Equilibrium sorting in a finite-horizon, two-sided matching market with heterogeneous agents is considered. It is shown that, if the match production function is additively separable in agent-types and if the division of match output is determined by the Nash bargaining solution, then an unraveling of the market obtains as the unique equilibrium in which all matches are formed in the first period.

Frederick Chen

English

Research Papers in Economics

Unraveling in a dynamic matching market with Nash
bargaining 
Frederick Chen
Department of Economics, Wake Forest University
Abstract
Equilibrium sorting in a finite−horizon, two−sided matching market with heterogeneous
agents is considered. It is shown that, if the match production function is additively separable
in agent−types and if the division of match output is determined by the Nash bargaining
solution, then an unraveling of the market obtains as the unique equilibrium in which all
matches are formed in the first period.
Citation: Chen, Frederick, (2003) "Unraveling in a dynamic matching market with Nash bargaining." Economics Bulletin, Vol.
3, No. 17 pp. 1−6
Submitted: July 14, 2003.  Accepted: July 28, 2003.
URL: http://www.economicsbulletin.com/2003/volume3/EB−03C70017A.pdf1. Introduction
Since the pioneering work of Becker (1973) on the subject, the study of
equilibrium sorting in two-sided matching markets with heterogeneous agents
has garnered much attention among economists. In particular, a primary focus
of recent research on the topic has been the analysis of equilibrium matching
patterns in the presence of search frictions. It has been shown that, in station-
ary environments with non-transferable utility, equilibria in these markets are
characterized by segregationist sorting, whereby agents are partitioned by types
into classes, and matching occurs only between agents from the same class (Mc-
Namara and Collins, 1990; Burdett and Coles, 1997; Bloch and Ryder, 2000;
Smith, 2002). Smith (1995) and Damiano et al (2002) have shown that such
segregation by types also obtains in ﬁnite-horizon matching markets with no
entry of new agents and non-transferable utility.
Here, equilibrium sorting in a ﬁnite-horizon market with side-payments is
considered.1 In particular, it is assumed that the division of match produc-
tion in a partnership is determined by the symmetric Nash bargaining solution.
It is shown that, if the match production function is additively separable in
agent-types, then an unraveling of the market results as the unique equilib-
rium outcome in which sorting by types no longer obtains and all matches are
consummated in the ﬁrst period.
2. The Result
The matching market is two-sided, and agents on each side of the market are
diﬀerentiated by a one-dimensional trait x b e l o n g i n gt ot h et y p e - s p a c e[a,b] ⊂
(0,+∞). The distribution of agent-types on either side of the market is given
by a twice continuously diﬀerentiable function F (·) with density f (·). Without
loss of generality, the measure of agents on each side is normalized to be 1.
The market is open for two time periods, with no entry of additional partic-
ipants in the second period. Matchings are one-to-one, voluntary transactions
between agents from each side, and agents exit the market immediately follow-
ing a match. In both periods, unmatched agents on either side of the market
randomly encounter those on the other side, and these meetings are generated
by a constant returns-to scale search technology.
The total output produced from a match between an agent of type x and
at y p e - y partner is given by x + y. Note that, by suitably redeﬁning the type-
space, other additively separable speciﬁc a t i o n si nw h i c hm a t c hp r o d u c t i o nt a k e s
the form h(x)+h(y),w h e r et h ef u n c t i o nh(·) is continuously diﬀerentiable
with h(x) > 0 and h0 (x) > 0 ∀x ∈ [a,b], can be reduced to this particular
speciﬁcation. As in Damiano et al (2002), it can be shown that, in equilibrium,
only agents of suﬃciently high types match in the ﬁrst period if match utility is
1A characterization of equilibrium sorting in inﬁnite-horizon matching markets with Nash
bargaining is given in Shimer and Smith (2000).
1non-transferable. In contrasts, it is assumed here that the division of output in a
match is determined by the (symmetric) Nash bargaining solution, with agents’
continuation payoﬀs serving as the “threat points”. A payoﬀ of 0 is received if
an agent is unmatched. All agents are risk-neutral and have a common discount
factor of 1.
In order to analyze the equilibrium outcome of this market, ﬁrst consider
