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An Equilibrium Analysis of Search and Breach of
Contract,
P.
A.
II:
A Non-Steady State Example
Diamond and Eric Maskin*
Number 237
Apri
I
1979
*
We gratefully acknowledge financial
from Joel Yel in.
I
support from the NSF and mathematical assistance
An Equilibrium Analysis of Search and Breach of Contract,
II:
A Non-Steady State Example
Peter Diamond and Eric Maskin
I
.
1
ntroduction
Consider
of
a
market where individuals meet pairwise and where
pa
a
i
individuals makes at most one trade (or carries out at most one project).
The market for waterfront summer rentals
is
one example.
there are fixed (and equal) numbers of potential
Suppose that tastes differ
of two houses is better but
in
When the market opens, potential
Poisson process.
is
In
the
For
the same for each.
good and poor.
renters search for houses to rent,
Searchers meet according to
renters.
When a meeting results
in a
v/here the renter's evaluation of the house
tiated, a
landlords and renters.
assume that houses are ex ante identical
simplicity, assume only two possible evaluations:
landlords seek potential
Assume that
that two renters may disagree about which
sense that the distribution of evaluations
and
'
.
is
a
good match (a meeting
"good"), a rental
lease is signed, and both parties stop searching.
is
nego-
When a
poor match occurs, the parties will also negotiate and sign a contract,
if
neither already has
partners, however,
costs.
If
if
a
partner.
Tl-.ey
may continue to search for new
the expected benefits of further search exceed the
neither partner finds
carry out the nepotiated rental.
a
If
better deal, the two will ultimately
,
through
search,
one
of
We gratefully acknowledge financial support from the NSF and mathematical
assistance from Joei Yell in.
- 2 -
them
f
'nds a better match,
he can break the original
his former partner for the loss borne.
lease and compensate
The process terminates when no
one wishes to continue search, at which point all contracts remaining
in
existence are carried out.
We consider two distinct meeting processes or search technologies.
In
the first, the probability of
a
(assuming they are both searching)
This process
searchers.
given renter's meeting
is
a
given landlord
independent of the number of other
designated the quadratic case since the rate
is
of meetings rises with the square of the number of searchers.
With the
second meeting process, the probability of an individual's meeting someone
at all
is
independent of the number of searchers.
This is the
I
i
near
technology.
With the linear process, the equilibrium pathsense of maximizing aggregate net output.
the equilibrium path may not
inefficiency.
b"5
efficient
in
the
With the quadratic process,
efficient.
Both arise from the effect of
to search of others.
is
There are two sources of possible
individual search on the return
Were all searchers identical, they would choose to stop
search at the same time.
Since aggregate net output is the sum of individual
net outputs, search would cease when the marginal
reached zero. When those
in
social gain from search
poor matches continue searching for a better. .match,
however, searchers are not all
identical
since only some are partnerless.
When
those with poor matches stop search, they lower the return to search of others.
Because they are not compensated for the external economy their search creates,
those with poor matches always stop searching too soon.
externality
is
created when two individuals
contracts to form a good match; that
is,
in
An additional
poLltive
separate poor matches break these
when a double breach occurs^
On each
side of the market, a double breach replaces two individuals with partners by
—
By 'fequi Ibrlum path" we mean the trajectory that the numbers of searchers
follow when at each instant each searcher maximizes his expected net gain, given
the behavior of others.
-3^
i
one partneriess person.
This change benefits others because a searcher prefers
the probability of meeting
a
partneriess person to twice that probability of
meeting an individual with
a
partner.
Because of this externality the equilib-
rium path may involve too little double breach.
Afrer setting up the model
(Section 2), the paper begins with con-
sideration of the quadratic process:
Section
3
examines the equilibrium
path assuming full compensation for breach of contract; Section 4, the
efficiency implications of small changes from the equilibrium path; Section
5,
the efficient path; and Section 6, the equilibrium path assuming no compensation for breach.
with a partial
The linear process
summary
is
presented
in
Section 7,
We conclude
.
The model just described
is
essentially that of our earlier paper ClU,
except that we no longer postulate the continuous arrival of new searchers
and so are no longer concerned with steady state behavior.
To avoid repeti-
tion, we do not describe the model as fully as in the earlier paper..
- 4 -
The Model
2.
We consider
a model
with two types of
are distinguished by type only
in
individuals.—
Individuals
that each partnership (contract) requires
exactly one partner of each type.
Individuals search for
a
partner (of
the opposite type) with whom to undertake a single project,
are wel -matched, the project
I
output
—
ing
is
and only then
completed.
is
We assume X
2X'.
—
is
> X'
worth 2X.
> 0.
If
if
partners
they are not well-matched,
After partners have stopped search-
the project corresponding to their partnership
Individuals are risk neutral and are able to make side pay-
ments with no bankruptcy constraints.
Each individual can engage
in
at most
one project and belong to at most one partnership,
can engage in at most one project and belong to at most one partnership.
Individuals can meet new potential partners only
and the cost of search is a flow, c, per unit time.
(of
if
they search,
For any two searchers
opposite types), the probability of their meeting, under the quadratic
technology,
is a
probability that
We assume a
is
per unit time.
a
Under the linear technology,
a
is
the
given searcher meets someone at all, per unit time.
sufficiently small so that we can ignore the possibility
that two partners who are both searching will simultaneously find new
2/
potential partners.—
their matching poorly
match.
All
so we shall
—
When two individuals meet, the probability of
is p,
with l-p, the probability they are a good
parameters are the same for individuals of both types, and
refer to just one type.
E.g., buyers and sellers or lessors and
2/
—
We
lessees.
have implicitly modeled contracting as instantaneous. Without
instantaneous contracting, the assumption of no simultaneous meeting
is an approximation.
-
Lot a partnerless
number of M's, and
a
poor contract,
the number of N's.
h
can disregard the number of
never search.
be designated by "M", and
individual
refer to an individual with
5
h,,
M
let "N"
represent the
will
For most of our analysis, we
individuals with good contracts, since they
At the start of the market period there are
h.,(0)
M
M's
and no N's.
We can classify search and breach behavior among four configurations.
As we shall
a
see,
two M's meet it will
if
be in their
contract regardless of the quality of match.
will
breach only to form good contracts.
interest to sign
N's, on the other hand,
Our interest
in
breaching behavior
centers on whether N's will breach to form good contracts with both N's
(double breaches) and M's or merely M's (single breaches).
pari bus ,
it
is
Ceteris
more advantageous, as we explain below, to form
with an M than with an N.
for an M as for an N.
Similarly, search
is
a
contract
at least as profitable
Thus there are three possibilities for search:
it
may be unprofitable for everyone, profitable for M's but not for N's, or
profitable for both M's and N's.
possibilities
—
In
in
two breaching
either both single and double breaches are advantageous or
only single breaches.
I.
The last case subdivides
The four possible behavior modes are shown
in
Table
The effects of various meetings on the numbers of M's and N's are shown
Table
2.
