Matching and Price Competition with Multiple Vacancies
Thomas Jungbauer∗†
January 11, 2016
Please check for a revised version @
www.kellogg.northwestern.edu/faculty/jungbauer
∗ PhD
candidate of Managerial Economics and Strategy at the Department of Managerial Economics and
Decision Sciences (MEDS) and the Strategy Department, Kellogg School of Management, Northwestern
University.
E-Mail: t-jungbauer@kellogg.northwestern.edu
† First and foremost I would like to express my gratitude to my dissertation committee Peter Klibanoff, Nicola
Persico, James Schummer and Rakesh Vohra for their time, effort and mentoring. I am also thankful
for comments in alphabetical order by Nemanja Antic, Lori Beaman, Nicola Bianchi, Eddie Dekel, David
Dranove, Georgy Egorov, Jeff Ely, George Georgiadis, Soheil Ghili, Thomas Hubbard, Daniel Martin, George
Mailath, Mike Powell, Mark Satterthwaite, Matt Schmitt, Rainer Widmann and everybody I forgot to
mention. Finally, I am grateful to my early advisor at Northwestern University, Dale Mortensen.
Abstract
This paper analyzes the effects of firm-size variation on the performance of central
clearing houses in high-skill labor markets such as the markets for medical interns
in Canada and the US. I find that strategic wage setting in centralized markets
governed by a deferred acceptance algorithm does not result in assortative matching. While firms compete with others of similar quality within market segments,
large firms face additional incentives to diversify their bidding behavior. As a
consequence, large firms tend to offer lower wages in expectation than their smaller
competitors. As for the distribution of surplus, firms gain from the introduction
of a central clearing house compared to a competitive outcome. These additional
profits are bought at the cost of workers, who earn wages well below the competitive
level. Finally, I show that in equilibrium firms do not gain from offering different
wages for different slots. Thus strategic wage setting in the presence of a central
clearing house is compatible with absence of wage variation within firms.
JEL-Classification: C78, D44, I11, J31, J44, K21, L44
1 Introduction
When a decentralized high-skill labor market switches to a central clearing house,
what are the consequences for welfare, profits, and wages? As an example of a central
clearing house, consider the National Resident Matching Program (NRMP), a non-profit
non-governmental organization allocating medical graduates to residency programs in
the United States since 1951. Preferences of hospitals and residents are solicited and
then a match, stable with respect to reported preferences, is found and implemented.1
An antitrust lawsuit filed in 2002 argued that the NRMP adversely affects wages and
working conditions of resident physicians.2 Although the suit was dismissed after heavy
political intervention, and the NRMP eventually granted immunity, questions remained.
What can economic theory tell us about the outcome of this process? Are the gains
from matching actually split in a way that is adverse to the workers?
The matching literature is well-established, of course, but to the extent that the question
of splitting the gains from matching has been addressed, the prices that split the gains
from matching have been primarily determined through a “competitive equilibrium”
type approach. That is to say, after an allocation has been identified, equilibrium prices
(wages) are imposed. Equilibrium prices are those that satisfy some kind of stability
conditions.
To analyze, however, whether a market fosters anti-competitive outcomes, it takes a
model where prices are formed by agent behavior. Bulow and Levin [2006] innovate in
1 The
currently used matching algorithm is described in Roth and Perranson [1999]. For an extensive treatment
of the NRMP and its history see Roth [1984], Roth and Sotomayor [1990], Roth [2002] and Roth [2003].
2 The punch line of the claim reads a follows:“Defendants [hospitals and NRMP] and others have illegally
contracted, combined and conspired among themselves to displace competition in the recruitment, hiring,
employment and compensation of resident physicians, and to impose a scheme of restraints which have
the purpose and effect of fixing, artificially depressing, standardizing and stabilizing resident physician
compensation and other terms of employment.”
1
this sense by featuring strategic wage setting in a clearing house model. Firms and
workers are both characterized by their quality. Each firm can only hire a single worker.
A firm’s output is the product of its quality with the quality of the worker it is matched
to.3 Firms prefer higher quality workers over lower ones and workers care exclusively
about their wages. Firms simultaneously announce wages and workers submit their
preferences in response. Then–using the deferred acceptance algorithm–a match is
found.4 Bulow and Levin [2006] find that in equilibrium firms offer wages that are
below their competitive level. In particular, the wage function is more compressed
compared to the wage function in any competitive equilibrium. This compression is
the result of localized competition, an equilibrium phenomenon whereby firms end up
competing with a relatively small subset of similarly-productive firms, as opposed to
competing with the whole set of firms in the market. However, the efficient (assortative)
allocation still obtains.5
I introduce a tractable model of competition among multi-unit firms combining strategic
wage setting with many-to-one matching. Firms post wages for each of their vacancies.
Workers submit their preferences of firms. Then, better workers are matched to firms
offering higher wages. The model yields a rich set of implications regarding the market
structure, conduct and performance. There are two main classes of implications: first,
if the market becomes more concentrated, wages decline. This is a direct consequence
of the lack of wage competition within a firm. Second, the efficiency of the outcome
depends on firm-size6 variation. Welfare loss is more severe–everything else equal–if
3 See
Bulow and Levin [2006] for an argument that the form of the production is innocuous as long as both
arguments exhibit increasing marginal products.
4 The worker- and firm-preferred outcomes coincide when there is perfect complementarity in the market.
5 This is true modulo risk introduced by mixed strategies.
6 As for readability, I refer to firm size as the number of vacancies throughout the paper.
2
there is higher correlation between firm quality and size.
The underlying mechanism at action–not present in the one-to-one setting–is that on
average, smaller firms pay higher wages and therefore hire better workers.7 The force
driving this result is quite robust: when bidding for a vacancy a firm rivals the rest
of its own vacancies. A larger firm is thus more likely to “steal” high quality workers
from itself when compared to a small firm. I call this force internal rivalry. The same
force, by the way, implies that firm mergers tend to lower market wages.
The fact that large firms bid less for workers has far-reaching consequences for efficiency
and for surplus distribution. The asymmetry in bidding behavior causes larger firms
to hire worse workers than smaller firms of similar quality. If the size differential is
substantial, this even holds true when the large firm is of significantly higher quality
than its smaller competitor. As a result, the outcome suffers from local inefficiencies.
While all firms gain from the introduction of a central clearing house when compared
to any competitive equilibrium, welfare loss is entirely born by the workforce.
The model yields a number of additional results. First, despite large firms hiring worse
workers, equilibrium wages are such that profits are increasing in firm size. Second,
although adding a worker to a firm always results in additional profits, the added profit
per worker decreases.8 Third, higher quality firms —holding firm size fixed— accrue
more profits. Since they can produce more with the same input, they outbid their
inferior competitors to hire a better set of workers.
7 This
result about matching markets governed by a clearing house stands in stark contrast to theoretical
and empirical findings concerning large decentralized markets. Numerous contributions in the labor market
literature show that larger firms tend to pay higher wages [see e.g. Krueger and Summers, 1988].
8 This suggests that a model with vacancy posting costs should find an optimal number of workers when
holding the market environment fixed.
3
I identify the entire set of Nash equilibria for the model. Across the equilibrium set,
the average wage within each firm is constant and efficiency is constant as well. An
interesting feature of all the equilibria is that a firm’s best response always contains a
strategy in which the firm offers the same wage to all its workers. This finding relates
to real world concerns about wage inequities. It is generally believed that firms in
high-skill labor markets prefer not to wage-discriminate between their workers at the
entry level. Equal wages avoid tension within the workforce and protect from potential
legal action. Also, firms might find themselves haggling with job candidates once a
discriminatory wage policy becomes common knowledge. By definition, the multitude
of one-to-one matching models can not take on this phenomenon. Within firm wage
variation cannot be analyzed in a model which restricts firms to hire a single worker.
My results suggest that paying equal wages when a central clearing house operates the
market comes as a benefit to the firm without any associated costs. As a consequence,
absence of wage variation is a result of this paper rather than an assumption.
As for the generality of results, the findings in this paper do not only apply to the
NRMP but to centralized high-skill labor markets in general. The crowding out effect
within the firm, which I call internal rivalry is a more general phenomenon. It applies
whenever agents bid strategically for non-unit amounts of items. As a consequence, the
formal results of this paper extend to multi-unit auctions, contests and tournaments.
