting another agent does not necessarily imply an opportunity
to trade, since the other agent may not want what you have. If both agents want
what the other has, then they swap inventories one-for-one; otherwise, they part
company. For now, we denote the probability
that a randomly located trader is
willing to accept good i at date t by O,(i). When a good is accepted in trade, there
is a disutility cost c > 0 (a transaction cost) that must be paid by the receiver. Let
u,(z) = u(z) - E; then we assume u,:(O) > 0, and define z, < I by u,(z,) = 0. Thus,
zr is the greatest distance from one’s ideal good that generates nonnegative
utility.
It may seem natural to assume that after trading an agent can consume and
return to production,
at any time, or stay and try to make an additional trade.
However, we assume that upon accepting a good in trade, the receiver must
decide immediately if he is going to consume it or try to trade it again for
something else. This assumption
is not restrictive in a steady state (since if the
agent is ever going to consume the good he should do so as soon as possible),
however, it could potentially be restrictive along a dynamic path. Nevertheless,
making this assumption
simplifies the presentation
considerably.”
Further, in this paper we consider only symmetric equilibria,
in which no
agent or commodity
type plays any special role. In particular, the decision to
accept good j by agent i will depend only on the distance between i and j and not
the values of i or j individually
[plus, of course, the probability
of future
meetings described by the paths of O,(i) and N, for all future t]. It turns out that,
“An alternative assumption
that generates the same results is that goods are only storable by the
agent who produces them. This would be natural if. for example, we interpret the goods as sewices.
M. Boldrin PI al.. Equilibrium
model.for
search, production,
72-l
and cwhange
under these conditions,
there is no indirect trade, and agents accept a good in
exchange if and only if they are going to consume it.
To understand
this, observe that our notion of a symmetric equilibrium
implies that the probability
of a randomly located individual accepting good i is
in fact independent
of i: 19,(i)= 8, for all i at every date r. Now consider an
individual holding good i deciding whether to accept a trade for good j. If he is
not going to consume good j, by accepting it he pays transaction cost E and gains
no benefit, since his trading position is not enhanced as long as 0,(j) = 0,(i) = 8,.
Hence, he will not accept unless he is going to consume j. This implies that all
trades are made for direct consumption,
and there is no indirect exchange.4
We proceed using dynamic programming.
Let the state variable j indicate
sector, j = 0 for production
and j = 1 for exchange, and let Vjl be the optimal
value function for a representative
agent in sectorj at date t. Note that VI, does
not depend on the type of good in storage for the same reason that 0, does not.
Then the continuous
time versions of Bellman’s equations that must be satisfied
by the Vj, are
s
cc
rV,,=
-ww,+cr
max[O,
VI, -
VO,- c] dF(c) + pot,
(2.1)
0
s
1
rV,, = -
w1
+
B(N,)B,
max[O, VO,-
Vrl + u,(z)] dz + p,‘,,,
(2.2)
0
where r is the constant discount
The maximization
problems
strategies: (i) accept a production
than k,, where
rate.5
in (2.1) and (2.2) are solved by reservation
project at date t if and only if the cost c is less
(2.3)
(ii) accept a trade offer at date t if and only if the distance
being offered and your ideal good is less than x,, where
r&(x,) = VI, -
vo*.
z between
the good
(2.4)
4We are not claiming there could not exist asymmetric equilibria where certain goods become
focal points for indirect trade-we
simply choose to concentrate
on symmetric outcomes. Obviously,
the assumption
described in the previous footnote - that goods are only storable by their producers
_ also implies there is no indirect exchange.
‘Formal derivations
are contained in the appendix. Intuitively, these equations set the return to
searching in sector j. r Vj, equal to the flow yield wj plus two ‘capital gain’ terms, an expected option
value plus a pure time change in Vj. In (2.1) the expected option value is the rate at which projects
arrive times the value of rejecting or accepting the opportunity,
whereas in (2.2). it is the rate at which
partners arrive times the probability
that they are willing to trade times the value of rejecting or
accepting their offer.
If we insert (2.3) and (2.4) into (2.1) and (2.2) we have
t-V,, =
- wn + rsO(k,) + r’,, .
t-V,, =
-
wl
+
B(N,)U,s,(x,)
where, to reduce notation,
s
0
s
1[u(z)
till,
(2.6)
the functions
s
k
(k - c) dF(c) =
sI(x) =
+
we have introduced
k
s,(k) =
(2.5)
F(C) dc ,
0
- u(x)] dz = - j’
zdu(z)
0
Because so and sr are important,
we catalog some of their basic properties
in the following lemma. The proofs are easy and left to the reader. Note,
however, that the convexity of s,(e) requires the assumption
u’(z) + zu”(z) I 0,
and then the inequality
E(x) 2 0 asserted in the lemma follows directly from
convexity.
Lemma
positive,
1.
For all k, s,(k)
increasing,
convex,
is positive, increasing, and (‘0nve.x. For all X, S,(X)
and satisjies C(.u) = xs; (x) - .Yl (.U) 2 0.
is
For arbitrary time paths of N, and 0, a solution to the representative
agent’s
decision problem is given by a set of functions {.Y~,k,, V,,, VI,} satisfying eqs.
(2.3)-(2.6) and the boundary conditions 0 5 x, I z,, k, 2 0 for all t. The boundary conditions
rule out the consumption
of goods yielding negative utility, and
also imply that V,, - V,, = u,(x,) 2 0 for all t, or, that the value to being in the
exchange sector is never less than the value to being in the production
sector
(which would conflict with an obvious free disposal condition).
As discussed
earlier, we concentrate
on symmetric outcomes, which means all agents use the
same reservation strategies. This implies the probability
that an agent chosen at
random will accept a trade for a given good at date t equals the probability
that
the distance between it and his ideal good is less than his reservation distance:
8, = pr(z I x,) = x, (given the uniform distribution
of agent and commodity
types).
Since the probability
that a representative
agent accepts a trade is x,, the
probability
of two agents trading in any particular
meeting is given by the
probability
that both have something that the other finds acceptable, x:. This
further implies that the measure of agents in the exchange sector evolves
M. Boldrin ei al., Equilibrium
according
model,for search, production,
129
and exchange
to the law of motion6
ni,= ctF(k,)(l
- N,) - m(N,)xf
Putting these results together,
equilibrium
for the model:
.
(2.7)
we have the following
definition
of a (symmetric)
DeJinition.
An equilibrium with initial condition
No at t = 0 consists of time
paths for {k,, x,, Vat, Vrt, B,, N,} defined for all t E [0, CC) and satisfying:
(a) N(O)= No;
(b) conditions
(2.3)-(2.6),
following maximizing
(c) condition
k, 2 0, and 0 5 x, I z, for all r, implying
strategies given N, and 0,;
(2.7) and 0, = x, for all t, implying
expectations
agents
are
are rational.
