A Search Model of Statistical Discrimination
aa r X i v : . [ ec on . T H ] A p r A Search Model of Statistical Discrimination ∗ Jiadong Gu † Peter Norman ‡ April 27, 2020
Abstract
We offer a search-theoretic model of statistical discrimination, in which firms treat identicalgroups unequally based on their occupational choices. The model admits symmetric equilibriain which the group characteristic is ignored, but also asymmetric equilibria in which a groupis statistically discriminated against, even when symmetric equilibria are unique. Moreover, arobust possibility is that symmetric equilibria become unstable when the group characteristicis introduced. Unlike most previous literature, our model can justify affirmative action since iteliminates asymmetric equilibria without distorting incentives.
Keywords:
Search, Statistical Discrimination, Inequality.
JEL Classification Number:
D43, L13. ∗ We thank Simon Alder, Luca Flabbi, Teddy Kim, Fei Li, Anja Prummer, and Huseyin Yildrim for commentsand discussions. The usual disclaimer applies. † Department of Economics, University of North Carolina at Chapel Hill, 107 Gardner Hall, CB 3305, Chapel Hill,NC 27599-3305. Email: [email protected] ‡ Department of Economics, University of North Carolina at Chapel Hill, 107 Gardner Hall, CB 3305, Chapel Hill,NC 27599-3305. Email: [email protected]
Introduction
Statistical discrimination refers to situations in which some agents use observable characteristics as a proxy for payoff relevant unobservable characteristics . The observable characteristics on whichstatistical discrimination may be based include gender, race, job separation rate, unemploymentduration, job leave duration, or anything else that may serve as a proxy for unobservables that themarket cares about. Models of statistical discrimination were initially developed as an alternativeto taste-based models to explain group inequalities (see Fang and Moro (2011)). There are manyvariants, but the main point with many of these models is that interactions between a signalextraction problem and human capital investments can generate equilibria in which some group(s)are worse off than others despite no fundamental differences between groups (see Arrow et al.(1973), Coate and Loury (1993), and Moro and Norman (2004)). We consider statistical discrimination in a frictional search framework. This allows us to explorean alternative channel in which self-fulfilling statistical discrimination can be sustained that iscomplementary to the existing literature. Instead of investments in human capital, inequalities arisein equilibrium due to search frictions and occupational choice. As demonstrated by Xiao (2020),differences in occupational choice account for much of the gender wage gap, but it is accountedfor as reflecting preferences for amenities. This paper provides a complementary explanation, as itshows that women may enter low paying occupations as a result of statistical discrimination.Our baseline model without group characteristics is a discrete time random search model inwhich there are two types of workers and two distinct technologies. We label the workers as qualified or unqualified and the technologies as high tech and low tech. Conditional on an appropriate match,a high tech firm is more productive than a low tech firm. However, only qualified workers areproductive at high tech firms, whereas the type of worker does not matter for low tech firms. Thistype of technology has been considered before in the labor-search literature by Albrecht and Vroman(2002), Gautier (2002), Dolado et al. (2009), and others. However, as far as we know, the only paperthat has combined a technology along these lines with asymmetric information about worker typeis Jarosch and Pilossoph (2019), but they have an equilibrium characterization very different fromours.When a high tech firm matches with a worker, they would like to hire the worker if and onlyif the worker is qualified. Because the firm cannot directly observe the type of worker, this idealhiring rule can not be implemented. Instead, the firm observes a noisy signal that is correlatedwith the type and may be interpreted as the result of a job interview. The signal is labeled so thata higher signal is good news, which implies that an optimal hiring rule is one in which a worker isoffered a job if and only if the signal exceeds some threshold.Importantly, the optimal hiring threshold depends not only on parameters of the noisy signalbut also on the prior probability that a worker is qualified . Since we consider random matching, thisprior probability is simply the proportion of qualified workers in the pool of unemployed, which is The first paper along these lines is Arrow et al. (1973). See also Coate and Loury (1993), Moro and Norman(2004), and the survey by Fang and Moro (2011). informational externalities that are novel to the search and matchingliterature.If the probability that a qualified worker accepts a low tech position increases, then the outflowof qualified workers from the pool of unemployed increases. In steady state, this reduces theproportion of qualified workers in the pool of unemployed, which increases the hiring threshold.Hence, a steady state increase in the probability that a qualified worker accepts a low tech jobmakes it harder to obtain a high tech job. With free entry, this also affects entry decisions and thedetails of how that works are somewhat intricate and also depend on whether workers are followinga pure or mixed acceptance rule. The main point, however, is that there is feedback from theworker acceptance rule in the low tech sector to the optimal hiring threshold for high tech firms.The baseline model may have a unique equilibrium, or there may be multiple equilibria. How-ever, it is important to note that nothing in our analysis with multiple groups rests on multiplicityin the single group model. We introduce a payoff irrelevant observable characteristic by assumingthat a worker either is from group m or group f , which can be observed by the firms. The twogroups are identical in the sense that the proportion of qualified workers is the same for both groups,but we allow the groups to be of different sizes. We demonstrate that the existence of asymmetricequilibria is a robust possibility, whether or not the baseline model has a unique equilibrium.If everyone ignores the observable characteristic, the equilibrium conditions in the case withmultiple groups are the same as in the baseline model, so there is always at least one equilibriumwith equal treatment of the workers. However, these symmetric equilibria may be very fragile inthe case of multiple groups. One possible symmetric equilibrium is when both sectors are active,and qualified workers randomize between accepting low tech jobs. In such an equilibrium, firmsare indifferent between entering as a high tech and as a low tech firm, and the proportion of hightech firms is determined so that qualified workers are indifferent between accepting and rejectinglow tech job offers. Imagine that the proportion of qualified m workers increases ever so slightlywhile the proportion of qualified f workers is held fix. Assuming no change in the fraction of hightech jobs, this would make the best response for m workers to reject low tech jobs for sure. In thefull equilibrium, the proportion of high tech job will in general change, but this does not changethe argument much. Either m workers have to reject low tech jobs for sure or f workers have toaccept low tech jobs for sure after the change. Therefore, an arbitrarily small exogenous change inthe proportion of qualified men creates a significant difference through the equilibrium conditions.Hence, symmetric equilibria of the form just described can not be stable.A related point is that there is a true interaction between groups. Unlike models like Coate and Loury(1993), in which discrimination is interpreted as one group coordinating on a good equilibrium andanother on a bad one, incentives for one group are affected by the behavior of the other. Thisnon-separability allows asymmetric equilibria to exist even if the baseline model has a unique equi-librium.Most existing models of statistical discrimination focuses on the interplay between incentivesfor human capital investments and hiring decisions. A group is discriminated in the labor market2ecause, in equilibrium, the group is less skilled on average. In our frictional model of the labormarket, the average skill level in the pool of unemployed is not the same as the skill level in thepopulation, so we are able to explain statistical discrimination between groups that in equilibriumare equally skilled.The standard model explains the lower skill level as the consequence of having less high poweredincentives to invest in skills. It has been argued that there is very little empirical evidence forthis. In fact, in the case of the black-white wage gap, Neal and Johnson (1996) and Neal (2006)argue that blacks have stronger incentives to acquire skills than whites, which is inconsistent withstandard models of statistical discrimination. In our model, skills are exogenous, but statisticaldiscrimination is still a possibility. Instead of feeding into incentives for skill acquisition, the labormarket responses feed into incentives to accept dead-end jobs. So, in a sense, the decision to turndown bad jobs in our model plays a similar role as skill investments in the standard model, andthere are similar free-riding considerations involved as individuals benefit from the total number ofpeople within the group that turn low tech jobs down.Another issue with standard models of statistical discrimination is that, in equilibrium, thediscriminated group is, on average, less productive than the dominant group. This is possibledespite there being no intrinsic differences between groups, but a theory that implies that womenare significantly less productive than men because of the lack of human capital investments maynot be the most plausible. After all, women are now acquiring more education than men. Whileeducation is not the kind of unobservable investments that are considered in models of statisticaldiscrimination, we find our alternative explanation quite plausible. There is also rather convincingevidence of various forms of mismatch between worker skills and jobs (for example, see Clark et al.(2017) and the references therein on worker over-education) suggesting that it seems reasonable tohave mismatch also with respect to unobservable skills.Affirmative action policies can be counterproductive in many conventional models of statisticaldiscrimination. In Coate and Loury (1993), the problem is that preferential treatment may reducethe incentives to acquire skills. This may also be true in Moro and Norman (2003), where, addi-tionally, the targeted group may be may worse off, and such perverse welfare effects are even moreprevalent in Fang and Norman (2006). In contrast, a hiring quota requiring firms to hire workersin accordance with population proportions in this model eliminates all asymmetric equilibria. Thisis because firms must have lower standards for women if there are fewer qualified women in thepool of unemployed. This unambiguously makes qualified women having a better chance than menat high tech jobs, which moves incentives in the desired direction. In contrast, in the previousliterature, the hiring quota may create disincentives to invest in human capital.We make many simplifying assumptions in order to make our analysis as transparent as possible.Most notably, we assume that wages are exogenous. While this is obviously unrealistic, we see noreason why endogenizing wages through posting or bargaining would qualitatively change anything.Our model is not the first dynamic model of statistical discrimination. However, existing dy- However, Glover et al. (2017) provides some evidence in favor of models of statistical discrimination. We consider a labor market where a unit mass of workers and a continuum of firms are matchedrandomly. Workers and firms are infinitely lived, forward-looking, risk-neutral, and have a commondiscount factor of β . Time is discrete, and we focus on steady state equilibria.Later on, we will add observable payoff irrelevant group characteristics that can be interpretedas race or gender, but to minimize notation we begin by introducing a benchmark model withno group characteristics. This baseline model also provides a full characterization of symmetricequilibria in the model with observable group characteristics.A proportion ψ of the workers are qualified , and the remaining fraction 1 − ψ are referred toas unqualified . Qualified workers are equipped with a skill that matters for some firms, but not forothers. Specifically, we assume that a firm can be one of two types. Some firms, referred to as lowtech firms, produce flow output y l and pays exogenous wage w l when matched with a worker ofeither type. We assume that y l − w l > high tech firm ismatched with a qualified worker the match produces flow output y h and the firm pays the exogenouswage w h , where y h − w h >
