Eliminating Persistent Statistical Discrimination: An Analysis of Several Policy Options

Rebecca M. Glawtschew

doi:10.1515/bejeap-2014-0005

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Eliminating Persistent Statistical Discrimination: An Analysis of Several Policy Options

Rebecca M. Glawtschew

Published/Copyright: November 21, 2014

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From the journal The B.E. Journal of Economic Analysis & Policy Volume 15 Issue 2

Abstract

I consider a model of endogenous human capital formation with competitively determined wages in a dynamic setting. In the presence of two distinguishable, but ex ante identical groups of workers, discrimination will be persistent in equilibrium. Using this framework, I then consider the effectiveness of three government policies designed to eliminate this discrimination. I determine the paths that workers will take after a policy is instated as well as how long a policy needs to be in place to guarantee the successful elimination of discrimination. The policies I consider are (1) a hiring subsidy that promotes the hiring of disadvantaged workers to the better job, (2) an investment voucher that defrays the monetary cost of human capital investment, and (3) an equal treatment policy under which firms are required to treat workers equally across groups. I find that all three policies have the potential to eliminate persistent discrimination if certain conditions are met. In a general equilibrium setting, I also address the welfare effects of the three policies for a parametric example.

Keywords: statistical discrimination; general equilibrium; asymmetric information; human capital

JEL Classification:: D58; D62; D82; J71; J78

1 Introduction

This paper explores the consequences of three government policies designed to eliminate existing persistent statistical discrimination. This analysis is done in the framework of a dynamic model of endogenous human capital formation with competitively determined wages and two distinguishable, but ex ante identical groups of workers. By examining these policies in a dynamic framework, I am able to characterize the gradual change in each group’s human capital investment decisions that occurs as a result of each policy and to determine when and if the government can safely remove each policy. I am also able to meaningfully analyze the welfare implications of the three policies, because wage is endogenously determined through competition between firms.

The dynamic model of statistical discrimination on which I rely to assess the implications of various policies has three wage steady states in equilibrium. The middle steady state is unstable and paths to the high and low steady states emanate from that point. These paths can take several forms depending on the parameter values. No matter what form the dynamic paths take, a group becomes “stuck” in a steady state as soon as it is reached. As such, if the groups end up at different steady states discrimination is present and persistent.

For the purpose of this analysis, I assume that the economy is experiencing discrimination, so that one group of workers, which I refer to as disadvantaged, is stuck in the low steady state and the other group of workers, which I refer to as advantaged, is stuck in the high steady state. I examine the impact of three government policies from this starting point. The goal of these policies is to move the disadvantaged group from the low steady state to the high steady state.

The first policy I examine is one in which the government offers a hiring subsidy to firms for correctly assigning skilled workers in the disadvantaged group to the more productive job. This policy is always effective at eliminating discrimination as long as a large enough subsidy is offered.

The second policy of interest is an investment voucher offered to workers in the disadvantaged group. This voucher defrays the monetary cost associated with skill acquisition. As long as the monetary costs are high, this policy can eliminate discrimination.

The last policy is one in which the government enforces equal treatment of workers across groups. This policy may be effective depending on what proportion of the total worker population is disadvantaged. The policy will successfully eliminate discrimination if a small proportion of the population is disadvantaged. If the groups are close to equal in size, then the policy may eliminate discrimination and the final outcome depends on the specific parameterization. If the disadvantaged group is relatively large, then this policy will move the advantaged group from the high to the low steady state, resulting in a population that is worse off on average.

Using a simplified version of the dynamic model, I examine the welfare effects of the three policies. I complete this analysis in a general equilibrium framework. Workers in the advantaged group are taxed in order to fund the policies. I find that all three policies create a net increase in welfare while they are in place. However, there are significant differences across policies with respect to how much the advantaged group is impacted over the course of the policy. The hiring subsidy results in a very large loss in welfare to the advantaged group, but it moves the disadvantaged group to the high steady state the fastest. The investment voucher reduces the welfare of the advantaged group the least of all the policies and for this reason it may be viewed as optimal by some policy makers. The equal treatment policy has a moderate impact of the advantaged group’s welfare and may be viewed as optimal by a policy maker who does not want to redistribute.

The paper is structured as follows. The related literature is discussed in the remainder of Section 1. The details of the general dynamic model are described in Section 2. Section 3 develops the equilibria of the dynamic model and addresses the conditions under which discrimination will be persistent. Section 4 describes the three policies and examines their impact on the general dynamic model. In Section 5 a simple version of the dynamic model is developed; using that model I consider the welfare effects of the three policies. Finally, Section 6 concludes.

1.1 Related literature

The model presented in this paper is closely related to Coate and Loury (1993) and Moro and Norman (2003). Coate and Loury (1993) present a model of statistical discrimination where two ex ante groups may end up in different, Pareto ranked, equilibria. This model formalizes many of the ideas that were originally presented loosely in Arrow (1973), but it makes the additional assumption that wages are set exogenously. They find that inequality will occur if the two groups of workers achieve different equilibria. In the context of their model, Coate and Loury discuss whether affirmative action or similar policies can be temporary policies. However, owing to the static nature of the model they are unable to make a formal determination. The model presented here allows for a formal discussion of whether or not these types of policies can be removed after sufficient time has passed.

Moro and Norman (2003) relax two of the assumptions in Coate and Loury’s model. In particular, they remove the linearity of the production technology and the exogeneity of wages. Allowing for curvature in the production function eliminates the separability of groups, thus allowing for true interaction effects between groups. Introducing endogenous wages allows for more meaningful welfare analysis. They find that under these relaxed assumptions the dominant group will actually gain from discrimination and, as a result, they will be motivated to ensure its continuation. The model presented here adopts the endogeneity of wages, which allows me to perform meaningful welfare analysis of the policies.

There is relatively little literature that addresses discrimination in a dynamic environment. Fryer (2007), Blume (2006), and Kim and Loury (2014) all develop dynamic models that are adaptations of the original Coate and Loury (1993) paper. These models all differ from the one presented here in two key areas: they do not incorporate endogenous wages nor do they consider the welfare effects of potential policies designed to eliminate or mitigate discrimination.

The paper that is most related to the analysis presented here is Kim and Loury (2014). They also add long-lived workers to a static model in order to incorporate dynamics. However, unlike the model presented here, they do not include endogenous wages. Moreover, the goals of the two papers differ significantly. They develop a dynamic model of statistical discrimination in order to analyze how group reputation determines the final outcome of a group. They also complete a brief analysis of two policies that may eliminate discrimination. One of these policies, a training subsidy, is similar to the investment voucher policy that I examine. My goal is to develop a dynamic model in order to undertake a thorough analysis of the effectiveness of three policies. Specifically, I determine in the potential paths that each group will follow once a policy is put in place, how long a policy needs to be in place to be effective and the potential welfare impacts of each policy while it is in place. All of these results go well beyond what is developed in Kim and Loury (2014).

2 The model

Consider a market where short-lived firms engage in Bertrand-style competition for workers in each period. Time is continuous and all changes occur at the beginning of each period of length Δ.

There are two identifiable groups of workers, B(lack) and W(hite), and the total population of each group is constant at λJ for J=B,W. Workers are long-lived and subject to a Poisson death process with parameter β. Consequently, at the beginning of each period Δβ of the existing workers are replaced by newborn individuals. Workers also discount future pay-offs at a strictly positive rate r. Prior to entering the labor market, each worker has the option to invest in human capital. Workers who make this investment become qualified and those that do not remain unqualified. These premarket investment decisions are final; a worker cannot choose to invest in human capital later in life.

A worker who invests in human capital incurs a total cost c. This cost of investment consists of two important parts: a monetary cost, cm, which is strictly positive and fixed across all workers and a nonmonetary cost, ce, which varies across workers. The nonmonetary cost represents both the effort required to become qualified and any value that a worker places on being qualified (outside of the potential increase in expected wage); it may be positive or negative.

Total cost of investment will vary across workers because of the differences in ce. I assume that the total cost of investment, c, is distributed over [c_,cˉ]⊆ℜ according to a continuous and strictly increasing distribution G(c). Workers are risk-neutral and care only about the wage that they receive and the cost they incur. Pay-offs are additively separable in income and cost of investment.

Output in each period is generated by the completion of two types of tasks, a simple task and a complex task. All workers can perform the simple task, but only those workers who are qualified can successfully perform the complex task. For a given period starting at t, the effective input of labor into the simple task, St, is the total number of workers employed in that task. The effective input of labor into the complex task, Ct, is the number of qualified workers employed in that task. Firms’ per period output is given by the linear production function YΔ(St,Ct)=Δ[αsSt+αcCt], where αc>αs>0. This condition implies that firms will always want to assign a skilled worker to the complex task.

Firms cannot observe whether a worker is qualified, nor are they able to distinguish between new and old workers. The inability to distinguish between old and new workers is a limitation of the model that is required to generate manageable dynamics. The only information a firm has about a worker in a given period, aside from his group identity, is his current signal θt∈[0,1], which is distributed according to fu if a worker is unqualified or fq if a worker is qualified. Both densities are bounded away from zero and, without further loss of generality, fq(θ)/fu(θ) is strictly increasing in θ. I denote the associated cumulative distributions by Fq and Fu. I assume a law of large numbers holds so that these are also the realized frequency distributions of signals for qualified and unqualified workers, respectively.

A firm’s prior belief about what proportion of the total J population is qualified at time t is represented by ΠtJ. The posterior probability that a worker from group J with signal θt is qualified given beliefs ΠtJ is

[1]p(θt,ΠtJ)≡ΠtJfq(θ)ΠtJfq(θ)+(1−ΠtJ)fu(θ).

The strict monotone likelihood ratio property implies that p(θt,ΠtJ) is strictly increasing in θt so that a high signal reflects positively on a worker and a low signal negatively.

The timing of each period of the dynamic game is as follows: In the first stage, Δβ new workers enter the workforce and decide whether or not to invest in becoming qualified. At the same time, firms post wage and task assignment rules. Firms may condition wage and task assignments on θt. Formally, a strategy for each firm is to select some wage schedule ωtJ:[0,1]→ℜ+ and a task assignment rule τtJ:[0,1]→[0,1] for each group J. In the second stage, nature assigns a signal θt to each worker according to the appropriate density. Workers observe the posted wage and task assignment rules and decide where to work. Then tasks are performed, wages are paid, and output is realized. In the last stage, Δβ of the workers exit the workforce.

Workers only care about potential wages when comparing job offers and their investment costs are sunk. Hence they will choose to work for the firm whose wage and task assignment rule provides them with the highest wage.

3 Equilibria

There are two main components to an equilibrium of this game. First, firms and workers must behave optimally in each period. This means that firms must employ both an optimal task assignment rule and an equilibrium wage rule in each period. While members of each new cohort of workers invest only if their expected lifetime benefit at the time of birth exceeds their cost, all workers choose firms in a sequentially rational way after any history of play. Second, firms have rational beliefs about proportion of qualified workers in each period. Formally, an equilibrium of this model is described by a sequence of rational firm beliefs regarding the proportion of qualified workers in each group, ΠtJt=0∞ for J=B,W.

Let (ΠtB,ΠtW) denote the group-specific investment rates in the period starting at time t. Given this arbitrary investment behavior by workers, I can address the optimal behavior of firms in each period. The probability that a worker is qualified, as defined in eq. [1], is strictly increasing in signal. This implies that threshold rules are optimal when assigning workers to tasks. Under this type of rule, workers with a signal above the threshold are assigned to the complex task and workers with lower signals are assigned to the simple task. Given a threshold θˉtJ for group J at time t the effective input of labor into the complex task is CtJ=ΠtJ(1−Fq(θˉtJ)). This is simply the number of qualified workers who are assigned to the complex task. Any unqualified workers with signals above the threshold will fail to complete the complex task. Consequently, these workers do not appear in the effective input of labor into the complex task. The effective input of labor into the simple task is StJ=ΠtJFq(θˉtJ)+(1−ΠtJ)Fu(θˉtJ). More specifically, this is all the workers, both qualified and unqualified, with a signal below the threshold, θˉtJ.

The optimal threshold is the one that maximizes a firm’s expected output conditional on investment decisions. Formally, the task assignment problem is

[2]∑J=B,WλJmaxθ tJ∈[0,1](Δ αc[ΠtJ(1−Fq(θ tJ))]+Δαs[ΠtJ Fq(θ tJ)+(1−ΠtJ) Fu(θ tJ)]).

This task assignment problem will always have a unique solution. See Lemma 1 of Glawtschew (2013) for a formal proof. Moreover, the linear production function allows for the task assignment problem to be solved separately for each group. Consequently, there is a unique optimal threshold for each group. It is

[3]θ^(ΠtJ)={1 ifαcp(1,ΠtJ)≤αs, 0 ifαcp(0,ΠtJ)≥αs,the unique solution toαc p(θ,ΠtJ)=αs ifαc p(0,ΠtJ)αsαc p(1,ΠtJ).

Under this task assignment rule, all workers are assigned to the simple task (a threshold of 1) if prior beliefs are such that a worker with a signal of 1 has higher expected productivity in the simple task. All workers are assigned to the complex task (a threshold of 0) if prior beliefs are such that a worker with a signal of 0 has higher expected productivity in the complex task. Otherwise, the optimal threshold is the θ at which a worker’s expected productivity in the complex task is equivalent to that of the simple task.

