Holdup and hiring discrimination with search friction

TL;DR: In this paper, a holdup problem on workers' skill investment can arise when employers adopt discriminatory hiring norm to extract higher than socially optimal profit, when hiring priority is determined by both productivity-dependent (skill level) and -independent characteristics (discrimination).

Abstract: A holdup problem on workers' skill investment can arise when employers adopt discriminatory hiring norm to extract higher than socially optimal profit. When hiring priority is determined by both productivity-dependent (skill level) and -independent characteristics (discrimination), skill investment decision becomes strategic between the discriminated and favored group. We consider frictional markets with either posted or bargained wage (fixed sharing rule). With posted wage, depending on market tightness there may be equilibrium or multiple equilibria on skill investment. With discriminatory hiring, if in equilibrium both groups stay high skilled, both are worse off and firms better off; In any equilibrium where one group underinvest, the other group remain high skilled and are better off, while firms are worse off with discrimination. With bargained wage, similar equilibrium where the favored group underinvest exists, and firms incur cost for an intermediate range of bargaining power when they discriminate.

Summary

1 Introduction

• A holdup problem arises when some investment is sunk ex ante by one party, and the payo is shared with that one party's trading partner.
• Since cost has no other use once sunk, that trading partner will have every incentive to squeeze the pro t at the ex post stage.
• On one hand, there is ranking by productivity-dependent type identity: workers are either high skilled (type H) or low skilled (type L); high skilled have priority to low skilled simply because such ranking gives rms higher pro t.

1.1 Relation to the literature

• Job search process is an important channel through which discrimination keeps functioning in the labor market.
• Search frictions are derived endogenously through agents' sequential strategic interactions.
• While the setup of wage bargaining (no information of level of wage before matching) is more prevalent, it neglects an important trade-o that the workers make to some extent in their search for jobs: the wage and the probability of obtaining it.
• Rosen (1997) is an exception and shows that discrimination can result even if there are no di erences across groups.

2 The model without discrimination

• The authors start with a context without hiring discrimination.
• Consider an economy populated by two kinds of agents, the workers and the rms.
• Firms observe skills of job seekers, and announce the wage (wL, wH), also known as Stage 1.
• 5Incompleteness of contract is the source of ine ciency for the holdup problem.
• See Acemoglu and Shimer (1999) for related literature.

2.1 Speci cation of the Strategies, matching probabilities, and payo functions

• This section provides a quick summary for the general understanding of the context.
• The authors should distinguish two terms: (1) each job seeker's expected payo from application, and (2) her expected market payo .
• By doing so, the authors are able to abstract from some equilibrium which only arises under theoretical rigor but at the same time yields insights to a limited extent and induces unnecessary complexity in analysis.
• Ut, solving wt from the constraint, substituting it into the objective function, and maximizing with respect to qt, the authors can obtain an optimal functional relationship between q ∗ t and Ut.

2.2 Decentralized Market Equilibrium without discrimination under Notion 2

• Firms' wage o ers are conditioned on job seekers' skill levels, so the authors rst consider the skill investment decision of job seekers at rst stage.
• Some invest in high skill, while the remaining in low skill.
• The authors now show that the rivalry between the 8The program could also be understood as a deviating rm's pro t maximization, taking all the other rms' best response as given.
• There is no incentive for them to deviate, and the output is highest among all the equilibria.
• When the value of yH−yLEH−EL is moderate, there exists an equilibrium where job seekers are indi erent from being high skilled or low skills; all rms nd it optimal to attract both skill types; the output is lower compared to the previous equilibrium.

2.2.1 Decentralized Market Equilibrium without discrimination under Notion 1

• Under Notion 1, there is only one skill type present in the market.
• It turns out that the threshold which makes workers indi erent is the same as β̂ established under Notion 2.

2.3 Constrained e cient allocations

• The objective of this section is to nd the e cient allocations in the centralized market, and evaluate whether the decentralized market attains its e ciency.
• More precisely, the social planner chooses the fraction of workers to be high-skilled, divides rms into di erent groups to attract distinct compositions of workers, and assigns workers to match with a certain group of rms.
• If αp ∈ (0, 1), there are both high and low skilled job seekers and it is optimal for the planner to assign all rms to post wages for both the high and low skilled (shown in Shi (2006)).
• The above objective includes all cases with di erent values of α.
• The equilibrium labor allocation and skill investment choice are socially optimal.