0. The firm would thus want to hire the worker if the worker is knownto be qualified. In contrast, if the worker is unqualified the flow output is (normalized to) zero,so the flow profit is − w h , so a high tech firm would never want to hire an unqualified worker. Weassume that w l < w h , implying that high tech matches are more desirable than low tech matchesfor the worker. The flow value of unemployment is b < w l .The firm cannot directly observe whether a worker is qualified or not. Instead, the firm observesa noisy signal θ ∈ [0 , f q ( θ ) if the worker is qualified and from f u ( θ ) See also Antonovics (2006), Eeckhout (2006), Glawtschew (2015), and Masters (2014) θ is pure information, there is no loss of generality to label the signalrealizations so that higher realizations correspond with a higher probability of the worker beingqualified, so we assume that the likelihood ratio f q ( θ ) f u ( θ ) is strictly increasing in θ . Let π be the firm’s prior probability that a worker is qualified. Using the monotone likelihoodratio property, it is immediate that the posterior probability after observing θP ( θ, π ) = πf q ( θ ) πf q ( θ ) + (1 − π ) f u ( θ ) , (1)is also monotone in θ, , which makes it very easy to describe optimal hiring rules.If a match forms we also assume that the qualification of the worker is revealed to the corre-sponding matched firm in every period with some probability r . Should a worker in a high skilledfirm be revealed to have low productivity, that worker is fired. There is also an exogenous separationprobability φ , which is ensuring that steady state conditions behave nicely.We parametrize the model so that the unqualified workers’ acceptance rules are trivial: unqual-ified workers accept any job offer they get. In contrast, a qualified worker will always accept a hightech offer as b < w l < w h , but may or may not accept low tech jobs depending on, for example,how likely it is to match with a high tech firm in the future. This is the only non-trivial workerchoice in the model, and we let α ∈ [0 ,
1] denote the endogenous probability that a qualified workeraccepts when offered a position at a low tech firm.There is free entry into both the high and the low tech sectors, and the flow cost of a vacancy is
K > α ∈ [0 ,
1] for a qualified worker to accept low tech jobs.(2) Entry decisions by firms in both sectors.(3) Given the noisy signal θ ∈ [0 , Assuming that the likelihood ratio is strictly increasing and assuming away mass points at the same time is notcompletely without loss. We make the assumption to assure a unique solution for the firm optimal hiring strategy.Weakening to weak monotonicity allows for signals replicating discrete support. This is less elegant, but can be dealtwith. Equilibria in the Baseline Model
We begin by considering the hiring decision for a high-tech firm . Let π ∈ [0 ,
1] be the endogenousstationary proportion of qualified workers in the pool of unemployed, which is also the prior proba-bility that the worker is qualified from the point of view of the firm. Also, let W (0 , π ) , W ( u, π ) and W ( q, π ) be the value of a vacancy, being matched with an unqualified worker, and being matchedwith a qualified worker respectively, as a function of π . The firm values of hiring an unqualifiedand qualified worker are W ( u, π ) = − w h + β [( φ + (1 − φ ) r ) W (0 , π ) + (1 − φ ) (1 − r ) W ( u, π )] (2) W ( q, π ) = y h − w h + β [ φW (0 , π ) + (1 − φ ) W ( q, π )] . For the unqualified worker, the probability of moving from employment at a high tech firm tounemployment is higher than for the qualified worker. This is because the type may be revealed inaddition to the common separation rate. Solving for these values in terms of the value of a vacancyand using the free entry condition W (0 , π ) = 0, we can express these values purely in terms of theexogenous parameters as W ( u, π ) = − w h − β (1 − φ ) (1 − r ) ≡ W u < W ( q, π ) = y h − w h − β (1 − φ ) ≡ W q > . Now, assume that a firm has matched with a worker and that the noisy signal θ ∈ [0 ,
1] has beendrawn. If the worker is hired, the expected continuation payoff is W q if the worker is qualified and W u if the worker is not. In contrast, if the worker is not hired, the firm moves on to the next periodwith a vacancy, which has value 0 . Hence, the firm is better off hiring the worker in expectation ifand only if P ( θ, π ) W q + (1 − P ( θ, π )) W u ≥ , (4)where P ( θ, π ) is defined in (1). Since P ( θ, π ) is strictly increasing in θ for any π ∈ [0 ,
1] we havethat:
Lemma 1.
For any given π ∈ [0 , the unique optimal hiring rule is a threshold rule where a workeris offered a job if and only θ ≥ s ( π ) . If πf q (0)(1 − π ) f u (0) ≥ − W u W q then s ( π ) = 0 and πf q (1)(1 − π ) f u (1) ≤ − W u W q then s (1) = 0 . If πf q (0)(1 − π ) f u (0) < − W u W q < πf q (1)(1 − π ) f u (1) the threshold s ( π ) ∈ (0 , satisfies πf q ( s ( π ))(1 − π ) f u ( s ( π )) = − W u W q . (5)6otice that one can also understand the threshold s ( π ) as the solution tomax θ π [1 − F q ( θ )] W q + (1 − π ) [1 − F u ( θ )] W u . (6)The interpretation of this problem is as the ex ante expected profit from a threshold rule. This isbecause π is the probability that a worker is qualified and 1 − F q ( θ ) is the conditional probabilitythat the signal is above θ, so π (1 − F q ( θ )) is the probability of hiring a qualified worker if thehiring threshold is θ. Symmetrically, (1 − π ) (1 − F u ( θ )) is the probability of hiring an unqualifiedworker if the hiring threshold is θ , so the objective function in (6) is the profit as a function of thehiring threshold. The condition (5) can be obtained from the first order condition for the problem(6).For a low tech firm , it always offers a job to any worker following a match. The continuationpayoff from hiring is W l = y l − w l − β (1 − φ ) > , (7)using the free entry condition as for high tech firms. For notational simplicity let A q ( π ) = 1 − F q ( s ( π )) (8) A u ( π ) = 1 − F u ( s ( π ))be the unique probabilities that qualified and unqualified workers are hired when matched witha high tech firm in an equilibrium in which the proportion of qualified workers in the pool ofunemployed in π. It should be intuitively clear that these probabilities will affect the incentives forqualified workers. We therefore write α ( π ) for the probability that a qualified worker accepts alow tech job explicitly as a function of π despite not yet having discussed the worker optimizationproblem. The free entry conditions for high each and low tech sectors are then0 = − K + βp f [ πA q ( π ) W q + (1 − π ) A u ( π ) W u ] (9)0 = − K + βp f [ πα ( π ) + (1 − π )] W l , where p f is the probability that a firm matches with a worker, which is equal to the ratio ofunemployment over vacancies, or the inverse market tightness rate. Note that Lemma 2.
The profit of entering high tech sector, πA q ( π ) W q + (1 − π ) A u ( π ) W u , is strictly in-creasing in π . The intuitive idea is that the profit is strictly increasing in π for any fixed hiring threshold, andadjusting the threshold only increases the gain. The proof is in the appendix. It follows that p f
7s strictly decreasing in π , given that the high tech sector is active. Hence, total entry is strictlyincreasing in π, except in the uninteresting case in which no high tech firms enter.However, a condition that will prove more useful for the equilibrium characterization is that(9) implies that when both sectors are active, firms must be indifferent between entering as highor low tech firms, or [ πα ( π ) + (1 − π )] W l = πA q ( π ) W q + (1 − π ) A u ( π ) W u . (10)Notice that the assumption that K is equal for the two sectors is just a normalization, as differencesin costs of maintaining a vacancy can be incorporated in W l . Next, consider the possibility thatqualified workers always reject low tech wage offers. In order for both sectors to be active in thiscase, it must be that the proportion of qualified workers among the unemployed is π , which wedefine as the unique solution to(1 − π ) W l = πA q ( π ) W q + (1 − π ) A u ( π ) W u , (11)The expected profit of being in the low tech sector decreases in π and the expected profit of beingin the high tech sector increases in π , so if π < π then only low tech firms would be willing to enter.Symmetrically, if qualified workers always accept low tech jobs the proportion of qualified workersthat ensure indifference, π , is W l = πA q ( π ) W q + (1 − π ) A u ( π ) W u . (12)Hence, if π > π the expected payoff of being a high tech firm is higher than being a low techfirm even if qualified workers always accept low tech offers. Summing this up we have that: Lemma 3.
Suppose that K is small enough so that there are firms that want to enter. Then(1) If π < π then firms will enter the low tech sector only regardless of what workers do.(2) If π = π then firms are willing to enter in each sector if and only if α ( π ) = 0 . If α ( π ) > firms are willing to enter the low tech sector only.(3) If π < π < π there is a unique α ( π ) ∈ (0 , such that firms are willing to enter in eachsector. For α > α ( π ) ∈ (0 , only low tech firms are willing to enter and if α < α ( π ) ∈ (0 , only high tech firms are willing to enter.(4) If π = π firms are willing to enter in each sector if and only if α ( π ) = 1 . If α ( π ) < onlyhigh tech firms are willing to enter.(5) If π > π firms will enter the high tech sector only regardless of what workers do. The firms’ optimality is summarized into the following Figure 1. The details of the proof are inAppendix A.3. 8 π π
Since we have trivialized the unqualified workers into agents who always accept every offer, we onlyneed to consider the qualified workers. Denote by V ( π ), V h ( π ) and V l ( π ) the value for a qualifiedworker from being unemployed, being employed in the high tech sector, and being employed in thelow tech sector, respectively. Also, let p ( π ) be the endogenous probability that the worker meets ahigh tech firm, so that 1 − p ( π ) is the probability of meeting a low tech firm. We use the conventionthat V ( π ) is the value of unemployment immediately prior to matching with the firm, so that V ( π ) = p ( π ) [ A q ( π ) V h ( π ) + (1 − A q ( π )) ( b + V ( π ))] (13)+ (1 − p ( π )) max α ∈ [0 , [ αV l ( π ) + (1 − α ) [ b + βV ( π )]] , where we can write the values of being employed in terms of exogenous parameters and the valueof unemployment as V h ( π ) = w h + φβV ( π )1 − (1 − φ ) β (14) V l ( π ) = w l + φβV ( π )1 − (1 − φ ) β . Hence, letting α ( π ) be the solution to the optimization problem in (13). Then, α ( π ) = w l + φβV ( π )1 − (1 − φ ) β > b + βV ( π )[0 ,
1] if w l + φβV ( π )1 − (1 − φ ) β = b + βV ( π )0 if w l + φβV ( π )1 − (1 − φ ) β < b + βV ( π ) . (15)Let V ∗ be the value of unemployment that makes the worker indifferent between accepting andrejecting, that is w l + φβV ∗ − (1 − φ ) β = b + βV ∗ . (16)Then, since the value of working in the low tech sector increases in the value of unemployment ata rate strictly slower than β we can express (15) as α ( π ) = V ( π ) < V ∗ [0 ,
1] if V ( π ) = V ∗ V ( π ) > V ∗ . (17)9ombining with (13) we have that for the worker to be indifferent it must be that V ∗ = p ( π ) A q ( π ) w h + φβV ∗ − (1 − φ ) β + (1 − A q ( π ) p ( π )) ( b + βV ∗ ) . Let p ( π ) A q ( π ) = Q ( π ) be the probability of meeting a high-tech firm and being hired for aqualified worker, we have V ∗ = Q ( π ) w h + φβV ∗ − (1 − φ ) β + (1 − Q ( π )) ( b + βV ∗ ) . (18)Hence, the probability for a qualified worker to be hired by a high tech firm that makes theworker indifferent between accepting and rejecting low tech jobs can be determined independentlyfrom π . We thus get the following simple characterization of the worker acceptance rule. Lemma 4.
