We show that in a search/matching model with endogenous participation in which workers are heterogeneous with respect to market productivity, satisfying the Hosios rule leads to excessive vacancy creation. The reason is that the marginal worker does not internalize the effect of his or her participation on average productivity.search, matching, participation, Hosios rule, efficiency

James Albrecht

Lucas Navarro

Susan Vroman

English

Research Papers in Economics

E¢ ciency in a Search and Matching Model with
Endogenous Participation￿
James Albrechty Lucas Navarroz Susan Vromanx
February 2009
Abstract
We show that in a search/matching model with endogenous par-
ticipation in which workers are heterogeneous with respect to market
productivity, satisfying the Hosios rule leads to excessive vacancy cre-
ation. The reason is that the marginal worker does not internalize the
e⁄ect of his or her participation on average productivity.
In the standard Pissarides (2000) search/matching model, equilibrium is
e¢ cient when wages are determined by Nash bargaining if the worker share
of the net surplus of the match equals the elasticity of the matching function
with respect to unemployment, i.e., if the Hosios (1990) rule is satis￿ed. As
discussed in Pissarides (2000, Chapter 8), this result holds more generally. In
particular, the Hosios rule implements the e¢ cient outcome when workers
are heterogenous with respect to their outside options, e.g., their leisure
values, and labor force participation is endogenous.
In this note, we also allow for endogenous labor force participation, but
assume that workers are heterogeneous with respect to market productiv-
ity1 rather than with respect to leisure values. In this setting, the Hosios
￿JEL classi￿cation: D3, J6. Keywords: search, matching, participation, Hosios rule,
e¢ ciency.
yCorresponding author: Department of Economics, Georgetown University, Washing-
ton, DC 20057, USA, and IZA, Bonn, Germany, albrecht@georgetown.edu
zDepartment of Economics, ILADES-Universidad Alberto Hurtado, Erasmo Escala
1835, Santiago, Chile, lunavarr@uahurtado.cl
xDepartment of Economics, Georgetown University, Washington, DC 20057, USA, and
IZA, Bonn, Germany, vromans@georgetown.edu
1A similar issue arises in the Albrecht, Navarro and Vroman (2009) model of developing
economy labor markets with an informal sector in which workers have di⁄erent formal-
sector productivities.
1condition fails because the participation decision a⁄ects not only labor mar-
ket tightness but also the average productivity of matches. This average
productivity e⁄ect is not present in the Pissarides version of the model with
endogenous labor force participation. In our model, an increase in participa-
tion causes average match productivity to fall, but the marginal participant
does not internalize this e⁄ect. As a result, when wages are determined by
the Hosios rule, the labor force participation rate is too high. Equivalently,
there is excessive vacancy creation.
To make our point as simply as possible, consider a one-period version
of the model with a continuum of workers of measure one. Each worker
chooses between searching for a job (participating) and engaging in home
production (not participating). A nonparticipant receives z with certainty,
but a participant￿ s expected payo⁄ depends on his or her type and on labor
market tightness. Productivity in market work is distributed across workers
according to a continuous distribution function F(y); 0 ￿ y ￿ 1 and F(0) =
0: All participants search and ￿nd a job with probability m(￿); where ￿ is
market tightness.2 A worker of type y who ￿nds a job gets a fraction ￿ of
the output that he or she produces; a worker who participates but fails to
￿nd a job gets a payo⁄ that is normalized to zero. The expected payo⁄ of a
participant of type y is thus m(￿)￿y: A worker participates i⁄ m(￿)￿y ￿ z,
so there is a cuto⁄ value of y;
y￿ =
z
￿m(￿)
(1)
such that all workers with y ￿ y￿ participate; the remaining workers are
nonparticipants. Call this function y￿ = h(￿); and note that
h0(￿) = ￿(
m0(￿)
m(￿)
)y￿ < 0:
The participation rate is 1 ￿ F(y￿); so labor market tightness is
￿ =
v
1 ￿ F(y￿)
=
v
1 ￿ F(h(￿))
;
2As in Pissarides (2000), the matching function M(v;u) is assumed to have constant
returns to scale so it can be written as m(￿)u: We assume that m(￿) is independent of
y; i.e., all participants have an equal chance of ￿nding a job. Our model can be thought
of as one in which employers search sequentially for candidates, e.g., an employer hires
the ￿rst worker who applies for the job. Villena-RoldÆn (2008) considers a model of
