Discrimination, Bayesian updating of employer beliefs, and human capital accumulation.

Farmer, Amy ; Terrell, Dek

I. INTRODUCTION

Can labor market discrimination persist over time? How do employer beliefs influence minority groups and their choices? Previous theoretical models of discrimination can be grouped into two distinct categories based upon the source of the discrimination. The most common method of modelling discrimination follows the work of Becker [1972] in which the employer, manager, or other employees receive disutility from associating with members of a particular group. In a second class of models referred to as statistical discrimination, initially proposed by Phelps [1972], discrimination results from differences in the groups' ability to signal output or capabilities; this result can be reproduced as a special case of the model presented here. This paper proposes an alternative model in which discrimination results initially from differences in the employer's prior opinion of average group ability. Using a Bayesian updating model, we analyze the dynamic effects of this prior belief on human capital acquisition and the potential for continued discrimination.

Previous research provides an abundance of evidence that society's initial perceptions about numerous characteristics, including those affecting productivity, differ by group. For example, Smith [1990] summarizes the results of the General Social Survey, which asked individuals to rank ethnic groups on a scale of one to seven in various categories.(1) The survey found a significant portion of those surveyed believed that African Americans, Hispanics, and white Southerners lagged behind the rest of society in intelligence and work effort. In fact, 53.2 percent of those surveyed ranked African Americans below whites in intelligence. Unless employers differ systematically from the rest of society, this evidence implies that the initial or prior employer assessment of ability will be lower for members of these groups.

In addition to strong evidence that such general priors exist, numerous studies find that prior beliefs influence the evaluation of employee ability. The Urban Institute recently conducted a study comparing the evaluation of black and Hispanic workers to that of equally qualified white workers.(2) Given equal resumes, the black and Hispanic job candidates trailed white candidates at every stage in the job seeking process, from receiving fewer initial interviews to a lower number of job offers. Similar biases have been found in experiments in which identical resumes were evaluated differently simply due to the race or gender of the applicant.(3) Furthermore, even if worker output is unambiguously observed and evaluated in an unbiased setting, studies find that success of males is attributed to ability while that of females is more likely to be attributed to luck.(4) This adherence to initial perceptions suggests that agents believe strongly in their initial beliefs, and hence would require substantial evidence to alter their perceptions. Our model incorporates the strength of the prior beliefs through a variance parameter surrounding the initial belief of mean ability level; a lower variance implies a decreased willingness to admit error in one's prior beliefs and update accordingly.

Discrimination, in our model, results from the entrepreneur's sincere, although perhaps incorrect, beliefs concerning the distribution of ability levels of a particular group. Although empirical results provide strong evidence of labor market discrimination persisting over many years [Cain 1986], most existing theoretical models of discrimination are static designs used to explain only within-period discrimination. We propose a dynamic model in which a Bayesian employer learns about an employee's ability over time based on output observed up to a stochastic error term, thus updating prior beliefs concerning both the individual and the group in each period. Viewing output as a signal of ability, it is clear that traditional statistical discrimination models are embedded within ours; the special case in which one group's signal is less reliable than another's can produce the statistical discrimination results.

Section II of this paper outlines a two-period overlapping generations model. This section describes the updating of the firm's belief about a single employee given output. Given an initial prior distribution on group ability, the employer chooses the first-period wage for the individual. At the end of the period the employer observes output, which is a function of the unobserved ability of the worker and the level of acquired human capital, plus a stochastic error term. The employer updates his prior beliefs of an individual's ability in the second period based on the observation of first-period output; these new beliefs determine the expectation of second-period output and, therefore, the wage rate. Employees know the prior beliefs of the employer and choose to acquire the level of human capital that maximizes the present value of expected lifetime income. Using comparative statics, we show that both lowering the initial expected level of ability for a particular group and raising the uncertainty surrounding output for undervalued employees diminishes the accumulation of human capital for members of the group.

The result that incorrect priors of group ability leads to lower human capital acquisition, even temporarily, has far-reaching implications. Lundberg and Startz [1993] propose a model in which lower ethnic capital for one group at any point in time results in wage differences persisting over time. Our model reveals that lower initial assessments of ability lead to the reduction in minority human capital assumed in the Lundberg and Startz model. Given their findings, our results imply that an incorrect prior can lead to persistent discrimination even if the prior rapidly becomes updated to the truth. Further, the process of adapting these priors may not be immediate, if it occurs at all; this time lag compounds the discrimination results implied by the initial diminished human capital acquisition. The impact of the prior assessment on wages over time requires a model of the evolution of prior beliefs of group ability over time.

