We study the theoretical properties that particular matching functions must satisfy to represent a frictional labor market within a general equilibrium random matching model. We analyze Cobb-Douglas, CES and other functional forms of the matching function. Our findings establish restrictions on the parameters of these matching functions to ensure that the equilibrium is interior. These restrictions provide both theoretical reasons to choose among several functional forms and model misspecification tests for empirical work.interior equilibrium, random matching, matching function, cobbdouglas

Ausias Ribo1 (Universitat de Barcelona)

Montserrat Vilalta-Bufi (Universitat de Barcelona)

English

Research Papers in Economics

DOCUMENTS DE TREBALL  
DE LA FACULTAT D’ECONOMIA I EMPRESA 
 
Col.lecció d’Economia 
E12/272 
 
 
 
Is the matching function Cobb-Douglas? 
 
 
 
Ausias  Ribó
1    Montserrat Vilalta-Bufí
2 
Universitat de Barcelona  Universitat de Barcelona 
 
 
 
 
 
 
 
 
 
1 Facultat d’Economia i Empresa and  CREB, Universitat de Barcelona.  
Avinguda Diagonal  690, 08034 Barcelona, Spain.  Email: ariboa@ub.edu 
 
 
2 Corresponding Author: Facultat d’Economia i Empresa and CREB, Universitat de  
Barcelona. Address: Avinguda Diagonal 690, 08034 Barcelona, Spain.  
Phone: (+34) 934020107.  Fax: (+34) 934039082.   E-mail: montsevilalta@ub.edu 
 
