Competitive Search Equilibrium
with Multidimensional Heterogeneity
and Two-Sided Ex-ante Investments
Belén Jerez
∗
†
April, 2017
Abstract
We analyze a competitive search environment where workers and rms make costly investments (e.g. in education and physical capital, respectively) before they enter the labor market.
The environment features transferable utility and symmetric information. A key novelty with
respect to existing work is that we allow for multidimensional heterogeneity on both sides of the
market. As in classical hedonic models, wages depend both on the job's and on the worker's
match-relevant characteristics. Yet the presence of search frictions implies that (unlike in those
models) markets do not clear. The hedonic wage function and the probabilities of nding and
lling dierent jobs are determined endogenously in a competitive search equilibrium. We show
that constrained ecient allocations can be determined as optimal solutions to a linear programming problem, whereas the wage function supporting these allocations and the associated
expected payos for workers and rms correspond to the solutions of the `dual' of that linear
program. We use this characterization to show that an equilibrium exists and is constrained
ecient under very general conditions. Jerez (2014) makes a similar point in the context of a
model where all the match-relevant characteristics of the traders are exogenous. Here we extend
the analysis to allow for two-sided ex-ante investments which are potentially multidimensional.
The fact that linear programming techniques are used for the structural estimation of frictionless
matching models suggests that our framework is potentially useful for empirical studies of labor
markets and other hedonic markets where search frictions are prevalent.
Journal of Economic Literature Classication Numbers: D50, D61, D83.
Key Words: search frictions; transferable utility; multidimensional two-sided heterogeneity;
ex-ante investments; competitive hedonic pricing; linear programming and duality theory.
∗
I thank an Associate Editor and two anonymous referees for very useful comments and suggestions. I have bene-
ted from conversations with Joe Ostroy, Raquel Carrasco, Antonia Díaz, Javier Fernández Blanco, Luisa Fuster and
Juan Pablo Rincón, and comments by participants at the Carlos III I.I.D. seminar. Financial support from Spanish
DGCYT (projects ECO2010-20614 and ECO2014-56676-C2-1-P) and the Bank of Spain is gratefully acknowledged.
Any errors are mine.
†
Departamento de
[email protected].
Economía,
Universidad
Carlos
III
de
Madrid,
28903
Getafe,
Spain.
E-mail:
1
Introduction
Competitive search models are known to deliver constrained ecient outcomes in a variety of settings.
12
In particular, they solve the familiar hold-up problem in several environments where
agents make productive investments before they search for trading opportunities.
3 This point has
been made by Acemoglu and Shimer (1999b) in a labor search model with transferable utility and
one-sided investments (in capital) by rms, and by Masters (2011) in a variant of that model where
workers also make investments (in education). While workers and rms are assumed homogeneous
in these models, Shi (2001) obtains the same result in a variant of Acemoglu and Shimer's model
where workers are heterogeneous in their skills. In all three papers the observability of the agents'
investments and of their match-relevant characteristics which rules out asymmetric information
problems is crucial for the eciency result. In this paper we extend the eciency and existence results in these papers to a general setting with transferable utility and symmetric information which
combines two-sided ex-ante investments (as in Masters 2011) with heterogeneity (as in Shi 2001). A
novelty with respect to Shi (2001) is that heterogeneity is two-sided and multidimensional, rather
than one-sided and one-dimensional; and a novelty with respect to all three papers is that we allow
investments to be multidimensional as well.
Our framework can be viewed as embedding the classical hedonic model of Rosen (1974) and
Lucas (1977) into the competitive search formulation. As noted by Lucas (1977), a distinguishing
feature of labor markets is that, just like the rms' payos depend on the match-relevant characteristics of the workers they hire, the workers' payos depend on the job's hedonic attributes. In
our setting workers dier in several exogenous characteristics, and may invest in a multidimensional
vector of productive attributes before they apply for a job. Firms can also make multidimensional
investments (e.g. in technology, equipment type, . . . ) prior to hiring their employees. In particular,
4
rm investments aect the value of employing dierent kinds of workers , and may also aect the
1
See Montgomery (1991), Moen (1997), Shimer (1996,2005), Acemoglu and Shimer (1999a, 1999b), Julien, Kennes,
and King (2000), Shi (2001), Eeckhout and Kircher (2010), Masters (2011), and Jerez (2014), among others.
2
By contrast, the classical Diamond-Mortensen-Pissarides random search and bargaining model features bilateral
monopoly power, so (in general) it does not lead to ecient outcomes. Neither do the price posting models introduced
by Burdett and Judd (1983) and Burdett and Mortensen (1998), which also feature imperfect competition.
See
Rogerson, Shimer, and Wright (2005) for a comprehensive survey of the labor search literature.
3
4
Again, this is unlike in random search and bargaining models; e.g. see Acemoglu (1996) and Masters (1998).
In certain jobs (involving routine or easy to automate tasks), investments in computerization make it easier to
replace workers with machines. This in turn increases the worker's returns from investing in attributes which are
complements (rather than substitutes) to the new capital (e.g. Deming 2015 nds evidence that socials skills have
been increasingly rewarded by the labor market in the past decades, specially when combined with cognitive skills).
1
hedonic attributes of the jobs rms create.
5 The hedonic wage function and the probabilities of
lling and nding dierent jobs are determined endogenously in a competitive search equilibrium.
The class of economies we study is large: we allow for general production, utility and matching
functions, general distributions of worker and rm types, and endogenous market participation.
While our presentation focuses on the labor market, the model applies to other hedonic markets
(like that for housing) where search frictions are prevalent.
There is a methodological dierence between this paper and the competitive search literature
cited above. Whereas the latter uses strategic (game-theoretic) models, here we adopt the Walrasian
(price-taking) approach in Jerez (2014).
In a nutshell, that paper shows that the Arrow-Debreu
equilibrium notion can be extended to environments with search frictions, essentially by replacing
market clearing with a trading technology that is not frictionless.
The key modeling choice is
to incorporate the uncertainty arising from rationing in the denition of a commodity. Prices of
commodities then depend not only on their physical characteristics, but also on the probability
that their trade is rationed. In equilibrium traders take prices as given, and have common rational
expectations about the rationing probabilities associated to each commodity. Prices then adjusts so
that the optimal decisions of the agents are consistent with the trading (i.e. matching) technology.
Jerez (2014) shows that this price-taking notion is a reduced form of the strategic notion used in the
literature. The point is made in the context of a related model where the traders' match-relevant
characteristics are exogenous. Here we extend the analysis to allow for investments by agents on
both sides of the market.