What follows is a brief characterization of the relevant literature. Section 2 introduces
internal rivalry and its effects by means of a numerical example. Sections 3 and 4
characterize the general problem, its solution and properties of equilibrium. Section
5 deals with the distributional consequences of my findings and section 6 shows that
firms do not prefer to wage discrimination over paying equal wages to the all incoming
4
workers. Section 7 concludes. Proofs not found in the main text are provided in the
appendix.
1.1 Background and related literature
The prominence of centralized matching algorithms stems primarily from their success in
stemming unraveling. Many decentralized high-skill labor markets tend to unravel, that
is, for employment negotiations to take place earlier and earlier relative to hiring date.9
Unraveling may lead to exploding offers and other disorderly hiring behavior which
may hurt efficiency. A central clearing house eliminates the unraveling by replacing
the strategic back and forth with an instantaneous allocation decision based on market
participants’ preferences. The consequences for the distribution of surplus among
market participants, however, are not well understood.
Gale and Shapley [1962] pioneer the study of stable outcomes in two-sided markets
and introduce a mechanism–called deferred acceptance–which always selects a stable
allocation. This early matching literature focuses on stability and efficiency. A second
generation of papers adds a focus on surplus distribution by introducing prices (wages).
Among others see Crawford and Knoer [1981], Kelso and Crawford [1982] and Hatfield
and Milgrom [2005] for a comprehensive treatment.
Kamecke [1998] introduces an early model of the NRMP with endogenous wage formation. In his model, hospitals sequentially set wages. After each round, hospitals
which have already specified their wage are allowed to withdraw from the centralized
mechanism to respond to unexpected offers by competitors. He finds that whereas
9 Unraveling
is also observed in other types of markets. For a detailed discussion of market unraveling and its
driving forces as well as an abundant collection of examples see Roth and Xing [1994].
5
the resulting allocation is efficient, wages fall significantly short of the competitive
level. Crawford [2008] innovates by introducing worker-specific wages and showing that
efficiency and competitive wages can thus be restored. Artemov [2008] analyzes the
effects of reporting errors under such a regime and shows that relatively small errors
may lead to highly distorted wages. Niederle [2007] shows that the low-wage equilibrium
of Bulow and Levin [2006] ceases to exist if firms have an option to personalize offers.
Niederle [2007] also argues that centralization may not be the driving force behind wage
compression. Agarwal [2015] indicates that firms’ capacity constraints, coupled with
medical graduates’ willingness to pay implicit tuition for desirable residency programs,
leads to compressed salaries. Kojima [2007] provides an example which shows that
non-unit and varying capacities of hospitals might not imply all workers to earn lower
wages in the match than in competitive equilibrium.
2 A numerical example
2.1 Equal wages within firms
Firms set wages simultaneously for all of their vacancies and then better workers are
matched with better paying jobs. Consider first a one-to-one setting in which every
firm is restricted to hire a single worker. In the unique wage setting equilibrium firms
compete only with firms of similar quality as opposed to market-wide competition.
Bulow and Levin [2006] provide a baseball analogy to explain the intuition behind this
localized competition: “...the Yankees have an easier schedule than the Tampa Bay
Devil Rays because they face all the same opponents, except that the Yankees get to
play the Devil Rays and the Devil Rays must play the Yankees.” Translated to the
matching model, good firms draw more benefit from competing with any set of firms
6
than weaker firms. Thus, the set of firms they compete with in equilibrium is similar
to themselves. This intuition–ceteris paribus–persists in a world in which firms have
multiple vacancies. However, another dimension, non-existent in the one-to-one setting,
comes into play. Provided below is a stylized numerical example of multi-unit firms. At
first, I will restrict firms to set a single wage for all its vacancies. Then, this restriction
will be lifted. The presentation of the general model follows the same path.
Before characterizing the equilibrium of a numerical example with three firms, I lay
out the basic idea of internal rivalry with two firms of equal quality. Firm 1 enters
the market with a single vacancy whereas firm 2 is interested in hiring two workers.
Consider both firms setting a wage w for one of their vacancies. Holding the behavior
of other market participants fixed, every firm faces a distribution over expected worker
quality given its wage offer. If firm 2 sets the same wage for its second vacancy, its
expected profit per vacancy at wage w declines since firm 2 cannot hire the best worker
available at wage w twice. Since larger firms are more likely to steal workers from itself,
in equilibrium they randomize over wider intervals than smaller firms. The maximal
wage offered by a firm, however, is independent of firm size. As a consequence, holding
quality fixed, larger firms pay lower wages than their smaller competitors in equilibrium
and thus, hire lesser qualified workers on average. In other words, if a large firm outbids
its smaller competitor, the former gains a small amount of better human capital but is
forced to up its wages for each of its workers. The latter, however looses relative more
human capital than the large firm gains and only benefits mildly from paying lower
wages to its small workforce. Thus, in equilibrium the small firm might outbid the
large one creating a better outcome for both firms. As a consequence, market efficiency
suffers and the burden is necessarily shouldered by workers.
7
In a two-firm model both firms have a single competitor. Thus, when optimizing, they
have to cover the same interval of wages. While this does not contradict the results of
this paper, it does not reveal the full story. Thus, I introduce a third firm to provide
an insightful numerical example. Let qm be the quality of firm m and sm its size.
Moreover, define a worker’s quality simply by her index and assume the number of
workers in the market to equal the number of total vacancies. Consider a problem
in which the firm’s quality is its index, i.e. qi = i, and assume firms 1 and 2 to have
a single opening whereas firm 3 has three. Firms compete for workers 1 through 5.
Initially, firms announce each a single wage simultaneously. Workers observe the wages
and are then allocated such that if the firm posting the highest wage has l openings
the top l workers will be paired up with this firm and so on and so forth.10
Observe first, that there cannot be a pure strategy equilibrium. Also there is no range
of bids b in equilibrium of the wage setting game at which firm 1 competes with the
other two firms since its required probability density to be indifferent over such an
interval would be negative. Thus, consider a price range over which only firms 2 and 3
are competing. To render each other indifferent, firm 3’s probability distribution over
bids is
1
s3 ·q2
=
1
6
and 2’s
3 has still 1 − 16 · 3 =
1
2
1
s2 ·q3
= 13 . Thus, the length of this segment has to be 3. Firm
of its bidding mass available and thus competes also with firm 1.
This competition has to take place at lower bids since otherwise firm 2 would join and
firm 1 drop out consequently. Firm 3 bids with a density of
the length of this interval is 1.5, leaving firm 1 with
10 As
1
2
1
3
as does firm 1. Thus,
of its bidding mass, which will
a tiebreaker assume workers to be assigned to the best firm bidding a particular wage. This mechanism
satisfies the properties of a deferred acceptance algorithm as discussed earlier. Due to perfect complementarity the worker-preferred and firm-preferred outcomes are identical. In fact the modeling approach is quite
robust. While it was designed to model a deferred acceptance algorithm, it is also compatible with serial
dictatorship by firms when wage-setting is the simultaneous tournament to decide succession and serial
dictatorship by workers when quality determines succession. For an alternative interpretation and way of
modeling wage formation in a centralized matching market see Kamecke [1998].
8
Gi
1
.5
G1
G3
G2
1.5
4.5
b
Figure 1: Cumulative densities over bids in the numerical example
be its atom at the worker outside option of 0. The graph below shows this strategies in
terms of cumulative densities of bids in equilibrium.
A brief glance at Figure 1 reveals that firm 2 outbids 3 in expectation and expects
to hire worker 5 whereas firm 3 hires workers 2, 3 and 4. Thus, the equilibrium
does not only introduce inefficiency by randomness due to mixed strategies but is
truly inefficient in expectation. Expected wages of workers can be calculated to be
be (w1 , ..., w5 ) = (.31, 1.62, 1.88, 1.94, 3.25) The lowest wages that support the unique
(assortative) competitive equilibrium allocation are (w1C , ..., w5C ) = (0, 1, 3, 5, 7).11 While
worker 1’s wage increases in the match when compared to competitive equilibrium all
other wages decrease. In fact, the average wage decreases by approximately 44%12
whereas at the same time average firm profit increases by 16%.13 Figure 2 below again
11 Section
5 discusses why firm 3 does not offer equal wages in competitive equilibrium.
the single worker replica of a model with multiple vacancies as the one-to-one matching model where
each vacancy keeps its quality but every firm demerges into single entities. When looking at the single
worker replica of this model and comparing it to competitive equilibrium the average wage only decreases
by around 33%.
13 In terms of efficiency an interesting benchmark –in particular in small markets– can be derived from a
random allocation model, consider a draft which calls upon vacancies in random order. Let W Fi be the
expected welfare for i = (M E, CE, RA), where ME=matching equilibrium, CE=competitive equilibrium
