# Revisiting the effect of search frictions on market concentration

Jules Depersin and Bérengère Patault Jules Depersin: University of Amsterdam,
<EMAIL_ADDRESS>; Berengere Patault: University of Amsterdam,
<EMAIL_ADDRESS>We thank Rémi Avignon, Simon Bunel, Pieter Gautier, Etienne
Guigue, Albert Jan Hummel, Isabelle Mejean, Horng Chern Wong for their
remarks.

(
Working paper)

###### Abstract

Search frictions can impede the formation of optimal matches between consumer
and supplier, or employee and employer, and lead to inefficiencies. This paper
revisits the effect of search frictions on the firm size distribution when
challenging two common but strong assumptions: that all agents share the same
ranking of firms, and that agents meet all firms, whether small or large, at
the same rate. We build a random search model in which we relax those two
assumptions and show that the intensity of search frictions has a non-
monotonic effect on market concentration. An increase in friction intensity
increases market concentration up to a certain threshold of frictions, that
depends on the slope of the meeting rate with respect to firm’s size. We
leverage unique French customs data to estimate this slope. First, we find
that in a range of plausible scenarios, search frictions intensity increases
market concentration. Second, we show that slopes have increased over time,
which unambiguously increases market concentration in our model. Overall, we
shed light on the importance of the structure of frictions, rather than their
intensity, to understand market concentration.

JEL Classification: F14, F16, M3, L2, L25.

##  1 Introduction

Market concentration has increased in the last decades in the US (Akcigit and
Ates,, 2021; Autor et al.,, 2020; Decker et al.,, 2016). Yet, economists are
still far from reaching a consensus on the causes of this increase. Two groups
of explanations prevail: differentiating them is crucial as they lead to
vastly different policy implications. On the one hand, the rise in
concentration might be due to a rise in barriers to entry: in this case,
market concentration is bad news, as it increases market power, and diminishes
innovation and efficiency (De Loecker et al.,, 2021; Gutiérrez and Philippon,,
2019). On the other hand, the rise in concentration might be due to an
increase in market competition, which enables the most-efficient firms to grow
faster: in this case, market concentration is good news (Autor et al.,, 2020).
For instance, the recent decrease in search frictions, with the advent of
digital platforms among others, has been advanced as one explanation for
increased market competition and thus increased market concentration (Albrecht
et al.,, 2022; Bai et al.,, 2020). The common view is that by preventing
agents from being matched with the most-efficient firm immediately, search
frictions allow the least-efficient firms to survive and thus decrease both
market concentration and market competition (Eaton et al.,, 2019; Lenoir et
al.,, 2022). In this paper, we revisit this theoretical link between search
frictions and market concentration. Our main result is twofold. First, we show
that the intensity of search frictions has a non-monotonic effect on market
concentration. In most realistic settings, market concentration increases with
the intensity of search frictions and thus increases while market competition
decreases. Second, we shed light on the importance of changing structures of
search frictions over time rather than their intensity as drivers of
concentration.

In a world of homogeneous preferences, all the agents - which could be either
consumers or workers - should ideally be matched with the firm that is
unanimously preferred - often referred to as the most efficient. All other
firms should optimally have a null size: frictions can only decrease market
concentration. In this scenario, an increase in market concentration is always
efficient, as it indicates that a larger fraction of agents have found the
most efficient firms and are matched with them. Yet, there is a substantial
evidence that agents have heterogeneous preferences: workers with different
skills rank jobs differently and climb different job ladders (Lindenlaub and
Postel-Vinay,, 2016; Moscarini,, 2001; Papageorgiou,, 2014), consumers have
different valuations of product characteristics (see the IO literature on
demand estimation, Berry et al., (1995)).111 In this paper, we are agnostic
about the reasons for heterogeneous agents’ preferences. These preferences can
arise from a variety of sources. One of them is that agents may have multi-
dimensional traits: for instance workers may have both a cognitive ability and
a manual ability ; and employers may have different needs for cognitive versus
manual skills. With such multi-dimensionality, different workers rank
employers differently (Lindenlaub and Postel-Vinay,, 2016). Another source of
heterogeneous preferences may be non-homothetic preferences: poor and wealthy
consumers may have different utility weights on the product’s quality and
price. As a result, consumers do not necessarily all minimize the quality-
adjusted price of products.  In the presence of heterogeneous preferences, the
optimal assignment changes, and each agent wants to be matched with their own
preferred firm. The efficient equilibrium has each firm matched with the
masses of agents that prefer them. In this case, frictions still induce
deviations between the optimal and the realized size distribution. Yet, we do
know whether the realized distribution is more or less concentrated than the
optimal one. Do search frictions smooth the distribution of matches or can
they concentrate it further? How does a decrease in search friction affect the
concentration of the market ?

We build the first theoretical model that studies the effect of frictions on
market concentration, allowing for heterogeneous preferences. We find that the
effect of search frictions on market concentration is not monotonic in
friction intensity. An increase in friction increases market concentration up
to a certain threshold of frictions, that depends on the slope of the meeting
rate with respect to firm’s size. Note that our theory encompasses the extreme
case in which the meeting rate is uniform across firms - the most common
assumption in the literature: in that case the threshold is null so that
frictions always decrease market concentration. The slope at which the meeting
rate increases with the firm’s size is therefore crucial to characterize the
effect of frictions for market concentration. We call this slope the
“structure” of the search frictions, which has been largely overlooked by the
literature. We use data from the international goods market to estimate this
structure empirically. We find that in a range of plausible scenarios, search
frictions increase market concentration, which casts doubt about the link
between rising market concentration and decreasing friction intensity.
However, using our model, we find that changes in the structure itself
throughout the years, which we observe empirically, can cause market
concentration while decreasing total welfare, making it a “bad concentration”.

To build intuition, we start with a toy model: two firms $A$ and $B$ match
with many agents. While firms have no capacity constraint and are always happy
to match with any agent that they encounter, agents can be matched with at
most one firm, but keep searching after their first match as they can switch
from one firm to another. We assume there are two types of agents: some have
an intrinsic preference for firm $A$, while others prefer firm $B$.222We are
agnostic about why different agents may prefer either firm $A$ or $B$. The
differences between the two companies could be due to a range of factors, such
as productivity, facilities, or quality, which we do not specify.  We first
show that when the meeting rate is the same for both firms, higher frictions
advantage the least preferred firm of the market: this firm’s market share is
higher in equilibrium than the share of agents who prefer this firm. Then, we
show what happens if a firm’s meeting rate depends on the firm’s market share.
We show empirically that this hypothesis can be more realistic in a number of
settings. In this case, an increase in friction intensity has a non-monotonic
effect, which is a unique feature of our model: it first advantages the most-
preferred firm of the market, then disadvantages it. The level of frictions at
which frictions start to disadvantage the most-preferred firm is an increasing
function of the slope at which meeting rates increase with firms size. While
frictions have been described in the literature as the force keeping small
firms alive, this shows that they can on the contrary lead to the
disappearance of small firms which would otherwise survive. We therefore claim
that the way meeting rates depend on the size of firms matters and can change
our representation of how frictions distort firm sizes and of who benefits
from them.

We then build a more general framework with a continuum of suppliers. We
develop a partial equilibrium model of the matching between atomistic agents
and firms. The model is a random search model that features two-sided
heterogeneity - agents and firms - and many-to-one matching: one agent can be
matched with one firm at most, and the equilibrium size of a firm is
determined by the number of agents to whom it is matched. In contrast with
most of the literature, we assume that the agents have intrinsic preferences
for certain firms, which we formalize building on Lindenlaub and Postel-Vinay,
(2016). The optimal firm size distribution is the distribution of agents’
preferences. We show that in this more general setup, basic features of our
toy-model are still present. Frictions tend to have an homogenizing (or
“deconcentrating”) effect on the firm size distribution when compared with the
optimal one _when the meeting rate is the same for all firms_. We show that in
other cases, where the meeting rate is allowed to depend on the market share
of the firm, frictions can lead to an increase of market concentration, which
is consistent with the findings of the two firms model and which is a unique
feature of our model. In such cases, an increase in friction increases market
concentration up to a certain threshold of frictions, that depends on the
slope of the meeting rate with respect to firm’s size. It is worth noting that
our model is applicable to any many-to-one matching framework, and is thus
readily applicable to labor markets or goods markets.

We model the meeting rate of a firm as a affine function of the firm’s size,
where the slope with respect to the firm’s size is equal to a parameter
$\alpha$.333One potential interpretation is that this parameter corresponds to
the share of agents that meet firms proportionally to their size, while the
other agents meet firms at a rate independent of their sizes. Agents may
indeed use different methods to meet firms: some might be using digital
platforms when other are not for instance. We then show that we can compute
with our theoretical model an optimal $\alpha$ that maximizes the matching
efficiency, defined as the sum of surpluses of all matches, for a given
friction intensity and for a given distribution of agents’ preferences. This
optimal $\alpha$ differs from the private optimal $\alpha$ that an individual
agent would choose if given the choice, as the agent does not internalize the
effect of her own match on the firm size distribution and thus on the meeting
rates of other agents. This paper thus introduces a new phenomenon of
concentration externality.

The conclusion of our theoretical study is that depending on the shape of
search frictions, higher frictions may either advantage the least or the most
preferred suppliers of the market. To gain further insights about what happens
in the real world, we study the shape of search frictions using French data.
We use a detailed firm-to-firm export dataset, provided by French Customs,
covering the universe of French firms and their exports to European Union
destinations. We confirm, regressing the flow of previously unmatched buyers
coming into a firm at time $t$ on the size of this firm at time $t-1$ that the
meeting rate is well approximated by a linear function of the market share of
suppliers. We estimate that $\hat{\alpha}=0.75$, which indicates that the
meeting rate increases close to one-to-one with the supplier’s market share.
We provide numerical evidence that with different scenarii of agents’
preferences and with such values of $\alpha$, frictions mainly have a
concentrating effect: search frictions tend to advantage the most preferred
firms of the market.

We study the evolution of $\alpha$ over the years and find that it has
increased significantly over the past 20 years: it rose from 0.63 in 1996 to
0.84 in 2016. This finding has important policy implications. Market
concentration has increased in the last decades, which is seen as worrisome by
both policy-makers and economists (Akcigit and Ates,, 2021; Autor et al.,,
2020; Decker et al.,, 2016) A potential explanation is a decrease in search
frictions, with the advent of digital platforms among others (Albrecht et
al.,, 2022; Bai et al.,, 2020; Lenoir et al.,, 2022; Mazet-Sonilhac,, 2021).
Our findings challenge and nuance this theory: we suggest that part of the
rise in market concentration can be due to a change in the _structure_ of
search frictions rather than a change in their _intensity_. This new
explanation has dramatically different policy implications than the standard
view: while a decline in frictions magnitude is welfare-enhancing, an increase
in the slope $\alpha$ is not. In the end, we show that the advent of digital
platforms may lead to what we call a “bad concentration” that decreases
efficiency.

