Costly Screening and Categorical Inequality Chapter 2

Chapter 2 This result is illustrated in the following example. Example 1. Consider identical and risk-neutral candidates with F (x) = 1 − e−x and θ = 0.05. Then rmin = θ/ (v1 − v0 ), and p̂ (r) = 1 − 1 θ · r v1 − v0 for all r > rmin . There are three equilibrium outcomes, two of which are active. The example is depicted in Figure 3, where the upward-sloping curve is the aggregate candidate response curve p̂ (r), and the quasi-concave curve is the screener’s response curve, r̂ (p). As is evident from the figure, there are at most two active equilibria if the function p̂ is concave wherever positive, that is, on the interval (rmin , +∞). To obtain a sufficient condition for this, define rmax = max rθ . θ∈Θ Above screening intensity rmax all candidates invest effort. We then have: Proposition 5. Suppose that the density function f is twice differentiable and log concave with f 0 < 0. If p̂ (rmax ) > pmin , then there exist at most two active equilibria. Many commonly used density functions are log-concave. Examples of transition probability functions F ∈ F with log concave densities are the exponential and normal, and, for certain parameter ranges, the gamma, chi-square, and Weibull distributions. We note that the claim in this proposition is robust in the sense that it holds also for less than perfectly homogeneous candidate pools. The condition p̂ (rmax ) > pmin ensures that all candidate types make positive effort. In sum: for generic parameter values, homogeneous candidate pools, and wide classes of state-transition probability functions, there are either no active equilibria or exactly two. One may ask which category an individual candidate would like to belong to. The model predicts that the candidate would like to belong to the category with the highest equilibrium qualification rate p̂. This follows from the fact that the accept rates, for both qualified and unqualified candidates, are both increasing in p̂. Thus the model can be used to explore issues of identity choice along the lines of Kim and Loury (2019). Furthermore, low cost members of a category may wish to expel those with higher costs, in order to raise the equilibrium qualification rate they face. While identity choice and social exclusion are beyond the scope of this paper, they are interesting avenues for future research. 18 3.5 Stability The set of equilibria may be refined by considering stability properties with respect to a dynamic process involving gradual behavioral adjustments by both the screener and the candidates. Suppose that the variable pair (p, r), the state of the system, is subject to a dynamic ( ṗ = φ(p − p̂ (r)) , (20) ṙ = ψ(r − r̂ (p)) where φ and are continuously differentiable functions (from R to R), such that φ (0) = (0) = 0, φ0 , ψ 0 ≥ 0 and φ0 (0) , ψ 0 (0) > 0. These monotonicity assumptions express the idea that the population share p of qualified candidates increases (decreases) over time when it is below (above) its equilibrium value, at any fixed and given screening intensity r, and that the screening intensity r increases (decreases) over time when below (above) its equilibrium value, at any fixed and given population share p. We call such dynamics regular. A state (p, r) is stationary if ṗ = ṙ = 0. A stationary state (p, r) is Lyapunov stable if every neighborhood B of (p, r) contains a sub-neighborhood such that the forward solution through every initial state in the sub-neighborhood remains forever in B. A state is unstable if it is not Lyapunov stable. A state (p, r) is (locally) asymptotically stable if it is Lyapunov stable and it has a neighborhood B 0 such that the forward solution converges to (p, r) from all initial states in B 0 . A closed set of states is forward invariant if the solution from any initial state in the set remains forever in the set. Without loss of generality we choose as state space for the dynamic any rectangle C = [0, p1] × [0, r1] that is forward invariant and large enough to contain all states (p, r̂ (p)) where r̂ (p) > 0. Now assume