Following Moro and Norman (2003), I call a strategy profile a continuation equilibrium if firms are implementing their optimal wage and task assignment rules given some arbitrary investment behavior by workers. This first result suggests that wages are given by expected marginal products and job assignments are constrained efficient in any continuation equilibrium.

Proposition 1. Suppose for a given period beginning at t that fractionsΠt=(ΠtB,ΠtW)of the workers invest and that(θˆt(ΠtB),θˆt(ΠtW))is as in eq. [3]. Then there exists a continuation equilibrium where both firms post wage schedules

[4]ωΔ(θt,ΠtJ)={Δαs if θt<θ^(ΠtJ)Δαcp(θt,ΠtJ) if θt≥θ^(ΠtJ),

and assign a worker with characteristics(J,θt)to the complex task if and only ifθt≥θˆt(Πt). Moreover, in any continuation equilibrium where fractionsΠt=(ΠtB,ΠtW)of the workers invest the posted wage schedule for group J, ωΔ(θt,ΠtJ)agrees with eq. [4] for almost allθt∈[0,1]for each firm.

Proof. See Appendix A.□

In a full equilibrium, investments must be best responses to wages. The optimal investment decision for a worker is to invest in human capital if and only if the lifetime gain in expected earnings is higher than the cost c. It is helpful to define what I refer to as the unit period gain from investment. This is a worker’s one-period expected gain from investment if periods have a length of one. For wages consistent with the continuation equilibrium, where a fraction ΠtJ of the J population is qualified, the unit period gain from investment is written as

[5]γ(ΠtJ)=1Δ(∫01ωΔ(θ,ΠJ)fq(θ)dθ−∫01ωΔ(θ,ΠJ)fu(θ)dθ) =αs[Fq(θ^(ΠtJ))−Fu(θ^(ΠtJ))]+αc∫θ^(ΠtJ)1p(θ,ΠtJ)[ fq(θ)−fu(θ)]dθ.

I can also define the per period gain from investment as Δγ(ΠtJ).

Using the per period gain from investment, I can define the lifetime gain from investment from the perspective of a worker at time t in group J. It is simply the discounted sum of all future per period gains from investment. It can be written as

[6]ΓtJ=∑g=0∞(1−Δβ−Δr)gΔγ(Πt+gΔJ)

[7]=(1−Δβ−Δr)Γt+ΔJ+Δγ(ΠtJ).

Rearranging eq. [6] yields

[8]Γt+ΔJ−ΓtJΔ=(β+r)Γt+ΔJ−γ(ΠtJ).

If I let Δ→0, then I can express how the lifetime gain from investment evolves over time with the following differential equation:

[9]Γ˙tJ=(β+r)ΓtJ−γ(ΠtJ).

I can also address how the investment rate of each group’s population will develop over time. In each period, workers in the new cohort invest if their lifetime gain from investment exceeds their cost. The proportion of new J population births that invest in the period starting at time t is G(ΓtJ), where G(⋅) is the distribution of costs.

Given the Poisson death process and the assumption that the total population is constant, the investment rate of the whole J population at time t is

[10]ΠtJ=(1−Δβ)Πt−ΔJ+ΔβG(ΓtJ).

Rearranging eq. [10] yields

[11]ΠtJ−Πt−ΔJΔ=β(G(ΓtJ)−Πt−ΔJ).

If I let Δ→0, then I can express how the overall investment rate of group J evolves over time with the following differential equation:

[12]Π˙tJ=β(G(ΓtJ)−ΠtJ).

I conclude that any equilibria of this model is characterized by a dynamic system consisting of eqs [9] and [12].

Proposition 2. An equilibrium of this dynamic model is fully characterized by the following two-variable differential equations:

Γ˙tJ=(β+r)ΓtJ−γ(ΠtJ),

[13]Π˙tJ=β(G(ΓtJ)−ΠtJ).

This proposition indicates that the difference between the investment rate of the newborn cohort and the overall investment rate of the group J determines the speed of the firms’ beliefs about the fraction of workers in group J that are qualified. The change in lifetime gain from investment is determined by the difference between the discounted lifetime gain from investment and the unit period gain from investment.

It is important to note that Kim and Loury (2014) develop a very similar dynamic system in their Theorem 1. However, unlike their result, this system incorporates competitively determined wages via the lifetime incentive to invest function, ΓtJ. The steady states of the dynamic system can be described as all beliefs satisfying

[14]ΠJ=Gγ(ΠJ)β+r.

The existence of a solution to eq. [14] is immediate since γ(ΠtJ) is composed of continuous functions. There may be a unique solution, in which case groups are treated identically when they are both in the steady state. For the purposes of this analysis, multiple steady states are desirable since persistent discrimination is only possible if this is the case.

Figure 1 shows an example of the dynamic system with three steady states. From now on I denote these three steady states as Σl(Γl,Πl), Σm(Γm,Πm), and Σh(Γh,Πh), referring to the low, middle, and high steady states, respectively. The hump-shaped curve is the Γ˙tJ=0 locus. The arrows indicate that to the right of the locus, when ΓtJ>γ(ΠtJ)/(β+r), ΓJ is increasing. To the left of the locus, ΓtJ is decreasing. The s-shaped curve is the Π˙tJ=0 locus. The arrows indicate that above this locus, when G(ΓtJ)<ΠtJ, ΠtJ is increasing and below the locus it is decreasing. The shape is from the assumption that G(⋅) is continuous and strictly increasing.

Figure 1:

Phase diagram of the dynamic system

The steady states in this model are a result of self-fulfilling human capital investment behavior. In particular, a group in the low steady state will have relatively low average investment rates and consequently they will be offered a less attractive wage schedule. These low wages result in low incentives to invest and as a result new generations will also have low average investment rates. In essence, a group in the low steady state is stuck in a cycle of self-perpetuating underinvestment in human capital. A group in the high steady state will have relatively high average investment rates, and as a result will be offered a more favorable wage schedule. These high wages result in large incentives to invest and as a result new generations will also have high average investment rates. So a group in the high steady state is stuck in a self-perpetuating cycle of high investment in human capital. A group in the middle steady state has a similar story, but with intermediate investment behavior and wage schedules.

The existence of multiple solutions is not guaranteed and depends on the shape of Γ and G(⋅). The possibility of multiple solutions can be proven by construction: if I fix the parameters fu, fq, αc and αs, then I can find an appropriate G(⋅) function such that eq. [14] has multiple solutions. Graphically, if I fix the Γ˙tJ=0 locus in Figure 1, I can ensure multiple equilibria simply by selecting a G(⋅) function that yields multiple intersections of the two loci. I can conclude that for any fu and fq that satisfy the monotone likelihood ratio and any technology parameters, I can find a nontrivial set of G(⋅) functions that ensure multiple steady states. These multiple steady states are Pareto rankable. In particular, it is simple to show that the steady state with the highest Π, Σh(Γh,Πh), is Pareto dominant. See Proposition 4 of Glawtschew (2013) for the formal proof.

It is necessary to establish which of these three steady states are stable and to determine the potential shapes of the paths leading to steady states. Understanding the behavior of the system outside of the steady states is important both for determining if discrimination is persistent and for analyzing the effectiveness of the policies in Section 4. The phase diagram indicates that both the high and the low steady states are saddle points, while the middle steady state is an unstable source. See Lemma 2 of Glawtschew (2013) for a formal proof. ^[1]

The existence of two saddle points means that members of a group may rationally conjecture that firms will eventually have positive beliefs about their group’s investment rate, Πh, or have negative beliefs about their group’s investment rate, Πl. The only reasonable steady states for a group to approach are Σh and Σl. It is possible to characterize the shape that the paths to these saddle stable steady states will take.

The analysis of path shape relies heavily on the conclusions drawn in Krugman (1991). In that paper Krugman develops a simple model of trade in which there are three steady states with different levels of specialization in two goods. The two extreme steady states are saddle stable and the third, intermediate steady state is unstable. He uses this model to show that depending on the structure of the economy, the steady state an economy will converge to may depend entirely on history, or it may be determined by expectations, which can be thought of as self-fulfilling prophecy. In particular, he shows that it is the shape of the paths to the steady states that determines whether history or expectations will direct an economy’s final resting place. If the paths are S-shaped, then history alone is the determining factor. If the paths form interlocking spirals, then there is a range of initial values such that an economy can converge to either steady state depending on the economy’s expectations. He refers to this range of initial conditions as the “overlap,” as it is where the paths to the two saddle stable steady states overlap one another.

Clearly the structure of the dynamic model of statistical discrimination presented here mirrors that of the model addressed in Krugman (1991). Specifically, I have a model with three steady states, in which the high and low steady states are saddle stable and the middle steady state is unstable. Consequently, I can use the same analysis to determine the parameter values that generate S-shaped paths, which I refer to as deterministic paths, and that generate paths with interlocking spirals, which I refer to as spiralling paths. When paths are deterministic, historical firm beliefs completely determine the final steady state of a group. On the other hand, when paths are spiralling, there is a region of initial firm beliefs that can lead to either of the saddle stable steady states depending on the expectations of the group. See Lemma 3 of Glawtschew (2013) for a formal discussion of the conditions that lead to each type of path.

Figure 2 depicts a case where there are three steady states and paths are spiralling. As determined in Krugman (1991), spiralling paths to the steady states create a range of initial firm beliefs that can lead to either the high or the low steady state, which is indicated by the purple region. I call this range of initial beliefs the uncertain region rather than the “overlap,” as in Krugman (1991), because it generates uncertainty about the final outcome of a group. This region is formed by the bottom bound of the path to the high steady state, ∏⌣, and by the upper bound of the path to the low steady state, Πˆ. If initial firm beliefs about the investment rate of a group are in this range, then that group can converge to either Σh(Πh,Γh) or Σl(Πl,Γl).

Figure 2:

Spiralling paths to the steady states

When initial firm beliefs are in the uncertain region, the steady state that a group converges to is determined by the expectations of the workers in that group. Again, expectations can be thought of as a self-fulfilling prophecy. So, if a group believes they will have high investment rates in the future and all the wage benefits that result from that, then they make their current investment decisions according to that belief. As a result, the average investment rate increases each period until the group reaches the high steady state. In other words, if a group develops and shares an optimistic view of the future, then they will converge to the high steady state. Conversely, if they develop a pessimistic view, they will converge to the low steady state. Consequently, I refer to the path that leads to the high steady state as the optimistic path and the path that leads to the low steady state as the pessimistic path. See Appendix C.1 for a discussion of how these paths are calculated.

If initial firm beliefs for a group fall outside the uncertain region, then the steady state the group converges to will depend entirely on history. In the case of spiralling paths, when initial beliefs about investment rate are such that Π0J∈(Πˆ,1], then group J will converge to the high qualification steady state with certainty. This is because the optimistic path is the only reasonable path to take. When initial firms’ beliefs are such that Π0J∈[0,Π⌣), then group J will converge to the low qualification steady state with certainty. Additionally, the high and low steady states will always lie outside of the uncertain region, which means that there is no way for a group to move from one steady state to another.

Figure 3:

Deterministic paths to the steady states

Figure 3 depicts a case where there are three steady states and paths are deterministic. In this situation there is no uncertain region, as a result initial firm beliefs are the only factor that determine the final steady state of a group. If initial beliefs about investment rate are such that Π0J∈(Πm,1], then group J will converge to the high qualification steady state with certainty. If Π0J∈[0,Πm), then the group will converge to the low qualification steady state with certainty. Again, when paths are deterministic, groups are unable to move from one steady state to another.

From now on, regardless of the shape of the paths, I refer to the range of initial firm beliefs that lead only to the high steady state as the certain region forΣh, and I refer to the range of initial firm beliefs that lead only to the low steady state as the certain region forΣl. These certain regions make it possible for a group to get stuck permanently at either the high or the low steady state given appropriate initial firm beliefs. This suggests that once present, discrimination will persist indefinitely without a structural change to the model. ^[2] If a policy moves firm beliefs into the certain region for Σh, then it can be removed and the group will converge to the high steady state.

4 Potential policies

The analysis of Section 3 suggests that, if present, statistical discrimination is persistent. One way to eliminate this discrimination is through appropriate government intervention. In this section, I consider three government policies that have the potential to eliminate existing statistical discrimination.

As a starting point for the analysis of the policies I assume that one group is in the high steady state and the other is in the low steady state, so that persistent discrimination is present. From now on I refer to the group in the low steady state as the disadvantaged group and the group in the high steady state as the advantaged group. I then determine under what circumstances the three policies are able to move the disadvantaged group from the low to the high steady state. Discrimination in this model will be eliminated if both groups are in either the high or the low steady state. Since the high steady state Pareto dominates the low steady state, I will only consider a policy successful if it is able to move both groups to the high steady state.

Specifically, in the analysis that follows, I determine whether each policy allows the disadvantaged group to get on the path to the high steady state. I also address if the policy needs to be in effect permanently or if there is a point at which it can be removed and the discrimination will not return. I assume that the government commits to employing each policy at least until it can be safely removed and still successfully eliminate the discrimination.

The policies that I investigate are (1) a subsidy paid by the government to the firms for every worker in the disadvantaged group that is correctly assigned to the complex task, (2) a government subsidization of investment costs for those workers in the disadvantaged group, and (3) a government policy that forces firms to assign workers in different groups with the same signal to the same task and pay them the same wage.