Section II describes the updating procedure for a single employer/employee relationship, but characterizing the process of developing group priors poses a different problem. While greater than expected output raises an employer's prior beliefs about a single employee's ability, it may have little effect on the employer's prior belief about the group. Higher than expected output of one worker provides much information about individual ability, but only a single data point to estimate the average ability of a population of millions. In addition to observation of workers, an employer receives an abundance of information on average group ability from other sources. Observations of average output, or perhaps occupations, of other members of the group influence the assessment of group ability.

Section III addresses the evolution of these beliefs about the group as a whole. In section III we develop a model in which the prior beliefs of group ability result from the employer's observation of average output for heterogeneous groups. Due to the complexity of the model no closed-form solutions for steady-state output and human capital investment exist, but simulation results examine the effects of prior assessments and other parameters on long-run solutions to the model.

II. THE TWO-PERIOD, INDIVIDUAL MODEL

As noted above, this section models behavior for a single employer and employee, given some prior belief by the employer about employee ability. The two-period model analyzes the impact of lower initial assessments of group ability on wages and human capital accumulation for an individual. We model the interaction between one employee and the employer for whom she expects to work. Clearly employees have incentives to move to firms holding the highest evaluation of their ability. Yet, the literature discussed in section I provides abundant evidence that in much of the U.S. minority employees seek employment with firm's holding lower assessments of their ability than that of white workers.(5) We take the match between employee and employer as given and consider the effect of the employer's prior opinions on the wages and human capital investment decisions of an employee.

The information available to agents plays a critical role in this model. The workers in the model are endowed with a set of traits affecting their job performance which are known to the worker but unobservable to the employer. Such traits may be inherent intelligence, motivation, family, cultural upbringing, or a host of other factors. In addition to such unobservable characteristics, in the first period of life workers purchase some level of education at a cost; educational attainment is observable by all employers. Workers live and supply one unit of labor for two periods, but education can only be purchased in the initial period.

We assume that employers observe individual output:

(1) Q = [A.sup.[Alpha]][H.sup.[Beta]][Epsilon]

where

A denotes inherent ability which represents the employee's unobservable characteristics.

H represents the educational level attained by the employee in the first period of life; it is observable by the employer.

[Epsilon] is a stochastic error term distributed lognormally, In [Epsilon] [similar to] N(-[[Sigma].sup.2]/2, [[Sigma]2]). This implies that the mean of [Epsilon] is 1. This distribution is known to both the employer and the employee.

The error term in the equation can be interpreted as an uncertainty in the production process itself, or simply as an imperfection in the employer's ability to precisely monitor an individual's output. Measurement difficulties may be present in fields such as academics, medicine, and management; unobservable output may present problems in industries in which employees work within a group and can shirk some responsibility, thus making individual output difficult to discern. Finally, there may be a number of industries in which output is observable ex post but the production process contains uncertainty, making it difficult to determine ability strictly from the observation of output. Even for a professional athlete, whose performance is easily measured and observed, ability may not translate directly into performance; players have slumps and hot streaks making it difficult to accurately predict output. The former interpretation concerning observational imperfections provides an opportunity to analyze the traditional statistical discrimination assumption.(6)

Each individual falls into a distinct group which may be differentiated from the remainder of society by the employer. Although the employer cannot observe the individual's true ability level, A, the employer does hold some prior beliefs concerning A conditional on the individual's group. These beliefs, which are known to the employee and conditional on the employee's group, are expressed as

[Mathematical Expression Omitted].

Employers choose labor to maximize expected profits; thus, assuming perfect labor markets, the wage in each period equals the expected value of the employee's output. If we assume that the employee supplies a fixed quantity of labor, the wage is simply the value of the output the employee is expected to produce. Normalizing the product's price to 1, the wages will be

(2) [w.sub.1] E ([Q.sub.1]) = [E.sub.1]([H.sup.[Beta]] [A.sup.[Alpha]]),

[w.sub.2] = E ([Q.sub.2]) = [E.sub.2]([H.sup.[Beta]][A.sup.[Beta]]).

