Assortative Matching And Search

Assortative Matching and Search
Author(s): Robert Shimer and Lones Smith
Source: Econometrica, Vol. 68, No. 2 (Mar., 2000), pp. 343-369
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/2999430
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Econloin7etrica, Vol. 68, No. 2 (March, 2000), 343-369
ASSORTATIVE MATCHING AND SEARCH'
BY ROBERT SHIMER AND LONES SMITH_
In Becker's (1973) neoclassical marriage market model, matching is positively assorta-
tive if types are complements: i.e., match output f(x, y) is suipermoddlar in x and y. We
reprise this famous result assuming time-intensive partner search and transferable output.
We prove existence of a search equilibrium with a continuum of types, and then
characterize matching. After showing that Becker's conditions on match output no longer
suffice for assortative matching, we find sufficient conditions valid for any search frictions
and type distribution: supermodularity not only of output f, but also of log f, and log f
Symmetric submodularity conditions imply negatively assortative matching. Examples
show these conditions are necessary.
KEYWORDS:Search frictions, matching, assignment.
1. INTRODUCTION
THISPAPER
REEXAMINES
a classic insight of the assignment literature when
matching is assortative in an environment with search frictions. We assume a
continuum of heterogeneous agents who can produce only in pairs. If two agents
form a match, they generate a flow of divisible output. We depart from the
neoclassical assignment literature (e.g., Becker (1973)) in assuming that match
creation is time consuming: each unmatched agent faces a Poisson arrival of
potential mates (Diamond (1982), Mortensen (1982), Pissarides (1990)). As
matching precludes further search, agents must weigh the opportunity cost of
ceasing to search for better options, against the benefit of producing immedi-
ately.
Individuals' behavior is described by their acceptance sets, which specify with
whom they are willing to match; only mutually acceptable matches are consum-
mated. When agents match, they evenly divide the match surplus, i.e., their
output flow in the match less their values while searching, as in the Nash
bargaining solution. Equilibrium requires that everyone's acceptance set maxi-
'This paper answers questions stemming from a 1994 version of our mimeo "Matching, Search,
and Heterogeneity." The current version of that paper solves a constrained social planner's problem
and focuses on the relationship between equilibrium and socially optimal matching patterns.
2We thank Daron Acemoglu, Susan Athey, Richard Boylan, Peter Diamond, Ed Schlee, Peter
S0rensen, and Charles Wilson, as well as a co-editor and two anonymous referees, for useful
comments on this paper. We also thank seminar participants at NYU, Princeton, Copenhagen, LSE,
Harvard-MIT, Chicago, Rochester, Michigan, Georgetown, Penn State, Iowa, and UCSB, the Yale
NBER GE Conference, the 1997 Canadian Theory Meetings, the 1997 Summer Meeting of the
Econometric Society, the 1997 Northwestern Summer Micro Theory Conference, and the 1998
Cleveland Fed NBER Conference. While we have since found an alternate proof, we are grateful to
Sheldon Chang (M.I.T. Math Dept.) for fixing a serious flaw in a continuity assertion needed for
existence. We acknowledge help received via sci.math.research. Marcos Chamon provided valuable
simulations assistance. Financial support from NSF Grants SBR-9709881 for Shimer and SBR-
9422988 and SBR-9711885 for Smith are gratefully acknowledged.
343

344 R. SHIMER AND L. SMITH
mizes her expected payoff, and the distribution of unmatched agents is in steady
state. We provide what we believe to be the first general existence theorem for
search models with ex ante heterogeneous agents. The proof is complicated by
the obselvation that agents' acceptance sets affect the steady state unmatched
distribution, and thus the agents' matching opportunities and their willingness to
accept matches.
We then turn to assortative matching. Becker's (1973) sufficient condition in
the frictionless model is well-known. Assume that types x and y in [0, 1] produce
f(x, y) when matched and nothing otherwise. With complementarity (fr, > 0),
the marginal product of a higher partner rises in one's type; therefore, in a core
allocation, matching is positively assortative-matched partners are identical.
The easy derivation of this famous result offers hope that it naturally extends to
a model with search frictions: Agents match with an interval of types around
their own. Examples in Figure 3 disprove this. First, for the complementary
function f(x, y) = (x + y - 1)2, type 4produces nothing in the core allocation.
With search frictions, nearby agents will match with ^. When type 9 agents meet
each other, they then prefer to wait for a profitable match, as matching produces
nothing, but precludes further search. By continuity, this argument extends to
types near l. These agents match with higher and lower types, but not among
themselves.
That (+, ) minimizes f is inessential to this critique. Let f(x,y)=(x?y)2
and suppose "high" types are willing to match with "middle" types. This
opportunity is wonderful for middle types, and so if two of them should meet,
they prefer to continue to search for high types. On the other hand, if they meet
a "low" type without such a valuable option, matching is mutually agreeable.
Despite these setbacks, we find restrictions on the production function alone
that ensure assortative matching for any search frictions or type distribution. By
this we formally mean that any two matches can be severed, and the greater two
and lesser two types agreeably rematched. Observe that our motivational fail-
ures of assortative matching in Figure 3 featured individuals with nonconvex
matching sets some agents only willing to match with higher and lower types.
In fact, we show that matching set convexity is logically necessary for assortative
matching, and-along with simple conditions that orient matching sets-suffices
as well. Convexity, in turn, follows if all agents' preferences over partners' types
are single-peaked.
This suggests an indirect attack on assortative matching. We show that
single-peaked preferences, and hence assortative matching, ask that not only the
production function be supermodular,3 but also its log first- and cross-partial
derivatives, log
gf and log f Supermodularity of f ensures that any high
enough type's utility rises in her partner's type; supermodularity of log f' yields
single-peaked preferences for low types; and supermodularityof log , provides
a single-crossing property which allows us to classify every type as either low or
high. Finally, we prove that negatively assortative matching-matching with
opposite types-obtains under symmetric submodularity conditions.
3We define supermodularity in Assumption Al-Sup. See Topkis (1998) for details.

MATCHING AND SEARCH 345
We are aware of two other papers that consider "transferable utility" (i.e.,
shared output) search models with ex ante heterogeneity.4 Sattinger (1995) does
not explore the link with models in the frictionless assignment literature. Lu and
McAfee (1996) establishes assortative matching when f(x, y) xy, a production
function that satisfies our sufficient conditions. However, they do not consider
other production functions, and so do not touch on the necessary and sufficient
conditions that are central to our paper. In addition, Lu and McAfee's (1996)
existence proof sidesteps the endogeneity of the unmatched distribution. While
Sattinger (1995) endogenizes the unmatched distribution, he does not prove
existence of a search equilibrium.
By way of overview, Section 2 summarizes Becker's two frictionless results,
and then introduces our search model. We define and characterize search
equilibria in Section 3, and establish their existence in Section 4. Section 5 first
defines assortative matching and shows that it requires convex matching sets.
We then prove that convex matching sets and Becker's condition ensure assorta-
tive matching. Finally, we prove that our three supermodularity condition imply
convex matching sets, and address necessity with counterexamples. Less intuitive
proofs are appendicized.
2. THE MODEL
There is an atomless continuum of agents, each indexed by her exogenously
given and publicly observable productivity type x E [0, 1]. Normalize the mass of
agents to unity, and let L: [0, 1] ->[0, 1] be the type distribution. Associated with
this is a type density function 1.For the existence of an equilibrium, we require
that 1 be positive and boundedly finite: 0 < 1 < 1(x) < I < c/c for all x. Our
language in the paper, as well as the interpretation we lend to it, implicitly
assumes a continuum of every type of agent. Agents then belong to the graph
{(x, i)Ix E [0, 1], 0 < i < l(x)} in R' with Lebesque measure, where i is an index
number of the type x agent.
Without loss of generality, normalize the flow output of an unmatched agent
to 0. When agents (types) x and y are matched together, their flow output
depends on their types, f: [0, 1]2 -> R. We later refer to a basic set of assump-
tions:
A0 (REGULARITY CONDITIONS): Theproductionfiunctionf(x, y) is nonnegative,
symmetric(f(x, y) =f(y, x)), continuous, and twice differentiable,with uniformly
boundedfirstpartial derivativeson [0, 1] x [0,1].
2.1. TheFrictionlessMatchingBenchmark
In the core allocation, prices allocate the scarce resource, high productivity
agents. When is there positively assortativematchinzg(PAM), where each agent
4Morgan (1995), Burdett and Coles (1997), and Smith (1998) study heterogeneous-agent search
models with nontransferable utility (NTU), i.e., with exogenous output sharing rules.

