Statistical Theories of Discrimination in Labor Markets Author(S): Dennis J

Statistical Theories of Discrimination in Labor Markets Author(s): Dennis J. Aigner and Glen G. Cain Reviewed work(s): Source: Industrial and Labor Relations Review, Vol. 30, No. 2 (Jan., 1977), pp. 175-187 Published by: Cornell University, School of Industrial & Labor Relations Stable URL: http://www.jstor.org/stable/2522871 . Accessed: 24/02/2012 10:02

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http://www.jstor.org STATISTICAL THEORIES OF DISCRIMINATION IN LABOR MARKETS

DENNIS J. AIGNER and GLEN G. CAIN

E CONOMIC discrimination has been diffi- perspective suggested by Kenneth Arrow, cult to explain by means of standard John J. McCall, Edmund S. Phelps, Mel- neoclassical economic models that assume vin W. Reder, and A. Michael Spence, all pervasive competition. Why, after all, of whom focused on certain implications should two groups of workers who have of employer uncertainty about the produc- the same productivity receive different re- tivity of racial (or sex) groups of workers, muneration? The challenge to explain this particularly in the context of hiring and phenomenon is posed most sharply by the placement decisions., This paper offers marked differentials in wages and earnings several models that clarify the meaning of between blacks and whites and between economic "statistical discrimination," sim- men and women differentials that remain plify the theory, and yield plausible em- substantial despite diligent efforts to con- pirical implications. On the other hand, trol for supply-side productivity traits. the paper also identifies several shortcom- This paper examines that issue from a ings of "statistical discrimination" models; shows that the often-cited Phelps model does not constitute economic discrimina- Economic discrimination in labor markets is con- ventionally defined as the presence of different pay for workers of the same ability. This paper analyzes iKenneth Arrow, "Models of Job Discrimination" that problem with the aid of a simple stochastic and "Some Mathematical Models of Race in the model in which employers hire, place, anad pay Labor Market," in Anthony H. Pascal, ed., Racial workers on the basis of imperfect information about Discrimination in Econ omic Life (Lexington, Mass.: their abilities. The available information consists Lexington Books, D. C. Heath and Co., 1972), pp. of both group membership (black, white; male, fe- 83-102 and 187-204; and "The Theory of Discrimi- male) and information about individlual perform- nation," in Orley Ashenfelter and Albert Rees, eds., ance on some fallible indicator of ability (e.g., a Discrimination, in Labor Markets (Princeton, N. J.: test). Several types of economic discrimination with- Princeton University Press, 1973), pp. 3-33; John J. in the context of competitive market assumptions McCall, Inicomie Mobility, Racial Discrimination, are examined by means of several models, anid the and Economic Growth (Lexington, Mass.: Lexington empirical plausibility anad implications of these Books, D. C. Heath and Co., 1972); and "The models are discussed. The authors conclude that Simple Mathematics of Information, Job Search, the statistical theories are unlikely to provide an and Prejudices," in Anthony H. Pascal, ed., Racial important explanation of labor market discrimina- Discrinsination7 in Economic Life (Lexington, Mass.: tion uender conventional neoclassical assumptions. Lexington Books, D. C. Heath and Co., 1972), pp. Dennis J. Aigner and Glen G. Cain are both Pro- 205-44; Edmund S. Phelps, "The Statistical Theory fessors of Econormics at the University of Wvisconsin. of Racism and Sexism," Amierican Economiic Review, They express their gratitude for the extensive com- Vol. 62, No. 4 (September 1972), pp. 659-61; Melvin ments of Arthur S. Goldberger. This research was W. Reder, "HunmanCapital and Economic Discrimi- supported iln part by funds granted to the Institute nation," in Ivar Berg, ed., Hu-nman Resources and for Research on Poverty at the University of Wis- Economic Welfare; Essays in Honor of Eli Ginzberg consin-Madison by the Office of Economic Oppor- (New York: Columbia University Press, 1972), pp. tunity plursuant to the Econoomic Opportunity .-\ct 71-88; and A. Michael Spence, "Job Market Signal- of 1964 (GGC), and by NSF grant GS-30005 (DJA). ing," Quarterly Journal of Economics, Vol. 87, No. 3 The opinions expressed here are those of the (August 1973), pp. 355-74 and Market Signtaling au thors-EDITOR (Cambridge, Mass.: Harvard University Press, 1974). 175 176 INDUSTRIAL AND LABOR RELATIONS REVIEW