Suppose that ≤ w l − b ≤ β (1 − φ )( w h − b ) . Then there exists a unique constant Q ∗ ∈ [0 , such that V ∗ = Q ∗ w h + φβV ∗ − (1 − φ ) β + (1 − Q ∗ ) ( b + βV ∗ ) , (19) where V ∗ is defined in (16). Moreover, α ( π ) = if p ( π ) A q ( π ) < Q ∗ [0 , if p ( π ) A q ( π ) = Q ∗ if p ( π ) A q ( π ) > Q ∗ . (20)The proof can be found in the appendix. The parameter restrictions are to avoid corner cases,which in terms of the statement of the Lemma would show up as Q ∗ being negative or larger thanone. First, if w l < b , then workers prefer unemployment to working in the low tech sector, so thelow tech sector would always be inactive. Secondly, if w l − b > β (1 − φ )( w h − b ) one can show thatlow tech jobs are always accepted even if p ( π ) A q ( π ) = 1. Let N be the total mass of unemployed workers, and N π be the mass of qualified workers in the unemployed workers, N (1 − π ) be the mass of unqualified worker in the unemployed workers. Let E k,s be the mass of workers with skill k ∈ { q, u } employed in sector s ∈ { h, l } . We can then writethe steady state conditions equalizing the outflow and inflow of each pool of qualified-high tech,qualified-low tech, unqualified-high tech, unqualified-low tech as N πp ( π ) A q ( π ) = E q,h φ (21) N π (1 − p ( π )) α ( π ) = E q,l φN (1 − π ) p ( π ) A u ( π ) = E u,h ( φ + (1 − φ ) r ) N (1 − π ) (1 − p ( π )) = E u,l φ. N π + E q,h + E q,l = ψ (22) N (1 − π ) + E u,h + E u,l = 1 − ψ. By substituting from Equations (21) into Equation (22) we obtain
N π + N πp ( π ) A q ( π ) φ + N π (1 − p ( π )) α ( π ) φ = ψ (23) N (1 − π ) + N (1 − π ) p ( π ) A u ( π ) φ + (1 − φ ) r + N (1 − π )(1 − p ( π )) φ = 1 − ψ. Finally, by eliminating N we can summarize the steady state condition in a single equation1 + p ( π ) A u ( π ) φ + (1 − φ ) r + 1 − p ( π ) φ = 1 − ψ − π πψ (cid:20) p ( π ) A q ( π ) φ + (1 − p ( π )) α ( π ) φ (cid:21) . (24)Equation (24) represents the steady state conditions for the labor market, given qualified worker’sacceptance rule α ( π ) and high-tech firms’ hiring strategy A q ( π ) and A u ( π ).For the analysis that follows it is useful to define real function G : [0 , → R , G ( π, α, p ) = 1 + pA u ( π ) φ + (1 − φ ) r + 1 − pφ − − ψ − π πψ (cid:20) pA q ( π ) φ + (1 − p ) αφ (cid:21) . (25)Using this notation, we can define an equilibrium compactly as: Definition 1.
An equilibrium is a triple ( π, α ( π ) , p ( π )) such that i) α ( π ) satisfies worker opti-mality condition (20), and p ( π ) is consistent with optimal entry and; ii) G ( π, α ( π ) , p ( π )) = 0 . There is also a choice of an optimal hiring threshold s ( π ), but this is built into the terms A q ( π )and A u ( π ) in the steady state conditions. Also note that if 0 < α ( π ) <
1, then both sectors mustbe active implying that any p ( π ) ∈ [0 ,
1] is a best response. In this case p ( π ) are thus set to makethe workers indifferent. Finally, note that the total mass of firms entering is also an equilibriumobject in principle. However, the matching technology is such that workers match with exactly onefirm in every period, so this does not interact with any other variable and can be left implicit inthe analysis. While the steady state equation (24), firm entry, and worker optimality conditions are all ratherstraightforward, the equilibrating mechanism is different in different ranges. For convenience, wehave summarized the various cases in the figure below.11 π π α ( π ) = 1 α ( π ) ∈ (0 , πp ( π ) A q ( π ) constant α ( π ) indeterminate α ( π ) = 0 p ( π ) α ( π ) = 1 p ( π )Figure 2: Equilibrium Candidates. Depending on parameters, an equilibrium can have only low tech firms active, only high tech jobsactive, or both sectors active. The necessary and sufficient conditions for a low tech equilibriumare straightforward and easy to understand:
Proposition 1.
An equilibrium in which only low tech firms are active exists if and only if ψ ≤ π. This is the unique equilibrium if ψ < π.
The intuition is simple. In a low tech equilibrium, workers become homogenous, so there isno longer any selection. Therefore π = ψ in steady state. Recalling that π defined in (12) is thecritical value for π that makes firms indifferent between the two sectors when all qualified workersaccept low tech jobs, we note that for π ≤ π firms are more profitable in the low tech sector thanin the high tech sector if all workers accept low tech jobs. Moreover, workers have a strict incentiveto accept low tech jobs since the probability of a high tech offer is zero. The fact that no otherequilibrium can exist when π follows directly from the definition of (11).The conditions for existence of equilibria with only the high tech sector active are a bit moreinvolved for two reasons. Firstly, there will be negative selection effects in such an equilibrium.Secondly, the worker optimality condition is now non-trival. We now consider the steady statecondition if only high tech firms enter. Since there are no low tech firms around, the determinationof π in such an equilibrium is independent of α ( π ) and simplifies to e G ( π ) = G ( π, α ( π ) ,
1) = 1 + A u ( π ) φ + (1 − φ ) r − − ψ − π πψ (cid:20) A q ( π ) φ (cid:21) = 0 . (26)Note that A u ( π ) φ +(1 − φ ) r < A q ( π ) φ for any π, which reflects that this type of an equilibrium generatesnegative selection as qualified workers are more likely to move from unemployment to employmentand less likely to move from employment to unemployment.However, while α ( π ) is irrelevant for the steady state condition we still have to ask what aworker would do if counter factually matching with a low tech firm. That is, if a worker wouldaccept an offer from a deviating low tech entrant and if π is strictly in between π and π , then thisis inconsistent with a high tech equilibrium. It thus follows:12 roposition 2. There is at least one solution π ∗ to (26), and;1. Suppose π ∗ < π solves (26). Then there is no high tech equilibrium corresponding to π ∗ .
2. Suppose π ∗ ∈ [ π, π ] solves (26) and A q ( π ∗ ) < Q ∗ . Then there is no high tech equilibriumcorresponding to π ∗ .3. Suppose π ∗ ∈ [ π, π ] solves (26) and A q ( π ∗ ) ≥ Q ∗ . Then ( π ∗ , α ( π ∗ ) = 0 , p ( π ∗ ) = 1) is anequilibrium.4. Suppose π ∗ > π solves (26). Then ( π ∗ , α ( π ∗ ) = 1 , p ( π ∗ ) = 1) is an equilibrium. One may note that π ∗ < ψ for any solution to (26), because if π ∗ solves (26) then1 > A u ( π ∗ ) φ +(1 − φ ) r A q ( π ∗ ) φ = 1 − ψ − π ∗ π ∗ ψ , (27)as A u ( π ∗ ) φ +(1 − φ ) r < A q ( π ∗ ) φ for any π ∗ , . Hence, being unemployed is correlated with being unqualified withbeing unqualified in an equilibrium with only the high tech sector active. Moreover, A q ( π ∗ ) φ → ∞ as φ → A u ( π ) φ +(1 − φ ) r stays finite provided that r > . Hence, π ∗ → φ → φ ∗ > π ∗ < π for any φ ≤ φ ∗ andany solution to (26). We conclude: Corollary 1.
There exists φ ∗ > such that no equilibrium in which only high tech firms enter canexist if φ ≤ φ ∗ . Having discussed the conditions with the less interesting single sector equilibria, we now considerequilibria in which both sectors are active. Such equilibria may take on different forms in differentparts of the parameter space. There are three distinct possibilities:1. All workers accept low tech jobs. This requires that π = π. Additionally, it must be that p ( π ) A q ( π ) ≤ Q ∗ to justify the decisions of the workers.2. All workers reject low tech jobs. This requires that π = π. Additionally, it must be that p ( π ) A q ( π ) ≥ Q ∗ to justify the decisions of the workers.3. Workers are indifferent and randomize in a way so that firms are indifferent between the twosectors. 13 .2.1 Low Tech Jobs Accepted First, consider the possibility that α ( π ) = 1 , which implies that π = ¯ π to ensure firms’ indifference.To be consistent with the steady state condition (24) it must be that p ( π ) is such that G ( π, , p ( π )) = 1 + p ( π ) A u ( π ) φ + (1 − φ ) r + 1 − p ( π ) φ − − ψ − π πψ (cid:20) p ( π ) A q ( π ) φ + (1 − p ( π )) φ (cid:21) = 0 , (28)holds and p ( π ) A q ( π ) ≤ Q ∗ . In the appendix we show that this is consistent with equilibrium underthe following conditions.
Proposition 3.