3 The model with hiring discrimination

• Consider an economy where workers can be partitioned into two groups, group a and group b, according to certain trait which is irrelevant to productivity.
• The two group of workers are ex ante identical in all other aspects.
• The parts 1−e −qa qa and 1−e −qb qb capture the within group competition, while the remaining parts with x capture the between group competition.
• Then the employment probability of workers from the preferred group (group a) is higher under discrimination than that in the case without discrimination.
• The closer x approaches to the extremes of the interval [0, 1], the more intensive the hiring discrimination is.

3.1 The case of strong discrimination: x = 1

• These assumptions help introduce some heterogeneity which is not related to productivity among the labor pool.
• Firms are not allowed to post wages which are dependent on the group identity, also known as Assumption 1.
• The authors denote in this section the expected market payo of high skilled job seekers from group a and group b as UaH and UbH respectively.
• The above assumptions imply UaH = FaH (qaH , qbH)×.
• In the next section, the authors review the results from LMD (2005), where they study the case with discrimination but no di erence in workers' skill levels (or productivity).

3.2 Existing results revisited and reinterpreted

• In a context where there are two groups of workers with identical productivity (skill level) and rms strongly prefer group a to group b. LMD (2005) show that any subgame-perfect competitive equilibrium (SPCE) is separating.
• LMD (2005) show that there is no wage to which both groups of job seekers apply.
• That is, there are some rms posting a higher level of wage attracting only the preferred group a, whereas the rest of rms o ering a lower wage which is applied only by the discriminated group b (see Proposition 2 in LMD (2005)).
• Following are several noteworthy properties of such an equilibrium.
• The authors enter more detailed discussions in the following section.

3.3 Analysis under our context

• In the last section, the authors interpreted the equilibrium of the wage posting subgame given that all workers choose to be high skilled.
• Whenever they contemplate to lower skill investment, they understand that they will be ranked behind the high skilled group b; then the term e−qbH which captures the competition from bH will appear in their payo s.
• The results on the equilibrium are summarized as follows: Proposition 3: Under the above assumptions, there exist two thresholds β̂2 and β̂1 with 0 < β̂2 < β̂1 < β̂, such that (1) When 0 < β < β̂2, there exists a unique equilibrium in which both group a and group b invest in high skill, (aH, bH).
• Since qP1aH does not depend on yH and yL, there exists always a pair of yL and yH such that this condition is satis ed. 17 Documents de travail du Centre d'Economie de la Sorbonne - 2016.02 rms turn out to earn lower expected pro ts compared to the case without discrimination.

4 Comparison with xed sharing rule (Wage Bargaining)

• The authors shut down the channel through which rms use wages to in uence workers' choices on applications, and examine whether the ine ciency can be alleviated.
• If the authors denote the bargaining power for all workers as ψ, then from the output yt, workers receive ψyt, and rms receive (1− ψ) yt.
• The authors focus on the case where ψ is the same for both skill levels, otherwise there is too much degree of freedom.
• Each group of workers invest in skills simultaneously.
• Their simple result suggest that rms are simply better o not discriminating when wages are principally bargained, since the loss in pro t from discriminating in the high skilled sector may surpass the gain from discriminating in the low skilled sector.

5 Discussion

• LMD (2005) have shown that their economy under discrimination with workers' identical in productivity can be generalized to take into account rms' free entry.
• Then rms which earn expected non-positive pro ts after the reduction of capital cost would simply not enter into the market.
• Shi (2006) shows that in such a model with multiple skill levels free of discrimination, the result that rms always rank the high skilled workers in priority to the workers with lower skills can be generalized.
• The di culty under the context with discrimination, as just stated, is on the extent of the game.

6 Conclusion

• The authors study a holdup problem where rms can use discriminatory hiring norms to extract higher than socially optimal pro ts.
• In case wages are posted, the authors suggest that depending on the market tightness there may be equilibrium or multiple equilibria on skill investment; in some equilibrium the discriminated group can obtain higher expected payo compared to the case without discrimination12 and rms can be worse o .
• The authors also consider xed sharing rule (bargained wage) and make a comparison.
• Firms' pro ts are piecewise monotone because the skill composition hence the average productivity of the market improves discretely with respect to the bargaining power, and pro t loss may be incurred with discrimination within an intermediate range of bargaining power.