 Abstract
We study the theoretical properties that particular matching functions must
satisfy to represent a frictional labor market within a general equilibrium
random matching model. We analyze Cobb-Douglas, CES and other func-
tional forms of the matching function. Our ￿ndings establish restrictions on
the parameters of these matching functions to ensure that the equilibrium is
interior. These restrictions provide both theoretical reasons to choose among
several functional forms and model misspeci￿cation tests for empirical work.
Estudiem les propietats te￿riques que una funci￿ d￿ emparellament ha de
satisfer per tal de representar un mercat laboral amb friccions dins d￿ un model
d￿ equilibri general amb emparellament aleatori. Analitzem el cas Cobb-
Douglas, CES i altres formes funcionals per a la funci￿ d￿ emparellament.
Els nostres resultats estableixen restriccions sobre els par￿metres d￿ aquests
formes funcionals per assegurar que l￿ equilibri Øs interior. Aquestes restric-
cions aporten raons te￿riques per escollir entre diverses formes funcionals i
permeten dissenyar tests d￿ error d￿ especi￿caci￿ de model en els treballs em-
p￿rics.
Keywords: matching function, random matching, interior equilibrium, Cobb-
Douglas
JEL: J61 Introduction
The use of matching functions in economic models allows for the introduction
of market frictions in a tractable fashion. This has proven important when
studying unemployment and its relationship with other phenomena. The
extensive body of literature on matching models shows its many applications
(see Petrongolo and Pissarides, 2001 and Rogerson et al., 2005 for surveys).
While labor market frictions might account for the coexistence of unem-
ployment and empty vacancies observed in reality, not all matching functions
are consistent with this result. For some speci￿cations, truncation is neces-
sary to avoid job ￿nding probabilities above unity, which would imply that
labor market frictions are not active. In this paper, we study theoretical
properties that particular matching functions must satisfy to be compatible
with labor market frictions. We analyze the Cobb-Douglas functional form
under di⁄erent returns to scale, and the constant elasticity of substitution
(CES) form to allow for di⁄erent degrees of substitution between vacancies
and unemployment. We also analyze the matching function that den Haan
et al. (HRW, 2000) propose to avoid corner equilibria in their numerical
exercises. Our results have implications for the empirical and theoretical
literature on matching.
The paper is organized as follows. Next section speci￿es the basic frame-
work of our analysis. Sections 3 to 5 show the analysis of a Cobb-Douglas,
two CES and the HRW matching functions, respectively. We discuss the
implications of our results in the concluding section.
2 Basic framework
Let us denote by M the number of matches in the economy, which is a
function of the number of job seekers (U) and the number of open vacancies
(V ), M = m(U;V ). We assume, as it is common in this literature, that the
matching function is increasing in its arguments and concave. In this set-up,
the average probability of ￿nding a job by a worker is the number of matches
per unemployed, p = m(U;V )=U, and the average probability of ￿lling in a
vacancy by a ￿rm is the number of matches per vacancy, q = m(U;V )=V .
We assume that the number of vacancies is endogenously determined.
This requires a positive cost of opening a vacancy and free entry in the
market of vacancies. The free-entry condition is the following:
qB = ￿; (1)
where q is the probability to ￿ll in a vacancy, B are the expected future pro￿ts
1of a ￿lled vacancy for the ￿rm and ￿ > 0 is the cost of posting a vacancy. This
condition applies to a large variety of random matching models, from those
with any type of separation rate to those with match-speci￿c productivity.
These models will di⁄er in their speci￿cation of B (Rogerson et al., 2005).
De￿nition 1 An interior equilibrium occurs when the number of matches
is bounded from above by the number of job seekers and the number of vacan-
cies in the economy (M < minfU;V g).
De￿nition 1 implies that the probability of ￿nding a job (p) and the
probability of ￿lling in a vacancy (q) are both below unity. Notice from
(1) that, in equilibrium, the probability of ￿lling in a vacancy is lower than
1 if and only if B > ￿. This result holds for any functional form of the
matching function. However, the matching function will be important in
determining whether the probability of ￿nding a job is lower than 1. In the
following sections, we study the conditions under which several matching
functions lead to an interior equilibrium. Throughout the paper we assume
that B > ￿ so that, in equilibrium, the condition q < 1 is automatically