tion, statistical or otherwise; and concludes By normal distribution theory, Equation that these models probably do not explain 2 is the least squares regression, expressing most labor market discrimination. q in terms of a group effect [(1 - .y)a] and an individual effect (yy). It is useful to think The Basic Model of Equation 2 as a conditional expectation We introduce the statistical model of from a linear population regression func- discrimination with the version by Phelps, tion: contained in an article with the imposing title, "The Statistical Theory of Racism (4) q = (1 -y)a + yy + u' and Sexism.'"'2 The essential features are where u' is the usual well-behaved error as follows. In the hiring and placement of term. In principle, the regression is opera- workers, employers base their decisions on tional, because employers could measure some indicator of skill, y, (such as a per- the actual q of a worker on the basis of a formance test) that measures the true skill post hoc evaluation of the worker's per- level, q. The terms "ability," "productiv- formance. ity," and "skill" will be used interchange- Now, consider two differentiated groups ably herein. In practice, y would un- of workers, say whites and blacks, with doubtedly involve a number of measures, possibly different means, aw and a B, and but the assumption here will be that a possibly different variances of q and u. single test score is all that is measured by (Although we use whites and blacks y. The measurement equation is throughout, our discussion is equally ap- plicable to males and females.) The em- (1) y = q + u, ployer is assumed to pay a worker an where u is a normally distributed error amount, q, based on the specific informa- term, independent of q, with zero mean tion available for each group and indi- and constant variance; q is also assumed vidual (see Equation 2): to be normally distributed with a mean equal to a and with a constant variance. (5a) qW= (I -yW ) W + WyW Employers can observe the test score, y, (5b) AB= (1 B)B + ByB. but they are interested in this only insofar as it gives them information about the The slope, y, will generally differ for the unobservable variable, q. Thus, the imme- two groups if the variances of q and u diate interest of the employer is the ex- differ, as shown by Equation 3.3 pected or predicted value of q, which we The nature of the hiring and placement shall label 97. process requires that the employer make a subjective assessment of a worker's skill. The expected value of q, given y (E(qjy)) is: We assume that this assessment of q, given y, will equal the expectation of q, condi- (2) q=E(q y) = (I - y) a + yy, tional on y. This assumption is in keeping with wage-maximizing behavior by work- where a is ithe group mean of q (and y) and 3Any random error in y as a measure of q is (3) ly Var(q) Cov(qy) represented by u. A systematic error in y as a meas- groups could Var(q) + Var(u) Var(y) tire of q for one or the other racial also be introduced, but this would not add substan- tively to our analysis. For example, if blacks scored = [ Cov(q,y)2 = r2] below whites by some constant amount for any q LVar(q) Va r(y) value, a negative intercept term could be added to where r2 is the squared coefficient of cor- Equation 1. However, a simple transformation in which this intercept difference was added to qB relation between q and y. In classical test would restore comparability in the q values for both score theory, y is the reliability of a test groups according to a new set of equations like score, y, as a measure of the true score, Equations 5a and 5b. This type of bias in the test q. Clearly, 0 < y < 1. instrument would not, by itself, affect the reliability of the instrument and is, therefore, inconsequential. The unreliability of y as a measure of q, however, 2American Economic Review (September 1972). is another matter, as we demonstrate later. STATISTICAL THEORIES OF DISCRIMINATION 177 ers and profit-maximizing behavior by em- interpret the "statistical theory of dis- ployers, since a job market function of crimination" as a theory of "erroneous" or employers is to assess (or predict) factor "mistaken" behavior by employers, as have productivity, given the costs of available some economists,6 is therefore without information, and to pay the factors of foundation. Furthermore, Andrew I. production accordingly. Employers who Kohen errs by claiming that, "Phelps are inefficient in this function will tend to [1972] demonstrates that irrespective of the be weeded out by the "market mecha- validity of using sex" as a proxy variable nism" of competition. As Spence con- for productivity characteristics of the job cludes in considering a similar model of applicant, "discrimination is the out- employer behavior, "In an equilibrium the come."7 Phelps demonstrates no such re- subjective distribution and the one im- sult. plied in the market mechanism are identical," assuming that neither group of Definitions of Economic Discrimination workers is completely isolated from em- Economic discrimination is said to exist ployers.4 when workers do not receive pay or remu- To anticipate a possible source of con- neration commensurate with their produc- fusion, we should emphasize that the tivity-when, in short, equal productivity assumed correspondence between the em- is not rewarded with equal pay. Our focus ployers' subjective conditional expectation is on labor market discrimination, which of q and the conditional expectation of means that we will generally assume that realized q in the market implies that the the worker's pre-labor market investments group means (the -as) are estimated without and endowments are given. We adopt the bias. In particular, employers will not prevailing convention of defining produc- persist in believing that aw > aB if, in tivity in terms of physical output or actual fact, aw aB. If employers mistakenly be- job performance, acknowledging, however, lieve aw > aB, then they will mistakenly that this definition can be ambiguous. As overpay whites relative to blacks, and we others have pointed out, discrimination may doubt that such mistaken behavior against a particular group of workers can will persist in competitive markets. In- always be explained away by attaching a deed, as an explanation of discrimination cost to some characteristic of the group against blacks, a theory of discrimination that is not directly related to their work based on employers' mistakes is even abilities. harder to accept than the explanation It is necessary to distinguish group dis- based on employers' "tastes for discrimi- crimination from individual discrimina- nation," because the "tastes" are at least tion that is independent of group mem- presumed to provide a source of "psychic gain" (utility) to the discriminator., To discriminating employers drive the discriminating employers out of business in the long-run competi- 4Spence, "Job Market Signaling," pp. 360-61. tive equilibrium. See Gary S. Becker, The Econom- 5It is more precise to say that the wage policy ics of Discrimination (Chicago: University of Chi- of employers whose subjective assessment of q per- cago Press, 1971), 2d ed.; and Arrow, "Models of sistently differs from the expected value of actual Jol) Discrimination" and "The Theory of Discrimi- q would not be viable unless all current and poten- nation." tial employers made the same error. Otherwise, the 6Cynthia B. Lloyd, "The Division of Labor Be- forces of competition would lead to an expansion tween the Sexes: A Review," in Cynthia B. Lloyd, of output by employers who erred the least (or not ed., Sex, Discrimnination, and the Division of Labor at all) at the expense of those who erred the most. (New York: Columbia University Press, 1975), pp. Thus, imposing a wedge between the subjective 1-24 and Harriet Zellner, "Discrimination Against expectation of q and the actual expected value of q Women, Occupational Segregation, and the Relative is analytically equivalent to imposing employers' Wage," Amierican EcoionomicReview, Vol. 62, No. 2 "tastes for discrimination" as a wedge between the (May 1972), pp. 157-60. employers' subjective evaluation of the worth (or 7Andrew I. Kohen, section entitled "Differentia- productivity) of a worker and his actual worth. tion in the Market," (pp. 1256-62) in Hilda Kahne, As both Gary S. Becker and Arrow have made clear, "Economic Perspectives of the Roles of Women in variance in tastes for discrimination among employ- the American Economy," Journal of Economic Lit- ers will lead to a situation in which relatively non- erature, Vol. 13, No. 4 (December 1975), pp. 1249-92. 178 INDUSTRIAL AND LABOR RELATIONS REVIEW bership. Race or sex discrimination is a Perhaps not so obvious is the fact that consequence of group discrimination; dis- group discrimination may be absent even crimination among individuals within a though the wages, q, of blacks and whites group, on the other hand, carries no pre- with the same ability, qo, are not generally sumption of group discrimination nor, equal. Generally, E(qB qo) # E(qtwqo)-ex- therefore, of race or sex discrimination. pressions obtained by taking expecta- Group discrimination in labor markets is tions conditional on q in Equation 2. Thus: evident when the average wage of a group E(q lq) = (1 -My)a + yE(yXq). But E(yq) = is not proportional to its average produc- q; so tivity. On this basis, our findings reveal (6) E(q Iq) = (1 - -y)a + 'yq, that even nondiscriminatory practices by employers may yield a discriminatory out- and -y and -a may differ for blacks and come: groups that have the same average whites. However, there need not be any ability may receive different average pay. average difference in compensation be- Within-group or individual discrimina- tween groups, because the individual in- tion is inevitable. The fact that within a equalities of the above expectation over group, all individual workers with the the range of q may be offsetting between same true ability will not receive the same whites and blacks. These points are dem- pay is clearly shown in Equation 4, in onstrated and clarified in our analysis of which q is not exactly predicted by y. To particular models. illustrate that this does not necessarily involve group discrimination, consider a case A Phelps Model in which all college graduates are offered The implications of Phelps's model, out- one wage, equal to their average produc- lined previously, depend on assumptions tivity and higher than the wage offered to about the average abilities, the variances of all high-school graduates. Although indi- ability, and the variances of measurement vidual discrimination occurs within each error for the two groups-blacks and schooling group (except in the unrealistic whites. Phelps makes three assumptions, case of zero variance in ability within each each of which we question at some point group), no presumption of between-group in our discussion. In most of his paper he discrimination is warranted. The distinc- assumes, first, that uIv and uB have the tion between these two types of discrimina- same variances; second, that the variance tion has not always been clear in the of qIv is less than the variance of qB; and, literatures third, the the average ability of blacks is lower than that of whites.9 SFor one example, see Francine B. Blau and Carol L. Jusenius, "Economists' Approaches to Sex Segre- gation in the Labor Market: An Appraisal," in tory" says no more than that the decision maker, Martha Blaxall and Barbara Reagan, eds., Womien like the rest of us in our decisions, lacks perfect and the Workplace (Chicago: University of Chicago knowledge! Finally, consider the following quotation Press), pp. 181-99. In discussing sex discrimination, hy Lester C. Thurow in Generating Inequality (New Blau and Jusenitus claim that "stereotyping, the York: Basic Books, Inc., 1975): treatment of each individual member of a group [Statistical discrimination] occurs whenever an indi- as if he/she possessed the average characteristics of vidual is judged on the basis of the average charac- the group" is "appropriately defined as a form of teristics of the group, or groups, to which he or discrimination even if the employers' perceptions of she belongs rather than upon his or her own char- the average [group] . . . differential are correct" acteristics (p. 172). (p. 194). In another source Michael J. Piore, in dis- The statement is meaningless unless the following cussing race discrimination, labels as statistical dlis- questions are answered: Are the group characteristics crimination a situation in which job candidates are correctly judged on average? Is a person's age or rejected because they do not possess traits that test score, for example, an "own" characteristic, or "tend to be statistically correlated with job per- does it also refer to the group (population) with formance" (p. 56). See Michael J. Piore, "Jobs and that particular value or score? Is statistical discrimi- Training," in Samuel H. Beer and Richard E. Bar- nation averted only when all relevant "own" char- ringer, eds., The State and the Poor (Cambridge, acteristics are known? Mass.: Winthrop Press, 1970), pp. 53-83. If the 9The lower mean value of qB is represented in decision rules are correct on average, however, then Phelps's paper by a dummy variable for race (1 if Piore's assertion that the employer is "discrimina- black) with a negative coefficient, but the presenta- STATISTICAL THEORIES OF DISCRIMINATION 179