matches with another of the same type? A sufficient condition is that types be
complementary:
Al -Sup (STRICT SUPERMODULARITY): Theproductionfunctionf is strictlysuper-
modular. Thatis, the own marginalproduictof anyx > 0 is strictlyincreasinig
in her
partner's type;or if x > x' and y > y', then f(x, y) + f(x', y') > f(x, y') + f(x', y).
If Al-Sup obtains, all agents have higher marginal products when they match
with high productivity agents. In the core allocation there must be PAM. The
proof of this well-known result of Becker is simple: Any allocation in which
some type x agents match with type x' o x agents admits a Pareto-improvement.
Output rises if all such agents rematch with another of their own type, since
Al-Sup implies f(x, x) + f(x', x') > 2f(x, x') whenever x 7&x'.Thus the unique
output-maximizing allocation entails PAM, and hence so must the core.
Al-SUB (STRICT SUBMODULARITY): Theproductionfunctionf is str-ictly
subnmod-
ular: if x>x' an1d) >>y',thenf(x, y) +f(x', y') <f(x, y') +f(x', y).
Under Al-Sub, the unique core allocation entails negativ)ely
assortatiVe
match-
ing (NAM): Each agent x matches with her "opposite" type y(x), where
L(x) + L(y(x)) 1. For Al-Sub implies that if there are four agents, z1 < Z2 <
Z3 < Z4, the allocation in which z1 and z4 are matched and z2 and z3 are
matched Pareto dominates the two other possible allocations in which these four
agents match in pairs.
Throughout, we maintain Al-Sup or Al-Sub. This excludes production func-
tions with nonmonotonic marginal products (like f(x, y) maxKx2y,xy2>, in
Kremer and Maskin (1995)), for which matching patterns are not easily charac-
terized.
2.2. MatchingwithSearch
We now develop a continuous time, infinite horizon matching model with
search frictions, in which meeting other agents is time-consuming and hap-
hazard.
Action Sets: At any instant in continuous time, an agent is either matched or
utinmatched.
Only the unmatched engage in (costless) search for a new partner.
When two unmatched agents meet, they immediately observe each other's type.
Either may veto the proposed match; it is only consummated if both accept.
Since in a steady state environment, a match that is profitable to accept is
profitable to sustain, we simplify our notation by ignoring the possibility of quits.
To maintain a steady state population of unmatched agents, we assume
exogenous match dissolutions. Thus, nature randomly destroys any match with a
constant flow probability (Poisson rate) S> 0, i.e., it lasts an elapse time of t

with chance e At the moment the match is destroyed, both agents re-enter
the pool of searchers.
Preferences:Each agent maximizes her expected present value of payoffs,
discounted at the interest rate r > 0. As in Becker, we assume that match output
f(x, y) is shared. Thus, x earns an endogenous flow payoff -T(xIy) when
matched with y. Because payoffs exhaust match output, 7T(xly)+
?7T(yx)-
f(x, y).
UnmatchedAgents and Search: Let u < 1 denote the unmatched densityflnc-
tion, i.e., fXu(x) dx is the mass of unmatched agents with types x E X c [0, 1].
Search frictions capture the following story. Were it possible, an unmatched
individual would meet a random unmatched or matched agent at the flow rate
p > 0. However, it is infeasible to meet someone who is already matched she is
engaged, and so misses any meeting. Thus, one simply meets any y E Y c [0, 1] at
a rate proportional to the mass of those unmatched in Y: pf, u(y) dy. Our
conclusion underscores that our descriptive theory extends well beyond this
search technology, but we use this assumption in our equilibrium existence
proof.
Strategies:A steady state (pure) strategy for agent of type x is a time-in-
variant5 Borel measurable set A(x) of agents with whom x is willing to match.
(That agents of the same type use the same strategy is not a restriction, as will
follow from ?3.1.) The strategy depends only on the unmatched density function
u, the sole payoff-relevant state variable.
Next, agent x's matchingset /,?(x) -A(x) n {y x eA(y)} is the set of accept-
able types y who are willing to match with her. Call a match (x, y) m.1utually
agreeableif y e 4//(x). By construction, matching sets are symmetric, y e //Y(x)if
and only if x ea-//(y). We shall sometimes consider the matching correspon-
dence, /4':[0,
1]0u [0, 1]. Finally, for use in our existence result, Proposition 1, we
define a match indicator function a: a (x, y) = 1 if y E.11(x) and 0 otherwise.
SteadyState. In steady state, the flow creation and flow destruction of matches
for every type of agent must exactly balance. The density of matched agents
x E [0, 1] is 1(x) - u(x); these agents' matches exogenously dissolve with flow
probability (. The flow of matches created by unmatched agents of type x is
pU(X)f/,(,)U(y) dy.Puttingthis together,in steadystate for all types x E [0, 1],
(1) 8(((x) - u(x)) = pu(x)f u(y) dy = pu(x)f ax(x,y)ut(y) dy.
LY/(
.x) 0
5In a stationary world, assuming stationary acceptance sets is without loss of generality: As the
strategy of no single agent affects the future state of the economy, if an acceptance set is optimal at
time s, it remains so at time t > s.

3. SEARCH EQUILIBRIUM
In a steadystate search equilibrium(SE), (i) eveiyone maximizes her expected
payoff, taking all other strategies as given; (ii) if matching weakly increases both
agents' payoffs, then they both accept the match;6 and (iii) all unmatched rates
are in steady-state. This section formalizes (i) and (ii) in a compact, recursive
way.
3.1. AnalyticDesc4iptionof SearchEquilibrium
The Two BellmnanEquations: Let W(x) denlote the expected value of an
unmatched agent x. Similarly, let W(xly) be the present value for x while
matched with y, and thus S(x Iy) W(xIy) - W(x) is her "personal" surplus
when matched. We begin by providing the Bellman equations solved by these
values.
While unmatched, x earns nothing, but at flow rate pf/(x)u(y) dy, she meets
and matches with some y E-(x), enjoying a capital gain S(xly). Summarizing:
(2) rW(x)= pf S(xly)u(y)dy.
Similarly, x gets an endogenous flow payoff -w(xly) when matched with y. With
Poisson rate S, her match is destroyed, and she suffers a capital loss S(xly).
Hence,
(3) rW(xIy) = 7T(xly) - S(xly).
Match Suiplus Division: Search frictions create temporary bilateral rents,
since an agreeable match now is generically strictly preferred to waiting for a
better future match. This shifts the determination of the flow payoffs 7- into the
realm of bargaining theory. We follow a number of authors, e.g., Pissarides
(1990), closing the model with the Nash bargaining solution: namely, S(xly)
S(ylx) for all (x, y). Using this, equation (3), and the resource constraint
d(xly) + -T(yIx)-f(x, y), we have
(4) S(xly)
f
f(x, y) -rW(x) -rW(y)
Personal surplus is half the excess of flow match output over both flow
unmatched values. Discounting accounts both for impatience and match imper-
manence.
MatchingSets: In a SE, an agent's strategy is to accept any match that weakly
exceeds her expected present unmatched value: S(xly) ? 0 if and only if
6In our 1996 working paper (available on request), we allow that agents may reject a match if they
are just indifferent. This generalization does not affect our conclusions. We omit it here, as it
complicates our analysis throughout the paper.

y EA(x). Since S(x Iy) =-S(y Ix), this implies y EA(x) if and only if x EA(y),
and so /Al(x) A(x). Thus by (4), the mutual optimality condition is
(5) S(x, y) f(x,y) - w(x) - w(y) ? 0 '= 'yE/f(x)
where w(x) rW(x) is the average present value of an unmatched agent, her
"reservation wage," and S(x, y) is the (flow)match sttplus.
The ValueEquation: Substituting (4) into (2) yields an implicit value equation
(6) w(x) = o (f(x,y) - w(x) - w(y))u(y) dy
x/(t)
o
0 s(x, y)u(y) dy
*1(x)
where 0 p/2(r + S). An agent's unmatched value is proportional to her share
of match surplus.
Summary: A SE is fully described by specifying: (i) who is matched with whom
(matching sets 40; (ii) the measure of types searching (unmatched density u);
and (iii) how much everyone's time is worth (unmatched value w).
DEFINITION(SE CHARACTERIZATION):
A SE can be represented as a triple
(w,/, u) where: w solves the implicit system (6), given ( u',
u); . is optimal
given w, i.e., it obeys (5); and u solves the steady state equation (1) given c'.
Figure 1 graphically depicts the matching sets for the production function
f(x,y) = xy as well as a particularchoice of search frictions, impatience, and type
distribution.7 Since this production function satisfies Al-Sup, in the frictionless
benchmark agents are only willing to match with their own type. With search
frictions, agents match with an interval of types, including their frictionless
partner.
3.2. Propertiesof ValueFunctions anidMatchingSets
We now summarize the critical properties of the value function.
LEMMA1: GivenAO, the valuiefunction w satisfies the value inequality
(7) w(x) ?> (f (x, y) - w(x) - w(y))u(y) dy
for arbitraiy MC [0,1]. In particular, w is nonnegative. Also, w is Lipschitz,
7Figures in this paper represent numerical approximations of the equilibrium matching sets. To
create them, we divided the type space into 500 discrete types. We posited a matching set, calculated
the associated steady state unmatched rates using (1), calculated the value function using (6), and
then calculated a new matching set using (5). This taitonnement process converged, and thus by
definition, to a SE. The program is available upon request.