This paper is organized as follows. Section 2 reviews the literature. Section
3 presents a toy model with two firms, which enables the reader to gain
intuition about the equilibrium effect of search frictions. Section 4 shows
that our results on the effects of frictions generalize to an environment with
a continuum of firms. Section 5 provides empirical evidence that the meeting
rate of a firm does increase with this firm’s market share. Section 6
concludes.

##  2 Related literature

Our paper, through analyzing the role of search frictions for the firm size
distribution - contributes to three large and distinct literatures: the Labor
literature, the Trade literature and the Industrial Organization literature.

A vast, but relatively recent, body of research describes the presence of
information frictions in international goods markets (Allen,, 2014; Bernard et
al.,, 2022; Chaney,, 2014; Dasgupta and Mondria,, 2018; Eaton et al.,, 2019;
Fontaine et al.,, 2022; Krolikowski and McCallum,, 2021; Lenoir et al.,, 2022;
Ornelas et al.,, 2021; Startz,, 2016; Steinwender,, 2018). Those papers
explore the magnitude of such frictions, and their effect on equilibrium
outcomes, such as the equilibrium number of buyers per firm (Lenoir et al.,,
2022; Eaton et al.,, 2019; Fontaine et al.,, 2022). Lenoir et al., (2022)
explain that search frictions distort the effectiveness of resource allocation
across heterogeneous producers, through allowing the least-efficient firms to
survive.444This increased competition due to a decline of search frictions has
also been emphasized on domestic product markets Albrecht et al., (2022) or
credit markets Mazet-Sonilhac, (2021). Yet, this result hinges on the
assumption of a common ranking of suppliers across buyers. We contribute to
this literature by including, for the first time, heterogeneous buyers’
preferences for suppliers in a model of international goods markets. In this
setting, search frictions may impact the effectiveness of matching through
novel mechanisms. Notably, we show that, taken together with the fact that the
rate at which foreign buyers meet suppliers increases with the supplier’s
market share, higher frictions may lead to a higher market concentration,
which is a unique feature of our model.

The Labor literature was the first to model search frictions. A vast number of
papers has analyzed the matching of workers to firms, to explain the
dispersion of wages as well as to study the assortativity of matching between
workers and firms (Postel-Vinay and Robin,, 2002). Formally, our paper builds
on a recent paper (Lindenlaub and Postel-Vinay,, 2016), which analyzes sorting
in a framework where workers and firms have multi-dimensional characteristics.
This multi-dimensionality generates sorting between workers and firms that
cannot be obtained in simpler settings.555More precisely, to generate sorting,
one needs either multi-dimensionality of agents’ characteristics, capacity
constraints on the firm side, or type-dependent peaks of the surplus function.
We re-interpret their model as different agents of type $x$ having
heterogeneous preferences for firms of type $y$. We contribute to this
literature by showing that, with meeting rates increasing in firms’ sizes on
top of heterogeneous preferences, the effect of search frictions on firm size
and on the efficiency of the matching process is significantly altered.

Our paper also closely relates to the existing literature on information
frictions in online marketplaces (Bai et al.,, 2020; Hui et al.,, 2022; Li et
al.,, 2020). Bai et al., (2020) show that digital platforms have a limited
ability to help small high-quality sellers overcome the demand frictions,
because of the too large market congestion. They emphasize in particular the
demand reinforcement effect, through which past sales cause the arrival of
future sales. This demand reinforcement effect is equivalent to our finding
that the rate at which an agent can meet a firm increases with the firm’s
market share. We contribute to this literature by showing that this demand
reinforcement effect has consequences on the effect of search frictions on the
firm size distribution. In particular, an increase in this effect could
explain part of the increase in market concentration in the recent decades.

Last, our paper relates to the macroeconomic literature on the misallocation
of resources in the economy, and its consequences for aggregate output and
growth (David et al.,, 2016; Hsieh and Klenow,, 2009; Restuccia and Rogerson,,
2017).

##  3 Two-firms model

To gain insight into the effect of search frictions on the allocation of
agents to firms, we present a toy model with two firms and heterogeneous
agents. We deviate from the common assumption that all agents prefer the most
productive firm of the market, and introduce heterogeneous agents’
preferences. We study the effect of frictions on the allocation of agents to
firms in this setup. In a world with homogeneous agents’ preferences, search
frictions always advantage the least preferred firms. We show that with
heterogeneous agents’ preferences this needs not be true. In particular we
show that how one models search frictions is not innocuous: in some cases
frictions advantage the least preferred firms in the market, and in other the
most preferred firms.

###  3.1 Main assumptions

We consider two firms $A$ and $B$. Agents enter the market at rate $\mu$. When
entering the market, the agents are unmatched and look for firms. They then
meet firm $A$ at a rate $\lambda_{A}$, and firm $B$ at a rate $\lambda_{B}$.
The agents match initially with the first firm they meet.666Making this
assumption is equivalent to assuming that all matches lead to a positive
surplus.
After this first match, agents keep on searching for new firms, and meet the
other firm, whether $A$ or $B$, at respective rate $\lambda_{A}$ or
$\lambda_{B}$.777This hypothesis is made in this section to keep the toy model
as simple as possible. It is however unrealistic that unmatched and matched
agents meet firms at the same rate. All results presented continue to hold
when allowing meeting rate for matched agents to be the one for unmatched ones
divided by a constant. Our multi-firm model in the next section uses this more
realistic hypothesis.  When they meet the other firm, they can switch firms or
remain with their current match, depending on their _personal preferences_. We
assume that firms have no capacity constraint and that all matches yield a
positive surplus. Therefore firms are always happy to match with any agent
that they encounter. There are two types of agents: type $a$ agents have an
intrinsic preference for firm $A$, while type $b$ agents prefer firm $B$. We
denote $p_{a}$ the proportion of agents which prefer firm $A$ while a
proportion $p_{b}=1-p_{a}$ prefers firm $B$. Note that in most existing models
of agent-firm matching with frictions, one assumes that all agents prefer the
same firm, the one with the highest productivity, and therefore
$p_{a}\in\\{0,1\\}$.

Agents exit the market at rate $\mu$, independently of whether they are
matched and whom they are matched with. We denote $u(i)$ the share of
unmatched agent of type $i$, $u$ the total share of unmatched agents,
$u=u(a)+u(b)$ and $m=1-u$ the share of matched agents. We define the total
meeting rate $\lambda_{tot}=\lambda_{A}+\lambda_{B}$.

We denote $h(i,A)$ and $h(i,B)$ the number of type $i$ agents matched with
respectively firm $A$ and with firm $B$. The number of agents matched with
firm $A$ writes: $h(A)=h(a,A)+h(b,A)$.

At steady state the inflows and outflows into and out of each state compensate
each other. For instance, the number of type $i$ agents entering the unmatched
state should equate the number of type $i$ agents leaving the unmatched state.
In each period, a proportion $\mu\times p_{i}\times dt$ of type $i$ agents
enter the market and become unmatched: these are the inflows into the
unmatched state. The outflows are composed of: a proportion
$(\lambda_{A}+\lambda_{B})$ of the unmatched agents of type $i$ which become
matched, and a proportion $\mu$ of the unmatched agents of type $i$ which exit
the market. At equilibrium, inflows into the unmatched state equal outflows,
therefore:

$(\lambda_{A}+\lambda_{B}+\mu)u(i)=\mu p_{i}.$

From this equation we can compute $u(i)$ and recover
$m=\lambda_{tot}/(\mu+\lambda_{tot})$.

The inflows of type $a$ agents into firm $A$ are equal to the unmatched agents
of type $a$ \- $u(a)$ \- which meet $A$, which happens at rate $\lambda_{A}$,
and the type $a$ agents matched initially with $B$, i.e. $h(a,B)$, which meet
A. The outflows are type $a$ agents matched with $A$ which exit the market.
Therefore at equilibrium:

$\displaystyle\mu h(a,A)=h(a,B)\lambda_{A}+u(a)\lambda_{A}.$

The inflows of type $a$ agents into firm $B$ are the unmatched agents of type
$a$ which meet $B$. The outflows are the type $a$ agents initially in $B$
which either exit the market, which happens at rate $\mu$, or meet $A$, which
happens at rate $\lambda_{A}$:

$\displaystyle\mu h(a,B)+\lambda_{A}h(a,B)=\lambda_{B}u(a)$

Writing those equations with type $b$ agents and re-arranging gives the
equilibrium number of agents matched with firm $A$ and firm $B$:

$\displaystyle h(A)$
$\displaystyle=\frac{\lambda_{tot}}{\mu+\lambda_{tot}}\left[\left(\frac{\lambda_{A}}{\mu+\lambda_{B}}\times\frac{\mu}{\lambda_{tot}}\right)+p_{a}\left(\frac{\lambda_{A}\lambda_{B}(2\mu+\lambda_{tot})}{\lambda_{tot}(\mu+\lambda_{A})(\mu+\lambda_{B})}\right)\right]$
$\displaystyle h(B)$
$\displaystyle=\frac{\lambda_{tot}}{\mu+\lambda_{tot}}\left[\left(\frac{\lambda_{B}}{\mu+\lambda_{A}}\times\frac{\mu}{\lambda_{tot}}\right)+p_{b}\left(\frac{\lambda_{A}\lambda_{B}(2\mu+\lambda_{tot})}{\lambda_{tot}(\mu+\lambda_{A})(\mu+\lambda_{B})}\right)\right].$

We can also write those results in term of the market share $s_{A}$ and
$s_{B}$ of both firms:

$s_{A}=\frac{h(A)}{m}=\left(\frac{\lambda_{A}}{\mu+\lambda_{B}}\times\frac{\mu}{\lambda_{tot}}\right)+p_{a}\left(\frac{\lambda_{A}\lambda_{B}(2\mu+\lambda_{tot})}{\lambda_{tot}(\mu+\lambda_{A})(\mu+\lambda_{B})}\right).$
(1)

The market share of each firm will be the main quantity investigated in this
study, we will sometimes call it the “size” of the firm.