The equations determi
by multiplying the
n
i
ng the numbers of searchers are calculated
induced changes by the frequency of different types of
meetings.
Under configuration A, both M's and N's search, and any good match
results
in a
equations of
—
contract being signed.
From Table 2 we can infer that the
satisfy i/
All differential equations are equations in the mean, to avoid stochastic
components.
Thus we are assuming that numbers are sufficiently large
to realize the expected number of meetings.
- 6 -
Table
Behavioral Configurations
1
Search by
M's
Search by
N's
yes
yes
yes
no
yes
yes
A
B
C
D
low
Change in numbers of each type:
without Dartners (h,,)
in
poor matches
in
good matches
(h^.)
yes
no
irre evant
irre evant
Numbers of searchers.
Quadratic technology
Action
f
Double
breaches
yes
yes
no
no
Table 2
Rate of
Single
breaches
poor
match
of M's
good
match
of M's
^P^M
a(l-p)h2
-1
+
single
.breach
2a(l-p)h^h^
1
1
3(l-p)hN
+
-1
+
double
breach
.
1
-1
-2
+1
+
1
- 7
2
The number of M's is diminished by matches between M's
by the
individuals left partnerless when two N's make
2
(a( l-p)h|^^)
2
(aph|^)
The number of N's
.
is
(2)
h.,
>
M ^
h^ >
increased by poor matches between M's
as
h.,
2
h..)
.
Notice that
< h,,(l-p)
M >
N
^
,^''i^,, „^^„-l
^
as h^ > h^((l-p)'^ + /,
(1+p) ')(l-p) ^p
,
Under Configuration B, behavior
double breach does not occur.
I
)
good match
I
I
(
a
and enlarged
and diminished by good matches between N's and M's (2a( -p)hj.h^)
and between N's (2a( -p)
as
(ahj.)
is
the same as under A except that
The equations are, therefore, the same
after deleting all terms involving double breaches:
(3)
(4)
2
h.
.
M
= -ah..
M
\
=
h,
<
M
>
N <
h,,
2a(l-p)VN
^P^M -
as
h,,
> 2h,,(-!^).
N
p
M <
Under Configuration C, only M's search.
are:
(5)
h^ = -ah^ <
\
=
^P^M >
0.
Therefore the equations
- 8 -
Finally, under Configuration D, there is no search at ail.
Let us define the positional values of being an M or an N, when
the numbers of M's and N's are
as
(h.,,h.,),
M
N
V.,(h,,,h.,)
M
M
N
or
respec-
V,,(h,,,h.,)
N
M
N
^
An individual's positional value is the payoff that he can expect,
tiveiy.
given that he and all others exhibit optimal search and breach behavior.
The equations of motion depend on the prevailing configuration and the
numbers of M's and N's.
The private decisions which determine the choice
of configuration depend on (correct)
and the values of positions.
solely on
forecast of the equations of motion
Thus positional values do,
indeed, depend
(h.,,h.,)
M
N
For positional value to be well-defined, we must describe how con-
Suppose that individuals
tracting works.
a
contemplate signing
j
contract which would yield them a combined positional value of 2V.
and
Let V
i
and
i
V"'
be the current (i.e. pre-contract) positional values of
and j, respectively.
Suppose further that
and
i
contracts which specify that they pay damages D
to their partners
if
they breach,—
if
and
i
j
and
'
-
V^^
- d'
-
2/
gain by signing the contract
if
division rule,
and
+
V''
is
S,
individuals
i
and only
j
if
S
2/
—
This
,
respectively,
D-^
We postulate
.
is
The individuals
positive.
attain positional values
Under our
V
+
^s
S
respectively, from the contract.
If
or j is currently partner ess, then D
respectively.
j_/
D"^
in
sign the contract, then they divide output and/or make
side payments so as to share the surplus equally.—
and
currently are
Then, we can define the surplus of
the contemplated contract as S = 2V - V
that
j
i
I
division rule
is
=
or D^
= 0,
also the Nash Bargaining solution to the problem.
9 -
sav,
If,
j
i
breaches
damages equal to
being partnerless,
V-^
)
contract with j, we assume that
a
- y^.
That
that V^
(Recall
is,
(I)
the positional value of
maintains the same expected payoff he
j
had before the contract was breached.
pensatory.
is
he pays
The damages are, therefore, com-
We focus on compensatory damage rules
in
this paper because
they constitute the basic principle for assigning damages under common
law and
(2)
they are efficient
that a breach occurs
if
in
and only
the limited sense that they ensure
if
there
is an
increase
positional value of the principal affected parties—
their original partners,
without damage payments.
if
any).
In
In
in
the combined
(the new partners and
Section 6 we consider equilibrium
our previous article we also consider liquidated
damages, thedamages that contracting parties themselves would choose.
our simple two-quality model, two M's always wish to form
In
contract
if
they meet, since
>
N -
V.,
V^,
M
and X>V.,.
M
a
Furthermore, with com•
pensatory damages an N will breach his current contract only when he
finds a good match, because only then can the surplus of his new contract
be positive.
Notice that the incremental benefit that an M receives from any
potential contract is
larger than it would be were he an N, since he does
not pay damages (which would diminish the surplus) and has
tional
value.
for N's.
a
lower posi-
Therefore, the benefits of search are greater for M's than
We need not consider, therefore,
a
behavior configuration
in
which N's but not M's search.
Finally, we observe that since an N would prefer to form a new
contract with an M rather than another N, we can exclude configurations
where double but not single breaches occur.
where at any moment, all
Thus Configurations A,
—
B,
We consider only equilibria
individuals in the same position behave identically.
C and D are,
indeed, collectively exhaustive.
For analysis of this effect of compensatory damages, see Mortensen [23.
10-
Positional Values, Trajectories and Boundaries
3.
Market:
Decentralized
in a
Quadratic Technology
We assume that an
whether to search
value when deciding
individual maximizes positional
Positional value, however, depends on the
and breach.
future evolution of the economy and so must be determined by working bacKwards. We are interested in Nash equilibrium time paths.
An equilibrium
path specifies a behavior Configuration at each instant of time and has
the property that each individual
finds the behavior prescribed for him optimal
given the specified behavior of the others.
To calculate the evolution
of an equilibrium path, we derive and solve differential
positional value in each of the Configurations.
Configuration at any instant
is
equations for
We then examine which
consistent with equilibrium.
configurations are all consistent at some instant
-
When several
i.e., when there are
multiple equilibrium paths - we select the Configuration involving the most
search and breach.
So,
for example, we select Configuration A over B, C, or D
We assume everyone is partnerless at the start,— i.e.
We begin the analysis with Configuration C.
of
(h|^,h^,)
and
arbitrary.
h
We refer to the set
pairs where Configuration C behavior occurs on an equilibrium
path as Region C.
trajectory
is zero,
h^,
in
In
Region C, only M's search
Region C,
h|^^^
is
and, therefore, along a
steadily declining, while
at only p times the rate of hj^'s decrease.
h|^
is
increasing
Because M's make better partners
than N's the gains from search monotonica ly decline for both M's and N's.