3. Equilibria with constant or decreasing returns to scale
Without any real loss in generality, we set w. = wr here to ease the presentation. Then, subtracting
(2.5) from (2.6), we obtain
r(V, -
Vo) = B(N)Bst(x) -
aso +
(time subscripts
will be omitted from now
8 = x, k = VI - V. = u,(x) and the condition
reduce this expression to
w,(x) = B(N)xs, (x) - ca,[u,(x)]
ri,- to
on). The equilibrium
conditions
PI - PO = u’(x)a can be used to
+ u’(x),?,
which implies
i = T(N, x) = (l/u’(x))
{ru,(x) +
m~Cu,(x)l
- B(W~I(X)I.
(3.1)
Also, k = u,(x) can be used to reduce (2.7) to
Pi =
S(N, x) - crF[u,(.x)] (1 - N) - m(N)x’
.
(3.2)
Eqs. (3.1) and (3.2) define a dynamical
system over the set d = {(N, x):
0 I N I 1,0 I x I z,}. Given an initial value of No E [0, l] and an initial value
6Note that, although individual histories are random, the aggregate variable N, evolves according
to a deterministic process. This is an application of the law of large numbers for atomless economies
derived in Uhlig (1990).
of x0 E [0, zE], any solution [N(t), .x(t)] to this system going through (N,, x,,) at
t = 0 and remaining in U for all t 2 0 constitutes an equilibrium
for the model.
Notice that the initial value of No is given by nature; however, there is nothing to
pin down x0, which can take on any value at date t = 0.
The boundaries
of U are: (i) U, = {(N, x): x = 0, 0 2 N 5 1); (ii)
UZ = {(N, x): x = z,, 0 I N I 1); (iii) U3 = ((N, x): 0 < x < z,, N = 0); (iv)
U4 = ((N, x): 0 < x < zC, N = 11. Any trajectory satisfying (3.1) and (3.2) and
starting in U1 or CJZwill leave U and never return, while any trajectory starting
in U3 or U4 will enter U, and remain in U at least for small t > 0. We now
characterize
the behavior of the system on the set U = int(il). We analyze
separately
the cases where B’ = 0, B’ < 0, and B’ > 0, as they have rather
different properties. We also sometimes make a distinction
between the cases in
which c = 0 and c > 0, where we recall that c is the greatest lower bound of the
production
cost distribution
F(c). The case where B’ > 0 for small N and B’ < 0
for large N (which may be the most realistic) can be understood
by combining
the results obtained in the two cases separately. In what follows, we call the set of
points in U satisfying S(N, x) = 0 and T(N, x) = 0 the S-locus and T-locus,
respectively, and a critical point or steady state of the system is a point (N, x)
solving S(N, x) = T(N, x) = 0.
3.1. Constant
returns
to scale
We begin with the simplest situation
where m(N) = PN, which implies
B(N) = /I, for all N > 0. The horizontal
line {(N, x): 0 I N I 1 and x = zp},
where zg is the unique solution to
m,(x)
+ aso[u,(x)]
-
pxs, (x) = 0,
(3.4)
describes the T-locus. With regard to the S-locus, it is easy to see that if (’ = 0
and F’(0) > 0, it begins at the point (0, z,) and slopes downward to the point
(0, 1). On the other hand, if c > 0 and we define z, by u,(z,) = g, then the S-locus
coincides with the vertical axis over the nondegenerate
interval [z,, z,] and
slopes downward from the point (0, z,) to the point (0, 1).
In either case, if we assume that c is small enough so that z, > zp, there exists
exactly one nondegenerate
steady state, with x = zfl and N = N, given by
N, = ctFCu,(~~)ll(crFCu,(zp)l
A straightforward
with consideration
fig. 1.
+
b;).
(3.5)
local analysis
of the system around
(N,, za), together
of its global behavior, yields the following result, depicted in
M. Boldrin et al., Equilibrium
model for search. production,
mrf ca.whange
731
Fig. I
Proposition I. Assume CRS and z, > zp; then there exists a unique equilibrium
path for any initial condition No E (0, 11. This equilibrium satisfies x, = zpfor all
tandN,+Ngast+
+ e.
Proof
See appendix.
3.2. Decreasing returns to scale
The assumptions
on B(*) in this case are: B(0) = /?, B(N) > 0 and continuous
for all N > 0, and B’(N) < 0 for all N > 0. This kind of functional
form may
seem unrealistic, although a negative relation between the arrival rate and the
number of searchers may well arise (due to ‘congestion’) at high levels of N. The
extreme case we are considering
is nevertheless
worthy of study because it
reveals that IRS in the matching
technology
is not necessary to generate
a multiplicity
of interior steady states or a continuum
of equilibria.
The T-locus slopes upward, while the S-locus slopes downward
when the
elasticity of B is less than 1, but may also slope upward if this elasticity is greater
than I. As long as zp < z, at least one nondegenerate
steady state exists. The
existence of others depends on the elasticity of B(N), but it is clear that if
a second intersection
of the S-locus and T-locus exists, then a third must also
732
M. Boldrin
et al., Equilibrium
model /iv scorch, production,
und e.whunge
exist since the T-locus must eventually reach the point (0, 1). In other words,
there is (generically) an odd number of steady states.
Proposition
2 describes the global dynamics for the case where the elasticity
of B(N) is everywhere less than 1, which guarantees the existence of exactly one
interior steady state. In all relevant respects it is the same as the CRS case (see
fig. 1). If the elasticity condition is violated there can be multiple interior steady
states, We will not pursue this here because it turns out to be formally equivalent
to one of the IRS cases analyzed in the next section.
Proposition 2.
Assume DRS and INB’IBJ I I ,fbr all N; then there exists
u unique equilibrium path ,for uny initial condition No E (0, 1). The equilibrium
path coincides with the global stable man$~ld qf‘the unique interior critical point
qJ’(3.1) and (3.2).
4. Equilibria with increasing
returns to scale
now the case in which B is increasing
with B(0) = 0, B’(N) > 0
begins at (0, z,) and slopes downward in the
(N, x) plane to the point (1, z,), where z, is the unique value of x E (0, z,) that
solves
Consider
VN > 0. In this case, the T-locus
ru,(x) + ~~s~[u,(x)] - B( l)xs,
(x) = 0.
The qualitative behavior of the S-locus is the same as above; it slopes downward
from (0, z,) to (1,0) and is coincident
with the vertical axis between z, and z,
when c > 0. It lies below the T-locus in a neighborhood
of N = 1 because
z1 > Oland, as long as c > 0, it will be below the T-locus at N = 0 also. Hence,
there exists an even number
of interior
steady
states,
Ej = (Nj, xj),
j = 0, 1,. . , 2n. Assuming n > 0, we will concentrate
most of our analysis on the
general situation shown in fig. 2, in which there are exactly three critical points,
E”, E’, and E2, with 0 = No < N’ < N2 and z, = x0 > x’ > x2.
4.1.
Local analvsis
Around
a critical
point
Ej the linear
part of (3.1)-(3.2)
is described
where a(*, -) is of order smaller than or at most equal
I/N - Nj 11’. The determinant
of the Jacobian matrix is
det = T,S,
- S,T,
= S,TX{dx/dN17=,,
- dx/dN,,=,)
by
to 1)x - .Xj11’ and
.