that the hypothesis of Proposition 5 holds. Then the system (20) has at least one and at most three stationary states. The passive equilibrium, (p, r) = (0, 0), is always a stationary state. For generic parameter values, there are either two active equilibria or none. Any stationary point of the system (20) is an equilibrium, though not all equilibria will be stable. In the case of two active equilibria, let (p∗ , r∗ ) and (p∗∗ , r∗∗ ), denote the lowest and highest active equilibria as before. Then the former is dynamically unstable, while the latter is asymptotically stable, as is the passive equilibrium: Proposition 6. Under the dynamics (20), the passive equilibrium is asymptotically stable, the low active equilibrium is unstable, and the high active equilibrium is asymptotically stable. The phase diagram in Figure 4 shows immediately that the passive equilibrium is stable and the lower active equilibrium is a saddle point (the stability of the high active equilibrium requires proof). Proposition 6 allows us to focus on a limited set of equilibria when consid- 19 Figure 4: Stability Properties of Equilibria. ering comparisons across categories. In particular, among active equilibria, we can ignore the unstable equilibrium and focus on the unique stable equilibrium. 4 Categorical Inequality We now turn to a comparison of outcomes across categories, with a focus on mechanisms sustaining inequality among social groups. Consider two categories or groups, A, B ∈ K, with screener bonuses and penalties (βA , γA ) and (βB , γB ), respectively, and with distributions µA and µB of the individual characteristic θ, candidates’ disutility of effort or cost of investment in human capital. Suppose that the response curves for the two groups are given by (p̂A , r̂A ) and (p̂B , r̂B ), and the screening intervals are denoted (pA min , pA max ) and (pB min , pB max ) respectively. Since there are no capacity constraints and the screener’s unit cost of information is constant (though possibly different across groups), the groups can be treated in isolation, and our earlier analysis applies independently for each group. Accordingly, we may use the term stable active equilibrium to mean that each of the groups is at its own stable active equilibrium, and let p∗A and p∗B denote the corresponding equilibrium qualification rates. 20 Figure 5: Categorical inequality when one group (shown using dashed lines) is costlier to screen. 4.1 Differential Screening Costs One branch of the statistical discrimination literature, descended from Phelps (1972), considers ex ante identical categories or groups, but allows for the possibility that the attributes of one group may be harder to ascertain that those of another, possibly because the screener herself belongs to the latter group. In the context of our model, the former group would be more costly to screen. As noted above, an increase in the cost of information acquisition is equivalent to a reduction in the parameters β and γ by the same proportion. Hence, if group A ∈ K is costlier to screen than group B ∈ K, this is represented in the model by setting βA = αβB and γA = αγB for some scaling factor α ∈ (0, 1). For the disadvantaged group A, this implies a lower qualification rate in equilibrium, more pessimistic beliefs about those who pass the test, and more optimistic beliefs about those who fail: Proposition 7. Suppose that group A is costlier to screen than group B, while the groups are ex ante identical in all other respects. Then p∗A < p∗B in the stable active equilibrium, and pA max < pB max and pA min > pB min . This result is illustrated in Figure 5, where the dashed lines correspond to the group that is harder to screen. This group faces a lower level of screening intensity at any given 21 qualification rate. Its members will be considered less productive conditional on passing the test, and more productive conditional on failing. This implies a smaller wage premium from passing, and hence a diminished incentive to invest in becoming qualified. Even if both groups have the same