I
Thus, a transition from Region C to either A or B is Impossible.
N's stop searching, they will
never wish to resume.
Consequently,
positional value of an M In Region C, depends only on h^.
M incurs search costs cAt in a small
In
In
Once all
Q
V|^,
Region C
the
arj,
Intervel of time At and finds a partner
the appendix we drop this assumption.
with probability ah^^At,
The value of
positional value at t+At,
where pX'
(l-p)X
-t
terms, defining
for h^ using
less search costs.
Thus
the expected output from a match.
is
=
IT
position at t equals the expected
-cAt+ah^t(pX' + (l-p)X) + (l-ah^t)V^(hj^(t+At));
-
V^(hj^(t))
(c)
a
pX'+(l-p)X,
Rearranging
letting At tend to zero, and substituting
yields
(5)
dV
9
(h
•
)
P
M
Equation (7) completes the first piece of the analysis:
the change
values
in
in
calculation of
Region C.
Because h^ declines steadily
in
Region C,
it
ultimately reaches
^
h^,
|,
where the gain from search for the next instant
is
zero.
At this
point the search cost equals expected gross gain:
(8)
c = ah^..
Thus, search ceases at
h|^,
and the line h^ =
boundary between Regions C and D.-/ To find
I
h|^
P
V^j,
serves as the transition
therefore, we merely solve
r
(7)
with terminal condition V.,(h')
1^
M
(9)
where G
-1/
,,
v^ = n
= |-
(
I
n
-
^
=
0.
We obtain
^-^ + g-,
-
I).
We refer to the locus of possible transitions as the transition
boundary,
Since part of this locus may be in Region A or, alternatively, may not
be
reachable from an initial position on the h axis, the actual boundary of
Region C is a proper subset of the transition boundary.
-12-
The transition
Next consider possible transitions from B to C.
boundary separating B from C falls at that critical number of M's h",
where an N finds search just barely profitable:
(10) c = ah|;;|(i-p)(X-X'),
where X-X'
is
one half the surplus of
N when further search by N's
a
contract between an M and an
unprofitable.
is
note first that the value of the N position
is
is X'
(2X-V^-V,,-D) = (2X-2V,,) = 2{X-X').
M
since N's do no search further
Thus, damages are X'-V|^, and the surplus from a new
once Region C is reached.
contract
To understand this equation,
N
N
Notice that (l-p)(X-X') <
Therefore h" > h', and transition borders appear as
in
Figu'-e
n.
where D borders
I,
only on C, and C only on B.
Next, consider positional values in Region B.
searches for time At beginning at time t.
he will
Suppose that an M
With probability ah^(i-p)At,
meet another M and form a good match, giving value X.
meets an M and forms
becomes V
.
If
poor match (probability
a
I
f
he
his positional value
^Phj^^Ai)
he encounters an N with whom he makes a good match (prob-
ability a( '"P^^n^At)
his positional value is V
of 2X-Vj^-Vj^-D = 2X-2V
.
plus one-half the surplus
Otherwise his positional value
is
Vj^(h|y|(t+At) ,h|^(t+At)
)
Thus, we have
(11) V^(h^(t),h„(t)) = -cAt+aAth,,(l-p)X+aAth,,pV^(h,,(t+At),h.,(t+At))
+ aAth„(l-p)(X-V,,(h,,(t+At),h,,(t+At))+V.,(h,,(t+At),h (t+At)
N
NM
N
MM
+ (l-aAth,,-aAth,,(i-p))V.,(h,.(t+At),h,,(t+At)).
M M
N
N
M
N
-
13
For an N, only a meeting with an M where
a
good match is made
Positional value becomes
changes positional value.
y
+h
(2X-2V
)
= X.
Thus we have
(12)
V^(h^(t),hj^(t)) = -cAt+aAth|^(l-p)X+(l-aAth|^(:-p))V^(h|^(t+At),h|^(t+At))
Taking limits and substituting from the
equations
of differential
•
equations we obtain the pair
h
d^M
B
B
-7^- = c - a(l-p)(h.,+h,,)X-a(ph^,-(l-p)h,,)V,, + ah^V,,
-^MN
dt
'^NN
^M
MM
(13)
3+—=
+ a(l-p)h,,V^.
c - a(l-p)h,,X
^
^
M
M N
dt
Using th^se equations, we can conclude that
from
The surplus
r
to A is impossible.
r
In
Region B,
is
S
a
transition from Region B
a
double breach
is S
= X+V,,-2V,,.
M
N
negative since double breaches are unprofitable.
Furthermore,
'
R rlV^
dS _ °'m
=
(14)
rr
dt
-rz
dt
HV^
°'n
,. ^ . ,^
c ^
,v uB,..,
< n
2 -rr- = -c-a(X-V.,)
-p)h.,+ph.,)+ah.,S
0.
(
dt
M
M
N
N
'
„
.
(
I
.
Therefore, the double breach surplus never becomes positive, and movement
from B to A is ruled out.
Because transition from B to A
is
impossible and since h" > h', any
trajectory crossing Region B must then move into Region C.
(See Figure I.)
Thus, once Region B is reached, N's never again contract with other N's and
so
v.,
N
becomes a function of
•
by solving the differential
Vp/h") and V^(hV) = X'
MM
N
I
.
h.,
M
alone.
We can calculate positional values
equation pair (13) with terminal conditions
For
p
/
Jj
we obtain
in
B
'^w^'^m''^m^~
14
X a(l-2p)h^
^M^^M'^N^ M
^
h":
M
2u2p-2
a(l-p)(l-2p)'^(h|;;|)^P"'
-c(l-p)h..
N
ap(l-2p)
chf.
ap(,.p)(h^)P
(15)
k2p-3
a(l-p)(l-2p)(h|;;|)^P
M
where H and
J
are chosen so that the initital conditions hold.
We now turn to an examination of Region A,
of new opportunities
in
Region
the same.
in
For an M, the arrival
Region A follows the same rules as their arrival
Thus the differential equation for the change
B.
in
value
is
For an N, the value of search falls more rapidly because of the
added opportunity of double breaches.
and add X-2V
A
+
A
V^^
These occur at the rate a(l-p)h
to value when they occur.
Thus the differential equations
for values in Region A satisfy
^=
^^M
A
A
c - a(l-p)(h^th^)X-a(ph^-(l-p)h^)vJ + ah^V^
A
^^N = c - a(l-p)h,, (X-vJ) - a(!-p)h„(X-2vS+ vfj
N
-rr—
M
N
N
M
dt
'^
dvj
,_A
"^
dvj
-a(n-p)h,,+ ph^))(v2-v2)
d^V^
^^2
^ = a(l-p)(h,+hjc+a2(l-p)ph,(h,+hj(v;-vj) > 0.