In order to determine these wages, we must consider the employee's problem and the information that employers have about this optimization process. A risk-neutral employee supplies one unit of labor each period; this individual chooses educational attainment in the initial period to maximize expected lifetime earnings:

[Mathematical Expression Omitted]

where C(H) is the educational cost and [Delta] is the subjective discount factor; both are unobservable to the employer.

Note that employers are unable to learn the true value of A, ability, simply by observing the employee's choice of H, education. Without knowledge of educational costs or the individual's discount factor, it is impossible to deduce the true value of A. Thus, the employer will simply offer wages based upon the observed value of H and the expectations of A. These expectations, in the first-period, are simply a function of the priors; in the second-period, the Bayesian employer updates his beliefs based upon the observed first-period output. This updating process is known to employees. Table I provides a list of variables and by which individuals they are observed.

In maximizing the present value of expected wages, employees recognize that the initial investment in education will increase expected output in both periods, thus raising the expected wages. In addition, increased education raises the expected first-period output and therefore the expected ability in the second period; this, in turn, increases the expected wage in the second-period. Given the initial prior on ability, A, employees can compute the expected wages in each period as a function of educational choice.

The first-period wage for an employee in group i is simply: [w.sub.i] = E([Q.sub.1]) = [H.sup.[Beta]]E([A.sup.[Alpha]]) where the expectation of A is derived from the prior for the group. Since A is assumed to be lognormal, E([A.sup.[Alpha]]) = exp([Alpha][Mu] + (1/2)[[Alpha].sup.2] [Mathematical Expression Omitted]). Also, given that [Mu] and [Mathematical Expression Omitted] are common knowledge, all employees know the wage they will be offered in period one as a function of their choice of education and, with knowledge of the Bayesian updating, can form an estimate of the expected wage in period two.

Aware that second-period wages will depend upon the updated beliefs of their ability based upon observed output, employees must consider the updating process when choosing education in the initial period. Given output in period one, the second-period wage is

(4) [w.sub.2] [where] [Q.sub.1] = [H.sup.[Beta]]E([A.sup.[Alpha]] [where] [Q.sub.1])E([Epsilon]).

The second-period wage depends on the employer's reaction to deviations from the expected wage. In general the employer attributes some portion of the deviation to the stochastic element of production with the remainder acknowledged as the result of higher or lower than perceived ability. We model this updating of ability through a Bayesian updating process. From the production function,

ln[Q.sub.1] - [Beta]lnH + [[Sigma].sup.2]/2 = [Alpha]lnA + ln[Epsilon] + [[Sigma].sup.2]/2.

Given the distributions of lnA and ln[Epsilon],

[Mathematical Expression Omitted],

[ln[Q.sub.1] - [Beta]lnH + [[Sigma].sup.2]/2 [where] [Alpha]lnA] [similar to] N ([Alpha]lnA, [[Sigma].sup.2]).

From the observation of output, Bayes rule implies the posterior belief of ability:

(6) P([Alpha]lnA [where] [Q.sub.1])

= P([Q.sub.1] [where] [Alpha]lnA)P([Alpha]InA)/P([Q.sub.1)].

Using the distributions of equation 5, we can determine the distribution P([Alpha]ln A [where] [Q.sub.1]):(8)

[Mathematical Expression Omitted]

where

[Mathematical Expression Omitted].

[TABULAR DATA FOR TABLE I OMITTED]

Since A is lognormal,

[Mathematical Expression Omitted].

From the equation above, we see that expected ability is a weighted average of prior beliefs and the unexplained portion of output. As [Mathematical Expression Omitted] rises, the weight on prior beliefs diminishes, and as [Mathematical Expression Omitted] approaches infinity, the influence of the prior ([Mu]) disappears. The same effect holds true for unexplained output; unexplained output's influence on expected ability increases as the variance of output declines. Thus the new assessment of ability depends critically on the strength of prior beliefs ([Mathematical Expression Omitted]) and the variance of the unobserved component of output ([[Sigma].sup.2]).

Substitution of expected ability into 4 yields the employee's expected wage in period 2:

[Mathematical Expression Omitted],

where

[Mathematical Expression Omitted].

[K.sub.1] denotes the constant dependent on the factors [Mu], [Mathematical Expression Omitted], [[Sigma].sup.2], H, which are deterministic and known to all parties.