It is disconcerting that Phelps assumed q a difference in average abilities at the out- set, because discrimination is defined as B differences in pay for workers of the same ability, or, equivalently, a difference in pay that is not related to a difference in ability. Sometimes, of course, the assumption of equal ability is facilitated by narrowing the race (or sex) comparisons to subgroups of workers of the same age, schooling, or Black experience. For expository reasons, we will .SWhite initially assume equal average abilities for the two groups: a B = aon = a. The case of 450 unequal average abilities will be examined a ~~~~~Y later. The other two assumptions in the basic Figure JA. Predictions of Productivity (q) Phelps model, Var(uB) = Var(uTV) and by Race and Test Score (y), Assuming a Var(qB) > Var(q'), dictate that the slope, Steeper Slope for Blacks. y, of the q-on-y regression in Equations 2 or 4 is steeper for blacks than for whites. The apparent definition of economic This is clear from Equation 3. It means discrimination revealed by Figure IA, and that the test score, y, is a more reliable which we must ascribe to Phelps, is "dif- predictor of q for blacks than for whites. ferent pay for the same y scores." But since Accepting this unusual result for the mo- y scores are intended only to indicate ex- ment, let us examine its consequences. pected productivity, it is discrimination As Phelps noted, "at some high test score with respect to q and not y that is eco- and higher ones the black applicant is pre- nomically relevant." Even a legal require- dicted by the employer to excel over any ment that payments be equal for equal y white applicant with the same or lower scores would contribute nothing to the test scores."'0 Figure 1A shows this-as well overall improvement of the status of as the corollary proposition that at low blacks, since, as is clear in Figure IA, what test scores the white worker is predicted to blacks would win at the lowest q values excel over a black worker with the same (relative to whites) they would lose at the test score. Low-scoring whites being paid highest q values. more than low-scoring blacks is offset by In any event, the assumption that yB > high-scoring whites being paid less than 71V or that the y score is a more reliable high-scoring blacks. In what sense, then, indicator of q for blacks than whites-is does this picture depict racial economic discrimination? Each worker is paid in ac- "Recall that the lefinition of economic (liscrirni- cordance with his expected productivity, nation as wage (lifferences among workers with the based on an unbiased predictor. Moreover, same produlctivity implies pervasive withlin -group the two groups, which have (by assump- discrimination, given the conditional variance in q. tion) the same mean ability, receive the Thus, some whites with a given test score, yo, who same mean are hired at a wage commensurate witlh E(q yo), wages. will have an actual q that is greater than E(q yo); others will turn out to have an actual q that is tion is not entirely clear. His equation (5') would less than the expected value. We could fairly say appear to represent a single regression model for that the former (positive residuals) are discrimi-inated workers of both races, in which z is an additive against and the latter (negative residuals) receive term that is equivalent to a dummy variable (O if preferential treatment. Presuimably, there is more white). However, the additivity of z gives the wrong of this sort of discrimination at the time of initial impression, because the all-worker regression re- hiriings than after the elapse of time, when the quires a zy interaction to capture Phelps's assump- experience of workers and employers will narrow tion that the slope of y on q is different for the the conditional variance of q, given what will then two races. be an augmented ). But, as stated earlier, a within- ?OPhelps,"The Statistical Theory of Racism and group conditional variance does not imply discrimi- Sexism," p. 661. nation between groups. 180 INDUSTRIAL AND LABOR RELATIONS REVIEW