6
The strength of the Walrasian formulation is that it allows us to apply the powerful tools of general equilibrium theory to derive the welfare and existence theorems under very general conditions.
Specically, both here and in Jerez (2014), we adopt the linear programming approach used by
Grestky, Ostroy and Zame (1992, 1999) to study a competitive frictionless matching environment.
7
Using a similar line of argument to the one used by these authors (to study the decentralization
5
Firm investments can aect the tasks performed by workers (e.g. Deming (2015) argues that advances in infor-
mation and communication technology tend to increase job rotation and the degree of worker multitasking). They
can also aect working conditions, and even job location (e.g. computerization increasing allows working from home
in some occupations).
6
Another dierence is that, in Jerez (2014), the valuations of the buyers depend on the type of seller they trade
with (as dierent seller types oer dierent goods), but seller valuations do not depend on the buyer's characteristics.
7
These authors study an hedonic endowment economy with indivisible goods. Chiappori, McCann, and Nesheim
(2010) analyze a similar model with production (see also recent work by Dizdar (2015) and Lindenlaub (2015), among
others). Makowski and Ostroy (1996, 2003) analyze a frictionless economy with divisible goods.
2
of ecient allocations), we show that constrained ecient allocations solve a linear programming
problem, whereas the hedonic wage function supporting these allocations and associated expected
payos for workers and rms correspond to the solutions of the `dual' of that problem. Our existence
theorem follows from the existence of optimal solutions to the two linear programs.
Linear programming techniques are not only useful to derive the properties of equilibrium allocations in dierent settings. They are increasingly being used in empirical work. In particular, a
recent and growing literature uses these techniques for the identication and estimation of frictionless matching models with transferable utility (e.g. see Galichon and Salanié (2012), Dupuy and
Galichon (2014), and Fox (2010)). The fact that the linear program in this paper is a close relative
of the one analyzed in the frictionless matching literature suggests that a similar methodology could
be used to take competitive search models to the data at a high degree of disaggregation.
Sections 2 and 3 describe the environment and the general equilibrium model in a setting with
homogeneous rms (for the sake of clarity). Section 4 describes the linear program and its dual,
characterizes constrained ecient allocations via the complementary slackness theorem of linear programming, and derives the welfare and existence theorems. The model with two-sided heterogeneity,
which is just a twist of the simplied model, is presented in Section 5. Section 6 concludes.
2
The Environment
Consider a static environment with a measure
ξˆ > 0 of ex-ante homogeneous rms, and a continuum
of heterogeneous workers. Worker types are indexed by
by a Borel measure
ξS
on
S.
s ∈ S , and the worker population is described
Firms are risk neutral and worker preferences are quasilinear in a
divisible numeraire. Each rm has a single job opening, and workers can have at most one job.
Before they enter the labor market, workers can invest in a list of attributes
c(s, h) > 0.8
Similarly, rms can invest in
a ∈ A
Agents may not invest at all; these choices are denoted by
c(·, h0 ) = C(a0 ) = 0.
8
Here
S, H
and
A
C(a) > 0
at cost
h0 ∈ H
at cost
before they open a vacancy.
and
a0 ∈ A,
respectively, where
are arbitrary compact metric spaces. Also,
These may include a description of the worker's education (e.g.
h ∈ H
c(s, h)
and
C(a)
years and quality schooling, skills acquired
through education,· · · ), the cost of which may depend on exogenous characteristics such as social background or
innate talent. It may also include exogenous characteristics (in
s)
which aect the productivity in some jobs. For
instance, some jobs impose requirements on attributes such as vision and hearing, or a maximum and/or minimum
age or hight. In this case,
c(s, h)
is assumed arbitrarily large for types who lack these characteristics.
3
are continuous (meaning that the costs of acquiring similar attributes are akin for similar worker
types, and the costs of similar investments are similar for rms).
A rm which invests in
Again,
f
a
and hires a worker with attributes
h
produces
f (h, a)
units of output.
is continuous (so workers with similar attributes have similar productivity at a given job,
and rms making similar investments have similar production technologies). The worker's disutility
of labor depends on the job's hedonic attributes, which we assume are determined by
a,
and also
9 This disutility is denoted
(possibly) on the worker's type and/or endogenously acquired attributes.
by
v(s, h, a),
where
v
is continuous (and is normalized to zero for unemployed workers). We assume
that, for each investment
a∈A
s∈S
rms can make, there is a worker type
by that type that generate a positive surplus:
and a choice of
h∈H
f (h, a) − C(a) − c(s, h) − v(s, h, a) > 0.
The agents' ex-ante investments are assumed observable, so rms know the kind of labor they
hire and workers know the kind of jobs they accept. As in the Arrow-Debreu model, the payos of
the agents are private information, and all the relevant information is transmitted through prices.
Jobs created in matches where the worker's and/or the job's attributes dier are regarded as
dierentiated (as in Lucas (1977)).
10
These jobs will be created in dierent search markets.
Whether or not type-(h, a) jobs are created (some workers choose
h,
some rms choose
a,
11
and some
of these workers end up working for some of these rms) will be determined endogenously.
Firms seeking to ll type-(h, a) job and workers searching for these jobs meet bilaterally and at
12 The probability of nding
random. In general, this random meeting process may dier across jobs.
a type-(h, a) job is
m(h, a, θ),
where
θ ∈ <+
is the ratio of rms to workers (or tightness level) in
the corresponding market. The probability of lling a type-(h, a) job is
As is standard,
α(h, a, θ)
higher
9
θ,
m(h, a, θ)
is decreasing in
is increasing in
θ
with
θ
with
m(h, a, 0) = 0
limθ→0 α(h, a, θ) = 1
and
and
α(h, a, θ) = m(h, a, θ)/θ.13
limθ→∞ m(h, a, θ) = 1,
limθ→∞ α(h, a, θ) = 0.
and
Intuitively, the
the easier it is for workers to nd a job and the harder it is for rms to ll a job in a given
For instance, female workers with children tend to prefer jobs which are compatible with child rearing (e.g. see
Erosa, Fuster, Kambourov, and Rogerson 2016). Blue-collar male workers are usually reluctant to work in jobs which
have been traditionally regarded as feminine (see The Weaker Sex, The Economist, May 30th 2015).
10
Admittedly, for some low-skilled jobs, the worker's investments may not aect output:
to devise a simple variant of the model where these jobs are described by
11
12
a
f (h, a) = f (a).