1.0 -
y 8.4'(x) also), then the;points (x, y) and (y, x) are shaded in the graph. The graph is therefore
.....
''.: ...............
X,.,.'.,.,,,,,--.-',-',''''"'"
''-.''
-'.
-.. ------ .......
this''"'
pictureis,,similar.to,t,cir,Fi e1 ( 12- .
9)..RR.R
continuous, anD a.e. differentiae in a- SE'. When differentiabl, its
derivative.is
;
:'
S''~~~~~~~~~~~~~~~~~~~~~~~~~."
.,--'-...........
,'
,-,:'............
,..,:., :... S. - - ,,,,-,,RRR
............... .......
n n
n
b
E
~~~~~~~~~~~~~~~~~~~~~~~~~
--.R-.''.""'-
/'--':,'.'~
~~ ~ ~~~~~
.., ...........g" ... ....
(8)r w'(x)r
an +
uiorm (X)riu(Y)n
dyaet,Lx= o x[,] fxX s
Here issoan intuitiveeoverviewno
the
points(x,y)a arof (d in Apeni
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valuetric inequality follows
bhecsaus maoductionfunchsn
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use; theyefore,
produc nongtv mac
surplus.t F or Lisht an cotnut, non and
allmostas we1Slml as nearby typres1 simply byimitatingtheirmatchingpattern,since
theproductiond
function
disfe ntiableinuouSE.
IfWthey
otim tize,
t willdobetatver
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diferentisabl
intxuiw'(x
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ofvalcuus Surplusvanishes
bealong mthes
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canigoelasnrb
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imiatcing st,eimandcsimpl
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iate
LEMMAe
2:lPositnAGioAllamatchingp.setsall(,)ware
nonemptcyiandsclosedsandathe
of Calculus. Surplus vanishes along the boundaw of the matching set;
therefore,~~
......... ............
.
matchingcorrespondenceX is upperhemicontinuous(u.h.c.).
These conclusionsare establishedin the proofof Lemma1 (Part3, Step 1).

4. EXISTENCE OF A SEARCH EQUILIBRIUM
PROPOSITION 1 (SE EXISTENCE): GivenA0 aid Al-Sup orAl-Sub, a SE exists.
Since values w play an analogous role to prices in Walrasian models, we look
for a fixed point in an appropriate map from the space of players' value
functions into itself. Even though a SE is a triple, this program works because
values encode all the information needed to recover matching sets .' and
unmatched densities u.
The proof demonstrates the continuity of three maps: condition (5) maps
value functions w into matching sets .7 (Lemma 3); equation (1) maps matching
sets //7 into steady state unmatched densities u (Lemma 4); and equation (6) is
the composite map .of values and induced matching sets and unmatched densi-
ties into new values (Proposition 1). SE's are fixed points of these mappings.
Previous existence theorems for heterogeneous agent search models exploit
an a priori known threshold structure of NTU matching sets to prove existence
by construction (Morgan (1995), Burdett and Coles (1997)). Some have also
assumed that the unmatched density u does not depend on agents' matching
sets, an interaction that we believe is of significant economic interest.
We consider value functions w as elements of the space C[O,1] of continuous
maps on [0,1] with the sup norm: llw1jK_
= supXE [0,l]lw(x)l. Instead of matching
sets ./, for Lemma 3, we work with the associated match indicator functions a,
and the norm a fL= l f a (x, y)l dxcly< oc, for a e1 ([0 1]2). However, we
restrict focus to the convex set V of a's with range in [0,1]. Finally, to satisfy
Lemma 4, unmatched densities u must be given the norm Ilull=
IfIIu(x)l clx.
LEMMA 3: PositAO and A1-Sup o0 A1-Sub. Any Borel measurablemap w I>a,,,
froM Valuefunctions to match indicatorfunctions in "Vsolving (5) is continluous.
Al-Sup or Al-Sub rule out an atom of zero surplus matches, a possibility that
would invalidate Lemma 3 and considerably complicate the proof of Proposi-
tion 1.
LEMMA 4: The map a- I > ua fr-om match indicator functionis in ,Wto the
unmatcheddensityimpliedby the steady-stateequation (1) is both well-definied
anld
continuous.
PROOF: Step 1: ao ua is well definied,so there exists a unique solution Ua to
(1). The critical idea is a log transformation of the unmatched density. Let us for
convenience define p- p78. Let F be the set of measurable maps L) of [0,1]
into [log I - log(l + p31),
log I]. For all x E [0,1] and tuE F, define
((x)
@v(x-- log 1
pOo'a(x, y)e' (Y)cly.

Here, u et? solves the steady-state condition (1) if and only if P,) = v. To
prove the steady state unmatched density is unique, we show that (P has a
unique fixed point.
As 0 < a((x, y) < 1 and I < I(x) < 1,one can verify that (Pamaps F into itself.
In this step alone, and not the continuity proof, we will use the sup-norm:
}II.=IK supE[O,jl]Iv(x)I, so that v belongs to the complete space -2"([0,1]) of
essentially bounded functions. By the Contraction Mapping Theorem, (Pahas a
unique fixed point if there is a XE (0, 1) with IyvPaU
- av2I. <xIIv1 - u211 for
any VL,
IV2 e F.
Use the definition of 'Pafor arbitraryx E [0,1] and L), V2
E F:
1 + Vfla (x, y)e`l'(Y) dy
,IP 2(X) - (PaL)(X) = log 1 + 23f c(x y)et (Y) dy
J
1(1 + plea Jcx (
x,y' e)ey dy
<logV 1 +
pJefu (xy)ea
(xt'' dy )
f
1 + I
<log +p J
The first inequality uses e'!tY) < ell-v' vII-eL2) for all y. Since ell"v2 > 1, the
resulting fraction is increasing in the integral. That the integral is less than I
yields the second inequality. To bound this final expression, observe
log(1 + 13le ) - log(1 + 1^1)
11r
V' -V211.z
_log( + 132(1 + 13)) log(i(1 ? P3)) XE (0 1)
log I - log / + log(1 + /31)
as the left hand side rises in flbv'- V2 log/ - log I+ log( + P) given v1, v2
eT.
Reversing the roles of v' and v2 proves that I(VP2(X) - PaV1(X)| <xII1 -
v2jKJ.
Since this holds for all x, we have proven that IvP2 - PIVll <xIIvl -
V21IJ..
Everywhere uniqueness of the solution to v = Pav follows.
Step 2: a - ua is continuous on the space Q/.Intuitively, (1) forms a system of
equations G(Ua, a) = 0, with G continuous. By some implicit function theorem,
at points ,3 near a, the unique solution to this equation u. must lie near u.
This logic requires that the derivative of G with respect to u be invertible and
continuous, which we address in Appendix B. Q.E.D.

PROOF OF PROPOSITION 1: Given values w in C[0, 1], we follow the Schauder
Fixed Point Theorem program in ?17.4 of Stokey and Lucas (1989).
* STEP 1: THE BEST RESPONSE VALUE: Consider the map T: C[0, 1] - C[0, 1]
(9) TW(X)_ OJ01max(f(x,y) -w(y),w(x))u"'(y)dy
where u" - ,
u, is the unmatched density implied by the value function w (by
Lemmas 3 and 4), and UV' f0uU'(z)dz is the implied mass of unmatched
agents. By definition, a fixed point of the mapping Tw= w is a SE.
* STEP 2: THE FAMILY S: To establish the existence of a fixed point of the
operator T, we need a nonempty, closed, bounded, and convex domain space
g c C[0, 1] such that (i) T: 9 --> ; (ii) T(9) is an equicontinuous family; and
(iii) T is a continuous operator. Let S be the space of Lipschitz functions w on
[0,1]satisfying0?<w(x) < sup,f (x, y) for all x and Iw(x2) - W(XI)l < Kix2 -X1X
for all xl, x, where K SUp,, 3fj(x, y)L,as in the proof of Lemma 1. This subset
of C[0, 1] is clearly nonempty, closed, bounded, and convex.
* STEP 3: T: ' - IS CONTINUOUS AND T(9) IS EQUICONTINUOUS: Quite
easily, if w E [0,sup,,f(, y)], then so is Tw. Next, ITw(x2) - Tw(x )! is at most
1 + '' f1ImaxKf(x2,y) -w(y),w(x,))
- maxKf(xl, y) - w(y), w(x,))Hu"'(y) dy
0f'imaxKf(x2, y) -f(x1, y), w(x) - w(x))Iu"'(y) dy
1 + 0uiTi
Since f(x, y) and w(x) are each Lipschitz in x with modulus K, Tw is Lipschitz
with modulus KOU V/(l + OR'") < K. A family of Lipschitz functions of the same
modulus is equicontinuous. Finally, Appendix B proves continuity of T alge-
braically. Q.E.D.
5. DESCRIPTIVE THEORY
5.1. AssortatiLeMatching
In the frictionless world, supermodularity(Al-Sup) ensures PAM-any type x
only matches with another type x. In a frictional setting, individuals are
generally willing to match with sets of agents, and so mismatch is the rule. Our
first step is to formulate a sensible generalization of assortative matching to this
environment.
DEFINITION: Take xi <x2 and YI <Y2. There is PAM if the matching sets
form a lattice in R2: Yi cG1(x1) and Y2 CG1(x2) whenever Y, cG.(x,) and
Y2 c /'(x1). There is NAM if Yi c/,(x,) and Y2 c
/e(xI) whenever Yi cv/1'(x,)

and Y2 e.,'(x2). This definition8 generalizes Becker's frictionless one in Section
2.1, x e M(x) for all x. The presumed contrary matches do not exist with
singleton matching sets, and so the implications are true. Moreover, with PAM
and symmetric matching sets, an agent who is willing to match with someone,
will match with her own type: y Gc,(x) implies x E4/(y) by symmetry, so
x e4Y(x) and y EY//(y) by PAM.
The definition also captures the intuition that PAM (NAM) describes a
preference for matching with similar (opposite) types. To understand why, we
must explore the links between assortative matching and matching set convexity.
First, we have the following proposition.
PROPOSITION 2 (ASSORTATIVE MATCHING =. CONVEXITY): Given PAMor-
NAM,
if all matchingsets are nonempt, then theyare coVwexas well.
PROOF: Since the two cases are symmetric, assume PAM. Take any x1 <x2 <
X3 and Y2 E [0, 1], with xl and X3 in ./(Y2). By assumption, there exists
y' Ek1(X2). If y' >Y9, like Y3 in Figure 2, then xXE ,'(y') and X3 C (Y2) imply
x2 E.1(y2), using PAM. If y'<Y2, like y1 in Figure 2, then x2 e/(y') and
y3 ------------- -9
Y2 -------0--------9
iI---. I
XI :2 I
FIGURE 2.-Proposition 2 Illustrated. If low and high types (xl and X3) match with some agent
Y2, then so must middle types (x,), given PAM 01 NAM.
S
'An equivalent formulation, valid in hiighierdimensions, is simply to say that the matching
indicator funiction a is affiliated:
ax(J8A 8')a( J8V
18') ? a( fla( 8') for any two matches 8 =(x, y)
and /3' =(x', Y'). As usual, V and A denote componentwise vector maxima and minima. Matching
is negatively assortative if the reverse inequality obtains (negative affiliation). This also extends our
formulation to probabilistic acceptance decisions, as would be necessai-y if there were atoms in the
type distribution; the probability-of-matching,function must be affiliated. Milgrom and Weber (1982)
is the classic (auction-theory) economic application of affiliation.

x1 E d'(y2) imply x2
e/4'(y2).
Finally, if y' =Y2' then obviously x,
.2/z(y).
Q.E.D.
Next, suppose matching sets are convex, closed, and nonempty; the last two
assumptions follow in particular from Assumption AO,by Lemma 2. Then they
are fully described by lower and upperboundfuinctions: a(x) min{yIy Ec/(x)}
and b(x) max{yly e.'(x)}. These easily-visualized functions provide an intu-
itive characterization of assortative matching:
PROPOSITION 3 (MATCHING SET BOUND FUNCTIONS): AssttmIe that miiatchinig
sets are closed and nonenipty. Then there is PAM (NAM) if anidonly ifmatching
sets are colvLexanidthe boundftuctions a anidb are inon1decreasinig
(noninc;-easinig).
(The proof is in Appendix C.) This confirms higher types have higher (lower)
matching sets under PAM (NAM). For example, higher types have higher mean
and median partners under PAM.
Establishing PAM or NAM requires a comparison of a 4-tuple of values, a
complex task that prevents us from directly finding conditions that ensure
assortative matching. Instead, we assault the problem indirectly. We first show
that convex matching sets and conditions that orient matching sets are sufficient
for assortative matching, essentially the converse of Proposition 2. In Section
5.2, we complete the argument by finding primitive sufficient conditions for
convexity.
PROPOSITION 4 (SUFFICIENT CONDITIONS FOR ASSOCIATIVE MATCHING): As-
suine symnmetric,
convex, and nio0senepty
mtatching
sets /1'(x) for all x, and an upper
hemicontinuousmatchintg
correspontdenceiii. Thereis PAM if antdonllyif 0 EC
A(O)
antd1 E/(1). Thereis NAMif antd
only if 0 E7(1) antd1
EI'(O).
PROOF: As the two cases are identical, we concentrate on PAM. We have
already argued that PAM and symmetric, nonempty matching sets imply x E
,@'(x)for all x. Hence, 0 Ec,7/(O)
and 1 Ed'(1).
To prove sufficiency of these conditions, take x1 <xX and Yi <Y2 with
x2 E ,/1f(y1)and x,1E/,1#(y,). We first prove x,1E ?/4Y(yl).Our 'Intermediate Value
Theorem' in Claim 1 of Appendix C then applies, since the matching correspon-
dence is nonempty- and convex-valued and u.h.c. Consequently, 0 eC/k'(O)
and
Y2 E-
G/(x1) (by symmetry) imply that there is x0 E [0,x1] with y1
eI-'(x0), and
thus x0 E #/(y1), again by symmetry. If x0 =x1, we are done. Otherwise, with
XO
<X1 < X2, convexity of C/11(y
1), x0 E .v1(y1), and x2 EIct'(y ), ensure that x l E
ik'(y1). A parallel construction uses 1 e/'(1) to prove that x2 ec//'(x,), establish-
ing PAM. Q.E.D.
5.2. SufficientConzditiois
for Convexity
Since we have shown that convex matching sets are necessary and, with
conditions that orient matching patterns, sufficient for assortative matching, it is
logical to attack assortative matching indirectly by finding conditions under

which matching sets are convex. As matches occur at points where the match
surplus function s is nonnegative valued, according to condition (5), we look for
conditions under which s is quasi-concave in each argument-i.e., each agent
has single-peaked preferences.
The logic of the frictionless model of Section 2.1 suggests that Assumption Al
might suffice. For assume PAM, so identical types match. Competition ensures
that output is evenly split: x earns w0(x) = f(x, x)/2, where w? is the 'zero
search frictions' value function, analogous to w. Define a frictionless surplus
function, s(x, y) -f(x, y) - wo(x) - w?(y). If so() <0 for all matches, then
the wages wo decentralize this allocation. For given x, an increase in her
partner's type yields marginal surplus
s?,(x, y) f (X, y) - f(y y) /2 - f,(y, y) /2 =(x, y) -
f,(y, y)
since f is symmetric. If Al-Sup obtains, then fj,(x, y) fj(y, y) as x Q y. So for a
given x, the surplus function is increasing or decreasing in her partner's type y
as x y. Thus, s?(x, ) is quasiconcave and maximized at x, with s?(x, x) = 0.
Parallel results obtain given NAM and Al-Sub.
If frictions reduce everyone's value equally, the surplus function would shift
up, but the set of points y with s(x, y) > 0 would remain convex, the result we
desire.
Examples disprove this conjecture. The production function f(x, y) = (x + y
- 1)2 in the top panel of Figure 3 obeys Al-Sup, and yet equilibrium matching
sets are not convex, given appropriate search frictions. For any x close enough
to l won't match with another x, but will match with both higher and lower
types. The intuition for this example is clear- produces nothing when
matched with her own type.
The bottom panel of Figure 3 uses the function f(x, y) = (x +y)2 to illustrate
that the nonconvexity is quite general, and does not require f nonmonotonic.
With enough search frictions, types near 1 are willing to match with intermedi-
ate types like 0.2. This opportunity offers a windfall for 0.2: When two of them
meet, they prefer to continue to search for types near 1. On the other hand, if
0.2 meets a sufficiently low type, with no such valuable outside option, match
surplus is once again positive, and the match is mutually agreeable.
We now introduce additional conditions on the production function f that
ensure convex matching sets. We hope the serious interplay of three forms of
supermodularity makes for an interesting application of the ongoing research
program here.
A2-Sup: Thefirst partial derivatitve
of the productionfunction is log-supermod-
ular:9for all xl x2 and y1 Y2, fx(x1I Y)f (X2,Y2)2 fJ(x1, Y9)fx(x2,Y1)
A3-Sup: The crosspartial derivativeof theproductionfunction is log-supernod-
ular: for all x1 ?x2I YIY2, fv JxI, y )fy,(x2, Y2)? fy(XI, y2)fy(x2, Y1)'
9Apositive function is log-supermodular if its log is supermodular. These definitions extend this
notion to possibly negative functions, in the way needed in this paper.

1.0 -
.................................
...... ..........
0.8- 0' =,4.=
1..0.......................
0.8
0.6
0.1
...,
.-i
-.-
-.j.
0.2 ..-.--,j
.i.-.....
....-
.-.----j;
--,i
, j;
.....,..,........ j.............
O.....jj.j:.g,::
....:.:
02
0 0.2 0.1 0.6 0.8 1.0
I
.0
.......
,...
......
....
..........
......... ..... .......... ............ ........
FIGURE 3.-Non-ConvexMatching.Th t pal d t m a t h
. g
. . r f
).. +
- 1
FIGUR,
E
lO. aond-L(x)exont[0,11.
Thebotom
panel
depicts
matching
setsforf(x, y)
-
(X +y-1)2,
8 =r, p=35r, and L(x) =x on [0,11].
A2-SUB: Thefirstpartialderivativeof theproductionfuxnction
is log-submodular:
for all x1 ?x2 andy1 ?Y2, fX(x1, y1)fX(x2, Y2) ?fX(X1, y2)fX(X2, Y1)
A3-SUB: T=he
crosspartial derivativeof the productionfunction is log-submod-
ular:for allx1 ?x2 andy ?l<Y2, fXY(x1,
y)fXY(x2,Y2) ?fXY(x1,
Y2)fXY(X2,y1).

While Assumptions Al, A2, A3-Sup are independent, as are Al, A2, A3-Sub,
they jointly are interrelated. For example, A3-Sup and A3-Sub simultaneously
hold for production functions of the form f(x, y) c1 + c2(g(x) + g(y)) +
c3h(x)h(y). Under the additional restriction g h, A2-Sup and A2-Sub obtain
as well. Thus, A2-Sup and A2-Sub jointly imply A3-Sup and A3-Sub, but not
conversely.
To better understand these assumptions, consider the constant elasticity of
substitution (CES) production function f(x, y) = (x(xa +ya))t/a with Cobb-
Douglas limit f(x, y) =(xy) I2 when a = 0. This satisfiesAl-Sup when the
elasticity of substitution is negative, a < 1. For a < 0-an elasticity of substitu-
tion between - 1 and 0-A2-Sup and A3-Sup are satisfied as well. When inputs
are more easily substituted, a > 0, A2-Sup is violated. Finally, A3-Sup obtains
when a < 1/2.
PROPOSITION 5 (CONVEX MATCHING): Posit AO. GiUenAl-Suip, A2-Sup, and
A3-Sup, or-Al-Sub, A2-Sub, and A3-Sub, all matchingsets are convex.
This is proven in Section 5.3. The importance of Al is clear from the
frictionless benchmark. The production functions in Figure 3 satisfy Al-Sup,
A2-Sub, and both A3-Sup and A3-Sub, proving A2 necessary. We postpone the
subtle issue of the necessity of A3.
As an important robustness check, both the primarypremises of Proposition 5
-Al, A2, A3-and the convexity conclusion in R, are scale- and order-indepen-
dent. Consider the cardinal specification of the type distribution. For instance,
relabel each agent x by her type's percentile L(x), and let f(L(x), L(y))
f(x, y). Then f satisfies the assumptions in Proposition 5 if and only if f does.
Likewise, we may reverse agents' ordering by relabeling each x as 1-x, and
letting f(l - x, 1- y) -f(x, y). This observation will play a key role in the proof
of Lemma 5.
Our main result follows from Propositions 4 and 5.
PROPOSITION 6 (ASSORTATIVE MATCHING CHARACTERIZATION): PositAO. Thenl
A 1-Sup, A 2-Sup, A 3-Suipandf,,(O,y) < 0 <
f?1, y) for-all y implyPAM; Al-Sub,
A 2-Sub, A 3-Sub and f(O, y) ? 0 for all y or 0 2 fy(1, y) for all y implyNAM.
We use Al, A2, and A3 to establish matching set convexity through Proposi-
tion 5. These assumptions do not guarantee assortative matching, as Figure 4
attests. The function f(x, y) = x + y + obeys all three supermodularity as-
sumptions, but matching sets violate the additional requirement from Proposi-
tion 4 that 0 E //'(0). With search frictions, some agents y > 0 match with 0,
since such matches are productive. Therefore, w(0) > 0 by (6), and the match
surplus of (0,0) is negative. By (5), 0 0 /./2(0).We preempt such difficulties via
the boundary conditions on the marginal product of f.
PROOF OF PROPOSITION 6: By construction, matching sets are symmetric. They
are convex by Proposition 5, and nonempty, with a u.h.c. correspondence by

0 0. 0. 0._ . .
1.0~~~~~~~~~~~~.
-""
-"""""
- -
-::':':
-
-"::'"'""""
........ ................ '- : "':'
FIGUE.4.A.Falu .o. .
. Ths t m s for.f(x, y)
i xyx
Indeed~~~~~~~~~~~~~~~{,,
thaf(, 0). 0-hut f-(,.y)> 0-fores
.. . .
....
.
.i .........'
f...........
..... v .......... ... .....--.,,.......... ....
"
'
'
'
Ei,
,,
i
E,
i
i!
E,
E
~~~~~.
....
........ .
Lemma 2. As all the assumptions of Propositn 4 ae s d, t e i
f....................
A.'--
,.,'
',.',.~~~~~~~~~~~~~---'--,-''.-''-'-
'........,.''.-
fa'' --
'----
,:--
- .. ..........
.f < . -: -;-i -- ~~
~~~~~~~...
.:................
: : . ..
.i-- ..
...--
.,.,.,.,.,~~~~~. . ......... , .. . . .. .... .. .... .,,
i
0 ZO)~~
and 1....) an thr is NAM if 0 -A (- an i e.f()
U.-
''.'","
"".<,.
f
,
~~~~~~~~~~~~.................
:. ,
,.,a-...,..,.., .--. .. . ........... . ....
0 O 0. . .6 ..8....0
We firstshowthat0 e A(0) underthe supermodularity
andassociatedbound-
ary assumptions.(That 1eXt(l) is similar.)For y with w'(y) ? 0, f$0, y) ? 0
triviallyimpliessy(O,
y) ? 0. Forall othery, by(8):
_
Ofw(y)f5(X,y)ul(X)dX f/,()fy(X,Y)Ul(X)dx
Y 1 + Of,(y,() z Ivty)() ?fy(Oy)
where the firstinequalityfollowsfromthe negativenumerator(as w'(y) <0O),
andthe secondfromAl-Sup. In eithercase, sy(O,
y) ? 0, so that type0 prefers
matchingwith lower types. Finally,0O A'(O),as matchingsets are nonempty
(Lemma2).
Nexttakethe submodularity
assumptions,
withfy(1 y) ? 0. If w'(y)> 0O
then
we trivially
haves/il, y) ? 0. Otherwise,use Al-Sub in the abovesupermodular-
ity argumentto prove w'(y) ?f/l, y). Hence, 1 prefersto matchwith lower
types;thus, 0 e.#(l) by nonemptiness,and 1 w.4(0) by symmetry.
The proo)f
withf$0O,
y) > 0 is similar. Q.E.D.
5.3. Convexity
Argument:Proofof Proposition5
We provethatmatchsurplusis quasiconcave
in each argument.
LEMMA5 (QUASICONCAVITY):
PositAO andfzx z. GivenAl1, A2, A3-SuporAl1,
A2, A3-Sub, the match surplusfunction s(z, y) is quasiconcavein y.

