Assortative Matching in Two-sided Continuum
Economies
Patrick Legros and Andrew Newman
February 2006 (revised March 2007)
Abstract
We consider two-sided markets with a continuum of agents and a finite
number or continuum of types. We shown that a single crossing condition
(GID) implies that any economy satisfies positive assortative matching.
1
Introduction
On most markets, the utility from transacting depends on the characteristics
of the matched trading partners: education, wealth or beauty in household
formation, education or productivity in firms. For this reason market equilibria
need to specify not only the payoffs but also the way agents match.
When agents are characterized by a one dimensional type, Becker (1973)
observed that if the product from the match is perfectly transferable between
the partners and if the marginal product of a type is increasing in the type of the
partner (there are complementarities in types), then necessarily the matching
will be assortative: higher types match with higher types.
This is a very useful result since it implies that the search for an equilibrium
can be limited to the search for payoffs that make the assortative match stable.
Moreover, the search for the equilibrium payoff is facilitated when there is a
continuum of agents on each side, when the type assignment is continuous and
when the production function is differentiable.
For instance consider marriage market with a continuum [0, 1] of men and
a continuum [0, 1] of women. Assume that the characteristic of agent i is i; if
the man i forms a household with the woman j, the production is y(i, j) = ij.
Since y12 = 1 > 0, the unique matching pattern is such that the match of the
man i is the woman i. Existence of an equilibrium implies that there exists a
payoff structure (u, v) where u(i) is the payoff to man i and v(i) is the payoff
to woman i such that assortative matching is stable. It is therefore enough to
verify that given that a woman j gets v(j), the payoff maximizing match of man
i is the woman i, that is that i ∈ arg max ij − v(j), and similarly that the best
match for the woman i is the man i, or i ∈ arg max ij − u(j). Because stability
implies that u and v are increasing, they are differentiable almost everywhere
1
and therefore for almost all i, we have v 0 (i) = u0 (i) = i : the marginal payoffs are
equal to the marginal productivities. It follows that by choosing an arbitrary
feasible payoff u(0) ∈ [0, 1] for the man 0, we have for each i,
Z
u(i) = u(0) +
i
xdx
(1)
1
2
= u(0) +
i
1
− .
2
2
and the women have payoff
v(i) =
i2
1
+ − u(0).
2
2
Note that these computations are facilitated because total production is independent of how it is allocated. This is no longer the case when there are
nontransferabilities, that is when the frontier is not linear with slope −1. In
Legros and Newman (2006), we consider situations where the frontier of the
feasible set of payoffs that two agents can achieve is a strictly decreasing frontier on the positive orthan. This is typically the case when risk averse agents
share risk in partnerships, whether or not incentive problems are present. In
an economy with finitely many agents we provide a condition (Generalized Increasing Differences or GID) that insures that all equilibria of the economy are
payoff equivalent to an equilibrium with assortative matching.
The purpose of this note is to show that this result extends to the continuum
economy. The extension is non trivial because the argument for the proof is different in the continuum and in the finite case. This extension is useful because,
as for the transferable case, the derivation of equilibrium payoffs is facilitated
when it is known that matching is assortative. We show that the traditional
result that agents are paid at their marginal productivity continues to hold,
which facilitates the search for equilibrium payoffs. However, while in the case
of transferability the marginal productivity is well defined, it is in the general
case a nontrivial function of the payoff of the partner. Hence while each partner
gets paid at the margin at their marginal productivity, one cannot use directly
a construction like in (1) but rather we need to solve a system of differential
equations to find the equilibrium payoffs. We illustrate this in section 4.
2
Preliminaries
We consider a two-sided market with a continuum of agents on each side of the
market. On one side of the market, we have principals indexed by i ∈ [0, 1] ; on
the other side of the market, we have agents indexed by j ∈ [0, 1] , where [0, 1]
is equipped with Lebesgue measure λ.
Principals have types in a compact interval P and agents in a compact
interval A and a type assignment specifies the type pi ∈ P of the i-th principal ,
2
and specifies a type aj of the j-th agent . We assume that types are taken from
a compact interval and that the orders on the indexes are chosen in such a way
that pi and aj are non-decreasing in i ∈ [0, 1] . We will abuse notation and call
alternatively pi the principal i and his type, and aj the agent j and his type.
We will also write pi < pj and ai < aj when i < j.
Since pi and aj are bounded and monotone, their set of discontinuity points
has measure zero. In most applications, pi and aj are either continuous or are
simple functions.