4.1 Hiring subsidy program

The first policy is a hiring subsidy that promotes the assignment of disadvantaged workers to the complex task. This subsidization is applied in the following way: For each worker in the disadvantaged group who is correctly assigned to the complex task, the government compensates firms with a subsidy of Δξ.

The government cannot actually observe whether workers are correctly assigned, instead this policy is implemented through a lump sum payment of ΔξΠtd[1−Fq(θˆ(Πtd))] to the firms. This is simply the value of the subsidy times the expected number of workers from the disadvantaged group that will be correctly assigned to the complex task. This type of subsidy requires that the government only be able to observe what the firms observe. Under the subsidy, the firms’ profit from the disadvantaged group, given an arbitrary threshold θtd, is

[15]λd((Δαc+Δξ)[Πtd(1−Fq(θtd))]+Δαs[ΠtdFq(θtd)+(1−Πtd)Fu(θtd)]).

Since there are two firms engaged in Bertrand competition for workers and the firms have linear production functions, all of the benefits of the subsidy program are passed on to the workers via an increase in continuation wage to workers assigned to the complex task. So the subsidy functions as a wage increase for those disadvantaged workers who are assigned to the complex task. This will clearly result in an increase in the gain from investment function for workers in the disadvantaged group. Workers in the advantaged group are not affected by this policy; they will remain at Σh. For simplicity of notation, I drop the group subscripts in this section.

The profit function in eq. [15] indicates that I can represent the hiring subsidy with a change to the productivity of qualified workers in the disadvantaged group. Given a subsidy of Δξ, the expected productivity of a disadvantaged worker with signal θt who is assigned to the complex task is Δ(ξ+αc)p(θt,Πt). The higher productivity of qualified workers in the disadvantaged group means that, given the same firm beliefs, the optimal threshold is reduced by the policy. The optimal threshold for the disadvantage group given a subsidy of Δξ is

[16]θ^ξ(Πt)={1 if (αc+ξ)p(1,Πt)≤αs,0 if (αc+ξ)p(0,Πt)≥αs,the unique solution to(αc+ξ)p(θ,Πt)=αs if (αc+ξ)p(0,Πt)<αs<(αc+ξ)p(1,Πt).

Since workers are paid their expected productivity, the wages of those workers assigned to the complex task will be higher as a result of the policy. The continuation wage under the hiring subsidy is

[17]ωΔξ(θ;Πt)={Δαs if θ<θ^ξ(Πt),Δ(αc+ξ)p(θ,Πt) if θ≥θ^ξ(Πt).

The gain from investment will also be impacted. The new unit period gain from investment is

[18]γξ(Πt)=αs[Fq(θ^ξ(Πt))−Fu(θ^ξ(Πt))]+(αc+ξ)[∫θ^ξ(Πt)1p(θ,Πt)[ fq(θ)−fu(θ)]dθ].

The change from γ(Πt) to γξ(Πt) also changes the lifetime gain from investment Γt.

Consequently, under the hiring subsidy, the equilibrium for the disadvantaged group is described by a new dynamic system.

Proposition 3. An equilibrium of the dynamic model under a hiring subsidy is fully characterized by the following two variable differential equations:

Γ˙t=(β+r)Γt−γξ(Πt),

[19]Π˙t=βG(Γt)−Πt.

It is clear that the hiring subsidy policy only impacts the Γ˙t=0 locus. The Π˙t=0 locus is unchanged.

In order for this policy to eliminate discrimination it must move the disadvantaged group into the certain region of Σh from the base model. This result is guaranteed if the dynamic system under the hiring subsidy has only one steady state that is within the relevant region.

Given that the baseline dynamic system has three steady states, the only way that the dynamic system under the hiring subsidy will have one steady state is if γξ(Πt) is either much larger or much smaller than γ(Πt) for every value of Πt. If γξ(Πt) is smaller, then there is only one steady state at a low investment level. If γξ(Πt) is larger, then the single steady state occurs at a high investment level.

The following lemma indicates that γξ(Πt) is nondecreasing in ξ (the unit period subsidy). This means that under the hiring subsidy policy, γξ(Πt) is larger than γ(Πt) when ξ is positive.

Lemma 1. IfΠtis such thatθˆξ(Πt)∈[0,1), then the unit period gain from investment is strictly increasing inξ. IfΠtis such thatθˆξ(Πt)=1, then the unit period gain from investment is equal to zero and unaffected by an increase inξ.

Proof. The partial derivative of the unit period gain from investment with respect to the hiring subsidy ξ is

[20]∂γξ(Πt)∂ξ=∂θ^ξ(Πt)∂ξ︸≤0[αs−(ξ+αc)p(θ^ξ(Πt),Πt)](fq(θ^ξ(Πt))−fu(θ^ξ(Πt)))+∫θ^ξ(Πt)1p(θ,Πt)[ fq(θ)−fu(θ)]dθ︸>0 unless Πt=0 or Πt=1.

If Πt=0 or 1, then γξ(Πt)=0 and consequently ∂γξ(Πt)/∂ξ=0. There are three additional ranges of Πt that need to be considered:

If Πt is such that θˆξ(Πt)∈(0,1), then αs−(ξ+αc)p(θˆξ(Πt),Πt)=0 so that ∂γξ(Πt)/∂ξ=∫θ^ξ(Πt)1p(θ,Πt)[ fq(θ)−fu(θ)]dθ>0..
If Πt is such that θ^ξ(Πt)=0, then γξ(Πt)=(αc+ξ)∫01p(θ,Πt)[ fq(θ)−fu(θ)]dθ and ∂γξ(Πt)/∂ξ=∫01p(θ,Πt)[ fq(θ)−fu(θ)]dθ>0.
If Πt is such that θˆξ(Πt)=1, then γξ(Πt)=0 and consequently ∂θ^ξ(Πt)/∂ξ=0.□

Corollary 1. IfΠsup≡supΠ|θˆξ(Π)=1, thenΠsupis decreasing inξ.

Proof. In order for θˆξ(Πt)=1, it must be that (αc+ξ)p(1,Πt)≤αs (see eq. [16]). As ξ increases, the highest Πt for which this inequality holds decreases.□

Lemma 1 and the associated corollary indicate that the unit period gain from investment is strictly increasing in ξ unless Πt is such that θˆξ(Πt)=1, and that the range of Πt values that induces θˆξ(Πt)=1 is decreasing in ξ. I conclude that, as ξ increases, the Γ˙=0 locus is stretched to the right and if there is a region of Πt values for which γξ(Πt)=0, its upper bound decreases as ξ increases. At some value of ξ, which i refer to as ξmin, only one steady state will remain and it will be at a relatively high investment rate. I refer to this steady state as Σξ(Γξ,Πξ).

Lemma 2. IfG(0)>0, then there exists aξminsuch that for allξ>ξminthe dynamic system under the hiring subsidy has only one steady state, which is a saddle point.

Proof. Lemma 1 indicates that γξ(ΠtJ) is strictly increasing in ξ. This result, along with the assumption that G(0)>0, implies that there exists a ξmin such that for all ξ>ξmin there is only one steady state.□

Figure 4 depicts a potential phase diagram for the dynamic system when the hiring subsidy is such that there is only one steady state. This steady state is a saddle point as indicated by the arrows. Consequently, when a hiring subsidy of sufficient size is put in place, the disadvantaged group will move from the low steady state to the path that leads to Σξ(Γξ,Πξ), the only remaining steady state. In each period that the policy is in place, the group will move along the path that approaches the steady state.

If the hiring subsidy is removed when the investment rate of the disadvantaged group is in the certain region for Σl, then the group will return to their initial steady state. If, on the other hand, the policy is removed after the disadvantaged group reaches the certain region for Σh, then they will converge to Σh and discrimination will be eliminated.

Figure 4:

Phase diagram of the dynamic system under the hiring subsidy

It is important that Σξ is not in the uncertain region, or else there is no way to guarantee the success of the hiring subsidy. The following lemma indicates that for a positive subsidy Σξ is always in the certain region of Σh.

Lemma 3. IfG(0)>0andξ>ξmin, thenΣξwill be in the certain region forΣh.

Proof. From Lemma 1, γξ(ΠtJ) is strictly increasing in ξ. By assumption, G(⋅) is strictly increasing in its argument. Together these imply that Σξ will always involve a higher investment rate than Σh. Also, the upper bound of the uncertain region is always below Πh. As a result, Σξ must be in the certain region of Σh because Πξ is greater than Πh, which is always in the certain region.□

Taking Lemma 3 into account, it is clear that the hiring subsidy policy will be successful as long as a large enough subsidy is offered.

Proposition 4. IfG(0)>0andξ>ξmin, then the hiring subsidy will eliminate discrimination in the general dynamic model.

Corollary 2. This hiring subsidy can be removed as soon asΠtis in the certain region forΣh.

$Figure 5: Path to Σξ$${\Sigma ^\xi}$$$

Figure 5:

Path to Σξ

Figure 5 depicts the path to the remaining steady state from the disadvantaged group’s starting point at the low steady state. See Appendix D.2 for a discussion of how these paths were generated. For comparison the figure also includes the original γ(ΠtJ) function. I do not show the path to Σh once the policy is removed, but as long as the policy is removed after Πt>Πˆ then the disadvantaged group will simply move to the path that leads to Σh as it is the only plausible path given the current investment rate of the group.

I conclude that the hiring subsidy will always effectively eliminate discrimination as long as a large enough subsidy is offered. This policy eliminates discrimination by changing the optimal behavior of the firms. Specifically, the subsidy makes hiring disadvantaged workers more attractive and this is passed on to workers with an increase in the gain from investment. The end result is that the workers in the disadvantaged group are able to move to the high steady state.

Now that it has been confirmed that the hiring subsidy policy will eliminate discrimination it is worth analyzing the costs associated with implementation of the policy. As mentioned previously, the unit period cost of the hiring subsidy policy is κtξ=ξλdΠtd[1−Fq(θˆ(Πtd))]. This is simply the value of the subsidy times the expected number of workers from the disadvantaged group that will be correctly assigned to the complex task. Analysis of this per period cost reveals that

[21]∂κtξ∂ξ=λdΠtd[1−Fq(θˆ(Πtd))]≥0,

[22]∂κtξ∂λd=ξΠtd[1−Fq(θˆ(Πtd))]≥0, and

[23]∂κtξ∂Πtd=ξλd(1−Fq(θˆ(Πtd)))−Πtdfq(θˆ(Πtd))θˆ′(Πtd)≥0.

These results indicate that the larger the subsidy, the more costly this policy becomes. Thus the government has incentive to offer the lowest effective subsidy. Also, the larger the disadvantaged group is, the more expensive the policy becomes. Finally, the cost of the policy is also increasing in Πtd, as the more qualified workers there are, the more frequently the subsidy will have to be paid out. Note that the longer the policy is in place, the higher Πtd gets, as the disadvantaged group converges to the only remaining steady state. Consequently, the longer this policy is in place the more expensive it becomes per period.

4.2 Investment voucher program

The second policy of interest approaches the elimination of discrimination more directly. Rather than providing hiring incentives to the firms, an investment voucher program is designed to affect the decisions of the workers. The voucher is applied in the following way: The government provides all workers in the disadvantaged group with an investment voucher of υ dollars. This voucher is only good toward the purchase of human capital, so it defrays the monetary cost of investment, cm. Consequently, a voucher larger than cm has no more impact than one equal to cm. By implementing the voucher in this way, the government does not need to observe whether an individual worker invested or not. This policy is applied only to the disadvantaged group, so for simplicity of notation I drop the group subscripts in this section.

I can represent this investment voucher with a change in the effective lifetime gain from investment since the voucher is only used if a worker invests. The impact of the policy on the dynamic system is a strict increase in the investment rate of new workers at time t. In particular, the proportion of investors is G(Γt+min{υ,cm}) under the voucher, rather than G(Γt), as in the baseline model.

Consequently, under the investment voucher the equilibrium for the disadvantaged group will be described by a new dynamic system.

Proposition 5. An equilibrium of the dynamic model under an investment voucher is fully characterized by the following two variable differential equations:

Γ˙t=(β+r)Γt−γ(Πt)

[24]Π˙t=β(G(Γt+min{υ,cm})−Πt).

The only resulting change to the phase diagram will be in the Π˙t=0 locus, which is updated to Πt=G(Γt+min{υ,cm}). Given that G(⋅) is strictly increasing, a negative voucher shifts the Π˙t=0 locus to the right, so that the investment rate is lower at every Γt. A positive voucher shifts the locus to the left, so that investment rate is higher at every Γt.

Just as in the analysis of the hiring subsidy, in order for this policy to eliminate discrimination it must move the disadvantaged group to the certain region of Σh. The original dynamic model assumes a G(⋅) function that generates three steady states. The investment voucher policy does not change γ or G(⋅) but instead affects how investment decisions are made by increasing the lifetime gain from investment by min{υ,cm}. The dynamic system under the investment voucher will have only one steady state if min{υ,cm} is sufficiently positive or negative.