Given the employer's wage offers which are based upon H and [Q.sub.1] (education and output), the employee chooses an educational level knowing how it will directly affect wages [w.sub.1] and [w.sub.2] as well as how it will interact with the true ability level to affect [Q.sub.1]. In the beginning of the first period of the worker's life, when the education level is chosen, the level of output in period one is unknown. Knowing their true level of unobservables, employees form expectations of [Q.sub.1] and therefore [w.sub.2] as a function of educational choice. Hence, the employee expects wages equal to

[Mathematical Expression Omitted].

Since output follows a lognormal distribution, ln[Q.sub.1] [similar to] N([Beta]lnH + [Alpha]lnA - [[Sigma].sup.2]/2,[[Sigma].sup.2]), it follows that

[Mathematical Expression Omitted],

and the second-period wage simply becomes

[Mathematical Expression Omitted].

Comparing the employer's expectations of output with unconditional expectations of output, we find that an employee is undervalued in periods one and two respectively if:

Period 1: [Mathematical Expression Omitted],

Period 2: [Mathematical Expression Omitted].

If ability exceeds the belief by enough to satisfy these conditions, then an employee will be undervalued and may have a diminished incentive to invest in human capital.(9) Similarly, lower ability employees will be overcompensated and may have some incentive to overinvest. Optimizing the present value of earnings yields the first-order condition:

[Mathematical Expression Omitted].

Equation 10 reflects the fact that the worker chooses a level of education such that the marginal benefit in terms of increased wages in periods one and two equals the marginal cost of education.

Totally differentiating this expression yields the following:

(11) [Delta]H/[Delta][Mu] [greater than] 0

[Delta]H/[Delta]A [greater than] 0

if [Mathematical Expression Omitted],

[Delta]H/[Delta][[Sigma].sup.2] [less than] 0.

Thus an employee's wages and, therefore, choice of education are increasing not only in her own ability but also in the employer's initial expectation; wages and education are decreasing, however, in the variability surrounding output when the employee is sufficiently undervalued in period one. In other words, when uncertainty surrounding output rises, the employer is less willing to attribute high or low levels of output to ability rather than chance. Consequently, employers adjust their beliefs more slowly as output variance rises. Thus, an employee that is initially undervalued, a high-ability employee, will have a diminished incentive to attain human capital as output variance rises.

The model also predicts human capital and wage compression for groups initially undervalued by employers. Analysis reveals that the impact of ability on human capital accumulation increases with [Mu] ([[Delta].sup.2]H/[Delta]A[Delta][Mu] [greater than] 0). This result implies that worker ability increases human capital accumulation by a smaller amount in groups facing a lower prior assessment of ability by employers. Because wages rise with human capital and [Mu], wages also rise less with increases in ability for undervalued groups.

Previous statistical discrimination models can be viewed as the special case of this model in which the initial distribution of beliefs for both groups is the same, but this output variance (or signal variance) is larger for the minority. Phelps's [1972] seminal paper on statistical discrimination concludes that high-ability minorities are underpaid with respect to their majority counterparts while low-ability minorities are overpaid. The literature provides two avenues for a lower average wage to emerge for the minority group. Aigner and Cain [1977] point out that if employers are risk averse, minorities are underpaid on average because less weight is attributed to a poorer signal and more uncertainty remains for the minority group. Lundberg and Startz [1981] argue that due to the educational incentives involved, even if the initial beliefs of the groups are identical, these effects do not necessarily translate into an equal average wage for each group. Because these models are nested within our model, these results apply.

The results of this section imply that lower expectations by the employer induce a lower level of worker productivity for each individual in the minority group by lowering the marginal benefit of investment in education. In the long run this lower productivity may serve to reinforce incorrect beliefs about the group as a whole. On the other hand, since the minority group's true ability is underestimated, despite diminished educational attainment relative to the majority, the group may still surpass expectations, and beliefs may be updated in a positive direction. Which effect dominates depends upon how employers process information pertaining to the group as a whole, as well as the relative sizes of the parameters. We consider this long-run dynamic process in section III.