By condition (5), Proposition 5 follows immediately from Lemma 5.
Each of the three assumptions plays a distinct role in our proof. In the
frictionless model, Al-Sup ensures that the highest type has an increasing
surplus function, s?(1, y) ? 0. Search frictions depress and flatten all value
functions w. As a result, the highest type's match surplus is strictly increasing in
her partner's type. By continuity, this argument extends to other high types.
For the lowest type, Al-Sup implies that the frictionless surplus function is
decreasing. Flattening the value function changes this in complex ways. A2-Sup
imposes that the percentage decline in productivity from 'mismatch' is smaller
for high types than for low types. As a result, low types suffer a larger
percentage decline in value. This ensures that low types continue to prefer to
match with relatively cheap low types. Finally, A3-Sup provides a single crossing
property (SCP), Lemma 6 below, which allows us to treat everyone as either a
high type or a low type.
The formal proof of Lemma 5 rules out local minima in any z's surplus
function, showing that if s(z, ) is falling at Yi, it is falling at Y2 >Y1 Here let us
assume w'(y) > 0. Let z solve f,(z,y)y w'(yI). By Al-Sup, f,,(z, Y1) >f,(i, Y1)
for z > i, so that s(z,) is increasing at Yi
Dealing with z < uses the other assumptions. By A2-Sup, fy(i, yY)fy(z, Y2)
<?f,(z, y )f,(i, Y2) for any Y2 >Yi* By the definition of z, the left-hand side is
equal to w'(y1)f5,(z, Y2) If we are in the interesting case where z's surplus
function is decreasing at Yi, the right-hand side is less than w'(y1)f(i, Y2)
Putting these together, fy,(z, Y2) <f,,(i, Y2) Now if we can prove that fy(, Y2) <
w'(y2) we will have shown that z's surplus function is decreasing at Y2'
completing the proof.
For this last step, we use the SCP. Diamond and Stiglitz (1974) showed
(Theorem 3, equivalence of (ii) and (iii)) that a gambler has a higher coefficient
of absolute risk aversion if and only if he requires a higher risk premium for any
gamble. Formally, let utility h depend on a prize x and preference parameter y.
Then the coefficient of absolute risk aversion -hxxl/hx is smoothly falling in y
if and only if the certainty equivalent of any gamble is smoothly rising in y. We
extend this result by assuming that utility is just once differentiable in
income but that log marginal utility log h, is supermodular in (x, y). As such,
the Arrow-Pratt risk aversion measure need no longer exist, yet alone its
derivative in y. We can find no way of patching their proof, as it includes
repeated integration by parts of a third derivative. Our proof in Appendix C
explicitly exploits the supermodular structure, our main focus.
LEMMA 6 (A SCP FOR GAMBLES): Consider h: [0, 1]2 -* R, differentiable,with
h. eitherpositive and log-supermodular,or negative and log-submodular. For all
Ye [0,1] and probability densities with support Mc [0,1], there is a unique
certainty equivalent z: h(z, y1) E,E Mh(x, y1). For all y' >y1, h(z, y') <
E Mh(x, y'). If h, > 0 is log-submodular or hx < 0 is log-supermodular, then
h(z.') > EXh(~mOx,y').