So far we have been agnostic on the meeting rate $\lambda_{A}$ and
$\lambda_{B}$. If these meeting rates do not depend on the market shares
$s_{A}$ and $s_{B}$, then Equation (1) gives:

$\frac{\partial s_{A}}{\partial
p_{a}}=\frac{\lambda_{A}\lambda_{B}(2\mu+\lambda_{tot})}{\lambda_{tot}(\mu+\lambda_{A})(\mu+\lambda_{B})}\leq
1,$

the equality being only attained when $\mu=0$. Therefore a change in the
agents preference $p_{a}$ will only partially affect the market share $s_{A}$,
and the frictions tone down changes in preferences. However, the meeting rates
may increase in each firm’s market share and be functions $\lambda_{A}(s_{A})$
of $s_{A}$. Bai et al., (2020) shows, using an experiment on an online
platform, that the the meeting rate does increase with the firm’s past sales,
and we provide evidence and a ground for this phenomenon in Section 5. We
encapsulate this idea by assuming from now on that the meeting rate might be
an increasing function of the firm’s market share. In this case,
$\frac{\partial s_{A}}{\partial p_{a}}$ becomes a much more complicated
function, because an increase in $p_{a}$ affects $s_{A}$ both directly
(conditionally on meeting both suppliers, more people prefer to match with
$A$) and indirectly through affecting the meeting rates. This indirect effect
can make $\frac{\partial s_{A}}{\partial p_{a}}$ greater than $1$ (depending
on the function $\lambda_{A}(s_{A})$) so that frictions accentuate changes in
$p_{a}$.

In the sub-sections thereafter we make different assumptions about the meeting
rates $\lambda_{A}$ and $\lambda_{B}$. We first study, in sub-section 3.2, the
case where the meeting rates are the same for both firms and do not depend on
the firms market shares. We then study the case in sub-section 3.3 where the
meeting rate is proportional to the firm’s market share. Finally, we study in
sub-section 3.3 the case in which the meeting rate is an affine function of
the firm’s market share. In all three cases search frictions distort the
allocation of agents to firms, as compared to the agents preferences. Yet, we
show that in some cases, an increase in search frictions advantages the least
preferred firm of the market, and in other cases the most preferred firm.

###  3.2 Constant meeting rate

We assume in this sub-section that agents meet firms at a constant rate, i.e.
a rate that does not depend on the firm. As a result, the agent meets $A$ and
$B$ at rate $\lambda_{A}=\lambda_{B}=\lambda$, so that
$\lambda_{tot}=2\lambda$. Let us define the ratio $r_{f}=\mu/\lambda_{tot}$,
which measures the intensity of search frictions. When this ratio is large,
agents meet firms at a very slow rate, whereas when it is low, agents meet
quickly both firms. Plugging those in Equation (1), we find that the market
share of firm $A$, among matched agents is:

$\displaystyle s_{A}=\frac{r_{f}}{1+2r_{f}}+\frac{p_{a}}{1+2r_{f}}.$

The first term grows with the intensity of search frictions and is uniform
across firms, while the second one decreases with the intensity of frictions
and is firm-specific. The market share of firm $A$ is not equal to the share
of agents who prefer $A$ in general: $s_{A}\neq p_{a}$, except in the non-
frictional case $r_{f}=0$. We also note that, when $p_{a}>1/2$, $s_{A}$ is
always smaller than $p_{a}$: _when meeting rates are constant across firms,
search frictions always disadvantage the most preferred firm_. In fact, this
effect gets stronger along with search frictions: the higher the search
frictions, the lower the market share of the most preferred firm in the
market. When the intensity of friction goes to infinity, both market shares
converge to 50%: the higher the frictions, the lower the impact of agents’
preferences on the equilibrium allocation.

We illustrate the results in Figure 1, where we display how the market share
$s_{A}$ of firm $A$ depends on the intensity of market frictions
$\mu/\lambda_{tot}$. Note that in existing International Trade papers, agents
all prefer the same firm, which provides the lowest quality-adjusted price on
the market.

Figure 1: Constant meeting rate

###  3.3 Meeting rate increasing with market share

An alternative way to model search frictions is to model a meeting rate which
increases in the market share of the firm. A rationale for this assumption is
that firms with many agents are more visible to new agents.

We begin by a simple example where the meeting rate is proportional to the
market share: $\lambda_{A}=\lambda s_{A}$ and $\lambda_{B}=\lambda s_{B}$.
This corresponds to the “balanced matching” assumption made in Labor
economics. In order to justify such an assumption, let us consider for example
the following mechanism: agents, in order to find a new firm, randomly pick an
other agent in the market, search for its firm and meet with it. With this
simple mechanism, the meeting rate of a given firm is directly proportional to
its market share.

Plugging those meeting rates in Equation (1), we find that the market share of
firm $A$, among matched agents is:

$s_{A}=\left\\{\begin{array}[]{ll}\displaystyle\displaystyle 0&\text{when
}p_{a}\leq\frac{r_{f}}{2r_{f}+1}\\\ p_{a}(2r_{f}+1)-r_{f}&\text{when
}\frac{r_{f}}{2r_{f}+1}\leq p_{a}\leq\frac{r_{f}+1}{2r_{f}+1}\\\ 1&\text{when
}\frac{r_{f}+1}{2r_{f}+1}\leq p_{a}.\\\ \end{array}\right.$

In this case, $s_{A}$ is piece-wise linear with respect to $p_{a}$. Figure 2
displays how the market share $s_{A}$ of firm $A$ depends on the intensity of
market frictions $\mu/\lambda_{tot}$. As in the previous section, we note that
$r_{f}=0$ leads to the frictionless equilibrium $s_{A}=p_{a}$. However, we
note important differences when compared with the previous section.

Result 1: when meeting rates are proportional to the market shares, search
frictions always advantage the most preferred firm. Indeed, when $p_{a}>1/2$,
$s_{A}$ is always greater than $p_{a}$. We see that better by writing
$p_{a}(2r_{f}+1)-r_{f}=r_{f}(2p_{a}-1)+p_{a}$. This effect gets stronger along
with search frictions: the higher the search frictions, the higher the market
share of the most preferred firm in the market. In the previous case,
frictions had an homogenizing effect, while in this case frictions accentuate
existing inequalities.

Result 2: when meeting rates are proportional to the market shares, a large
enough friction intensity leads to a winner-takes-all scenario.

Indeed we notice that when the intensity of friction reaches a certain level
$r_{f}^{*}=p_{b}/(p_{a}-p_{b})$, the firm $B$ disappears and the firm $A$ gets
all the agents.

Intense enough frictions can even cause one firm to disappear. While frictions
have been described in the literature as the force keeping small firms alive
or advantaging them as in the previous section, we show that they can have the
opposite effect and lead to the disappearance of small firms which would
otherwise survive. We therefore claim that the way meeting rates depend on the
size of firms matters and can change our representation of how frictions
distort firm sizes and of who benefits from them.

Figure 2: Meeting rate proportional to market share

###  3.4 Meeting rate increases in market share non-proportionally

Now we allow for a more general model in which the meeting rate is not
directly proportional to the market share of the firm. We assume that the
meeting rate of each firm writes: $\lambda_{i}=\left((1-\alpha)/2+\alpha
s_{i}\right)\lambda_{tot}$ for $i\in\\{A,B\\}$. $\alpha=0$ corresponds to the
case, described in sub-section 3.2, where the meeting rate is constant and
equal across firms. $\alpha=1$ corresponds to the case, described in sub-
section 3.3, where the meeting rate is proportional to the firm’s market
share.

We have shown that in those two extreme cases, the effects of search frictions
are opposed: for $\alpha=0$, higher search frictions advantage the least
preferred firm, while for $\alpha=1$, higher search frictions advantage the
most preferred firm. We want to show what happens in the case of
$\alpha\in]0,1[$. In this case, Equation (1) has no simple closed form since
it is a third-degree equation in $s_{A}$. However, using an analytical solver
allows to show how frictions affect the market shares for different values of
$\alpha$.

Figure 3 illustrates how the market share of a firm varies with the proportion
$p_{a}$ of agents preferring this firm for different values of $\alpha$, for a
given value of frictions intensity (here $\mu/\lambda=0.7$). We first note
that the market share of the most preferred firm is increasing with $\alpha$:
the most preferred firm benefits from a higher $\alpha$. The second thing we
note is that for some values of $\alpha$, the effect of frictions is
ambiguous: it does not _always_ advantage the most preferred agent nor always
advantage the least favorite one. For instance, for $\alpha=0.85$, and for
$\mu/\lambda=0.7$, frictions advantage firm $A$ for $0.5\leq p_{a}\leq 0.9$:
in this range the yellow curve is above the grey line, so $s_{A}>p_{a}$.
However, we note that for $p_{a}\geq 0.9$, frictions advantage the least
preferred firm - firm $B$ in this example. The effect of friction is thus
ambiguous in that case. Theorem 1 characterizes the cases where the effect of
frictions is ambiguous.

######  Theorem 1.

Take $\alpha\in[0,1[$.
When $r_{f}\geq\frac{\alpha-1/2}{1-\alpha}$, then $s_{A}\leq p_{a}$ for all
$p_{a}\geq 0.5$: frictions have a homogenizing effect.
When $r_{f}<\frac{\alpha-1/2}{1-\alpha}$, then $s_{A}>p_{a}$ for at least some
$p_{a}\geq 0.5$: frictions may accentuate existing inequalities.

We note that, for $\alpha=0$, we always have
$r_{f}\geq\frac{\alpha-1/2}{1-\alpha}$, thus we recover the result from
Section 3.2 that frictions always have a homogenizing effect in that case,
while for $\alpha\rightarrow 1$ one always have
$r_{f}<\frac{\alpha-1/2}{1-\alpha}$, which is coherent with the results from
section 3.3. One important insight of this Theorem is the following: for any
$\alpha\leq 1/2$, _frictions always have a homogenizing effect_. Frictions
only amplify inequality when $\alpha>1/2$, and we argue in Section 5 that this
is often the case in practice.

Figure 4 illustrates how the market share of a firm varies with the intensity
of frictions, for different $\alpha$. We take the case of a firm that is
preferred by 10% of agents. Once again we notice that the market share of the
least preferred firm decreases with $\alpha$. For small $\alpha$ the market
share of the firm increases with the frictions’ intensity (like in the case
described in sub-section 3.2), but for large $\alpha$ the market share
decreases with $\mu/\lambda$ (like in sub-section 3.3). We again see that for
some values of $\alpha$ (for instance for the yellow curve representing
$\alpha=0.85$), the market share is non-monotonic in the intensity of
frictions, thus the effect of frictions is ambiguous. The very direction of
the effect (whether it is an advantage or disadvantage for the firm) depends
on the intensity of the friction: the curve is below the line $s=0.1$ for some
values of $r_{f}$ and above for others.