agents’ behavior in the second period. Since the value of remaining unmatched
is 0, all agents in the last period will choose to match. Therefore, applying the
Nash bargaining solution, the expected payoﬀ to a type-x agent in the second
period is 1
2 (x + µ),w h e r eµ ∈ [a,b] i st h em e a na g e n t - t y p ei nt h a tp e r i o d .
Now, proceed to the ﬁrst period, and consider the decision problem faced
by a type-x agent who meets a type-y agent from the other side of the market.
The continuation payoﬀs to agents x and y of not matching with each other are
1
2 (x + µ) and 1
2 (y + µ), respectively. Therefore, the total surplus generated by
a match between them is
S (x,y;µ)=x + y −
1
2
(x + µ) −
1
2
(y + µ)=
1
2
(x + y) − µ.
Since the symmetric Nash bargaining solution allocates match surplus equally
between the two agents, the values of matching are 1
2 (x + µ)+1
2S (x,y;µ) and
1
2 (y + µ)+1
2S (x,y;µ) for agents x and y, respectively. Consequently, match
formation is optimal if the match surplus S (x,y;µ) is non-negative2, i.e.x+y ≥
2µ.L e t t i n g M (x;µ) denote the set of y ∈ [a,b] for which S (x,y;µ) ≥ 0,t h e
optimal matching strategy in the ﬁrst period as a function of agent-type and µ
is given as follows:
(i) if µ ∈
£
a, 1
2 (a + b)
¤
,t h e n
M (x;µ)=
½
[2µ − x,b], if x ∈ [a,2µ − a)
[a,b], if x ∈ [2µ − a,b] ;( 1 )
(ii) if µ ∈
¡1
2 (a + b),b
¤
,t h e n
M (x;µ)=
½
∅, if x ∈ [a,2µ − b)
[2µ − x,b], if x ∈ [2µ − b,b] .( 2 )
Thus, for ﬁxed µ, the optimal strategy in period 1 is a reservation rule whereby
m a t c h i n go b t a i n sa sl o n ga sp a r t n e r - t y p ei sa b o v eat h r e s h o l d .M o r e o v e r ,t h i s
lower bound for acceptable partner-type is decreasing in agent-type, so that
agents of lower types are more selective in terms of whom they choose to match
with. Note in particular that M (x;a)=[ a,b] ∀x ∈ [a,b].
Using the above characterization of ﬁrst period action, the expected agent-
type in period 2, µ, can be derived. Taking µ ∈ [a,b] as given for now, (1) and
2Since the distribution of agent-type is atomless, it can be assumed without loss of gener-
ality that agents choose to match in the case of indiﬀerence.
2(2) imply that the measure of (unmatched) agents in the second round is given
by
R(µ)=
( R 2µ−a
a f (x)F (2µ − x)dx, if µ ∈
£
a, 1
2 (a + b)
¤
F (2µ − b)+
R b
2µ−b f (x)F (2µ − x)dx, if µ ∈
¡1
2 (a + b),b
¤ .( 3 )
Therefore, the distribution of agent-type in the second period as a function of
the given µ, G(·;µ),s a t i s ﬁes:
(i) if µ ∈
£
a, 1
2 (a + b)
¤
,t h e n
G(y;µ)R(µ)=
½ R y
a f (x)F (2µ − x)dx, if y ∈ [a,2µ − a)
R(µ), if y ∈ [2µ − a,b] ;( 4 )
(ii) if µ ∈
¡1
2 (a + b),b
¤
,t h e n
G(y;µ)R(µ)
=
½
F (y), if y ∈ [a,2µ − b)
F (2µ − b)+
R y
2µ−b f (x)F (2µ − x)dx, if y ∈ [2µ − b,b] .(5)
An equilibrium of the market is thus given by a value µ∗ ∈ [a,b] such that
µ∗ =
Z b
a
xdG(x;µ∗),
where Z b
a
xdG(x;a) ≡ lim
µ→a
Z b
a
xdG(x;µ).
The proposition below states that, for all distribution functions F (·),t h e r e
exists a unique equilibrium in which all agents are matched in the ﬁrst period.
Proposition A unique equilibrium exists, with µ∗ = a.
Proof. The proof proceeds as follows. It will be shown that, ﬁrstly, a =
limµ→a
R b
a xdG(x;µ), establishing the existence of the equilibrium µ∗ = a,a n d ,
secondly, µ>
R b
a xdG(x;µ) for all µ ∈ (a,b].
Now, for µ ∈
¡
a, 1
2 (a + b)
¤
, (3) and (4) give
Z b
a
xdG(x;µ)=
R 2µ−a
a xf (x)F (2µ − x)dx
R 2µ−a
a f (x)F (2µ − x)dx
.( 6 )
3L’Hôpital’s Rule then yields
lim
µ→a
Z b
a
xdG(x;µ)
=l i m
µ→a
R 2µ−a
a xf (x)F (2µ − x)dx
R 2µ−a
a f (x)F (2µ − x)dx
=l i m
µ→a
2
R 2µ−a
a xf (x)f (2µ − x)dx
2
R 2µ−a
a f (x)f (2µ − x)dx
=l i m
µ→a
4
h
(2µ − a)f (2µ − a)f (a)+
R 2µ−a
a xf (x)f0 (2µ − x)dx
i
4
h
f (2µ − a)f (a)+
R 2µ−a
a f (x)f0 (2µ − x)dx
i
= a.
To show that µ>
R b
a xdG(x;µ) ∀µ ∈ (a,b], consider the following two cases.
• Case 1: µ ∈
¡
a, 1
2 (a + b)
¤
In this case, (6) gives
µ −
Z b
a
xdG(x;µ)=µ −
R 2µ−a
a xf (x)F (2µ − x)dx
R 2µ−a
a f (x)F (2µ − x)dx
.
To establish that this term is strictly positive, it suﬃces to show that
Z 2µ−a
a
(µ − x)f (x)F (2µ − x)dx > 0.
Now,
Z 2µ−a
a
(µ − x)f (x)F (2µ − x)dx
=
Z µ
a
(µ − x)f (x)F (2µ − x)dx +
Z 2µ−a
µ
(µ − y)f (y)F (2µ − y)dy
=
Z µ
a
(µ − x)f (x)F (2µ − x)dx +
Z µ
a
(x − µ)f (2µ − x)F (x)dx
=
Z µ
a
(µ − x)
·
d(F (x)F (2µ − x))
dx
¸
dx
=
Z µ
a
F (x)F (2µ − x)dx > 0,
where the second equality follows from the change-of-variable x =2 µ−y,
and the last equality obtains using integration-by-parts. This is the desired
result.
4• Case 2: µ ∈
¡1
2 (a + b),b
¤
In this case, (3) and (5) give
µ−
Z b
a
xdG(x;µ)=µ−