satis￿ed. In the following sections it is only left to check whether p < 1 holds
in equilibrium.
3 Cobb-Douglas matching function
In this section, we consider the Cobb-Douglas functional form, which allows
us to investigate the parameter restrictions that lead to an interior equilib-
rium under di⁄erent returns to scale on the matching technology.
Proposition 1 In a general equilibrium random matching model with a Cobb-
Douglas matching function, m = AU￿V ￿, where A > 0 is a scale parameter,
￿ 2 (0;1) and ￿ 2 (0;1), an interior equilibrium exists if and only if B > ￿
and:
a) ￿ + ￿ = 1 and A < (￿=B)￿.
b) ￿ + ￿ > 1 and U < A1=(1￿￿￿￿)(B=￿)￿=(1￿￿￿￿).
c) ￿ + ￿ < 1 and U > A1=(1￿￿￿￿)(B=￿)￿=(1￿￿￿￿).
Proof. Let us de￿ne the matching function to be Cobb-Douglas, m =
AU￿V ￿, where A > 0, ￿ 2 (0;1) and ￿ 2 (0;1). An interior equilibrium
exists if M < U.1
1Recall that since we assume B > ￿ the condition M < V is satis￿ed in any equilibrium.
2To prove that M < U, we substitute q in (1) for its expression, m for
its assumed functional form and solve for V . Then we substitute V for this
expression in the condition M < U, where M is de￿ned by the Cobb-Douglas
matching function. Re-arranging we obtain the following inequality:
U
1￿￿￿￿ > A
￿
B
￿
￿￿
: (2)
When there are constant returns to scale (CRS), the left-hand side (LHS)
of equation (2) is equal to 1 (U0 = 1). Then the inequality is satis￿ed if and
only if A < (￿=B)￿.
In the case of increasing returns to scale (IRS), inequality (2) determines
an upper bound to unemployment (U < A1=(1￿￿￿￿)(B=￿)￿=(1￿￿￿￿)) and con-
sequently to the number of matches.
In the case of decreasing returns to scale (DRS), inequality (2) determines
a lower bound to unemployment, U > A1=(1￿￿￿￿)(B=￿)￿=(1￿￿￿￿).
Proposition 1 states the necessary conditions for any Cobb-Douglas match-
ing function to lead to an interior equilibrium. Results depend on the returns
to scale. In the case of CRS, a Cobb-Douglas matching function with scale
parameter (A) larger or equal to 1 cannot be a good representation of labor
market frictions, since it would imply that any job seeker ￿lls a vacancy with
probability one.2
In the case of IRS, proposition 1 speci￿es an upper bound to the number
of unemployed individuals, which at the same time, sets an upper bound
to the number of matches created. This represents an arbitrary restriction
on endogenous variables that can only be checked for a posteriori. When
the number of job seekers is above this threshold, increasing returns ensure
that everybody ￿nds a job. Note that this result holds for any population
size. The larger the population size (L), the lower the upper bound on
unemployment rate (U=L) that ensures an interior equilibrium. This result
raises questions on the adequacy of a Cobb-Douglas form for a matching
function with IRS.
The case of DRS also implies an a priori restriction on the value of equi-
librium unemployment. However, this is less problematic in the sense that
it does not bind the number of matches. Moreover, there is little empirical
evidence supporting this case.
2Recall that since B > ￿ and ￿ > 0, then (￿=B)￿ < 1.
34 CES matching function
Proposition 2 In a general equilibrium random matching model with a match-
ing function represented by a CES function m = A(￿U￿ + (1 ￿ ￿)V ￿)
1=￿,
where A > 0 is a scale parameter, ￿ 2 (0;1), ￿ < 1 and 1=(1 ￿ ￿) is the
elasticity of substitution between unemployed people and vacancies, an inte-
rior equilibrium exists if and only if B > ￿ and:
a) ￿ < 0 and (1 ￿ ￿)￿1=￿(￿=B) < A < ((￿=B)￿￿ + 1 ￿ ￿)
￿1=￿ (￿=B) or,
b) ￿ > 0 and A < ((￿=B)￿￿ + 1 ￿ ￿)
￿1=￿ (￿=B).
Proof. Let us de￿ne the matching function to be CES such that
m = A(￿U
￿ + (1 ￿ ￿)V
￿)
1
￿ ;
where A > 0, ￿ 2 (0;1) and ￿ < 1. To prove that M < U, we substitute
q in (1) for its expression, m for its assumed functional form and derive the
following equation:
￿
￿
U
V
￿￿
=
￿ ￿
AB
￿￿
￿ (1 ￿ ￿): (3)
Notice that for the market tightness (V=U) to be positive, the RHS of the
previous equation must be positive. Therefore, in any equilibrium (￿=AB)
￿ >
(1 ￿ ￿). This translates into A < (1 ￿ ￿)￿1=￿(￿=B) if ￿ > 0 and A >
(1 ￿ ￿)￿1=￿(￿=B) if ￿ < 0.
Using equation (3) we solve for V and use this expression to substitute V
in the condition U > M, where M is de￿ned by the CES matching function
above. We obtain the following inequality:
￿￿ ￿
AB
￿￿
￿ (1 ￿ ￿)
￿1=￿
>
￿
B
￿
1=￿: (4)
This inequality is satis￿ed if and only if A < ((￿=B)￿￿ + 1 ￿ ￿)
￿1=￿ (￿=B)
for any ￿ < 1. Putting all the conditions together, we ￿nd the result in
proposition 2.
Proposition 3 In a general equilibrium random matching model with a match-
ing function represented by the following CES function, m = (U￿ + V ￿)
1=￿,
where ￿ < 1 and 1=(1 ￿ ￿) is the elasticity of substitution between unem-