Figure 3: Market share depending on proportion of type $a$ agent, for
different $\alpha$. Graph built with $\mu/\lambda=0.7$. Figure 4: Market share
depending on friction parameter, for different $\alpha$

##  4 Multi-firms model

We now show that our results on the effects of frictions generalize to an
environment with a continuum of firms. We characterize how the firm size
distribution is affected by search frictions in presence of heterogeneous
agents’ preferences, and according to the shape of frictions, when there are
more than two firms. We describe which intuitions of our toy model remain
valid in that set-up. We use a terminology close to Lindenlaub and Postel-
Vinay, (2016), studying what happens when one allows meeting rates to vary
across firms.

###  4.1 Setting

There is a continuum of firms where each firm is indexed by its type $y$ and
which is uniformly distributed on the interval $[0,1]$. We assume that firms
have no capacity constraint and that all matches yield a positive surplus.
Therefore firms are always happy to match with any agent that they encounter.

We also have a continuum of agents, which can be either matched to a firm or
unmatched. Each agent is indexed by its type $x$. We denote $\ell(x)$ the
total measure of type $x$ agents, and we impose the normalisation
$\int\ell(x)dx=1$. We denote $u(x)$ the measure of unmatched type $x$ agents.
Whether matched or unmatched, agents exit the market at Poisson rate $\mu$. If
the agent is unmatched, she meets firms of type $y$ at rate $\lambda_{0}(y)$
and match with them. We allow for on-the-job search: if the agent is already
matched with a firm $y$, she meets an offer from firm $y^{\prime}$ at rate
$\lambda_{1}(y^{\prime})$, and accepts this offer only if it is better than
her current offer. In what follows, we will make the hypothesis that, for all
$y$, $\lambda_{0}(y)=K\lambda_{1}(y)$ for some constant $K$.888This hypothesis
allows us to more clearly show the impact of the way we model meeting rates on
the distribution of firm’s sizes. Moreover, having similar distributions of
meeting opportunities across firms regardless of whether you are matched or
unmatched is realistic. Lindenlaub and Postel-Vinay, (2016) make a similar
assumption when stating that employed and unemployed workers sample job offers
from the same sampling distribution, but for them this distribution is fixed
and exogenous while we will allow meeting rate to depend on the size of firm
$y$.
We therefore can drop the subscript and write $\lambda_{1}(y)=\lambda(y)$.

We denote $\sigma(x,y)$ the value that agents get from a type-$(x,y)$ match. A
agent $x$ matched with a firm $y$ accepts an offer from $y^{\prime}$ if and
only if $\sigma(x,y)<\sigma(x,y^{\prime})$. We are agnostic about the form of
this function $\sigma(\cdot,\cdot)$, which is exogenous and given in our
model. The fact that $\sigma$ varies across $x$ means that different types of
agents have different preferences: constant $\sigma(\cdot,y)$ for all $y$
would imply homogeneous preferences.

In much of what will follow we will take the following example: we will take
$x$ and $y$ in $[0,1]$ and we will consider $\sigma(x,y)=f(d(x,y))$ where
$d(x,y)=\min_{k\in\mathbb{Z}}|x-y+k|$ and $f$ is any strictly decreasing
function so that agents of type $x$ have a preference for firms that have a
type close to $x$. 999The modulo in the distance is added to take into account
the corner problem described in Gautier et al., (2006) and to avoid an
asymmetry between firms placed in the corners (near $0$ and $1$) and the other
ones. $x$ and $y$ can therefore be thought of as a location on a circle of
circumference $1$, so that $x=1$ is equivalent to $x=0$ and the same for $y$
(see Gautier et al., (2006)).

Our analysis is in partial equilibrium: the distribution of types is fixed
once and for all and is exogenous; $\ell(x)$ is given and firms type are
uniformly distributed on the interval $[0,1]$. In this model, there is neither
an endogenous entry of firms nor an adaptation of firms to agents’
preferences.

Just as in the previous section, we want to compute the expected number of
agents that are matched with a given firm $y$ that we denote $h(y)$. In order
to compute $h(y)$ we first study the steady-state equilibrium density of type
$(x,y)$ matches, denoted by $h(x,y)$, which indicates how many agents of type
$x$ match with firms of type $y$.

###  4.2 Steady-State distribution of matches and firm sizes

The inflow into the stock of type $(x,y)$ matches is composed of two groups:
unmatched type $x$ agents who draw a type $y$ firm with Poisson rate
$\lambda_{0}(y)$ and accept it; and type $x$ agents already matched in any
type $y^{\prime}$ job who draw a type $y$ offer at rate $\lambda_{1}(y)$ and
accept it.

The outflows from the stock of type $(x,y)$ matches are: the agents which
leave the market, which happens at rate $\mu$, and the agents which meet a
‘better’ firm. Therefore in equilibrium:

$\displaystyle
h(x,y)\times\underbrace{\left(\mu+\int\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y)]\lambda_{1}(y^{\prime})dy^{\prime}\right)}_{G(x,y)}=$
$\displaystyle\lambda_{0}(y)u(x)+\lambda_{1}(y)\int\mathbbm{1}[\sigma(x,y)>\sigma(x,y^{\prime})]h(x,y^{\prime})dy^{\prime}.$
(2)

We denote $G(x,y)$ the rate at which type $(x,y)$ matches are destroyed,
either due to agent leaving the market or to another offer being accepted.

Writing similar inflow/outflow equations for the stock of type $x$ unmatched
agents gives:

$u(x)(K\lambda_{tot}+\mu)=\ell(x)\mu,$

where
$K\lambda_{tot}=K\int_{0}^{1}\lambda_{1}(y)dy=\int_{0}^{1}\lambda_{0}(y)dy$.

After a few computations, we show that:

$h(x,y)=\frac{u(x)\lambda_{1}(y)K(\mu+\lambda_{tot})}{G(x,y)^{2}}=\frac{\mu
K(\mu+\lambda_{tot})}{K\lambda_{tot}+\mu}\frac{\ell(x)\lambda_{1}(y)}{G(x,y)^{2}}.$
(3)

Note that when taking a constant $\lambda_{1}$ we recover the result from
Lindenlaub and Postel-Vinay, (2016).

The mass $h(y)$ of agents matched with firm $y$ writes:

$h(y)=\int h(x,y)dx=\frac{\mu
K(\mu+\lambda_{tot})}{K\lambda_{tot}+\mu}\lambda_{1}(y)\int\frac{\ell(x)}{G(x,y)^{2}}dx.$

As in the previous section we will denote $m$ the mass of matched agents and
we want to find the “market share” of the firm indexed by $y$. As we are in a
continuum, each firm has a market share of measure $0$, but we can still
measure relative market shares by considering the following differential:

$s(y)=\lim_{dy\rightarrow
0}\frac{1}{m}\frac{\int_{y}^{y+dy}h(u)du}{dy}=\frac{\mu(\mu+\lambda_{tot})}{\lambda_{tot}}\lambda_{1}(y)\int\frac{\ell(x)}{G(x,y)^{2}}dx.$
(4)

This will be the main quantity of interest for the rest of this section and
that can be seen as an infinitesimal rescaled equivalent of the market share.
We sometimes call this quantity “size” of firm $y$.101010Informally, when the
number of firms is finite and equal to $N$, $s(y)\simeq\tilde{s}(y)\times N$
where $\tilde{s}(y)$ is the true market share of $y$. As $\tilde{s}(y)$ goes
to zero when $N$ goes to infinity, we have to consider its rescaled version
$s(y)$. Note that $s(y)$ can be greater than 1.

Remark: The market share of firms does not depend on the factor $K$.

Result 1: When the frictions are negligible, the size of a firm is
proportional to the fraction of agents who rank this firm as their favorite.

When the ratio $r_{f}=\mu/\lambda_{tot}$ which measures the intensity of
frictions goes to zero, then $s(y)\rightarrow\ell(y)$. We prove this result in
Annex C. This result is consistent with the results of the toy model with two
firms. Without any friction, the firm size distribution simply corresponds to
the preferences’ distribution.

###  4.3 Constant meeting rates

In this section we take $\lambda_{1}(y)=\lambda_{tot}$ so that $\lambda_{1}$
does not vary across firms. Then Equation (4) simplifies into:

$\displaystyle s(y)$
$\displaystyle=\int\frac{r_{f}(r_{f}+1)\ell(x)}{(r_{f}+\int_{y^{\prime}}\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y)]dy^{\prime})^{2}}dx$
$\displaystyle=\int\frac{r_{f}(r_{f}+1)\ell(x)}{(r_{f}+2d(x,y))^{2}}dx=\ell*K(y).$

$s(y)$ is the convolution of $\ell$ and $K$ a kernel
$K(t)=\frac{r_{f}(r_{f}+1)}{(r_{f}+2d(t))^{2}}$ with
$d(t)=\min_{k\in\mathbb{Z}}|t+k|=\min(t,1-t)$. Therefore $s(y)$ behaves as a
weighted moving average of $\ell$ around $y$, with a characteristic bandwidth
proportional to $r_{f}$. Since it is an average, it behaves more smoothly than
$\ell$.

When $r_{f}$ goes to $0$, the bandwidth shrinks. The kernel $K$ becomes close
to a Dirac function and $s(y)\rightarrow l(y)$. This result does not come as a
surprise and is in line with results from Section 3. When there is no
friction, each agent is matched to its favorite firm, and thus the size of a
firm is proportional to the fraction of agents who rank this firm as their
favorite.

Result 2: When the friction intensity goes to infinity, firms sizes are
uniformly distributed.

On the contrary, when $r_{f}$ goes to infinity, $K(t)\rightarrow 1$ for all
$t$, the kernel becomes constant, and each firm has the same size in the
limit. Once again this is no surprise: in this setup, when frictions are too
strong, agents only have time to meet one firm, that they pick uniformly among
all firms. This is consistent with the results of section 3.2, with constant
meeting rates and two firms.

Result 3: With constant meeting rates, frictions have an homogenizing effect.

In presence of heterogeneous preferences, what we mean by having an
homogenizing effect needs to be defined formally. We find that, in presence of
search frictions:

  * •

$\max_{y}s(y)\leq\max_{y}l(y)$: the biggest firm has a market share that is
lower than it would have without frictions. The same way,
$\min_{y}s(y)\geq\min_{y}l(y)$: frictions have an averaging effect that smooth
extreme values.

  * •

$\operatorname{Var}_{y}(s(y))\leq\operatorname{Var}_{y}(l(y))$: this smoothing
leads to a reduction of the variance as compared to the frictionless case.