M E −∆RA
and RA=random allocation. Then ∆RA = ∆
indicates the fraction of potential welfare possible
∆CE −∆RA
over a random allocation realized in the match. For the numerical example above ∆RA = .4 whereas
for its single worker replica ∆RA = .95. This indicates the potentially significant differences in expected
12 Define
9
Firm 3
Firm 2
Firm 1
0
1.5
4.5
b
Figure 2: Bidding supports in the numerical example
visualizes that the expected allocation is inefficient (not assortative).
2.2 Wage discrimination within firms
Consider the above described problem and now lift the restriction of equal wages within
firms. Since a pure strategy equilibrium is still infeasible, we would expect negligible
inefficiency due to randomness of equilibrium but better firms to hire —on average—
better workers due to their dominant value creation process. This intuition, however,
is flawed. Suppose firms 1 and 2 to stick to their strategies and consider firm 3’s best
response. Since firm 3 does not compete with itself, consider it to offer the maximal
market wage of 4.5 for each of its vacancies. Its resulting payoff is
3 · (3 + 4 + 5) − 3 ∗ 4.5 = 36 − 13.5 = 22.5,
(1)
which doesn’t come as a surprise since this is exactly the payoff achieved by setting
any wage between 0 and 4.5 for all its vacancies. It is straightforward to verify that
firm 3 earns the same profit for any combination of bids within 0 and 4.5. As a result
the non-discriminating matching equilibrium presented above persists when firm 3 is
performance when one considers firms with multiple vacancies.
10
allowed to set different wages for its vacancies. Since the intuition presented under equal
wages within firms does not apply here, this finding has to be put into perspective.
The reason why firm 3 does not do better with differentiated prices is as follows. If
firm 3 bids for all its vacancies above 1.5, it appears that all three of its vacancies have
a shot at hiring the top worker. This intuition, however, is deceiving. Upon relabeling,
firm 3’s second and third vacancy do only compete for the second respectively third
best worker due to rivalry within the firm. In equilibrium, firm 3 has to be necessarily
indifferent between bidding in the top bracket and bidding below for all of its openings.
Additionally, firm 3 is equally well off when splitting the bidding of its vacancies over
the entire interval. One can observe that there are multiple discriminating equilibria.
In order to keep firm 2 indifferent over the interval [1.5, 4.5] firm 3 bids with a total
density of
1
2
over this interval, whether it does so with two vacancies and densities of
over disjoint intervals or with all three of them and a density of
1
6
1
2
for each of them. An
analogous condition has to hold for competition of firms 3 and 1 over [0, 1.5]. Firms 1
and 2 randomize as before. Whereas the expected allocation might vary depending on
equilibrium selection, efficiency, profits and the average wage offered by each and every
firm do not.
3 The model
3.1 The multiple vacancy problem
The agents in a multiple vacancy model are M firms and N workers. Every firm m is
characterized by a pair (qm , sm ), qm denoting its quality and sm its number of vacancies
11
(firm size). Worker n’s quality is simply defined by her index n.14 The total number of
job-seeking workers is assumed to equal the total number of vacancies in the market.
This assumption has no bearing on the interpretation of results and is purely employed
to increase the transparency of the model.15 Since rearrangement is always possible,
firms are ordered according to their quality, i.e. m > l ⇒ qm ≥ ql . Throughout the
paper production of a worker n at firm m will be the product of their qualities qm · n
16
and total firm production is additive.
Definition 1. A multiple vacancy problem is how to allocate N workers, defined
by their index, to M firms, each defined by its quality qm and size sm (number of
vacancies) when production of firms and workers is multiplicative and the market of
vacancies and workers is balanced.
Due to increasing marginal products of both firms and workers, the efficient allocation
of a multiple vacancy problem necessarily results in assortative matching. Allocation
14 Both
magnitude and distribution of worker qualities are inessential for qualitative results as long as no
consecutive pair of workers exhibits a significant relative quality differential. The exact crucial differential
is hard to pin down due to the number of possible permutations of firm size but the restriction becomes
negligible when the number of firm grows. Without this restriction, results as presented in Kojima [2007] are
possible. Kojima [2007] was the first to introduce the multiple vacancy idea into this literature. He shows
that wage compression does not necessarily persist in general if firms hire a different number of workers.
This is true if there is a pair of consecutively ranked workers exhibiting a significant difference in quality.
This potentially causes some workers to earn wages in excess of the competitive level in the match, in
particular if the market is very small. I rule out large gaps in quality between consecutively ranked workers
to circumvent this problem which becomes quickly negligible, even in fair sized markets. It is noteworthy
that this assumption exclusively affects results about surplus distribution.
15 Restriction of the number of workers to equal the number of open seats in the market simplifies analysis. It
rules out both excess demand and excess supply of workers. All allocation mechanisms analyzed in this
paper would be affected by a relaxation thereof in the same way. In expectation, excess demand leads to
the lowest ranked firms not filling their vacancies whereas excess supply causes the lowest ranked workers
to remain unemployed. While the first case causes a general increase in wages throughout the second case
implies the opposite. None of the qualitative results to follow are altered by relaxing this assumption.
16 Results are independent of this assumption as long as the production of a worker at a firm exhibits strictly
increasing marginal products in both arguments and the the firm’s total production is the sum of its
production with workers. The assumption of separability of the production function in workers is essential
for the result. The existence of non zero cross partial derivatives of production in workers’ qualities would
contradict the optimality of a deferred acceptance algorithm even in the absence of prices. For a matching
theory with non-separable production in the absence of a central clearing house see the O-Ring literature
around Kremer [1993]. I am thankful to Tom Hubbard for pointing this out to me.
12
of workers among firms with equal quality does not affect efficiency.
Remark 1. The efficient allocation of a multiple vacancy problem is assortative
and unique up to firms of equal quality.17
3.2 Matching equilibrium
As outlined in the introduction, this paper introduces a model of a centralized matching
mechanism with strategic wage formation. At first, I restrict firms to pay a single wage
to its incoming workforce. Workers’ preferences are independent of a firm’s quality and
are solely based on wages. Following Bulow and Levin [2006] the time line of events
is:
1. Firms engage in a simultaneous auction each posting a single wage which it
commits to pay to all incoming workers.
2. Better workers are matched with firms setting higher wages by a central clearing
house.
Firms’ actions in the simultaneous wage setting game are referred to as bids b to
preserve the notion of wages for realized outcomes. A strategy for firm m is a probability
distribution gm over bids b. Let Vm (·) be firm m’s expected payoff in the match.
Definition 2. A (non-discriminatory) matching equilibrium of a multiple vacancy
problem is a firm strategy gm for every firm m such that gm maximizes Vm (gm , g−m )
given the other M − 1 firms’ strategies g−m = (g1 , ..., gm−1 , gm+1 , ..., gM ).
Before matching equilibrium can be characterized, some of its features are established
ex ante:
17 See
the assignment game in Shapley and Shubik [1971].
13
Proposition 1 (Features of matching equilibrium). (a) There is no matching
equilibrium in pure strategies. (b) No bid b is offered by a only a single firm18 and (c)
P
some firm always bids arbitrarily close to 0. (d) Aggregate offering
sm · gm (b) is
m≤M
non-increasing in b. (e) There are no gaps in the support of a firm’s strategy and (f )
or, in the aggregate support of all firms. (g) No firm’s strategy can assign atoms except
at 0.19
If firm m bids b in equilibrium, b has to be an optimal choice, i.e.
dVm (b, g−m )
= 0.
db
(2)
Firm m’s expected profit from bidding b depends on the magnitude of b and its expected
rank among all firms’ bids. Let Gm (b) be the probability of firm m bidding below b
and Sm be the total vacancies of all firms i ≤ m. Then,
P
SM
Y
Vm (b, g−m ) =
Gi (b) · qm ·
j
SM −sm +1
Fi ∈F \Fm
Y
+
X
Gi (b)(1 − GM (b)) · qm ·
SM
−sM
X
SM −sM −sm +1
Fi ∈F \{Fm ,FM }
j
(3)
+ ...
+
Y
(1 − Gi (b)) · qm ·
sm
X
j
1
Fi ∈F \Fm
− sm b.
18 excluding
irrelevant measure zero cases;
of those properties are inherent to auctions with heterogeneous valuations and full information in
general or generalize their counterparts of the special case presented in Bulow and Levin [2006]. Thus, no
originality of their proofs in the appendix is claimed.
19 Many
14
Derivation of Equation (3) with respect to b reveals the following optimality condition:
Proposition 2 (Optimality condition). For all b in the support of firm m’s equilibrium strategy
X
si gi (b) =
Fi ∈F \Fm
1
.
qm
Proof of Proposition (2). If b is an optimal bid for firm m in equilibrium
(4)
dVm (b,g−m )
db
=
0. Thus,
dVm (b, g−m )
= qm s m
db
X
si gi (b) − sm
(5)
Fi ∈F \Fm
and as a result
X
si gi (b) =
Fi ∈F \Fm
1
.
qm
(6)
Propositions 1 and 2 imply that in a matching equilibrium every firm is randomizing
over an interval of bids:
Corollary 1 (Interval support). In equilibrium, supports are intervals, that is there
are bids bm and bm for all firms m ≤ M such that firm m randomizes over [bm , bm ].
As a consequence, firm m’s equilibrium behavior over [bm , bm ] is characterized by