It is easy
only.
This segmented market structure is standard in competitive search models.
E.g. the use of digital platforms (which allow to match workers with jobs more eciently) is more widespread in
some markets than in others.
13
Implicit is the assumption that the total number of trading meetings is determined by a matching function with
constant returns to scale, and that the Law of Large Numbers holds.
4
market. Both
3
m
α
and
are continuous, so the random meeting process is similar for similar jobs.
The general equilibrium model
A market is represented by a triple
the market tightness level
θ.
x = (h, a, θ) ∈ H × A × <+ , describing the job created (h, a) and
The value of
θ
determines the job nding and job lling probabilities,
and thus the degree of trading uncertainty that workers and rms face. In the spirit of ArrowDebreu theory, this uncertainty is included in the description of the commodities traded.
An allocation is an assignment of workers and rms across markets. Consistency requires that
h
only workers (rms) investing in
(resp.
a)
be assigned to
x = (h, a, θ)
(see below). On the other
hand, agents who do not participate in the labor market will be assigned to a ctitious market,
x0 .14
X
m and α to the set X ≡ (H × A × <+ ) ∪ {x0 } by setting m(x0 ) = α(x0 ) = 0.
We extend
x0 ).
is the set of all markets that can potentially be active (including
An allocation is formally described by a pair of Borel measures with compact support,
Mc+ (X) × Mc+ (S × X).
to a market
x
set of markets
X,
in
The measures
respectively.
Ω ⊆ X|x0
on
X
µSX
Ω).16
(i.e.,
and
15 Specically,
µS
D ⊆S
µB (Ω)
is the measure of rms assigned to a Borel
Ω).
who are assigned to
Finally, the aggregate labor supply in
µSX (Ω)
Ω
is
In turn,
Ω
is the measure
(the labor supply of these worker
µSX (Ω),
gives the total measure of workers assigned to
µS (D × Ω)
where
Ω).
µSX
is the marginal of
Since the supports of
describe the sets of markets where rms and workers participate, a market
whenever
(µB , µS ) ∈
describe the assignment of rms and workers
(the aggregate labor demand in
of workers with types in the set
types in
µB
Thus
x 6= x0
µB
µS
and
is active
x ∈ suppµB ∩ suppµSX .
Because an allocation must assign (almost) all the workers/rms who live in the economy to a
market
x ∈ X , (µB , µS )
must be equal to
14
We take
ξS ,
must satisfy two adding-up conditions. Namely, the marginal of
and
x0 = (h0 , a0 , θ0 )
µB (X)
must be equal to
for some arbitrary
θ 0 < 0.
µS
on
S
ξˆ.
So, by assumption, agents who do not participate make no
costly investments. Since we focus on constrained ecient allocations, this is without loss of generality. Note that a
market with
θ=0
is equivalent to
x0
from the worker's perspective if
So, we assume (without loss of generality) that markets with
15
θ=0
h = h0
(and is dominated by
x0
if
h 6= h0 ).
are inactive.
Y , Mc (Y ) denotes the space of signed regular Borel measures on Y with compact support
Y is compact, this is just the space M (Y ) of signed regular Borel measures on Y .
16
S
B
On the other hand, the measure of these workers (rms) who do not participate is µ (D×{x0 }}) (resp. µ ({x0 })).
Given a metric space
(with the weak-star topology). If
5
Let
w(x) ∈ <+
denote the wage in a market
x.17
As in classical hedonic models,
both on the attributes of the worker and the attributes of the job.
models, they
depend on the market tightness level.
wages depend
Moreover, as in competitive search
We follow Mas-Colell's (1975) description of the
price system for economies with a continuum of dierentiated commodities, and assume that
continuous (so similar jobs pay similar wages). Also, we extend
w
to
x0
by setting
w
is
w(x0 ) = 0.
All agents behave as price-takers, and have common rational expectations about the tightness
levels prevailing in each active market (see below). The expected prot of a rm which invests in
attributes
a
and then enters market
x = (h, a, θ)
is
π(x; w) = α(h, a, θ) [f (h, a) − w(h, a, θ)] − C(a).
The rm rst pays the cost of investing in
α(h, a, θ),
in which case it produces
a.
f (h, a)
(3.1)
Once in the market, it lls a job with probability
units of output and pays the wage
w(h, a, θ)
to its
employee. With complementary probability, the rm is rationed and remains inactive. Similarly,
the expected utility of a type-s worker who invests in
h
and then enters market
x = (h, a, θ)
u(s, x; w) = m(h, a, θ) [w(h, a, θ) − v(s, h, a)] − c(s, h)
The worker pays the cost of investing in
probability
m(h, a, θ),
h,
is
(3.2)
and then enters the market. There, she nds a job with
in which case she receives the wage net of the disutility of working. With
complementary probability, the worker is unemployed and her (ex post) utility is zero.
We are now ready to dene a competitive search equilibrium.
Denition
17
1.
A competitive search equilibrium is an allocation (µB∗ , µS∗ ) and prices w∗ such that:
All jobs created in market
x
pay wage
w(x).
This is unlike the directed search model of Coles and Eeckhout
(2003). These authors consider an economy (with a nite number of traders and no ex-ante heterogeneity) where each
seller can meet more than one buyer, and can charge a price contingent on the number of buyers she meets. Their
work thus extends that of Montgomery (1991), Peters (1991, 1997) and Burdett, Shi, and Wright (2001) (where sellers
charge the same price regardless of the number of buyers they meet) to allow for more general trading mechanisms
such as auctions. See also Kultti (1999) and Julien, Kennes, and King (2000). In models with a continuum of agents
(e.g. a la Moen (1997)) it is standard to assume that each seller meets at most one buyer, and vice versa.
6
(i) Firms and workers optimize taking w∗ as given:
Π(w∗ ) ≡ sup π(x; w∗ ) = π(x∗ ; w∗ ) for almost all x∗ ∈ suppµB∗ ,
(3.3)
x∈X
υs (w∗ ) ≡
sup u(s, xs ; w∗ ) = u(s, x∗s ; w∗ ) for almost all (s, x∗s ) ∈ suppµS∗ .
(3.4)
xs ∈X
S
(ii) All traders are assigned to a market: µB∗ (X) = ξˆ and µS∗
S =ξ .