unappealing. The Scholastic Aptitude Test q has been found, for example, to be a less reliable indicator of college grades for blacks than for whites.12 At the same time, B we see no reason to assume the variance in true ability differs for the two races, although arguments can be made for a difference in either direction.'1 Moreover, an B1ack implication of the hypothesis that yB > yW is that the white-black differential in pay- .White. reflecting a differential in expected q- narrows (and eventually becomes negative) 0 ______as the y indicator increases. The bulk of 4 5 the empirical evidence points, however, to l x ~~~Y the opposite result: if y is measured by Figure lB. Predictions of Productivity (q) years of school completed or by years of by Race and Test Score (y), Assuming a experience-two of the most important Steeper Slope for Whites. and commonly used indicators of productivity-the empirical relation between y fore, each worker is paid according to his and earnings (or wages) shows blacks faring expected productivity, resulting in equal worse relative to whites as y increases.14 average wages for the two racial groups. A model that reflects this evidence and The only difference shown by I B is that assumes that the testing process is less re- whites with y scores above the mean reliable for blacks is shown in Figure 1B. ceive higher wages than blacks, and the We will examine implications of this speci- reverse is true for y scores below the mean. fication below, but the point here is that economic discrimination is no more evi- An Alternative Model dent in Figure lB than it is in lA. As be- Up to this point we have assumed ex- plicitly that the employers know E(qly) l2Robert L. Linn, "Fair Test Use in Selection," and implicitly that the dispersion of q Y Review of Educational Research, Vol. 43, No. 2 is costless. This is equivalent to assuming (1973), pp. 139-61. that q enters the profit function linearly, l3Blacks confront environmental restrictions on fulfilling their capacities, and this may lead to a or that the employer is risk-neutral with smaller variance of qB. On the other hand, perhaps respect to q. It is more realistic to permit whites face a more homogeneous set of environ- q to enter the profit function (or the mental determinants of q, which would make the "utility of profit" function) nonlinearly, variance of qTV smaller. Any number of possibilities suggest themselves. which would allow the correct decision 14For example, Finis Welch in "Education and rule for hiring labor to involve higher Racial Discrimination," in Orley Ashenfelter and moments of q. In the simple model Albert Rees, eds., Discrimination in Labor Markets adopted below, only the conditional vari- (Princeton, N.J.: Princeton University Press, 1973), ance of q, written = Var(q)(l -y), pp. 43-81, remarks: "It is well known that, on bal- Var(qly) ance, the ability of schooling to boost Negro earn- is required to reflect risk aversion and to ings has been less than for whites, at least for yield a theoretical explanation for eco- males" (p. 43). Weiss supports this finding and also nomic discrimination.'l finds that "scholastic achievement" (as measured by test scores) was a better predictor of earnings for white males than for black males. See Randall 15Theexpression for the conditional variance of q D. is Weiss, "The Effect of Education on the derived from normal distribution theory. It is an Earnings of analogue of the expressionfor the residual variance Blacks and Whites," Review of Economic Statistics, in a simple linear regressionas: (1 - r2) Var(depend- Vol. 52, No. 2 (May 1970), pp. 150-59. Finally, a flatter age ent variable). Thus, in the population regression, (experience)/wage profile is shown for Equation 4, we have Var(qy) = Var(u') = Var(q)- black males and women than for white males in Robert E. Hall, "Why Is the Unemployment Rate 72 Var(y) = Var(q) - [Va(q) 12 Var(y), using So High at Full Employment?" Brookings Papers on LVar(y)l Economic Activity, No. 3 (Washington, D.C.: The an expressionfor -yfrom Equation 3; so Var(qy) = Brookings Institution, 1970), p. 394. Var(q)(1- y). STATISTICAL THEORIES OF DISCRIMINATION 181