distributions of investment costs, the group that is costlier to screen will therefore invest at lower rates at any given value of screening intensity. This further depresses the equilibrium qualification rate. 4.2 Prejudice Prejudice, in the sense of Becker (1957), refers to a willingness to give up material rewards in order to discriminate against a targeted group. In the present model, screener prejudice of this form against a group A can be represented by reducing the screener’s benefit βA of accepting qualified candidates and/or a higher penalty γA for accepting unqualified candidates. A lower benefit or higher penalty both have the same qualitative effect on the active screening interval, shifting it to the right. However, under a certain condition, they have opposite effects on the incentives of the targeted group to invest in becoming qualified. This condition may be stated as follows. Suppose that the screener’s function G is thrice differentiable, and call G regular if 1 G000 (p) 1 − < 00 < p G (p) 1−p ∀p ∈ [0, 1]. These inequalities are satisfied in the special case of Shannon mutual information, and hence in standard rational inattention models. First consider prejudice in the form of a reduced benefit of accepting qualified candidate. This causes the active screening interval to be shifted to the right, and in the case of regular G, depresses the qualification rate at each level of screening intensity: Proposition 8. Suppose that groups A and B are ex ante identical, except that βA < βB . In the stable active equilibrium, pA min > pB min and pA max > pB max . If G is regular, then the response functions satisfy p̂A (r) < p̂B (r) at every value of r. This result is illustrated in Figure 6. The smaller benefit makes the screener more selective, so that candidates who would otherwise have been screened are rejected outright, while those who would have been unconditionally accepted are screened. This has the effect of raising the posterior belief that those who pass the test are qualified. Despite this, candidates’ premium from passing the test can fall, because the posterior belief that those who fail the test are qualified also rises. In this case, the incentive to invest is reduced, and the net effect on the qualification rate is ambiguous. 22 Figure 6: Effects of prejudice: dashed lines correspond to a group with a lower value to the screener conditional on high productivity. Next consider the case of a larger penalty for accepting an unqualified candidate. Again prejudice shifts the active screening interval to the right but now the incentive effect is reversed: Proposition 9. Suppose that groups A and B are identical, except that γA > γB . Then pA min > pB min and pA max > pB max . If G is regular, then the response functions satisfy p̂A (r) > p̂B (r) at every value of r, and the qualification rates at the stable active equilibrium satisfy p∗A > p∗B . This result is illustrated in Figure 7. As in the case of a lower benefit, the screener becomes more selective and the screening interval is shifted to the right. But this time (in regular cases) the posteriors are adjusted in such a manner as to cause the expected wage premium to rise, and this increases candidates’ incentives to make effort or invest. The result is a higher qualification rate for the disadvantaged group in the stable active equilibrium. Regardless of whether prejudice operates through a smaller benefit or greater penalty, a prejudiced screener is more selective when facing the group subject to discriminatory prejudice, rejecting candidates without screening for a greater range of qualification rates, and screening candidates at qualification rates that would result in acceptance without screening for a group not subject to prejudice. As Becker (1993) observed, prejudice results in greater 23 Figure 7: Effects of prejudice: dashed lines correspond to a group facing a higher penalty conditional on low productivity. expected performance, conditional on acceptance, for the group subject to discriminatory treatment. In fact, this prediction has