(iii) The equilibrium allocation is consistent with the random matching technology:
Z
Z
B∗
for all Borel Ω ⊆ X|{x0 }.
m(x)dµS∗
X (x)
α(x)dµ (x) =
(3.5)
x∈Ω
x∈Ω
Recall that the support of
µB∗
describes the markets where rms participate. Equation (3.3),
together with the rst adding-up condition in (ii), says that all the rms in the economy choose
their ex-ante investments and the markets they enter so as to maximize their expected prots at
the given prices
Π(w∗ ).
w∗ .
Since they are symmetric, in equilibrium, all the rms make identical prots,
Equation (3.4) and the second condition in (ii) describe a similar optimization condition for
the workers.
18 There,
υs (w∗ )
denotes the equilibrium indirect utility of a type-s worker.
The only non-standard condition in Denition 1 is (iii). This aggregate consistency condition
says that the
number of workers who nd a job in each active market is equal to the number of
vacancies lled by rms in that market.19
x ∈ X,
and
Since
µB∗
describes the measure of rms assigned to each
α(x) is the fraction of rms who ll a vacancy in market x,
the left-hand side of (3.5) is
the measure of rms which ll a vacancy in an arbitrary set of markets
the measure of workers assigned to each
x
and
m(x)
Ω.
Similarly,
dµB∗ (h, a, θ) =
(h,a,θ)∈Ω
18
19
Recall that market
x ∈ Ω.
Z
θdµS∗
X (h, a, θ)
Note that (3.5) can be written as
for all Borel
Ω ⊆ X|{x0 },
20
(3.6)
(h,a,θ)∈Ω
x
attracts type-s workers in equilibrium if
(s, x) ∈ suppµS∗ .
This is in contrast to the market clearing condition of a frictionless economy, according to which the total numbers
of rms (demand) and workers (supply) are equal in each active market.
20
x,
is equivalent to a rational expectations condition on the traders' beliefs
about the tightness levels prevailing in active markets.
Z
describes
is the fraction of workers who nd a job in
so the right-hand side of (3.5) represents the measure of workers who nd a job
Crucially, condition (iii)
µS∗
X
Since
m(h, a, θ) = θ α(h, a, θ),
and
α(h, a, θ) > 0
for all
7
θ ∈ <+ .
Condition (3.6) says that the total measures of workers and rms who enter each active market in
equilibrium generate the market tightness levels that the traders take as given when they choose
which market to join.
markets. In this case,
Take an allocation which implies an atomless assignment of agents across
dµB (h, a, θ)
is the density of rms and
dµSX (h, a, θ)
in the set of active markets. If the traders' conjectures about
equal to
θdµSX (h, a, θ)
in this set. This is what (3.6) says.
θ
is the density of workers
are correct,
dµB (h, a, θ)
should be
21
In frictionless models, prices are pinned down by the market clearing condition, once the aggregate demand and supply curves are derived from the individual's optimization conditions. Here,
they are pinned down in a similar way, except that market clearing is replaced by (iii). For a given
w,
we may calculate the optimal decisions of workers/rms in (i), and aggregate these decisions
using (ii) to calculate the candidate equilibrium allocation. The function
w
is an equilibrium price
system whenever that allocation satises the consistency (rational expectations) condition in (iii).
In Section 4 we show that such a price system exists.
22
An important feature of our framework is that prices in inactive markets are indeterminate.
Consider rst those markets
x 6= x0
which are active.
For each such
x,
the rm's optimality
condition in (i) implies
Π(w∗ ) = α(h, a, θ) [f (h, a) − w∗ (h, a, θ)] − C(a).
Since there is a worker type
s̃
who participates in
x,
for that type, (ii) implies
υs̃ (w∗ ) = m(h, a, θ) [w∗ (h, a, θ) − v(s̃, h, a)] − c(s̃, h).
Equations (3.7) and (3.8) imply that
f (h, a) −
21
w∗ (x)
(3.8)
satises
C(a) + Π(w∗ )
υs̃ (w∗ ) + c(s̃, h)
= w∗ (x) = v(s̃, h, a) +
.
α(h, a, θ)
m(h, a, θ)
µB
(3.7)
(3.9)
H ×A×<+ is absolutely continuous with respect to the restriction
f (h, a, θ) = θ. The interpretation is the same if these
measures have atoms, except that in this case we are talking about masses rather than densities. If x attracts no
B
S
traders, (3.6) is vacuous since dµ (x) = dµX (x) = 0. That is, (3.6) is a restriction on active markets only.
of
Formally, (3.6) says that the restriction of
µS
X
22
to
to the same set, the Radon-Nikodym derivative being
This is a standard feature in general equilibrium models with a continuum of commodities (e.g. Mas-Colell and
Zame (1991) and Gretsky, Ostroy, and Zame (1999)), where it is common practice to use conventions to select a
unique supporting price system (see below). A related issue arises in directed search models where out-of-equilibrium
beliefs are indeterminate and renements are imposed to pin down these beliefs (e.g. see Peters 1997). A detailed
discussion of the relation between the two equilibrium concepts and the two indeterminacies appears in Jerez (2014).
8
w
w * (x s*ˆ )
 ŝ*
w * (x s*ˆ' )
'
 ŝ*
*
x s*ˆ
x s*ˆ
three possible
equilibrium wages
for inactive market x
x
'
x
Figure 1: Active and inactive markets.
In words, as illustrated in Figure 1, the value of
w∗ (x) can be obtained from the indierence contour
the worker attains in equilibrium, and from the isoprot contour that rms attain in equilibrium
(e.g. when these contours are smooth, they are both tangent to the wage function
space
(x, w),
w∗
at
x
on the
23
and thus tangent to each other).
On the other hand, for inactive markets, (i) implies
Π(w∗ ) ≥ α(h, a, θ) [f (h, a) − w∗ (h, a, θ)] − C(a),
υs (w∗ ) ≥ m(h, a, θ) [w∗ (h, a, θ) − v(s, h, a)] − c(s, h),
(3.10)
∀s ∈ S,
(3.11)
the weak inequality signs being strict in the case of markets which rms (workers) strictly prefer
not to join. So (3.10) and (3.11) imply that
f (h, a) −
w∗ (x)
satises:
υs (w∗ ) + c(s, h)
C(a) + Π(w∗ )
≤ w∗ (x) ≤ inf {v(s, h, a) +
}.
s∈S
α(h, a, θ)
m(h, a, θ)
As illustrated in Figure 1, the value of
(3.12)
w∗ (x) in this case lies above the isoprot contour of the rms
and below the indierence contours of the workers. The term in the left-hand side of (3.12) is the
23
Since
x
has three components, two of which are potentially multidimensional, a graph of these contours and of
the wage function on the space
(x, w)
would require more than three dimensions. Yet the intuition is the same as in
a standard two dimensional graph (e.g. see Moen 1997, where a market is described by
9
θ
only, so
x = θ).
highest wage rms would be willing to pay to participate in
x in equilibrium.
hand side of (3.12) is the lowest wage that a worker would accept to join
The term in the right-
x.