(4.2)
M. Boldrin
et al., Equilibrium
model,for
search. production.
and e.rchange
733
N
Fig. 2
Hence, det is positive or negative depending on whether the T-locus is steeper or
flatter than the S-locus at their intersection.
At the steady state with the greatest value of N (E2 in fig. 2), the T-locus is
flatter than the S-locus, and so det < 0. At the steady state with the next highest
value of N (El in fig. 2) the T-locus is steeper than the S-locus, and therefore
det > 0. Finally, at the steady state E” with N = 0 and x = z, the local structure
of the vector field depends on the nature of our assumptions on c and F. We
need to distinguish two cases:
(i) c > 0, and therefore F(0) = F’(0) = 0;
(ii) c = 0, F(0) = 0, and F’(0) > 0.
In case (i) both SN and S, are zero at E”, which implies det = 0; the steady
state is a degenerate critical point with one eigenvalue being zero. In case (ii) SN
is zero but S, is positive, which implies the determinant is positive at E”. By the
implicit function theorem, the S-locus is flat at E” while the T-locus is downward-sloping. When c = 0, one cannot exclude the situation in which the T- and
S-loci intersect only once in U. It is therefore possible to have a unique interior
steady state even in the presence of IRS when c = 0.
In summary, under the assumption of IRS m the matching technology the
following two situations are possible: (i) when c > 0, either there are no interior
134
M. Boldrin et 01.. Equilibriunl
model for srurch. production.
md r.uchungr
X
N
Fig. 3
stationary
states or there is an even number of them; (ii) when c = 0 and
F’(0) > 0, there always exists at least one interior steady state and when there
are more than one there are an odd number.
We begin with the case where c = 0 and there are only two steady states:
E” = (0, z,) and E’ = (N’, x1), as in fig. 3.
The global characterization
of the dynamic equilibria for this case is:
Proposition 3. Assume IRS, c = 0, and u unique interior steudy state. Then ,for
every N, in [0, l] there is a unique equilibrium. When No = 0, such an equilibrium
coincides with the stationary point (0, z,) ,for all t 2 0. When N > 0, No # N’ ,
such an equilibrium starts ut the unique value x(0) such that (No, x(0)) belongs to
W”(E’) (the global stable marCfold of E’) and converges to E’ .’
Proof:
See appendix.
We now proceed to the case where c > 0. In this case there is an even number
of interior steady states. We will concentrate
on the case in which there are two
‘To be complete we have to add that, in exceptional and structurally unstable cases, the stable
manifold of the saddle pomt E’ may go all the way up to E” and therefore connect the two critical
points. The fact that the S-locus is horizontal at E” prevents W”(E’ ) (the global unstable manifold of
E’) from getting into E”.
M. Boldrin et al.. Equilibrium
modelfor
search, production,
and e.uchange
735
N
Fig. 4
of them (fig. 2): E” = (0, zE), E’ = (N’, xl), and EZ = (N’, x2) in order of increasing values of N.
The local characterization
is simple:
Proposition 4: Assume IRS and c > 0. Then the boundary steady state E” is
a degenerate saddle-node (or saddle-focus) at which a transcritical bifurcation
occurs. TheJirst interior steady state, E’, is (generically) either a sink or a source
depending on parameter values, whereas the second one, E2, is a regular saddle.
Proof
See the appendix.
Fig. 4 depicts the local behavior of the flow around the boundary steady state
for the case in which the slope of the T-locus at E” is, in absolute value, less than
r/TN, which is the slope of the (local) center manifold. This occurs when TN > r,
whereas the opposite is true in the other case.
4.2. Global dynamics
The presence of a center manifold at E” can complicate our analysis purposelessly. We will therefore restrict attention
to the case in which the center
manifold [ Wc(Eo)] is unique with the motion on it converging to E”. Even with
736
M. Boldrin et al., Equilibrium
model ftir srarch. production,and r.whangr
this simplification
the structure of the equilibrium
set remains quite complicated, as a number of different configurations
are generated by different combinations of parameter values.
One has to distinguish, first of all, between the case in which E’ is a sink and
the case in which it is a source. Second, the stable [Ws(E2)]
and unstable
[ W”(E2)] global manifolds of the saddle point E2 may or may not be representable as graphs of functions in (N, x) space. When the latter situation occurs for
either of them, dynamic equilibria that are limit cycles may originate and, under
the circumstances
detailed below, they may become the attractors
of other
equilibria beginning nearby.
From an economic point of view, the relevant features of the set of equilibria
can be summarized
as follows:
(i)
To a given initial condition
No, one can often
(indeed, a continuum)
of dynamic equilibria.
associate
a multiplicity
(ii) The latter need not converge to the same attractor. Depending on essentially arbitrary expectations,
indexed by the initial value x0, equilibria beginning at a common value No may converge to very different asymptotic
positions.
(iii) As the initial condition
different attractors.
No moves
in [0, 11, the equilibria
converge
to
(iv) Expectations
may matter more than initial conditions.
Under certain configurations, even when No is very close to one, an equilibrium
path exists
that asymptotically
converges to E” if expectations
are pessimistic enough
and, conversely, very optimistic expectations produce equilibria converging
to E2 from initial conditions
near No = 0.
(v) Cyclic equilibria exist and a cycle may be the global attractor
initial conditions.
for almost all
Propositions
5 and 6 are dedicated to a precise mathematical
description of the
various cases. They are inevitably tedious and the reader should make use of
figs. 5a-5e and 6a-6e to facilitate understanding.
Shaded areas describe ‘basins
of attraction’ for the attractor contained within, and correspond to a continuum
of equilibria having common asymptotic behavior.8
Proposition
5.
Assume IRS, 5 > 0, and that E’ is a sink. The global dynamics of
the system can be as in either one qfjgs.
configurations of the equilibrium set.
Sa-Se.
These produce
three distinct
‘To simplify our statements we identify each orbit with one equilibrium. The reader will recognize
that more than one equilibrium can be extracted from a given orbit if the latter is oscillatory.
M. Boldrin et al.. Equilibrium
model,for
search, production.
and exrhange
737
Fig. 5a. There exists a repelling limit cycle y around Et , and WU(E2) and
Ws(E2) are the graphs of two functions from N to x. Denote by N < N’ and
N B N’ the westmost and eastmost points of y, respectively. Then:
a) for NO < N, there are two equilibria given by Wc(Eo) and W(E2);
b) for N < No < 15, there are four types of equilibria: Wc( W’), Ws(E2), y, and
the continuum of paths converging to E’ ;
c) for IV < No I fl, where I? denotes the eastmost point on Wc(Eo), there are
again two equilibria given by Wc(Eo) and W”(E’);
d) for No > l?‘, there is a unique equilibrium given by Ws(E2).