commonly been used as an outcome-based test for discriminatory treatment. Prejudice can also take other forms. For instance, there may be a bias in the screener’s belief about the distribution µA of individual effort costs in group A, and/or in the screener’s belief pA concerning the qualification rate in the group. Such biases, if anticipated by candidates, may remain unfalsified if negative. A negative bias against group A may induce candidates to make no effort and hence the equilibrium qualification rate may be zero, even though there may exist a stable active equilibrium in the absence of bias. In other words, negative biases may remain undetected indefinitely. By contrast, a positive bias towards a group that induces the screener to actively screen candidates who would have been rejected, or to blindly accept candidates who would have been screened, will eventually generate data on qualification rates that would serve to undermine the bias. 24 Figure 8: Categorical inequality when opportunities differ; the dashed line corresponds to the category with the less favorable distribution of investment costs. 4.3 Social capital An important mechanism through which categorical inequality is sustained over time is differential access to quality schooling and social networks rich in human capital, perpetuated through residential and social segregation (Loury, 1977; Bowles et al., 2014). In the present model this may be represented by differences in the distributions of effort or investment costs, µκ in different groups κ ∈ K. We say that a group A ∈ K is disadvantaged in comparison with group B ∈ K if the probability distribution µA first-order stochastically dominates the distribution µB . This implies a lower level of effort or investment in group A, at any given screening intensity, and hence a lower qualification rate at the group’s stable active equilibrium. However, since the screener’s response function is independent of the group’s distribution µ, equilibrium posterior beliefs, conditional on the screening outcome, do not differ across groups: Proposition 10. Suppose that group A is disadvantaged in comparison with group B. At the stable active equilibrium, (pA min , pA max ) = (pB min , pB max ), but p∗A < p∗B . This result is illustrated in Figure 8, where the dashed curve corresponds to the disadvantaged group. In the stable active equilibrium, the disadvantaged group will have a smaller 25 qualification rate, reflecting the diminished opportunities its members face. However, the screening intensity will adjust in such a manner as to leave unchanged the posterior beliefs about accepted and rejected candidates. That is, even though a randomly selected individual in the disadvantaged population will have a lower likelihood of being qualified, this would not be true of candidates in the post-screening market. If the differences in opportunity are sufficiently large, there may not exist any equilibrium in which the disadvantaged group is screened and selected. In this case any initial disparity in opportunity will be further entrenched and perpetuated. And even if equalization of opportunity results in the emergence of a stable active equilibrium, transition to such a state may be blocked because the passive equilibrium is also dynamically stable. 5 Discussion We have proposed a canonical model describing interactions between heterogeneous categories of candidates and a rationally inattentive screener seeking to accept those who are likely to be qualified, while rejecting those unqualified. Candidates choose how much effort to invest, and the screener chooses how much costly information to acquire. The payoff to candidates depends on a post-screening market. This basic structure can be found in a variety of environments. While the effort of candidates increases monotonically with screening intensity, the screener is only active when facing a candidate category with a qualification rate believed to lie within a certain interval. Outside this interval, the screener is passive and accepts or rejects everybody depending only