^^ M M N N M
M N
"^
-i:?-
see that trajectories can move from Region A directly to any
We shall
of the other three regions.
However, there are two distinct patters.
For
one set of parameter values, the equilibrium path moves from an initial
in
For the remaining values, movement from A is directly
A to B then C then D.
to either C or D and Region B does not exist.
two
We first show that which of these
patterns aoplies depends on which region, A or
Then we consider
position
B,
contains the line
h
M
=h"
M
•
turn, transitions from A to B, C, and D.
in
We observe from (2) that the direction of movement is as shown in Figure 2.
For an
initial
position on the
of the
line
=
h,,
while
h,,(l-p)
axis, the trajectory can never be to the left
h^^
In
Region A:
movements across the line are not
possible with Configuration A behavior, nor are transitions from B or C to A.
In
the appendix we briefly consider initial positions in this area.
In
the
text we consider Region boundaries only to the right of this line (although
this restriction is often unstated).
Consider the point
either this point
)
negative,
)
(h" h
point cannot lie
this point,
f
'
v.,
M
)
for some positive
bi
I
ity,
is
in
equals
N
I
values
on the B-C border.
Region
If
B.
V,,(h'').V,,
M
M
is
h
part of Region A.
N
'
(h"
h^,^^)
equals X'
in
if
is
Region
I
To
If
the surplus
Is
on the B-C border then, at
and we have
-2D = 2X-4V„+2V,,
M
N
M
B.
the surplus is positive, the
=2(X-2X'+n-f-l^.^(ln|n-- D)
=
From Figure
.
we need to evaluate the surplus from a double
using positiona
S = 2X-2V,
(17)
^iljh
the B-C border, or it
Is on
check the former possi
breach at (h",h
(
M
2(X-X'H|.(|-p)|n ^i|;Pl^^;;;:^,) )<_.0
-16-
cither all the points (h",h,,)
Sincc (17) decs not depend on
h,,,
B-C border or they are all
Region A.
In
(17)
thus
is
lie on the
necessary and suf-
a
ficienl condition for the existance of a B-C transition border.
indicates that for X-X' sufficiently small
(17)
the surplus at (hILh
)
is
X is relative to X', the
Sufficiently small
p
in
negative, and a B-C border exists.
also lead to
p
makes good matches easy to find,
equations
Decreasing
increasing the benefit to an N of remaining
(relative to X') or small
p
a
double breach opportunity.
implies that Region B exists.
holds, we know that Region B exists and can
(17)
The smaller
negative surplus.
a
the search market rather than taking advantage of
If
X'
less valuable is a good match and hence a double breach,
values of
Therefore, small X-X'
relative to
integrate the value
backward, ultimately reaching the A-B transition border.
border Is defined as the locus where there
+ V^(h^,h.,)
^-^"^l^^J
M M N
N
M
('8)
^
is
This
zero gain from a double breach:
= 0.
Using the equations (15) and (16), we can write (18) as:
\
= h^C-(l-p)2+(l-2p)(h^/h^)P+p2(h^/h;^)2P-']-'x
Cp(i-p)2|n(h^/h;;;)-2(i-2p)(h^/h;;;)P
(19)
V(i-2p)"'((h^/h;;)2p-'-i)
+p(l-p)(l-2p)in((pX'+(l-p)X)/(l-p)(X-X'))
+(l-2p)(2-p)].
This locus is shown in Figure 4
limit since
2
p
.
+(l-2p) equals (l-p)
As
2
.
h^^^^
approaches h"
h^^^
increases without
- 71
In
A
Figure
we show the transition boundary.
4
lies to the right and B to the
that this is correct.
left of
We have not confirmed
locus (19).
To see the potential complication, consider the pos-
sibility that an equilibrium trajectory, when
more than once
This suggests that Region
in
Region A
could cross (19)
see Figure Al and the discussion in the Appendix).
(
this
If
were possible, only the last crossing would be a bonafide A to B transition,
since we assumed the occurrence of the equilibrium path with the most breach.
It
would mean,
furthermore, that Region A protrudes to the left of (19).
have not been able to rule out such multiple crossings.
We can claim with
accuracy, therefore, only that the A-B transition border
is a
locus (19).
We
subset of
.
II
If
as defined by
S
A transition from B to A
of
rlqht
3
S
h
MM
=
h",
(17)
i
s
impossible.
and Regions C and D
1
Region B,
ie to
h^
=
positive, the line h^
is
in
exists,
it
if
lies
Therefore
the left.
Region A.
lies to the
a
positive
implies that no equilibrium path can go from Region B to another Region, thu s
precluding the existence of
B.
When B does not exist, we need to determine the A-C or A-D borders.
The A-C border is described by the curve showing indifference to continued
search by N's:
(20)
c = a(l-p)h^(X-X')
+ a(l-p)h|^(X-2X'+V^(h^))
The shape of the curve (20) depends on the
c>a(l-p)h' (X-XM.
h'
h!l'>
"y^,
^
sigii
of X-2X'
Note that
.
M
define
""""" h"'
by V^(h"')
"'^
Jf X <2X',
"M "^ 'M^"M
the
rhe
A-C border lies to the right of the line
mw h^
'
toward the line as
i
h
increases.
This possibility
is
=
2X'-X.Then
i
shown
=
in
n
h"',and
tends
Figure
5
-18-
If
X
>2X', the A-C border reaches the line
border exisfs and
is
This case
i
s
shown
of direction
+.
Figure 6,
In
In
this case an A-D
a(l-p)h|^(X-2X').'
In
Figure 6 we have omitted indication
of movement in Region A.
lines separating the areas
h'.
-
given by
c = a(l-p)h^(X-X')
(21)
=
h,,
One may readily verify that the straight
of different direction of movement of Figure 2
may bea-- any relation to the point of
intersection of Regions A, C and
D.
By analogy with the A-B border, there is the question of whether an
equilibrium trajectory
Region A can cross the A-C ((20)) or A-D ((21))
in
transition borders more than
or;:e.
Multiple crossings can be ruled out
by the fact that an equilibrium trajectory
(
is
flatter than
N>-l) (since the aggregate number of searchers
the transition borders are steeper than a 45
is
a
45
line
decreasing), whereas
from implicit differen-
line:
tiation of (20) we have
(22)
'\"
"=
.
-
'-'
^-^''Vlf
<-l
X-2X'+V^
M
^^M
"c
The Inequality follows since
is
not worthwhile).
There
is
in A,
it
As one moves backward
must remain worthwhile to search -and to double
Movinga backwards, both
Is
c
(since further search
M
V.,
M
The same conclusion holds for the A-D border (21),
'
therefore
>0 (see/Q)) and X* >
one remaining loose end to check.
along trajectories
breach.
v'
h.,
M
the return to search.
and
(h,,
N
+
h,,)
M
are increasing
^ and so too
Moving backward along a path, the
surplus from a double breach, cannot change sign (see (16)) since both
terms are negative and dominate the term in
S If
S
approaches zero.