To simplify the problem, assume that conditional variance in q, given y, is labor is the only factor of production, that larger or smaller for black or white work- output is fixed, and that prices and wage ers is, therefore, crucial in determining the rates are exogenously determined. Thus, direction of discrimination. Assuming ra- profits, H, are solely a function of labor cial equality in Var(q), this question hinges services. Given the number of workers re- on the reliability of y as a predictor of q, quired to maximize the utility-of-profits namely y, and we have already suggested function, U(ni), the employer need only that the tests are less reliable for blacks. choose the type of labor-here the assumed Thus, yB < yW (as depicted in Figure 1B) equally productive B or W groups-to follows from the assumption that Var(uB) maximize U(q). > Var(uW).18 Several well-known utility functions re- Figure 2 shows the new y, q relationships sult in a decision rule that depends on the incorporating the risk factors, RB and Rw, variance of the argument. We adopt a and the assumptions, yB < 7W and aB = function used by Michael Parkin,16 which aw = a. The lines W and B are from for our purposes may be written: Equation 2 in conjunction with the assumption that aw = aB and yB < yW. The (7) U(q) = a becq b, c > 0, parallel lines, W-RW and B-RB are the risk-discounted counterparts representing whence E(q y) - k Var(qjy) or E(qjy) - R. To (8) E[U(qly)] = see that the figure reveals economic dis- a - bec E(qly)+c2/2 Var(qly),17 crimination against blacks in the sense that they receive lower pay on average for where a, b, and c are parameters of the the same expected ability, simply note the utility function and e is the base of the lower black value of (q-R), given y =a. natural logarithm. The expected value of q for both whites It is easily seen that maximizing and blacks equals a for y = a, and therefore E[U(qjy)] is equivalent to maximizing its the lower pay for blacks is entirely attrib- logarithm, which in turn is equivalent to utable to the larger R factor for blacks. In maximizing [E(qjy) - k Var(qjy)], where Figure 3, which graphs the conditional k = c/2. Let R = k Var(qly), which may distribution of q for the point were be interpreted as a risk factor. y = a = E(y) and where E(q) = a for both It follows that an employer with this races. The observed smaller conditional utility function will attempt to hire from variance of qw in Figure 3 (which would the group of workers that maximizes ex- be the same at any value of y) is, of course, pected productivity, q, discounted for risk. precisely the source of the larger size of This risk can arise from differing variances YW, given that Var(qB) = Var(qW). in the distribution of q, or u, or both. Sub- The risk discount borne by black work- stantively, the risk costs of variance in ers in the form of a lower relative wage worker abilities may stem from variance in could be interpreted in terms of the extra output within homogeneous jobs or from search costs employers would have to bear the costs of mistakes in assigning workers to reduce the conditional variance of qB to to heterogeneous job slots. For the sake of convenience we have adopted a conditional variance that does not depend on 18A lower slope for blacks would also result from the level of y, so that the risk factor is the assumption that Var(UB) = Var(uW) and Var(qB) constant over the range of test < Var(qW). Indeed, the risk discount, Var(qly), is scores (see symmetric with respect to Var(u) and Var(q) since, footnote 15). by manipulations of relations in Footnote 15, we see The empirical question of whether the Var(q~y = Var(q) Var(u) Var(q) + Var(u) 16Michael Parkin, "Discount House Portfolio and In our comparisons between blacks and whites we Debt Selection," Review of Economic Studies, Vol. cannot, however, interchange the terms "reliability" 37, No. 4 (October 1970), pp. 469-98. (= -y = r2) and "risk-discount" [= Var(qly)] unless 17Since q is normally distributed, e-cq is lognor- we hold equal either Var(q) or Var(u) for the two mal, and its expected value is e-cE(q)+(c2/2)Var(q). groups. 182 INDUSTRIAL AND LABOR RELATIONS REVIEW

Probability of qlE(y) =a - W

Black - Black ...... White ...... White

BRB E(q) =o qlE(y) =o R c,-RW - -- Figure 3. The Distribution of Productivity o,-RB_ _ RB (q), Given the Test Score y = E(y), by Race.

Note: = E(q) = E(qB) = E(qW) = q12 E(qjE(y)) E(yB) = E(yl).