Figs. 56, SC. There exists a repelling limit cycle y around E’, and either W”(E2)
or W(E2) or both are not the graph of a function from N into x. Let N, N, and
fi be as in configuration 1 above, and denote by & < @ the westmost point on
WS(E2). Then:
a) for No c fl there exists only one equilibrium: Wc(Eo);
b) for u I No < N and N < No I fl, there exist two equilibria: wC(E’) and
W=(E2) ;
c) for N I No I N, there are four types of equilibria: Wc(Eo), Ws(E2), y, and
the continuum of paths converging to E’ ;
d) for No > i?‘, there exists only one equilibrium: WS(E2).
X
Shaded areas denote continua of equilibria.
Fig. Sa
738
\
Shaded areas denote continua of equilibria.
Fig. Sb
Z&
WUG2)
ZC
&
--f
WC(EO)
@\
WS(E2)
E2
Shaded areas denote continua of equilibria.
Fig. 5c
\
M. Boldrin CI al.. Equilibrium
model
fbr
search, produ&m,
and exchange
739
x
Shaded areas denote continua of equilibria.
Fig. 5d
Shaded areas denote continua of equilibria.
Fig. 5e
N
740
M. Boldrin et al., Equilibrium
model,ftw search. producrion.
md e.uchonge
3. Figs. 5d, 5e. There exists no limit cycle around E' , and Wc(Eo) is the graph of
a function from N into x. Let & be as defined in configuration 2. Then:
a) if 0 5 No < 8, there exists a unique equilibrium: Wc(Eo) ;
b) for No 2 ly, there exist three types of equilibria: Wc(Eo), Ws(E2), and the
continuum of equilibria converging to E’.
Proof:
See appendix.
Proposition 6. Let the hypotheses of Proposition 5 be true, but assume that E’ is
now a source. The global dynamics of the system in this case can be as in either one
offigs. 6a-6e. These produce three distinct con$gurations ?f the equilibrium set:
1. Fig. 6a. There are no limit cycles around El, and both W”(E2) and W”(E’) are
graphs offunctions from N into x. Then:
a) for all 0 I No I I?, where i? is as in Proposition 5, there are two equilibria:
Wc(Eo) and W”(E’);
b) if No > 13, there is a unique equilibrium: Ws(E2).
2. Figs. 6h, 6c. There are no limit cycles around EL, but either W”(E2) or Ws(E2)
or both are not the gruph qfa,function. Then.
a) ,for 0 2 No < @, there is only one equilibrium: Wc(Eo);
Fig. 6a
M. Boldrin CI al.. Equilibrium
model,for
search, production.
and e.uchange
741
X
Z&
zc
N
Fig. 6b
Fig. 6c
d areas denote continua of equilibria.
Fig. 6d
laded areas denote continua of equilibria.
Fig. 6e
M. Boldrin et al., Equilibrium
model,for
search. production.
and exchange
743
b) for ly I N$ < fi, there are two equilibria: Wc(Eo) and W(E’);
c) for No > N, there is a unique equilibrium: Ws(E2).
3. Figs. 6d, 6e. There is an attracting limit cycle y around E' , and Wc(Eo) is the
graph of a .function. Then:
a) for all 0 I No < N, there is a unique equilibrium: Wc(Eo);
b) for all & I No < N and N < No, there are three types of equilibria:
Wc(Eo), Ws(E2), and the continuum of equilibria converging to y;
c) for all p I No I N, there are four types of equilibria: the three in b) and the
limit cycle y.
Proof
See appendix.
From a practical point of view one is obviously interested in finding computable conditions that can be used to discriminate between the various cases.
Checking the local stability-instability of E’ is only a matter of linearizing and
computing the associated eigenvalues. The interesting thing is to have a simple
set of conditions that could detect the existence of the (attractive or repulsive)
limit cycles. The Andronov-Hopf
bifurcation theorem turns out to be quite
useful for this purpose.
Proposition 7. Assume that the trace of (4.1) equals zero, i.e., that
B’(N,)x:N,
+ B(N,)s, (xI)/u’(.xl) = r 2 0. Then there exists a p > 0, such that
for all r in either (r + u) or (f - p) the system (3.1) (3.2) has a limit cycle around
the critical point E’ . Jf such a cycle is unique, it will be locally asymptotically
stable when it exists,for r E (r + p) and unstable when it exists for r E (f - u).
Proof:
See appendix.
The following example illustrates explicitly the existence of stable limit cycles.
Assume that u,(x) = U - x, which implies sr(x) = x2/2, and m(N) = N’. Let
c be uniformly distributed on [d, 11, where d E (0, 1):
F(k) =
0,
k < d,
(k - d)/(l - d),
dskrl,
1,
k>l.
From this, we can compute
s,(k) =
0,
k < d,
(k - d)‘/2(1 - d),
dlkll,
(2k - 1 - d)/2,
k>l.
144
M. Boldrin
Then the dynamical
hi =
aF(U
et al., Equilibrium
nmlel
for .rrarch. production.
and cwhonge
system becomes
- x)(1 - N) - N8x2,
1 = - r(U - x) - srsO(U - x) + N@- ‘~~12,
where F(U - x) and sO(U - x) can be derived from the above expressions.
Consider the parameter values r = 1, 0 = 2, d = 0.5, and U = 1.05. By varying r and analyzing the system numerically on a differential equation solver (we
use PHASEPLAN),
one can check the following. At r = 0.039, the situation is
as depicted in fig. 6b, where the left branch of the stable manifold
Ws(E2)
emerges from E’, which is a source, while the left branch of the unstable
manifold W”(E2) loops around the outside of W*(E2) and heads south. As r is
increased to r = 0.040, the situation changes to that shown in fig. 6e, where the
unstable manifold W”(E2) now loops inside of the stable manifold Ws(E2) and
heads toward Et. Since E’ is a source at r = 0.040, there exists a limit cycle
y such that orbits beginning near y converge to y, as shown in the diagram. As
r increases further, the size of the cycle y shrinks. At r = 0.043, the cycle has
shrunk to zero and EL becomes a sink, as shown in fig. 5d.
5. Welfare
We now turn to a discussion of steady state welfare. In the IRS case with
multiple steady states, it is easy to see that those with greater values of N are
Pareto superior.’ Thus, for the IRS case, consider two steady states (N 1, x,) and
(2.5) implies
with
N2 > N,
therefore,
x2 < x1. Since
and,
(N2,
x2)>
in x. Since (2.4)
r V. = - w. + LYS~[U,(X)]
in steady state, V, is decreasing
implies VI = V,, + u,(x), VI is also decreasing in X. Hence, all agents are better
off in the high N-low x outcome. We now argue that even the steady state with
the greatest N is not generally Pareto optimal in our model.
To make the point in a simple way, we assume that c = 0 with probability
1 (production
is free, although agents still have to search for projects). We begin
with the case of CRS, and consider IRS below. With CRS there exists a unique
nondegenerate
steady state. With c = 0 with probability
1, this steady state
would be efficient in the standard model, but will not be in our model. This is
one sense in which the introduction
of multiple commodities makes a difference.
We also set w0 = w, = 0 to reduce notation.
‘The same is not true in the case of DRS: in this instance higher values of N go together with
higher values of x (the T-locus is upward-sloping) and the simple argument we give below may not
go through or may even be reversed.