A market is inactive
whenever the latter term exceeds the former (since opening such a market would imply negative
gains from trade). In this case,
as high as
w∗ (x)
is indeterminate: it could be as low as
f (h, a) −
υs (w∗ )+c(s,h)
m(h,a,θ) }, or anything in between. That is to say,
inf s∈S {v(s, h, a) +
C(a)+Π(w∗ )
α(h,a,θ) ,
w∗ (x)
could lie
in the rm's isoprot contour, in the indierence contour of the worker who is willing to accept a
lower wage, or anywhere in between. All these prices are consistent with
4
x
being inactive.
Welfare and existence theorems via Linear programming
An allocation is feasible if it satises the adding-up and aggregate consistency conditions specied
in the previous section. A constrained ecient allocation is a pair
(µB , µS ) ∈ Mc+ (X)×Mc+ (S ×X)
that maximizes total welfare in the set of feasible allocations; i.e., it solves
R
sup
− C(a)]dµB (h, a, θ) −
X [α(h, a, θ)f (h, a)
s.t.
R
S×X [m(h, a, θ)v(s, h, a)
+ c(s, h)]dµS (s, h, a, θ)
ˆ
µB (X) = ξ,
(4.1)
µSS = ξ S ,
Z
Z
α(h, a, θ)dµB (h, a, θ) =
m(h, a, θ)dµSX (h, a, θ)
(4.2)
Ω
for all Borel
Ω ⊂ X.
(4.3)
Ω
The objective function above represents total welfare under allocation
(µB , µS ).
The rst integral
in this function gives the economy's aggregate output net of the total cost of the rms' ex-ante
investments, whereas the second one aggregates the workers' disutilities and the costs of their exante investments across all markets. Since the objective function and all the constraints are linear
in
(µB , µS ),24
Let
the above problem is linear. We shall refer to it as the
qf ∈ <
primal LP problem, (P ).
be the dual variable associated with constraint (4.1). The dual variables associated
with the constraint systems (4.2) and (4.3) are given by two continuous functions:
and
24
w ∈ C(X),
25 Here,
respectively.
qf
q S ∈ C(S)
measures the shadow value of having (a small mass of )
All the non-constant terms in these equations are integrals with respect to
between (4.3) and (3.5), in that (4.3) extends (3.5) to
x0 .
µB
and
µS .
There is a slight dierence
This is convenient for our purposes and does not change
the problem (since (4.3) is vacuous when x = x0 given that m(x0 ) = α(x0 ) = 0).
25 S
q and w lie in the topological duals of the spaces M (S) and Mc (X) (where the measures
is endowed with the uniform norm topology and
C(X)
ξS
and
µB
lie).
C(S)
is endowed with the topology of uniform convergence on
10
additional rms enter the economy, whereas
26 In turn,
type-s.
θ.
q S (s)
is the corresponding shadow value for workers of
w(h, a, θ) is the shadow value of a type-(h, a) job created in market with tightness
We abuse notation slightly by denoting this shadow price by
w,
which is also how we denote
market wages. This is to emphasize the relationship between both functions.
The Lagrangian associated with problem
Z
(P )
is
Z
B
[α(h, a, θ)f (h, a) − C(a)]dµ (h, a, θ) −
[m(h, a, θ)v(s, h, a) + c(s, h)]dµS (s, h, a, θ)
S×X
Z
Z
+q f [ξˆ −
dµB (h, a, θ)] +
q S (s) dξ S (s) − dµSS (s)
X
S
Z
Z
S
+
w(h, a, θ)m(h, a, θ)dµX (h, a, θ) −
w(h, a, θ)α(h, a, θ)dµB (h, a, θ)
X Z
X
Z h
i
= q f ξˆ +
q S (s)dξ S (s) −
q f − α(h, a, θ)[f (h, a) − w(h, a, θ)] + C(a) dµB (h, a, θ)
S
X
Z
S
−
q (s) − m(h, a, θ)[w(h, a, θ) − v(s, h, a)] + c(s, h) dµS (s, h, a, θ),
L =
X
S×X
where we set
w(x0 ) = 0
denitions of
π(x; w)
L = q f ξˆ +
Z
u(s, x; w)
and
q S (s)dξ S (s) −
Z
S
The
dual
q f ξˆ +
inf
(D)
R
S
is to nd
It is direct to see that problem
qf
more compactly as:
S
q (s) − u(s, x; w) dµS (s, x).
(4.4)
S×X
to solve
π(x; w)
and
x ∈ X,
for all
(4.5)
(s, x) ∈ S × X.
(4.6)
(D) is linear, once it is noted thatbecause utility is transferablew
u(s, x; w).
expected prots in each market
to (4.5),
Z
(q f , q S , w) ∈ < × C(S) × C(X)
for all
q S (s) ≥ u(s, x; w)
enters linearly in
[q f − π(x; w)]dµB (x) −
L
We may use the
q S (s)dξ S (s)
q f ≥ π(x; w)
s.t.
in (3.1) and (3.2) to write
X
problem
m(x0 ) = α(x0 ) = 0).
without loss of generality (since
x∈X
The term in the right-hand side of (4.5) gives the rms'
as a function of the shadow price of job
x, w(x).
According
ought to be an upper bound for the rms' expected prots in all markets given these
compact sets, so
M (S)
and
Mc (X)
are their respective topological duals. Since
M (S)
and
Mc (X)
are endowed with
the weak-star topology, the converse statement also holds. See Jerez (2017).
26
f
In the terminology of Gretsky, Ostroy, and Zame (1992) and Makowski and Ostroy (1996), q is the marginal
S
product (or contribution to social welfare) of a rm, and q (s) is the marginal product of a type-s worker.
11
shadow prices. Similarly, (4.6) says that
q S (s) is an upper bound for the expected utility of a type-s
worker in the dierent markets (given the shadow price of the jobs created in those markets).
It can be shown that problems
(P )
(D)
and
27
have optimal solutions, which can be characterized
by the complementary slackness theorem of linear programming (see Theorem 3.20 in Anderson and
Nash 1987).
Theorem
28
1.