450 ence is to retain the term "economic" in y Yo describing this type of discrimination, Figure 2. Predictions of Productivity (q) although such discrimination stems from by Race and Test Score (y) with "Risk- inadequate test instruments rather than Discounts" and Flatter Slope for Blacks. employers' acting upon their tastes for discriminating against black or female workers. equal Var(qlvly). However, the model does A second realistic feature is that the dif- not require any ad hoc assumptions about ferential wage or income advantage of the direct hiring costs being larger for white male workers increases as the indi- blacks compared to whites, although cost cator variable increases. In Figure 2, how- differentials may exist. The geographic ever, the wages of B workers exceed those segregation of black workers away from of 147workers with the same y scores for white employers (firms), for example, may Y < yo, and only if the risk penalty were as well impose extra search and informational large as q2- q would every W worker be costs upon black workers.19 paid more than every B worker for a Figure 2 represents a hypothetical given y score, over the whole range of y. model, of course, but it is consistent with We are not aware of any actual data re- our view of reality in two important re- vealing a smaller wage for Ws compared spects. First, as a consequence of actual to Bs for low scores of productivity indi- employer practices, economic discrimina- cators. Furthermore, although we have not tion against blacks, women, and other expressed the units in dollars, the em- groups does exist, resulting in group dif- pirical magnitudes of the differential ferences in pay despite equal group abili- (RB - RIV) borne by black workers-per- ties to perform on the job. However, if the haps 10 to 30 percent for the same number definition of ability includes reliability in of years of schooling completed-seem too test-taking-on grounds, perhaps, that this large to be rationalized by risk aversion. aptitude conveys useful information to em- Indeed, three reasons are suggested for ployers-then one could deny that eco- skepticism about the size of R.20 First, nomnic discrimination exists. Our prefer- large firms have some capacity to self- insure against risks of output variability 19For an analysis of this particular disadvantage or mistaken job assignments. In perfect to black workers, see McCall, "The Simple Mathe- capital markets, even small firms could matics of Information, Job Search, and Prejudices," and John F. Kain, "Housing Segregation, Black Em- "purchase" such insurance through various ployment, and Metropolitan Decentralization: A pooling devices. Second, dispersion in risk Retrospective View," in George M. von Furstenberg, Bennett Harrison, and Ann R. Horowitz, eds., Pat- ternes of Racial Discrimination, Vol. I: Housing 2OWc are grateful to Orley Ashenfelter, H. Gregg (Lexington, Mass.: Lexington Books, D. C. Heath Lewis, Donald A. Nichols, and Melvin AV. Reder and Co., 1974), pp. 5-20. for helpful suggestions about this section. STATISTICAL THEORIES OF DISCRIMINATION 183 aversion among employers leads to a situa- to multiple equilibria, and it then is a tion in which those with the least aversion small step to show that there may be dif- reduce the R discount by "bidding up" the ferent equilibria for two distinguishable wages of black workers, in the same man- racial groups of workers, even though the ner as employers with the lowest "tastes two groups have identical productivity dis- for discrimination" serve to equalize the tributions and face the same signaling cost wages of black and white workers of the per unit of y. In particular, if the thresh- same productivity. Finally, a large R fac- old level (y*) is higher for blacks than tor should activate a market for "test in- whites, but still low enough so that it pays struments" that are tailored to the separate high-ability blacks to acquire the y* signal, groups to achieve more nearly equal re- then high-ability white workers will obvi- liability. (For example, just as tests in their ously earn a higher net wage-the em- native language have been prepared for ployer's wage offer minus the costs of foreign workers, so test developers could attaining y*. devise ways to communicate more clearly The question is, however, whether this with members of minority groups.) The type of discriminatory situation is stable. wage differential should not exceed the If workers know their own productivity "signaling cost"-to use Spence's suggestive (or, equivalently, their costs of attaining term.2I y*), then the high-ability black workers will know they are being underpaid (or, Other Models of Discrimination more accurately, overtaxed for y signals), A second model that attempts to explain relative to high-ability white workers. It is the wage differential comes from A. not difficult to construct examples of ar- Michael Spence.22 He provides a dynamic rangements between individual high- equilibrium analysis in which group dif- ability black workers and employers, ferences in wages persist in competitive whereby the former agree (to the benefit labor markets. In his model employers are of both parties) to accept lower wages in uncertain about the workers' q values and return for signaling with a lower y * base their wage offers (qs) on the workers' eventually a y* as low as that for white y scores-which, like our y scores, do not high-abiilty workers.24In equilibrium there represent productivity skills, per se. Work- would be common wage offers, y attain- ers attain y scores by investing their time ments, and, therefore, common net wages and resources, and the Spence model as- for similarly productive white and black sumes that the cost of y is negatively cor- This assumption drives the related with q. 24A simple example of a breakdown in the Spence wage system, under plausible conditions, model of cliscrimination may 1)e obtained from to an equilibrium in which q is positively Glen G. Cain, Department of Economics, University correlated with y, so that the employers' of Wisconsin-handout entitled, "Criticisms and expectations are self-confirming and the Counter-Examples of the Spence Model of 'Market Failures' in a World of Job Market Signaling," workers' signaling behavior reproduces April 27, 1976. See also the suggestions of Stiglitz itself.23 and Thurow about "trial-period" wage offers by In the Spence model, however, the level workers as mechanisms by which a unique equilib- of y (call this y*) that distinguishes high- rium (or, at least, a nondiscriminatory equilibrium) use his can be achieved. See Joseph E. Stiglitz, "Theories from low-ability workers (to simple of Discrimination and Economic Policy," in George case of just two levels of productivity) is M. von Furstenberg, Ann R. Horowitz, and Bennett arbitrary over a certain range. This leads Harrison, eds., Patterns of Racial Discrimnination, Vol. II, Emnployment and Income (Lexington, Mass.: Lexington Books, D. C. Heath and Co., 1974), pp. 21Spence, Market Signaling. 5-26 and Thurow, Generating Inequality, p. 173. 22Spence, Market Signaling and "Job Market Sig- Still another source of possible attainment of a naling." unique equilibrium-this time from actions initiated 23Indeed, the informational feedback loop that by employers-is suggestecd by Spence, Market Sig- Spence describes provides an effective counterargu- naling in Appendix E (especially pp. 174-76), and ment to the claim, mentioned earlier, that employ- this possibility is given credence and analyzed by ers' "erroneous" or "mistaken" behavior will sustain John G. Riley, "Competitive Signaling," Journal of discrimination. Economic Theory, Vol. 10 (April 1975), pp. 174-86. 184 INDUSTRIAL AND LABOR RELATIONS REVIEW