X 1
1 
1
1
−
,
gm (b) =
sm
|K(b)| − 1
qi
qm
(7)
i∈K(b)
which is the only solution to the system of equations defined by Proposition (2), where
K(b) denotes the set of firms competing at b. The expression within the square brackets
determines whether firm m is good enough to compete with the remainder of K(b).
15
Thus, the highest bid of a firm in equilibrium is independent of its size. After firm m
sterns this test, sm scales its bidding strategy, its probability density at b. This implies
that –ceteris paribus– bigger firms will randomize over wider intervals than smaller
firms. This confirms our intuition from the numerical example. To retrospectively
legitimize the matching equilibrium in the numerical example of section 2.1 observe
the following implication of Equation (7):
Corollary 2. If only two firms m and l are competing over an interval [b, b] in a
matching equilibrium, their respective strategies over this interval are gm (b) =
gl (b) =
1
sl qm
1
sm ql
and
for all b ∈ [b, b].
Thus, a firms strategy decreases locally in its opponents quality and its own size.
Smaller firms will therefore –ceteris paribus– on average, offer higher wages than larger
firms and expect better worker quality. The maximal bid offer of firm is independent
of firm size and increasing in quality:
Proposition 3 (Maximal bids). In any matching equilibrium firms of higher quality
bid up to weakly higher amounts, i.e. qm > ql ⇒ bm ≥ bl .
Based on these results an algorithm to identify matching equilibrium can be presented.
In general, one assumes a maximal bid b in the simultaneous wage setting game and
determines the set of firms willing to compete. This set is uniquely determined.20
Thereafter, the first firm to exit bidding can be determined. Keeping track of probability
offer mass this step is repeated until there is a single firm with positive probability
mass left. This is the probability with which this firm offer the workers’ outside option
of 0.21 Since this algorithm satisfies all features of matching equilibrium and each
20 As
long as low-quality firms are willing to compete, so are firms of higher quality. The cutoff must be unique.
detailed algorithm is provided in the appendix.
21 The
16
of its steps has a unique solution, matching equilibrium satisfies both existence and
uniqueness in the class of all multiple vacancy problems:
Theorem 1 (Matching equilibrium). Every multiple vacancy problem has a unique
matching equilibrium in which every firm m randomizes over an interval of bids
[bm , bm ]. Bidding strategies (probability densities) over bids are characterized by:
gm (p) =
 "