-19-
4.
Inefficiency of Equilibrium
In
Section
5,
examine the change
we describe efficient paths.
in
In
this section we
aggregate net output from perturbations of the
equilibrium transition boundaries.— We show that the C-D boundary
is
efficient, but that shifts to the left of the B-C, A-C, and A-D borders
(Implying increased search) raise aggregate net output.
breach resulting from
a
shift to the left
in
The increase
in
the A-B border also raises
net output.
First consider the border between Regions C and D.
If
2
search, the social
soc
gain per unit time is ah^(pX'+( -p)X)
I
cost
2/
is
ch^.—
expresions,
is
,
M's
all
while the social
Thus, the efficient C-D border, obtained by equating these
the same as the competitive border.
This coincidence may
seem suprising, since an additional searcher creates an externa ity (an
I
improvement
in
the positional
value of other searchers) that does not seem
to be captured by compensatory damages.
artifice of the model's
of the
individual
symmetry.
gains.
The coincidence however
is an
The social gain from search is the sum
Since, under Configuration C, all
searchers
are identical, the social gain becomes zero precisely at the point
where any individual gain vanishes.
Thus the social and private Incentives
for search are the same.—
In doing these perturbations, we assume they do not affect earlier
behavior.
_[/
2/
—
Aggregate output and search costs are twice these figures, but we
continue to focus on one of the two types that make a pair.
If individuals differed In search cost, all
lowest cost would search too little.
3/
but the searchers with
-20-
Next consider a slight shift to the left of the equilibrium B-C
border.
If
the N's cease searching when
from those still
in
-
h
the market is h^,X'+h"Vj^^(K^)
Suppose that N's continue
.
The presence of N's following Configuration
to search an instant longer.
B behavior does not alter the time path of
do not affect
aggregate net output
h"
h,,,
since single breaches
the number of M's (equations (3) and (5) are the same).
Thus after the N's cease searching, the trajectory
3.
The cost of the N's additional search
ditional output per unit time
good matches, 2a(
l-p)hh(X-X'
)
is ch,,
N
is
the same as
per unit time.
in
Section
The ad-
is
the aggregate surplus from the resulting
.
The N's receive only half of this surplus.
Thus the private incentive to search
is
smaller than the social gain.
A
shift to the left of the B-C border (increasi ng search) raises net aggregate
output since c
< 2a(
I
-p)h''(X-X'
M
).
We next consider perturbation of the equilibrium A-D border (which
exists when X
The border
> 2X').
is
the locus of points where N's are
just willing to search given that they receive half the surplus from
bothsingie and double breaches and where M's are willing
only
if
N's do so.
to search
M's find search (more than) worthwhile at the
A-D border only because they receive part of the surplus from single
breaches.—
Again there
is
—
too little search
the A-D border raises aggregate net ouput.
a
shift to the left of
The net gain from continued
search is the surplus from matches bewteen M's, from single breaches, and
from double breaches, minus the search costs:
—
M's find search at least as profitable as N's.
Thus when N's are
indifferent to search, M's are more than willing to continue searching.
-21-
(23) ah^((l-p)X+pX')+2a(l-p)h^h^(X-X')+a(I-p)h^(X-2X')-c(h^+h^)
At the equi
I
ibriijm border,
search implies
(24)
a
(21),
indifference of the N's to continued
net social gain from continued search of
h^(ah^((l-p)X+pX')+a(l-p)h|^(X-X')-c)
But from the indifference of N's to search ((21), again)
(24)
becomes
(25) ah,.(h,,+ (l-p)h„)X',
which
is
positive.
To obtain the efficient A-D border,
(23)
is
set
equal to zero.
Consideration of the A-C border introduces a new element, not present
in
discussions of the other borders: the effect of double breach on the
search environment as
effect
is
a
result of changing the number of M's.
The external
irrelevant at the A-D border because all search ceases there.
V* be the aggregate value of continued search.
nave V* equal to h,,V„(h.,).
Q
Since
is
V.,
Once region C
increasing
the A-C border generates an external economy.
in
h,,
a
is
Let
reached we
double breach at
Thus both single and double
breaches have social values that differ from their private values to N's.
The increase
is
in
aggregate value from continued search by N's of both types
the full value of single and double breaches plus the increased value of
the search process for M's:
(26)
2a(l-p)h
h
1^
N
(X-X')+a(l-p)h^(X-2X'+V^(h
+a(l-p)h.^h^v'^(h..)-ch,
N M M
M
N
N
.
MM
))
-22-
(26)
Using (20),
(2^)
which
Thus
in
is
a
becomes
a(l-p)h^j^(X-xM=a(l-p)h^h/^^,
positive.
leftward shift in the A-C border, resulting in more search, is desirable
part because of the externalities from double breaches.
to zero.
cient border we set (26) equal
We turn, finally,
For the effi-
to the A-B border.
A slight shift of the equilibrium
border induces no change in search but affects breaching behavior.
We show
that the continuation of double breaches beyond the A-B border is worthwhile,
assuming the rest of the equilibrium process
is
unchanged.
At the A-B
border, a double breach yields no private gain; nevertheless, there is a
social
gain.
(28)
The value of net output of continued search in Region B is
V* =
Vm^^^M'^^
'-^1%'^^'
A double breach creates a good match, adds one M and subtract two N's.
The impact of these changes on aggregate value is
(29, AV« = X .
ji- (h„V^.h„vB,
- 2
|-
M
,
KN,
^^
i-p
'-P
^M
MM
^M
TT^) ^TT")
since K. > h" on the A-B border.
M
M
derivative.)
(h^/h")^P"'-l
,
h|^
'
+ (-) C-r^ +
a
vS^h^V^)
-N
,o
>
,c.,p(2-p)
a
<
>
X
P"
>
^M
(We have used (15) to. calculate This
-25-
To understand the externalities created by breacii, we can examine the
impact of
a
(the two breachers and their partners).
from a good match with an N.
X-2V,,+V.,
the principals remain N's.
If
An M gains X-V
if
is
the double breach does not occur,
now we
I
and an N, X-V
,
party left partnerless, while an M gains
from a poor match with this party.
meeting the breacher (who
whereas an N gains
,
breach occurs, an M gains X-V
from a good match with the principal
Vj,-V
parties
double breach on individuals other than the four principal
I
Neither M nor N gains anything from
-matched)-.
Thus the sign of an M's
net gain from double breach is the same as that of
(30)
P(V^-V^) + (l-p)(X-V^-2(X-V^))
whereas the sign of the N's gain
(31)
is
Thus, when the principal
carrying out
breach
P(V^-V^) + (l-p)(2V^-X-V^);
the same
as that of
(|-p)(X-V^-2(X-V^+V^)).
At the A-3 border, X-2V,+V„ is zero.
M
N
there.
=
is
a
Hence both (30) and (31) are positive
parties are themselves
'
indifferent about
double breach, the overall externality induced by such
positive.
a
-24-
5.