on whether the perceived qualification rate is high or low. The location and size of the screening interval is completely determined by the screener’s payoff function. For candidates that are actively screened, the posterior qualification rates of accepted and rejected candidates correspond exactly to the endpoints of the screening interval and do not depend on the qualification rate among the candidates. This is intuitive: the screener would maintain her decision if she were to screen the same candidates again. In a generic setting, there is one passive and two active equilibria, of which the passive and the high active equilibria are stable. This relatively simple model has allowed us to reconsider and extend the literature on categorical inequality within a unified framework that allows for endogenous information acquisition. When disparities in social capital exist, a disadvantaged group will invest at lower rates, but the intensity of screening adjusts in such a manner as to result in posterior beliefs about productivity that are uniform across categories. That is, the screening process is calibrated to eliminate stereotypes among those accepted. However, when disparities across groups exist in the costs of screening, posterior beliefs about productivities are no longer 26 uniform across groups: accepted candidates drawn from a group facing higher screening costs will be thought to be of lower productivity, on average, than accepted candidates from categories that are easier to screen. This blunts the incentive to invest in the first place, resulting in lower equilibrium qualification rates in the disadvantaged group. The model also allows for an examination of the implications of prejudice in the sense of Becker (1957). In this case posterior beliefs conditional on selection are more favorable for the group subject to bias, consistent with outcome tests for discrimination in the empirical literature. That is, the expected productivity conditional on selection by the screener is greater for members of the category facing prejudice. Several useful extensions are possible, we mention two. Individuals who are identical but belong to different categories will face different degrees of screening and different outcomes, so there is an incentive to switch categories. This is possible in some situations, though costly, and the effects of this on categorical inequality could be explored within our framework. Additionally, one could explore the effects of affirmative action on the intensity of screening, equilibrium investment rates, and posterior beliefs about candidates drawn from various categories. Such affirmative action policies may be sighted or color-blind, and the consequences of one or the other policy on the level of screening intensity and the resulting outcomes would be worth examining within the parsimonious framework developed here. 27 Appendix Lemma 1. The screener’s objective (8) has a unique maximum. When the maximum is interior, then the screener is actively acquiring information, the conditional posteriors are independent of p and satisfy q̂0 < p < q̂1 . Otherwise the posterior is equal to the prior. Proof. We solve the screener’s problem in two steps. First, we fix y and carry out the maximization with respect to q0, q1. This problem has a concave objective and convex constraints. The solution is a function of y, which turns out to be strictly concave and hence has a unique maximum. Fix y and solve for q0 , q1 . Let 0 < y < 1. The Lagrangian is Λ (q0 , q1 , λ) := βyq1 − γy (1 − q1 ) − I (q0 , q1 ) + λ (yq1 + (1 − y) q0 − p) . The objective is seen to be concave and corner solutions are ruled out by the properties of G. The first-order conditions for the maximization of Π are  ∂Λ  q̂0 , q̂1 , λ̂ = (β + γ) y − yG0 (q̂1 ) + λ̂y ∂q1  ∂Λ  0 = q̂0 , q̂1 , λ̂ = − (1 − y) G0 (q̂0 ) + λ̂ (1 − y) ∂q0  ∂Λ  0 = q̂0 , q̂1 , λ̂ = y q̂1 + (1 − y) q̂0 − p. ∂λ 0 = Solving these leads to G0 (q̂1 ) − G0 (q̂0 ) = β + γ λ̂ = G0 (q̂0 ) p 1−y q̂1 = − q̂0 . y y (21) (22) (23) Now, G0 is increasing and β + γ > 0 and then (21) implies that q̂0 < q̂1 . From (23) and q̂0 < q̂1 we obtain that q̂0 < p < q̂1 . We note also for later use that y= p − q̂0 . q̂1 − q̂0 We now view q̂0, q̂1 as functions of y and insert these into the objective to obtain a 28 (24) concentrated objective. Π (y) = βy q̂1 − γy (1 − q̂1 ) − yG (q̂1 ) − (1 − y) G (q̂0 ) + G (p) . The constraint (23) is automatically satisfied at each value of y. We denote derivatives with respect to y as q̂00 and so on. In order to compute the first- and second-order derivatives of Π (y), we derive from (21) and (23) that G00 (q̂0 ) q̂00 = G00 (q̂1 ) q̂10 , p − q̂0 1 − y 0 q̂ . q̂10 = − 2 − y y 0 Combining these equations and solving for q̂10 shows that q̂10 = − p − q̂0 y2 1 + 1 0 1−y G 0 (q̂1 ) y G00 (q̂0 ) < 0. Then, using (21) and 0 = q̂1 − q̂0 + y q̂10 + (1 − y) q̂00 , Π0 (y) = β q̂1 − γ (1 − q̂1 ) − G (q̂1 ) + G (q̂0 ) + (β + γ) y q̂10 − yG0 (q̂1 ) q̂10 − (1 − y) G0 (q̂0 ) q̂00 = β q̂1 − γ (1 − q̂1 ) − G (q̂1 ) + G (q̂0 ) − G0 (q̂0 ) (y q̂10 + (1 − y) q̂00 ) = β q̂1 − γ (1 − q̂1 ) − G (q̂1 ) + G (q̂0 ) + G0 (q̂0 ) (q̂1 − q̂0 ) . (25) Differentiating again leads to Π00 (y) = (β + γ) q̂10 − G0 (q̂1 ) q̂10 + G0 (q̂0 ) q̂00 + G00 (q̂0 ) (q̂1 − q̂0 ) q̂00 + G0 (q̂0 ) (q̂10 − q̂00 ) = (β + γ) q̂10 + (G0 (q̂0 ) − G0 (q̂1 )) q̂10 + G00 (q̂0 ) (q̂1 − q̂0 ) q̂00 = G00 (q̂1 ) (q̂1 − q̂0 ) q̂10 < 0. This means that Π has a unique maximum, either given by the first-order condition or at the boundary. We note that the first-order condition for ŷ does not involve p. The same is true for the first-order conditions (21) and (23) that determine q̂0 , q̂1 . We conclude (as expected, see Caplin et al., 2017) that the conditional posteriors are independent of p when the screener is active. 29 Lemma 2. Since q̂0 (0) = p, we have from (21) −1 q̂1 (0) = (G0 ) (β + γ + G0 (p)) ∂ q̂1 (0) G00 (p) = >0 ∂p G00 (q̂1 (0)) ∂ q̂1 (0) ∂ q̂1 (0) 1 = = 00 > 0. ∂γ ∂β G (q̂1 (0)) (26) (27) (28) Similarly q̂1 (1) = p and −1 q̂0 (1) = (G0 ) (G0 (p) − (β + γ)) G00 (p) ∂ q̂0 (1) = ∂p G00 (q̂0 (1)) ∂ q̂0 (1) ∂ q̂0 (1) 1 = = − 00 < 0. ∂γ ∂β G (q̂0 (1)) (29) (30) (31) Proof of Proposition 1. Lemma 1 establishes that the amount of information acquired will be such that the conditional posteriors are independent of the prior. Define for convenience H (y) = G (y) − yG0 (y) and note that H 0 (y) = −yG00 (y) < 0. Use this and (21) to rewrite (25) as Π0 (y) = −γ + H (q̂0 (y)) − H (q̂1 (y)) . Note that q̂0 (0) = q̂1 (1) = p, since the posterior equals the prior when the screener either accepts or rejects all candidates. Optimality at an interior value of y requires that Π0 (0) > 0 and Π0 (1) < 0. which is equivalent to 0 < Π0 (0) = −γ + H (p) − H (q̂1 (0)) , (32) 0 > Π0 (1) = −γ + H (q̂0 (1)) − H (p) . (33) Differentiating the bounds of the screening interval with respect to p and using Lemma 2 eqs. (27) and (30) yields ∂Π0 (0) = (q̂1 (0) − p) G00 (p) > 0, ∂p ∂Π0 (1) = (p − q̂0 (1)) G00 (p) > 0. ∂p We find that both Π0 (0) and Π0 (1) are monotone functions of p such that the equations Π0 (0) = 0 and Π0 (1) = 0 have unique solutions. Then (32) determines the lower bound 30 of the screening interval while (33) determines the upper bound. The screening interval is nonempty due to the strict concavity of Π (y). Define pmin as value of p that achieves equality in (32) and similarly pmax as the value of p that achieves equality in (33). The screening interval is then [pmin , pmax ]. We have noted that the conditional posteriors are independent of p. It follows that pmin = q̂0 (ŷ) and pmax = q̂1 (ŷ). The assertion that ŷ varies linearly as a function of p follows from (24). The expressions for the conditional accept rates follow from Bayes’ law. p − pmin 1 − pmax , pmax − pmin 1 − p p − pmin pmax = . pmax − pmin p ŷ0 = ŷ1 It is straightforward to verify that these are strictly convex and concave, respectively, on the screening interval. Proof of eq. (12). In the case of the Shannon mutual information: G (y) = y ln y + (1 − y) ln (1 − y) y G0 (y) = ln 1−y H (y) = ln (1 − y) We know from the first-order conditions that G0 (q1 (y)) − G0 (q0 (y)) = β + γ. The screening interval is determined by the equations γ = H (q0 (0)) − H (q1 (0)) , q0 (0) = pmin γ = H (q0 (1)) − H (q1 (1)) , q1 (1) = pmax . Solving for pmin leads to two equations: γ = ln (1 − pmin ) − ln (1 − q1 (0)) q1 (0) pmin − ln . γ + β = ln 1 − q1 (0) 1 − pmin 31 Solve the first equation for q1 (0) = 1 − (1 − pmin ) e−γ and insert into the second: pmin 1 − (1 − pmin ) e−γ −(γ+β) = e ⇔ 1 − pmin (1 − pmin ) e−γ eγ − 1 . pmin = β+γ e −1 Similarly, solving for pmax , yields q0 (1) = 1 − (1 − pmax ) eγ and pmax 1 − (1 − pmax ) eγ γ+β = e ⇔ 1 − pmax (1 − pmax ) eγ 1 − e−γ pmax = . 