(Complementary Slackness) Feasible solutions (µB , µS ) and (q f , q S , w) for problems
(P ) and (D) are optimal if and only if they satisfy the complementary slackness conditions:
qf
for almost all x ∈ suppµB ,
= π(x; w)
for almost all (s, x) ∈ suppµS .
q S (s) = u(s, x; w)
According to the Theorem, (µ
exists shadow prices
(q f , q S , w)
(4.10)
(4.11)
B , µS ) is constrained ecient if satises (4.1)(4.3), and there
satisfying the dual feasibility constraints (4.5) and (4.6) and the
complementary slackness conditions (4.10) and (4.11). Note that (4.1)(4.2) are the conditions in
part (ii) of Denition 1, and (4.3) is the condition in (iii). Also, (4.5) and (4.10) are equivalent to
the rms' optimization condition in (i) (equation (3.3)): they say that
only if
x
is a prot maximizing choice given
they say that
given
w.
assigns type-s workers to
x
assigns rms to
x
if and
Similarly, (4.6) and (4.11) are equivalent to (3.2):
if and only if
x
is an optimal choice for these workers
This establishes the equivalence between constrained ecient allocations and competitive
equilibria.
27
µS
w.
µB
29 The optimal shadow price
w
describes the transfers that rms need to pay workers in
Following Makowski and Ostroy (1996), we may write (4.5) and (4.6) as
qf
≥
Π(w)
q (s)
≥
υs (w)
S
where
(4.7)
for all
s ∈ S,
(4.8)
Π is the rms' prot function, and υs Ris the indirect utility function of type-s workers (in Denition 1). Because
(D) is to minimize q f ξˆ + S q S (s)dξ S (s), (4.7) and (4.8) bind at an optimum. Thus, the optimal
of w minimizes the sum of rms' prots and the indirect utilities of all the workers who live in the economy:
o
Z
n
wo = arg
min q f Π(w) +
(4.9)
υs (w)dξ S (s) .
the objective in
value
w∈C(X)
The optimal values of
28
(P )
and
(D)
q
f
S
and
q
S
in turn give rms' prots and the workers' indirect utilities at prices
wo .
have the same optimal value (so the dual variables are the shadow prices of the primal constraints,
and vice versa). The proofs rely on the assumption that
f , C , c, v , m and α are continuous,
and the sets of attributes
and agent types are compact. See Jerez (2017) .
29
To be precise, one also needs to show is that the function
12
υs (w∗ )
is continuous in
s,
so it lies in the same space
each market in order to decentralize (µ
B , µS ), and the optimal values of
qf
and
q S (s) give the rms'
prots and the indirect utility of type-s workers given these transfers.
Theorem
2.
(Decentralization of constrained ecient allocations)
(I) Let (µB∗ , µS∗ , w∗ ) be a competitive equilibrium. Dene q f ∗ = Π(w∗ ) and q S∗ (s) = υs (w∗ ) for
each s ∈ S . Then (µB∗ , µS∗ ) solves problem (P ), and (q f ∗ , q S∗ , w∗ ) solves problem (D).
(II) Suppose (µBo , µSo ) and (q f o , q So , wo ) are optimal solutions for problems (P ) and (D). Then
(µBo , µSo , wo ) is a competitive equilibrium. Moreover, q f o gives the rms' equilibrium prots
and q So (s) gives the equilibrium indirect utilities of type-s workers.
The existence result follows directly from the existence of optimal solutions to the LP problems.
Theorem
5
3.
A competitive equilibrium exists.
Extensions
Several variants of the model involve straightforward variations of the LP formulation in Section
4, where virtually the same arguments allow to establish the welfare and existence theorems. We
briey discuss two: a variant with free entry of rms, and one with two-sided heterogeneity.
5.1
Free entry
With free entry, constraint (4.1) disappears (as the mass of rms in the economy is endogenous).
Also, an entry cost
κ
is typically introduced, so the rms' expected prot in market
π(x; w) = α(h, a, θ) [f (h, a) − w(h, a, θ)] − C(a) − κ.
as the dual variable
qS .
≡
sup
u(s, x; w∗ ) = u(s, x∗ ; w∗ ),
x∈suppµS∗
X
u
is
(5.1)
This follows trivially from Berge's Maximum Theorem since (3.4) implies
υs (w∗ )
where
x
is continuous and
suppµS∗
X
is compact.
13
(4.12)
Apart from this, the only other change in Denition 1 is that
Π(w∗ ) is set to zero in equation (3.3).30
The zero prot condition simplies the model because it allows to pin down the wages in active
markets from the zero isoprot curve of the rms. The rst equality in (3.9) in this case becomes
w∗ (h, a, θ) = f (h, a) −
C(a) + κ
α(h, a, θ)
for all
x ∈ suppµB∗ ∩ suppµS∗
X .
(5.2)
Moreover, one way to tackle the price indeterminacy in this case is to assume that the zero prot
condition holds also in inactive markets. In this case, (5.2) pins down the entire function
not just the prices in active markets).
w∗
(and
31
Example. One-dimensional investments by homogeneous workers and rms.
Consider the static
version of Masters' (2011) stationary environment. There is a measure one of homogeneous workers,
who may invest in human capital,
of leisure,
b > 0.
f (h, k).
k ∈ <+ ,
at cost
k.
m
C 2,
with
c(0) = c0 (0) = 0
and
limz→∞ fz = 0
for
z = h, k .
Also,
f (h, k) = 0
The expected payos of rms and workers in market
π(x; w) =
The function
∗
w∗
m(θ)
[f (h, k) − w(x)] − k;
θ
if
h=0
or
x = (h, k, θ) ∈ <3+
and
Second,
limh→∞ c0 (h) = ∞.
is homogeneous of degree one, increasing and strictly concave, with cross-derivatives
and
m(θ)
is strictly increasing,
C 2 , and its elasticity is decreasing and bounded in the interval (0, 1).
is strictly increasing, strictly convex and
limz→0 fz = ∞
The output
The job nding and job lling probabilities are given by
in all markets. Some additional assumptions are imposed. First,
strictly concave and
f
The disutility of work is the foregone utility
Prior to opening a vacancy, rms invest in capital,
produced in a match is
m(θ)/θ
h ∈ <+ , at cost c(h).
c
Finally,
fhk > 0
and
k = 0.
are the following:
u(x; w) = m(θ) [w(x) − b] − c(h).
(5.3)
is given by the zero isoprot curve of the rms:
w (x) = f (h, k) −
θ
m(θ)
k.