Feasible payoffs are described by a frontier φ (p, a, v) specifying the maximum
payoff to type p principal matched to a type-a agent who requires a payoff of v.
The maximum payoff to this principal is then φ (p, a, 0) . The “inverse” function
ψ (a, p, u) satisfies φ (p, a, ψ (a, p, u)) = u when u ≤ φ (p, a, 0) . We assume that
φ is continuous, strictly decreasing in v on [0, ψ (a, p, 0)], and that φ (p, a, 0) > 0
for all (p, a) ∈ P × A.
2
A pair (u, v) ∈ R2+ is feasible for (i, j) ∈ [0, 1] when u ≤ φ (pi , aj , v) and
v ≤ ψ (aj , pi , 0) .
Definition 1 Payoffs (u, v) are “on the frontier” for (i, j) when u = φ (pi , aj , v)
and v ≤ ψ (aj , pi , 0) . We write in this case (u, v) ∈ Φ (i, j) .
A match is a 1-1 measurable map m : [0, 1] → [0, 1] satisfying measure
consistency: for every open interval K ⊆ I, λ (K) = λ (m (K)) .1
Definition 2 Let I ⊆ [0, 1] be a set of full measure. An I-equilibrium is a triple
(m, π, ω) , where m : I → I is the match function, π : I → R and ω : I → R are
measurable payoff functions for principalsand agents, satisfying
(i) For all i ∈ I, πi , ωm(i) ∈ Φ pi , am(i)
(ii) For all (i, j) ∈ I 2 , πi ≥ φ (pi , aj , ωj ) .
(i) is the feasibility condition, (ii) is the stability condition.2 . Existence of
an I-equilibrium for some set I ⊆ [0, 1] of full measure is implied by KanekoWooders (1996). Note however that with GID one can prove existence by construction (as in section 4).3
Definition 3 An I-equilibrium (m, π, ω) satisfies positive assortative matching
(PAM) if for all i ∈ I, , i0 ∈ I, i0 < i implies that m (i0 ) ≤ m (i) .
Definition 4 An economy satisfies PAM if whenever (m, π, ω) is an I-equilibrium
for a full measure set I ⊆ [0, 1] , there exists a full measure set I 0 ⊆ I and an
I 0 -equilibrium (m∗ , π, ω) satisfying PAM.
1 This avoids matches of the type m (i) = i2 since for i = 1/2, λ (i) = 1/2 > λ (m (i)) = 1/4
which means that half the men are allocated to a fourth of the women.
2 Feasibility would require only that π (p ) ≤ φ (p , a , ω (a )) and ω (a ) ≤ ψ (a , p , 0) .
i
i j
j
j
j i
However if π (pi ) < φ (pi , aj , ω (aj )) , there would exist ε > 0 such that π (pi ) + ε ≤
φ (pi , aj , ω (aj ) + ε) , contradicting the stability condition. Hence there is no loss of generality in assuming that feasible payoffs for equilibrium matches are on the the frontier.
3 They establish nonemptiness of the f -core, where this allows for the possible existence of
a null set of individuals who receive infeasible payoffs. The restriction of this equilibrium to
the set of individuals with feasible payoffs is an equilibrium in our sense.
3
Note that any “a.e. identity” map, m(i) = i almost everywhere, satisfies
PAM. In general, the exceptional set will be finite, and for continuous type
assignments, the PAM condition can hold everywhere.
When there are finitely many agents, Legros and Newman (2002) show that
the following condition is necessary and sufficient for an economy to satisfy
PAM.
Definition 5 φ satisfies GID if for all i0 < i, j 0 < j, u ≤ φ (pi0 , aj , 0) ,
φ (pi , aj , ψ (aj , pi0 , u)) ≥ φ (pi , aj 0 , ψ (aj 0 , pi0 , u)) .
3
GID implies PAM
To show that the economy satisfies PAM, it is necessary to show that all equilibria can be made payoff equivalent to an equilibrium satisfying PAM almost
everywhere. This is stated below.
Proposition 6 If φ (p, a, v) satisfies GID, the economy satisfies PAM.
When the GID condition is satisfied strictly, Proposition 6 is immediate: if
there exist i > i0 with i, i0 ∈ I such that m (pi ) < m(pi0 ). Then,
φ pi , am(i0 ) , ω (m(i0 )) = φ pi , am(i0 ) , ψ am(i0 ) , pi0 , πi0
> φ pi , am(i) , ψ am(i) , pi0 , πi0
≥ φ pi , am(i) , ωm(i) ,
the strict inequality is GID, the
weak inequality follows payoff monotonicity
and ωm(pi ) ≥ ψ am(i) , pi0 , πi0 . But then, there exists v > ωm(i0 ) such that
φ pi , am(i0 ) , v > π (i) , contradicting stability.