1 − e−(β+γ) Proof of Proposition 2. It is easily verified that the screening intensity (13) is a strictly concave function of p. Treating (pmin , pmax ) as parameters, we find as required that ∂r̂ (p − pmax )2 1 = − < 0, 2 ∂pmin (pmax − pmin ) p (1 − p) ∂r̂ (p − pmin )2 1 = > 0. 2 ∂pmax (pmax − pmin ) p (1 − p) Lemma 3. Increasing the reward β for accepting a qualified candidate shifts the screening interval to the left. Increasing the penalty γ for accepting an unqualified candidate shifts the screening interval to the right. ∂pmin ∂pmax < 0, <0 ∂β ∂β ∂pmin ∂pmax > 0, > 0. ∂γ ∂γ 32 Proof. By the proof of Proposition 1, the conditional posterior for rejected candidates is determined by the equation 0 = Π0 (0) = −γ + H (pmin ) − H (pmax ) . Differentiating this equation, we find that ∂pmin ∂pmax = −pmax G00 (pmax ) , ∂β ∂β ∂pmin ∂pmax −pmin G00 (pmin ) = 1 − pmax G00 (pmax ) . ∂γ ∂γ −pmin G00 (pmin ) From (21), we obtain similarly that ∂pmax ∂pmin − G00 (pmin ) = 1, ∂β ∂β ∂pmax ∂pmin − G00 (pmin ) = 1. G00 (pmax ) ∂γ ∂γ G00 (pmax ) Solving the first set of equations yields ∂pmin pmax G00 (pmax ) ∂pmax = , ∂β pmin G00 (pmin ) ∂β 1 pmax G00 (pmax ) ∂pmax ∂pmin + = − . ∂γ pmin G00 (pmin ) pmin G00 (pmin ) ∂γ Inserting into the second set and solving yields ∂pmin ∂β ∂pmax ∂β ∂pmin ∂γ ∂pmax ∂γ 1 pmax < 0, (pmin ) pmax − pmin 1 pmin = − 00 < 0, G (pmax ) pmax − pmin 1 1 − pmax = > 0, 00 G (pmin ) pmax − pmin 1 1 − pmin = > 0. 00 G (pmax ) pmax − pmin = − G00 Proof of Proposition 5. By Proposition 2, the screening intensity function, r̂ is concave on the screening interval. We will show that the aggregate candidate response function p̂ is concave. In that case, there can be at most two equilibria. 33 Suppose that r > 0. The condition that p̂ (rmax ) > pmin implies that x̂θ (r) > 0 for all p in the screening interval and for all θ ∈ Θ. We then have p̂ (r) = X µ (θ) Fθ (x̂θ (r)) , θ∈Θ where the first-order condition (16) holds for each type θ, and where the wages are determined by pmin and pmax . It is then sufficient to show that ∂ 2 Fθ (x̂θ (r)) /∂r2 < 0 for all θ ∈ Θ. From the first-order condition we obtain (with simplified notation) that for all θ ∈ Θ: 0 = f 0 · x̂0 · r · (v1 − v0 ) + f · (v1 − v0 ) ⇔ x̂0 = − f ·r f0 and  0 2 f 00 · (x̂0 )2 · r + f 0 · x̂0 0 00 0 0 00 0 = f · x̂ · r + f · x̂ · r + f · x̂ ⇔ x̂ = − . f0 · r 00 Then ∂F (x̂ (r)) = f · x̂0 ∂r and ∂ 2 F (x̂ (r)) 2 = f 0 · (x̂0 ) + f · x̂00 2 ∂r f 00 · (x̂0 )2 · r + f 0 · x̂0 2 = f 0 · (x̂0 ) − f · f0 · r h i (x̂0 )2 x̂0 0 2 00 = (f ) − f · f · 0 − f · . f r But this is negative since ln f is concave iff (f 0 )2 > f 00 f, f 0 < 0 and x̂0 > 0. Proof of Proposition 6. In order to pin down the stability properties of the three stationary states, we linearize the vector field (φ, ψ) around each of them. The Jacobian of the vector field at any point (p, r) ∈ C where both p̂ and r̂ are differentiable is φ0 (0) φ0 (0) p̂0 (r) ψ 0 (0) r̂0 (p) ψ 0 (0) ! . The associated eigenvalues are the algebraic roots λ to the equation (λ − φ0 (0)) (λ − 0 (0)) − φ0 (0) ψ 0 (0) p̂0 (r) r̂0 (p) = 0. 34 The eigenvalues of the vector field’s Jacobian are thus 0 0 φ (0) + ψ (0) λ= ± 2 s φ0 (0) + ψ 0 (0) 2 2 + [p̂0 (r) r̂0 (p) − 1] φ0 (0) ψ 0 (0). By assumption, φ0 (0) , ψ 0 (0) < 0. Hence, φ0 (0) + ψ 0 (0) < 0 and φ0 (0) ψ 0 (0) > 0. We now examine the three stationary states in turn, using standard techniques(see e.g. Theorem 4.2 in Hale, 1969): 1. At the passive equilibrium, p̂0 (r) = 0. Thus, both eigenvalues have negative real parts. Hence, this stationary state is asymptotically stable. 