(5.4)
Following Masters (2011), we focus on symmetric allocations, where all workers/rms join the same
market
30
31
x∗ = (h∗ , k ∗ , θ∗ ).
Since beliefs about
θ
are rational, the mass of entering rms is then
q f = 0 in the dual.
∗
so w coincides with the
In terms of the LP formulation, eliminating (4.1) from the primal is equivalent to setting
This amounts to selecting the
lowest
prices that support the equilibrium allocation (i.e.,
rm's isoprot curve). This is precisely the assumption that is used to pin down out-of-equilibrium beliefs in directed
search models with free entry (e.g. Shi 2001 and Menzio and Shi 2011).
14
θ∗ .
Given that workers choose
x
taking the wage function in (5.4) as given, the equilibrium lies
32 (it is constrained ecient). Substituting (5.4) into
in the contract curve
u(x; w),
the worker's
optimization problem becomes
max
x=(h,k,θ)∈<3+
m(θ)(f (h, k) − b) − θk − c(h) .
(5.5)
So it amounts to choosing a (feasible) symmetric allocation that maximizes total welfare. One can
33 The rst order conditions with respect to
check that a maximum to problem (5.5) exists.
h and k
ensure that the marginal value of the agents' investments equals the corresponding marginal cost:
m(θ)fh (h, k) = c0 (h),
m(θ)
fk (h, k) = 1.
θ
θ
The rst order condition with respect to
(5.6)
(5.7)
in turn species the ecient division of the ex-post
bilateral surplus (i.e., the Hosios rule), since it can be combined with (5.4) to yield
η(θ)
f (h, k) − w∗ (x)
=
.
∗
w (x) − b
1 − η(θ)
In words, rms appropriate a fraction
5.2
(5.8)
η(θ)
of the bilateral surplus, and the rest goes to the worker.
Two-sided ex-ante heterogeneity
In the general model, rms are of dierent types
space), and are described by a measure
(where
Ĉ
is continuous and
worker who invest in
rm type
b ∈ B,
h
ξB
in
Ĉ(·, a0 ) = 0).34
is again
f (h, a),
there is an investment
B.
b∈B
(where
B
is an arbitrary compact metric
Type-b rms can invest in
a∈A
The output of a rm which invest in
the worker's disutility of labor being
a
by the rm, a worker type
s∈S
at cost
a
Ĉ(b, a)
and hires a
v(s, h, a).
For each
and a choice of
h∈H
by the latter that generates a positive bilateral surplus.
32
The equilibrium characterization in Masters (2011) is similar, in that it boils down to choosing a stationary
(symmetric) allocation in the contract curve where rms make zero prots.
33
The argument is essentially that in Masters (2011) (so it is omitted). As he notes, the rst order conditions are
not sucient as the objective function in (5.5) need not be jointly concave in
(h, k, θ).
Yet, whenever the system of
rst order conditions has a single solution, that solution corresponds to the equilibrium. (Also, since the workers'
optimal choice is unique in this case, the symmetric equilibrium is the only possible equilibrium).
34
As in the case of the workers,
a
may include exogenous characteristics (in
15
b).
Jobs, markets and wages are described as before, and so are the workers' expected payos. The
expected payo of a type-b rm which invests in
a∈A
and then joins market
x
π(b, x; w) = α(h, a, θ) [f (h, a) − w(h, a, θ)] − Ĉ(b, a).
is
(5.9)
In describing an allocation, workers' assignment is as before. The rms' assignment across markets
is now represented by a measure
type
b∈F
µB ∈ Mc+ (B × X);
which are assigned to a market in
labor demand) in
Ω
is
µB
X (Ω),
where
µB
X
Denote the equilibrium allocation by
i.e.,
Ω ⊆ X.
µ̂B (F × Ω)
Hence, the total measure of rms (aggregate
is the marginal of
(µB∗ , µS∗ ).
is the measure of rms with
µB
on
X.
In the denition of equilibrium, the worker's
optimization and adding-up conditions are as before, but the rms' optimization condition is now
Πb (ŵ∗ ) ≡ sup π(b, x; ŵ∗ ) = π(b, x∗b ; ŵ∗ ),
for almost all
x∈X
and the rms' adding up condition in (ii) is replaced by
(b, x∗b ) ∈ suppµB∗ ,
B
µB∗
B =ξ .
(5.10)
These equations say all rm types
make prot maximizing choices. Finally, the aggregate consistency condition in (iii) now reads
Z
α(x)dµB∗
X (x)
Z
m(x)dµS∗
X (x)
=
x∈Ω
for all Borel
Ω ⊆ X|{x0 },
(5.11)
x∈Ω
35
and has essentially the same interpretation as before.
35
The (equivalent) rational expectations condition in this case is
Z
(h,a,θ)∈Ω
dµB∗
X (h, a, θ) =
Z
θdµS∗
X (h, a, θ)
for all Borel
(h,a,θ)∈Ω
16
Ω ⊆ X|{x0 },
(5.12)
5.2.1 LP formulation and main results
The primal LP problem in this model is to nd
Z
(µB , µS ) ∈ Mc+ (B × X) × Mc+ (S × X)
to solve
[α(h, a, θ)f (h, a) − Ĉ(b, a)]dµB (b, h, a, θ)
sup
B×X
Z
[m(h, a, θ)v(s, h, a) + c(s, h)]dµS (s, h, a, θ)
−
S×X
B
s.t. µB
B =ξ ,
(5.13)
µSS = ξ S ,
Z
Z
B
α(h, a, θ)dµX (h, a, θ) =
m(h, a, θ)dµSX (h, a, θ)
Ω
(5.14)
for all Borel
Ω ⊆ X,
(5.15)
Ω
where the objective function describes the economy's total welfare. The shadow price of (5.13) is a
continuous function
q̂ B ∈ C(B),
where
q̂ B (b)
is the marginal contribution to social welfare of type-b
rms. The spaces where the other dual variables lie are as before.
The dual problem is to nd
Z
inf
B
B
Z
to solve
q S (s)dξ S (s)
q (b)dξ (b) +
B
s.t.
(q B , q S , w) ∈ C(B) × C(S) × C(X)
S
q B (b) ≥ π(b, x; w)
for all
(b, x) ∈ B × X,
(5.16)
q S (s) ≥ u(s, x; w)
for all
(s, x) ∈ S × X.
(5.17)
Again, it can be shown that the two LP problems have optimal solutions, and that the complementary slackness theorem implies an equivalence between competitive search equilibria and the
optimal solutions to these problems. So, in particular, a competitive search equilibrium exists.