All of this is due to the fact that $s$ behaves as a moving average of $\ell$.
We take an example to get a sense of the quantitative effect of frictions in
that set up. Let us take for instance
$l(x)=5\times\mathbbm{1}[x\in[0.4,0.6]]$. Figure 6 shows how the distribution
$s(y)$ of firm size changes with the intensity of frictions $r_{f}$. The green
line represents the distribution of agents’ preferences. When frictions are
non-existent -i.e. when $r_{f}=0$ \- the size distribution is simply the
preferences distribution. The figure shows that the higher the frictions -
i.e. the higher $r_{f}$ \- the smoother the distribution. Figure 6 completes
this picture by showing that the variance of the firm size distribution $s(y)$
decreases with the intensity of frictions.

This homogenizing effect of frictions is in line with the results of the toy
model with constant meeting rates. Introducing heterogeneous preferences
therefore does not affect the impact that search frictions have on the firm
size distribution, when meeting rates are constant. We show in the next sub-
section that when meeting rates increase in the firms’ market shares, the
results change dramatically.

Figure 5: Market share depending on firms types, for different $r_{f}$.

Figure 6: Variance of the market share depending on $r_{f}$.

###  4.4 Meeting rate proportional to the market share

We now consider the case where the meeting rate of a firm is directly
proportional to its size, as in Section 3.3. We assume that for a given firm
$y$: $\lambda(y)=\lambda_{tot}\times s(y)$.

The main equation on $s(y)$ is more complicated than in the previous case
because it is implicit. Indeed Equation 4 gives, for all $y$ so that $s(y)>0$,

$\int\frac{r_{f}(r_{f}+1)\ell(x)}{(r_{f}+\int_{y^{\prime}}s(y^{\prime})\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y)]dy^{\prime})^{2}}dx=1.$

Let us first note that, like in the previous case, when the frictions are
negligible, the size of a firm is proportional to the fraction of agents who
rank this firm as their favorite. We will now present a first result that
claims that, as in the two firms case, a winner-takes-all scenario will
prevail when the friction intensity is high enough.

Let us denote $\mathcal{S}$ the set of every interval of size $1/2$ on the
interval $[0,1]$ regarded as a circle. Let us denote
$\mathcal{S}_{y}=\\{I\in|y\notin I\\}$ the set of intervals in $\mathcal{S}$
that do not contain $y$.

######  Theorem 2.

If there is $y^{*}$ so that $\int_{I}\ell(x)<1/2$ for every interval $I$ in
$\mathcal{S}_{y^{*}}$, then, when $r_{f}$ goes to infinity, the distribution
of $s(y)$ converges to the Dirac function $\delta_{y^{*}}$.

What does Theorem 2 say ? Theorem 2 means that for all interval $I$ that does
not contain $y^{*}$, $\int_{I}s(y)dy=0$ for large enough frictions. We have
just as in Section 3.3 a concentration of the market in one particular point
while all other points are being dried out. Theorem 2 characterizes the point
of concentration $y^{*}$. It is not in general the point $x_{m}$ where $\ell$
reaches a maximum. We note that $x_{m}$ is the point where the size reaches a
maximum in frictionless cases, meaning that in this setup the _friction
intensity changes the ranking of firm sizes._ The point $y^{*}$ is the “median
point” of the distribution $\ell$: the intervals $[y^{*},y^{*}+1/2]$ and
$[y^{*},y^{*}-1/2]$ (regarded as intervals over a circle) both contain half of
the mass of the distribution $\ell$.111111 In which cases doesn’t Theorem 2
apply ? In some degenerate cases, there might not be any points so that
$\int_{I}\ell(x)<1/2$ for every interval $I$ in $\mathcal{S}_{y^{*}}$. This
the case for instance when $\ell$ is constant or when
$\ell(x)=1+\cos(2x/(2\pi))$. In the first case, the perfect symmetry of the
problem ensures that firms sizes remain equal regardless of the frictions. The
second case is not covered by the theorem, and there again the perfectly
similar conditions on points $x$ and $x+1/2$ would certainly prevent the
concentration of market share in only point. If there are strong oscillations
in $\ell$, Theorem 2 may not be applicable. Simple examples where Theorem 2
applies include cases where $\ell$ is unimodal.

The most important take-away of Theorem 2 is that, even when considering a
continuum of firms, the way meeting rates depend on firms size matters. As in
the case with two firms, a concentration phenomenon as well as a winner-takes-
all phenomenon happen in this setting when frictions go to infinity. This is
to our knowledge the first time that such a phenomenon is derived as the
consequence of search frictions.

This is not possible to obtain a closed-form expression for $s(y)$ in this
case. In the next sub-section, we go beyond Theorem 2 with the help of
numerical approximations and provide further evidence on the effect of
frictions on the firm size distribution.

###  4.5 Numerical evidence

We now take $\lambda(y)=\lambda_{tot}(\alpha s(y)+(1-\alpha))$.121212We
normalize here the mass of firms to one. We compute numerically the value of
$s$ for different values of $\alpha$ (see Annex D).

First, we study a variation of the previous example with
$\ell(x)=2\times\mathbbm{1}[x\in[0.25,0.75]]$. This example is one of the
simplest one can think of as agents’ preferences are single-peaked. We showed
in Figure 6 that with constant meeting rates, higher frictions lead to a
smoother firm size distribution. We exhibit in Figure 7(a) the equilibrium
firm size distribution, when we take $\alpha=0.8$. The effect of search
frictions is non-motonic: search frictions first concentrate the preferences’
distribution - blue and red curves - then attenuate it - yellow curve. This
result is in line with results from Section 3: when $0<\alpha<1$, an increase
in frictions does not always have the same effect: it can concentrate the
distribution (for small values of friction intensity) or smooth it (for higher
values). The threshold between those two behaviours depend both on the value
of $\alpha$ and on the shape of the preference distribution $\ell$.

In Figure 7(b), we display a similar similar Figures with a higher dependency
of the meeting rate on the firm’s market share: $\alpha=0.99$. We see that
with this higher $\alpha$, the firm size distribution is more concentrated for
every level of friction intensity $r_{f}$ when compared with $\alpha=0.8$. For
instance for $r_{f}=3$, the effect is smoothing for $\alpha=0.8$ while it is
concentrating for $\alpha=0.99$. From a number of numerical analysis
conducted, we formulate the following hypothesis: higher values of $\alpha$
lead to more concentrated distributions of $s$, everything else remaining
equal. While this seems hard to prove formally as $s$ has no closed-form
expression, it is an outcome that arises repeatedly in our numerical studies.

Second, we take the example of double-peaked preferences to illustrate the
fact derived in the previous Section that the firms benefiting from frictions
can be different from the most preferred firms where $\ell$ reaches a maximum,
even in cases where frictions concentrate the distribution (meaning they
decrease its variance). Theorem 2 states that the maximum of $s$ is reached at
the median firm, where all agents tend to concentrate when $\alpha=1$. Figure
8 shows such a phenomenon arises even for smaller values of $\alpha$. Here the
median firm has a size bigger than $\max_{y}\ell(y)$, which can never occur
when $\alpha=0$ (as mentioned in Section 4.3).

(a) Market share depending on firms type, for different $r_{f}$, with
$\alpha=0.8$.

(b) Market share depending on firms type, for different $r_{f}$, with
$\alpha=0.99$.

Figure 8: Market share depending on firms type, for different $r_{f}$, with
$\alpha=0.95$.

###  4.6 Optimal $\alpha$ and externalities

After observing that the value of $\alpha$ has a crucial impact on the firm
size distribution, it is natural to ask if $\alpha$ has an impact on the
efficiency of the matching procedure. We define the efficiency of matching as
the sum of all the surpluses of matched pairs. This surplus is maximal when
all agents are matched to their preferred firm.

From Equation (3), we can compute the total efficiency of the matching:

$\displaystyle\text{Eff}(\alpha)$ $\displaystyle=\int
h_{\alpha}(x,y)\sigma(x,y)dxdy$ $\displaystyle=\int\frac{\mu
K(\mu+\lambda_{tot})}{K\lambda_{tot}+\mu}\frac{\ell(x)\lambda_{\alpha}(y)}{G_{\alpha}(x,y)^{2}}\sigma(x,y)dxdy$
$\displaystyle\propto\int\frac{\ell(x)(\alpha
s_{\alpha}(y)+(1-\alpha))}{\left(r_{f}+\int\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y)](\alpha
s_{\alpha}(y^{\prime})+(1-\alpha))dy^{\prime}\right)^{2}}\sigma(x,y)dxdy.$

where $s_{\alpha}(y)$ is the market share of firm $y$ when the meeting rate is
equal to:
$\lambda(y)=\lambda_{tot}\left(\alpha s(y)+1-\alpha\right)$.

Taking $\sigma(x,y)=1-d(x,y)$ and using our numerical approximation of $s$, we
plot the efficiency of the matching as a function of $\alpha$, for a given
level of frictions ($\lambda$, $\mu$ and $K$ are fixed). The graph is
presented in Figure 9(a), for $r_{f}=2$ and for a Gaussian $\ell$. We notice
that $\text{Eff}(\alpha)$ is non monotonic in $\alpha$, which means that, for
a given level of frictions, there is an optimal $\alpha$ that maximizes the
efficiency and that is neither $0$ nor $1$. This implies that the firm size
distribution can be either too concentrated or too flat for efficient matching
to occur. The optimal value of $\alpha$ changes with the level of frictions,
as illustrated by Figure 9(b).

Let us assume that one agent wants to maximize her own match surplus, taking
the $\tilde{\alpha}$ chosen by other agents as given, not knowing the position
of its preference $x$ in the distribution (but knowing the global shape of the
preference distribution $\ell$). Then her optimization program is:

$\displaystyle\alpha(\tilde{\alpha})$
$\displaystyle=\operatorname{argmax}_{\alpha}\int\frac{\ell(x)(\alpha
s_{\tilde{\alpha}}(y)+(1-\alpha))}{\left(r_{f}+\int\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y)](\alpha
s_{\tilde{\alpha}}(y^{\prime})+(1-\alpha))dy^{\prime}\right)^{2}}\sigma(x,y)dxdy.$
$\displaystyle:=\operatorname{argmax}_{\alpha}\mathcal{U}_{\tilde{\alpha}}(\alpha)$

The Nash equilibrium is then given by the equation
$\alpha(\tilde{\alpha}^{*})=\tilde{\alpha}^{*}$. In general
$\tilde{\alpha}^{*}\neq\operatorname{argmax}_{\alpha}\text{Eff}(\alpha)$, as
illustrated in Figure 10, where we plot both $\text{Eff}(\alpha)$ and
$\mathcal{U}_{\tilde{\alpha}}(\alpha)$. For a single-peaked $\ell$ function
for instance, numerical simulations indicate that the best response to any
value of $\tilde{\alpha}$ is always $1$. This does not come as a surprise: it
is in the agent’s interest to get the most information out of the firm size
distribution and to meet first firms $x$ with high values of $\ell(x)$, since
they are most likely to be the agent’s favorite - because they are the
favorite of a large share of agents. This means that a social planner would
choose a different $\alpha$ than individual maximizing agents.

