4.3 The model with hiring discrimination
4.3.3 Analysis under our context
In the last subsection, we interpreted the equilibrium of the wage posting subgame given that all workers choose to be high skilled. In this subsection, we study how discrimination leads to di↵erent incentives of skill investment for the two groups respectively, and attempt to find the corresponding equilibrium.
An important observation is that the skill decision for group a and group b workers is strategic, and this is a direct consequence of the coexistence of ranking through the productivity-dependent (skill) and productivity-independent (discrimination) traits. Ranking by skills requires that the high skilled worker has the priority; while ranking by productivity-independent traits means that group a has the priority. Although multidimensional characteristics are involved, these two ranking schedules yield a unique market hierarchy:
aH bH aL bL.
It reads as follows: high skilled group a (aH) is preferred to high skilled group b (bH), who is preferred to low skilled group a (aL), who is then preferred to low
4. HOLDUP, DISCRIMINATION WITH SEARCH FRICTION
skilled group b (bL).16 The matching probability 1 e q
q captures the intensity of
competition within the same type (within type), while the probability e q captures
the intensity of competition from the higher ranked type (between type).
Under Assumption 4.1, all workers choose to be high skilled in the nondiscrimi- natory regime. Now we see that with hiring discrimination they may have incentive of deviating to low skill. Let ↵s be the fraction of group s choosing to be high
skilled, for s = a or b. To make the analysis simplified (but without losing the attraction of the model), we assume
Assumption 4.4. Whenever indi↵erent, all workers with the same group identity (as a group) choose either high or either low skilled.
For workers in the same group, whenever indi↵erent between the two alternatives (L and H), they all randomize towards the same direction: that is to say, we consider the group of workers as a whole, or there is perfect correlation on their skill choices. As a result, ↵ does not represent an individuals probability of choosing high skill, and either ↵s = 1 or ↵s= 0. Thus, we have the following four possibilities:
(P1) ↵a = 1 and ↵b = 1: group a - high, group b - high;
(P2) ↵a = 1 and ↵b = 0: group a - high, group b - low;
(P3) ↵a = 0 and ↵b = 1: group a - low, group b - high;
(P4) ↵a = 0 and ↵b = 0: group a - low, group b - low.
To decide the skill investment, workers take into account firms’ best response in the wage posting stage to infer the expected income from the application, and compare the payo↵s net of the cost of skill investment. In the wage posting subgame, when facing all workers with identical skill level (as in case (P1) and (P4)), firms’ optimal strategy is the same as stated in LMD (2005); that is, some firms post a higher wage to attract group a, whereas the remaining firms post a lower wage targeting merely group b. When there are both low and high skilled workers (as in case (P2) and (P3)), firms post wages conditional on skill levels, and it is optimal for firms to attract both skill levels and rank the high skilled in priority to low skilled, as in Shi (2006). We then proceed to find workers’ best response
16Quantitatively, in terms of employment probability, we have F
aH > FbH > FaL> FaLfor
any qst> 0, where s2 {a, b} and t 2 {H, L}. That is,
1 e qaH qaH > e qaH 1 e qbH qbH > e qaH qbH 1 e qaL qaL > e qaH qbH qaL 1 e qbL qbL . (4.21)
The inequality comes from the fact that 1 eq q > e q and 0 < 1 e q
q < 1 for any q > 0. Note that
by defining 1 eq q = 1 for q = 0, we can extend (4.21) for all qst 0, s2 {a, b} and t 2 {H, L}
4.3 The model with hiring discrimination
in the skill investment stage, and in turn the equilibrium in this discriminatory context with skill investment. We will use P1, P2, P3, P4 as the superscript for corresponding equilibrium allocations.
When ↵a = 1 and ↵b = 1, workers are composed of type aH and bH. Firms
post wages separately, and by (4.13) and (4.17), we have the payo↵ of group a (i.e. aH) and group b (i.e. bH) are, respectively,
VaHP 1 = e qP 1aH⇤ y H EH, (4.22) and VbHP 1 = 1 e qP 1⇤ bH qP 1⇤ bH e qaHP 1⇤ y H EH, (4.23)
and from (4.19) we have qP 1⇤
aH > and qP 1bH⇤ < .