6
36
Concluding Remarks
To the best of our knowledge, ours is the rst model in the competitive search literature which
37 In this sense, it is related
combines two-sided heterogeneity and two-sided ex-ante investments.
36
37
Since the argument is similar to that in Jerez (2014), the proofs are omitted (but can be found in Jerez 2017).
Competitive models with these features do exist in the frictionless matching literature.
Cole, Mailath, and
Postlewaite (2001), Peters (2009), and Felli and Roberts (2016) study the eciency and existence properties of
competitive equilibria in models with one-dimensional characteristics and investments.
analysis to a multidimensional setting.
Dizdar (2015) extends the
Makowski (2004) claries the meaning of perfect competition (e.g.
17
the
to the models of Burdett and Coles (1997,2001), the key dierence being that the latter feature
non-transferable utility, random search and one-dimensional heterogeneity and investments.
A key novelty of our work is the introduction of multidimensional heterogeneity in the competitive search formulation. This feature, which is the norm in applied work (e.g. studying how labor
market outcomes vary across dierent groups of workers),
38 is also standard in general equilibrium
theory. The trick is to write down the competitive search model as one of general equilibrium.
Our framework points at a connection between two growing literatures.
The rst is that on
competitive search an equilibrium notion which is widely used in applications ranging from labor,
nance, housing, monetary economics, and industrial organization (in most of which ex-ante investments are potentially important). The second is a recent literature which uses linear programming
for the identication and estimation of frictionless matching and hedonic models (see Galichon and
Salanié (2012), Dupuy and Galichon (2014), Fox (2010), and Lindenlaub (2015), among others).
Much of this work exploits the availability of fast algorithms for computing equilibria, and nds
evidence that matching is actually multidimensional in many markets. Particularly interesting is
the work of Galichon and Salanié (2012), who study a frictionless marriage market where some
characteristics of the agents seeking to form matches are unobservable to the econometrician. They
show that, while the social surplus is linear in the assignment of dierent agent types, it is non-linear
in the assignment of
observable
types (which is what is available in the data).
They proceed by
imposing identifying assumptions that render this non-linear problem convex, and use the tools of
convex analysis to identify and estimate the social surplus of matching dierent observable types,
and the agents' equilibrium utilities. Using a similar estimation strategy on a broader set of data
on attributes of spouses, Dupuy and Galichon (2014) investigate which attributes are relevant for
the sorting of agents in the marriage market.
The LP problem analyzed in the frictionless matching literature which is known as the optimal
transportation problem is a close relative to the LP problem in this paper (which is not surprising,
conditions under which the price-taking assumption is valid) in these models.
38
The literature is too vast to provide a review, but (by way of example) it is worth mentioning a couple of
contributions. Dustmann, Ludsteck, and Schnberg (2009) challenge previous work claiming that (unlike in the US)
the wage distribution in Germany was fairly stable in the 1980s and the 1990s. Using a large administrative data set,
they document a substantial increase in wage inequality, the main drivers of which include changes in the workforce
composition (in terms of age and education), and demand factors such as the polarization of labor demand across
occupations requiring dierent skill levels (which is linked to computerization). Bover, Arellano, and Bentolila (2002)
study the inuence benet duration on the exit rate from unemployment into employment, controlling for observed
worker characteristics, unobserved worker heterogeneity, and sectoral dummy variables.
18
since we are essentially replacing the frictionless matching technology by one with search frictions).
Therefore, it seems quite likely that the empirical strategies used to estimate frictionless matching
models can be adapted to take competitive search models to the data. In particular, in labor market
applications, there is data on the assignment of workers and rms with dierent characteristics, as
well as on transfers (wages) and rationing probabilities (e.g. unemployment duration). We believe
39
that empirical research along these lines has the potential to be fruitful.
There are very few papers which develop search models with multidimensional heterogeneity
(all of which are recent). Lindenlaub and Postel-Vinay (2016) present a dynamic random search
model along these lines which features one-the-job search (though they assume that the matchrelevant characteristics of the agents are exogenous and their focus is rather dierent). An important
conclusion of their analysis is that one can make substantial qualitative and quantitative errors
regarding sorting patterns and mismatch by approximating the traders' characteristics by a onedimensional index. Postel-Vinay and Lise (2016) conduct a structural estimation of a variant of the
model which incorporates on-the-job learning.
40 In both papers the assumption of random search
implies that the equilibrium is not constrained ecient. We view our work as complementary to the
line of research initiated by these authors, in that it is another step in the direction of building a
bridge between search theoretic models and applied work studying labor markets and other markets
where search frictions are prevalent using highly disaggregated data.
Our environment is static. In dynamic extensions of the model, the distributions that describe
41 It is well-known that the analysis of
the endogenous attributes of the agents will change over time.
dynamic search environments with endogenous heterogeneity is challenging (even if heterogeneity is
one-dimensional). Recent (highly inuential) work by Menzio and Shi (2011) shows that competitive
search models with free entry of rms facilitate such an analysis because unlike random search
models they have a block recursive structure (meaning that the agents' value and policy functions
42 In their setting, workers and rms are ex-
do not depend on the distribution of heterogeneity).
ante homogeneous, but they are heterogenous ex post due to the existence of idiosyncratic shocks
39
It would be interesting by itself to see how the predictions of Galichon and Salanié (2012) (and other related
papers) change when one acknowledges the fact that the markets they analyze are characterized by frictions.
40
41
See Sanders and Taber (2012) for a related model with multidimensional skills and human capital accumulation.
In stationary settings, for instance, one must establish the existence of the relevant stationary distributions
(e.g.
of searching workers/rms across markets, of employment across dierent jobs, and of the attributes of the
unemployed). In addition, employment relations are inherently dynamic, so (at least in the case of the labor market)
the role of dynamic contracts is important (e.g. see Shi (2009) and Menzio and Shi (2011)).
42
See also Shi (2009).
19
to match output. The block recursive structure allows them to prove that an equilibrium exists and
is constrained ecient in the presence of additional uncertainty at the aggregate level, and to study
43 In
not only steady states (as in the case of random search models) but also transitional dynamics.
related work, Menzio, Shi, and Sun (2013) establish the existence of a block recursive equilibrium
in a monetary search model where (ex-ante homogeneous) traders dier in their endogenous money
holdings, and analyze the properties of the stationary distribution of money holdings. We believe
that the general equilibrium approach in this paper can further contribute to the literature initiated
by these authors.
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