R 2µ−b
a xf (x)dx +
R b
2µ−b xf (x)F (2µ − x)dx
R 2µ−b
a f (x)dx +
R b
2µ−b f (x)F (2µ − x)dx

,
which is strictly positive if it can be shown that
Z 2µ−b
a
(µ − x)f (x)dx +
Z b
2µ−b
(µ − x)f (x)F (2µ − x)dx > 0.
Now, using the same steps as in the previous case,
Z b
2µ−b
(µ − x)f (x)F (2µ − x)dx = −(b − µ)F (2µ − b)
+
Z b
µ
F (x)F (2µ − x)dx.
In addition,
Z 2µ−b
a
(µ − x)f (x)dx =( b − µ)F (2µ − b)+
Z 2µ−b
a
F (x)dx,
which follows from integration-by-parts. Therefore,
Z 2µ−b
a
(µ − x)f (x)dx +
Z b
2µ−b
(µ − x)f (x)F (2µ − x)dx
=
Z 2µ−b
a
F (x)dx +
Z b
µ
F (x)F (2µ − x)dx > 0.
If the market is open for any ﬁnite number of rounds, then it follows from the
stated result and backward induction that a unique equilibrium obtains in which
all agents are matched in the ﬁrst period. Therefore, equilibrium matching in
these markets is random, and all agents behave as if the market is open for a
single round.
3. Concluding Remarks
Although the focus here on the separable production function is admittedly
restrictive, this speciﬁcation nevertheless provides some insights into equilibrium
matching behavior in ﬁnite-horizon search environments. In particular, the
result given here demonstrates how allowing for bargaining can dramatically
alter the pattern of equilibrium sorting, and it shows that the introduction of
5side-payments leads to an unraveling of the market. Note that this unraveling
mechanism diﬀers from that analyzed in Li and Rosen (1998) and Li and Suen
(2000), both of which focus on production uncertainty and agents’ risk-aversion
as the driving force behind early contracting and unraveling in labor markets. In
contrast, the analysis here indicates that unraveling can obtain as an equilibrium
outcome even with risk-neutral agents when utility is transferable. Finding
general conditions on the match production function that lead to unraveling
of markets with ex ante heterogeneous agents is a task that can be tackled in
future research.
References
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Bloch, F., and H.Ryder (2000) “Two-Sided Search, Marriages, and Matchmak-
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Burdett, K., and M.Coles (1997) “Marriage and Class” Quarterly Journal of
Economics 112, 141-168.
Damiano, E., H.Li, and W.Suen (2002) “Unraveling of Dynamic Sorting” Work-
ing paper, University of Toronto.
Li, H., and S.Rosen (1998) “Unraveling in Matching Markets” American Eco-
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Li, H., and W.Suen (2000) “Risk-sharing, Sorting, and Early Contracting” Jour-
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McNamara, J., and E.Collins (1990) “The Job Search Problem as an Employer-
Candidate Game” Journal of Applied Probability 28, 815-827.
Shimer, R., and L.Smith (2000) “Assortative Matching and Search” Economet-
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Smith, L. (1995) “Cross-Sectional Dynamics in a Two-Sided Matching Model”
Working paper, Massachusetts Institute of Technology.
Smith, L. (2002) “The Marriage Model with Search Frictions” Working paper,
University of Michigan.
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Unraveling in a dynamic matching market with Nash bargaining

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