MATCHING
AND SEARCH 361
In our case, w'(yl) is proportionalto the certaintyequivalent Exfy(x,yl),
accordingto (8), and is also equal to fy$, yl) by definition.Similarly,
w'(Y2) is
proportionalto the certaintyequivalent Exfy(x,Y2). Thus A3-Sup,log-super-
modularityof fxy,ensuresW'(Y2)2f y(i, Y2)*
Our formal proof of Lemma 5 tightens this argumentand uses a slightly
differentcharacterization
of quasiconcavity.
A well-behavednon-quasiconcave
function o- has a local minimumx. The characterization
in the next lemma
(proofappendicized)
focuseson a criticalpropertyof pointsYiandY2 to the left
andrightof x.
LEMMA7: Assume a continuous and a.e. differentiablemap o-: [0,1] R obeys:
O(Y1) <O(Y2) for 0 <Y1 <Y2 implies f'(y1)2 0 when defined. Then o- is
quasiconcave.
Nowwe have the tools to provethat the matchsurplusfunctionis quasicon-
cave.
PROOFOFLEMMA
5: Sincef is continuousanddifferentiablebyA0, andw is
continuousand a.e. differentiableby Lemma1, surpluss is continuousanda.e.
differentiable.We will use Lemma 7 to prove that an arbitrarys(z, y) is
quasiconcavein y under the supermodularity
assumptions.We omit the sym-
metricsubmodularity
proof.
Fix 0 <Yj < 1. If w'(yl) is defined,we mayassumewithoutloss of generality
that w'(y1)is nonnegative.Forif w'(y1)< 0, we couldinsteadworkin theworld
with a reversed type ordering,recalling the discussion after Proposition5.
Lettingf(x, y) f(1 - x, 1 - y) denote the production function and L(x) 1-
L(1 - x) denote the typedistributionin such a world,therewouldbe a SE with
value functionwi(1-y) w(y), and in particularw(1 -y1) = -w'(y1) > 0. We
could instead proceed with the analysis of type 1 - Y1,proving that s(1 - z, 1 - yi )
is quasiconcavein its second argument.By construction,this would establish
quasiconcavity
of s(z, y), as desired.
Next, fix Y2 E (yl, 1) and z. We must prove that if s(z, Yl) <s(z, Y2), then
SY(z,
Y) ? 0 whendefined.If w'(y1)is undefined,then so is sy(z,yl) andwe are
done. Otherwise,implicitlydefine 2 so fy(, yl) is the expectedvalue of the
gamble fY(x,yd):
(10) fy($2 Y ) f,f(y1)f$X, y)u(x) dx
* STEP 1: Sy(Z,yj) 2 0 FORALL Z 2z.
0O?<w'(y1)
=
X
O(Y,)fY(x, yl)u(x) dx fy($2,yl)Of,(Yl)u(x) dx
1 + OI(5,)u(x) dx 1 + Of(5)u(x) dx
?fy(2, Yi)
by (8) and (10), so sy(2, yl) 2 0. By Al-Sup, sy(z, y1) > sy(Z,y1) 2 0 for all z >2.