M. Boldrin et ul., Equilibrium
When c E 0, eq. (2.1)
and thus V,, - Vi = the reservation
trade,
value of x, then, by eq.
model,for search. producrion, and e.whunge
745
immediately implies r V, = a( V, - Vo) at a steady state,
rV,/(cc + r). Further, if we let any value of x determine
and not necessarily
the individual
utility-maximizing
(2.2) in steady state we have
s
x
rV,
=
j3ex(Vo -
VI) + /RI
0
u,(z)dz
x
u,(z) dz .
= - (@~/(a + r)) t-V1 + /IO
(5.1)
s 0
Differentiating
and setting aV,/ax = 0 yields the first-order
value of x that maximizes
V1. If we substitute
this into
equilibrium
condition
0 = x, we arrive at
u,(x) = (Bl(~ + r)bsI C-4.
condition
(5.1) and
for the
use the
(5.2)
The x that satisfies (5.2) is the unique nondegenerate
steady state equilibrium
for
this case. We claim it is not Pareto optimal.
To see why, consider a social planner’s problem of choosing x to maximize V1
and Vo.io If we insert 8 = x into (5.1) and then differentiate, we find that the
first-order condition for the planner’s choice of x satisfies
s
s
x
2xrV,
=
(LX+ r)xu,(x)
+ (0I + r)
u,(z) dz .
0
Substituting
this into (5.1) and simplifying
yields
x
u,(x) = (P/(~+ r))xhW - U/x)
u,(z) dz .
(5.3)
0
The solution to this equation is the socially optimal reservation good. Comparing (5.2) and (5.3), the right side of the latter has an additional
term, which is
negative, and thus the planner chooses a greater x than the equilibrium
value.
In equilibrium,
therefore, there are too many people in exchange and too few
in production.
Agents searching for production
opportunities
are often identified as unemployed in this literature, and so we could say that there is too little
“There is no ambiguity as to the correct welfare criterion for a social planner in the special case
under consideration (CRS and free production), because VI and VO are proportional. However, we
are considering only a limited class of options for the planner (and note, in particular, that agents are
assumed to consume after every trade). Nevertheless, we can verify the inefficiency of equilibrium by
dominating the equilibrium strategies with strategies from this class.
unemployment
in laissez-faire. The intuition
here is somewhat different from
existing theories. In this model, individuals are only willing to trade when they
meet a partner who has something they want, which means a good within the
distance x of their ideal. In order for trade to occur, both partners have to have
something the other finds acceptable, and therefore individuals often get denied
things they want because their partners do not want what they have. All agents
would be better off if they would all lower their standards. This would reduce the
amount of time spent in exchange and increase the time spent in the production
process. ’ ’
We now consider the IRS case. An argument similar to the one used above
implies the decentralized
steady state is characterized
by (5.2), except that /? is
replaced by &N(x)), where N(x) solves iii = ~(1 - N) - B(N)Nx2 = 0. Now
the steady state equilibrium
satisfies
u,(x) = (B(N(x))/(a + Y))Y.S,(x) .
(5.4)
There can, of course, be more than one solution to (5.4); so we consider the best
steady state (with the lowest x and greatest N). As above, it can be shown that
the socially optimal reservation level satisfies
u,(x) = (B(N(x))/(a
+ r))xs,(x)
-
[l/u
+ B’N’/B]
-‘u,(z) dz.
s0
(5.5)
By comparing
(5.4) with (5.5) we see that whether the optimal .Y is above or
below the equilibrium .Ydepends on the sign of the expression in square brackets
in (5.5).
The first term in the bracketed expression
represents the effect discussed
above for the case of CRS - individuals neglect the effect of x on the frequency of
trade. However, the second term inside the square brackets in (5.5) is negative,
since N’ < 0 along the I\i = 0 locus. This effect tends to make .Ytoo big, which
reduces N and therefore the arrival rate /3. The net result obviously depends on
which effect dominates.
In any case, our generalization
does point out an
important
qualification
to the prediction of the standard model which is that
there are too many agents in production
and too few in exchange.”
’ ’ This gives some insight into the welfare-improving
role of fiat currency in the versions of this
model studied in Kiyotaki and Wright (1991~1, b). A generally accepted fiat medium of exchange
increases the frequency of trade, and since there is too little trade in laissez-faire. the introduction of
fiat money can make all agents better off.
“Such a qualification becomes more relevant in the DRS case, even when only one stationary
state exists. In this instance the sign of B’ is negative and the whole expression within square brackets
becomes positive. Therefore, the equilibrium level of .Y is unequivocally too low and N too high.
M. Boldrin PI al., Equilibrium model. for starch, production. and cwAangc
6. Comparative
141
statics
In this section we study the effects of changes in parameters, concentrating
on
the case of CRS. We are interested in how exogenous changes impact on agents’
decisions about how demanding they should be in exchange (as measured by x)
and in production (as measured by k). In the CRS case, the unique nondegenerate equilibrium
is characterized
by
4(x) = ru,(x) + tU~[U,(X)] - flxs,(.x) + w, - wg = 0.
(6.1)
If we let d = - l/d’ > 0, then (6.1) and k = u,(x) yield:
axjaw,
aklaw, = dd c 0,
= A >o,
axlaw,=
aklawo = - u’A > 0)
-A<o,
0,
aklar = u’du, c 0,
ax/ax = dso > 0,
aklaa = ddso c 0,
axlap = - ds, < 0,
aklap = - WAS, > 0,
a.qa& = - A(r + ctF) < 0)
ak@
a.@% =
AU, >
=
@(x~u'
-
St)<
0.
All of these are intuitive, and interpretation
is left to the reader.
We would also like to consider changes in the production
cost distribution,
F(c). An efficient way to parameterize
things is to suppose that F belongs to
a family of CDFs indexed by CT,{F(c, o)}, with the property that o2 > 6, implies
F(c, 02) second-order
stochastically
dominates F(c, 0,); that is, c2 > 0, implies
KIF( c, 02) - F(c, a,)] dc I 0
s0
Dividing
by ~7~- c1 and taking
for all
K 2 0.
the limit as c2 -+ 0, reduces
this to
K
Fz(c, CT)dc
I 0
for all
K 2 0,
s0
where F2(c, CT)= aF/ao.
result
Differentiating
(6.1), we now immediately
obtain
the
K
ax/au = A
F,(c, g) dc I 0.
s0
Also, since k =
U,(X), we
have ak/aa = u’axjao 2 0.
(6.3)
We conclude that changing F(c) so that it second-order
stochastically
increases makes agents more demanding
in exchange and less demanding
in
production. Several interesting effects are special cases of this general result. For
example, suppose that the mean of F(c, a) is independent
of a; then reducing 0
implies an increase in risk in the standard mean preserving spread sense. Hence,
more risk raises x and lowers k, so that agents become less demanding in trade
but more demanding
in production.
For other cases of interest, recall that
second-order
is implied by first-order
stochastic
dominance.
In particular,
reducing all costs by a fixed amount (a translation
of F), or proportionally
(a
scale transformation
of F), first-order and therefore second-order
stochastically
decreases F, which reduces 0 and thereby raises x and lowers k.