III. LONG-RUN CONVERGENCE AND THE EVOLUTION OF GROUP PRIORS

In section II, we examined the method by which employers update their beliefs about a particular employee given prior beliefs conditional on the employee's group. The two-period model allows employers to use the observation of an employee's output to update beliefs about that individual's ability, but the model states nothing about the source of priors conditional on group ability and long-run equilibria. Theory suggests two factors, competition and updating of priors of group ability, might eliminate discrimination in the long run. We examine both factors assuming that the prior of lower ability exists for a minority group. Based on analysis of competition and updating of priors of group ability, this section attempts to determine whether discrimination can persist in the long run, and if so under what conditions it will persist. We begin with the issue of competition.

Competition

Thus far, we have considered only a representative employer. Suppose instead that there exist two types of employers: [n.sub.1] employers believe the mean ability level of the minority group to be [[Mu].sub.1] while [n.sub.2] hold a belief of [[Mu].sub.2]. The total number of firms is [n.sub.1] + [n.sub.2], and [[Mu].sub.1] [greater than] [[Mu].sub.2], or firms of type one value the minorities more highly. Suppose also that type-one firms have a lower marginal cost due to their more accurate assessment of the minority group. Define M[C.sub.1] = a[Q.sub.1] and M[C.sub.2] = b[Q.sub.2] where a [less than] b. Then in a competitive output market, type-one firms each produce P/a units while those of type two produce only P/b; assume this level of production requires [l.sub.1] and [l.sub.2] employees respectively.

If the labor market is competitive with a market wage dictated by type-two firms (found from equation 2), then all type-one firms will employ minorities and earn positive profits. However, if [n.sub.1][l.sub.1] [less than] total minority population, then these firms need only to offer the market wage which is less than their belief of true marginal product. In the short run, type-one firms earn positive profits and both groups of employers pay lower wages to minority workers. It is likely that the firms with more accurate beliefs (type one) are owned by members of the minority group. With no barriers to entry, type-one firms enter over time until they employ the entire minority population. However, if barriers prevent entry or limit the size of type-one firms, entry will not lead to convergence of wages across groups.(10)

Updating Prior Assessments of Group Ability

Just as employers update prior assessments of individual ability, priors of group ability should adjust over time based on the labor market performance of that group. The updating of group priors provides another opportunity to eliminate discrimination in the long run. Unfortunately, group priors likely depend on an enormous variety of factors, and modelling this process poses a daunting challenge. In this section, we present two potential methods of updating based on extensions of the model presented in section II. The long-run updating models yield no closed-form solution, so simulation results are used to check for cases where discrimination persists over long periods or even indefinitely.

The updating of priors on group ability depends on more than simply output of a single employee. Even if the employer increases his expectation of an employee's ability based on higher than expected output, it could easily be concluded that the observed individual is simply an outlier in the group. If the population of the group is large, observing output and education of a small number of employees provides negligible information and contributes little to the evolution of group priors. Instead, observations of the group as an entire unit determine this belief of mean ability level. This process depends crucially upon the information individuals have concerning the members of any particular group.

If employers knew the output and educational attainment of each member of the group, they could arrive at an updated mean level of group ability, and eventually employers would arrive at an accurate estimate. However, since employers cannot observe output and educational information for every member of the group, this updating process must rely upon observations of group averages. Two alternative cases in which employers have different degrees of information concerning group performance are considered; if less information is available, the updating process becomes more lengthy. Both cases assume that the employer updates his beliefs based on observations of mean levels of group output.

The long-run model assumes a large number of employers and employees, all with behavior characterized by the model described in section II. Consistent with that model, both employees and employers live two periods, with the employee purchasing education in the first period. Thus the initial priors of the employer determine wages in period one and simultaneously influence expected wages in period two and the employee's level of education. In period two and all periods afterward a new generation of employers and employees is born.

Given the limited information available, the new generation of employers forms a prior for a group's ability based on observation of average output and average educational attainment from the last period. Using the Cobb-Douglas production function from section II, new employers deduce the updated ability level to be

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] represent group averages derived in the appendix.

The updating process above is always imperfect due to the lack of information, however simulations reveal that the error is quite small if initial beliefs are accurate. We assume the new employers always observe and use the average output by each group to form the group priors. For CASE 1 updating, the new employer also observes the average level of education for each group and forms priors on group ability by substituting average output and education of the group into equation 12.