When the GID condition is satisfied weakly, the proof of Proposition 6 is
provided below.
We first invoke a key lemma: if i > i0 , j > j 0 ,and (pi , aj 0 ) , (pi0 , aj ) are
such that equilibrium payoffs are on the frontier of these matches, then the
equilibrium payoffs are also on the frontiers of (pi , aj ) , (pi0 , aj 0 ).
Lemma 7 Suppose that GID holds. Let (m, π, ω) be an equilibrium. If i > i0
and j > j 0 , are such that (πi , ωj 0 ) ∈ Φ (i, j 0 ) and (πi0 , ωj ) ∈ Φ (i0 , j) , then
(πi , ωj ) ∈ Φ (i, j) and (πi0 , ωj 0 ) ∈ Φ (i0 , j 0 ) .
Proof. Legros and Newman (2006) prove this lemma when j = m (i0 ) and
j 0 = m (i) ; their argument generalizes straightforwardly to j, j 0 as in the Lemma.
We next show that at continuity points i of the type assignments, it is always
feasible for pairs (pi , ai ) to match and generate their equilibrium payoffs.
Lemma 8 Suppose GID and consider an equilibrium (m, π, ω) . If a or p is
continuous at i, then (πi , ωi ) ∈ Φ (i, i) .
4
Proof. Consider an I-equilibrium and suppose that there exists i ∈ I, i ∈
/
{0, 1}, such that m (i) < i (the case m (i) > i is similar and is omitted). Since
(πi , ωi ) ∈
/ Φ (i, i) , we must have either ωi < ψ (ai , pi , πi ) or ωi > ψ (ai , pi , πi ) .
By stability, the first inequality is not possible since it would imply that there
is a beneficial deviation by (i, i) . Therefore for all i0 < i
ωi > ψ (ai , pi , πi )
= ψ ai , pi , φ pi , am(i) , ωm(i)
(2)
≥ ψ ai , pi0 , φ pi0 , am(i) , ωm(i)
≥ ψ (ai , pi0 , πi0 ) .
(3)
The equality is by feasibility of payoffs in equilibrium, the next inequality is by
GID applied to i0 ≤ i, the last inequality is by payoff monotonicity and the fact
that by stability, πi0 ≥ φ pi0 , am(i) , ωm(i) . Hence, for all i0 ≤ i,
ωi > ψ (ai , pi0 , πi0 ) .
(4)
This implies that the match of ai is pk , where k > i. Suppose
that there exists
i0 < i such that m (i0 ) > i, then (pk , ai ) and pi0 , am(i0 ) are in a NAM pattern
and by Lemma 7, we must have (πi0 , ωi ) ∈ Φ (i0 , i) which contradicts (4). Hence,
for all i0 < i, m (i0 ) < i.
But then, by measure consistency, principals in the interval (p0 , pi ) have their
match in the interval (a0 , ai ). Let m (i) = j, where aj < ai by assumption.
If ai is a continuity point of the agent type assignment, choose a sequence
T = {jn } ⊂ (j, i) converging to i; by continuity, ajn → ai . Lemma 7 implies
that for all jn ,
πi = φ (pi , ajn , ωjn ) , for all jn ∈ T.
(5)
Since ψ is continuous, ωjn = ψ(ajn , pi , πi ) → ψ(ai , pi , πi ) ≡ ω̂ < ωi , where the
inequality follows (4).
Remember that the match of ai is pk with k > i. Then,
πk = φ (pk , ai , ωi )
< φ(pk , ai , ω̂)
where the strict inequality follows (2) and (πk , ωi ) ∈ Φ (k, i) . Since ajn → ai
and ωjn → ω̂ there exists jn ∈ T such that πk < φ (pk , ajn , ωjn ) ,implying that
(pk , ajn ) can profitably deviate. This contradicts stability of the (pk , ai ) match,
and we therefore must have (pi , ai ) ∈ Φ (i, i) , as claimed.