Another way to show this last result is to define net cost as cn = (1 - ;I)(, - T.
where we could interpret y and T as proportional
and lump sum production
subsidies. An increase in r is equivalent to a translation
of the CDF, while an
increase in y is equivalent to a scale transformation.
Generalizing
the arguments
leading to (6.1) the steady state condition becomes
b(x) =
WI -
w. + t-u,(x) + ct(1 - y)s,(k)
- flxs, (x) ,
(6.4)
where
now k = (u,(x) + ~)/(l - y). Differentiation
yields &/a~ > 0 and
axlay > 0, as argued above using stochastic dominance.
However, we also find
after simplification
that
ak/as
= (A/( i - ~))cpS1 - (r + 0X2) U'] > 0,
ak/a;,
= (A/( 1 - i’)) [k/h,
(6.5)
- (kr + k/3 + zso) u’] > 0
The results in (6.5) may appear to contradict
our earlier conclusion,
that
reducing c by a fixed amount or proportionally
should lower k. Fortunately,
the
contradiction
is only apparent. Let k, = (1 - 7) = r define the net reservation
cost. Then k, = u,(x), and hence we immediately have ak,l& = u’ax/ar < 0 and
ak,/ay = u’ax/ay < 0. In other words, the net reservation
cost definitely falls
when all costs are reduced by a fixed amount or proportionally,
consistent with
our earlier results, and it is the ‘after-subsidy’
value of k that is relevant. As
shown in fig. 7, a lump sum cost subsidy increases k, but by less than the amount
of the subsidy, so that k, falls, This makes agents more willing to take on
expensive projects, in gross terms, but not in net terms. Fig. 7 also shows the
effect of a translation
of F(m), which results in exactly the same net effect.
We now investigate how the steady state value of N depends on the parameters. Let W. = S(k)
be the hazard rate in production
(the rate at which
projects arrive times the probability
they are accepted), and let HI = /3x2 be the
hazard rate in the exchange sector (the rate at which agents meet times the
M. Boldrin et al., Equilibrium
model.for
search, production, and e.whange
749
1. Translating f to the left a distance z reduces k, but by less than 2.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
kz-r
kl
I
I
I
I
I \
I
k2
2. Subsidizing costs by a lump sum z increases k, but by less than 2.
Fig. 7
\
c
probability
they agree to trade).r3 Notice that an increase in .Y or a fall in k,
holding c( and p constant, will raise HI, lower No, and lower N. Thus, for
example, our earlier results imply that an increase in u‘, - u’,, or r reduces N. An
increase in e is more complicated
since it lowers both HI and Ho, and the net
effect cannot be determined
in general. The same is true for changes in the
distribution
of production
costs.
The impact of changes in the arrival rates, r and B, are complicated, because
they affect N directly as well as indirectly through x and k, but we can make
some progress. To develop our intuition, we begin with the case where F(k) = 1,
which means that all projects are acceptable (this would certainly be true, e.g., if
c = 0 with probability
1). We also assume w. = 111rfrom now on to reduce the
notation. Then N = z/((x + fix’), and after simplification
we find
aN/&t = @fi[2rk
- (r + z)xu’ + fiZ_] > 0,
i?N/Clfi = CpctX[(r + Y.)u’ - PC] < 0.
(6.6)
where @ = dX/(cx + fix’)’ > 0 and Z= xs’, - s, 2 0 by Lemma 1. At least
when F(k) = 1, then we have the reasonable
result that an increase in s( or
a decrease in fi increases the number of agents in the exchange sector.
Returning to the general case, one can easily verify aH,/i% > 0, aH, /afl > 0,
and aH,/ag > 0, but, perhaps surprisingly,
the effect of IXon Ho = crF[u,(x)]
gives us trouble. Differentiating,
we have
s
k
aHo/&
=
F + stF’ 11’ ax@
=
F + xF'u' A
F(c) dc
0
The sign of (6.7) cannot be determined in general. This is analogous to a result in
the partial equilibrium
job search model, where an increase in the offer arrival
rate need not increase the probability
of leaving unemployment
because the
reservation wage increases. Burdett (1981) was the first to show the counterintuitive case, where more frequent offers increases the duration of unemployment,
can be ruled out by assuming that the wage offer distribution
is log-concave, and
our goal is to derive some similar results here.14
“As mentioned earlier, agents looking for an opportunity in the production sector are sometimes
referred to as unemployed. On this interpretation, I/H0 is the average duration of an unemployment
spell. l/H, is the average time between such spells. and I - N = H,/(H, + H,) is the aggregate
unemployment rate.
is a concave function, where /‘= F’ is the density. This
“Log-concavity
means that log[j(c)]
restriction has been used frequently in search theory since Burdett’s work [see Wright and Loberg
(1987). e.g., for an extended discussion and references].
M. Boldrin et cd., Equilibrium
model,for search, production,
and e.uchange
751
The important fact about log-concavity for our purposes is that it implies the
left and right truncated mean functions,
s
0
s
K
K
.MW =
cdW/(l
- F(K)),
P,(K) =
0
c dF(c)lW) ,
have slopes that are less than one [see, e.g., Goldberger (1983)]. In particular,
p: = F(c) dcF’/F’ I 1. Inserting &. into (6.7) and simplifying,
aHo/au= dF[fis,
- (r + PX') U’ + uFu’(~; - l)] .
Since log-concavity implies p: < 1, it guarantees aH,/acc > 0, and therefore
increasing the arrival rate c1will necessarily raise the hazard rate and lower the
average length of a spell in the production sector.15
Even if an increase in c1raises Ho, this does not mean that it will necessarily
lower the steady state proportion of agents in the production sector, since an
increase in CIalso raises H L. After simplification, the net effect is given by
aN/ar = @[Bsr - (I + px’) u’ - aFu’(1 - p:) - 2crsolx] ,
where @ > 0. The sign of this expression cannot be determined in general. But if
we assume log-concavity, then p: I 1, and after some algebra, we have
aNlaM 2 dfixF[Bsr
- (r + 0x2) UT- 20rsojx]
= APxF [ /3X - ru’ + 2rk/x]
> 0,
where, again, .Z 2 0 by Lemma 1. We conclude that log-concavity not only
guarantees that an increase in GI raises Ho, but by more than enough to
compensate for the increase in H, , so that the net effect is to increase N.
7. Conclusion
This paper has analyzed the steady state behavior of an equilibrium search
model with heterogeneous commodities. Some of the results are similar to the
standard one-good model, like the qualitative dynamics, while others are rather
different, like the welfare properties. We point out that the dynamic behavior of
this class of models is very robust: as indicated by Propositions 5 and 6, two
economies starting at identical initial conditions, as described by No, can
15Without
and r.
log-concavity,
we could have p; > I, and hence Sf,,/ax
< 0, for small values of 1
converge to remarkably different attractors depending on the initial value of x,,,
and the model in no way pins down x0. The welfare properties of the model with
multiple commodities
give us some insight into why simple versions of the
model have a role for fiat currency. Potentially
interesting
future work may
consider the dynamic behavior of monetary versions of this model with alternative assumptions
about the meeting technology.