A two sector equilibrium with workers accepting low tech jobs exists if and only ifthe following conditions hold:1. ψ > π G ( π, , ≤ p ( π ) A q ( π ) ≤ Q ∗ for the solution to (28).There can be at most one such equilibrium. Notice that the equilibrating variable that balances the steady state condition is no longer π for these types of equilibrium. Instead, the steady state is achieved by finding the right mix oflow tech and high tech firms. Also note that in this type of equilibrium, the probability of beingqualified must be lower for unemployed workers than in the population as a whole, explaining thefirst condition. Finally, the steady state condition (3) is linear in the proportion of high tech firms,explaining why there can be at most one equilibrium in this form. Suppose first that both sectors are active and all qualified workers reject low tech jobs. Then π = π and α ( π ) = 0. We then seek solutions p ( π ) to the steady state equilibrium condition (24) that isconsistent with qualified workers rejecting low tech jobs. This is true if and only if G ( π, , p ( π )) = 1 + p ( π ) A u ( π ) φ + (1 − φ ) r + 1 − p ( π ) φ − − ψ − π πψ (cid:20) p ( π ) A q ( π ) φ (cid:21) = 0 (29) p ( π ) A q ( π ) ≥ Q ∗ . As G ( π, , p ) is linear in p , and strictly decreasing in p since A u ( π ) φ + (1 − φ ) r − φ − − ψ − π πψ A q ( π ) φ < . It follows that:
Proposition 4.
A two sector equilibrium with workers rejecting low tech jobs exists if and only ifthe following conditions hold: . G ( π, , > G ( π, , < p ( π ) A q ( π ) ≥ Q ∗ for the solution to (29).There can be at most one such equilibrium. Again, the proportion of high tech firms is the equilibrating variable, and, again, the steadystate condition is linear in this proportion. Assuming that ψ ≥ π we have that G ( π, ,
0) = 1 + 1 φ − − ψ − π πψ ≥ φ , (30)so a sufficient condition for G ( π, , > ψ ≥ π. It is also the case that for any ψ and π thereexists φ sufficiently small for G ( π, , > . A small enough φ also ensures that G ( π, , < . Finally, we consider the possibility of an equilibrium in which both sectors are active and π < π < π, which requires indifference on behalf of the workers. For indifference at some π ∈ ( π, π ) it isnecessary that p ( π ) = Q ∗ A q ( π ) ∈ [0 , . (31)Moreover, provided that π ≤ π ≤ π there is a unique α ( π ) ∈ [0 ,
1] that makes firms indifferentacross sectors, α ( · ) is strictly increasing, continuous and satisfies α ( π ) = 0 and α ( π ) = 1 . Hence,we seek a solution so π ∗ ∈ ( π, π ) to G (cid:18) π ∗ , α ( π ∗ ) , Q ∗ A q ( π ∗ ) (cid:19) = 0 . (32)Equilibria in which both firms and workers randomize may co-exist with other types of equilibria.However, we next establish that there is always at least one steady state equilibrium of the model. Proposition 5.
There is always at least one equilibrium.
The proof is in the appendix.
To illustrate various possibilities, we now consider a pair of numerical examples. We set f q ( θ ) = 2 θ , f u ( θ ) = 2(1 − θ ), which is not only consistent with the monotone likelihood rate property, but alsoguarantees interior hiring thresholds as the likelihood ratio is 0 at θ = 0 and approaches infinity as θ converges to one. We also assume that w h , y h , β , φ , and r are set so that W q = 1 and W u = − s ( π ) = 1 − π . Pluggingthis threshold into F q ( θ ) = θ and F u ( θ ) = 2 θ − θ and using the definitions in (8) we obtain15 q ( π ) = π (2 − π ) and A u ( π ) = π , respectively. All our numerical examples use these signals andfirm values.The purpose of the examples is to illustrate the difference between the model with and withoutgroup characteristics. In this section, in which have yet to introduce the group characteristic, weshow that we may have unique of multiple equilibria. However, the example in which the equilibriumis unique is the exact same parametrization as the numerical example with multiple groups . Withmultiple groups, asymmetric discriminatory equilibria appear, demonstrating how discriminationin this model is driven by spillover effects between groups as opposed to pure coordination. Figure 3 plots the relevant steady state equation as a function of π for the three ranges in which π is the variable that adjusts to support the steady state. For π < π we know that the onlycandidate solution is a low tech equilibrium with π = ψ , but we nevertheless plot G ( π, ,
0) forcompleteness (the formula is in (47)). For π ∈ ( π, π ), the equilibrium π is pinned down by thesteady state equation (32), and for π ∈ ( π, π is pinned down by the steady stateequation (26). From Figure 3 we see that there is neither a mixed, nor a high tech nor a low techequilibrium. Only Low-tech Two sectors Only High-tech S t ead y S t a t e F un c t i on G G-functionLower/UpperZero
Figure 3: The bounds π = 0 . π = 0 . β =0 . , φ = 0 . , r = 0 . , ψ = 0 . , b = 0 .
2; and W q = 1, W u = − y l = 0 . w l = 0 . π = π or π = π . Solvingfor the unique p ( π ) ∈ [0 ,
1] that is consistent with steady state in the left side of Figure 4, we findthat under the parameter of the example p ( π ) A q ( π ) − Q ∗ = 0 . × . × (2 − . − . > π = π , the steadystate value p ( π ) ∈ [0 ,
1] shown in the right side of Figure 4 is not consistent with workers’ acceptinglow tech offers as p ( π ) A q ( π ) − Q ∗ = 0 . × . × (2 − . − . >
0. Hence there is aunique equilibrium in this example. 16 p -6-4-202468101214 G ( , , p ) G-functionZero p -6-4-202468101214 G ( , , p ) G-functionZero
Figure 4: The left side shows the steady state value p ( π ) and the right side shows the steady statevalue p ( π ) The latter is not consistent with worker optimization. We now change the parameters so that the model admits multiple equilibria. See the appendixfor details. Note that both sectors are active in every equilibrium of the example. The firstequilibrium is a fully mixed equilibrium π ∗ ∈ ( π, π ) shown in Figure 5. Here both sectors areactive, and qualified workers are indifferent between accepting and rejecting, so α ( π ∗ ) ∈ (0 ,
1) isset to create indifference firms that enter.
Only Low-tech Two sectors Only High-tech S t ead y S t a t e F un c t i on G G-functionLower/UpperZero
Figure 5: The bounds are π = 0 . π = 0 . π ∗ = 0 . β = 0 . , φ = 0 . , r = 0 . , ψ = 0 . , b = 0 .
2; and W q = 1, W u = − y l = 0 . w l = 0 . π and π , respectively. At π low tech jobs must be rejected by workers to keep firmsindifferent, so α ( π ) = 0. The equilibrium p ( π ) characterized by (29) is p ( π ) = 0 . ∈ (0 ,
1) shown17n the left side of Figure (6). And p ( π ) A q ( π ) − Q ∗ = p ( π ) π (2 − π ) − Q ∗ > α ( π ) = 0.At π low tech jobs must be accepted. The equilibrium p ( π ) characterized by (28) is p ( π ) =0 . ∈ (0 ,
1) shown in the right side of Figure (6). And p ( π ) A q ( π ) − Q ∗ = p ( π ) π (2 − π ) − Q ∗ < α ( π ) = 1. p -6-4-202468101214 G ( , , p ) G-functionZero p -6-4-202468101214 G ( , , p ) G-functionZero
Figure 6: The left side shows the steady state equilibrium p ( π ) at π , and the right side shows thesteady state equilibrium p ( π ) at π . We now add a payoff irrelevant group characteristic to the model. Each worker belongs to group j ∈ { f, m } and we denote by λ j the fraction of workers in the population that belongs to group j . For notational brevity, we let π = (cid:0) π m , π f (cid:1) denote the endogenous stationary proportions ofqualified workers from each group. For the same reasons as in the baseline model, the values of employing qualified and unqualifiedworkers are independent of π = (cid:0) π m , π f (cid:1) and given by W u and W q defined in (3). Hence, there isno change in the optimal firm hiring decision. Given group-specific prior π j the and signal θ thefirm is better off hiring the worker from group j ∈ { f, m } in expectation if and only if P (cid:0) θ, π j (cid:1) W q + (cid:0) − P (cid:0) θ, π j (cid:1)(cid:1) W u ≥ . (33)Hence, the optimal hiring rule is derived just like in the symmetric model, so for any j = m, f andany π j ∈ [0 ,
1] the optimal hiring threshold is characterized as in Lemma 1. We write s ( π m ) and s (cid:0) π f (cid:1) for the thresholds as they can be determined independently. We also keep the shorthandnotation A q (cid:0) π j (cid:1) = 1 − F q (cid:0) s (cid:0) π j (cid:1)(cid:1) for the probability that a qualified worker will get an offer whenmatched with a high tech firm and A u (cid:0) π j (cid:1) = 1 − F u (cid:0) s (cid:0) π j (cid:1)(cid:1) for the corresponding probability for18n unqualified worker. As in the symmetric model these are uniquely determined and π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u is strictly increasing in π j for exactly the same reasons as the baseline model.The free entry conditions are the obvious extensions of the ones for the symmetric model,namely K = βp f X j = f,m λ j (cid:2) π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u (cid:3) (34) K = βp f X j = f,m λ j (cid:2) π j α j ( π ) + (1 − π j ) (cid:3) W l , where p f is the probability that a firm matches with a worker, which, since we use simple urnball matching, is equated with the ratio of unemployment over vacancies, or the inverse markettightness rate. Notice that α j ( π ), the randomization probability for a worker in group j is writtenas a function of the proportion of qualified unemployed for both groups . This is because of feedbackeffects between groups that are explained below.The argument is identical to the argument for the symmetric model. Also, we note that anecessary condition for existence of an equilibrium with both sectors active is X j = f,m λ j (cid:2) π j α j ( π ) + (1 − π j ) (cid:3) W l = X j = f,m λ j (cid:2) π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u (cid:3) , (35)generalizing the condition in (10) from the baseline model. This condition illustrates that if startingfrom a mixed strategy equilibria, an increase in π j no longer needs to be accompanied by anincreased probability of accepting low tech jobs for group j to keep firm indifference, as the othergroup can adjust their behavior instead. Indeed, the worker optimization problem will createincentives for such cross groups effects to be present. As in the symmetric model, we only need to consider the qualified workers. Denote by V j ( π ), V jh ( π ) and V jl ( π ) the value for a qualified worker from being unemployed, being employed in thehigh tech sector, and being employed in the low tech sector, respectively. Also, let p ( π ) be theprobability that the worker meets a high tech firm, so that 1 − p ( π ) is the probability of meetinga low tech firm. Following the same steps ( e.g., from the obvious extensions of (13) and (14) as inthe symmetric model we have that V jh ( π ) = w h + φβV j ( π )1 − (1 − φ ) β (36) V jl ( π ) = w l + φβV j ( π )1 − (1 − φ ) β , V ∗ is defined in (16) as the value of unemployment that makes the worker indifferent betweenaccepting and rejecting. For the same reasons as in the symmetric model we have that α j ( π ) = V j ( π ) < V ∗ [0 ,
1] if V j ( π ) = V ∗ V j ( π ) > V ∗ . (37)By substituting into the analog of (13) it follows that if the worker is indifferent, then it must bethat V ∗ = p ( π ) A q (cid:0) π j (cid:1) w h + φβV ∗ − (1 − φ ) β + (cid:0) − A q (cid:0) π j (cid:1) p ( π ) (cid:1) ( b + βV ∗ ) . (38)Hence, the characterization of the optimal worker behavior is almost identical to the symmetricmodel. Using the exact same steps as in the baseline model, we have that: Lemma 5.