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Holdup and hiring discrimination with search friction
Sheng Bi, Yuanyuan Li
To cite this version:
Sheng Bi, Yuanyuan Li. Holdup and hiring discrimination with search friction. 2016. �halshs-
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Documents de Travail du
Centre d’Economie de la Sorbonne
Holdup and hiring discrimination with search friction
Sheng BI, Yuanyuan LI
2016.02
Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647 Paris Cedex 13
http://centredeconomiesorbonne.univ-paris1.fr/
ISSN : 1955-611X

Holdup and hiring discrimination with search friction
∗
BI Sheng
†
, LI Yuanyuan
‡
30th, December
Abstract
A holdup problem on workers' skill investment can arise when employers adopt discriminatory hir-
ing norm to extract higher than socially optimal prot. When hiring priority is determined by both
productivity-dependent (skill level) and -independent characteristics (discrimination), skill investment
decision becomes strategic between the discriminated and favored group. We consider frictional markets
with either posted or bargained wage (xed sharing rule). With posted wage, depending on market tight-
ness there may be equilibrium or multiple equilibria on skill investment. With discriminatory hiring, if
in equilibrium both groups stay high skilled, both are worse o and rms better o; In any equilibrium
where one group underinvest, the other group remain high skilled and are better o, while rms are worse
o with discrimination. With bargained wage, similar equilibrium where the favored group underinvest
exists, and rms incur cost for an intermediate range of bargaining power when they discriminate.
JEL Classications:
J7, J42
Key words:
Discrimination; Directed Search; Pre-matching Investment
∗
We are very grateful for the discussions and encouragement from Professor Bernhard Eckwert, Professor François Langot,
Professor Anna Zaharieva, Professor Bertrand Wignoille, Nikolai Brandt, Dr. Andreas Szczutkowski, as well as participants in
BIGSEM Colloquium, BIGSEM Workshop, EDEEM Annual meeting, and GAINS seminar. Yuanyuan LI would like to thank
the EDEEM program for the nancial support, and BI would like to thank for the nancial sponsorship from University of
Paris 1 and University of Bielefeld, and Deutsch-Französishe Hoshschule.
†
University of Paris 1, and University of Bielefeld; E-mail: brbisheng@gmail.com
‡
University of Bielefeld, and University of Paris 1
1
Documents de travail du Centre d'Economie de la Sorbonne - 2016.02

1 Introduction
A holdup problem arises when some investment is sunk ex ante by one party, and the payo is shared
with that one party's trading partner. Since cost has no other use once sunk, that trading partner will have
every incentive to squeeze the prot at the ex post stage. In an important study on such a problem in a
labor market with search friction, Acemoglu and Shimer (1999b) show that with rms' sinking capital and
ex post wage bargaining, the equilibrium is always inecient, since wages paid ex post can be so high such
that rms' ex ante incentive of investment is harmed; while if rms are able to post wages to direct workers'
search, then the holdup problem to rms' investment no longer appears; the eciency can be achieved,
because wage posting allows workers to observe oers and choose where to apply, and it induces workers
to optimize their expected payo from application by making trade-o between every wage they observe
and the probability of obtaining it. Within conventional wage posting framework, we spot another source of
ineciency in a holdup problem where workers sink skill investment cost: when the market is crowded for the
rms, by adopting a discriminatory hiring norm rms are able to expropriate higher than socially optimal
level of prot, and this has the consequence of discouraging the investment incentives for both the favored
and discriminated groups. We analyze the impact of such rent seeking behavior of rms on the structure of
market segmentation, and on the workers' skill investment incentives.
When discrimination is absent, the wage posting economy with workers' ex ante skill investment attains
eciency in the equilibria, and we show which equilibrium emerges depends on the rivalry between the
log return to skill and the market tightness (workers/rms ratio) which measures the degree of market
competition. The fundamental reason behind this eciency result is that skill achievement is a quality which
can be legally written into the wage contracts. It is a dierent story when other (binary) characteristics which
are not closely related to productivity, such as gender, race, height, origin etc. enter also into rms' preference.
Under equal pay legislation, posted wages can not be conditioned explicitly on these characteristics; however,
if rms still select workers according to their preference on these characteristics, a separating equilibrium
can result where separate rms post dierent levels of wages, and workers of dierent groups sort themselves
and apply to dierent wages: the market is then endogenously segregated. On the side of rms, they have
incentive to adopt such discriminatory hiring norm, when workers' return to skill investment is suciently
high; in that case discrimination allows them to grasp higher than the socially optimal level of operating
prot. On the side of the workers, it proves that both the discriminated group and favored group are worse
o: for the former, it is because discrimination discretely reduces the labor market opportunity of these
workers, who anticipate discrimination, then demand lower wages, which makes them cheaper to hire; for
the latter, it is so because when rms are able to hire the discriminated workers cheaply, it is as if rms
enjoy larger market power, which allows them to suppress further the undiscriminated workers' expected
payo. Naturally, anticipating discrimination, all groups expect lower payo from search, jeopardizing their
skill investment incentives.
A key feature of our study is the multidimensionality of characteristics based on which workers are
ranked. On one hand, there is ranking by productivity-dependent type identity: workers are either high
skilled (type
H
) or low skilled (type
L
); high skilled have priority to low skilled simply because such ranking
gives rms higher prot. On the other hand, there is ranking by productivity-independent group identity:
workers belong either to the favored (group
a
) or the discriminated group (group
b
). The resulting ranking
schedule has the following order:
aH  bH  aL  bL
. It reads: given any skill level, group
a
are preferred
to group
b
; the high skilled are always preferred to low skilled. Under such an intertwined ranking order,
the skill investment decision for dierent groups becomes strategically interdependent. Focusing on Nash
2
Documents de travail du Centre d'Economie de la Sorbonne - 2016.02