Appendix
Derivation
qf (2.1) and (2.2)
Here we discuss a simple way to derive the continuous
time dynamic programming equations (2.1) and (2.2). Consider a discrete time model with periods
of length At. Then the value of being in state 0, for example, is the instantaneous
search cost plus the discounted value of next period, which is the probability
of
finding a project times the expected maximum value of rejecting it or accepting it
and switching to exchange, plus the probability
of not finding one times the
value to remaining in production,
Vat =
-
w,At + e-*I’
max CVO.,
+ ,I,I K., + ,I~ -
(,IW4
0
+ (1 - ~At)Vo.r+.~r+ MAtI
I
,
where o(At) represents
the probability
of two or more Poisson arrivals in
a period, and hence satisfies o(At)/At + 0 as At + 0. Rearranging
the previous
expression,
(( 1 - em’“‘)/At) Vo,
+ (Vo.r+nt Letting
At -+ 0, we arrive
Vo,)lAf + Wr)lAt
I
at (2.1). The derivation
of (2.2) is similar.
Proof qf Proposition 1. We consider in detail only the case of < = 0. The same
proof goes through when c > 0 with a minor adjustment for the behavior of the
flow on the upper portion of the boundary segment US. The only critical point is
ED = (N,,, za). Standard linearization
techniques show that EB is a regular saddle
with one negative
eigenvalue
i, = SN < 0 and one positive
eigenvalue
M. Boldrin et al., Equilibrium
nmldfor
search. production.
and e.whange
753
,I2 = TX > 0. The associated (local) stable and unstable manifolds are tangent to
the eigenspaces spanned by (0, 1> and { 1, (TX - S,)/S,) , respectively. Global
stable Ws(EB) and unstable W”(,!?) manifolds also exist [see Irwin (1980)]. The
first is easily seen to coincide with the straight line x = zp, as the latter is an
invariant
set of the flow and IV’(@) is tangent to it around (Np, za). As for
W”(I?), the fact that, around .I?, it is steeper than the S-locus (because TX - SN
is larger than - S,) implies that it will never cross the S-locus again and it will
therefore point outward as indicated in fig. 1. It is representable
as a function,
as points on the stable or unstable
manifolds
have to satisfy dx/dN =
T(N, x)/S(N, x) and W”(,!?) cannot cross the S-locus. The remaining
part of
the global behavior of the system can be filled in using our knowledge of the
boundary dynamics.
Q.E.D.
Proqf qf Proposition 3.
Linearization
around E” gives the following two
eigenvalues:
A1+2 = i{r f (r’ + 4T,S,)“‘}.
Therefore, Re(&) > 0 for i = 1, 2,
and E” is a source. The position of the two local, unstable manifolds is given by
span{,II/S,,
1) and span{A2/S,, 1). This, together with the fact that the S-locus is
flat in a small neighborhood
of E” fully characterizes
the local behavior of the
solution to (3.1)-(3.2). To check that the other stationary
state is a saddle is
trivial. The same type of argument used in the, proof of Proposition
1 applies
here to W”(E’) and W”(E’).
Q.E.D.
Proqf qf Proposition 4. That the flow around E2 has a saddle point structure is
immediate from linearization.
Denote by A1 the positive and by A2 the negative
root. A little algebra shows that Ws(E2) and W”(E2) are negatively sloped, with
the unstable one being steeper than the stable one. Moreover, the unstable
manifold is steeper than the S-locus, which is steeper than the T-locus, which is
steeper than the stable manifold at E2.
As for E’, the Jacobian of the linearized system has a positive determinant
there, so both roots will have real parts of the same sign. The trace of the
Jacobian is
SN+ TX = r - B(N)s, (x)/u’(x)
- x2NB’(N),
and therefore the two roots may have either a positive (source) or a negative
(sink) real part. They are complex when (S, - T,.)’ < - 4S,T,.
Now let us consider the boundary
point E O. Linearization
shows that there
exists one positive root (AI) and one zero root (A2). The associated eigenspaces
are, respectively,
E”(E’)
= span(l,O}
and
E’(E’)
= span{ 1, - r/TN}.
The (local) unstable manifold will therefore coincide initially with the vertical
axis. In fact, we know already that the whole segment {No = 0, z, _< x I z,} is
invariant for our flow, so it will coincide with the initial portion of the global
unstable
manifold
W”(E’), which then extends from (0, z,) in a southeast
direction.
To understand
the effect of the zero root we have to appeal to the center
manifold theorem [see, e.g., Guckenheimer
and Holmes (1983, p. 127)]. This
guarantees the existence of a (differentiable) center manifold Wc(Eo) tangent to
Ec(.Eo) at (0, z,) which will be invariant for the flow generated by (3.1) and (3.2).
Such a manifold, however, need not be unique (contrary to w” and W”, which
are). Moreover, the direction of the motion over W’(E’) cannot be inferred from
the first-order approximation,
but requires higher-order
derivatives of S and
T that cannot be signed under our assumptions.
Q.E.D.
Proqf qf Proposition 5. Two simple facts are key to understanding
our argument:
(1) As the center manifold, Wc(Eo) , of E” exists and we have assumed it to be
unique, with the flow on it pointing toward E”, it has to extend backward all the
way from E” toward the boundary line U4. It may or may not reach such a line
depending on the behavior of the unstable manifold of E2, W”(E2) : when the
latter is the graph of a function it will exit I? through U2 and therefore bound the
extent to which Wc(Eo) may be prolonged in the southeast direction.
(2) The stable and unstable manifolds of E2 exist but need not be the graphs
of two distinct functions from N into x. Recall that both wS(E’) and W”(E’) are
solutions
to our system of differential
equations.
Recall also that they are
differentiable
manifolds:
points on Ws(E2) and W”(E’) have to satisfy the
relation dx/dN = T(N, x)/S(N, x). Note also that, in a neighborhood
of E2, they
are tangent to the eigenspaces associated with the Jacobian of the linearization
of (3.1) and (3.2) at E2.
Consider the partition of the domain given in fig. 2. W(E2) belongs to Szi on
the right of E2 and 523 on the left of E2, whereas W”(E’) belongs to a, on the left
side of E2 and to s23 on the right of E2. Consider first the backward extension of
the right portion of Ws(E2). Could it come from Sz,? Clearly not, as the direction
of motion in Q2 and on its upper boundary
(the T-locus) contradicts
the
direction of motion on wS(E2). It has therefore to come from outside the phase
plane and enter the latter through U4. The southeast portion of Ws(E2) is
therefore the graph of a function from N into x lying completely in Szi . Similarly,
it is easy to see that the southeast portion of W”(E’) is also the graph of
a function from N into x lying completely in Q3.