ployed people and vacancies, the equilibrium is interior if and only if B > ￿
and ￿ < 0.
4Proof. Let us de￿ne the matching function to be CES such that
m = (U
￿ + V
￿)
1
￿
where ￿ < 1. In order to prove that M < U, we substitute q in (1) for its
expression, m for its assumed functional form and solve for V . Rearranging
we obtain the following:
￿ ￿
B
￿￿
￿ 1 =
￿
U
V
￿￿
:
Notice that, given B > ￿, the market tightness (V=U) is positive if and only
if ￿ < 0. Therefore, we need ￿ < 0 in any equilibrium.
To continue the proof, we solve for V and substitute it in the condition
U > M, where M is de￿ned by the CES matching function above. We obtain
the following inequality:
1 >
￿
B
1
((￿=B)
￿ ￿ 1)
1=￿;
which, given B > ￿, is satis￿ed for any ￿ < 0.
Propositions 2 and 3 report the necessary conditions for two di⁄erent CES
matching functions to lead to an interior equilibrium. The CES function
speci￿ed in proposition 2 requires restrictions on the scale parameter, as in
the Cobb-Douglas case with CRS. In contrast, the only requirement for the
CES matching function studied in proposition 3 is to have an elasticity of
substitution between vacancies and unemployed below unity.
5 The HRW matching function
Den Haan et al. (2000) study the propagation of aggregate shocks in a
dynamic general equilibrium with labor frictions represented by the following
matching function:
m(U;V ) =
UV
(U￿ + V ￿)
1=￿:
They recognize that with a Cobb-Douglas matching function "truncation
is necessary to rule out matching probabilities greater than unity" (den Haan
et al. 2000, p. 485). In section 2, we showed under which conditions this is
true. In this section, we analyze the conditions under which the matching
function they propose leads to an interior equilibrium.
5Proposition 4 In a general equilibrium random matching model with a match-
ing function represented by m(U;V ) = UV=(U￿ + V ￿)
1=￿, the equilibrium is
interior if and only if B > ￿ and ￿ > 0.
Proof. Let us de￿ne the matching function to be such that
m(U;V ) =
UV
(U￿ + V ￿)
1=￿:
To prove that M < U, we substitute q in (1) for its expression, m for its
assumed functional form and solve for V . Then we substitute V for this ex-
pression in the condition M < U, where M is de￿ned by the HRW matching
function. Re-arranging we obtain the following inequality:
B > (B
￿ ￿ ￿
￿)
1=￿:
Given ￿ > 0, this inequality is only satis￿ed when ￿ > 0.
Proposition 4 states that the parameter in the HRW matching function
must be positive in order to obtain an interior equilibrium. Although den
Haan et al. (2000) do not explicitly state this condition in their paper, their
calibration is consistent with our results.
6 Conclusions
We show theoretical properties that particular matching functions must sat-
isfy in a general equilibrium random matching framework to allow for la-
bor market frictions in equilibrium. Our results show that a Cobb-Douglas
function with increasing returns to scale is not a good representation of the
matching process, since it implies an upper bound to the number of matches
independent of the population size. In contrast, we ￿nd that all the match-
ing functions analyzed with constant returns to scale (Cobb-Douglas, CES,
HRM) lead to an interior equilibrium as long as they satisfy some restrictions
on their parameters.
These results have several implications. First, most studies that esti-
mate aggregate matching functions use a Cobb-Douglas functional form (see
Petrongolo and Pissarides 2001 for a review). Our results provide tests to
check whether the assumed functional form is miss-speci￿ed in a model with
endogenous vacancies. Similarly, calibration exercises using a matching func-
tion can avoid truncation problems by taking into account our results. Gen-
erally, our results provide restrictions on Cobb-Douglas and other functional
forms that any aggregate matching function should satisfy.
6Acknowledgements
The authors wish to thank Lorenzo Burlon, Aleksander Kucel, Xavier Rau-
rich and seminar participants at University of Barcelona for helpful comments
and suggestions. Financial support from the Spanish Ministry of Educa-
tion through grant ECO2009-06953 and from the Government of Catalonia
through the Barcelona GSE Research Network and grant 2009SGR1051 is
gratefully acknowledged.
References
den Haan, W. J., Ramey G., Watson J., 2000. Job Destruction and Propa-
gation of Shocks. Am. Econ. Rev. 90(3), 482-498.
Petrongolo, B., Pissarides C. A., 2001. Looking into the Black Box: A
Survey of the Matching Function. J. Econ. Lit. 39(2), 390-431.
Rogerson, R., Shimer R., Wright R., 2005. Search-Theoretic Models of
the Labor Market: A Survey. J. Econ. Lit. 43(4) 959-988.
7

Is the matching function Cobb-Douglas?

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