The Efficient Path
Since the C-D border
is
efficient, we can straightforwardly derive
the efficient A-C and A-D borders from the perturbation analysis above.
To complete the analysis, we verify that an efficient path can never cross
from Region C to either A or B
and that
it
never entails Configuration B
behavior.
To describe the efficient A-C and A-D borders, we
the perturbations of these borders with zero.
(26)
equal
to zero yields
equate value of
Thus setting (23) and
the border equations.
In
Figures
7,
8,
and 9
we compare the efficient and equilibrium borders.
When X > 2X' an A-D border exists and
(23) equal
is
obtained by setting
to zero:
(32) a(l-p)X(h^+h^)2-2a(l-p)h^(h^+h^)X'+aph;^X'
=
c(h^+h^)
or
^2
"'
(33) c = a(l-p)h^(X-X')+a(l-p)h^(X-2X')+aX'(p
The equilibrium equation,
term on the right.
(21)
differs from (33);
+
j^^
M N
it
(l-p)h^).
does not contain the last
Thus for any value of h^, there is a smaller value of
for the efficiency border than for the equilibrium border.
h
For the A-C
border, setting (26) equal to zero gives
(34)c = 2a(l-p)h^.(X-X')+a(l-p)h„(X-2X'+vJ:ih,J)
M
MM
N
.
+c(l-p)h^h-'ln(h^/h^).
When X > 2X', the coefficients for
hand side of
(34) exceeds that of
h
(20)
and
h
are both positive and, the right
the equilibrium border equation.
-25-
Thus the efficient border lies to the left of the equ
This relation and -the A-D border are shown
Figure 8 we illustrate the case where
In
equilibrium border and X
that the coefficient of
,,qN
^
'
_M_
=
h'
M
< 2X'
in
h.,
vanishes at
(34)
is
In
the line
values of
h.,==h.,
M
h
positive at the
(rrom (9)) one sees
\'
V
is
satisfying
h.,
is
where the market equit'ibrium has
a
this case the efficient A-C border has the same characteristics
its equilibrium counterpart:
-s
7.
X-2X*-i-V,,(h,,)
i
border.
asymptotic to this line.
The remaining case to consider
B.
Figure
Substituting for
.
ibriurr:
I
PX'+(l-p)X
2X'-X
Thus the efficient border
Region
in
i
M
.
it
intersects the
h^^
axis and
is
asymptotic to
Of course the two borders do not coincide (the particular
are different).
This
is
shown
in
Figure
9.
As the figures show, we have too little search in equilibrium unless
h
(0)
is
in
Region C or Region D.—
This discussion at efficient borders implicitly assumed that N's
never
return to search after having stopped,
i.e., that the efficient
path does not enter Regions A or B from Region C.
such transitions are impossible.
In
We now prove that
Region C, the dynamic programming
We have assumed throughout that all N's stop searching at the same time.
This assumption is justified since the social value of search by an N decreases
That is, just to the right of the hypothesized
when another N stops searching.
borders, 3 V* exceeds X', and, when these values are equal, it Is socially worth-
J_/
while for all N's to stop searching simultaneously.
-26-
value equation for aggregate net output is
V*=3V*
8V*
+
h,,
^2 3V*
h,,
2 3 V*
^
"'=^5-^,
Jh^
"
,—^
<36)
'''"'-„
=ch„ - a(l-p)h^ X,
where
is
V*(h.,,h,,)
M N
aggregate value from those not
addition, we know that,
good matches.
in
In
Region C, the marginal social value of N
in
is
constant, since N's simply accumulatef
^(3^1
(37)
^
dt
=
3h,/
From these two equations we can contradict the rise
the value of
in
search by N's which would necessarily accompany a transition from C to
2a( -p)h^(t) (X-
Search by an N at time t has net value
either B or A.
I
—
3 V*
)>c.
^-r
N
in
region C,h|^(t)
3 V*
i5-7-—
-\
< X
is
Furthermore
decreasing and all other terms are constant.
because an N can, at best, make
a
good match.
Thus the value of
search declines, and so a C to B transition is impossible.
To consider
available N's.
a
move from C to A, we must consider search by
The aggregate return to search by N's
2a(l-p)h,,h,,(X- 3V*
5-1—
M N
9h.,
N
Using (36) to eliminate
(38)
3
)
2
+ 2a(l-p)hf-,
^
N
V*/ 3h
3V*
.
,
,
!3V*
3 h.,
ch,
we derive the return to search per N,
2a(l-p)h|^ (X-3V^) + 2a(l-p)h^ (X-2 3V*
Sir
'^
N
+
(ah^)"' (aph^^
TTTN
""
'^
(l-p)h (X- 3V*) + 2a(l-p)h„(2-p)(X-3V*
-c<2(i-p)h„hj;;'+h^).
^*
"""N
•
+a(l-p)h,, X - c)) - ch„
M
N
2a
the
[
M
N
I
is
,., ^
+^-^)(X-2-^^
3 h.,
al
)
-
-27-
Di
•
ferentiating with respect to time (and using (37)) we have
2a(l-p)
(39)
(X - d\i*)
+ (2-p)h„)
(h,.
-c(2(l-p)((h^h^
VMVh^)+
-
2a2(l-p)(X-|^)(-h^
=
+
h^)
(2-p)p h^)
N
-ac(2(l-p) (ph^+h^) + ph^)
since (2-p)
in
p<l.
Region C and
,
0,
to search by N's decreases per N while
Thus the return
therefore,
<
can never become positive.
We now turn to the proposition that the efficient trajectory never
We show that there is a higher net output
involves Configuration B behavior.
flow from either Configuration A or C behavior
the efficient trajectory leaves Region B—
if
there
a
border where
This contradicts the possibility
of Configuration B behavior on the efficient path.
border.
is
First consider the B-C
The excess of aggregate net ouput flow that accrues from Configuration
B behavior above that yielded by Configuration C is the additional
output
from single breaches less the search cost of N's,
'^»'2="-P'VN<'<-|f'--''N-
,
N
If
a
B-C efficient border exists,
Since N's search no further 3V*
'g-pj— =
is
defined by the equation
(41)
—
h.,
M
MM
h,,
it
is
the locus where (40) equals zero.
^j
,
X' at^ ^,
the ,border.
= h..,
where
=
2a(|-p)(X-X')
Nor can it stay indefinitely
in
Region B.
Therfore, the border
-28-
The difference between the Configuration A and B rates of aggregate net
output
is
the social gain from double breaches:
(42) a(l-p)h'^(X-2
|^+ ^)
At the B-C efficient border, |Vl . x' and
|^M
N
C
given by (9).