nonsequential employer search in which a worker￿ s chance of getting a job depends on his
or her productivity.
2where v is the measure of vacancy creation. Di⁄erentiating implicitly with
respect to v,
d￿
dv
=
@￿
@v
+
@￿
@y￿h0(￿)
d￿
dv
; (2)
where
@￿
@v
=
1
1 ￿ F(y￿)
@￿
@y￿ =
￿f(y￿)
1 ￿ F(y￿)
:
Note that d￿=dv > 0:
Equilibrium requires that workers optimally choose whether to partici-
pate or not and that vacancies are created until the value of a vacancy equals
zero. The value of a vacancy is
V = ￿c +
m(￿)
￿
(1 ￿ ￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy;
where c is the cost to open the vacancy, m(￿)=￿ is the probability that the
vacancy hires a worker, and y￿ is determined by worker choice. Setting this
value to zero gives
m(￿)
￿
(1 ￿ ￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy = c: (3)
Equations (1) and (3) can be solved for the equilibrium values of ￿ and y￿:
Next we consider the e¢ ciency problem. The social planner chooses v
to maximize
￿ = zF(y￿) + (1 ￿ F(y￿))m(￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy ￿ cv
= zF(y￿) + m(￿)
Z 1
y￿
yf(y)dy ￿ cv;
taking into account that workers choose whether or not to participate. The
￿rst term on the RHS is the value for nonparticipants, the second term is the
value of market output, and the third term subtracts the costs of vacancy
creation. The derivative of the social planner￿ s objective with respect to v
is
d￿
dv
= (z ￿ m(￿)y￿)f(y￿)h0(￿)
d￿
dv
+ m0(￿)
Z 1
y￿
yf(y)dy(
d￿
dv
) ￿ c
= (z ￿ m(￿)y￿)f(y￿)h0(￿)
d￿
dv
+ m0(￿)
Z 1
y￿
yf(y)dy
￿
@￿
@v
+
@￿
@y￿h0(￿)
d￿
dv
￿
￿ c:
3Substituting for
@￿
@v
and
@￿
@y￿ and rearranging gives
d￿
dv
=
￿
z ￿ m(￿)y￿ + ￿m0(￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy
￿
f(y￿)h0(￿)
d￿
dv
+m0(￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy ￿ c: (4)
The Hosios rule sets the worker share of the net surplus equal to the elasticity
of the matching function with respect to unemployment.3 It can be written
as
1 ￿ ￿ =
￿m0(￿)
m(￿)
:
If this condition holds, then
m0(￿)(
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy) =
m(￿)
￿
(1 ￿ ￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy;
so from equation (3),
m0(￿)(
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy) = c;
and the ￿nal two terms in equation (4) cancel. That is,
d￿
dv
=
￿
z ￿ m(￿)y￿ + ￿m0(￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy
￿
f(y￿)h0(￿)
d￿
dv
: (5)
Note that we are evaluating d￿=dv at the equilibrium level of vacancy cre-
ation when ￿ equals the Hosios value. If d￿=dv > 0 at the ￿Hosios level￿
of vacancy creation, there is not enough vacancy creation in equilibrium; if
d￿=dv < 0; too many vacancies are being set up.4 Since f(y￿)h0(￿)(
d￿
dv
) < 0;
the sign of d￿=dv at the Hosios level of v is the opposite of that of the
term in parentheses in equation (5). The net output of the marginal par-
ticipant is m(￿)y￿ ￿ z: The marginal participant also reduces the average
productivity of matches in the market sector. This is re￿ ected in the term
￿m0(￿)
R 1
y￿
yf(y)
1 ￿ F(y￿)
dy: If workers were homogeneous with respect to market
3As discussed in footnote 2, the matching function is m(￿)u:
4When z = 0; the equilibrium level of vacancy creation is e¢ cient, i.e., d￿=dv = 0:
The reason is that when z = 0; all workers participate, i.e., y
￿ = 0; which in turn implies
h
0(￿) = 0:
4productivity, then this term would be independent of y￿; i.e., participation
would have no e⁄ect on average productivity.
To sign d￿=dv; note that
z ￿ m(￿)y￿ = ￿(1 ￿ ￿)m(￿)y￿
and, by the Hosios condition,
￿m0(￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy = (1 ￿ ￿)m(￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy:
We then have
d￿
dv
=
￿
￿(1 ￿ ￿)m(￿)y￿ + (1 ￿ ￿)m(￿)
Z 1
y￿
yf(y)
1 ￿ F(y￿)
dy
￿
f(y￿)h0(￿)(
d￿
dv
)
= (1 ￿ ￿)m(￿)
￿Z 1
y￿
(y ￿ y￿)f(y)
1 ￿ F(y￿)
dy
￿
f(y￿)h0(￿)(
d￿
dv
) < 0:
We have thus shown that when ￿ equals the Hosios value, too many vacancies
are created in equilibrium. Equivalently, y￿ is too low ￿the rate of labor
force participation is ine¢ ciently high.
References
[1] Albrecht, J., L. Navarro and S. Vroman, 2009, The E⁄ects of Labour
Market Policies in an Economy with an Informal Sector, Economic Jour-
nal, forthcoming.
[2] Hosios, A., 1990, On the E¢ ciency of Matching and Related Models of
Search and Unemployment, Review of Economic Studies, 57, 279-98.
[3] Pissarides, C. 2000, Equilibrium Unemployment Theory, 2nd edition,
MIT Press, Cambridge, Mass.
[4] Villena-RoldÆn, B., 2008, Aggregate Implications of Employer Search
and Recruiting Selection, mimeo, University of Rochester.
5

Efficiency in a Search and Matching Model with Endogenous Participation

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