The case in which i is a point of continuity of p proceeds similarly. By measure
consistency, for almost all pl ∈ (pi , pk ) , m (l) > i. Choose a sequence S = {in } ⊂
(i, k) converging to i. Then pin → pi , and since (pin , am(in ) ) and (pk , ai ) are in a
NAM pattern, (πin , ωi ) ∈ Φ(in , i) so that πin = φ(pin , ai , ωi ) → φ (pi , ai , ωi ) ≡
π̂ < πi , where the inequality is by 2. Then for j = m(i), ωj = ψ (aj , pi , πi ) <
ψ (aj , pi , π̂) , and thus for n sufficiently large, ωj < ψ (aj , pin , πin ) , contradicting
stability of (pi , aj ). Thus, (pi , ai ) ∈ Φ (i, i) if i is a continuity point of p.
5
Suppose that GID holds and that (m, π, ω) is an I-equilibrium. Let Ep be
the set of points of discontinuity of the type assignment p and Ea be the set of
points of discontinuity of the type assignment a. Let E = Ep ∩Ea . and I 0 = I\E;
then I 0 is of full measure. By Lemma 8 for all i ∈ I 0 , (πi , ωi ) ∈ Φ (i, i) . Defining
m∗ (i) = i, i ∈ I 0 , we have that (m∗ , π, ω) is an I 0 -equilibrium.
4
Computing Equilibrium Payoffs
Since under GID, all equilibria are essentially equivalent to one with PAM, we
can restrict attention to matches where the man i is matched with the woman
i. This can be part of an equilibrium if there exist payoff maps u and v such
that the following conditions are satisfied almost everywhere:
u(i) = φ (pi , ai , v(i))
u(i) = max φ (pi , aj , v(j))
(6)
(7)
v(i) = max ψ (ai , pj , u (j))
(8)
j
j
(6) is feasibility: in a match payoffs must be on the frontier; (7) and (8) are the
incentive compatibility conditions: a man i must prefer to match with woman
i given that women ask payoffs v. Note the similarity between the incentive
conditions and those obtained when we consider screening problems. Like in
this literature, the search for equilibrium payoffs is facilitated by looking at local
conditions, something that is acceptable as long as a single crossing condition is
satisfied: if the local incentive compatibility condition holds, then single crossing
implies that larger types have even less incentives to deviate downward, insuring
global stability. The GID condition is a single crossing condition. It can be
equivalently expressed as: 4
Definition 9 Let i0 < i, j 0 < j, and suppose that v and v 0 are such that
φ (pi0 , aj 0 , v 0 ) = φ (pi0 , aj , v) , then, φ (pi , aj , v) ≥ φ (pi , aj 0 , v 0 ) .
However a local incentive condition in the continuum is useful if it is known
that the payoff is differentiable in the type of the partner. Here since φ is differentiable, it is enough to show that the payoffs u and v must be differentiable. In
screening models, when payoffs are quasi-linear, incentive compatibility indeed
implies that payoffs are monotonically increasing in types.
In our case, we do not have quasi-linarity of payoffs (otherwise the frontier
would have slope −1!) but as long as φ (p, a, v) is increasing in p and ψ (a, p, v)
is increasing in a (higher types are ”more productive”), incentive compatibility
indeed implies that payoffs must be non-decreasing. For any j the incentive
compatibility of the man i requires u (i) ≥ φ (pi , aj , v (j)) ; since φ is increasing,
`
´
see this note that if v, v 0 satisfy φ pi0 , aj 0 , v 0 = φ (pi0 , aj , v) , then letting u be the
`
´
`
´
0
common value we have v = ψ aj , pj 0 , u and v = ψ aj 0 , pi0 , u , and substituting this into
the terms in the inequality gives the GID condition in Definition 5.
4 To
6
when j < i, we have φ (pi , aj , v (j)) ≥ φ (pj , aj , v (j)) , but the right hand side
is by (6) equal to u (j) proving that payoffs are non-decreasing. A similar
logic shows that v is increasing in i. Since increasing functions are differentiable
almost everywhere, the global incentive compatibility conditions can be replaced
almost everywhere by the first order conditions
dφ (pi , aj , v (j)) =0
dj
j=i
dψ (ai , pj , u (j)) = 0.
dj
j=i
Proposition 10 Suppose that the type assignments p and a are continuously
differentiable and that GID holds. Then there exist payoffs u, v that stabilize the
assortative matching in which man i is matched with woman i and these payoffs
solve the system
p0i ψ2
ψ3
0
a
φ2
v 0 (i) = a0i ψ1 = − i
φ3
u0 (i) = p0i φ1 = −
(9)
(10)
Proof. If the type assignment maps p and a are differentiable, the first order
conditions can be written
a0i φ2 + v 0 (i) φ3 = 0
p0i ψ2 + u0 (i)ψ3 = 0,
(11)
(12)
where for k = 2, 3, φk = φk (pi , ai , v (i)) and ψk = ψk (ai , pi , u (i)) . Hence
incentive compatibility requires solving a system of differentiable equations in
u, v. However, contrary to the screening problem there is an additional feasibility
constraint to take into account. Since (6) is an identity, we can differentiate with
respect to i and get
u0 (i) = p0i φ1 + a0i φ2 + v 0 (i) φ3 .