* STEP 2: w(Y1) < W(Y2) AND f(Z,YI) <f(Z,Y2) WHENEVERS(Z,Y1) < S(Z,Y2)
AND z <2. At Y2 with s(Z,YI) ? s(z, Y2), there is nothingto verify.Otherwise:
(11) f(z, Y2) -f(z, Y1)> W(Y2)- W(Y1)
>OfO(y
1)((X IY2) -f(X,I YI)) u(x) dx
1 + Ofj(y,)u(x) dx
wherethe secondinequalityappliesequation(6) and inequality(7).
ByAl-Sup andA3-Sup,h =fy meets Lemma6'sconditions.So for all y' > Yi,
f-f,(Y)fY(x,y')u(x) dx
M2 )
f.(,Y,)u(xY) dx(zyt
by(10).For Y2>Y1, integratethis overy' E [Yi, Y2], anduse Al-Sup with 2 > z:
(12) fJ(y()(f(X, Y2) -f(X, yl))u(x) dx
f..(yl)u(x) dx
?f(f,Y2) -f(2,y1) >f(Z,Y2) -f(Z,Y1).
By transitivity,
the last termin (11) is smallerthanthe firsttermin (12):
JJ(yj)(f(X, Y2) -f(X, Y1))u(x) dx
Lf,(Y)u(x) dx
> Off(y,)(f(X9 Y2) -f(X, Yl))U(X) dx
1 + OJ,(Y()u(x)dx
Hence, fe(y()(f(X,Y2)-f(x,yl))u(x)dx>0. Using this, (11) implies w(y2)>
w(y1) and f(Z, Y2)>f(z, y1). Note that(12) then impliesf(, Y2)> f(2, Y1)
STEP 3: sY(Z,Y,) 2 0 WHENEVER S(Z,YI) < S(Z, Y2) AND z <2. Divide (8) by
(11), andthen (10) by (12),verifyingthat the relevanttermsare positive:
(13) w Yl <r(y,)fy(x,yl)u(x) dx
W(y2) - W(yd) fj(yj)(f(X Y2) -f(x, yl))u(x) dx
< y(il Yd)
f(Z, Y2) -f(Z, Yl)
IntegratingA2-Supimpliesthat for Yi<Y2 and z <z,
fy(29 Yd)(f(Z9 Y2) -f(Z, Y)) <fy(Z' Yd)(f(2, Y2) -t(, Yd))
Divide by f(Z, Y2)-f(Z, Y) > 0 and f(, Y2)-f(2, Y1)> 0, and combinewith
(13):
w,(yl) < y(zg Yd)

As 0 < W(y2) - W(y1) <f(Z,Y2) -f(z, Y1), multiplication
yieldsw'(y1) <?f(z, y1)
or sY,(z,yI) ? 0, completing the proof. Q.E.D.
One can prove that Al-Sup and A2-Sup preclude all but holes in the center of
acceptance sets, a pattern we are unable to produce. Instead, we offer an
example showing that A3 is necessary for surplus quasiconcavity, and therefore
critical to any general proof of matching set convexity. Consider the production
function f(x, y) = 1000 - (xy)3 + 9(xy)2 + 3(x + y), obeying Al-Sup, A2-Sup,
and A3-Sub (but not A3-Sup). Let p=50000r, =r, and L(x) = x on [0,1].
Since a partner's type has a negligible effect on output, not surprisingly all
matches are acceptable. In Appendix C, we show that this model is analytically
solvable: w(x) =495.1144 + 2.9472(x +x2) - 0.2456x3. Thus, the surplus func-
tion of x E [0.560,0.562] is not quasiconcave. For example, s(O.560,y) is mini-
mized at y = 0.913.
6. CONCLUSION
This paper has pushed the assortative matching insights into a plausible
search setting. We have generalized PAM and NAM for this frictional environ-
ment, and have identified the three supermodularity (submodularity) assump-
tions under which matching sets are convex and, with additional boundary
conditions, increasing (decreasing). We have also developed a general existence
theorem for search models with endogenous type distributions, and general
matching sets.
We have investigated this model because it can be fully solved. In proving
existence of a SE, we made assumptions on the search technology to avoid
significant complications; however, our descriptive theory in Section 5 applies
for any anonymotts search technology, where the rate that searchers meet is
independent of their types. This includes, for example, a linear search technol-
ogy, in which the meeting rate is independent of the measure of unmatched
agents.
Likewise, we have assumed that a steady-state is maintained through deaths
of existing agents. We could alternatively posit an inflow of entrants, and assume
that all matches are permanent. While our assortative matching and convexity
conclusions do not depend on this modeling choice, our existence proof does.
This model may be generalized in several important ways. For example, so as
not to double the notation, we have assumed one class of agents. All our results
extend to a model with workers and firms, or men and women. Our definition of
assortative matching generalizes to multiple dimensions, while convex coordi-
nate slices (biconvexity) is the scale-independent extension of convexity in one
dimension.
Department of Economics, Princeton Unitlersity,204 Fisher Hcll, Princeton,NJ
08544 U.S.A.;shimers@princeton.
edu;www.princeton.
edu/ shimer/