Given the goal of moving the disadvantaged group to Σh, an effective investment voucher must be positive. Moreover, assuming that the monetary cost of investment, cm, is high enough, at some value of υ, which I refer to as υmin, only one steady state will remain. I refer to this steady state as Συ(Γυ,Πυ). If the monetary cost of investment is low, then it is possible that even the largest effective voucher, υ=cm, is not able to eliminate the two lower steady states. In this case, the voucher policy is not effective at eliminating discrimination and no matter what υ is employed there will be three steady states.

Lemma 4. If the dynamic system under the largest effective investment voucher, υ=cm, has a single steady state, then there exists aυminsuch that for allυ>υminthe dynamic system under the investment voucher has only one steady state, which is a saddle point.

Proof. G(Γt+min{υ,cm}) is strictly increasing in υ when υ<cm. As υ increases, the Π˙t=0 locus shifts to the left. Given a large enough cm, there must exist a υmin such that for all υ>υmin there is one steady state.□

Figure 6:

Phase diagram of the dynamic system under the investment voucher

Figure 6 depicts a potential phase diagram for the dynamic system when the investment voucher is large enough to induce a single steady state. The arrows indicate that this steady state is a saddle point. Just as in the case of the hiring subsidy, the existence of a single steady state means that when the policy is put in place the disadvantaged group moves from the low steady state to the path that leads to Συ(Γυ,Πυ). In each period the policy is in effect, the group moves along the path approaching that steady state.

In order to successfully eliminate the discrimination, the policy can only be removed when the disadvantaged group is in the certain region for Σh. Again it is important that Συ is not in the uncertain region, or else there is no way to guarantee the success of the investment voucher. The following lemma indicates that for any voucher greater than υmin, Συ will be in the certain region of Σh.

Lemma 5. Ifυ>υmin, thenΣυwill be in the certain region forΣh.

Proof. G(Γt+min{υ,cm}) is strictly increasing in υ when υ<cm. So, the Π˙t=0 locus is always higher under the policy. This implies that Συ will involve a higher investment rate than Σh. Also, the upper bound of the uncertain region is always below Πh. Συ must be in the certain region of Σh because Πυ is greater than Πh, which is always in the certain region.□

Taking Lemma 5 into account, it is clear that the investment voucher policy will be successful as long as monetary costs make up a large enough proportion of total costs and a sufficient voucher is offered.

Proposition 6. If there exists aυmin, then for allυ>υmin, the investment voucher will eliminate discrimination in the general dynamic model.

Corollary 3. This investment voucher policy can be removed as soon asΠtis in the certain region forΣh.

Figure 7 depicts the path to the remaining steady state from the disadvantaged group’s starting point at Σl. See Appendix D.3 for a discussion of how these paths were generated. For comparison, the figure also includes the original Π˙=0 locus. As long as the policy is removed after Πt>Πˆ, the disadvantaged group will simply move to the path that leads to Σh as it is the only plausible path given the firms’ current beliefs about the group’s overall investment rate.

$Figure 7: Path to Συ$${\Sigma ^\upsilon}$$$

Figure 7:

Path to Συ

The investment voucher approaches the discrimination issue in a different manner than the hiring subsidy. It works by changing the incentives of the workers and making it easier for a worker’s gain from investment to exceed his total cost. By adjusting the optimal behavior of workers, this policy allows the firms and workers to move to the high steady state.

Now that it has been confirmed that the investment voucher policy will eliminate discrimination it is worth analyzing the costs associated with implementation of the policy. The per period cost of the investment voucher policy is κtυ=υλdβG(Γ(Πtd)+υ). This is simply the voucher times the fraction of new births that choose to invest under the policy. Analysis of this per period cost reveals that

[25]∂κtυ∂υ=λdβG(Γ(Πtd)+υ)+υλdβG′(Γ(Πtd)+υ)≥0,

[26]∂κtυ∂λd=υβG(Γ(Πtd)+υ)≥0,

[27]∂κtυ∂Πtd=υβG′(Γ(Πtd)+υ)Γ′(Πtd)≥0,and

[28]∂κtυ∂β=υλdG(Γ(Πtd)+υ)≥0.

The cost of the voucher policy in each period is clearly increasing in the size of the voucher offered as well as in the size and average investment rate of the disadvantaged group. Also note, that under this policy the faster the population turns over, the higher the per period costs become. Just as in the case of the hiring subsidy policy, the longer the investment voucher policy is in place the more expensive it becomes per period, since the voucher must be paid out to an increasing number of new investors as time progresses.

4.3 Government-enforced equal treatment

The final policy of interest involves intervention by the governing body with respect to the optimal behavior of the firms. Specifically, under what I call the equal treatment policy, the government requires that both firms ignore a workers’ group identity when assigning tasks and wages. This policy affects both disadvantaged and advantaged workers.

The equal treatment policy can be formally stated as: Firms are forbidden to condition task and wage assignments on group identity. I assume that the government is able to observe the firms’ task and wage assignments and if they observe a deviation from the policy, they will charge the firm a substantial fine. As long as this fine is large enough, the firms will never have incentive to deviate.

Under the equal treatment policy, the firms’ optimal behavior can no longer depend on group identity. So there will be a single θˆ(Πt) and a single ωΔ(θt;Πt) in each period, rather than one for each group.

The updated optimal threshold at time t is the one which maximizes a firm’s output conditional on investment decisions. Formally, the task assignment problem is

[29]maxθt∈[0,1]Δαc[(λdΠtd+λaΠta)(1−Fq(θt))]+Δαs[(λdΠtd+λaΠta)Fq(θt)+(1−(λdΠtd+λaΠta))Fu(θt)].

Just as in the original dynamic model there is a unique solution to the task assignment problem. But, in this case, it is not solved separately for each group. If I define a firm’s beliefs about the entire population of workers as Πpop=λdΠtd+λaΠta, then the unique optimal threshold is

[30]θ^(Πtpop)={1 if αc p(1,Πtpop)≤αs,0 if αc p(0,Πtpop)≥αs,the unique solution toαcp(θ,Πtpop)=αs ifαc p(0,Πtpop)>αs>αc p(1,Πtpop).

The continuation wage under the equal treatment policy will also depend on Πpop rather than the group investment rates. The new optimal wage schedule for both firms is

[31]ωΔ(θt;Πtpop)={Δαs if θt<θ^(Πtpop),Δαcp(θt,Πtpop) if θt≥θ^(Πtpop).

This change in the task assignment rule and the associated alteration of the continuation wage results in a new unit period gain from investment (and per period gain from investment). The updated unit period gain from investment is

[32]γ(Πtpop)=αs[Fq(θ^(Πtpop))−Fu(θ^(Πtpop))]+αc∫θ^(Πtpop)1p(θ,Πtpop)[ fq(θ)−fu(θ)]dθ.

Under the policy, the lifetime gain from investment for each group will be determined by the population investment rate rather than the group investment rate. This causes a subsequent change in the dynamic system.

Proposition 7. An equilibrium of the dynamic model under the equal treatment policy is fully characterized by the following differential equations:

Γ˙tpop=(β+r)Γtpop−γtpop(Πtpop).

[33]Π˙tpop=β(G(Γtpop)−Πtpop).

The dynamic system in eq. [33] is identical to that of the original dynamic model. The only difference is that it describes the entire population rather than each individual group. In order to talk about when the equal treatment policy can be removed, it is also necessary to determine how a firm’s beliefs about the investment rate of each group will change over time. Under the policy these beliefs are described by Π˙tJ=β(G(Γtpop)−ΠtJ).

Since I have already assumed that the original dynamic model has three steady states, with the high and the low steady states being saddle points, I know that the dynamic system under the equal treatment policy will as well. I call these three steady states Σlpop(Γlpop,Πlpop), Σmpop(Γmpop,Πmpop), and Σhpop(Γhpop,Πhpop) referring to the low, middle, and high steady states, respectively.

Lemma 6. Under the equal treatment policy the dynamic system of the general model has three steady states. ΣlpopandΣhpopare saddle points andΣmpopis a source.

The uncertain region, if present, is identical to that of the original dynamic model. With that in mind, it is straightforward to determine that this policy will always be able to eliminate discrimination. However, in some cases discrimination will be eliminated by moving both groups to the high steady state, in other cases both groups will end up in the low steady state. Clearly this is not a preferable outcome as the high steady state Pareto dominates the low steady state.

Given that the starting point of this analysis is one in which discrimination is present, when the policy is put in place the investment rate of the population is Π0pop=λaΠh+λdΠl. If this Π0pop is in the certain region for Σh, then the policy will be successful as the entire population will move to the only feasible path, which happens to lead to the high steady state. However if Π0pop is in the uncertain region of firm beliefs, then the policy will move the population to the high steady state if workers are optimistic and to the low steady state if workers are pessimistic. The policy is guaranteed to result in both groups at the low steady state if Π0pop is in the certain region for Σl.

Proposition 8. IfΠpopis in the certain region forΣhor ifΠpopis in the uncertain region and the population is optimistic, then the equal treatment policy will eliminate discrimination by moving both groups to the Pareto-dominant steady state. Otherwise, the policy will move the two groups to the Pareto-dominated steady state.

Corollary 4. This equal treatment policy can be removed as soon asΠtaandΠtdare both in the certain region forΣhor forΣl.

I can restate the results of Proposition 8 in terms of the size of the disadvantaged population. Specifically, if λd<(Πh−Π^/Πh−Πl), then the equal treatment policy is guaranteed to move both groups to the high steady state. I conclude that this policy will have a preferable outcome if the disadvantaged population is small or if the difference between Πl and Πh is small. ^[3]

This policy is different from the previous two because it impacts workers in both the advantaged and the disadvantaged groups. It is attractive to consider this policy because there is no direct cost associated with it. However, the policy has the potential to have the opposite of the intended effect if the initial population investment rate is in the certain region for Σl, or in the uncertain region and the population is pessimistic.

Now that it has been confirmed that the equal treatment policy can eliminate discrimination it is worth analyzing the costs associated with implementation of the policy. As stated earlier, the policy involves no direct monetary costs. However, it will result in a reduction in welfare for those in the advantaged group.

Welfare for those in the advantaged group when there is no equal treatment policy is

[34]W(Πta)=Πta︸Proportion ofqualified workers[∫01ωΔ(θ,Πta)fq(θ)dθ−︸Expected wageof qualified workersβ︸Fraction ofnew births(∫c¯Γ(Πta)cg(c)dcG(Γ(Πta)))︸Expected costgiven investment] , +(1−Πta)︸Proportion ofunqualified worker∫01ωΔ(θ,Πta)fu(θ)dθ︸Expected wageof unqualified workers

where given the assumptions about the starting point of the economy when the policy is put in place Πta=Πh. Under the equal treatment policy, all wages and task assignments are determined by the average investment rate of the population, Πpop, rather than the individual group investment rate. So the welfare of the advantaged group under the equal treatment policy is

[35]Wa(Πta,Πtpop)=Πta∫01ωΔ(θ,Πtpop)fq(θ)dθ−β(∫c¯Γ(Πtpop)cg(c)dcG(Γ(Πtpop))) +(1−Πta)∫01ωΔ(θ,Πtpop)fu(θ)dθ.

The true cost of the equal treatment policy is the difference between welfare in the high steady state and the welfare of the advantaged group under the policy: κtet=W(Πh)−Wa(Πta,Πtpop). In this general dynamic environment, there are no conclusions that I can draw regarding the impacts that λd,λa,Πtd,Πta,orβ have on the cost of the equal treatment policy. In Section 5 I examine a simple version of the dynamic model, within that structure I am able to be more specific regarding the costs of the equal treatment policy.

5 Analysis in the simple dynamic model

In this section I examine the welfare effects of the three policies in a simplified version of the dynamic model. First, I briefly describe the simple dynamic model. Next, I determine that all results from Section 3 and Section 4 hold for the simple dynamic model. Lastly, I use the simple model to analyze the welfare effects of each policy.

5.1 The simple dynamic model

Firms face the same linear per period production function as in the original model. For simplicity workers have only two possible signals: θH and θL where θH>θL. If a worker is qualified, they receive a signal of θH with probability Pq>1/2 and a signal of θL with probability 1−Pq. If a worker is unqualified, they receive a signal of θL with probability Pu>1/2 and a signal of θH with probability 1−Pu. The cost of investment is distributed uniformly over [c_,cˉ]. Time is discrete, so that Δ=1.

The equilibria of this example are characterized by the same dynamic system as in Proposition 2. Given the parameterization, I can find more specific forms of the relevant functions, θˆ(ΠtJ)andγ(ΠtJ). See Appendix B.1 for the updated functions for the threshold rule and unit period gain from investment.

In this simplified version of the dynamic model it is possible for there to be one or three steady states depending on the parameter values. For the analysis in the following sections I assume that there are three steady states. ^[4] Just as in the general dynamic model, the high and low steady states are saddle points and the middle steady state is an unstable source. See Appendix B.1 for a sample phase diagram for the simple dynamic model as well as a figure demonstrating the paths to the high and low steady states.

5.2 Policies in the simple dynamic model

If I analyze the three policies using the simple dynamic model described above I find that all major results from Section 4 hold. In particular, Propositions 3, 4, 5, 6, 7, and 8 are all confirmed. Using the simple dynamic model I can determine more specific functions for the dynamic systems in Propositions 3, 5, and 7 as well as for the minimum required hiring subsidy and the minimum required investment voucher mentioned in Propositions 4 and 6.