Suppose that ≤ w l − b ≤ β (1 − φ )( w h − b ) and let Q ∗ ∈ [0 , be defined in Lemma 4.Then, the optimal worker choice correspondence for group j is, α j ( π ) = if p ( π ) A q (cid:0) π j (cid:1) < Q ∗ [0 , if p ( π ) A q (cid:0) π j (cid:1) = Q ∗ if p ( π ) A q (cid:0) π j (cid:1) > Q ∗ . (39)At this point, it may appear that the two group model collapses to the baseline model, but thisisn’t so because of a subtle difference. We assume that workers from the two groups are randomlymatched with firms. Hence, both groups are faced with the same probability of matching a hightech firm, implying that firm entry cannot adjust to keep both groups willing to randomize unless π m = π f . In the symmetric model, should π change slightly from a fully mixed equilibrium, an associatedsmall change in the proportion of high tech firms can restore indifference for the workers. Incontrast, when there are multiple groups and π j is perturbed from a symmetric mixed strategyequilibrium, there is simply no way for the proportion of high tech firms to adjust so as to makeboth groups indifferent. This is immediate from the acceptance rules in (39). It follows that theresponse must involve at least one group to reject low tech offers for sure or one group to accept lowtech offers for sure. Hence, equilibria in which both groups are mixing is a knife edge possibility,whereas mixing in the symmetric model may be a robust possibility.20 .4 Steady State Conditions Refer to the symmetric model and note that because the steady state conditions in terms of p , α j and π j are just like the symmetric model, we now how a pair of steady state conditions G ( π f , α f ( π ) , p ( π )) = 0 (40) G ( π m , α m ( π ) , p ( π )) = 0 , where G is defined in (25). Derivations are identical to the baseline model. Hence: Definition 2.
A steady state equilibrium is an object (cid:0) π , α f ( π ) , α m ( π ) , p ( π ) (cid:1) such that (40) holds, (cid:0) α f ( π ) , α m ( π ) (cid:1) satisfy the worker optimality condition (39) and p ( π ) is consistent with optimalentry. It is immediate from (40) that if ( π, α ( π ) , p ( π )) is an equilibrium in the baseline model,then ( π, π ) satisfies (40). Moreover, the optimality condition (39) and firm entry conditions re-duce to the ones in the baseline model. Hence, any equilibrium in the baseline model corre-sponds to a non-discriminatory equilibrium in the model with observable group characteristics (cid:0) π , α f ( π ) , α m ( π ) , p ( π ) (cid:1) = (( π, π ) , α ( π ) , α ( π ) , p ( π )). We now argue that even in the case with a unique equilibrium in the symmetric model, there maybe asymmetric equilibria in the model with observable group characteristics.Assume that π < Π ∗ < π is such that A q (Π ∗ ) = Q ∗ . Since Q ∗ can be set in any way wewant without affecting the incentives for the firms by simultaneously changing the wage and theproductivity this is always possible. Also assume that ψ > π, which assures that G ( π, , > . Moreover, let φ be small enough so that any π ∗ < Π ∗ for any π ∗ such that e G ( π ∗ ) = 0 and so that G ( π, , < p ( π ) < Q ∗ A q ( π ) for the unique solution to G ( π, , p ( π )) = 0 . This is possible as π ∗ ( φ ) → φ → e G ( π ∗ ( φ )) = 0 , G ( π, , → −∞ as φ → p ( π ( φ )) → φ → G ( π, , p ( π ; φ )) = 0 . Together theseassumptions rule out any kind of symmetric equilibrium except for a fully randomized equilibrium.Moreover, we have that e G (Π ∗ ) = G (Π ∗ , α (Π ∗ ) , < < G (cid:18) π, , Q ∗ A q ( π ) (cid:19) (41)where the first inequality is because there is no high tech equilibrium and the second becausethere is no two sector equilibrium at π. By continuity, there exists some fully mixed symmetricequilibrium π ∗ ∈ (Π ∗ , π ) . In general, it may or may not be unique, but the case with uniqueness(which is possible) is the more interesting case. 21 roposition 6.
Suppose that (41) is satisfied. Then there exists a fully mixed symmetric equilib-rium (cid:16) π ∗ , α ( π ∗ ) , Q ∗ A q ( π ∗ ) (cid:17) . Moreover, for any such symmetric equilibrium there exists an interval (cid:0) p, p (cid:1) containing Q ∗ A q ( π ∗ ) and π f ( p ) < π ∗ < π m ( p ) such that each p ∈ (cid:0) p, p (cid:1) , (cid:0) π f ( p ) , π m ( p ) (cid:1) cor-responds to an asymmetric equilibrium in which α f ( p ) = 1 and a m ( p ) = 0 for some populationproportions (cid:0) λ f ( p ) , λ m ( p ) (cid:1) with λ f ( p ) + λ m ( p ) = 1 . Moreover, if φ is small enough λ m ( p ) isstrictly increasing in p implying that there is a generic set of population fractions such that anasymmetric equilibrium exists. The idea is straightforward, but some of the details of the proof in the appendix are somewhattedious. The first step simply notes that if the proportion of high tech firms stays the same andwomen accept low tech jobs for sure and men reject them for sure, then the steady state proportionsof qualified men and females diverge. Men are now more likely to be qualified and women are lesslikely to be qualified than in the randomized equilibrium. All else equal the profitability of meetingmen (women) increases (decreases) in the high tech sector and decreases (increases) in the low techsector, so the population proportion that leaves the firms indifferent at the original high tech firmprobability Q ∗ /A q ( π ∗ ) are uniquely determined. However, p can be perturbed around Q ∗ /A q ( π ∗ )while still having men with a strict incentive to reject low tech jobs and women having a strictincentive to accept. This can be used to show that there is a robust set of (cid:0) λ f , λ m (cid:1) for which anasymmetric equilibrium exists.It is important to notice that nowhere is the proof of Proposition 6 relying on multiplicity inthe underlying one-group model. Instead, the result is driven by the fact that men and womencompete in the same market , which allows the two groups to specialize in equilibrium. All elseequal, if men get pickier, which increases the fraction of qualified unemployed men, the high techsector gets more profitable. To restore equal profits, it is thus necessary for women to get lesspicky. One could, of course, object that instead the high tech sector should compete away the lowtech jobs, but this would drive down the proportion of qualified workers in both groups to such anextent that only low tech firms would like to enter when φ is small. We consider the two group version of the model in Section 5.1, assuming groups are of equal size.Recall that in the baseline model, this is a parametrization in which the equilibrium is unique. Thisequilibrium corresponds to a symmetric equilibrium with two groups, but as we show below, thereare now also discriminatory equilibria, illustrating that the model creates potential incentives forspecialization.We know from from (39) that at most group can randomize. There may be equilibria in whichno group randomizes, but we will consider the case in which π m > π f and α f ( π ) ∈ (0 , This is somewhat similar to Bardhi et al. (2019) where learning dynamics can create sizable inequality from smalldifferences between groups
22e have that α m ( π ) = 0 from (39): p ( π ) A q ( π m ) > p ( π ) A q ( π f ) = Q ∗ | {z } hiring probability of high tech firm and α m = 0 < α f ∈ (0 , | {z } acceptance of low tech job . This can be interpreted as a cross group effect that comes from both groups searching in the samelabor market. They therefore share the same probability of matching with a high tech firm p ( π ),which drives the spillovers across groups.In the discriminatory equilibrium p ( π ) A q ( π m ) > Q ∗ = p ( π ) A q ( π f ) to justify the worker accep-tance rules. Moreover, the indifference condition for two active sectors (35) evaluated at α m ( π ) = 0simplifies to λ m (1 − π m ) W l + λ f h π f α f ( π ) + (1 − π f ) i W l = X j = f,m λ j (cid:2) π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u (cid:3) . (42)This condition pins down the female group’s acceptance decision α f ( π ) as an increasing functionin both π f and π m , α f ( π ) = P j = f,m λ j (cid:2) π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u − (1 − π j ) W l (cid:3) λ f π f W l . (43)The steady state conditions (40) evaluated at α m ( π ) = 0 and p ( π ) = Q ∗ A q ( π f ) are then1 + Q ∗ A q ( π f ) A u ( π m ) φ + (1 − φ ) r + 1 − Q ∗ A q ( π f ) φ = 1 − ψ − π m π m ψ (cid:20) Q ∗ A q ( π f ) A q ( π m ) φ (cid:21) , (44)1 + Q ∗ A q ( π f ) A u ( π f ) φ + (1 − φ ) r + 1 − Q ∗ A q ( π f ) φ = 1 − ψ − π f π f ψ Q ∗ φ + 1 − Q ∗ A q ( π f ) φ α f ( π ) , (45)where α f ( π ) is determined in (43). These two equations pin down the equilibrium (cid:8) π m , π f (cid:9) . If thegroup m workers’ condition p ( π ) A q ( π f ) = Q ∗ < p ( π ) A q ( π m ) hold, then the equilibria candidate isan equilibrium.Substituting the parametric assumptions into equations (44) and (45) we plot the result inFigure 7. The plot shows the discrimination equilibrium beliefs about the qualification of group f and m . The steady state condition for male group (44) is captured by the blue line, and the steadystate condition for female group (45) is represented by the red line. The equilibrium beliefs aboutthe qualification of the male group and female group π m > π f , confirming that we have constructedan equilibrium. 23 m f G-function MaleG-function Female
Figure 7: Discrimination equilibria with π m > π f . The red line captures the steady state functionfor women, the blue line captures the steady state function for men. Parameters for this exampleare the same as Section 5.1: β = 0 . φ = 0 . r = 0 . ψ = 0 . b = 0 . f . Hence, high tech firms offer ajob to the workers from group f with a lower probability than workers from group m . As a result,group f workers incentives to accept low tech jobs are reinforced. One may notice that this looksa lot like women are less self-confident than men, so our model could be viewed as an instrumentalmodel of effects of confidence on the gender gap studied by Kamas and Preston (2018) and others. Most applied theory papers on affirmative action are aimed at pointing out potential unintendedeffects of the policy. Coate and Loury (1993) argue that the policy may need to be permanentbecause incentives to acquire human capital may be perversely affected. Moro and Norman (2003)point out that if wages are endogenous, affirmative action may not even benefit the targeted group.Chan and Eyster (2003) show that banning affirmative action in college admissions may backfire be-cause universities create more randomness in the admission process. Finally, Fershtman and Pavan(2020) argue that requiring a larger proportion of minority candidates to be considered can backfireas employers may respond by increasing the pool of candidates.In our model, when comparing steady states before and after the policy, none of these unintendedconsequences occur. Assuming that the policy mandates that the proportion of men and womenshould be equalized in the high tech sector, the only possible steady state equilibrium is a symmetricequilibrium. Should π f < π m , the only way the firms could satisfy the quota is to be less demandingon female applicants. Hence, women would have weaker incentives to accept low tech jobs, which isinconsistent with π f < π m . While we have not worked out the adjustment dynamics, this suggeststhat gender quotas could be useful if statistical discrimination driven by occupational choice is24mportant for the gender wage gap.Our model and Coate and Loury (1993) interpret affirmative action as a relative hiring quota,and the two models have a similar externality. In Coate and Loury (1993), the more your groupinvests the higher is the prior belief about the worker, while in our case, the pickier your group isthe higher is the prior belief about the worker. So, why are the results so different? Besides ourmodel being a full-fledged dynamic model, the key difference is that making it easier to get themore attractive jobs unambiguously creates the right incentives to reject low end jobs. In contrast,Coate and Loury (1993) focuses on the possibility that making it easier to get the attractive jobsdistorts incentives for pre-market investments in human capital. We propose a search model of statistical discrimination. The interaction between occupationalchoice, search externalities, and a signal extraction problem makes it possible that identical groupsspecialize in equilibrium, resulting in cross-group inequality. Groups share the same labor market,creating spillover effects between groups, and discriminatory equilibria may exist also when thebaseline model without group characteristic has a unique equilibrium. For the same reason, it ispossible that the introduction of group characteristics destabilizes a symmetric equilibrium. Unlikethe previous literature, affirmative action is an appropriate remedy to eliminate discriminatoryequilibria in our model. 25
Appendix A: Omitted Proofs
A.1 Proof of Lemma 1
Proof.