 s1m
#
!
1
|K(b)|−1
P
i∈K(b)
1
qi
−
1
qm



0
if b ∈ [bm , bm ],
(8)
else,
where K(b) indicates the set of firms competing at b.
On top of the intuition about effects due to firm quality and size, two observations
are noteworthy. First, firm m’s strategy at b is independent of the magnitude of the
bid itself. It rather depends on the quality of firms competing at b as well as firm m’s
characteristics. Secondly, small changes in firm quality will only cause small changes of
equilibrium strategies:
Remark 2. Strategies in a matching equilibrium are piecewise continuous in firm
quality. Thus, small changes in firm qualities do not alter competing sets of firms.
4 Properties of matching equilibrium
The unique source of inefficiency in a one-to-one matching equilibrium is randomness
of equilibrium strategies. The expected allocation in equilibrium, however, is always
guaranteed to be efficient. Naturally, the one-to-one setting is a special case of a
multiple vacancy problem with equal-sized firms. If the number of vacancies does not
vary among firms the following statement holds true:
17
Firm M
Firm M-1
Firm M
Firm M-1
Firm 2
Firm 1
Firm 2
Firm 1
b
b
Figure 3: Supports with variable vs. constant firm size
Corollary 3. Fix a multiple vacancy problem (q, s) with equal-sized firms, i.e. sm = s
for all firms m. Firms of better quality submit higher maximal bids bm and higher
minimal bids bm .
As a consequence better firms expect to hire better workers and thus:
Corollary 4. The expected allocation in the matching equilibrium of a multiple vacancy
problem with equal-sized firms is efficient (assortative).
Firm size variation induces a a fundamental feature which cannot be observed in the
one-to-one setting. The expected allocation in matching equilibrium turns inefficient.
An example of this feature is provided by section 2.
Theorem 2 (Inefficiency in expectation). The expected allocation in the matching
equilibrium of a multiple vacancy problem is, in general, inefficient.
Figure 3 shows potential supports with firm size variation on the left and the typical
pattern for equal-sized firms on the right. The following theorems provide deeper
insights into matching equilibrium:
18
Theorem 3 (Decreasing returns to scale). Fix a multiple vacancy problem (q, s)
with two firms m and l of equal (or sufficiently similar) quality q ≡ qm = ql . Without
loss of generality assume firm m to be bigger than l, i.e. sm > sl . Then,
(1) firm l pays on average higher wages than firm m and thus,
(2) firm l expects to hire a better average worker than firm m.
(3) Firm l’s profit per worker exceeds firm m’s but
(4) firm m’s profit exceeds firm l’s.
Proof of Theorem 3. The average worker quality firm m expects to hire when bidding b comes from the expected number of workers hired at bids below b and firm m’s
number of vacancies sm . Firm m’s expected payoff at b, Vm (b, g−m ) is thus:

Vm (b, g−m ) = qm sm 

1
Gj (b)sj  + (sm + 1) − sm b.
2

X
Fj ∈F\Fm
(9)
(1) Since sm > sl by Equation (7) we have that gm (b) < gl (b) for all b ∈ [bm , bm ] ∩ [bl , bl ]
and bm < bl .22 This proves the claim.
(2) is implied by (1).
22 This
holds true except bm = 0 in case of which firm m’s strategy would attach an atom to 0 whereas firm l’s
strategy does not.
19
(3) Now by Proposition (3) define b ≡ bm = bl . Exploiting Equation (7)



X
Vm (b, g−m )
1
= qm 
Gj (b)sj  + (sm + 1) − b
sm
2
Fj ∈F\Fm



X
1
= q 
Gj (b)sj  + sl + (sm + 1) − b
2
Fj ∈F\{Fm ,Fl }
 


X
sm 1




+
= q
Gj (b)sj − b + q sl +
2
2
Fj ∈F\{Fm ,Fl }
 


X
sl 1




< q
Gj (b)sj − b + q sm + +
2
2
(10)
Fj ∈F\{Fl ,Fm }
=
Vl (b, g−l )
.
sl
(4)



1
Gj (b)sj  + (sm + 1) − sm b
2
Fj ∈F\Fm
 


X
1
= sm q 
Gj (b)sj  − b + qsm sl + qsm (sm + 1)
2
Fj ∈F\{Fm ,Fl }
 


X
1
> sl q 
Gj (b)sj  − b + qsl sm + qsl (sl + 1)
2
Vm (b, g−m ) = sm qm 
X
(11)
Fj ∈F\{Fl ,Fm }
= Vl (b, g−l ).
Remark 2 implies the above theorem to hold true as well if qm − ql > 0 is sufficiently
small.
Thus, smaller firms pay –ceteris paribus– on average better wages in the matching
20
equilibrium to obtain better workers. A valid intuition for this result is as follows. If
firms are restricted to offer a single wage, it is immediate that higher bids are more
costly to larger firms. Section 6 —analyzing the case in which firms are allowed to
post one wage per vacancy— puts this intuition in perspective. To the extent to which
the number of vacancies can be considered a firm’s choice in real markets, Theorem 3
hints at the existence of an optimal number of workers, holding everything else fixed,
if relevant costs are considered. The subsequent lemma confirms our intuition that
–ceteris paribus– better firms accrue higher profits:
Lemma 1 (Profits increase in firm quality). Fix a multiple vacancy problem (q, s)
with two firms m and l of equal size s ≡ sm = sl . Without loss of generality assume
firm m to be better than l, i.e. qm > ql . Then, firm m’s profits exceed firm l’s.
Proof of Lemma 1. Let firm m hypothetically offer bl . Then,