Employers easily obtain some measure of average output by a group through observing the status of group members in the workforce. However, employers may not observe a good measure of the average level of education in the minority group and could evaluate a group's ability levels without recognizing the differences in average educational attainment. Subgroup information is often more difficult to acquire, and even if it is available, psychological studies suggest that it may never be received by an individual. Studies reveal that information is processed differently across groups; information that is consistent with beliefs is valued more highly than that which is not.(11) In the case of education, employers observe laws guaranteeing equal access to education among all groups and educational opportunities targeted specifically at minority groups. The cumulative effect of such information may lead employers to discount lack of education as a source of low output. CASE 2 updating allows employers to ignore subgroup information on education and assumes that the new generation of employers form priors on minority ability using the average education level for the entire population rather than that of the minority group. We assume that the minority group is small relative to the population and thus employers generate CASE 2 minority group priors by substituting average minority output and average majority education into equation 12.

Long-Run Equilibria with Updating of Group Priors

Both proposed methods of updating group priors allow the employer to update through observations of average output and education; thus the priors of ability and output will vary over time. The model in section II provides a solution for current output and education conditional on the priors of employers in a given period. The appendix describes the equations for determining the priors of group ability formed by a new generation of employers conditional on current output and education. No analytic solution exists for the integrals described in the appendix although numerical solutions can be calculated. For this reason, we use simulations to determine output for updating procedures in both CASE 1 and 2. Intuition suggests that the updating of priors of group ability should move to eliminate discrepancies in pay between equally able groups. We varied the parameters in the simulations to check for cases where discrimination persists for long periods or possibly even indefinitely.(12)

The extent to which discrimination persists in the long run, (i.e. minority output lies below that of the majority) provides the fundamental question we seek to answer with the simulations. Although the choice of parameter values is largely arbitrary, such a finding for any set of parameters forces economists to consider the possibility that discrimination has long-run consequences. Simulations were performed by varying all parameters at different times to analyze the impact of each of the variables on convergence. For both CASE 1 and CASE 2 updating, we find that minority output converges to a level below that of the majority group for many parameter settings and at times slowly drops to zero. For other values of the parameters, the minority eventually converges to the majority, but only after a potentially long number of periods and thus a substantial output cost for society.

Since true ability is equal across groups, intuition suggests that updating should lead minority output and education to converge to that of the majority group; however, this is not necessarily the case. Table II summarizes the effects of a change in a single variable, holding others constant. Column 1 contains the change in variable considered. A yes in column 2 indicates that the rise or fall in the variable described in column 1 can slow convergence with CASE 1 updating. A yes in column 3 indicates that, in some cases, the change in column 1 can cause minority output to converge to a level permanently below that of the majority with CASE 1 updating. Columns 4 and 5 provide similar information for CASE 2 updating.

The results above provide several interesting findings common to both priors. Lowering [[Mu].sub.min] implies a larger initial gap between perceived and actual ability, and thus leads to a longer period of time before convergence to the long-run equilibrium. However, simulations reveal no case where lowering the prior assessment of ability alone leads to a long-run level of output for the minority group lower than that of the majority group.

Although the initial beliefs concerning expected group ability affect only the length of time until discrimination is eliminated, the strength of these convictions determines whether it will be eliminated at all. The prior belief concerning the variance of the group's ability ([Mathematical Expression Omitted]) provides an indication of how rigid an employer's initial assessment is. Simulations reveal that the more inflexible the beliefs, not only the longer discrimination persists, but the more likely it is to remain permanently.

If the parameters are such that minority output eventually converges to majority output with equal output variance ([[Sigma].sup.2)], [TABULAR DATA FOR TABLE II OMITTED] then an increase in the variance of minority output generally results in the convergence of minority output to a level permanently below that of the majority. Statistical discrimination models explain discrimination through differences in signalling ability. We find that as output variance (a signal of ability) rises, the minority may suffer discrimination permanently; the beliefs of minority ability converge to a level below that of the majority. An equal increase in [[Sigma].sup.2] for both groups slows the speed at which minority output converges, but results show no tendency for simultaneous changes in the equal variance of output for both groups to have any permanent impact on output.

The simulations also reveal significant differences between CASE 1 updating and CASE 2 updating. CASE 1 is more likely to converge and do so more quickly in every instance than is CASE 2. The greater the level of information that employers have concerning the minority group, the less likely discrimination is to persist in the long run. Whether this information directly enhances the perceptions employers hold about ability, or if it is simply the recognition that the group's educational incentives are diminished, such information will raise the likelihood that discrimination will be eliminated in the long run.