This phenomenon comes from the fact that agents do not internalize their
effect on the firm size distribution. They typically want a higher value of
$\alpha$ than the optimal one, since they do not take into account that their
choice and their match will concentrate the firm size distribution beyond the
optimal level. We discuss in Section 5 how this phenomenon can relate to the
increase in $\alpha$ over the years.

We have seen in this Section that, in presence of heterogeneous preferences,
the impact of search frictions crucially depends on the shape of meeting
frictions. In the following section, we study the empirical shape of the
meeting rates in the case of Firm to firm export. In particular we ask if
modelling meeting rates as constant matches the data. If so, then frictions
always have an homogenizing effect and a decrease in friction intensity causes
a concentration of the market as stated in Bai et al., (2020); Albrecht et
al., (2022); Lenoir et al., (2022); Mazet-Sonilhac, (2021). If not, then this
needs not be the case.

(a) Matching efficiency depending on $\alpha$, $r_{f}=0.2$.

(b) Matching efficiency depending on $\alpha$, $r_{f}=0.5$.

Figure 10: Matching efficiency and utility depending on $\alpha$, for
$\tilde{\alpha}$=1.

##  5 A case for heterogeneous meeting rates

In the previous Sections, we have shown that, in presence of heterogeneous
agents preferences, the way the meeting rates $\lambda(y)$ depend on the
firms’ market shares $s(y)$ is decisive for the equilibrium firm size
distribution. In particular, the parameter $\alpha$, which governs the
dependency between the meeting rates and the market shares, determines whether
frictions tend to favor the least preferred or the most preferred firms of the
market. In this Section, we evaluate the value of $\alpha$, to determine which
of these regimes is closest to reality. We first review what the previous
literature has shown. Second, we estimate $\alpha$ for the international goods
market, where foreign buyers match with French firms. We find that the meeting
rate is close to proportional to the firms’ market share, and that $\alpha$
has increased over the years.

###  5.1 Literature

In Labor economics, two extreme benchmark cases are often analyzed: random
matching in which all firms have an equal probability of being sampled - this
corresponds to the constant meeting rate case $\alpha=0$ \- and balanced
matching, in which the probability of being sampled is proportional to firm
size - this corresponds to the case where the meeting rate is proportional to
the market share $\alpha=1$. A few papers have investigated the theoretical
implications of having either random or balanced matching. For instance,
Burdett and Vishwanath, (1988) use a wage-posting model, and study
specifically the implications of balanced matching, with respect to random
matching on the wage distribution. Mortensen and Vishwanath, (1994) analyze a
form of matching that mixes random matching and balanced matching. Postel-
Vinay and Robin, (2002) write and estimate a sequential auctions model, which
does not specify any a priori functional form for the meeting rates. One
theoretical result is that with balanced matching, the most productive firm
hires all the workers of the economy. The estimation of the model leads them
to reject both the assumptions of random and balanced matching.131313More
precisely, they find that firm’s size is not monotonic in the hiring effort,
and conclude that both the assumptions of random and balanced matching are
rejected by the data. To the best of our knowledge, the precise estimation of
the way meeting rates depend on the size of firms has not been undertaken
before in the labor literature.

The International Trade literature has adapted the Labor literature to study
the matching of buyers to suppliers. Lenoir et al., (2022) assume in their
baseline model that the meeting rate of buyers is constant across suppliers,
which corresponds to $\alpha=0$ in our setting. Then, they extend the model to
a case where the meeting rate increases with the supplier’s productivity. They
show that frictions have qualitatively the same distortive impact in this
extended setting. Yet, they do not estimate the shape of the meeting rate.
Eaton et al., (2019) also allow for the meeting rate with suppliers to depend
on the supplier’s productivity. They model congestion effects on the sellers’
side: the matching intensity for a given supplier decreases with the mass of
sellers which are more productive. Therefore the more productive a seller, the
higher the meeting rate of a given buyer with this seller. As their paper
estimates this congestion parameter assuming complex parametric functional
forms and using the hypothesis of homogeneous preferences, it is hard to
relate this parameter to $\alpha$ and to our work.

###  5.2 Empirical evidence

Theory Using the notations of Section 4, we formulate the flows $f_{t}(u,y)$
of unmatched agents into the firm indexed by $y$ at time $t$ as follows:

$f_{t}(u,y)=\lambda_{0t}(y)\times U_{t}$

where $U_{t}$ is the number of unmatched agents at time $t$, and
$\lambda_{0t}(y)$ is the meeting rate, by unmatched agents, of a firm of type
$y$ at time $t$. Recall that we assumed that the meeting rate of firm $y$
takes the following linear form:

$\lambda_{0t}(y)=\left(\frac{1-\alpha_{t}}{N}+\alpha_{t}s_{t}(y)\right)\lambda_{tot,t}^{0}$

with $N$ the number of firms in the market, $s_{t}(y)$ the market share of
firm $y$ among matched agents, and, by definition,
$\lambda_{tot,t}^{0}=\sum_{y}\lambda_{0t}(y)$.

From these equations, we can write the share $s_{t}(u,y)$ of flows from the
unmatched state that each firm $y$ gets at time $t$ as:

$s_{t}(u,y)=\frac{f_{t}(u,y)}{\sum_{y^{\prime}\in
Y}f_{t}(u,y^{\prime})}=\frac{1-\alpha_{t}}{N}+\alpha_{t}s_{t}(y)$

Regressing $s_{t}(u,y)$ on the market share $s_{t}(y)$ enables to recover an
estimate of $\alpha_{t}$. Note that our estimation strategy does not rely on
using steady-state outcomes given by our model. Our model is applicable to any
setting with many-to-one frictional matching.141414More precisely, our model
applies to any partial equilibrium model with many-to-one matching, with no
capacity constraint on the firm side, transferable utility and on-the-job
search. We choose to apply it the international goods market.

Matching on the international goods market We exploit a French firm-to-firm
trade dataset which records the identity, for each French exporter, of its
non-domestic buyers in the European Union. The dataset records for each
transaction the identity of the exporting firm (its SIREN identifier), the
identification number of the importer (an anonymized version of its VAT code),
the date of the transaction (month and year), the product category (at the
8-digit level of the combined nomenclature) and the value of the shipment. The
data cleaning and product harmonization over time for this data are described
in Bergounhon et al., (2018). We perform the analysis over the period
1996-2016. We define a buyer as a buyer $\times$ 8-digit product combination,
and select each year the buyers which import from at most one French firm this
year and the year before.151515The 8-digit products are harmonized over the
period 2005-2017, as explained in Bergounhon et al., (2018). This eliminates
6.4% of buyers. Our final dataset includes 20,925,116 foreign buyers and
92,310 French firms. We then define markets as year $\times$ country $\times$
6-digit product cells. One foreign buyer can thus belong to only one market.
We have 39,549 markets in total.161616In what follows, we are able to estimate
$\alpha$ for 22,969 markets. Note that declaration rules for exports change
over our period of analysis.171717From 1993 to 2000, the declaration threshold
was set at 250,000 euros of exports in a given year. In 2001, it was set at
650,000. From 2002 to 2006, it was set at 100,000. From 2007 to 2010, it was
set at 150,000. From 2011 to 2018, it was set at 460,000. See Bergounhon et
al., (2018). We discuss this issue in the robustness tests below.

First, we compute for each year $t$, each French firm and each market the
number of buyers which were not matched in year $t-1$ to a French supplier and
to which the firm starts selling in year $t$. Note that this measure is not
exactly the empirical counterpart of $f_{t}(u,y)$. In our dataset, we only
observe the connections of foreign buyers with French firms. We therefore do
not know whether a foreign buyer not in our dataset is matched with a non-
French firm or whether it is unmatched. What we observe is
$f_{t}(u,y)+f_{t}(\bar{F},y)$, i.e. the flows into a supplier $y$ of buyers
coming from non-French suppliers and the flows into a supplier $y$ coming from
the unmatched state, but we cannot disentangle those two elements with our
dataset.

To deal with that issue, we assume that among the flow into $y$ of buyers that
were already matched, a fraction $\beta_{1}$ of them were matched with a
French supplier. In other words, we assume that there exists
$\beta_{1}\in[0,1]$ such that for each agent $x$ and each firm $y$:

$\displaystyle\underbrace{\int_{y^{\prime}\in
F}\mathbbm{1}[\sigma(x,y)>\sigma(x,y^{\prime})]h(x,y^{\prime})dy^{\prime}}_{f(F,y)}=$
$\displaystyle\beta_{1}\left(\underbrace{\int_{y^{\prime}\in
F}\mathbbm{1}[\sigma(x,y)>\sigma(x,y^{\prime})]h(x,y^{\prime})dy^{\prime}}_{f(F,y)}+\underbrace{\int_{y^{\prime}\in\bar{F}}\mathbbm{1}[\sigma(x,y)>\sigma(x,y^{\prime})]h(x,y^{\prime})dy^{\prime}}_{f(\bar{F},y)}\right)$

$\beta_{1}$ is the share of buyers who, conditionally on arriving to a firm
$y$ and on being previously matched with a supplier in the EU, were previously
matched with a French supplier. An approximation for $\beta_{1}$ is the share
of France in the European Union’s GDP. We use $\beta_{1}=0.15$ for our
estimation, but we show that our estimated $\hat{\alpha}$ is not very
sensitive to $\beta_{1}$.

We can therefore reconstruct from the data the flows of buyers from the
unmatched state into firm $y$ as follows:

$f_{t}(u,y)=\underbrace{f_{t}(u,y)+f_{t}(\bar{F},y)}_{observed}-\frac{1-\beta_{1}}{\beta_{1}}\underbrace{f_{t}(F,y)}_{observed}$
(5)

Second, we compute the total number of previously-unmatched buyers, the
empirical counterpart of $\sum_{y}f_{t}(u,y)$, to obtain the share
$s_{t}(u,y)$. Third, we compute the market share $s_{t}(y)$ of each French
firm in each market: we divide the number of buyers a firm has in a market in
a given year by the total number of buyers of this market importing from
France this year.