Vj..is
(
43)
.
exist.
where,
)
-^
afi^
M
= X-2X'+n-2(l-p)(X-X')
= p(X-X')
is
V^(h^)+h V^(h
Thus, at the border,
X-2 4r- + ii- = X-2X'+nN
^^M
Formula (42)
=
>
therefore positive, and so an efficient B-C border does not
That is, when the social
return to search by N's
and that of M's is positive, double breach
An efficient B-D border
value of double breaches.
from double breach,
is
is
socially worthwhile.
similarly ruled out by the positive social
At a B-D border,
(42),
close to zero
is
yJ—
-
n
Z—
and the aain
= x'
is
,
(44)
a(l-p)h^, (X-2X).
N
At this border the additional
net output from search is just zero.
(45) ah^n+2aCl-p)h^h^(X-X')
= c(h^+hj^)
That is,
.
Furthermore, search by N's must be socially worthwhile at the border.
is.
Configuration B must be more efficient than C,
(46) 2a(l-p)h|^(X-X')
Subtracting
h.,
N
> c
.
times (46) from (45), we obtain
(47)ah2n< ch^
implying
h
>_
h
That
or
.
-29-
From the condition
>
h
c > en. ,11 =
(47) becomes
h^,
2(t-p)(X-X')
'M
or
(48)
2(l-p)(X-X') > (l-p)X+X'
This last expression implies X > 2X',
breaches ((44)), which
is
a
implying a positive gain from double
contradiction.
The last remaining possibility
transition from B to A.
is a
At the
A-B border, the surplus from double breaches must be zero; hence
.1^
X-lpl
(49)
Under configurate
the dynamic programming value equation
B,
V*(h,,(t),h„(t))
-4x
M
N
dt
(50)
= 3V*h,
TT
'
=
-ah/-^+
^
8
3V*
Tr~
(aph^ - 2a(l-p)h^h^-) 9V* =
h,,
..
N
c(h^ + h^)
a(l-p)(h^ + 2h^h^)x.
-
h
9 h.u
Solving for
+
is
V* using
(49) and
(50)
w& obtain
3h,
2
(51)
-
(a(2-p)h
1^
+ 2a(l-p)h^h.,)3V* = c(h.^ +
M N r-r—
M
9h,
h^,)
N
9
- a((2-p)hf,
M
+ 2h,,h,,(|-p)X,
^
M N
or
(52)
^
av*
^^N
_
^^^M
"^
^N^
a((2-p)h./ + 2(l-p)h,,h,,)
M
M N
2a(l-p)h,,
^
M
-30-
If
'
configuration B
(53)
(I.e.,
,,
is at
8V*
3h^
^
-
least as efficient as C,
however, then
2a(l-p)
non-negative value of search by N's),
Since fneqaulities (52) and (53)
are mutually contradictory, we conclude that there is no efficient
Region B.
-3I'
6.
Equilibrium Without Damages
We have considered equilibrium assuming an
law's provision of compensatory damages.
themselves of these damages.
that individual
sdo
One way of modelling this behavior
individual
on that match when further search
has found a poor match^ he can
unprof itab
is
le,
to assume
Instead,
fallback
provided the fallback partner
and that individuals do not replace an earlier fallback with a
long as the earlier one is available.—
in
is
We assume that only one fallback contact is preserved,
available.
still
Often individuals do not avail
not sign contracts unless they stop searching.
we assume that after an
is
idealization of the common
Sections 2 and 3
if
two individuals make
a
complete the project
is
later one as
This behavior is captured by the model
damages are always set at zero.
We assume that after
poor match, the decision to stop searching and to
jointly made.
If
the search decision were not joint,
each partner would find search individually profitable, assuming his fallback
partner did not search, at the point where search becomes jointly unprofitable.
The C-D border is the same with or without damages since only M's are InThus
volved.
V
(h
)
is
the same
in
Region C as previously.
Region B does not
exist, because a double breach for a good match is always profitable with zero
damages.
Thus we
that the
absence
are interested
O'f
in
A-C and A-D transitions.
damages lowers the incentive to search.
come out of surplus before its division.
half the damages to one's old partner.
Thus
a
We shall
see
Damage payments
new partner effectively pays
This monopoly power over new partners
serves as a further incentive to search for the original
partners when damaces
are positive.—
In
Configuration A search by
each cAt.
—
2/
—
a
pair of N's for additional time At costs
Each has probability a(l-p)h Atof forming a good match with an
These two assumptions are discussed below,
This theme is explored
..
in our previous paper which examines the
damages which partners would choose to set (liquidated damaoeiLJ
.
-32-
•
\
M,
yielding
a
gain to the pair of one-half the surplus, Js(2X-V -V
(but not paid) of V -V
less the damages suffered
.
Each partner has
probability a(l-p)h|^At of a double breach with surplus h(.2X-2\l^)
the same damages, V -V
),
,
and
When the pair is just willing to search, V
.
equals X', and we have the A-C border equation
(54) c = a(l-p)h|^(X +
7
-
V|^
I
Because new partners do not share
X' )+a( l-p)h^(X-2X'+V|^)
in
damage payments, equation (54) differs
from the A-C border with compensatory damages,
(Symmetry implies no change
is
less
•
(18),
by -a(l-p)h|^ '^(X'-V
the gain from double breaches.)
in
).
Thus there
incentive to search without damages and the A-C border lies to the
right of its position with damages."
A-C border by setting
V
We can derive the A-D border from the
to zero.
equal
Again, there is less search than
with compensatory damages.
The behavioral assumptions
in
modification of the basic model.
incentive inherent
in
this section have permitted a simple
This modification illustrates the search
With a further modification, we can
damages.
illustrate the role of compensatory damages
joint maximization by partners.
in
We assumed that there were no single breaches for the sake of replacing one
poor match with another.
individuals have an Incentive to do so.
Yet
When
two M's form a poor match, they plan to evenly divide the output^ 2X', should
they carry out the nratch.
If
oneof these N's meets a new M just as he is
about to stop searching, the N can gain from breach even
a
poor match.
the M receives
Forming a new partnership, the N receives
V^.
+ '5(X'-V|^).
previous partner from X' to V„,
—
Is
if
—
his new partner is
X'+ ^^(X'-Vj.) while
The breach reduces the positional value of the
Thus the aggregate positional value of the
The surplus from breach, V -V
Just as N's are due. to stop searching, V =X'
positive everywhere, however, not just at the A-C border.
.
-33-
partners has declined by one-half the compensatory damages that are
original
not being paid.
If
the original partners have no
efficient breaching behavior, they will
order condition
model,
(18),
find search
for the erd of search will
'MM
way to control this inless profitable. The first
differ from that
by - %ah,,(X'TV,,) rather than the factor -
5^
a
(
in
the basic
MM
l-p)h.,(X'-V,J
in
the
first modification.
It
is
artificial to assume that individuals keep track of only one fall-
back partner.
So too,
in
the basic model
it
is
artificial toassumethat in-
dividuals do not keep track of potential partners they have met with whom they
do not form partnerships.
would be interesting
the model
Introducing
a
more complicated information structure
but would add considerably to the difficulty of analyzing
-34-
Linear Technology
7.