1
Gj (bl )sj  + (s + 1) − sbl
2
Fj ∈F\Fm



X
1
= sqm 
Gj (bl )sj  + (s + 1) − sbl
2
Fj ∈F\{Fm ,Fl }



X
1
> sql 
Gj (bl )sj  + (s + 1) − sbl
2
Fj ∈F\{Fl ,Fm }



X
1
= sql 
Gj (bl )sj  + (s + 1) − sbl
2
Vm (bl , g−m ) = sqm 
X
(12)
Fj ∈F\Fl
= Vl (bl , g−l ),
which proves the claim.
Moreover, bigger firms –ceteris paribus– draw more benefit from a quality increase than
21
their smaller competitors:
Theorem 4 (Comparative statics in firm quality). Fix a multiple vacancy problem
(q, s) with two firms m and l of equal (or sufficiently similar) quality q ≡ qm = ql .
Without loss of generality assume firm m to be bigger than l, i.e. sm ≥ sl . Then, firm
m’s benefits more than firm l if
(1) all firms enjoy an equal relative increase in quality or
(2) only firms m and l enjoy an equal increase in quality or
(3) firms m or l face an equal quality increase one at a time.
Theorem 4 suggests that bigger firms –ceteris paribus– enjoy additional incentives
for self-improvement than their smaller competitors. Likewise they enjoy additional
motivation to improve firms’ production technology in the market. These results
can be cautiously interpreted as a stylized argument of a static model explaining a
dynamic market feature. It appears increased incentives to self-improve and improve
the structural environment are compatible with a positive correlation of firm quality
and size over time.
The maximal total surplus in a multiple vacancy problem is achieved by strictly
assortative matching. The performance of any allocation mechanism can be measured
in relation to the optimal total surplus:
Definition 3. In a multiple vacancy problem, the performance of an allocation
mechanism is the ratio of its expected total surplus in equilibrium to the total surplus
achieved by an assortative (efficient) allocation.
22
If markets grow large we expect the performance of a matching mechanism to improve.
To put this intuition to the test I analyze the implications of a market being large.
Markets can be perceived as large for different reasons. First, the number of workers
increases while the set of expanding firms is fixed:
Theorem 5 (Replication of workers). Fix a multiple vacancy problem (q, s). If
every firm grows at the same rate k ∈ N performance of the matching equilibrium is
(roughly) constant in k.23
Thus, increasing the market by expanding the set of workers and simply creating
additional openings at existing firms does unsurprisingly not improve the performance
of a match. As a consequence, the number of allocated workers is a sub-optimal measure
when referring to the size of a matching market. A very different conclusion can be
reached if the number of firms becomes large:
I analyze the limiting case of infinitely many firms with dense quality. Consider the
following variation of a multiple vacancy problem. The set of firms is given by a
continuum [0, M ] representing firm quality. An arbitrary density function24 s over
this interval represents firm size/firm type frequency.25 Continuing to rule out excess
demand respectively supply of workers the set of workers is given by a an interval
RN
RM
[0, N ] with a frequency density of η 26 such that η(x)dx = s(x)dx to ensure market
0
0
balance.
23 Performance
is constant ignoring a negligible integer problem caused by discreteness of a multiple vacancy
problem. This is explained in the proof.
24 The density function is arbitrary up the the point that it is a smooth positive bounded function with a
bounded first derivative.
25 Firm size does not have any bite in this definition. If firms and workers are dense the workers hired by every
firm are a set of measure 0 and infinitely similar (equal).
26 Assume η to satisfy the same assumptions as s.
23
Theorem 6 (Dense firms). If firm quality is dense the unique matching equilibrium
is efficient (assortative).
Proof of Theorem 6 by example.
(1) In what follows Theorem 6 is proved for a particular dense multiple vacancy problem.
The proof for the general case is provided in the appendix.
Assume F = [0, 4], s(x) = 1 ∀x ∈ [0, 4] and likewise W = [0, 4] with a frequency of
η(x) = 1 ∀x ∈ [0, 4]. Let f ∈ F denote a particular firm and w ∈ W a particular
worker. Bids are —as is standard throughout the paper— denoted by b.
Assume the efficient outcome in matching equilibrium. Formulating the firm’s problem
as a worker choice problem –equivalent to a bid choice problem if there is a one-to-one
relation between bids and workers– the firm chooses a worker w to maximize f ·w −b(w),
b(w) being the bid necessary to hire worker w. Thus, to support the efficient allocation
it has to be true that b0 (f ) = f. As a result, b(w) =
w2
2
+ c where c denotes a constant
determined by the fact that b(0) = 0 and thus, c = 0 in this example.27
Thus, every firm f bidding
f2
2
constitutes a matching equilibrium of this dense multiple
vacancy problem since no firm has a unilateral incentive to deviate if others don’t.
Further, it is the clearly the only equilibrium in pure strategies.
Thus, if the number of firms becomes large and average quality differential shrinks,
Theorem 6 indicates (near) efficient outcomes in markets if there are many firms.
Increasing the number of firms by quality extension has a similar effect on efficiency.
This is straightforward to show in a market with equal-sized firms. In this special case
it also easy to see that while welfare loss vanishes, the assortative allocation is not
27 There
are always infinitely many functions satisfying the firms’ incentive compatibility constraint. The unique
price funcyion can always be identified by normalization of b(0) = 0.
24
restored.28 In the general case it is however unclear how to increase the number of
firms by quality extension while holding firm-size variation fixed. As a consequence of
these findings, the notion of a large matching market should be reserved to markets
with an abundant number of competing firms.
A single worker replica of a multiple vacancy problem is the one-to-one matching
problem obtained by separating each firm of a multiple vacancy problem into separate
entities each posting a single vacancy. The firm quality of the spin offs equals the
original parent company’s quality.
Definition 4. A single worker replica of a multiple vacancy problem (q, s) is itself
a multiple vacancy problem (q R , sR ) in which each firm of the original problem is divided
into single entities each posting a single vacancy.
A single worker replica of a multiple vacancy problem admits an analysis of increased
competition in the market while holding the total number of vacancies and their
underlying quality fixed. In an analogy to a goods market a single worker replica of
a multiple vacancy problem features increased competition without the increase of
an analogue to supply. Since a single worker replica is a multiple vacancy problem
with equal-sized firms by definition, Corollary (4) establishes the unique matching
equilibrium to be assortative in expectation. Call the original multiple vacancy problem
of a single worker replica its total merger.
Corollary 5 (Total merger). Fix a multiple vacancy problem and consider its single
worker replica. The expected allocation of the unique matching equilibrium in the single
worker replica is assortative (efficient). This is not true for its total merger.
28 Consider
adding firms on top of the quality range. The effect on the bidding between low-quality firms is
ambiguous but will not overturn the internal rivalry effect.
25
One can derive from equation (7) that firm size variation is particularly harmful from
an efficiency perspective if better firms are bigger.
Corollary 6 (Merger of top firm). Fix a multiple vacancy problem (q, s). If the
top firm M demerges into single entities, welfare loss in the match decreases and vice
versa.
5 Wage compression
As a benchmark for wage comparison I introduce a counterfactual competitive equilibrium.
5.1 Competitive equilibrium
Definition 5. A competitive equilibrium of a multiple vacancy problem is defined
C ) ≥ 0 satisfying (1) individual rationality for each
as a wage vector wC = (w1C , ..., wN
vacancy and (2) incentive compatibility for every firm m:
(1) qm · j − wjc ≥ 0 for all Wj ∈ W (Fm ) and
(2) qm · j − wjc ≥ qm · k − wkc for all Wj ∈ W (Fm ) and Wk ∈ W (−Fm ).
I refrain from unnecessarily complicating this definition by introducing the notion of an
allocation since any competitive equilibrium is necessarily efficient (assortative) if firms
can discriminate between workers (see e.g. Shapley and Shubik [1971], Agarwal [2015]).
The above definition requires a wage per worker, does however not imply wages paid
within a firm to be necessarily different. To show, in general, that this definition does in
fact not admit an equilibrium in which firms do not discriminate between their workers
consider a simple example with ten firms of equal size:
26
Example 1. Let q = (q1 , ..., q10 ) and s = 2 for all firms. Let wi be the wage offered
by firm firm i for both workers it hires in the market. Incentive compatibility implies
prices to be increasing in quality and existence to necessitate
qm
ql
≥
s[m−l]+sl +sm −1
s([m−l]+1
for
m > l, s[m − l] denoting the total number of vacancies of firms better than l but worse
than m. Since this inequality has to be satisfied also for every pair of consecutive firms
there exists a non-discriminating equilibrium satisfying Definition 5 if among other
conditions F10 is at least about 20, 000 times better than F1 .
This result shows that, in general, firms do not offer equal wages in competitive
equilibrium according to Definition 5. To mirror the assumption of matching equilibrium
that every firm is bound to offer a single wage to all its incoming employees we can
weaken the incentive compatibility condition in Definition 5. Modeling firms’ preference
for non-discrimination we instead require competitive wages to be such that no firm
has an ex post incentive to lower or increase its one and only wage to obtain a different
set of workers.
Definition 6. A non-discriminating competitive equilibrium restricts firms to
pay equal wages to all workers it hires in the market. It consists of a wage vector
N D ) ≥ 0. wND satisfies individual rationality of firms and no firm
wND = (w1N D , ..., wM
wants to change its wage to hire a different set of workers.
Observe that this definition does not necessitate efficiency.
Lemma 2. Non-discriminating competitive equilibrium fails existence in the
class of all multiple vacancy problems.
Since there is no competitive equilibrium in which firms remunerate all their workers
equally, I resort to the general competitive equilibrium in 5 as benchmark. As mentioned above, the unique allocation supportable by wages in competitive equilibrium
27
is assortative. There is a range of wages supporting the efficient allocation in the
one-to-one setting. In particular, there is a upper bound, the worker-preferred wages
P
P
wnc =
qi as well as a lower bound, the firm-preferred wages wnc =
qi . Arguing
i<n
i≤n
about wage compression the firm-preferred wages being the smallest possible wages
in competitive equilibrium are the benchmark of interest. Since firms’ vacancies do
not have a direct strategic meaning if all firms are equal-sized, Kojima [2007] argues
that the argument of Bulow and Levin [2006] extends to the class of multiple vacancy
problems with equal-sized firms. However, competitive equilibrium in the multiple
vacancy case does in general not equal competitive equilibrium of its single worker
replica.
Proposition 4 (Upper bound of firm-preferred wages). The generalization of
the firm-preferred competitive equilibrium wages in a single worker replica is an upper
bound of the firm-preferred competitive equilibrium wages in its total merger.
Proof of Proposition 4. Let F (Wm ) be the firm hiring worker m in the assortative
allocation. Then, the generalization of the firm-preferred wages from the one-to-one
P
setting is wnC =
qF (Wi ) . Since every incentive compatibility constraint in a multiple
i<n
vacancy problem persists in its single worker replica, every competitive equilibrium of
the one-to-one problem is an equilibrium of its total merger.
Consider the following example showing that in general the firm-preferred equilibria do
not coincide:
Example 2. Consider a multiple vacancy problem with two firms (q, s) = ((1, 2), (2, 1)).
Solving the system of incentive compatibility constraints, the firm-preferred equilibrium
wages are wC = (0, 0, 2) whereas their counterparts in the single worker replica are
wC = (0, 1, 2).
28
The reason for this discrepancy is the deletion of incentive compatibility constraints
between vacancies posted by the same firm in a multiple vacancy problem. In turn, this
implies that the lowest wage paid by every firm has to be equal in both equilibria since
inter-firm incentive constraints persist. Thus, in general, wage differences between firms
persist in a multiple vacancy problem whereas wages within firms are compressed.29
5.2 Wage compression with multiple vacancies