Suppose that s ( π ) solves P ( s ( π ) , π ) W q + (1 − P ( s ( π ) , π )) W u = 0 , which can be rear-ranged as (5). By the monotone likelihood property it follows that P ( θ, π ) W q +(1 − P ( θ, π )) W u > θ > s ( π ) and P ( θ, π ) W q +(1 − P ( θ, π )) W u < θ < s ( π ), so the firm has a strict incentiveto hire a worker with θ > s ( π ) and a strict incentive not to hire a worker with θ < s ( π ) . Thecorner cases are immediate.
A.2 Proof of Lemma 2
Proof.
Applying the envelope theorem to πA q ( π ) W q + (1 − π ) A u ( π ) W u = max θ π [1 − F q ( θ )] W q +(1 − π ) [1 − F u ( θ )] W u , we have ddπ [ πA q ( π ) W q + (1 − π ) A u ( π ) W u ]= [1 − F q ( θ ( π ))] W q − [1 − F u ( θ ( π ))] W u > , where the strict inequality follows from 1 − F q ( θ ) > − F u ( θ ) , which is an implication of f q f u beingstrictly increasing in θ , and W q > W u . A.3 Proof of Lemma 3
Proof.
To show (1).
For π < π , − K + βp f [ πA q ( π ) W q + (1 − π ) A u ( π ) W u ] < − K + βp f [ πA q ( π ) W q + (1 − π ) A u ( π ) W u ]= − K + βp f (1 − π ) W l < − K + βp f (1 − π ) W l ≤ − K + βp f [ πα + (1 − π )] W l where the first inequality follows from πA q ( π ) W q + (1 − π ) A u ( π ) W u is strictly increasing (inLemma 2), the equality follows from (1 − π ) W l = πA q ( π ) W q + (1 − π ) A u ( π ) W u , and the lastinequality hold for any α ∈ [0 , α ∈ [0 ,
1] when π ∈ [0 , π ). To show (5).
For π > π , − K + βp f [ πA q ( π ) W q + (1 − π ) A u ( π ) W u ] > − K + βp f [ πA q ( π ) W q + (1 − π ) A u ( π ) W u ]= − K + βp f W l ≥ − K + βp f [ πα + (1 − π )] W l . πA q ( π ) W q + (1 − π ) A u ( π ) W u is strictly increasing (inLemma 2), the equality follows from W l = πA q ( π ) W q + (1 − π ) A u ( π ) W u , the second inequalityholds for any α ∈ [0 , α ∈ [0 ,
1] when π ∈ ( π, To show (2) (3) (4).
The workers’ acceptance rule α matters only when two sectors are active.When both sectors are active, the indifference condition holds[ πα ( π ) + (1 − π )] W l = πA q ( π ) W q + (1 − π ) A u ( π ) W u , from which we have a unique α ( π ) = πA q ( π ) W q +(1 − π ) A u ( π ) W u − (1 − π ) W l πW l .From Lemma 2, πA q ( π ) W q + (1 − π ) A u ( π ) W u is strictly increasing. It is thus immediate that α ( π ) must be strictly increasing to keep firms indifferent between the two sectors. A.4 Proof of Lemma 4
Proof.
For the first part, equation (18) is simply a linear function of Q ( π ). It has solution Q ∗ = w l − (1 − (1 − φ ) β ) b (1 − φ ) β − w l w h − w l ∈ [0 , , provided that 0 ≤ w l − b ≤ β (1 − φ )( w h − b ). For the second part, suppose that α ( π ) = 1 is optimal.Then, by (17) V ( π ) ≤ V ∗ and, directly from (13) V l ( π ) ≥ b + βV ( π ). Equation (13) becomes V ( π ) = p ( π ) (cid:20) A q ( π ) w h + φβV ( π )1 − (1 − φ ) β + (1 − A q ( π )) ( b + βV ( π )) (cid:21) + (1 − p ( π )) V l ( π ) ≥ p ( π ) (cid:20) A q ( π ) w h + φβV ( π )1 − (1 − φ ) β + (1 − A q ( π )) ( b + βV ( π )) (cid:21) + (1 − p ( π )) ( b + βV ( π ))= p ( π ) A q ( π ) w h + φβV ( π )1 − (1 − φ ) β + (1 − A q ( π ) p ( π )) ( b + βV ( π ))= p ( π ) A q ( π ) (cid:20) w h + φβV ( π )1 − (1 − φ ) β − ( b + βV ( π )) (cid:21) + ( b + βV ( π )) (46)or p ( π ) A q ( π ) ≤ V ( π ) − ( b + βV ( π )) w h + φβV ( π )1 − (1 − φ ) β − ( b + βV ( π )) , where the RHS is increasing in V ( π ) and V ( π ) ≤ V ∗ . Hence, p ( π ) A q ( π ) ≤ Q ∗ . A symmetricargument shows that α ( π ) = 0 is optimal if and only if p ( π ) A q ( π ) ≥ Q ∗ and the derivation of(18) shows that the worker is indifferent if and only if p ( π ) A q ( π ) = Q ∗ .27 .5 Proof of Proposition 1 Proof.
Suppose an equilibrium with p ( π ) = 0 exists, which implies that α ( π ) = 1 from (20). Forthis to be consistent with the steady state conditions G ( π, ,
0) = 1 + 1 φ − − ψ − π πψ (cid:20) φ (cid:21) = 0 . (47)The unique solution to (47) is π = ψ is the unique solution to this equation. Since p ( π ) = p ( ψ ) = 0it follows that p ( ψ ) A q ( ψ ) = 0 < Q ∗ so this is an equilibrium as ψ ≤ π. A.6 Proof of Proposition 2
Proof.
We note that e G is continuous, that e G (0) > , and e G ( π ) → −∞ as π → . Hence, thesteady state condition (26) has at least one solution π ∗ . From Lemma 3 it is then immediatethat ( π ∗ , p ( π ∗ )) = ( π ∗ ,
1) is inconsistent with equilibrium if π ∗ < π. For π ∗ ∈ [ π, π ] is consistentwith equilibrium if and only if workers are willing to reject low tech offers, which is if and only if A q ( π ∗ ) ≥ Q ∗ . Finally, if π ∗ > π it is direct from Lemma 3 that only high tech firms are willing toenter regardless of what workers do. A.7 Proof of Proposition 3
Proof. G ( π, ,
0) = ψ − π (1 − π ) ψ (cid:20) φ (cid:21) > ψ > π , and G ( π, , p ) = G ( π, ,
0) + p (cid:20) A u ( π ) φ + (1 − φ ) r − φ − − ψ − π πψ (cid:18) A q ( π ) φ − φ (cid:19)(cid:21) . (49)Hence, if ψ > π then G ( π, , > . So a unique solution to (28) exists if G ( π, , ≤ , whichis possible since A u ( π ) φ + (1 − φ ) r − φ − − ψ − π πψ (cid:18) A q ( π ) φ − φ (cid:19) < ψ − π (1 − π ) ψ (cid:18) A q ( π ) φ − φ (cid:19) ≤ . If instead ψ < π , then G ( π, , p ) is strictly increasing in p ∈ [0 ,
1] since A u ( π ) φ + (1 − φ ) r − φ − − ψ − π πψ (cid:18) A q ( π ) φ − φ (cid:19) > ψ − π (1 − π ) ψ (cid:20) A u ( π ) φ + (1 − φ ) r − φ (cid:21) > . But G ( π, ,
1) = 1 + A u ( π ) φ + (1 − φ ) r − − ψ − π πψ (cid:20) A q ( π ) φ (cid:21) < ψ − π (1 − π ) ψ (cid:20) A q ( π ) φ (cid:21) < , ψ < π there is no solution p ( π ) ∈ [0 ,
1] to (28) .