When the density o^ potential trading partners
technology may be reasonable approximation.
is
the quadratic
low,
However,when the density
high
is
or the information about location is good, a searcher's problem Is less one of
finding
a
potential partner than of finding one who yields
a
high surplus.
Such a situation can be approximated by assuming a constant probability of
meeting someone at all,
independent of the numbers of potential partners
(although constancy is improbable
IS
If
the numbers of searchers are sma
I
I
)
.
Ana lysi
quite different from that above since the market possibilities do
not alter as time (and the numbers of searchers) changes.—
entire
hi^-h^,
space
is
characterized by
2/
or D, depending on parameters.—
a
single configuration. A, C,
That is,
find a good match, or search until
individuals search until they
their first match, or do not search
What is more, the equilibrium path
at all.
Thus the
is
efficient.
There are three possibilities:
c >
all
search is not worthwhile (Region D)
air
> c >
a(l-p)(X-X') search is worthwhile for M's but not for
N's (Region C)
c < a(l-p)(X-X')
search is worthwhile for N's,
no bad matches are made;
—
V,,
N
implying that
= V.,
M
(Region A).
^
This result would change if those with poor contracts searched and
wefe changed so that poor contracts were sometime carried
the.imodol
,
out.
2/
—
If
search continues until a good match is made a poor match s of no
additional value over being partnerless. Thus double breaches are profitable and Configuration B does not occur.
i
s
-35-
With this technology and behavior positional values are independent of
the numbers of searchers, giving
M
V.^
M
''
= V? = X -
Thus aggregate net output, V*,
process
—
is
a
N
is
"
ad-p)'
linear in
h
implying that the competitive
efficient.L'
The efficiency of the competitive process under the linear technology
robust to generalizations of the model.
(See our earler paper.)
is not
36-
Brief Summary
8.
We have studied an allocation mechanism in which a searcher's
meeting opportunities arrive according to
a
Poisson process.
Conf igurations A and B under the quadratic technology,
In
for example,
poor opportunities arrive at the rate aph (t) and good ones at
a( l-p) (h,.(t )+h,,(t)
endogenousi
y;
)
The values of these opportunities are determined
.
they depend on the evolution of the allocation process.
The first part of the paper examines equilibrium evolutions:
time
paths where search and breach decisions are individually optimal.
An equilibrium time path consists of a sequence of behavior con-
figurations determined by the parameteres of the search technology (a,c),
of tastes(p,X ,X
'
)and of
possible sequences.
involving all
paths
It
initial
position {h|^(0)).
Section
3
enumerates all
demonstrates for example that the only equilibrium
four behavior Configurations are paths beginning
Region A and proceeding
in
The heart of the paper
turn to B,C, and
is
in
D.
the demonstration that under the quadratic
technology, search and breach give rise to externalities that generally
cause inefficiency
a
in
in
positive economy for other searchers.
equi
I
Search by an individual creates
the market process.
Because this economy
ibr um, N' s stop searching too soon
for efficiency.
i
is
uncompensated,
Double breach
also creates external economies; it alters the search envi ronment by replacing
two N's by an M on each side of the market.
breach is individually just worthwh
i
I
e)
,
a
Since (at
a
point where double
searcher prefers the probability
of meeting an M to twice that probability of meeting an N,
is a
positive externality.
such a replacement
Therefore, equilibrium paths entail too little
breach; i.e., the transition from Region A to B occurs too soon for efficiency.
-A
I
APPENDIX
In
this appendix we discuss two issues:
first, the possibility
of multiple equilibria and, second, the nature of equilibrium paths from Initial
positions that are not on the
Whenever
h^.
axis.
Nash equilibrium is the solution concept, as
the question of possible multiple equilibria arises.
multiplicity
in
everyone else stops searching, the remaining
If
individual must obviously find search unprofitable.
equilibrium path and altering
it so that,
individuals switch toConf iguratlon
if
D^
Thus, taking any
at some arbitrary point, all
we trlvally generate a new equilibrium.
this change may require modification of earlier transitions.)
Similarly, to the left of the line
only
We avoided discussing
the text by considering only the path with the maximum
search and breach.
(Of course,
this paper,
In
other N's
a
1
h.,
=
=
M
h''
M
an N's search
h'",
Thus an equ
so search.
uration A between the lines
h,,
I
1
I
br
I
urn
path fol lowing Conf ig-
and the A-C transition
at any time, switch to Conf iguration C behavior and still
equilibrium trajectory.
worthwhile
Is
locus could,
remain an
Indeed, an equilibrium path between these two
curves could oscillate between Configurations A and C arbitrarily.
More Interesting
A-B transitions.
If
Is
the possibility of mulltple equilibria
involving
We have neither confirmed nor ruled out this possibility.
multiplicity were possible,
a
Configuration A trajectory would
necessarily cross the A-B transition border as
In
Figure
Al
.
Anywhere
on A-B transition border (more precisely, just to the left of the border),
an N finds double breach unprofitable
uration B behavior.
If
everyone else follows Config-
Therefore, any equilibrium path
an equilibrium continuation
In
in
Region A has
Region B beginning at the A-B transition
-A2-
Suppose, however, that when an equilibrium path reaches the
border.
individuals persist with A behavior.
border,
such behavior can be in equilibrium.
The question
The answer is yes
if
whether
is
and only
if
the Configuration A trajectory from this point crosses the A-B transition
border again.
Now let us turn to equilbrium paths with initial
the
h.,
axis.
h.,
=
h.,(
I
N
in
As long as the initial
-p) '(see
position' lies below the
the analysis
(2))
is
as before.
fore, the question of A-C transitions when the
this line and when Region B does not exist.
ition border derived
border
in
in
positions not on
line
Consider, there-
initial
position is above
Figure A2 shows the A-C trans-
the text (not yet shown, however, to be the region
the present case) and a family of Configuration A trajectories.
Moving backwards on one of these trajectories, the surplus from double
breach remains positive as does an N's gains from search (see (16)).
fore, Region A consists of all
There-
points to the left of the A-C transition
border that lie above the trajectory just tangent to this border (see
Figure A-2).
This analysis implies that positional values are not continuous
positions.
initial
V
= X'
until
As the initial
Region A
The reason for this
d
i
is
position moves up the h^ axis,
reached, at which point V
increases
scronti nu ity is that equilibrium paths
A are impossible below Region A.
Starting from
a
in
d
i
scontinuousi y
Configuration
point just below this
Region, for example, an N's gain from search and double breach would be
positive for awhile.
However,
if
ever the Configuration switches from A
to C or D (as it must on an equilibrium path), the gains from search would
be negative just before the transition,
an equi
I
i
br ium.
preventing such
a
path from being
References
Diamond, P. and E. Maskin, "An Equilibrium Analysis of Search and Breach
Journal
Steady States, " Bel
10 (1979)
of Contract, I:
I
Mortensen, D., "Specific Capital, Bargaining, and Labor Turnover", Bel
(1978), 572-586
I
Journa
9
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