(13)
Hence, (11) and (13) imply that
u0 (i) = p0i φ1 .
Men i’s marginal payoff is indeed equal to his marginal product when he is
matched with woman i. Using the incentive compatibility condition of the
woman (12) we obtain (9); (10) is obtained in a similar fashion.
These equalities translate two effects present in models with non-transferability:
the usual type-type transferability in transferable utility cases and a type-payoff
transferability. Having type-type transferability is not enough for PAM if higher
types find it also more difficult to transfer surplus to their partner. Note that
the rate at which a man can transfer to a woman is given by the ratio p0 ψ2 /ψ3 ,
7
and incentive compatibility requires that this equates the marginal productivity
of the man.
Consider first a situation where there is perfect transferability, that is when
φ(p, a, v) = y(p, a) − v. Then, φ3 = ψ3 = −1 and φ2 = ψ1 and ψ2 = φ1 and the
second equality in conditions (9) and (10) are trivial; we have u0 (i) = p0i y1 (pi , ai )
and v 0 (i) = a0i y2 (pi , ai ) .
With non-transferabilities φ2 is not equal to ψ1 in general, and the second
equality in (9) and (10) is not redundant. As an illustration of these conditions,
√
consider φ (p, a, v) = pa − v, then ψ (a, p, u) = pa − u2 . This frontier satisfies
GID.5
√
a
Note that φ1 = 2√pa−v
, ψ2 = a, ψ3 = −2u; since u = pa − v, we have
0
indeed p0 φ1 = − pψψ32 . The same is true in (10) and therefore we need to solve
the system
ai
u0 (i) = p0i p
2 pi ai − v(i)
v 0 (i) = a0i pi
u(i)2 + v (i) = pi ai .
Suppose for instance that pi = ai = i for all i ∈ [0, 1]. Then, the conditions are
i
u0 (i) = p
2 i2 − v(i)
v 0 (i) = i
2
u (i) + v (i) = i2 .
At i = 0, the unique feasible payoff consistent with individual rationality is
Ri
2
u (0) = v(0) = 0. Then v (ı́) = i implies that v (i) = 0 xdx = i2 . The feasibility
= √12 .
condition implies that u (i) = √i2 . We verify that u0 (i) = √ 2i
2
i −v(i)
We represent below a typical frontier for a match {i, i} and the equilbrium
payoffs of men and women.
If one is interested in total surplus maximization, the payoffs should be
u(i) = i and v(i) = 0 when i < 1/2 since the slope of the frontier at v = 0
is smaller than −1. This obviously illustrates the fact that non-transferability
implies that surplus division and surplus maximization are conflicting with each
other. Since the payoff to men is linear while the payoff to women is quadratic
in type, women ”catch up” with men when i is large enough;
moreover, while
√
the degree of inequality is an inverted U for i less than 2/ 2, it is increasing for
larger values of i : women not only catch up but leave men behind. Obviously
these results are specific to the example.
5 In fact this economy admits a TU representation: use the change of variable F (u) = u2
and consider the new frontier φ∗ (p, a, v) = φ (p, a, v)2 . Then F (u) = pa − v and there is
transferability; clearly φ∗ satisfied GID. Legros and Newman (2006) show that in this case φ
also satisfies GID.
8
u
u, v
v(i)
i
u(i)
−
1
2i
i
i2
2
2
v
Frontier for a match {i,i}
Equilibrium payoffs
Figure 1: Frontier and Equilibrium Payoffs when φ (pi , ai , v) =
9
√
i2 − v.
References
[1] Becker, G. (1973), “A Theory of Marriage, Part I, Journal of Political Economy, 81: 813-846.”
[2] Kaneko, M., and M. Wooders (1996), “The Nonemptiness of the f -Core of
a Game Without Side Payments,” International Journal of Game Theory,
25: 245-258.
[3] Legros, P. and A.F. Newman (2006), “Beauty is a Beast, Frog is a Prince: Assortative Matching with Nontransferabilities,” forthcoming, Econometrica.
10