Since the additional dimension of firm size variation significantly complicates proving
wage compression in the general multiple vacancy problem, I provide a proof for the
special case of equal-sized firms together with an argument that it’s logic extends to
almost all cases of relevance.
Theorem 7 (Profit gains in matching equilibrium). Fix a multiple vacancy
problem with equal-sized firms. Every firm has higher expected profits in the matching
equilibrium than in any competitive equilibrium.
Proof of Theorem 7.
30
Consider two firms with successive indices m − 1 and m
and define the sum of all vacancies of firms hiring below firm m as Sm . Then, firm m’s
profit in the firm-preferred competitive equilibrium is equal to
Vm =
s·m
X
qm · j − wjC .
(13)
s·m−1+1
29 An
algorithm to identify the worker-preferred competitive equilibrium can be found in the online appendix
of Agarwal [2015]. The process of finding the firm-preferred one is analogous.
30 This proof is inspired by the proof in the one-to-one setting presented in Bulow and Levin [2006]. The
extension, however, is not straightforward.
29
C
We know that ws·(m−1)+1
=
P
s · qi . Thus let Vm be a proxy for Vm :
i<m
V m = qm
s·m
X
j − s2 ·
X
qi
i<m
s·(m−1)+1
X
s−1
− s2 ·
qi .
= qm s · m −
2
(14)
i<m
Due to assortativeness of the matching equilibrium there has to be a bid b such that
firm m expects to hire workers s · (m − 1) + 1 through s · m when bidding b. Moreover,
by the properties of matching equilibrium firm m − 1 is bidding b as well. As a firm’s
profit is equivalent for all bids supported by its strategy, we can express the profit
difference in the unique matching equilibrium between firms of consecutive rank with
respect to quality as follows:
s−1
Vm (b) − Vm−1 (b) = qm · s · s · m −
2
s−1
− qm−1 · s · s · m −
+ Gm (b) − Gm−1 (b)
2
= Vm − Vm−1 + qm−1 · s · (Gm−1 (b) − Gm (b)).
(15)
Thus, the profit difference in the unique matching equilibrium between firms of consecutive rank with respect to quality exceeds the difference between the proxys for
competitive equilibrium. Observing that this difference itself exceeds the true difference
between profits in competitive equilibrium completes the proof.31
Theorem 8 (Wage compression). A matching equilibrium of a multiple vacancy
problem with equal-sized firms exhibits wage compression. That is, every worker,
31 In
the firm-preferred competitive equilibrium of a multiple vacancy problem with equal-sized firms the wages
differentials of better workers to the worst worker within a firm necessarily increase in firm quality.
30
except the ones hired by firm 1 in competitive equilibrium, i.e. workers 1 to s1 , is paid
less in matching equilibrium (in expectation) than in any competitive equilibrium.
Bulow and Levin [2006] show that in the one-to-one setting all firms gain from the
introduction of a central clearing house. Whereas the clearing house profit of the
lowest quality firm is identical to its competitive equilibrium payoff, every other firm
gains strictly. As a consequence, every worker except the lowest quality worker earns
below their competitive equilibrium income. Since those wage differentials accumulate,
the better the quality of the worker the bigger her wage cut. By Theorem 8 and
by extension of this argument, this holds necessarily true for an environment with
equal-sized firms. Moreover, inspection of the proof of Theorem 7 reveals that these
results hold true with greater slack if firms are larger. As a consequence, the gain of
firms under the presence of a clearing house over competitive equilibrium is increasing
in firm size and so is the wage cut of workers.
Due to the fact that all those results hold with slack, and even more so for high quality
firms respectively workers, minor changes to the firm size distribution do not overturn
any of these results by a simple argument in the vein of continuity. In fact, firms gain
and workers lose from the introduction of a clearing house if locally firm quality and
size do not abruptly increase at the same time. Thus, the workforce typically carries
the welfare loss. In fact, it takes quite substantial differences in firm size and quality
between consecutively ranked firms to to induce inefficiencies which hurt both workers
and firms. Since firm qualities can vary along two dimensions is it is impossible to
pin down a condition in closed form for every multiple vacancy problem. For instance,
in a two firm model, in which a single vacancy firm 1’s quality is normalized to 1,
a ten times bigger and more productive firm 2 would provoke enough efficiency loss
31
to make firm 2 worse off than in competitive equilibrium. In such cases, while the
entire workforce always loses, it is possible for some of the workers to expect higher
wages under a clearing house. In fact, this might hold true for the marginal group of
workers at this decisive point. To be more precise, consider a two firm model with
firms of substantial quality and size differential. If sufficiently different, the expected
wage gap between workers s1 and s1 + 1 in the matching equilibrium exceeds their
wage differential in any competitive equilibrium. While wages among workers 1 to
s1 and s1 + 1 to s1 + s2 will still be compressed, the wages of worker s1 + 1 and its
immediate neighbors of better rank might potentially exceed their competitive level.
At the same time the better firm potentially earns below its competitive level due to
severe mismatch. The wage sum of the entire workforce, however, always falls weakly
short of its competitive level. Due to the accumulation of profit gains and wage loss
respectively, however, the relevance of these special cases vanishes in even fair sized
markets. Simulation shows that markets with 10, 50, 100 firms require extreme jumps
in the firm-size distribution together with extreme firm quality differences to induce
merely a few firms to lose respectively workers to gain from a central clearing house.
6 Discriminatory matching equilibrium
This section rises a single question: Are the results of this paper entirely driven by the
stark assumption that firms prefer a single wage over freedom to set multiple wages to
target different workers? Naturally, we would expect better firms to set higher wages if
firms were able to wage-discriminate between its employees. This intuition, however, is
deceiving. Call an equilibrium of the matching game in which firms are allowed to set
different wages for their vacancies a non-discriminatory matching equilibrium.
32
Theorem 9 (Robustness of matching equilibrium). Fix a multiple vacancy problem (q, s). The unique non-discriminatory matching equilibrium remains an equilibrium
if firms are allowed to set one wage per vacancy.
However, this is not the only equilibrium in a discriminatory setting. In fact, there are
multiple equilibria in which firms post wages for all their vacancies between bm and bm .
i (b) be the offer density of firm m’s ith vacancy.
Let gm
Lemma 3 (Multiple equilibria). Every strategy combination such that
sm
P
i (b) =
gm
i=1
sm · gm (b) for all firms m and all bids b–gm (b) being firm m’s non-discriminatory
matching equilibrium strategy–constitutes a discriminatory matching equilibrium of a
multiple vacancy problem.
Proof of Lemma 3. If
sm
P
i (b) = s · g (b) for all firms except m and all bids b,
gm
m
m
i=1
firm m faces the same probability distribution as in the discriminatory problem. That
is, if the lowest bids of all vacancies by firm m is b, it expects to hire the same worker
with its lowest ranked bid as in the non-discriminatory equilibrium. Thus, Theorem 9
proves the claim.
As a consequence, every multiple vacancy problem has multiple equilibria in which
firms bid over identical intervals. Even more striking, these are the only discriminatory
matching equilibria of a multiple vacancy problem since no firm gains by conditioning
their vacancies’ strategies on each other.
Theorem 10 (Uniqueness of equilibrium type). The equilibria described in Lemma
3 are the only discriminatory equilibria of a multiple vacancy problem.
The mechanics behind this surprising result are briefly discussed in the description
of the numerical example in section 2.1. There cannot be any discriminatory pure
33
strategy equilibrium for the same reasons as discussed when restricting firm to pay
a single wage to all it workers. A discriminatory matching equilibrium of a multiple
vacancy problem does not resemble the matching equilibrium of its single worker replica
because vacancies of a firm do not consider each other as competitors. Thus, if at any
bid b only vacancies of one firm are competing, this firm is not optimizing.
The payoff of a vacancy declines due to internal rivalry if additional vacancies are
competing over the same interval. Thus, a large firm will disperse its bidding over a
wider interval than a small firm. In equilibrium, every vacancy has to be indifferent
over the firm’s entire support.
It is important to note that all results provided in this paper hold qualitatively true
for any matching equilibrium. If one considers the set of all equilibria there are two
extremal ones. The non-discriminatory and one where no pair of vacancies within a
firm overlap in bidding. Call the latter the perfectly discriminatory equilibrium. There
are two minor differences between those equilibria. First, the perfectly discriminating
equilibrium marginally reduces wage compression by adding wage intra-firm variation.
And secondly, the expected allocation in the former is marginally less efficient although
overall expected efficiency does not change. To provide intuition consider the following
example:
Example 3. Consider an example with 2 firms and 3 workers. Firms’ qualities and
sizes are given by their index. The non-discriminatory equilibrium produces a wage
vector of ( 23 , 1, 34 ) and assigns worker 2 with certainty to firm 2. It’s other worker is
basically determined by a coin flip. The discriminatory equilibrium produces a wage
5
19
vector of ( 12
, 1, 12
) and assigns workers 1 and 3 in expectation to firm 2. The bigger
the market the lesser the difference.
34
7 Conclusions
In real-world matching markets firms typically post multiple vacancies. Incorporating
this fact as a feature of a central clearing house model has far-reaching consequences.
When multi-unit firms bid strategically for their vacancies, two forces are simultaneously
at play:
1. Localized competition leads firms to compete with a relatively small set of similarlyproductive firms, as opposed to market-wide. This force exists in the singlevacancy setting as well.
2. Internal rivalry leads larger firms to offer, on average, lower wages. Thus, wages
decrease in market concentration. This force is unique to the multiple-vacancy
setting.
While both forces affect the distribution of surplus among firms and workers, the second
force also impacts the efficiency of the allocation. Internal rivalry causes large firms to
not compete aggressively for workers, leading to an inefficient allocation where large
firms end up with worse workers on average, conditional on their quality. Furthermore,
because internal rivalry creates downward pressure on wages, wage compression becomes
even worse in multiple-vacancy settings.32
The equilibrium of my model features limited intra-firm variation in wages. This feature
is consistent with real-world evidence from the NRMP [Niederle et al., 2006] that firms
prefer not to discriminate between workers which hold equal positions in the firm.
32 As
such, this paper argues that the example presented in Kojima [2007], while intriguing, relies on strong
assumptions.
35
I find that local inefficiencies do not vanish when markets become large, but they
vanish as a fraction of total market surplus. In this sense, the potential efficiency loss
is of greater concern in smaller markets.33 In this connection, it is worth noting that
high-skill labor markets are frequently quite small, including many NRMP specialties
markets. Therefore, the efficiency loss studied in this paper has the potential to be
quantitatively relevant.
33 An
extremal example for this claim is provided by the numerical example in section 2, in which a central
clearing house only slightly outperforms a random allocation process.
36
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39
Appendix
Proofs omitted from the text
Proof of Proposition 1. (a) In case of pure strategies, there must be a firm with an incentive
to reduce its bid or outbid an opponent marginally. (b) Suppose there was an interval in
which only one firm bids. This firm cannot be optimizing. (c) Suppose b > 0 is the smallest
bid by any firm in equilibrium. Then, at least one firm bidding b would do better when
bidding 0 instead. (d) Fix a bid b and identify the set of firms H offering a price b just
below b. In order for one such firm, say firm h, to offer a price b just above b we have to
P
P
have
sm · gm (b) ≥
sm · gm (b). The sum over all h ∈ H establishes the claim. (e)
m∈H\h
m∈H\h
Suppose m were to make offers just below b and above b but not in between. Thus, for every
b ∈ (b, b)