2. At the low active equilibrium, the p̂-curve intersects the r̂ curve from above: [p̂0 (r)]−1 < r̂0 (p). Hence, p̂0 (r) r̂0 (p) > 1. Thus one eigenvalue (the one with the plus sign in the equation) has a positive real part. Hence, this stationary state is unstable (a saddle point). 3. At the high active equilibrium, the p̂-curve intersects the r̂ curve from below: [p̂0 (r)]−1 > r̂0 (p). Hence, p̂0 (r) r̂0 (p) < 1. Thus, again both eigenvalues have negative real parts, so also this stationary state is asymptotically stable. Proof of Proposition 7. Replace (β, γ) by (β̃, γ̃) = (β/α, γ/α). Increasing α corresponds to a higher screening cost. Find from (21) that G00 (p̃max ) ∂ p̃min β+γ ∂ p̃max − G00 (p̃min ) = − 2 , ∂α ∂α α (34) which is equivalent to ∂ p̃max G00 (p̃min ) ∂ p̃min 1 β+γ = 00 − 00 . ∂α G (p̃max ) ∂α G (p̃max ) α2 Write (25) as 0=− γ + H(p̃min ) − H(p̃max ), α 35 (35) such that differentiating and solving leads to γ ∂ p̃max ∂ p̃min 0 0 − H (p̃ ) + H (p̃ ) max min α2 ∂α ∂α  00  G (p̃min ) ∂ p̃min 1 β+γ γ ∂ p̃min 00 00 + p̃max G (p̃max ) − 00 = − p̃min G (p̃min ) α2 ∂α G00 (p̃max ) ∂α G (p̃max ) α2 γ ∂ p̃min β+γ ∂ p̃min = + p̃max G00 (p̃min ) − p̃max 2 ⇔ − p̃min G00 (p̃min ) 2 α ∂α ∂α α 1 γ − p̃max (β + γ) 1 = − 2 . α p̃max − p̃min G00 (p̃min ) 0 = ∂ p̃min ∂α Inserting back into (35) leads to G00 (p̃min ) ∂ p̃min β+γ ∂ p̃max = − 2 00 00 ∂α G (p̃max ) ∂α α G (p̃max ) β+γ 1 γ − p̃max (β + γ) 1 − 2 00 = − 2 00 α G (p̃max ) p̃max − p̃min α G (p̃max ) 1 (β + γ) p̃min − γ = . 2 00 α G (p̃max ) p̃max − p̃min This implies that γ ∂ p̃min > 0⇔ < p̃max ∂α β+γ γ ∂ p̃max > 0 ⇔ p̃min < . ∂α β+γ γ γ min If p̃min > β+γ then also p̃max > β+γ , which implies that ∂ p̃∂α < 0 for all α > 0.We γ γ conclude that p̃min ≤ β+γ . Similar reasoning leads to the conclusion that β+γ ≤ p̃max . Altogether we conclude that increasing the scale of the information cost makes the screening interval narrower. Proof of Proposition 8. We recall Lemma 3. The screener’s chosen screening intensity r̂ changes as 36 ∂r̂ (p) ∂pmin ∂r̂ (p) ∂pmax ∂r̂ (p) = + ∂β ∂pmin ∂β ∂pmax ∂β 2 ∂pmin (p − pmin )2 ∂pmax (p − pmax ) 1 1 + = − 2 2 (pmax − pmin ) p (1 − p) ∂β (pmax − pmin ) p (1 − p) ∂β   (p − pmax )2 1 1 pmax = − − 00 G (pmin ) pmax − pmin (pmax − pmin )2 p (1 − p)   1 pmin (p − pmin )2 1 − 00 + G (pmax ) pmax − pmin (pmax − pmin )2 p (1 − p) ! 1 (p − pmax )2 pmax (p − pmin )2 pmin 1 = − , p (1 − p) (pmax − pmin )3 G00 (pmin ) G00 (pmax ) which is positive iff pmax G00 (pmax ) (p − pmin )2 < . pmin G00 (pmin ) (p − pmax )2 Note that LHS increases from 0 to ∞ as a function of p, while the RHS is constant. There r̂(p) changes from positive to negative. is thus a unique threshold value of p at which ∂ ∂β Consider the wage differential between accepted and rejected candidates as β increases: 1 1 ∂ pmin pmax (pmax − pmin ) = − 00 + 00 ∂β G (pmax ) pmax − pmin G (pmin ) pmax − pmin   pmin 1 pmax − = . pmax − pmin G00 (pmin ) G00 (pmax ) this is positive iff pmin G00 (pmin ) < pmax G00 (pmax ) . It is straightforward to show that if G is regular then pG00 (p) is increasing, which implies the desired inequality. Similarly, ∂r̂ (p) ∂r̂ (p) ∂pmin ∂r̂ (p) ∂pmax = + ∂γ ∂pmin ∂γ ∂pmax ∂γ 2 (p − pmax ) 1 ∂pmin (p − pmin )2 1 ∂pmax = − + 2 2 (pmax − pmin ) p (1 − p) ∂γ (pmax − pmin ) p (1 − p) ∂γ 1 1 = p (1 − p) (pmax − pmin )3 (p − pmin )2 (1 − pmin ) (p − pmax )2 (1 − pmax ) − G00 (pmax ) G00 (pmin ) 37 ! , which is positive iff (p − pmin )2 (1 − pmax ) G00 (pmax ) > , (1 − pmin ) G00 (pmin ) (p − pmax )2 and we note again that there is a unique value of p where the sign changes. The wage differential changes according to ∂ 1 1 − pmin 1 1 − pmax (pmax − pmin ) = − 00 00 ∂γ G (pmax ) pmax − pmin G (pmin ) pmax − pmin   1 1 − pmin 1 − pmax = − , pmax − pmin G00 (pmax ) G00 (pmin ) which is positive iff (1 − pmax ) G00 (pmax ) < (1 − pmin ) G00 (pmin ) . We note that regularity of G implies that (1 − p) G00 (p) is decreasing in p, which in turn implies the desired inequality. 38