workers-in other words there would be Same Distribution q of q for B and W no economic discrimination. Another model of economic discrimination is suggested by the work of the Proportion Which "Merits" Hiring psychologist R. L. Thorndike (reported by Linn).25 It incorporates reliability dif- B ferences but without recourse to risk aver-

sion. As we present it, this "selection- - Black truncation model" contains one feature of ...... White Spence's model-namely, a lower-bound threshold value of y, below which all y X--45 .. scores signal what is, in effect, a single q value. Above the lower-bound threshold, Figure 4. Predictions of Productivity l(q) the y, q relation is positive and continuous by Race and Test Score (y), Assuming Per- as it was previously in our models. As in fect and Zero Predictive Relations for the preceding case, black workers have Whites and Blacks, Respectively. equal average ability but receive a lower average wage; here, however, employers of course, greater relevance to, say, college are not practicing racial discrimination. admissions than to the labor market, but The crucial assumption of the selection- it may be at least suggestive of some eco- truncation model is that the hiring or se- nomic situations. In fact, theories of the lection decision is confined to the upper labor market that emphasize pervasive end of the y distribution. Given their ad- long-run wage rigidities, such as the "job vantage in test reliability, whites tend to be competition" theory of Thurow, are fertile preferred on the basis of expected values soil for the type of selection-truncation of q, given y, even though their q distri- model of discrimination presented here.26 bution is actually identical to that of blacks. The potential preference for blacks at the lower end of the y distribu- Unequal Average Abilities tion is nullified because workers with low We return now to the model in Phelps's y scores are not hired. Clearly, a higher paper in which the variances of q and u average value of q (or wage rate) for are equal for the two groups and the whites would emerge-evidence of eco- mean ability of blacks is less than the nomic discrimination in market outcomes- mean ability of whites. Here, the system- despite the fact that employers are not atic effect of blackness, aB < alv, leads to race-biased in their hiring process: that is, a lower predicted value of q for blacks they hire workers solely on the basis of than whites, even if the y scores are equal, E(q Iy). because y is, by assumption, a fallible indi- Figure 4 shows this result in an extreme cator. Phelps remarks that a B < aWv might form. Perfect reliability for Ws and zero reflect "disadvantageous social factors."27 reliability for Bs are assumed. The distribution of q is identical for Ws and Bs (as indicated on the vertical axis). Only values 26Thurow, Genierating Inequality, pp. 173-75. of q > -a are eligible for hire. (Assume 27Th-e three words in quotes are used by Phelps but are not written in a single phrase, although that a represents a comprehensive, legal the expression fairly conveys his meaning. Without minimum wage, here unrealistically set more information, however, the interpretation of equal to the wage corresponding to the this expression-and of aJ-B < aI'V could be am- overall average value of productivity.) biguous. Does the lower value of aB reflect a real deficiency in skills, as would be the case if the Given the costs of hiring and associated social factors were less schooling, less training, and costs of making a mistake, all blacks, but poorer health? Or does a B < aWv reflect merely a only half the whites, would be unemployed misconception or a false stereotype held by employ- or not in the labor force. The model has, ers? In accordance with our earlier expressed preference for believing that employers will not persist in erroneous behavior, we assume the first interpre- 25Linn, "Fair Test Use in Selection." tation. STATISTICAL THEORIES OF DISCRIMINATION 185