A.8 Proof of Proposition 5
Proof. If ψ ≤ π a low tech equilibrium exists. Hence, consider the case in which ψ > π. Assumefirst that A q ( π ) ≤ Q ∗ . Then, if e G ( π ) = G ( π, ,
1) = G ( π, , ≥ π ∗ ≥ π to(26) which corresponds to a high tech equilibrium (even if π ∗ = π because A q ( π ) ≤ Q ∗ ). Hence, forno high tech equilibrium to exist e G ( π ) = G ( π, ,
1) = G ( π, , < . Since we are also assuming ψ > π and A q ( π ) ≤ Q ∗ , Proposition 3 guarantees existence of a pure strategy equilibrium at π. Hence, the only possibility for non-existence is that A q ( π ) > Q ∗ . Then, (assuming away thetrivial case with A q (0) ≥ Q ∗ ) there exists Π ∗ < π such that A q (Π ∗ ) = Q ∗ . Assume that Π ∗ ≤ π. Then for no high tech equilibrium to exist e G ( π ) = G ( π, , < π ∗ such that e G ( π ∗ ) = 0 exists and corresponds to a high tech equilibrium . From (30) we know that G ( π, , > ψ ≥ π, so for no two sector equilibrium to exist at π it must be that p ( π ) A q ( π ) < Q ∗ for the unique solution to G ( π, , p ( π )) = 0 . Since G ( π, , p ) in linear in p and p ( π ) < Q ∗ A q ( π ) it follows that G (cid:16) π, , Q ∗ A q ( π ) (cid:17) < . Hence, for a mixed strategy equilibrium not toexist G (cid:16) π, , Q ∗ A q ( π ) (cid:17) ≤ π on [ π, π ] . We also have that G ( π, , > ψ > π, so, by linearity, if there is no fully mixed equilibrium there exists p ( π ) ∈ (cid:16) , Q ∗ A q ( π ) i such that G ( π, , p ( π )) = 0 . Hence, we have established existence of at least one equilibriumwhenever Π ∗ ≤ π. The final possibility is that π < Π ∗ < π. Then, for no high tech equilibrium toexist e G (Π ∗ ) = G (Π ∗ , α (Π ∗ ) , < . For the same reason as above, for no mixed equilibrium toexist it must be that G (cid:16) π, , Q ∗ A q ( π ) (cid:17) ≤ . Repeating the argument above it follows that there existsan equilibrium at π. A.9 Proof of Proposition 6
Proof.
Consider an alternative candidate equilibrium in which α f ( π ) = 1 and α m ( π ) = 0 . In suchan equilibrium, the following conditions must hold G (cid:16) π f , , p ( π ) (cid:17) = 1 + p ( π ) A u ( π f ) φ + (1 − φ ) r + 1 − p ( π ) φ − − ψ − π f π f ψ (cid:20) p ( π ) A q ( π f ) φ + 1 − p ( π ) φ (cid:21) = 0 ,p ( π ) A q ( π f ) ≤ Q ∗ ,G ( π m , , p ( π )) = 1 + p ( π ) A u ( π m ) φ + (1 − φ ) r + 1 − p ( π ) φ − − ψ − π m π m ψ (cid:20) p ( π ) A q ( π m ) φ (cid:21) = 0 ,p ( π ) A q ( π m ) ≥ Q ∗ . Note that G (cid:18) π ∗ , , Q ∗ A q ( π ∗ ) (cid:19) − G (cid:18) π ∗ , α ( π ∗ ) , Q ∗ A q ( π ∗ ) (cid:19) = ( α ( π ∗ ) −
1) 1 − ψ − π ∗ π ∗ ψ " − Q ∗ A q ( π ∗ ) φ < . G (cid:16) π ∗ , α ( π ∗ ) , Q ∗ A q ( π ∗ ) (cid:17) = 0 it follows that G (cid:16) π ∗ , , Q ∗ A q ( π ∗ ) (cid:17) < . Symmetrically, G (cid:18) π ∗ , , Q ∗ A q ( π ∗ ) (cid:19) − G (cid:18) π ∗ , α ( π ∗ ) , Q ∗ A q ( π ∗ ) (cid:19) = α ( π ∗ ) 1 − ψ − π ∗ π ∗ ψ " − Q ∗ A q ( π ∗ ) φ > , so G (cid:16) π ∗ , , Q ∗ A q ( π ∗ ) (cid:17) > G (cid:16) , , Q ∗ A q ( π ∗ ) (cid:17) > π → G (cid:16) π, , Q ∗ A q ( π ∗ ) (cid:17) = −∞ . Hence, there exists (cid:0) π f , π m (cid:1) with π f < π ∗ < π m such that G (cid:18) π f , , Q ∗ A q ( π ∗ ) (cid:19) = 0 (50) G (cid:18) π m , , Q ∗ A q ( π ∗ ) (cid:19) = 0 . It is thus immediate that A q (cid:0) π f (cid:1) Q ∗ A q ( π ∗ ) < Q ∗ and A q ( π m ) Q ∗ A q ( π ∗ ) > Q ∗ , justifying the incentivesto accept and reject low tech jobs for the two groups. In contrast, the indifference condition forthe entrant firms is not necessarily satisfied. However,1 > π ∗ α ( π ∗ ) + (1 − π ∗ )= π ∗ A q ( π ∗ ) W q + (1 − π ∗ ) A u ( π ∗ ) W u > π f A q ( π f ) W q + (1 − π f ) A u ( π f ) W u , and (1 − π m ) < − π ∗ < π ∗ α ( π ∗ ) + (1 − π ∗ )= π ∗ A q ( π ∗ ) W q + (1 − π ∗ ) A u ( π ∗ ) W u < π m A q ( π m ) W q + (1 − π m ) A u ( π m ) W u , so there exists a unique (cid:0) λ f , λ m (cid:1) with λ f + λ m = 1 such that h λ f + λ m (1 − π m ) i W l = λ f h π f A q ( π f ) W q + (1 − π f ) A u ( π f ) W u i + λ m [ π m A q ( π m ) W q + (1 − π m ) A u ( π m ) W u ] . By continuity of G there is an interval (cid:0) p, p (cid:1) around Q ∗ A q ( π ∗ ) such that pA q ( π f ) < Q ∗ and pA q ( π m ) >Q ∗ and G (0 , , p ) > > G ( π ∗ , , p ) G ( π ∗ , , p ) > > lim π → G ( π, , p )for each p ∈ (cid:0) p, p (cid:1) . For each p in the interval there is a corresponding solution π f ( p ) , π m ( p )30uch that π f ( p ) < π ∗ < π m ( p ) and for each such solution there is a unique (cid:0) λ f ( p ) , λ m ( p ) (cid:1) with λ f ( p )+ λ m ( p ) = 1 that makes firms indifferent across sectors. Moreover, if we pick (cid:0) π f ( p ) , π m ( p ) (cid:1) as always being the smallest solutions differentiability of G ( π, α, p ) on (0 ,
1) implies that withinthe interval ∂G (cid:0) π f ( p ) , , p (cid:1) ∂π f < , (51) ∂G ( π m ( p ) , , p ) ∂π f < . By a direct calculation ∂G ( π, α, p ) ∂p = A u ( π ) φ + (1 − φ ) r − φ − − ψ − π πψ (cid:20) A q ( π ) φ − αφ (cid:21) . Hence, ∂G ( π m ( p ) , ,p ) ∂p < ∂G ( π f ( p ) , ,p ) ∂p is ambiguous. However, if φ is large enough we canmake sure that ∂G ( π f ( p ) , ,p ) ∂p ≤ p ∈ (cid:0) p, p (cid:1) in which case a standard implicit differentiationimplies that π f ( p ) and π m ( p ) are both decreasing in p . It then follows that for firms to keepindifference it is necessary that λ m ( p ) is strictly increasing in p . B Appendix B: Numerical Examples
B.1 Description of Function (25)
We describe the G function (25) piecewisely as follows. For π < π , (25) evaluated at p ( π ) = 0 and α ( π ) = 1 G ( π, ,
0) = 1 + 1 φ − − ψ − π πψ (cid:20) φ (cid:21) . (52)For π > π , p ( π ) = 1, α ( π ) is indetermined, (25) evaluated at p ( π ) = 1 G ( π, α ( π ) ,
1) = 1 + A u ( π ) φ + (1 − φ ) r − − ψ − π πψ (cid:20) A q ( π ) φ (cid:21) . (53)For π ∈ ( π, π ), α ( π ) ∈ (0 , p ( π ) ∈ [0 , p ( π ) A q ( π ) = Q ∗ G (cid:18) π, α ( π ) , Q ∗ A q ( π ) (cid:19) = 1 + Q ∗ A q ( π ) A u ( π ) φ + (1 − φ ) r + 1 − Q ∗ A q ( π ) φ − − ψ − π πψ " Q ∗ A q ( π ) A q ( π ) φ + 1 − Q ∗ A q ( π ) φ α ( π ) , (54)where α ( π ) = πA q ( π ) W q + (1 − π ) A u ( π ) W u − (1 − π ) W l πW l . π = π , α ( π ) = 0, (25) evaluated at π and α ( π ) = 0 G ( π, , p ( π )) = 1 + p ( π ) A u ( π ) φ + (1 − φ ) r + 1 − p ( π ) φ − − ψ − π πψ (cid:20) p ( π ) A q ( π ) φ (cid:21) . (55)At π = π , α ( π ) = 1, (25) evaluated at π and α ( π ) = 1 G ( π, , p ( π )) = 1 + p ( π ) A u ( π ) φ + (1 − φ ) r + 1 − p ( π ) φ − − ψ − π πψ (cid:20) p ( π ) A q ( π ) φ + 1 − p ( π ) φ (cid:21) . (56)In sum, we simplify function (25) into a function of one dimension. For π ∈ [0 ,
1] except π and π , (25) is a function of πG ( π ) = φ − − ψ − π πψ h φ i if π ∈ [0 , π )1 + Q ∗ Aq ( π ) A u ( π ) φ +(1 − φ ) r + − Q ∗ Aq ( π ) φ − − ψ − π πψ " Q ∗ Aq ( π ) A q ( π ) φ + − Q ∗ Aq ( π ) φ πA q ( π ) W q +(1 − π ) A u ( π ) W u − (1 − π ) W l πW l if π ∈ ( π, π ) . A u ( π ) φ +(1 − φ ) r − − ψ − π πψ h A q ( π ) φ i if π ∈ ( π, π and π , (25) is a function of pG ( p ) = p ( π ) A u ( π ) φ +(1 − φ ) r + − p ( π ) φ − − ψ − π πψ h p ( π ) A q ( π ) φ i if π = π p ( π ) A u ( π ) φ +(1 − φ ) r + − p ( π ) φ − − ψ − π πψ h p ( π ) A q ( π ) φ + − p ( π ) φ i if π = π. In all the numerical examples, we use the signal distributions f q ( θ ) = 2 θ , f u ( θ ) = 2(1 − θ ),which satisfy the property of monotone likelihood ratio. Calibrating W q = 1 and W u = −
1. Thenthe hiring probability for qualified and unqualified workers are A q ( π ) = π (2 − π ) and A u ( π ) = π ,respectively. Proof.