We also find that differences in technology can lead to persistent discrimination only if new employers form priors using CASE 2 updating and ignoring educational differences among groups. Higher values of [Beta] or lower values of [Alpha] (in other words, a larger importance of education relative to true ability) may push both minority output and wages to a lower level in CASE 2 or slow the speed of convergence when minority output converges to majority output. Since [Beta] represents the importance of education in the production function, it follows that the diminished educational incentives are more costly to the group as [Beta] rises. If discrimination causes early generations to purchase lower levels of education, early output tends to be low. This lack of education is amplified in production functions where education plays a larger role in production. For equality to occur, higher than expected ability must raise output above the expected level to continue the adjustment process. In an economy with a more education-intensive production function, ability's effect on output is small and thus convergence to the level of the majority is less likely and slower if it occurs at all.

The results in this section suggest that updating of group priors eliminates incorrect priors in many cases, but fails to lead minority output to converge to that of the majority group in other cases. Convergence also appears sensitive to prior and technological parameters and the form of updating. These results question the ability of updating to rapidly eliminate incorrect priors.

IV. FURTHER APPLICATIONS AND CONCLUSIONS

This paper provides an alternative explanation of discrimination in which differences in wage rates arise from different prior assessments of worker ability by the employer based on an employee's group type. Data from Urban Institute audits and the National Opinion Research Center provides evidence that society believes that black and Hispanic workers possess lower ability than do white workers. Section II models the impact of this prior assessment on output and human capital accumulation by minority workers. Results of the two-period updating process demonstrate that lower prior beliefs of worker ability reduce the worker's wage rate and level of investment in education. With no real difference in ability across races, the model generates lower wages and human capital accumulation in groups facing a lower initial assessment of ability.

The extension of the model in section III includes a process to update priors on average group ability and examine the evolution of these initial priors throughout generations. Simulations reveal that the employer's assessment of a minority's average group ability may or may not converge over time to that of the majority. We also find that even if discrimination is ultimately eliminated, the process may converge very slowly resulting in substantial output losses for society. From the various cases in the simulation, it is clear that greater information minimizes discrimination. The value is not only in offering information that might directly enhance the expected ability level; informing employers of the diminished educational incentives of an undervalued group also expedites the updating process.

Further, the model predicts more rapid elimination of discrimination when output is easily observed. For example, the model predicts more rapid convergence for minority athletes than for other minority workers whose output is more difficult to measure. The model also cautions policymakers that perceptions may have real effects on wages. Affirmative action may be warranted to assist in raising initial returns to human capital investments and to accelerate the updating process. Note, however, if affirmative action causes employers to increasingly discount achievements of minority groups, the policy may have an unintended adverse effect; further research is needed to understand the net impact of affirmative action on the evolution of priors.

The introduction of perceptions and Bayesian learning offers a new approach to understanding the influence of prior beliefs on the persistence of discrimination. This analysis has applicability beyond an enhanced understanding of discrimination. For example, the model predicts depreciation of credentials over time as employers use output observations to update initial perceptions; this minimizes the usefulness of tracking graduates for any significant length of rime beyond their initial employment. There is also some suggestion that individual career paths may be affected by expectations. Although discrimination may not be present in the form of differential wages for similar occupations, prior beliefs may induce individuals to sort themselves toward occupations in which they are expected to excel. Finally, any time an individual's contribution is undervalued due to an incorrect assessment, diminished incentives might have long-run effects. Certain types of education or labor market skills may be undervalued, measures to reduce risk for insurance purposes may not be appropriately rewarded, or consumers' prior assessments may underestimate the quality of some brands; the model in this paper can assist in the analysis of a variety of these problems.

Applying the model to other problems may explain recent empirical findings. In a sample of young workers, Bratsberg and Terrell [1994] find that black workers earn much smaller returns to general labor market experience than white workers, but returns to tenure on the job appear higher for black workers. In the context of this paper, this result reflects the fact that workers staying with the same employer benefit from an updating of the employer's assessment of ability. Adapting the model to examine the influence of an individual's appearance on priors may also explain Hammermesh and Biddle's [1994] results indicating a premium for beauty in the labor market.