The minimum required subsidy is ξmin=αs1−1−PuPqcˉc_−αc. Note that ξmin is decreasing in the precision of signals (Pu and Pq) as well as in αc; it is increasing in αs. The minimum required voucher is υmin=(cˉ−c_)αs(1−Pu)αs(1−Pu)+(αc−αs)Pq+c_. This minimum voucher is decreasing in the precision of signals (Pu and Pq) as well as in αc, and it is increasing in αs. For a full description of the effects of the three policies in the simple dynamic model see Appendix B.

5.3 Welfare analysis of the three policies in the simple dynamic model

I am now able to look at the welfare effects of the three policies. The numerical and graphical results detailed in this section are for one specific parameterization of the dynamic model: Pq=2/3, Pu=2/3, αs=1, αc=2, β=0.2, r=0.05, c_=−0.1, and cˉ=0.9. However, the qualitative results hold for all parameterizations that generate three steady states with the high and low being saddle points.

I complete the welfare analysis in a general equilibrium framework. The hiring subsidy and investment voucher policies require that the government has funds to distribute to the firms and to the workers in the disadvantaged group, respectively. Given the general equilibrium framework, these funds are collected by the government via a tax levied on workers in the advantaged group or to state it more generally, a tax levied on workers who are offered the better continuation wage schedule. ^[5]

For the purpose of more meaningful comparison between the three policies, I apply the minimum necessary voucher and minimum necessary subsidies. For the parametric example υmin=7/30 and ξmin=7/2. I also assume that the population is optimistic so as to guarantee that the equal treatment policy is successful. Furthermore I assume that the population will follow the shortest possible path to the high steady state. The government also commits to employing each policy until the impacted groups reach their new steady state.

Table 1 describes the conditions of the advantaged and disadvantaged groups before a policy is applied. See Appendix D.1 for a discussion of how these values were calculated.

Table 1:

Details of the discriminatory equilibrium

Pu=Pq=2/3, λa=3/4, c:U[−0.1,0.9]	Discriminatory equilibrium
αs=1, αc=2, β=0.2, r=0.05	Disadvantaged	Advantaged
Steady-state investment	Πd=0.1000	Πa=0.7631
Lifetime gain from investment	Γd(Πd)=0	Γa(Πa)=0.663
Wages	ω(θH,Πd)=1	ω(θH,Πa)=1.7313
	ω(θL,Πd)=1	ω(θL,Πa)=1.2340
Average per period expected welfare	1.0010	1.4833

Figures 8–10 depict the average expected welfare of each group and of the population as a whole under the subsidy, voucher, and equal treatment policies, respectively. Each figure shows the welfare before the policy is in place, how welfare changes when the policy is applied, and then when the policy is removed. All three policies move those in the disadvantaged group to the high steady state, so the end result of each policy is that the welfare across groups is equal and the discrimination is eliminated. See Appendices D.2, D.3, and D.4 for a discussion of how these values were calculated.

Comparing across figures indicates that the hiring subsidy is the most costly of the three policies. The reason for this is twofold. First, the transfer that is necessary to fund the policy is very large. Second, the hiring subsidy is provided for each worker correctly employed in the complex task in every period the policy is in place. The result is that the advantaged group loses a significant amount of welfare over the course of the policy. However, this policy will get the disadvantaged group to the steady state in the shortest amount of time.

Figure 8:

Welfare effects of the hiring subsidy

Figure 9:

Welfare effects of the investment voucher

Figure 10:

Welfare effects of the equal treatment policy

The voucher policy is the least costly of the three policies. This is because the transfer necessary to support the policy is very low and in each period it is only paid to new workers that choose to invest. This policy needs to be in place for slightly longer than the hiring subsidy policy. The equal treatment policy also has a relatively small negative impact on the advantaged group, but it has the additional benefit of not requiring the government to redistribute income across groups. However, the equal treatment policy must be in place for the longest amount of time.

Table 2 shows the total change in welfare for each group and the population while each policy is in place. These numbers are simply the sum of the welfare gains of the disadvantaged group and the sum of the welfare losses of the advantaged group for each period that the policy is in place. The net welfare effect is the weighted sum of these two values, taking into account the relative sizes of the two groups. This table indicates that, while the subsidy does increase the disadvantaged group’s welfare greatly, the net result over the course of the policy is the least favorable. The most favorable, in terms of net change in welfare, is the equal treatment policy and the voucher policy falls somewhere in the middle.

While the numerical values of the results above are specific to the chosen parameters, the qualitative results are not. For all parameterizations of the simple dynamic model that generate three steady states and allow for all three policies to be successful, the hiring subsidy is the most costly to implement but leads to the fastest convergence. Similarly, the investment voucher is always the least costly policy while the equal treatment policy needs to be in place the longest, but results in the most positive net change in population welfare. See Appendix D.5 for welfare results for a sampling of parameterizations.

I can conclude that depending on policy makers’ preferences, it may be that the investment voucher, equal treatment policy, or hiring subsidy is preferred. If the policy maker wants to inconvenience the advantaged group the least, then the investment voucher is best. If his goal is to ensure that net welfare is improved the most, then the equal treatment policy is best. If the wants the policy in place for the shortest amount of time then the hiring subsidy is preferred.

Table 2:

Total change in welfare while the policies are in effect

	Group d welfare gain	Group a welfare loss	Net change in pop. welfare
Hiring subsidy	61.908	17.855	2.086
Investment voucher	12.251	0.329	2.816
Equal treatment	18.495	0.887	3.959

It is important to note that both the investment voucher and equal treatment policies can fail to be successful for some parameterizations. In particular, if the monetary cost of investment is low, then it may be that there is no voucher that can eliminate the low steady state. Similarly, if the population is pessimistic, then the equal treatment policy will have the suboptimal outcome of moving both groups to the low steady state. The subsidy policy, on the other hand, will always have the intended effect given that a large enough subsidy is applied.

I can also draw a few conclusions about how the different parameters impact welfare gains, losses, and policy length. The larger the disadvantaged group is, the longer the equal treatment policy will need to be in effect. This is because the larger the disadvantaged group is, the lower the Πpop becomes and the further the population is from the steady state once the policy is put in place. Across all three policies, the welfare loss to the advantaged group is increasing in the size of the disadvantaged group. Additionally, for all policies, the lower β is, the longer a policy needs to be in place and the smaller the welfare losses are.

6 Conclusion

The main contribution of this paper is to analyze the effectiveness of several government policies designed to eliminate discrimination in a dynamic environment. Unlike policy analysis in a static environment, I am able to determine the paths that each group will take once a policy is put in place and to address how long a policy needs to be in effect. The base model allows for competitive wages, and as a result I am also able to perform meaningful welfare analysis of the three policies in a general equilibrium setting. These results are a significant and important addition to the current literature.

The dynamic environment I develop and analyze in Sections 2 and 3 is what makes these results possible. If I were to analyze the same policies in a static setting, then I would not be able to draw any conclusions about whether each policy would be successful, as it would be impossible to predict which equilibrium the two groups would recoordinate on after the policy was put in place. Furthermore, if a policy happened to cause both groups to recoordinate on the Pareto-dominant equilibrium, there would be no way to predict how the groups would react to the removal of the policy. A static environment offers no information about the paths each group would take when a policy is put in place or when it is removed. Consequently, without the addition of dynamics to the model, I would not be able to draw any conclusions about the welfare effects of the policies.

Using the dynamic model developed in Sections 2 and 3, I find that the hiring subsidy policy will always effectively eliminate discrimination but it results in a significant loss in welfare to the advantaged group while in place. However, this policy does have the advantage of needing to be in effect for the shortest amount of time. I also find that the investment voucher is a successful policy as long as a large enough proportion of the investment costs are monetary in nature. This policy may be attractive to policy makers because it involves very little loss in welfare to the advantaged group.

The equal treatment policy may or may not be successful depending on several conditions. The policy is safe to implement in industries where a majority of the workers are in the advantaged group or if the population as a whole is optimistic. If this is the case, then it may be the preferred policy, as it requires no transfers from advantaged to disadvantaged workers and welfare losses are relatively small. However, the equal treatment policy should not be applied if a large proportion of workers are disadvantaged or if the population as a whole is pessimistic.

I also determine that all three policies need only be in effect until the investment rates of both groups are in the certain region of the high steady state. Once this occurs the policies can be safely removed and the groups will converge to the high steady state on their own.

Acknowledgments

I thank my dissertation advisor, Peter Norman, for his direction and support during this project. I am also grateful to the remainder of my committee Gary Biglaiser, R. Vijay Krishna, Sergio Parraries, and Helen Tauchen for their time and effort.

Appendix

A Proof of Proposition 1

The following proof is a slightly modified version of one found in Moro and Norman (2002, 2003). I make only slight adjustments to suit my purposes, as such all credit should be given to them. Due to the linear nature of the firms’ production function, I will address each group separately. For simplicity in notation I drop all group indicators J.

Proof (Sufficiency). Given π∈(0,1], let θ(π) be the unique solution to the task assignment problem, τ:[0,1]→[0,1] be the threshold rule with cut-off θ(π), and (C(π),S(π))=(πFq(θ),πFq(θ)+(1−π)Fu(θ)) be the associated aggregate factor inputs. Suppose that each firm posts the wage schedule ω:[0,1]→ℜ given by ω(θ;π) in eq. [4]. Moreover, suppose that workers facing indifference between working for firm 1 and firm 2 based on wage and task assignments (in equilibrium this will be all workers) simply flip a fair coin. This implies that

C1(π)=C2(π)=C(π)2,

[36]S1(π)=S2(π)=S(π)2.

The profit for firm 1 is (firm 2 is equivalent)

[37]Π1=αsS1(π)+αcC1(π)−12∫01ω(θ;π)[πfq(θ)+(1−π)fu(θ)]dθ =αsS1(π)+αcC1(π)−12∫0θ(π)αs[πfq(θ)+(1−π)fu(θ)]dθ −12∫θ(π)11αcπfq(θ)πfq(θ)+(1−π)fu(θ)︸p(θ,πJ)[πfq(θ)+(1−π)fu(θ)]dθ

[38]Π1=αsS1(π)+αcC1(π)−12[αs∫0θ(π)[πfq(θ)+(1−π)fu(θ)]dθ]+∫θ(π)1πfq(θ)dθ =αsS1(π)+αcC1(π)−12[αsS(π)+αcC(π)]=0.

Suppose one firm deviates to (ω′,τ′)≠(ω,τ). Let C′ and S′ denote the implied factor inputs and let a(θ)∈[0,1] denote the fraction of workers with signal θ that accept a job at the deviating firm. Since ω′(θ;π)≥ω(θ;π) for all θ such that a(θ)>0 the profit for the deviating firm, Πi′, satisfies

[39]Πi′≤αsS′+αcC′−∫01ω(θ;π)a(θ)[πfq(θ)+(1−π)fu(θ)]dθ,

where (here the assumption that ties are broken the same way by qualified and unqualified workers is used)

[40]C′=∫τ′(θ)πfq(θ)a(θ)dθ and S′=∫(1−τ′(θ))a(θ)[πfq(θ)+(1−π)fu(θ)]dθ.

Moreover ω(θ;π)=maxαcp(θ,π),αs so,

[41]∫ω(θ;π)a(θ)[πfq(θ)+(1−π)fu(θ)]dθ =∫τ′(θ)ω(θ;π)a(θ)[πfq(θ)+(1−π)fu(θ)]dθ +∫(1−τ′(θ))ω(θ;π)a(θ)[πfq(θ)+(1−π)fu(θ)]dθ ≥αs(1−τ′(θ))a(θ)[πfq(θ)+(1−π)fu(θ)]dθ+αc∫τ′(θ)πfq(θ)a(θ)dθ =αsS′+αcC′.

This implies that Πi′≤αsS′+αcC′−[αsS′+αcC′]=0.□

To prove necessity of the conditions in Proposition 1 I proceed by proving a sequence of intermediate results:

Lemma 7. Each firm earns zero profit in equilibrium.

Proof. Let Π1 and Π2 denote the profits for firm 1 and firm 2 and assume for contradiction that Π1>0. If Π2<0 there would be a profitable deviation, so I assume without loss that 0≤Π2≤Π1. Total industry profits are

[42]Π1+Π2=αsS1+αcC1+αsS2+αcC2 −∫θmax{ω1(θ),ω2(θ)}[πfq(θ)+(1−π)fu(θ)]dθ.

I observe that αsS1+αcC1+αsS2+αcC2≤αsS(π)+αcC(π). This simply means that aggregate output cannot exceed what a planner could achieve. Suppose firm 2 deviates by offering ω2′ given by ω′2(θ)=max{ω1(θ),ω2(θ)}+ ε for some ε>0, implying that firm 2 attracts all workers. In addition suppose firm 2 assigns as in the solution to eq. [2]. The corresponding profit is

[43]Π′2=αsS(π)+αcC(π)−∫θmax{ω1(θ),ω2(θ)}[πfq(θ)+(1−π)fu(θ)]dθ−ε = αsS(π)+αcC(π)−[αsS1+αcC1+αsS2+αcC2]+Π1+Π2−ε ≥Π1+Π2−ε.

Hence for ε sufficiently small Π2′>Π2 is a profitable deviation, thus proving the statement by contradiction.

Lemma 8. ω1(θ)=ω2(θ)for almost allθ∈[0,1]in any equilibrium.