Results Figure 11 displays the correlations between the share $s(u,y)$ of the
flows of unmatched buyers with the market share $s(y)$ of firms. The
relationship seems fairly linear, which validates the functional form we used
in our theory for the meeting rate. We then perform the regression of $s(u,y)$
on $s(y)$ market by market, in order to estimate distinct values of $\alpha$
for each market. We find that the average $\hat{\alpha}$ is equal to $0.75$.
All standard test reject at any level the hypothesis that $\alpha=0$,
hypothesis under which most of the literature operates as we saw in the
previous section. With this estimation of $\alpha$ in hand, we show in the
Appendix, see Figure 13, that with plausible values of search frictions and
preferences’ heterogeneity, a rise in frictions leads to a higher
concentration of the market, rather than a lower concentration of the market.
The rise in market concentration in the last decades thus cannot plausibly be
explained by the decrease in search frictions.

In Figure 12, we show that the estimated $\alpha$ has risen over time. It went
from 0.63 in 1996 to 0.84 in 2016. This is one of the main result of our
study. This rise can be a novel explanation for the rise in market
concentration over the years. Indeed, we showed in Section 4 that a rise in
$\alpha$ causes in most cases a concentration of the market for a fixed
friction intensity. Therefore, Figure 12 proposes a new diagnosis on the
market concentration: it might - at least partially - be due to a change in
the structure of frictions, and not only by a reduction of friction intensity.

This rise in the dependency of the meeting rate with respect to the size of
the firm could be explained, among other, by the rise of digital platforms.
Indeed, in a world where suppliers or jobs are mainly found locally and
through word of mouth, agents first meet firms that are geographically close.
If agents and firms are distributed the same way across the territory, agents
meet all firms at the same rate, regardless of their size. On the contrary,
digital platforms allow agent to meet bigger firms first, _because it is what
the agents find optimal for themselves_ , as mentioned in Section 4. It is
therefore possible that digital platforms have two different effects that can
both lead to concentration and do not have the same consequence in terms of
efficiency: digital platforms can allow agents to meet more firms - reducing
the intensity of friction $r_{f}$ \- and also allow agents to meet _bigger
firms_ \- increasing the value of $\alpha$.

Note that the shape of the preferences’ distribution is likely to vary a lot
across markets. Such an estimation is beyond the scope of the paper, and is
left for future work. We devise, in a companion paper, a methodology to
estimate jointly the heterogeneity of preferences and the search frictions,
and apply it to the international goods market.

Figure 11: Correlation between $s(u,y)$ and $s(y)$

Note: the market share of each firm is determined for each market, where a
market is a EU country x 6-digit product x year cell. We include market fixed
effects in the regression.

Figure 12: The evolution of the structure of meeting rates over the years

###  5.3 Robustness tests

Declaration rules Declaration rules for exports change over our period of
analysis. In particular, they become more stringent in 2011, when the
threshold is set at 460,000 euros of exports per year per firm. We perform
here two exercises to test the robustness of our results to these changing
thresholds. First, we assess whether the evolution of the structure of meeting
rates over the year, exhibited in Figure 12, comes partly from the declaration
thresholds. We check this is not the case by estimating for each market and
year the $\alpha_{t}$, conditional on French exporters selling more than
650,000 euros abroad. This thresholds corresponds to the most stringent
threshold over the period - set in 2001 - and therefore the $\alpha_{t}$ are
more directly comparable across years. Second, our baseline estimation might
be an over-estimation of $\alpha_{t}$, due to non-zero declaration thresholds.

##  6 Conclusion

In this paper, we study the effect of search frictions on the firm size
distribution in presence of heterogeneous preferences. The literature has so
far analyzed the role of search frictions in presence of homogenous agents’
preferences: it was assumed that all workers for instance prefer the same
employer. In this context, the higher the search frictions, the less
concentrated the firm size distribution: search frictions advantage the least-
productive firms in the market. With heterogeneous preferences, we assume
there is no longer a common ranking of firms by agents, so that the optimal
firm size distribution corresponds to the agents’ preferences distribution. In
this context, frictions induce deviations between the optimal and the realized
size distribution, but they can now do so in diverse ways: they can have
either an homogenizing effect or a concentrating effect.

In this paper, we provide a theoretical framework, inspired by Lindenlaub and
Postel-Vinay, (2016), to study this question. We develop a partial equilibrium
model of the matching process between atomistic agents and firms, which can be
applicable to many settings, such as the labor markets or the goods markets.

We show that frictions have an homogenizing effect if meeting rates are
constant across firms. This effect is the same as in the homogenous
preferences case. However, when the meeting rate is proportional to the market
share, higher frictions may lead to a more concentrated market, as compared to
agents’ preferences. In other words, frictions may now advantage the most
preferred firm in the market.

The shape of meeting frictions is therefore crucial to understand the impact
of frictions on the firm size distribution. We explore empirically which shape
of the meeting rate is the most likely to prevail. We study the international
goods market - and show that the meeting rate is close to proportional to the
firms’ market share. We show that with plausible values of search frictions
and preferences’ heterogeneity, a rise in frictions leads to a higher
concentration of the market, rather than a lower concentration of the market.

Our results have important policy implications. First, our results have
implications for the redistributive effects of some policies, especially
policies aiming at reducing frictions on the labor or the good markets.
Second, the change in the structure of search frictions can be one plausible
explanation for the rise in market concentration, occurring in the last
decades. It is commonly thought that search frictions have decreased in the
last decades, and that could explain partially the rise in concentration. We
show that the structure of search frictions might matter as much as the
intensity of search frictions.

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##  Appendix A Annex to Section 3

Theorem 1 characterizes the cases where the effect of friction is ambiguous.

######  Theorem.

When $r_{f}\geq\frac{\alpha-1/2}{1-\alpha}$, then $s_{A}\leq p_{a}$ for all
$p_{a}\geq 0.5$ (frictions have a homogenizing effect).
When $r_{f}<\frac{\alpha-1/2}{1-\alpha}$, then $s_{A}>p_{a}$ for at least some
$p_{a}\geq 0.5$ (frictions may accentuate existing inequalities).

###### Proof.

From Equation (1), we can write, with
$\lambda_{A}(s_{A})=\left((1-\alpha)/2+\alpha s_{A}\right)\lambda_{tot}$ and
$\lambda_{B}(s_{A})=\left((1-\alpha)/2+\alpha(1-s_{A})\right)\lambda_{tot}$,

$\displaystyle p_{a}(s_{A})$
$\displaystyle=\lambda_{tot}(\mu+\lambda_{A}(s_{A}))(\mu+\lambda_{B}(s_{A}))\frac{s_{A}-\left(\frac{\lambda_{A}(s_{A})}{\mu+\lambda_{B}(s_{A})}\times\frac{\mu}{\lambda_{tot}}\right)}{\lambda_{A}(s_{A})\lambda_{B}(s_{A})(2\mu+\lambda_{tot})}$
$\displaystyle=\lambda_{tot}(\mu+\lambda_{A}(s_{A}))\frac{(\mu+\lambda_{B}(s_{A}))s_{A}-\left(\lambda_{A}(s_{A})\times\frac{\mu}{\lambda_{tot}}\right)}{\lambda_{A}(s_{A})\lambda_{B}(s_{A})(2\mu+\lambda_{tot})}.$

We denote $f(s_{A})=p_{a}(s_{A})-s_{A}$. We note that $f(x)=P(x)/Q(x)$ where
$P$ is a third degree polynomial with a positive leading coefficient and $Q$
is a second degree polynomial with a negative leading coefficient. Because $A$
and $B$ are symmetric, we know that $f(1-x)=-f(x)$ and we notice that
$Q(1-x)=Q(x)$, so $P(1-x)=-P(x)$. As $P$ is anti-symmetric about $1/2$ and a
third degree polynomial, we know its derivative reach a minimum in $1/2$.
Therefore, if its derivative in $1/2$ is positive, then its derivative is
positive in all points and then $P(x)>0$ for all $x>1/2$. However if
$P^{\prime}(1/2)<0$, as $Q^{\prime}(1/2)=0$, then $f^{\prime}(1/2)<0$ and
$f(x)<0$ for some $x>1/2$, meaning that $s_{A}>p_{a}$ for some $s_{A}>1/2$.

The only thing left is to compute the value of $P^{\prime}(1/2)$.
Straightforward computations give:

$P^{\prime}(1/2)/\lambda_{tot}^{3}=r_{f}^{2}-\alpha r_{f}^{2}-\alpha
r_{f}+r_{f}/2.$

This in turn, leads to the result of Theorem 1. ∎

######  Theorem.

When $p_{a}\geq(\frac{1-\alpha}{2}+\alpha
p_{a})(\frac{1-\alpha}{2}+p_{a}+2\alpha p_{a}(1-p_{a}))$, and $p_{a}>0.5$ then
$s_{A}\leq p_{a}$ for all $p_{a}\geq 0.5$ (frictions have a homogenizing
effect).

##  Appendix B Annex to Section 4

Let us denote, as in Lindenlaub and Postel-Vinay, (2016) $F_{\sigma\mid x}$
the conditional sampling cdf of value $\sigma$, given $x$ (with density
$f_{\sigma\mid x}$) so that

$\bar{F}_{\sigma\mid
x}(s)=\mathbb{E}_{y}\left[\mathbf{1}\left\\{\sigma\left(x,y\right)\leq
s\right\\}\right].$

We also consider the following functions

${H}_{\sigma\mid
x}(s)=\mathbb{E}_{y}\left[h(x,y)\mathbf{1}\left\\{\sigma\left(x,y\right)<s\right\\}\right],$

with density $\chi_{x}(s)$. We note that
$\chi_{x}(s)=\sum_{y|\sigma(x,y)=s}h(x,y)\times\left(\frac{\partial\sigma(x,y)}{\partial
y}\right)^{-1}$. We also introduce

${\Lambda}_{\sigma\mid
x}(s)=\mathbb{E}_{y}\left[\lambda_{1}(y)\mathbf{1}\left\\{\sigma\left(x,y\right)<s\right\\}\right],$

with density
$\lambda_{x}(s)=\sum_{y|\sigma(x,y)=s}\lambda_{1}(y)\times\left(\frac{\partial\sigma(x,y)}{\partial
y}\right)^{-1}$.