and
Department of Economics, University of Michigan, 611 Tappan Street, Ann
Arbor,MI 48109, U.S.A.;econ-theorist@earthling.net;
www.umich.edu/ lones7.
ManuscriptreceivedDecember, 1996;final revisionreceivedFebruary,1999.
APPENDICES: OMITTED PROOFS
A. VALUE FUNCTIONPROPERTIES
PART 1 OF LEMMA 1: VALUE INEQUALITY.If (7) wereviolated,(6) wouldyield eithery e (x),
y 4 M, and f(x, y) - w(x) - w(y) <O; or y e M, y 4AX(x), and f(x, y) - w(x) - w(y) >O. Either
possibility contradicts (5). Q.E.D.
PART 2 OF LEMMA1: LIPSCHITZ
AND THUSCONTINUITY.
Since fx is continuouson [0,1]2,
K maxX,yIf(x,y)lis well-defined. Also, by (6) and (7), for all xl <x2,
of X(x)(f(x2,y)-f(xl,y) - W(X2)+ W(X))u(y) dy
2 W(X2)- W(X1)2 of ((x2, y) -f(xI ,y) - W(X2)+ W(X0))U(Y)dy.
Solvingeachinequalityfor w(x2) - w(x1) andusing If(x2,y)
-f(xI,y)l ?<K(x2 -x1),
K(X2-_XI)'OL,r(X2)U(y)
dy - K(X2-x1
X OffA(X1)u(Y)
dY
y( dy W(X2) -W(Xd
>
(~
I
O
1 + OJ'f(X2)U(y) dy (x? 1 + 6IXl)u(y) dy
So w is Lipschitz, Iw(x2) - W(X1)l/(X2 -x1) < K, and thus continuous. Q.E.D.
PART 3 OF LEMMA
1: DIFFERENTIABILITY.
Let D(B, C) be the Hausdorff distance between sets B
and C: namely, D(B, C) = inf{dlV(b,c) E (B, C), 3(b', c') E (C, B), with lb - b'l< d and Ic- c'l < d}.
Call a correspondence M continuous at x if for all e > 0, there is a neighborhood Nx of x, such that
D(M(x), M(x')) < e if x' E Nx.
* STEP 1: AXIS CONTINUOUS
AT A.E. x. First, ,f is nonempty-valued: if X'(x) = 0 for some x,
then w(x) = 0 by (6). Thus f(x, x) - 2w(x) 2 0 by A0, and (5) implies x e .A(x), a contradiction.
Next,X' is u.h.c.:Takeanysequence(xv,yY)-- (x, y), withynE A'(xn)forall n. Thens(x",yn)? 0
for all n, and so s(x, y) ? 0 as well, since s is continuous by AO and Part 2 of this Lemma. Thus,
y E:A'(x), establishing u.h.c. Fixing xn = x for all n, X.(x) is closed too. Also, A'(x) 5 [0,1] is
bounded, whence X' is compact-valued.
We call x an --continuitypoint of a correspondence M, and say x belongs to WM(e), if for all x'
sufficiently close to x, D(M(x), M(x')) < e. Since A' is u.h.c. and compact valued, Theorem
I-B-III-4 in Hildenbrand (1974) implies that (given our nonatomic density u) for all ? > 0, a.e. x is
an e-continuity point of X'. Then for all n = 1,2,..., a.e. x E FM(1/n), and the countable
intersection n, FM(l/n) contains a.e. x as well. That is, for a.e. x, for all n, if x' is sufficiently close
to x, D(A'(x), X'(x')) < 1/n. So A' is a.e. continuous.
* STEP 2: DECOMPOSITION
OFTHEVALUE FUNCrION'SSLOPE.Take any sequence x, -. x. Add and
subtract OJr.tx)(f(xn,y) - w(xn) - w(y))u(y) dy from w(xn) - w(x) for each n, and divide through
by x -x:
W(Xn) -WWX f(Xn, Y) -W(X,) -W(Y)
(14) x = 0 - u(y) dy
Xn -x X') .-4X(x3 Xn -x
(f(xn, y) -f(x,y)) - (w(xn) -w(x)) ( d
f Xn
-xuyd)

MATCHING
AND SEARCH 365
where JA- B JAIA-IB.
The first term in brackets vanishes if Xf is continuous at x, because
continuityof f and w andcondition(5) implysurplusvanishesat changesin X'(x). The remaining
termstendto the desiredexpression
for w'(x) at a.e.x. Q.E.D.
B. EQUILIBRIUM
EXISTENCE
CONTINUITY
OF Wc4 a(W): PROOF
OFLEMMA
3.
* STEP1: SURPLUS
FUNCTIONISRARELY
CONSTANT
INONEVARIABLE.
Define ZW(x)
= {Y: s(x, y)
= 0} and Z, = {(x, y): s(x, y) = 0}, and let ,.t be Lebesgue measure on [0,1]. We claim that under
Al-Sup or Al-Sub, g(Zs(x)) = 0 for a.e. x. Let x # x' and y # y', with s(x, y) = s(x', y) = s(x, y') = 0.
Under Al-Sup or Al-Sub, f(x, y) +f(x', y') #f(x', y) +f(x, y'), from which s(x', y') 0
Ofollows.
Thus,Z,(x) n Z,(x') containsat mostone pointwheneverx ox'.
Assume A(Zs(x)) > 0 for an uncountable number of x.10 Then for some k, there are infinitely
many (xv >with tL(Zs(xn))> 1/k, whereupon En = I /(Z,(xn)) = oo? Since xij = Z,(xi) n Z,(xj) con-
tains at most one point, N = U j1=xij is countable, and so ,u(N) =0. Also, Z,(xi)N and
Zs(xj)N are disjoint for all i j. Thus
1 2
I.L U Z,(x,)N E
, .(Z,(xn)N) = (Z,(x"))= ??.
n=1 n=1 n=1
Giventhis contradiction,
there are onlycountablymanyx with 11(Z(x)) > 0. So pu(Z,(x))= 0 for
a.e. x, andbyFubini'stheorem(A X uXXZs)
= 0.
* STEP2: CLOSE
SURPLUS
FUNCTIONSRARELY
DIFFER
INSIGN.As q 0, the set l(ti) = {(x, y)
with Is(x,y)1G[0,1J]} shrinks monotonically to nk=l, s(l1/k)=s-1(0)=Zs. By the countable
intersection property of measures,
00
lim ( A x t,u)(.Xsq)) = lim ( ,u x a) ( (1/k)) = (,u x u) ns(llk)
,q- 0 k oo k=/ 1
-( x
X L)(Z5) = O.
Finally,let w, andw2 be valuefunctions,with IIwI(x) - w2(x)II <?i/2, and a1,a2 corresponding
match indicator functions. If sl(x, y) af(x, y) - wl(x) - wl(y) > 71,then s2(x, y) =f(x, y) - w2(x)
- w2(y) > 0, and so al(x, y) = a2(x, y) = 1, while if sl(x, y) < - -, then s2(x, y) < 0, and al(x, y) =
a2(x, y) = 0. Consequently, {(x, y)l al(x, y) ?#a2(x, y)} c Xs1(1), whose Lebesgue measure vanishes
as 1
-* 0. This implies the desired continuity lim IWl-W211 Dolal1- a211L' = 0-
CONTINUITY
OFa C-!u(a): FINISHING
THEPROOF
OFLEMMA
4. Normalize 8= 1.
For fixed a0, let uo be the associated unique unmatched density: i.e., 1(x) - uo(x) =
pu0(x)Jgca0(x,y)uo(y) dy. Defining G(a, u) = u(xXl + pJa(x, y)u(y) dy) - 1(x), we see that U = ua
solves(1) given a iff G(a, u) = 0. As proven,u,yis unique.
* STEP1: AN INVERTIBLE
STEADY-STATE
DERIVATIVE
OPERATOR.
The derivative Gu(a, U) is a
bounded linear operator on 22[0, 1], defined as the following limit:
Gu(a, u)(g) = lim(G(a, u + tg) - G(a, u))/t = g (l + p au) + puf1ag
that is also clearly continuous in u. Write GU(a,u) I + pH. Since a is symmetric (a(x, y)=
a(y, x)), the operator H is self-adjoint, and positive-definite on the space 2 2([0, 1],1/u) of
10Toby
Gifford(MathDept.,WashingtonU. in St. Louis)tightenedthe logicof thisparagraph.

functions square-integrable with respect to density 1/lt, because'1
2Kg, Hg)= 2 g(x)Hg(x)/ii(x) clx
0
=2f (g(x)ti (y)/l r(x) +g(x)g(y)) a(x,y) dxdy
fo o
0 0
flfI[g(X)2t(y)/1(x) + 2g(x)g(y) +g(y)2 u(x)/It(y)]cIa(x, y) dxdy
0 -0
f1f1[g(x)Vit(y)/-i(x) +g(y) it(x)/It(y) ]a(x,y) dxdy>0.
Since it is boitindedly positive (given p < sc, I > 1> 0) and finite (11 < 1 < cG), y2([o 1], 1/u) =y2[0, 1].
So H is a self-adjoint and positive definite linear operator in 52[0, 1], and its spectrum is thus real
and nonnegative; the spectrum of Gja, it) is then contained in [1,cc), and excludes 0. Hence,
G(a, it) is invertible in 22[0, 1].
* STEP2: G is LIPSCHITZ
IN a. First, we have the subtraction:
G(a, it)(x) - G( /, it)(x) = pit(x)f 1[a(x,y) - f(x,y)]ui(y) dy
0
f)[G(a,li)(x) - G( f X,,2)(x)]2 dx?p2i4fl( f[a(x,y)Y- (Xy)]dy) dx
? p9if'f '[a(x,y) -ff3(x,y)]axdydx
fo o
0 0
since t(x) ? 1.Then IIG(
a, ii) - G( :, ti)ll< pl11a - 11,
both norms in y2
* STEP3: CONTINUITYOF a ->
it.. Since G(a, it) has Lipschitz constant L = pl2 > 0 in a, we
have IIG(a',ita)|I = IIG(a',it.) - G(a, ")11< Llla- all, and so
11
G(a', ilHa ) + (ll a - U) 1 = 1I
G(a', ilt. + lick
- uta') =G(a', 'la)) < Ll a - a'll
for some J, = G(a', tula + (1 - O)uit,')and t E [0,1] by the mean value theorem, given continuity of
Gi, in it. By Step 1, J, is an invertible linear operator in y2[0, 1] with spectrum contained in [1,oc).
So its inverse K, exists, and IIK,11
< 1 for all t. Hence,
|lltia- lia'||
=
||K,J,(lta
,
-taC')II
< IIKt II lljt(tla
- la,)l < Lll a-a'11
since IIK,II ? IIKtIllull
<l|ull by the norm inequality and IIK,i ? 1. So when a is near a' in
y ([0 1]2), t is close to itu in 52[0,1]. Finally, continuity holds for both norms in Yl: The
Cauchy-Schwarz inequality implies ll_'(a
- ua'IK" ? llu ?
- ua,IK2; and lla' - a'll,,_2 <?I I-a- Il"
holds in our restricted domain, as Ia - a''l < 1.
CONTINUITYOF THE OPERATORT: FINISHINGTHE PROOF OF PROPOSITION
1.
For all w1,wE2 3 and xE=[0,1], the triangle inequality implies lTw2(x) - Tw1(x)l < D(x) +
D2(x),where
D(x)_|oo?
max<f(x,y) - w(y),w(x))it ,(y) dy
1+ 06W"'
6j0 max<f(x,y) - w2(y),w2(y),w2(x))l,v(y) dy
1+
w W)
w
l
l1We thank Robert Israel (UBC Math Dept.) for remarking on this fact.

