# Holdup and hiring discrimination with search friction

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 (4 min read)

### 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 wi..." refers background in this paper

...Define the priority rule x ∈ [0, 1], which is the probability of choosing group type H job applicants when both types are present....

[...]

...See Acemoglu and Shimer (1999) for related literature....

[...]

...ηa = 1 if 1−e −qaH qaH × ψyH − EH > e−qaH−qbH 1−e −qaL qaL × ψyL − EL = 0 if 1−e −qaH qaH × ψyH − EH < e−qaH−qbH 1−e −qaL qaL × ψyL − EL ∈ [0, 1] if 1−e −qaH qaH × ψyH − EH = e−qaH−qbH 1−e −qaL qaL × ψyL − EL...

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...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…...

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...ηb = 1 if e−qaH 1−e −qbH qbH × ψyH − EH > e−qaH−qbH−qaL 1−e −qbL qbL × ψyL − EL = 0 if e−qaH 1−e −qbH qbH × ψyH − EH < e−qaH−qbH−qaL 1−e −qbL qbL × ψyL − EL ∈ [0, 1] if e−qaH 1−e −qbH qbH × ψyH − EH = e−qaH−qbH−qaL 1−e −qbL qbL × ψyL − EL...

[...]

512 citations

### Additional excerpts

...(5) For ψ ∈ ( ψ̂aH,b, 1 ) , the unique pure...

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