Proof. For contradiction, suppose that ω1(θ)>ω2(θ) for all θ∈Θ′⊂[0,1] and let

[44]β=∫θ∈Θ′(ω1(θ)−ω2(θ))[πfq(θ)+(1−π)fu(θ)]dθ,

where Θ′ has a positive measure, which implies that β>0. Let C1,C2,S1,S2 be the effective factor inputs in the hypothetical equilibrium and suppose that firm 1 deviates and offers ω1′ given by ω1′(θ)=ω2(θ)+ε for all θ and assigns all workers (the deviation attracts all workers) in accordance with eq. [2], the solution to the task assignment problem. The implies profits are

[45]Π′1(ε)=αsS(π)+αcC(π)−∫θω2(θ)[πfq(θ)+(1−π)fu(θ)]dθ−ε=αsS(π)+αcC(π)−∫θ∈Θ′ω1(θ)[πfq(θ)+(1−π)fu(θ)]dθ+β −∫θ∈[0,1]\Θ′ω2(θ)[πfq(θ)+(1−π)fu(θ)]dθ−ε≥αsS1+αcC1+αsS2+αcC2 −∫θmax{ω1(θ),ω2(θ)}[πfq(θ)+ (1−π)fu(θ)]dθ+β−ε=β−ε,

where the last inequality follows from Lemma 7. The deviation is thus profitable if ε is small enough, thus the equilibrium where ω1(θ)>ω2(θ) cannot be optimal.

Lemma 9. αsS1+αcC1+αsS2+αcC2=αsS(π)+αcC(π).

Proof. By feasibility αsS1+αcC1+αsS2+αcC2≤αsS(π)+αcC(π), so assume for contradiction that αsS(π)+αcC(π)−αsS1−αcC1−αsS2−αcC2=β>0. Suppose firm 1 offers ω1′(θ)=ω2(θ)+ε for all θ and assigns all workers(the deviation attracts all workers) in accordance with eq. [2], the solution to the task assignment problem. The implied profits are

[46]Π′1(ε)= αsS(π)+αcC(π)−∫θω2(θ)[πfq(θ)+(1−π)fu(θ)]dθ−ε >αsS1+αcC1+αsS2+αcC2−∫θω2(θ)[πfq(θ)+(1−π)fu(θ)]dθ−ε.

But (Lemma 8) ω1(θ)=ω2(θ) almost everywhere so ∫θω2(θ)[πfq(θ)+(1−π)fu(θ)]dθ is the sum of wages paid out by firms 1 and 2 before the deviation. By zero profits (Lemma 7) this implies that Π′(ε)=β−ε, so for ε small enough the deviation is profitable.□

Lemma 10. Suppose(ω1,ω2)is a pair of equilibrium wage schedules and letθ(π)be as in eq. [3]. Then there is a pair(ks,kc)such that (1)ωi(θ)=ksfori=1,2and for almost allθ<θ(π), (2)ωi(θ)=p(θ,π)kcfori=1,2and for almost allθ≥θ(π).

Proof. The two parts have almost identical proofs, I will prove only part (2), which may appear as less obvious. Let ω(θ)=max{ω1(θ),ω2(θ)} and (C(π),S(π)) be the factor inputs corresponding to eq. [2] the unique solution to the task assignment problem. For contradiction suppose there is a set A⊂[θ(π),1], where m=∫Aπfq(θ)dθ>0, and for some β>0 such that for all θ∈A,

[47]ω(θ)p(θ,π) ≤ 11−Fq(θ(π))∫θ(π)1ω(θ)p(θ,π)fq(θ)dθ−β= 1π(1−Fq(θ(π)))∫θ(π)1ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ−β+ 1C(π)∫θ(π)1ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ−β.

By continuity there exists a set B∈[0,θ(π)) such that ∫B[πfq(θ)+(1−π)fu(θ)]dθ=S(π)C(π)m and

[48]ω(θ)≤ 1πFq(θ)+(1−π)Fu(θ)∫θ(π)1ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ = 1S(π)∫0θ(π)ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ

for every θ∈B. Consider a deviation by firm i where it offers ωi′(θ)=ω(θ)+ε to workers with θ∈A∪B and ωi′(θ)=0 for all other θ, and assigns workers from A to the complex task and workers from B to the simple task. The profit from this deviation is

[49]Π′= αc∫Aπfq(θ)dθ+αs∫B[πfq(θ)+(1−π)fu(θ)]dθ−∫θ∈A∪B(ω(θ)+ε)[πfq(θ)+(1−π)fu(θ)]dθ≥[αcC(π)+αsS(π)]mC(π)−(1C(π)∫θ(π)1ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ−β)∫θ∈Ap(θ,π)[πfq(θ)+(1−π)fu(θ)]︸=πfq(θ)dθ−∫θ∈B[πfq(θ)+(1−π)fu(θ)]dθ[1S(π)∫0θ(π)ω(θ)[πfq(θ)+(1−π)fu(θ)]]−ε∫θ∈A∪B[πfq(θ)+(1−π)fu(θ)]dθ

[50]=[αcC(π)+αsS(π)]mC(π)−(∫θ(π)1ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ−β)mC(π)− S(π)C(π)m[1S(π)∫0θ(π)ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ]−ε∫θ∈A∪B[πfq(θ)+(1−π)fu(θ)]dθ=mC(π)(αcC(π)+αsS(π)−∫θ∈[0,1]ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ)︸= 0 by Lemmas 7 and 8+ βmC(π)−ε∫θ∈A∪B[πfq(θ)+(1−π)fu(θ)]dθ.

Hence, Π′≥βmC(π)−ε∫θ∈A∪B[πfq(θ)+(1−π)fu(θ)]dθ>0 for ε small enough, which together with Lemma 8 establishes part (2) of the claim. The proof for the other half is symmetric. It’s simply a matter of removing β from eq. [47] and inserting a β in the inequality in eq. [48] and again constructing an A and B such that the factor ratio is as in the unique threshold in eq. [2]. The rest of the argument is unaltered.

Proof (Necessity). It remains to be shown that ks=αs and kc=P(θ,π)αc. Firms would make positive profits if ks<αs and kc<P(θ,π)αc and negative profits if the inequalities go the other way. Thus I must only consider cases where the inequalities are of opposing directions. The arguments are symmetric so I only consider the case with ks>αs and kc<P(θ,π)αc. If θ(π)=0,(1) each firm makes positive profits (loss), so the only case to consider is when θ(π) is interior. A necessary condition for optimality for problem (2) is that αcP(θ(π),π)=αs. Hence there must be an interval (θ(π),θ∗) where ωi(θ)=p(θ,π)kc<ks for all θ∈(θ(π),θ∗). Consider the deviation

[51]ω′i(θ)={ω(θ)for θ∈(θ(π),θ*)0otherwise and τ′i(θ)={0for θ∈(θ(π),θ′)1for θ∈(θ′,θ*),

where θ′ is set so that the factor ratio is as in the solutions to eq. [2],

[52](∫θ′θ*πfq(θ)dθ∫θ(π)θ′[πfq(θ)+(1−π)fu(θ)]dθ=C(π)S(π)).

The profit is

[53]Π′=αc∫θ′θ*πfq(θ)dθ+αs∫θ(π)θ′[πfq(θ)+(1−π)fu(θ)]dθ−∫θ(π)θ′ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ−∫θ′θ*ω(θ)[πfq(θ)+(1−π)fu(θ)]dθ>(αcC(π)+αsS(π))∫θ′θ*πfq(θ)dθC(θ)−αc∫θ(π)θ′[πfq(θ)+(1−π)fu(θ)]dθ−αs∫θ′θ*πfq(θ)dθ= 1C(π)∫θ′θ*πfq(θ)dθ[αcC(π)+αsS(π)−αsS(π)−αcC(π)]=0.

which completes the proof of Proposition 1.□

B Simple dynamic model

B.1 Phase diagram of the simple dynamic model

The equilibria of the simple dynamic described in 5.1 are characterized by the same dynamic system as in Proposition 2. Given the parameterization, I can find more specific forms of the relevant functions. The updated optimal threshold rule is

[54]θ^(ΠtJ)={1ifΠtJ≤Π¯θHifΠ¯<ΠtJ<Π¯0ifΠtJ≥Π¯,

where Π_ and Π¯ are calculated by updating θˆ(ΠtJ) in eq. [3]; they are

[55]Π_=αs(1−Pu)αs(1−Pu)+(αc−αs)Pq and Π¯=αsPuαsPu+(αc−αs)(1−Pq).

The unit period gain from investment is updated to

[56]γ(ΠtJ)={0ifΠtJ≤Π_(Pq+Pu−1)(αcP(θH,ΠtJ)−αs)ifΠ_<ΠtJ<Π¯αc(Pq+Pu−1)(P(θH,ΠtJ)−P(θL,ΠtJ))ifΠtJ≥Π¯.

The posterior probability that a worker with signal θH is qualified is

[57]P(θH,ΠtJ)=ΠtJPqΠtJPq+(1−ΠtJ)(1−Pu),

and the posterior probability that a worker with signal θL is qualified is

[58]P(θL,ΠtJ)=ΠtJ(1−Pq)ΠtJ(1−Pq)+(1−ΠtJ)Pu.

The two loci of the dynamic system are now ΓtJ=γ(ΠtJ)β+r, where γ(ΠtJ) is as in eq. [56], and ΠtJ=G(ΓtJ), where G is the uniform distribution over [c_,cˉ]. Using parameters Pq=2/3, Pu=2/3, αs=1, αc=2, β=0.2, r=0.05, c_=−0.1, and cˉ=0.9, these generate the phase diagram in Figure 11.

Figure 11:

Phase diagram for the parametric example

Figure 12:

Spiralling paths to the steady states in the parametric example

Figure 12 demonstrates the spiralling paths to the two saddle points. The steady-state beliefs for this particular parametric example are Πl=0.1, Πm=0.47, and Πh=0.76. The paths are generated by inducting backward from the high and low steady states using eqs [6] and [10]. See Appendix C.2 for further discussion of how these paths were generated. The figure indicates that there is an uncertain region (designated by the purple region) that results from the spiralling paths. There are also certain regions for both Σh (the pink area) and Σl (the blue area). These certain regions will help determine how long a potential policy needs to be in effect to guarantee the elimination of discrimination.

B.2 Effects of the hiring subsidy in the simple dynamic model

The equilibria of this simple model are described by the same dynamic system as in Proposition 3. The updated forms of the relevant functions are

[59]γξ(Πt)={0ifΠt≤Π_ξ(Pq+Pu−1)[(αc+ξ)P(θH,Πt)−αs]ifΠ_ξ<Πt<Π¯ξ(Pq+Pu−1)(αc+ξ)[P(θH,Πt)−P(θL,Πt)]ifΠt≥Π¯ξ,

where Π_ξ and Π¯ξ are calculated by updating θˆ(Πtξ) in eq. [16]. They are

[60]Π_ξ=αs(1−Pu)αs(1−Pu)+(αc+ξ−αs)Pq and Π¯ξ=αsPuαsPu+(αc+ξ−αs)(1−Pq)

Figure 13 depicts the phase diagram of the new dynamic system given that ξ=6. All other parameter values are as in Section 5.1. The result of this hiring subsidy is a dynamic system with only one steady state, which I refer to as Σξ(Γξ,Πξ). The arrows indicate that Σξ is a saddle point. Consequently, all results from Section 4.1 hold for this parametric example.

Figure 13:

Phase diagram for the parametric example under the hiring subsidy

$Figure 14: Path to Σξ$${\Sigma ^\xi}$$ in the parametric example$

Figure 14:

Path to Σξ in the parametric example

The minimum subsidy necessary for the dynamic system to have only one steady state is ξmin=αs1−PuPq1−G(0)G(0)+1−αc. Note that ξmin is decreasing in the precision of signals (Pu and Pq) as well as in αc; it is increasing in αs.

Figure 14 demonstrates the path the disadvantaged group will take once the policy is put in place as well as the group’s starting point. Figure 15 demonstrates the path the group will take to Σh assuming that the policy is removed after the group reaches Σξ, see Appendix D.2 for a discussion of how these paths were generated.

$Figure 15: Path to Σh$${\Sigma _h}$$ once the hiring subsidy policy is lifted$

Figure 15:

Path to Σh once the hiring subsidy policy is lifted

B.3 Effects of the investment voucher program in the simple dynamic model

The equilibria of this model are described by the same dynamic system as in Proposition 6. The relevant functions are just as in Appendix B.1. The only change resulting from the policy is that the Π˙=0 locus is updated to Πt=G(ΓtJ+min{υ,cm}). In this simple model, G(⋅) is uniform so the policy shifts the Π˙=0 locus to the left by the value of the voucher.

Figure 16 depicts the updated phase diagram given that υ=0.3 and cm=0.5. All other parameter values are as in Section 5.1. The result is a dynamic system with only one steady state that is a saddle point. Consequently, all results from Section 4.1 hold for this parameterization.

Figure 16:

Phase diagram for the parametric example under the investment voucher

The minimum voucher necessary to produce only one steady state, assuming that cm is large enough, is υmin=(cˉ−c_)Π_+c_. This minimum voucher is decreasing in the precision of signals (Pu and Pq) as well as in αc, and it is increasing in αs. Note that if υmin>cm then the policy will not be effective.