qn · 
X
Fj ∈F \Fm
X
sj · Gj (b) −
sj · Gj (b) ≤ b − b.
(16)
Fj ∈F \Fm
sj ·gj (b) = q1m for every b in the interval.
P
P
sj · gj (b0 ) >
This must also be true for a b0 marginally above b. But then
sj · gj (b)
P
As firm m does not bid in the interval (d) implies
Fj ∈F \Fm
Fj ∈F
Fj ∈F
in contradiction of (d). (f ) Suppose that no firm bids in (b, b).Then, there is a lowest bid
b0 above b. At least one firm cannot be optimizing at b0 . (g) Suppose firm m bids b with
positive probability. By (b) and (f) there is some firm just bidding below b. This firm cannot
be optimizing.
Proof of Proposition 3. If firm l is good enough to compete at b, so is firm m. Firm size
will –conditional on quality– determine the length of the interval over which a firm randomizes
but not its upper bound.
Proof of Corollary 3. Assume firm Fm ’s quality to exceed Fl ’s, i.e. qm ≥ ql . Then, by
Proposition 3 bm ≥ bl . Now, since gm (b) ≥ gl (b) ∀b ∈ [bm , bm ] ∩ [bl , bl ] by Equation (7) and the
fact there is no b > bm at which Fl bids, the claim has to be true.
40
Proof of Corollary 4. The statement follows directly from Corollary 3 and the fact that
Equation 7 implies that gn (b) ≥ gm (b) if gn (b) > 0 and qn ≤ qm .
Proof of Theorem 2. A single pair of firms, similar enough in quality, with the better firm
being the bigger one, implies that result. See the numerical example in section 2.
Proof of Theorem 4.
(1) Create a new multiple vacancy problem (q 0 , s) with q 0 = k · q for k > 1.34 By Equation (7)
an equal relative increase in the quality of all firms causes all strategies to shrink by a factor
of k. Thus, the resulting matching equilibrium of (q 0 , s) resembles the matching equilibrium
of (q, s) in so far that everything which holds true for (q, s) at b, holds true for (q 0 , s) at k · b.
Thus, every firm’s profit will increase by the factor k. Point (4) of Theorem 3 then proves the
claim.
(2) Create a new multiple vacancy problem (q , s) with qi = qi ∀Fi ∈ F \ {Fm , Fl }. Let all
parameters and variables with an refer to the new problem whereas standard notation refers
to the original problem. Let q ≡ qm
= ql = q + for a sufficiently small > 0 and b ≡ bm = bl
and b ≡ bm = bl . For all firms Fi with Gi (b) = 0 in (q, s) it will hold true that Gi (b ) = 0 in
(q , s). Thus, Fm ’s and Fl ’s expected lowest-ranked worker when offering their maximal bid is
the same as in the original problem. Since the increase of profits for Fm and Fl is a first order
change whereas there is a second order change in prices due to decreased strategies of all firms
except Fm and Fl competing at b both firms expect higher profits in the matching equilibrium
of the new problem. Point (4) of Theorem 3 now proves the claim.
(3) By the argument in (2) either firm’s profit increases if it is the only firm facing an increase
in quality. Point (4) of Theorem 3 now proves the claim.
Proof of Theorem 5. Consider a multiple vacancy problem (q, s) and its k-th multiple with
respect to firm size (q, sk ) with sk = k · s. The stated result is independent of how we generate
a sufficient number of workers to fill all vacancies. Consider 2 cases:
34 The
proof goes through with k < 1 as well.
41
(1) (Replication by quality extension) Multiplying the number of workers by k and extending thereby the top worker’s quality from N to k · N in line with Definition 1 of a multiple
vacancy problem will by Equation (7) divide each firm’s strategy by k and not change the sets
of firms competing with each other. To be more precise, if firm m bids b in the original problem
it will bid k · b in the new problem. If every firm hires k times as many workers with a k times
better average we would expect production to increase by a factor of k 2 . Due to an integer
problem caused by discreteness this is not true. Consider firm F1 hiring workers W1 to Ws1
in the original problem and consequently 1 to k · s1 in the new problem. It’s production will
increase from 12 q1 (s1 + 1) s1 to k 2 · 21 q1 s1 + k1 s1 . This is simply caused by our convention
of setting the quality of the lowest-ranked worker in the market to 1. Consider worker skills
to be measured as intervals on a continuum, i.e. the lowest worker’s productivity is uniformly
distributed over [0, 1] and so on. Thus in expectation W1 ’s quality is 12 , W2 ’s is
3
2
and so on
and so forth. Then if firm F1 were to hire W1 to Ws1 in the original problem and W1 to k · Ws1
in the new problem its output would increase from
q1 ·s1
2
to k 2 · q12·s1 . Since this logic extends to
all other firms output in the matching equilibrium increases by k 2 . By the same token maximal
output increases by k 2 and thus performance is constant in k. This result does not depend
on the uncertainty introduced over worker skills. It holds true if worker k’s quality is simply
defined to be k − 12 .
(2) (True replication of workers) Consider a multiple vacancy problem and extend by
increasing each firms’s size by a factor k and by cloning each worker k times in contradiction
to Definition 1 of a multiple vacancy problem. This is of course at odds with all our findings
about equilibrium strategies. The changes however would be negligible. If firm F1 were to hire
workers W1 to Ws1 with qualities (1, 2, ..., s1 ) in the original problem it would hire k workers of
each quality between 1 and s1 in the replicated problem. Looking at the proof of Proposition
(2) it comes clear that cloning workers and increasing firm size cancels each other out. Thus,
firms would play the same bidding game. Thus, every firm hires k times as many of the same
workers at the same wage as before. This increases every firm’s output both in the matching
equilibrium and the efficient allocation by a factor of k and thus the claim holds true without
42
an integer problem.
Proof of Theorem 6. The problem is defined by F = [0, M ], s, W = [0, N ] and η with
RN
RM
η(x)dx = s(x)dx. Impose the efficient (assortative) allocation. In this allocation f hires
0
0
w(f
R )
w(f ) such that
η(x)dx =
0
Rf
s(x)dx. Due to the fact that both η and s are continuous,
0
bounded and positive over their support w(f ) exists and is increasing. Now construct a price
function p(w) such that p0 (f (w)) = f . Such a function exists and is increasing by definition.
Determine p(w) uniquely by setting p(0) = 0. Clearly, no firm has an incentive to deviate
unilaterally.
Proof of Lemma 2. First I show that
qm
sm
>
ql
sl
ND
implies wm
> wlN D .
Suppose this is not true and in particular suppose firms m and l to offer consecutive wages
ND
< wlN D but
such that wm
qm
sm
>
ql
sl .
Incentive compatibility requires now
!
P
i∈F m
si +sm
X
qm
i=
P
i∈F m
qm
j−
ND
sm wm
2
P
si + sm + 1 sm
i∈F m
=
ND
− sm wm
2
si +1
!
P
i∈F m
si +sl +sm
X
≥qm
i=
P
i∈F m
qm
j − sm wlN D =
2
P
2
si +sl +1
ND
⇔wlN D ≥ wm
+ q m sl
43
(17)
si + 2sl + sm + 1 sm
i∈F m
− sm wlN D
and
!
P
si +sl
ql
i∈F l
X
ql
i=
P
j−
sl wlN D
2
P
si + sl + 1 sl
i∈F l
=
− sl wlN D
2
si +1
i∈F l
!
P
si −sm +sl
ql
i∈F l
X
≥ql
i=
P
P
2
ND
j − sl w m
=
(18)
si − 2sm + sl + 1 sl
i∈F l
2
si −sm +1
ND
− sl wm
i∈F l
ND
⇔wlN D ≤ wm
+ q l sm .
Now
wlN D ∈ [wlN D + qm sl , wlN D + ql sm ] = ∅
and henceforth, there cannot be two firms with
ND
wm
< wlN D . Thus wiN D is increasing in
qi
si
qm
sm
>
ql
sl
(19)
offering consecutive wages such that
as otherwise there would be at least a single
consecutive pair as described.
Now consider a multiple vacancy problem with (qi , si ){i=1,2,3} = ((1, 1), (1.5, 2), (2, 3)). The
above logic tells us that w1N D ≥ w2N D ≥ w3N D implying 6 − w1N D ≥ 1 − w3N D and 12 − 3w3N D ≥
30 − w1N D and as a consequence w1N D ∈ [w3N D + 6, w3N D + 5] = ∅.
Proof of Theorem 8. Theorem 8 follows immediately from Theorem 7.
Proof of Theorem 9. Consider all firms except firm m to stick to their strategies in the
problem without restriction. Firm m enjoys now freedom to set one wage per vacancy. If firm
m is bidding the maximal market bid in non-discriminatory equilibrium it cannot bid more in
any discriminatory equilibrium since it would not be optimizing. Thus, assume firm m not to
bid the maximal market bid in non-discriminatory equilibrium. By Equation (7) the highest bid
of a firm is independent of its size. Thus, firm m cannot bid above bm for any of its vacancies
in discriminatory equilibrium.
44
Consider firm m to bid for all its vacancies within [bm , bm ]. The question is, can it improve
by differentiation within this interval? By Theorem 1 firm m is indifferent between any bid in
[bm , bm ] if it were to bid b for all its vacancies. Then if W e (b) is the expected lowest ranked
worker firm m hires if bidding b, its profit is
qm · [W e (b) + W e (b) + 1 + ... + W e (b) + sm − 1] − sm · b
(sm − 1)(sm − 2)
e
= qm sm · W (b) +
− sm · b.
2
(20)
This profit being the same for every b in the interval, for b1 < b2 we have that
(sm − 1)(sm − 2)
] − s m · b1
2
(sm − 1)(sm − 2)
] − sm · b2
= qm [sm · (W e (b2 ) +
2
(21)
qm · W e (b1 ) − b1 = qm · W e (b2 ) − b2 .
(22)
qm [sm · W e (b1 ) +
and thus,
Taking into account that any vacancy posting a price below another vacancy of the same firm
increases the latter’s expected worker rank by one we have that that the profit of firm m from
posting sm (potentially) different bids b1 , b2 , ..., bsm within [bm , bm ] is
qm · [W e (b1 ) + W e (b2 ) + 1 + ... + W e (bsm ) + sm − 1] −
sm
X
i=1
bi
(23)
(sm − 1)(sm − 2)
− sm · b
= qm sm · W e (b) +
2
for all b in [bm , bm ]. Thus, firm m is indifferent among all prices in this interval for all its
vacancies.
Finally, we have to show that firm m does not profit from bidding below bm . If this were
true, one of the vacancies posting below bm at b would be necessarily better off than its
counterpart in the non-discriminatory matching equilibrium. By the logic above, then bidding
45
b for all vacancies must be strictly better than any b in [bm , bm ]. But this contradicts the
non-discriminatory matching equilibrium in the first place.
Since firm m was chosen arbitrary this proves the claim. This proof depends on the assumption
that worker qualities are distributed uniformly. There is, however, no indication that a similar
argument with more complicated density functions would fail in the general case.
Proof of Theorem 10. Fix a multiple vacancy problem (q, s) and consider a strategy combination of all firms for all vacancies and assume it to be a discriminatory matching equilibrium.
Consider firm m submit different bids for its vacancies. Denote the vacancies of firm m as υ1
to υsm and let the index indicate the rank of the expected bid of a vacancy, i.e. the expected
bid of vacancy υi exceeds the expected bid of a υj weakly if i > j. I claim that the expected
profit for each vacancy must be the related in equilibrium in the following sense. Consider υsm
and υsm −k . The expected profit of υsm must exceed the expected profit of υsm −k by exactly
qm · k (upon relabeling if bidding overlaps) in equilibrium for every k < sm .
If this were not the case, firm m could do better by either adjusting υsm ’s or υsm −k ’s bid. Since
this has to hold true for every action in the support of firm m’s bidding strategy and no vacancy
can employ a pure strategy in equilibrium, every vacancy by itself (upon relabeling if there is
overlap) has to be indifferent over its support. Thus, each vacancy υi of firm m has to face a
probability distribution in equilibrium such that
X
si
X
i
gm
(b) =
Fi ∈F \Fm i=1
1
1
=
i
qm
qm
(24)
at every bid b in its support.
Consider gaps in vacancies’ or a total firm’s bidding strategy. By the analogue of condition (e)
in Proposition 1 this cannot be true in equilibrium. Hence all firms randomize over intervals.
Now it follows from the uniqueness of non-discriminatory equilibrium that there can not be any
46
equilibrium other than the ones characterized in Lemma 3. Again, this proof depends on the
assumption that worker qualities are distributed uniformly. There is, however, no indication
that a similar argument with more complicated density functions would fail in the general
case.
Algorithm to obtain matching equilibrium strategies
Algorithm 1.
1. Define a mass vector µ = (µ1 , ..., µM ).
2. Assume p1 to be the highest price offered by any firm.
3. Find the set K(p1 ) by identifying the firm with the lowest quality willing to compete at
p1 by determining arg min gm (p1 ) > 0 .
F
4. Determine the firm with the greatest positive offer density at p1 Fk and find the lowest
price offered by that firm p2 .
5. Update the mass vector by calculating µm = 1 −
gm (p1 )
gk (p1 ) .
6. Find the set K(p2 ) by identifying the firm with the lowest quality willing to compete at
p2 by determining arg min gm (p2 ) > 0|µm > 0 .
F
7. Determine the firm with the lowest ratio of remaining probability mass to offer density at
p2 Fl and find the lowest price offered by that firm p3 .
8. Update the mass vector by calculating µm = µm −
gm (p2 )µl
gl (p2 ) .
9. Repeat steps 6 to 8 with consecutive labeling of prices until there is a single firm Fi left
such that µi > 0. Employ the equilibrium strategy of firm Fi with an atom of µi at 0.
10. Calculate p1 to pM recursively.
Observe that every step of the above algorithm leads to a unique outcome and is always feasible.
Thus every multiple vacancy problem has a unique (non-discriminatory) matching equilibrium.
47