Now consider the northwest portion of Ws(E2); it is the graph of a function as
long as it lies in Qj. This is the case represented in figs. 5a and Se. On the other
hand, its backward extension cannot enter 52, through the T-locus, coming from
&. Once again, this would conflict with the direction of motion on W”(E’) and
on the T-locus. Nonetheless, it is clear that the directions of motion on Ws(E2)
and the S-locus are consistent with Ws(E2) entering s23 from n, through the
M. Boldrin et ul., Equilibrium
mo&for
search, production, and exchange
755
S-locus, which is the lower boundary of the latter region. This amounts to saying
that Ws(.E2) need not enter R3 from the boundary
line Us. When this occurs,
wS(E’) enters Q3 from .Q5. Proceeding backward, Ws(E2) has, in turn, to enter
Q5 from Qr . At this point, with w”(E2) in .Qr, two new possibilities arise: it may
have entered QI from Q4 and Q4 from Q3. This case we have represented in figs.
5a, 5b, and 5c. Alternatively,
it may have entered Q, from outside of the phase
plane, cutting through the boundary
line U4. This is the case we have represented in fig. 5d. It should be now possible for the reader to carry on the same
exercise for the northwest portion of the backward extension of W”(E2). Three
cases, perfectly analogous to those for Ws(E2), will arise. Obviously, not all the
possible combinations
of these two groups of individual
configurations
are
admissible as solutions to our system. For example, it is clear that uniqueness of
the solution to (3.1) and (3.2) for any given initial condition will rule out the case
in which Ws(E2) is the graph of a function ‘coming from’ U3 and at the same
time the backward extension of the northwest portion of W”(E2) comes from the
boundary
line U, .
These considerations
prove that only the five, qualitatively
distinct, phase
planes given in figs. 5a-5e are admissible. Notice that various nongeneric
and
structurally
unstable cases have not been considered here, as, for example, the
heteroclynic case in which Ws(E2) and W”(E2) coincide on the left side of E2.
In order to complete the proof we need only to justify the asserted existence of
(at least one) unstable limit cycle in the cases of figs. 5b and SC. This can be easily
done by a straightforward
application
of the following:
PoincarP-Bendixson
Theorem.
A nonempty compact LX-or w-limit set of a twodimensionaljlow
which contains no Jixed points qf‘the powwill contain a closed
orbit.
For a proof the reader may consult Lefschetz (1957). Consider first the case of
fig. 5b. Let E > 0 be small enough to guarantee
that all orbits with initial
conditions in the open ball of radius E around E’ will remain in such a ball and
converge to E’. Such an E has to exist because E’ is a sink. Define as fl the
largest value of N for which there exist at least two values of x; call them 2, < .Y2
such that (E, Zi) E Ws(E2), i = 1,2. Such an g < N2 exists by construction.
Now consider the closed curve y2 beginning at (f, Zr), continuing
from there
along Ws(E2) in direction opposite to the flow until the point (fi, T2). and then
coinciding with the vertical line connecting (f, Z2) to (m, 2,). Denote by yI the
closed curve which is the boundary
of the s-ball around E’ we constructed
before. The anular region r contained between y1 and y2, boundaries
included,
has the following properties: (a) it is compact; (b) it contains no critical point of
the flow;(c) it contains the a-limit set of all the orbits beginning in it. Therefore it
has to contain at least one closed orbit which belongs to the flow. In fact, it may
contain
more than one closed orbit. Nevertheless,
the outermost
and the
innermost among these cycles have to be repulsive and equilibria will leave them
as time goes forward. If only one such limit cycle exists, as in the figures, it has to
be unstable.
A brief inspection of figs. Sa and 5c will reveal that a similar construction
can
be carried out also in those cases. Our proof is therefore complete.
Q.E.D.
Proqf’qf’Proposition
6. There is very little to prove. The five types of configurations for the three manifolds
Wc(Eo), Ws(E2), and W”(E2) have already been
justified in the proof of Proposition
5. The existence of the limit cycles in the case
of figs. 6d and 6e also follows from the Poincari-Bendixson
theorem. The inner
boundary y, of the compact region I- can be constructed
here also by choosing
an E-ball around E’ such that all the orbits beginning on y1 point toward r and
remain there for large values oft. Such a ball exists because E’ is a source. The
outer boundary y2 will now be given by an appropriately
chosen portion of the
unstable manifold W”(E2) together with a vertical segment at fi < N2 connecting the lower branch of W”(.E*) to its upper branch. In this way a region r is
created that satisfies the theorem. Notice that it is the o-set, as opposed to the
cc-set, of the trajectories starting inside r that is now contained in f. For this
reason the outermost
and the innermost
among the cycles will be attractive.
Hence, the asymptotic stability statement for the case in which there is only one
such cycle.
Q.E.D.
Proof?/’ Proposition 7. We only need to show that the condition
text is sufficient to guarantee that all the hypotheses of the following
satisfied:
given in the
theorem are
Andronov-Hopf
j = g,(x,
Bijiircation Theorm.
Suppose that the system J? =f;(x,
y),
- y),,for x, y, and p in R has a critical point at (x, y)jor p = p. Assume that:
(H. 1) D( &, g,)(X, jj) has a simple pair of pure imaginary eigenvalues, i.e., assume
one can write the Taylor approximation (in deviation form) ofdegree three
around (2, j) as
.t=
j =
-tuy+a(x2+y2)x-h(x2+y2)y.
(ox
+
b(x* + y2)x +
a(x2 + y2)y.
(H.2) The real part oj’the two eigenvalues
d(Re4d)/W,=~
varies with cc around p = j, i.e.,
# 0.
(H.3) The element a of the Taylor approximation
when evaluated at (X, j, fi).
in (H.1) is diferent
from zero
Then there is a unique three-dimensional center manifold ofperiodic solutions to
the original system of ODE. If a < 0, these periodic solutions are stable limit
cycles, while if a > 0, the periodic solutions ate repelling.
The reader may consult Marsden and McCracken (1976) for a proof.
Denote by J the Jacobian matrix associated with the linear approximation
of
(3.1), (3.2) around E’. The two eigenvalues of J are: 2A1,2 = Trace(J) Z!I(A)“‘,
with d = (Trace(J))2 - 4Det(J). When B’(N)x2N + B(N)s,(x)/u’(x)
is larger
than zero at E’, the value f, defined in the proposition,
can be chosen for the
interest rate. It implies Trace(J) = 0 and A < 0. This yields the two purely
imaginary eigenvalues at the bifurcation value r = F. In fact, continuity
implies
that the two eigenvalues will remain complex, i.e., A will remain negative, for
values of r in a small neighborhood
of r. The p > 0 in the statement of the
proposition will have to be chosen appropriately
to guarantee that (r - ,u, r + cc)
contains the neighborhood
in which A is negative. (H.l) is then satisfied and
(H.2) is also satisfied as Re(1) = Trace(J) at the bifurcation
point and Trace(J)
varies smoothly with the parameter r.
Finally, condition
(H.3) is known to be generically
true for smooth twodimensional
vector fields like ours [see Arnold (1980, ch. 6)].
We cannot verify the sign of a. Given our assumptions,
the stability- instability of the closed orbit follows from our discussion of the global behavior of the
flow around E’ (see Propositions
5 and 6). Q.E.D.
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