With F q ( θ ) = θ , F u ( θ ) = 2 θ − θ , firms’ hiring threshold from problem 6 is s ( π ) = arg max θ π (cid:2) − θ (cid:3) − (1 − π ) (cid:2) − θ + θ (cid:3) , (57)so s ( π ) = 1 − π , and A q ( π ) = 1 − F q ( s ( π )) = 1 − (1 − π ) = 2 π − π , A u ( π ) = 1 − F u ( s ( π )) = π . All the graphs are plotted by plugging A q ( π ) and A u ( π ) into the function (25) under differentcalibrated parameters. 32 eferences James Albrecht and Susan Vroman. A matching model with endogenous skill requirements.
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External Appendix: Not Intended For Publication
C.1 Other Examples in Baseline Model
There exist other unique equilibria and multiple equilibria in the baseline model. We keep the sameparameters for W q = 1, W u = − y l = 0 . w l = 0 . β , φ , r , ψ and b . Other unique equilibria examples .Unique Equilibria β φ r ψ b
In EquilibriumExample (a) .99 .15 .75 .075 .2 only low techExample (b) .99 .15 .75 .75 .2 only high techExample (c) .9 .06 .75 .25 .2 two sectors with low tech jobs rejectedExample (d) .8 .06 .75 .25 .2 two sectors with low tech job accepted(a) With β = . , φ = . , r = . , ψ = . , b = .
2, there is a unique equilibria in which only lowtech sector is active. An important change is that the mass of skilled workers is decreasedrelative to the other examples. From Figure 8 we see that there exists a low tech equilibria,and there is neither a mixed, nor a high tech equilibrium.
Low-tech Two sectors High-tech S t ead y S t a t e F un c t i on G G-functionLower/UpperZero
Figure 8: β = 0 . φ = 0 . r = 0 . ψ = 0 . b = 0 . π = π , solving for the steady state value p ( π ) = 0 . ∈ [0 , α ( π ) = 0 does not hold because p ( π ) A q ( π ) − Q ∗ = − . <
0. At π = π , thesteady state value p ( π ) = 2 . / ∈ [0 , β = . , φ = . , r = . , ψ = . , b = .
2, there is a unique equilibria in which only hightech sector is active. From Figure 9 we see that there exists a high tech equilibria, and thereis neither a mixed, nor a low tech equilibrium.35 ow-tech Two sectors High-tech S t ead y S t a t e F un c t i on G G-functionLower/UpperZero
Figure 9: β = 0 . φ = 0 . r = 0 . ψ = 0 . b = 0 . π = π , the steady state value p ( π ) = 1 . / ∈ [0 , π = π , the steady state value p ( π ) = 1 . / ∈ [0 , β = . , φ = . , r = . , ψ = . , b = .
2, there is a unique equilibria in which bothsectors are active and the low tech jobs are rejected (accepting is not an equilibrium). Thisis the case of uniqueness in the baseline model in Section 5.1.(d) With β = . , φ = . , r = . , ψ = . , b = .
2, there is a unique equilibria in which bothsectors are active and the low tech jobs are accepted (rejecting is not an equilibria). FromFigure 10 we see that there is neither a mixed, nor a high tech nor a low tech equilibrium.
Low-tech Two sectors High-tech S t ead y S t a t e F un c t i on G G-functionLower/UpperZero
Figure 10: β = 0 . φ = 0 . r = 0 . ψ = 0 . b = 0 . p ( π ) ∈ [0 ,
1] that is consistent with steady state we find that under36he parameter of the example p ( π ) A q ( π ) − Q ∗ = − . <
0, so rejecting low tech offers isinconsistent with worker optimality. At π = π , the steady state value p ( π ) = 0 . ∈ [0 , p ( π ) A q ( π ) − Q ∗ = − . < Other Multiple Equilibria Examples in the baseline model.Multiple Equilibria β φ r ψ b
Two equilibria .99 .06 .75 (.6) .25 .2 (1) two sectors with low techjobs rejected; (2) two sectorswith workers mixingThree equilibria .99 .08 .75 .25 .2 (1) two sectors with low techjobs accepted; (2) two sectorswith workers mixing; (3) twosectors with low tech jobs re-jectedThree equilibria .99 .06 .75 .2 .2 (1) low tech; (2) two sectorswith workers mixing; (3) twosectors with low tech jobs re-jected
C.2 Single Sector Equilibria with Groups
Only low tech firms.
In the case of only low tech firms p ( π ) = 0, so α m ( π ) = α f ( π ) = 1, thesteady state functions for j = f, m are G ( π j , ,
0) = 1 + 1 φ − − ψ − π j π j ψ (cid:20) φ (cid:21) . So π f = π m = ψ .Moreover, firms’ entry optimality for p ( π ) = 0 requires X j = f,m λ j (cid:2) π j α j ( π ) + (1 − π j ) (cid:3) W l ≥ X j = f,m λ j (cid:2) π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u (cid:3) , degenerating into the baseline model W l ≥ [ ψA q ( ψ ) W q + (1 − ψ ) A u ( ψ ) W u ] , so p ( π ) = 0 requires ψ ≤ π . Lemma 6.
There is a symmetric equilibrium in which only low tech firms are active and π m = π f = ψ if and only if ψ ≤ π . nly high-tech firms. In the case of only high-tech firms p ( π ) = 1, so workers’ optimality ofacceptance rule doesn’t matter. The steady state functions for j = f, m are G ( π f , α f , p ( π ) = 1) = 1 + A u ( π f ) φ + (1 − φ ) r − − ψ − π f π f ψ (cid:20) A q ( π f ) φ (cid:21) = 0 ,G ( π m , α m , p ( π ) = 1) = 1 + A u ( π m ) φ + (1 − φ ) r − − ψ − π m π m ψ (cid:20) A q ( π m ) φ (cid:21) = 0 , which pin down π m and π f . Moreover, firms’ entry optimality for p ( π ) = 1 requires X j = f,m λ j (cid:2) π j α j ( π ) + (1 − π j ) (cid:3) W l ≤ X j = f,m λ j (cid:2) π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u (cid:3) , which is easily valid because α f and α m are free if p ( π ) = 0.From the baseline analysis, we know G ( π, α, p ( π ) = 1) = 1 + A u ( π ) φ +(1 − φ ) r − − ψ − π πψ h A q ( π ) φ i = 0has at least one solution. Lemma 7. If G ( π, α, p ( π ) = 1) = 0 has a unique solution, then there is a unique symmetricequilibrium in which only high tech firms are active and π f = π m . Otherwise, either π f = π m or π f = π m . C.3 Other Asymmetric Equilibria
Similarly, we can construct equilibria in which π m > π f , α m ∈ (0 , α f = 1 under some parame-ters. Only group m is indifferent : If α m ( π ) ∈ (0 , α f ( π ) = 1. To make { π m > π f , α m ∈ (0 , , α f = 1 } an equilibrium, π f and π m should satisfy:(1) Firms’ indifference condition: λ m [ π m α m ( π ) + (1 − π m )] W l + λ f W l = X j = f,m λ j (cid:2) π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u (cid:3) ,α m ( π ) is increasing in both π f and π m .(2) Steady state conditions: G ( π m , α m ( π ) , p ( π )) = 0, G ( π f , α f ( π ) = 1 , p ( π )) = 01 + p ( π ) A u ( π m ) φ + (1 − φ ) r + 1 − p ( π ) φ = 1 − ψ − π m π m ψ (cid:20) p ( π ) A q ( π m ) φ + (1 − p ( π )) α m ( π ) φ (cid:21) p ( π ) A u ( π f ) φ + (1 − φ ) r + 1 − p ( π ) φ = 1 − ψ − π f π f ψ (cid:20) p ( π ) A q ( π f ) φ + 1 − p ( π ) φ (cid:21) (3) Group m workers’ indifference: Q ∗ = p ( π ) A q ( π m )38hen, { π m , π f , α m ( π ) , p ( π ) } are pinned down by the four equations. Substituting out p ( π ) and α m ( π ), the equilibrium π m , π f are determined by1 + Q ∗ A q ( π m ) A u ( π m ) φ + (1 − φ ) r + 1 − Q ∗ A q ( π m ) φ = 1 − ψ − π m π m ψ " Q ∗ φ + 1 − Q ∗ A q ( π m ) φ P j = f,m λ j (cid:2) π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u (cid:3) − λ f W l − λ m (1 − π m ) W l λ m π m W l , A u ( π f ) A q ( π m ) Q ∗ φ + (1 − φ ) r + 1 − Q ∗ A q ( π m ) φ = 1 − ψ − π f π f ψ " Q ∗ A q ( π m ) A q ( π f ) φ + 1 − Q ∗ A q ( π m ) φ . Lastly, check the group f workers’ conditions: p ( π ) A q ( π f ) < Q ∗ . Let m = min p ( π ) A q ( π f ) subjectto π f , π m are solutions to the above two equations. Then if m < Q ∗ there exists some equilibriumin which group m workers are indifferent and group f workers reject the low tech job offers. Neither group is indifferent.
There are four cases: { α m = 0 , α f = 1 } , { α m = 1 , α f = 0 } , { α m = 1 , α f = 1 } , { α m = 0 , α f = 0 } .With π m > π f , the only interesting one is: { α m = 0 , α f = 1 } . To make { π m > π f , α m =0 , α f = 1 } be an equilibrium, it should satisfy(1) Firms’ indifference: λ m (1 − π m ) W l + λ f W l = X j = f,m λ j (cid:2) π j A q ( π j ) W q + (1 − π j ) A u ( π j ) W u (cid:3) (2) Steady state condition: G ( π m , α m ( π ) = 0 , p ( π ))) = 0, G ( π f , α f ( π ) = 1 , p ( π ))) = 01 + p ( π ) A u ( π m ) φ + (1 − φ ) r + 1 − p ( π ) φ = 1 − ψ − π m π m ψ (cid:20) p ( π ) A q ( π m ) φ (cid:21) p ( π ) A u ( π f ) φ + (1 − φ ) r + 1 − p ( π ) φ = 1 − ψ − π f π f ψ (cid:20) p ( π ) A q ( π f ) φ + 1 − p ( π ) φ (cid:21) . Then { π m , π f , p ( π ) } are pinned down by the three equations, and the workers’ conditions hold: p ( π ) A q ( π f ) ≤ Q ∗ ≤ p ( π ) A q ( π mm