MATCHING
AND SEARCH 367
We will show that D1 and D2 convergeto zero when w1 convergesto w2. First,supxD1(x)<
sup, Iw(z) - w2(z)I,since
D (x) OfoImax<f(x,y) - w2(y),w2(x)> - max<f(x,y) - w1(y),wl(x)>Iuw (y) dy
1 + Oii
0fo'Imax(w1(y)
- w2(y),w2(x) - w1(x)>Iuwl(y) dy
< - 1 <
~~~~~~~~sup
Iwt(z)- w2(z)I
1 + oi-w,
Second,we canboundD2(x) througha seriesof algebraicmanipulations.
I
- W2(y)W2(X)>
OU2 O(y) WIY
D2(x) <fmax<f(x, y)w2(y),w2(x)> |+O -
2 |dy
21 Xuw(y)uwf(y)
< supf(x,y)f -
OU 2 -
W
dy
y 0 1 + ouw 1 + ouw
O
OUW2(y) OuW'(y) I Ouwl(y) 0Wu'(y) I
supf(z,y) 1 + OW2 1 + oUw2 | + |1+ W2 1 + O
dy
< Osupf(z,y)(fIIuw2(y) - uwl(y)l dy + IiuW2
_
-WI,
Accordingto Lemmas3 and4, fJ10uW2(y)
- uw'(y)l dy converges to zero when w2 convergesto w'.
An immediateimplication
is thatthe differencebetweenthe massof unmatchedagents,Ilw -w1iWI
mustconvergeto zero as well.
C. DESCRIPTIVE
THEORY
PROOFS
MATCHING
SETBOUNDFUNCTIONS:
PROOF
OFPROPOSITION
3.
* STEP 1: MONOTONIC
BOUNDFUNCTIONS
>ASSORTATIVE
MATCHING.
As the two cases are
analogous,we provethatnondecreasing
boundsa,b implyPAM.Choosexl <x2 and Yl<Y2 with
YiE X(x2) and Y2EA'(xj). Then b(x2) 2 b(xl) 2 Y2, where the firstinequalityfollowsfrom b's
monotonicity,
andthe secondfromthe factthatY2E2f(xX). Also,Y2>Y1 2 a(x2),wherethe second
inequalityfollowsfrom Yl
I E`(x2). In summary,b(x2) ?Y2 > a(x2), and since matchingsets are
closed and convex,Y2E. f(x2). Similarly,a nondecreasinglower boundfunction a ensuresthat
YiE.#(xd), whencematchingis positivelyassortative.
* STEP2: ASSORTATIVE
MATCHING
=> MONOTONIC
BOUND
FUNCTIONS.
Proposition 2 provedthat
assortative matching implies convex matching sets. To avoid tedious repctition among four similar
cases, we simply prove that b is nondecreasing with PAM. If not, b(x1) > b(x2) for some pair
x1 <x2. Then since b(xl) c#(xj) and b(x2) e'f(x2) (matching sets being closed), PAM implies
b(x1) EJ((x2), contradicting
thatb(x2) is the upper bound of x2's matching set. Q.E.D.
AN INTERMEDIATE
VALUE THEOREM
CLAIM
1: Let the correspondenceX(: [0,1] [0,1] be upper hemicontinuous (u.h.c.), and convex-
and nonempty-valued.Take yO 2, Zo E 1(yo), and Z2E#(Y2). If ZO
<Z2, then for all z1 E [ZO,Z2],
thereexistsy1 E [Yo,Y2] withZ1e fO(Y1)-
PROOF:Define Yl= sup{ye [YoY2]L(y) n [0,z1] # 0}, the largestpointwhoseimageincludes
points less than z1. Since X' is u.h.c., A'(yj) n [0,z1] 0 as well. If Yl=Y2' then convexityof
-X(Y2) implies z1 XE
A(Y2). Otherwise,
takea convergent
sequenceof pointsyf Y withY2?yn >Yi
for all n. Associatewith this anothersequence of points {zn}with Ene A(Yn) for all n. By the

368 R. SHIMERAND L. SMITH
construction
of Yl, zn E (z1,1]forall n. Since Xf is u.h.c.,thereis a convergent
subsequenceof {z"}
whoselimitpoint z E [z1,1]is in X(y1). We haveshownthat A(y1) includespointsweaklygreater
thanandweaklyless thanzl. Sinceit is convex,z1E f(yl). Q.E.D.
SINGLE
CROSSING
PROPERTY:
PROOF
OFLEMMA
6. We need a preliminary
result:
CLAIM
2 (DENSITY-FREE
INTEGRAL
COMPARISON):
TakeMc [0,1] andu: M R+t
+. Let4, f':[0,1]
lR be increasing(decreasing)functions with a nondecreasing
(nonincreasing)ratio i/fl. If
fM0f(x)u(x) dx = 0, then mIM
(x)u(x) dx 2 0.
PROOF:
Assume withoutloss of generalitythat i and qi are increasing,as in Figure 5. If
M # [0,1], extend u to [0,1] by definingu(x) 0 for all x -M. Define A(x) Jfxi(x')u(x') dx'.
ClearlyI(O)= 0, while I(l) = 0 by assumption.Then I is quasiconvex,
as if is increasing,and so
I(x) <0 for all x E[0, 1].
Substituting
with I andthe nondecreasing
quotientq(x) +'(x)/+f(x) yields
M (x)u(x) dx-f q(x)I'(x) dx= q()I() - q(O)I() - I(x) dq(x)
wherewe haveintegratedbyparts.The firsttwotermsin the last expressionarezero,andthe final
termnonnegative,
as I() <0, dq(-)? 0. So fMP(x)u(x)dx2 0. Q.E.D.
PROOF
OFLEMMA
6: We just establishthe supermodular
case. First, z is uniquelydefined,as
h(2,yl) is strictlyincreasingin 2. Next, integratinghx(x2,y')hx(xl,yl) > hx(x2,yl)hx(xl,y') over
xI E [2,x2] andthen x2 E [xI, 21,we discoverthat
hx(X,y')(h(x,yl) - h(2, yl)) 2 hx(x, y1)(h(x, y') - h(2, y'))
for all y' >yl, andfor all x > 2 and cruciallyalso all x < z. Thisis equivalentto
d h(x,y')-h(2,y')
Ax h(x, yl))-h(:z, y)
-
Let +(x) h(x,y') - h(2,y') and +i(x)- h(x,Yl)- h(2, yl), increasing.Since q5/If is nondecreas-
ingandExe M
if(x) =0 byconstruction,
Ex m
M+(X)0 byClaim2. Q.E.D.
FIGURE
5.-Illustration of Claim2. Thisillustratesthe increasing
case.As Jmif'(x)u(x)dx=0, a
z existssuchthat qif(x)
><
0 forall x ~tz. Andsince 4O/ifis nondecreasing,
40mustequalzeroat the
same point. Then the ratio conditionimplies that, since the positive and negative areas of if'
balance,the positiveareaof 4 mustoutweighthe negativearea.

PROOFOF LEMMA
7: We will prove the contrapositive: If u- is not quasiconcave, there exist
0 <yj <y2 such that u'(v1) < 0 and o-(yl) < o(y2). Since u- is not quasiconcave, there are three
points Yo<Yj <Y2 with o-(9,) < min<o(y0), (d(Y2))>
First take the case where u(y) < 0(Y2) for all y E [yo,91]1Then as J'Y$Q'(z)odz ,)- (yO)
< 0, there exists Yi E (yo,.91) with o-'(yl) < 0. By construction, 0(y1) < o(y,) as well, the desired
counterexample.
Alternatively, if max,,, ]o-(y) > (y,), define 90 =sup{y <911jo(y) ? 0(y1)}. Note 90<9
because o- is continuous and U(91) < u(y2). Then we can proceed as before, with 90 playing the
role of yo, to prove that there is )' E
(9I,9E ) with o'(yi) < 0 and, by construction, (y) < (y,).
Q.E.D.
ANALYTICAL
SOLUTIONFORTHE VALUE FUNCTION:If all imatches are acceptable, it is possible to
solve analytically for the value function. First, equation (1) implies the unmatched density function
satisfies ii(x) -tl(x) for all x, where Tiis the aggregate unemployment rate, the larger root of the
quadratic equation 8(1 -Ky)= pl2:
-8+ 82+48p
2 p
Next, (8) yields an analytical solution for w'(x) for all x. Finally, to pin down the level of the value
function, use the implicit value equation (6) for type 0:
W(0) = Of'(f(O,Y) -w(O) -w(O) -J
YW/ (z) dz)u(y) dy
where we use w(y) -w(O) + Jj'w'(z)cz. This can be solved explicitly for w(O).
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