$Figure 17: Path to Συ$${\Sigma ^\upsilon}$$ for the parametric example$

Figure 17:

Path to Συ for the parametric example

$Figure 18: Path to Σh$${\Sigma _h}$$ once the investment voucher policy is lifted$

Figure 18:

Path to Σh once the investment voucher policy is lifted

Figure 17 demonstrates the path that the disadvantaged group will take to the new steady state from their starting point at the low steady state. Figure 18 depicts the path that will be taken if the policy is removed after the group reaches Συ. See Appendix D.2 for a discussion of how these paths were generated.

B.4 Effects of the equal treatment policy in the simple dynamic model

In the simple dynamic model under the equal treatment policy the optimal threshold rule for both groups will be

[61]θˆt(Πtpop)={1ifΠtpop≤Π_θHifΠ_<Πtpop<Π¯0ifΠtpop≥Π¯,}

where Π_ and Π¯ are just as in the original parametric example. They are

[62]Π_=αs(1−Pu)αs(1−Pu)+(αc−αs)Pq and Π¯=αsPuαsPu+(αc−αs)(1−Pq).

Given the optimal task assignment rule in eq. [61], the unit period gain from investment is

[63]γ(Πtpop)={0ifΠtpop≤Π_(Pq+Pu−1)(αcP(θH,Πtpop)−αs)ifΠ_<Πtpop<Π¯αc(Pq+Pu−1)(P(θH,Πtpop)−P(θL,Πtpop))ifΠtpop≥Π¯.

The two loci of the dynamic system under the equal treatment policy are Γtpop=γ(Πtpop)β+r, where γ(Πtpop) is as in eq. [63], and Πtpop=G(Γtpop), where G(⋅) is the uniform distribution over [c_,cˉ].

I can draw the same conclusions for the parametric example under the equal treatment policy as I did for the general dynamic model. If the original parameterization has three steady states, so will the parameterization under the equal treatment policy. Similarly, if Π0pop is in the certain region for Σh, then the policy will definitely eliminate discrimination. On the other hand, if Π0pop is in the uncertain region, then discrimination will be eliminated only if the population is optimistic.

$Figure 19: Path to Σh$${\Sigma _h}$$ given an optimistic population$

Figure 19:

Path to Σh given an optimistic population

Figure 19 shows the paths to the steady states for the same parameter values as in Section 5.1. I make the additional assumption that λa=0.75 and λd=0.25. From the analysis in Section 5.1 I know that the steady-state beliefs in this example are Πl=0.1, Πm=0.47, and Πh=0.76. So the resulting initial beliefs about the proportion of the population that is qualified are Π0pop=0.60; this starting point is indicated in the figure by the purple dotted line. For this parameterization, the uncertain region of initial firm beliefs is Πt∈[0.32,0.65]. It is clear that Π0pop is in this region. As such, the policy may or may not be successful depending on whether the population is optimistic or pessimistic. The purple dots in Figure 19 indicate the three points that the population may move to after the policy is put in place. Note that there are two points on the pessimistic path and one on the optimistic path.

Figure 20 shows the path that the population will take if it is optimistic. I also include the paths that the two groups will take from their individual starting points. See Appendix D.4 for a discussion of how these paths were generated. The disadvantaged group starts with an investment rate of Πl because that is the steady state that they were previously operating in. The advantaged group starts with an investment rate of Πh for the same reason. Figure 21 shows one of the paths that the population may take if it is pessimistic.

$Figure 20: Path to Σh$${\Sigma _h}$$ given an optimistic population$

Figure 20:

Path to Σh given an optimistic population

$Figure 21: Path to Σl$${\Sigma _l}$$ given a pessimistic population$

Figure 21:

Path to Σl given a pessimistic population

Just as in the general dynamic model under the equal treatment policy, if the proportion of the population that is in the disadvantaged group is very small, then the policy will be successful at eliminating the discrimination. For this particular set of parameter values if λd<0.17, then the policy is guaranteed to eliminate the discrimination.

C Calculating paths to the steady states

C.1 Paths to the steady states in the general dynamic model

Calculating the paths to the steady states is relatively straightforward. First, it is necessary to determine all the steady-state values by finding the solutions to the following system of equations:

[64]Π∗=G(Γ∗),

[65]Γ∗=γ(Π∗)β+r.

Once Σh(Πh,Γh), Σm(Πm,Γm), and Σl(Πl,Γl) are known, I can determine the path to the high steady state by evaluating

[66]Πt−1J=ΠtJ−βG(ΓtJ)1−β

and

[67]Γt−1J=(1−β−r)ΓtJ+γ(Πt−1J)

at ΠT=Πh−ε and ΓT=Γh−ε. This process yields ΠT−1 and ΓT−1, which can then be used to find ΠT−2 and ΓT−2. I repeat this process until the middle steady state is reached. The result is a series, Γt,Πtt=nT, that maps out the path from the middle steady state to the high steady state.

I can determine the path to the low steady state by evaluating eqs [66] and [67] at ΠT=Πl+ε and ΓT=Γl+ε. This process yields ΠT−1 and ΓT−1, which can then be used to find ΠT−2 and ΓT−2. I repeat this process until the middle steady state is reached. The result is a series, Γt,Πtt=nT, that maps out the path from the middle steady state to the low steady state.

C.2 Paths to the steady states in the simple dynamic model

For the parametric example used in Section 5.3 the steady states are the solutions to the following system of equations:

[68]Π∗=Γ∗+0.10,

[69]Γ∗=4γ(Π∗),

where

[70]γ(ΠtJ)={0 if ΠtJ≤1/313(3ΠtJ−1ΠtJ+1) if 1/3<ΠtJ<2/323(2ΠtJΠtJ+1−ΠtJ2−ΠtJ) if ΠtJ≥2/3.

Solving this yields the following three steady states: Σh(Γh=0.6631,Πh=0.7631),Σm(Γm=0.3687,Πm=0.4687), and Σl(Γl=0.0,Πl=0.1000).

In order to generate the optimistic path pictured in Figure 12, I induct backward from ΠT=Πh−0.0001 and ΓT=Γh−0.0001 using the following equations:

[71]Πt−1J=1.25ΠtJ−0.25ΓtJ−0.025,

[72]Γt−1J=.75ΓtJ+γ(Πt−1J).

The resulting series Γt,Πtt=nT describes the optimistic path. Similarly, I generate the pessimistic path by inducting backward from ΠT=Πl+0.0001 and ΓT=Γl+0.0001. The resulting series, Γt,Πtt=nT, describes the pessimistic path.

D Welfare calculations

D.1 Welfare in the base model

In order to understand the welfare effects of the three policies, it is necessary to determine the average expected welfare in both the high and low steady states of the base model for the parameterization described in Section 5.3. The general equation for the average expected welfare in one period is

[73]W(ΠtJ)=ΠtJ︸Proportion ofqualified workers[∫01ωΔ(θ,ΠtJ)fq(θ)dθ︸Expected wageof qualified workers−β︸Fraction ofnew births(∫c¯Γ(ΠtJ)cg(c)dcG(Γ(ΠtJ)))︸Expected costgiven investment] +(1−ΠtJ)︸Proportion ofunqualified worker∫01ωΔ(θ,ΠtJ)fu(θ)dθ︸Expected wageof unqualified workers.

For the parameterization used in Section 5.3 the average expected welfare can be simplified to

[74]W(ΠtJ)=ΠtJEt[ωΔ|q]−βΓ(ΠtJ)−0.102+(1−ΠtJ)Et[ωΔ|u],

where

[75]Et[ωΔ|u]={1ifΠtJ≤1/3233ΠtJ+1ΠtJ+1if1/3<ΠtJ<2/3232ΠtJ2−ΠtJ+2ΠtJ1+ΠtJifΠtJ≥2/3

and

[76]Et[ωΔ|q]={1ifΠtJ≤1/3139ΠtJ+1ΠtJ+1if1/3<ΠtJ<2/323ΠtJ2−ΠtJ+4ΠtJ1+ΠtJifΠtJ≥2/3.

Evaluating eq. [74] at the high and low steady states provides the baseline welfare level for workers in the advantaged and disadvantaged groups, respectively. In this case, W(Πh)=1.4833 and W(Πl)=1.0010.

D.2 Welfare under a hiring subsidy

The first step to determining the welfare effects of the hiring subsidy policy is to calculate the path that the disadvantaged group will take when the policy is put in place as well as the path that they will take to the high steady state when the policy is lifted. Once these paths are established, I can calculate the welfare of each group for each period of interest.

I begin by determining the single steady state of the dynamic system under the policy. This steady state is the solution to the following system of equations:

[77]Πξ=G(Γξ),

[78]Γξ=γξ(Πξ)β+r.

I refer to the single steady state as Σξ(Γξ,Πξ). Once this is determined I find the disadvantaged group’s path to the steady state by inducting backward from ΓT=Γξ−ε and ΠT=Πξ−ε using the following equations:

[79]Πt−1d=Πtd−βGΓtd1−β,

[80]Γt−1d=(1−β−r)Γtd+γξ(Πt−1d).

To determine the path taken after the policy is removed I use eqs [66] and [67] and induct backward from ΠT=Πh+ε and ΓT=Γh+ε.

For the parameterization of interest the steady state is Σξ(Γξ=0.8159,Πξ=0.9159). This is determined by finding the solution to the following system of equations:

[81]Πξ=Γξ+0.10,

[82]Γξ=4γξ(Πξ),

where

[83]γξ(ΠtJ)={0ifΠt−1J≤0.101310ΠtJ−1ΠtJ+1if0.10<ΠtJ<0.311162ΠtJΠtJ+1−ΠtJ2−ΠtJifΠtJ≥0.31.

In order to find the path that the disadvantaged group will take to this steady state, I induct backward from ΓT=Γξ−ε and ΠT=Πξ−ε using the following equations:

[84]Πt−1d=1.25Πtd−0.25Γtd−0.025,

[85]Γt−1d=.75Γtd+γξ(Πt−1d),

where ε=0.00000001.

I use eqs [71] and [72] and induct backward from ΠT=Πh+ε and ΓT=Γh+ε to find the path taken after the policy is removed.

To determine the welfare effects while the subsidy is in place, I look at the Γt,Πt pairs on the path to Σξ starting from Πl, the disadvantaged group starting point, through the steady-state values under the policy. To find how welfare changes after the policy is lifted I look at the Γt,Πt pairs starting from Πξ until the high steady state is reached. With these values I am able to calculate each group’s welfare during each period that the policy is in place as well as during the periods after it is removed.

The policy is applied in a general equilibrium framework so that the workers in the advantaged group are taxed in order to fund the policy. As such, the welfare of both groups will be impacted by the policy, even though the behavior of the advantaged group is not affected. The welfare of the advantaged group while the policy is in place is the following

[86]Wa(Πta,Πtd)=W(Πh)︸Welfare in thehigh steady state−ξ︸Value of thesubsidyλdλa︸Fraction of thesubsidy that eachworker providesΠtd︸Proportion ofqualified workers∫θ^(Πtd)1fq(θ)dθ.︸Probability a qualifiedworker is assigned to C︸Per worker transfer in each period

This is simply the average welfare in the high steady state minus the average expected transfer. Note that the transfer each worker in the advantaged group provides depends on the relative sizes of the two groups. Clearly, if more of the population is disadvantaged, then the loss in welfare to those in the advantaged group is higher.

The average expected welfare of the disadvantaged group while the policy is in place is the following

[87]Wd(Πtd)= Πtd[∫01ωΔξ(θ,Πtd)fq(θ)dθ︸Expected wage ofqualified workers underthe subsidy policy−β(∫c¯Γ(Πtd)cg(c)dcG(Γ(Πtd)))] +(1−Πtd)∫01ωΔξ(θ,Πtd)fu(θ)dθ︸Expected wage ofunqualified workers underthe subsidy policy.

The only way that this differs from eq. [73] is that it depends on the expected wage under the subsidy policy, ωΔξ(θ,Πtd); all other components are the same.

For the parameterization, the advantaged group’s average expected welfare while the policy is in place is

[88]Wa(Πta,Πtd)=1.4833−7Πtd6P[θt>θˆ(Πtd)|q],

where

[89]P[θt>θˆ(Πtd)|q]={0ifΠtd≤0.1023if0.10<Πtd<0.311ifΠtd≥0.31.

The average expected welfare for the disadvantaged group is

[90]Wtd(Πtd)=Πtd{Et[ωΔξ|q]−βΓ(Πtd)−0.102+(1−Πtd)Et[ωΔξ|u],

where

[91]Et[ωΔξ|u]={1ifΠtd≤0.101313Πtd+2Πtd+1if0.10<Πtd<0.31113Πtd2−Πtd+Πtd1+ΠtdifΠtd≥0.31

and

[92]Et[ωΔξ|q]={1 if Πtd≤0.1013(23Πtd+1Πtd+1) if 0.10<Πtd<0.31116(Πtd1−Πtd+4Πtd1+Πtd) if Πtd≥031.

To generate Figure 8 I evaluate eqs [88] and [90] at all Γt,Πt pairs along the path starting from Πl=0.10 till the group is within 0.0001 of the single steady state of the dynamic system under the hiring subsidy. To calculate the welfare after the policy is removed I evaluate eq. [74] at all Γt,Πt pairs along the path from Πξ till the group is within 0.0001 of the high steady state.