Equation (4.2) can be written as

$\displaystyle h(x,y)\times\left[\mu+\bar{\Lambda}_{\sigma\mid
x}(\sigma(x,y))\right]=\lambda_{0}(y)u(x)+\lambda_{1}(y){H}_{\sigma\mid
x}(\sigma(x,y))$

Summing over $y$ such that $\sigma(x,y)=s$ leads to:

$\left[\mu+\bar{\Lambda}_{\sigma\mid
x}(s)\right]\chi_{x}(s)=K\lambda_{x}(s)u(x)+\lambda_{x}(s)\int_{0}^{s}\chi_{x}(s^{\prime})ds^{\prime}.$
(6)

We denote $G(s)=\left[\mu+\bar{\Lambda}_{\sigma\mid x}(s)\right]$. Dividing by
$\lambda_{x}(s)$ and differentiating in $s$ we get:

$\frac{G(s)}{\lambda(s)}\chi^{\prime}_{x}(s)-\frac{\lambda_{x}(s)^{2}+G(s)\lambda_{x}^{\prime}(s)}{\lambda_{x}(s)^{2}}\chi_{x}(s)=\chi_{x}(s).$

Therefore

$\frac{\chi^{\prime}_{x}(s)}{\chi_{x}(s)}=-2\frac{G^{\prime}(s)}{G(s)}+\frac{\lambda_{x}^{\prime}(s)}{\lambda_{x}(s)}.$

This ODE solves as:

$\chi_{x}(s)=C_{x}\frac{\lambda_{x}(s)}{G(s)^{2}},$

where $C_{x}$ does not depend on $s$. We find $C_{x}$ by plugging this form in
Equation (6), leading to

$\frac{\lambda_{x}(s)}{G(s)}=Ku(x)+\int_{0}^{s}\chi_{x}(s^{\prime})ds^{\prime}.$

Taking $s=0$ gives $C_{x}=Ku(x)(\mu+\lambda_{tot})$.

We can write this as

$\sum_{y,\sigma(x,y)=s}h(x,y)\left(\frac{\partial\sigma(x,y)}{\partial
y}\right)^{-1}=\frac{C_{x}}{G(s)^{2}}\sum_{y,\sigma(x,y)=s}\lambda_{1}(y)\left(\frac{\partial\sigma(x,y)}{\partial
y}\right)^{-1}$

As we know from Equation (4.2) that among the different $y$ such that
$\sigma(x,y)=s$, $h(x,y)/\lambda_{1}(y)$ is constant, we conclude that

$h(x,y)=C_{x}\frac{\lambda_{1}(y)}{G(s)^{2}}.$

In conclusion

$h(x,y)=\frac{Ku(x)(\mu+\lambda_{tot})\lambda_{1}(y)}{G(s)^{2}}.$

##  Appendix C Annex to Section 4 part 2

First we look at the case where $r_{f}$ goes to $0$. We use the fact that for
$s(y)>0$

$s(y)=\frac{\lambda(y)}{\lambda_{tot}}\int\frac{r_{f}(r_{f}+1)\ell(x)}{(r_{f}+\int_{y^{\prime}}\frac{\lambda(y^{\prime})}{\lambda_{tot}}\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y)]dy^{\prime})^{2}}dx$

We know that, in that set up,

$\tilde{G}(x,y)=(r_{f}+\int_{y^{\prime}}\frac{\lambda(y^{\prime})}{\lambda_{tot}}\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y)]dy^{\prime})^{2}=(r_{f}+2\frac{\lambda(y)}{\lambda_{tot}}|y-x|+o(|y-x|))^{2}$

when $x$ goes to $y$ and that $\tilde{G}(x,y)$ goes to a non-zero value
otherwise. We can integrate between $y-\eta$ and $y+\eta$ as the rest of the
integral goes to zero when $r_{f}$ goes to $0$. Using a change of variable in
the integral, we see that

$\int_{y-\eta}^{y+\eta}\frac{r_{f}(r_{f}+1)\ell(x)}{(r_{f}+2\frac{\lambda(y)}{\lambda_{tot}}|y-x|)^{2}}dx\rightarrow\frac{\ell(y)\lambda_{tot}}{\lambda(y)},$

which gives the result $s(y)\rightarrow\ell(y)$.

For the case where $r_{f}$ goes to infinity, we work in analogy with Section
3.

Let us denote

$\mathcal{S}=\left\\{[x,x+1/2]\ |\
x\in[0,1/2]\right\\}\bigcup\left\\{[0,x]\cup[1/2+x,1]\ |\
x\in[0,1/2]\right\\}$

the set of intervals of size $1/2$ on the interval $[0,1]$ regarded as a
circle and $\mathcal{S}_{y}=\\{I\in|y\notin I\\}$ the set of intervals in
$\mathcal{S}$ that does not contain $y$. Let us take $y^{*}$ such that
$\int_{I}\ell(x)<1/2$ for every interval $I$ in $\mathcal{S}_{y^{*}}$ . Then
let us consider $I_{\epsilon}=[y^{*}-\epsilon,y^{*}+\epsilon]$ and
$I_{-\epsilon}=[0,1]\backslash I_{\epsilon}$ for some small $\epsilon$.

When $r_{f}$ goes to infinity, the number of agents that have meet $n$
suppliers if of order $\mathcal{O}(r_{f}^{-n})$. We keep only the first two
orders for matched agents: we assume that all matched agents have encountered
either $1$ (first order) or $2$ (second order) suppliers.

Then, we group together all firms in $I_{\epsilon}$ and see them as a “super-
firm” denoted $A$ and all firms in $I_{-\epsilon}$ as a “super-firm” denoted
$B$. This analogy only works in regime where $r_{f}$ goes to infinity because
we exclude agents that would go from $B$ to $A$ and back to $B$ as this is a
third order fraction of all agents.

Following notations from Section 3, we denote
$s_{A}=\int_{I_{\epsilon}}s(y)dy$ and $\lambda_{A}$, $\lambda_{B}$
accordingly. We now from Section 3 that $s_{A}$ either goes to one or to zero
depending on $p_{a}$ the proportion of agents that prefer firm $A$. Here the
first order (in $\epsilon$) approximation of $p_{a}$ is given by:

$\int\int_{y^{\prime}\in
I_{-\epsilon}}\mathbbm{1}[\sigma(x,y^{\prime})<\sigma(x,y^{*})]\frac{\lambda(y^{\prime})\ell(x)}{\lambda_{B}}dy^{\prime}dx,$

approximation made assuming all $\sigma(x,y)\simeq\sigma(x,y^{*})$ for $y\in
I_{\epsilon}$. We give a lower bound on that quantity by giving an upper bound
on $p_{b}=1-p_{a}$:

$\displaystyle p_{b}$
$\displaystyle\leq\sup_{\lambda(y^{\prime})|\int\lambda(y^{\prime})=\lambda_{B}}\int_{y^{\prime}\in
I_{-\epsilon}}\int\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y^{*})]\frac{\lambda(y^{\prime})\ell(x)}{\lambda_{B}}dxdy^{\prime}$
$\displaystyle=\sup_{y^{\prime}\in
I_{\epsilon}}\int\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y^{*})]\ell(x)dx$

Taking $\sigma(x,y)=f(d(x,y))$, the set
$\\{x|\sigma(x,y^{\prime})>\sigma(x,y^{*})\\}$ is a set of $S_{y^{*}}$. By
hypothesis, we have

$\sup_{y^{\prime}\in
I_{\epsilon}}\int\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y^{*})]\ell(x)dx<1/2,$

so $p_{a}>1/2$ and, when $r_{f}$ goes to infinity, $s_{B}$ goes to 0.

##  Appendix D Annex to Section 4.5

We now that $s$ is the solution to the following equation:

$s(y)=(\alpha
s(y)+(1-\alpha))\int\frac{r_{f}(r_{f}+1)\ell(x)}{(r_{f}+\int_{y^{\prime}}s(y^{\prime})\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y)]dy^{\prime})^{2}}dx.$

We approximate it using the fixed-point iteration method. We compute
$s_{t+1}=F(s_{t})$ for the following functional $F$:

$s_{t+1}(y)=(\alpha
s_{t}(y)+(1-\alpha))\int\frac{r_{f}(r_{f}+1)\ell(x)}{(r_{f}+\int_{y^{\prime}}s_{t}(y^{\prime})\mathbbm{1}[\sigma(x,y^{\prime})>\sigma(x,y)]dy^{\prime})^{2}}dx.$

This sequence can only converge to a fixed-point of $F$, thus it can only
converge to $s$.

##  Appendix E Annex to Section 5

In Section 5, we provide novel empirical evidence on the shape of meeting
frictions for the international goods market. We complement this estimation
with a similar estimation for the French labor market.

Empirical methodology We use matched a French employer-employee dataset - DADS
Postes \- which describes every firm for which every French employee worked
during the calendar year, the number of hours worked, and the wage and
occupation. Workers have individual identifiers, which allows us to track them
across firms over time, and crucially identify flows into each firm of
previously unmatched workers. We perform the analysis for the year 2015. We
select workers who are working in a given year but were not the year before.
For each firm, we compute the share of those previously-unmatched workers
getting matched to the firm, the empirical counterpart to $s(u,y)$, and
regress it on the share of workers that this firm has in 2014, the empirical
counterpart to $s(y)$. Our final dataset includes 21.5 million workers and
1,340,482 French firms.

Results First, the relationship seems, as for the international goods market,
quite linear. Second, the slope of the line gives us: $\hat{\alpha}=0.957$.
The estimated $\alpha$ is therefore quite close to what we estimated for the
international goods market.

##  Appendix F Plausible scenarii for the effect of frictions

We perform simulations to study the likely impact of search frictions with our
estimated values of $\alpha$, around 0.75. The effect of frictions on the firm
size distribution depends on $\alpha$, but also on the range of likely values
for frictions, and on the heterogeneity of preferences.

Search frictions in the labor market are usually found to be between 0.1 and
0.2 (Postel-Vinay and Robin,, 2002).181818Postel-Vinay and Robin, (2002)
estimates that $\kappa_{1}$, which is the meeting rate $\lambda_{1}$ divided
by the exogenous destruction rate $\delta+\mu$ is equal to: 5.84 for unskilled
manual workers, 7.07 for skilled manual workers, 7.61 for executives, managers
and engineers, 8.87 for technical supervisors and technicians. This yields, in
our notation, an approximate of the intensity of search frictions
$\frac{\mu}{\lambda_{tot}}$ between 0.11 and 0.17. Based on such values, we
compute the equilibrium firm size distribution for frictions increasing from 0
to 0.3. One aspect we do not have much information on is how heterogeneous
preferences are. In Figure 13(a), we show that even with a mildly-
heterogeneous preferences’ distribution, higher frictions concentrate the
distribution around the median preference - 0.5 in our example. Figure 13(b)
displays a similar pattern with stronger heterogeneous preferences.

(a) Mildly-heterogenous preferences, with normal-shaped preferences

(b) Strongly-heterogenous preferences, with normal-shaped preferences

Figure 13: Market share depending on firms type, for different intensity
$r_{f}$ of search frictions, and with $\alpha=0.75$