pure strategy equilibrium on skill investment, in the wage posting economy, we nd that depending on the
value of market tightness there can be equilibrium or multiple equilibria on skill investment due to that
interdependence. Compared to the case without discrimination, when the market is very crowded (market
tightness is small) for the rms, discrimination is protable for rms and all the workers are worse o;
as the tightness further increases, both group can choose low skill and in equilibrium whenever one group
underinvest, the other group remain high skilled and are better o, while the rms are worse o with
discrimination. In particular, the equilibrium where the favored group underinvest, while the discriminated
group choose to remain high skilled exists; And in this case rms' prots drop since workers' underinvestment
in skill leads to lower average productivity in the economy compared to the case where discrimination is
absent.
In the economy where wages are bargained (determined according to a xed sharing rule) after matching
hence do not direct search, we nd similar equilibrium where the favored group underinvest, hence earn lower
expected payo compared to the case without discrimination within a certain region of bargaining power;
in such an equilibrium, surplus is transferred from rms and favored group to discriminated group. Firms'
prots are piecewise monotone, because increase of workers' bargaining power can increase workers' incentive
of skill investment, hence discretely improves the market skill composition and average productivity. We
also nd that there is an intermediate range of workers' bargaining power for values of which rms are
worse o by discriminating, due to discouraged skill investment from discriminated group. All in all, the
key dierence between wage posting and wage bargaining is that the actual wage now exogenously pegs on
the productivity, and rms can no longer manipulate their market power by translating their discriminatory
preference into constantly lower wages.
1.1 Relation to the literature
Job search process is an important channel through which discrimination keeps functioning in the labor
market. Several papers have highlighted the impact of discrimination through job search channel to the
wages gaps. To name a few, Pendakur and Woodcock (2010) show that the existent glass ceilings for the
immigrant and minority workers may be attributed by large measure to their poor access to the jobs in
high-wage rms; As well, in an important article from Ritter and Taylor (2011), they show that most of
the disparity in unemployment rate could not be explained by cognitive skills that emerge at an early stage,
although for wage gap it could be the case. This result concerning the unemployment disparity is conrmed
by the nding that this disparity is still signicant even for workers of similar skill levels.
Our work is most closely related to the directed search literature
1
. In this literature, search frictions
are derived endogenously through agents' sequential strategic interactions. Taking into account strategic
interaction allows search externality to be internalized. The resulting economy remains competitive, albeit
with a non-Walrasian market structure, and prices play an allocative role to achieve eciency. To the best
of our knowledge, among the discrimination literature with search friction, only two of them are built upon
wage posting context. Lang, Manove, and Dickens (2005, hereafter LMD) show that a discriminatory hiring
rule could lead to labor market segmentation and signicant wage gap with even a negligible dierence in
productivity; however, the discriminated group turn out to have lower unemployment rate, which is in sharp
contrast with evidence. Merlino (2012) aims at improving the result of LMD (2005). He considers further the
1
This literature is sometimes also termed as wage posting game with coordination friction
3
Documents de travail du Centre d'Economie de la Sorbonne - 2016.02

Frequently asked questions

Q1. What are the contributions in "Holdup and hiring discrimination with search friction" ?

The authors consider frictional markets with either posted or bargained wage ( xed sharing rule ).