The relevant y, q relation for Ws and terms, the discriminated groups are faced Bs is shown in Figure 5. The B line is with higher costs of signaling.) Given below and parallel to the W line; the these handicaps, however, the differential equality of slopes is a consequence of the pay (or difference in employer demand) assumptions of equal variances of q, u, appears to be no more economically dis- and, therefore, y. As noted earlier, com- criminatory than are the lower wages that petitive forces in the market lead em- would be paid to workers with less experi- ployers to pay workers according to their ence, other factors (like the y score) being expected productivity; thus, white workers equal. (Of course, "legal discrimination," will be preferred to (and get higher wages or other definitions of discrimination, than) black workers with the same y score. need not be bound by economic terms, nor Unlike Phelps, we do not believe that dif- are wage payments in accordance with ex- ferent pay for the same y score demon- pected productivity necessarily synony- strates economic discrimination. Indeed, mous with good social policy.) were Bs to get paid the same as Ws when Finally, although Figure 5 demonstrates both had the same y scores, there would nondiscrimination by the "outcome" cri- manifestly be discrimination against Ws, terion of a proportional relation between since by Phelps's assumption the latter are average compensation for the groups and more productive (i.e., they have a higher their respective average productivities, at average q). every ability level a black will be paid less One could argue, of course, that the very than his white counterpart. This is seen existence of different average ability, aB < also in Equation 6, E(qIq)==(I -y) a +yq. aw, demonstrates a type of discrimination For the same ability (q value) and regres- in which workers are not paid in accord- sion slope (y) but lower black mean ability, ance with their innate abilities, and we (1-y)aB < (1-y)aW. would agree. But we would generally view This apparent paradox is resolved sim- its source in premarket discrimination- ply by recalling that here we assume less discrimination in schooling or in the ac- average ability for blacks and imperfect quisition of other forms and amounts of information. Thus, blackness is assumed human capital that workers possess when to provide information that E[4BIy] ex they enter the labor market. (In Spence's ante is lower than E[q6vly] ex ante for all y. But the employer's ex ante expectation q that a black worker will be less productive than a white worker, given the same y scores, must result in a black worker with a given q ability receiving a lower wage than a white worker with that same q ability, on average, because EQ01q) and E(q'Iy)involve exactly the same parameters. In the presence of perfect information (y,/1), the systematic difference in E(qIy) or E(qlq) for the two racial groups disap- pears, of course, and the lines for both color groups coincide with the 450 line. 01 In light of known premarket discrimination against blacks, the assumption by employers of unequal average abilities is real- aB aW istic, and so is the assumption of imperfect - Black information. The systematic inequality in VWhite E(?j1q) that results is, therefore, pro- foundly disturbing, and perhaps this in- Figure 5. Prediction of Productivity (q), equality is what is referred to by the by Race and Test Score (y), Assuming the economists mentioned in footnote 8. One Slopes Are Equal. consolation is that this inequality should 186 INDUSTRIAL AND LABOR RELATIONS REVIEW decline as employers assimilate more the y variable. The q^ variable itself has knowledge over time, thereby reducing been assumed to represent a wage rate Var(u) and raising -y. throughout, implicitly relying on the prop- We end this discussion with the cau- osition that wages measure productivity tionary remark that assuming a B < aIV and that competition will, on average, along with yB = yWv may not be very real- match equal productive abilities with istic. If the "raw-labor" abilities of blacks equal wages. Finally, one may argue that and whites are equal, a higher average there is no discrimination when produc- ability of white workers must reflect an tivity is defined either to include the in- advantage in human capital acquisitions. formational content of "signaling" or in Under these conditions, there is no basis terms of contributions to an employer's for assuming that white workers have the utility function that allows for risk aver- same variance in q, and, therefore, no pre- sion. sumption that the y values for the two In Spence's model of market signaling, groups are equal. discrimination may result if the costs of signaling differ for different groups or if Conclusions different groups have different initial signals. The former case represents a type of Several economists have heretofore ad- pre-labor market discrimination that re- vanced statistical theories of discrimination sults in the discriminated group having in labor markets. Although the Phelps lower average abilities in the labor market. model does not, in our opinion, explain or The latter case is not clearly indicative of describe racial or sex discrimination, it sustained discrimination. A number of provides a useful point of departure for variations of the Spence model are cur- several models that do. The models focus rently being investigated by various econo- on differential reliability in productivity mists, however, so final judgments should indicators among identifiable groups of be withheld. workers. On empirical grounds the differ- Other real world influences that affect ential is more plausibly introduced when economic discrimination have also been blacks (or women) are assumed to have less ignored. We have not dealt with monopoly reliable scores. When we combine this re- or monopsony.28 Tastes for discrimination liability differential with risk aversion by by employers or their systematic subjective employers, our model depicts economic underevaluations of the abilities (q values) discrimination that is qualitatively con- of the discriminated groups have been sistent with empirical evidence. In another model the combination of lesser reliability -"As the reader may know, however, the evidence for blacks (or women) on tests with trun- for these anticompetitive sources for sustained eco- of also re- nomic discrimination is meager. Orley Ashenfelter cation lower-scoring applicants produces evidence against a net antiblack discrimi- veals a kind of economic discrimination. nation effect of unions in "Racial Discrimination Both examples call attention to the poten- and Trade Unionism," Journal of Political Economy, tial inequities that may stem from lower Vol. 80, No. 3 (May/June 1972), pp. 435-64. Becker, test reliabilities for minority groups. The Economnics of Discrimination, 2d ed., pp. 7-8, has disputed the claim made by Thurow, Poverty We are reluctant, however, to claim too and Discrimination (Washington, D.C.: The Brook- much for these models. In the model that ings Institution, 1969), that monopsony power by uses risk aversion, there are grounds for employers is an important explanation. And Armen questioning whether the size of the risk A. Alchian and Reuben A. Kessel have argued that premium would be very large, and we monopoly power in the product market is consistent with long-run economic discrimination only (or know of no empirical support for a "cross- mainly) when there are constraints on the employ- over" point at the lower end of the indi- ers' ability to maximize money profits, as in regu- cator scale, where blacks earn more than lated monopoly industries. (See "Competition, Mo- whites for comparable indicator scores. nopoly, and the Pursuit of Pecuniary Gain," we have made no thorough Universities-National Bureau Committee for Eco- Obviously, nomic Research, Aspects of Labor Economics: A attempt to test the model, or even to give Conference of the Committee (Princeton, N.J.: more satisfactory empirical definitions of Princeton University Press, 1962), pp. 156-75. STATISTICAL THEORIES OF DISCRIMINATION 187 downplayed. While neither of these latter prophecies.29 It is fair to say, however, that modes of behavior, by itself, is consistent most of the explanations of discrimination with long-run economic discrimination in that rely on noncompetitive, disequilibria, a competitive model, introducing addi- and "noneconomic" forces have been tional factors may provide consistency. It offered very tentatively and leave a num- would take a more extensive discussion to ber of unanswered questions. deal with Arrow's list of additional considerations, which includes capital market im- 2Arrow, "The Theory of Discrim-nination," pp. 26-32. See Stiglitz, "Theories of Discrimination and perfections, wage rate rigidities, discontin- Economic Policy" for insightful and sometimes skep- uities in hiring decisions, and self-fulfilling tical comments about these considerations.