D.3 Welfare under an investment voucher

In order to find the welfare effects of the investment voucher policy, I must determine the path that the disadvantaged group will take once the policy is put in place as well as the path that they will take to the high steady state once the policy is lifted. Once these paths are established, I can calculate the welfare of each group for each period that the policy is in place.

The first step to calculating the path that the disadvantaged group will take once the investment voucher policy is implemented is to determine the single steady state of the dynamic system under the policy. This steady state is the solution to the following system of equations:

[93]Πυ=G(Γυ+υ),

[94]Γυ=γ(Πυ)β+r.

I refer to it as Συ(Γυ,Πυ). Once this is determined, I can find the disadvantaged group’s path to the steady state by inducting backward from ΓT=Γυ−ε and ΠT=Πυ−ε using the following equations:

[95]Πt−1d=Πtd−βGΓtd+υ1−β,

[96]Γt−1d=(1−β−r)Γtd+γ(Πt−1d).

I then use eqs [66] and [67] and induct backward from ΠT=Πh+ε and ΓT=Γh+ε to find the path taken after the policy is removed.

For the parameterization, the steady state is Συ(Γυ=0.5058,Πυ=0.8391). This is determined by finding the solution to the following system of equations:

[97]Πυ=Γυ+13,

[98]Γυ=4γ(Πυ),

where γ(⋅) is as in eq. [70].

In order to find the path that the disadvantaged group will take to this steady state, I induct backward from ΓT=Γυ−ε and ΠT=Πυ−ε using the following equations:

[99]Πt−1d=1.25Πtd−0.25Γtd−0.0833¯,

[100]Γt−1J=.75Γtd+γ(Πt−1d),

where ε=0.00000001.

To determine the path taken after the policy is removed I use eqs [71] and [72] and induct backward from ΠT=Πh+ε and ΓT=Γh+ε.

I can calculate the welfare effects while the voucher is in place by looking at all the Γt,Πt pairs along the path to Συ starting from Πl, the disadvantaged group starting point. To find how welfare changes after the policy is lifted, I look at all the Γt,Πt pairs along the path to Σh starting from the Πυ. With these values, I can calculate each group’s welfare during each period that the policy is in place as well as during the periods after it is removed.

The policy is applied in a general equilibrium framework so that the workers in the advantaged group are taxed in order to fund the policy, as such the welfare of both groups will be impacted by the policy, even though the behavior of the advantaged group is not affected.

The welfare of the advantaged group while the policy is in place is

[101]Wa(Πta,Πtd)=W(Πh)︸Welfare in thehigh steady state−υ︸Value of thevoucherλdλa︸Fraction of thevoucher that eachworker providesβ︸Fraction ofnew birthsG(Γ(Πtd)+υ)︸Proportion ofworkers that investunder the voucher︸Per worker transfer in each period.

Note that the transfer that each worker in the advantaged group provides depends on the relative sizes of the two groups. Clearly if more of the population is disadvantaged, then the loss in welfare to those in the advantaged group is higher.

The welfare of the disadvantaged group while the policy is in place is

[102]Wd(Πtd)=Πtd∫01ωΔ(θ,Πtd)fq(θ)dθ−β(∫c¯Γ(Πtd)+υcg(c)dcG(Γ(Πtd)+υ)︸Expected cost giveninvestment under thevoucher policy−υ︸Value ofthe voucher) +(1−Πtd)∫01ωΔ(θ,Πtd)fu(θ)dθ.

The only way that this differs from eq. [73] is that workers are more likely to invest (as a result of the voucher) and the costs associated with this investment are reduced by υ.

For the parameterization, the average expected welfare for the advantaged group while the policy is in place is

[103]Wa(Πta,Πtd)=1.4833−0.0156(Γ(Πtd)+0.1333).

The average expected welfare for the disadvantaged group while the policy is in place is

[104]Wd(Πtd)=ΠtdEt[ωΔ|q]−βΓ(Πtd)−0.332+(1−Πtd)Et[ωΔ|u],

where Et[ωΔ|u] and Et[ωΔ|q] are as in eqs [75] and [76], respectively.

To generate Figure 9, I evaluate eqs [103] and [104] at all Γt,Πt pairs along the path to Συ starting from Πl, the disadvantaged groups starting point, till they are within 0.0001 of Συ. To calculate the welfare after the policy is removed I evaluate eq. [74] at all Γt,Πt pairs along the path to Σh starting from Πυ till they are within 0.0001 of the high steady state.

D.4 Welfare under the equal treatment policy

Finding the welfare effects of the equal treatment policy is slightly more involved than the process for the previous policies. Two steps are necessary. First, I must find the path that the entire population takes to the high steady state once the policy is put in place. Using this path I can then find the individual paths of the two groups. It is important to consider the fact that the population as a whole will reach the steady state before the two groups do, as such I must account for several periods where the population investment rate is not changing but the group investment rates still are. Once the two groups reach the steady state, then the policy can be safely removed.

The dynamic system is the same under the equal treatment policy as it is without the policy, but it depends on the firms’ beliefs about the average investment rate of the entire population. As such the first step is to calculate the path that the population as a whole will take once the policy is put in place. I assume that the population is optimistic so that they will take the path to the high steady state. I determine the population’s path to the high steady state by inducting backward from ΓTpop=Γh−ε and ΠTpop=Πh−ε using the following equations:

[105]Πt−1pop=Πtpop−βGΓtpop1−β,

[106]Γt−1pop=(1−β−r)Γtpop+γ(Πt−1pop).

After this process I have all the Γtpop, Πtpop pairs that form the path to the high steady state. I am only interested in the path starting at Πpop=λaΠh+λdΠl. Once I calculate the population path, I can find the paths that the individual groups will take by forward inducting from Π0d=Πl and Π0a=Πh using the following equations:

[107]Πtd=(1−β)Πt−1d+βGΓtpop,

[108]Πta=(1−β)Πt−1a+βGΓtpop.

Note that I know the relevant Γtpop values from the previous calculation of the population’s path of the high steady state.

For the parameterization of interest the high steady state is Σh(Γh=0.6631,Πh=0.7631). In order to find the path that the population will take to this steady state I induct backward from ΓTpop=Γh−ε and ΠTpop=Πh−ε where ε=0.00000001 using the following equations:

[109]Πt−1pop=1.25Πtpop−0.25Γtpop−0.025,

[110]Γt−1pop=.75Γtpop+γpop(λaΠt−1a+λdΠt−1d).

After this process I know all the Γtpop, Πtpop pairs that form the path to the high steady state. I am only interested in the path starting at Πpop=0.5973. With this path I am able to determine the paths that the two groups will take by forward inducting from Π0d=0.1000 and Π0a=0.7631 using the following equations:

[111]Πtd=0.80Πt−1d+0.20Γtpop+0.02,

[112]Πta=0.80Πt−1a+0.20Γtpop+0.02.

In this parameterization it is also necessary to include ten periods where Πpop is at the steady-state level but the individual groups are still approaching it.

In order to determine the average expected welfare of the disadvantaged group while the equal treatment policy is in place, I look at both the Γtpop,Πtpop pairs that form the population path and the Γtpop,Πtd pairs that form the disadvantaged group’s path starting from Πd=Πl and Πpop=λaΠh+λdΠl until Σpop is reached. For the advantaged group I consider both the Γtpop,Πtpop pairs that form the population path and the Γtpop,Πta pairs that form the advantaged group’s path starting from Πa=Πh and Πpop=λaΠh+λdΠl until Σpop is reached. With these values I can then calculate each group’s welfare during each period that the policy is in place.

The welfare of each group while the policy is in place is

[113]WJ(ΠtJ,Πtpop)=ΠtJ︸Proportion ofqualified workers[−6∫01ωΔ(θ,Πtpop)fq(θ)dθ︸Expected wageof qualified workers−β︸Fraction ofnew births(∫c¯Γ(Πtpop)cg(c)dcG(Γ(Πtpop)))︸Expected costgiven investment] +(1−ΠtJ)︸Proportion ofunqualified worker∫01ωΔ(θ,Πtpop)fu(θ)dθ︸Expected wageof unqualified workers for J=a,d.

For the parameterization the average expected welfare of each group while the policy is in place is

[114]WJ(ΠtJ,Πtpop)=ΠtJEt[ωΔpop|q]−βΓ(Πtpop)−0.102+(1−ΠtJ)Et[ωΔpop|u],

where Et[ωΔpop|u] and Et[ωΔpop|q] are as in eqs [75] and [76], respectively. They are simply evaluated at Πpop rather than ΠJ.

To generate Figure 10, I evaluate eq. 114 at all relevant pairs along the paths from the starting point Πpop=0.5973,Πd=0.1000,Πa=0.7631 till both groups are within 0.0001 of the high steady state.

D.5 Welfare results for a sampling of parameterizations

All these results assume starting parameter values of Pq=2/3, Pu=2/3, αs=1, αc=2, β=0.2, r=0.05, c_=−0.1, cˉ=0.9, λd=.25, and λa=0.75, only those parameters mentioned are varied from this point.

Table 3:

Welfare effects of varying λd

	Parameter value	Group d welfare gain	Group a welfare loss	Net change in pop. welfare	Policy length
Hiring subsidy	λd=0.05	61.908	3.124	0.128	20
	λd=0.25	61.908	17.855	2.086	20
	λd=0.67	61.908	106.550	6.317	20
Investment voucher	λd=0.05	12.251	0.053	0.563	25
	λd=0.25	12.251	0.329	2.816	25
	λd=0.67	12.251	2.028	7.539	25
Equal treatment	λd=0.05	19.460	0.189	0.794	41
	λd=0.25	18.495	0.887	3.959	43
	λd=0.67	21.817	2.622	13.752	52

Table 4:

Welfare effects of varying Pu and Pq

	Parameter value	Group d welfare gain	Group a welfare loss	Net change in pop. welfare	Policy length
Hiring subsidy	Pu=0.8	49.516	13.299	2.404	21
	Pq=0.5
	Pu=2/3	61.908	17.855	2.086	20
	Pq=2/3
	Pu=0.6	65.543	18.777	2.303	19
	Pq=0.75
	Pu=0.8	11.508	0.266	2.678	27
	Pq=0.5
Investment voucher	Pu=2/3	12.251	0.329	2.816	25
	Pq=2/3
	Pu=0.6	14.331	0.370	3.306	25
	Pq=0.75
	Pu=0.8	17.238	0.859	3.665	48
	Pq=0.5
Equal treatment	Pu=2/3	18.495	0.887	3.959	43
	Pq=2/3
	Pu=0.6	21.129	1.558	4.114	41
	Pq=0.75

Table 5:

Welfare effects of varying αs and αc

	Parameter	Group d welfare n	Group a welfare loss	Net change in pop. welfare	Policy length
Hiring subsidy	αs=0.95	63.427	17.634	2.63	21
	αc=2
	αs=1	61.908	17.855	2.086	20
	αc=2
	αs=1	59.647	16.434	2.587	19
	αc=2.1
	αs=0.95	11.883	0.289	2.754	24
	αc=2
Investment voucher	αs=1	12.251	0.329	2.816	25
	αc=2
	αs=1	13.937	0.294	3.264	24
	αc=2.1
	αs=0.95	21.237	1.540	4.154	48
	αc=2
Equal treatment	αs=1	18.495	0.887	3.959	43
	αc=2
	αs=1	25.274	1.026	5.549	46
	αc=2.1

Table 6:

Welfare effects of varying β and r

	Parameter	Group d welfare gain	Group a welfare loss	Net change in pop. welfare	Policy length
Hiring subsidy	β=0.1	87.994	29.918	–0.440	35
	r=0.1
	β=0.2	61.908	17.855	2.086	20
	r=0.05
	β=0.05	195.706	54.867	7.776	63
	r=0.2
	β=0.1	26.318	0.322	6.331	48
	r=0.1
Investment voucher	β=0.2	12.251	0.329	2.816	25
	r=0.05
	β=0.05	39.153	0.321	9.547	126
	r=0.2
	β=0.1	51.824	–0.724	13.499	91
	r=0.1
Equal treatment	β=0.2	19.495	0.887	3.959	43
	r=0.05
	β=0.05	90.728	–3.183	25.069	186
	r=0.2

Table 7:

Welfare effects of varying c_ and cˉ

	Parameter	Group d welfare gain	Group a welfare loss	Net change in pop. welfare	Policy length
Hiring subsidy	c_=−0.01	706.875	233.596	1.522	23
	c¯=0.81
	c_=−0.1	61.908	17.855	2.086	20
	c¯=0.9
	c_=−0.2	31.980	7.600	2.295	21
	c¯=1
	c_=−0.01	13.896	0.383	3.187	25
	c¯=0.81
Investment voucher	c_=−0.1	12.251	0.329	2.816	25
	c¯=0.9
	c_=−0.2	11.469	0.278	2.658	26
	c¯=1
	c_=−0.01	22.847	0.902	5.035	36
	c¯=0.81
Equal treatment	c_=−0.1	18.495	0.887	3.959	43
	c¯=0.9
	c_=−0.2	17.733	0.784	3.845	42
	c¯=1

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Published Online: 2014-11-21

Published in Print: 2015-4-1

Articles in the same Issue

https://doi.org/10.1515/bejeap-2014-0005

Keywords for this article

statistical discrimination; general equilibrium; asymmetric information; human capital