When ↵a = 1 and ↵b = 0, workers are composed of type aH and bL. All firms
post two wages to attract both types of workers at the same time. By (4.5) and (4.6), we have the payo↵ of group a and b are, respectively,
VaHP 2 = e qaHP 2⇤(y H yL) + e q P 2⇤ aH qP 2bL⇤ y L EH, (4.24) VbLP 2 = e qaHP 2⇤ qP 2bL⇤ y L EL, (4.25)
and note that qP 2⇤
aH = , qbLP 2⇤ = (1 ) , and qP 2aH⇤+ qbLP 2⇤ = .
When ↵a = 0 and ↵b = 1, workers are composed of di↵erent skill levels, aL and
bH, and we have similarly
VaLP 3 = e qaLP 3⇤ qbHP 3⇤ yL EL, (4.26) VbHP 3 = e qbHP 3⇤(y H yL) + e q P 3⇤ aL qP 3⇤bH y L EH, (4.27)
and similarly here we have qaLP 3⇤ = , qbHP 3⇤ = (1 ) , and qP 3⇤aL + qbHP 3⇤ = . When ↵a= 0 and ↵b = 0, workers are composed of type aL and bL. Both are
of the same skill level, firms discriminate and post wages as in LMD (2005), and workers payo↵s are
VaLP 4 = e qaLP 4⇤ y L EL, (4.28) and VbLP 4 = 1 e qP 4⇤ bL qP 4⇤ bL e qP 4aL⇤ y L EL, (4.29)
4. HOLDUP, DISCRIMINATION WITH SEARCH FRICTION
with qP 4⇤
aL > > qbLP 4⇤.
A pure-strategy Nash equilibrium consists of a profile of skill investment with the property that no single group can achieve a higher payo↵ by unilateral deviation. The existence of equilibrium depends on the value of . The payo↵s are summarized in table4.1 in the appendix. By comparing the payo↵s under di↵erent strategies, we can find the best response of workers in the skill investment. For example, responding to group b choosing to be high skilled, it is optimal for group a to choose high skill if VP 1
aH VaLP 3, i.e. e q
⇤
aH yH EH e yL EL, which is not
always true. Note that although both VP 1
aH and VaLP 4 are decreasing in , there
could be more than one critical value which equates VP 1
aH and VaLP 4. In order to
ensure a single threshold and to avoid unnecessary technical complexity, we further assume that
Assumption 4.5. For q⇤
aH( ) and qbH⇤ ( ) that are solved by (4.18),
1. 9! ˆ1 such that e q ⇤ aH( ˆ1) yH EH = e ˆ1 yL EL; 2. 9! ˆ2 such that 1 e q⇤bH (ˆ2) q⇤ bH( ˆ2) e qaH⇤ ( ˆ2) y H EH = e ˆ2 yL EL.
In fact, if one group chooses to be low skilled, the best response of the other group is always to be high skilled, while the best response to the other’s high skill choice depends on the two thresholds (see Figure 4.1). Furthermore, the rise of market tightness makes workers have stronger incentive to deviate to low skill, and group b is more prone to deviate compared to group a, in the sense that the threshold at which group b begins to contemplate to invest in low skill is lower that for group a.
aL
aH aH
bH ˆ2 bL ˆ1 bL ˆ
Figure 4.1: Best responses given the other group choosing H.
Focusing only on the pure strategy equilibrium, we formalize the results regard- ing the existence of equilibrium in the following proposition.
Proposition 4.3. Under Assumption 4.1 - 4.5, there exist two thresholds ˆ1 and
ˆ2, with 0 < ˆ2 < ˆ1 < ˆ.
1. When 0 < < ˆ2, there exists a unique pure strategy equilibrium in which