Finally, extensions of the model may further explain discrepancies in wages between groups. An analysis of the incentives for employers acting under a variety of market structures to acquire such information would offer a valuable extension of this work. Search theory models of the match between minority worker and employer or models analyzing the incentives to shirk across races when priors differ offer an interesting query. Empirical tests comparing the cost of changing employers across race may also lead to an increased understanding of the role of priors in discrimination. Although many questions remain, the paper demonstrates that prior opinions of employers about groups influence the human capital decisions of employees that may provide gaps between races lasting far into the future.

APPENDIX

Expected group averages can be found as a function of both the true distribution (with correct mean and variance) as well as the initial beliefs. According to equation 10, the employee's educational decision is found as a function of her true ability level as well as the employer's beliefs about the group to which she belongs.

[Mathematical Expression Omitted]

Thus, the expected output of employee i with true ability level [A.sub.i] is

[Mathematical Expression Omitted]

Given these individual levels of education and expected output, the averages for the group depend upon the distribution of true ability levels denoted f(A).

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted]

1. The General Social Survey consists of personal interviews of randomly selected households in the United States conducted by the National Opinion Research Center at the University of Chicago.

2. See Turner, Fix, and Struyk [1991] and Cross [1991] for details.

3. See Firth [1982].

4. See Almeida and Kanekar [1989] and Wong, Derlega and Colson [1988] among others.

5. Although analysis of the match of employer and employee poses an interesting related question, that problem is beyond the scope of this paper. There may simply be too few employers with accurate beliefs, or it may be that search costs prevent minority employees from avoiding employers with low prior opinions entirely. The evolution of group priors is addressed in section III.

6. Note that if employers compare random effects across employees to identify shocks common to the firm, the variance may be reduced, but it seems unlikely that the distribution would become degenerate in general.

7. The lognormal distribution implies that ability is always positive and is negatively skewed.

8. See Berger [1980, 92-96] for details and examples conjugating normal distributions.

9. The employee may be undervalued in one period and overvalued in another. The employee's decisions depend on the present value of total wages.

10. Evans and Jovanovich [1989] conclude that liquidity constraints serve as a significant barrier to most entrepreneurs who are likely to start a business. Bates [1989] finds that nonminority firms locating in minority areas and minority firms in general have a lower probability of obtaining loans than other firms. These empirical facts suggest that firms employing minority workers do face additional barriers to entry in the U.S. economy. Search costs, incomplete or costly information, and oligopoly or monopoly markets, discrimination by consumers, or perceptions of inferior quality in minority-produced goods may also serve to slow or prevent the elimination of firms with incorrect prior beliefs.

11. Numerous psychological studies including Kahneman, Slovic and Tversky [1982] indicate that individuals use stereotypic information rather than all available information to arrive at an impression. For example, Martin and Grubb [1990] find that African American juvenile offenders are more likely to be punished than their white counterparts who are more likely to receive counseling. The offense of the white youth is attributed to a psychological disorder deserving of treatment rather than a cultural bias toward violence. Without controlling for income, education and other social factors, individuals often conclude that members of a particular group are naturally more violent.

12. We simulate the model by sampling 100,000 individuals in each period t, with ability drawn with In [A.sub.i] [similar to] N(-.5, 1) for the majority group. After choosing the parameters, we calculate [H.sub.maj0] and [Q.sub.maj0] for individual i in the first period based on the solution to equation 10. The sample averages, [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are then calculated for use in the updating procedures. The procedures are repeated to obtain [Mathematical Expression Omitted] and [Mathematical Expression Omitted] for the minority group using the same distribution for true ability, ln [A.sub.i] [similar to] N(-.5, 1). Using the averages at time zero, the new generation of employer's group priors for period 2 are calculated for both updating procedures. Given these new priors, the method is repeated to determine average output and education over the first fifty generations.

13. It was mentioned above that the beliefs of the majority's ability may not necessarily converge to the true mean. Due to the imprecise nature of society's adaptation process, it is unlikely that employers will accurately determine average abilities. However, the beliefs about the majority (a group about whom the initial beliefs are correct) stabilize almost immediately at a level very near the true mean. It is also true that this long-run point of convergence is very robust to changes in the speed of adjustment.

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AMY FARMER and DEK TERRELL, Assistant Professor, University of Tennessee, and Assistant Professor, Kansas State University. We wish to thank Bernt Bratsberg, Hae-Shin Hwang, James Ragan, Richard Startz, and two anonymous referees for valuable comments.