Oxford University Press is collaborating with JSTOR to digitize, preserve and extend access to The Quarterly Journal of Economics. http://www.jstor.org Discrimination, Nepotism, and Long-Run Wage Differentials Author(s): Matthew S. Goldberg Source: The Quarterly Journal of Economics, Vol. 97, No. 2 (May, 1982), pp. 307-319 Published by: Oxford University Press Stable URL: http://www.jstor.org/stable/1880760 Accessed: 18-03-2015 10:00 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions DISCRIMINATION, NEPOTISM, AND LONG-RUN WAGE DIFFERENTIALS* MATTHEW S. GOLDBERG The wage discrimination model developed by Becker has been criticized for predicting that competitive forces will lead to the disappearance of racial discrimination in the long run. We have reformulated the model in terms of nepotism toward white workers rather than discrimination against black workers.In this new framework, both nepotistic and taste-neutral firms are expected to survive the competitive struggle in the long run. Therefore, the new framework is consistent with long-run as well as short-run racial wage differentials. I. INTRODUCTION The economic theory of discrimination is largely due to Becker [1971b]. His approach treats wage differentials between otherwise identical workers differing only in race as being the observable manifestation of "tastes for discrimination," that is, disutility associated with market contact between the races. While several authors have recently proposed an alternative theoretical frameworkdesigned to explain wage differentials [Arrow,1972, 1973;Borjas and Goldberg, 1978;Phelps, 1972], the Becker model retains its position as the most influential one in the literature. Despite its great impact, Becker's model has been severely criticized, since one of its predictions is apparently at odds with empirical evidence. According to Becker: If all firms had the same linear and homogeneous production function, firms that discriminated would always have larger unit net costs than firms that did not. The smaller (in absolute value) the discrimination coefficient of any firm, the less would be its unit net costs. The firm with the smallest discrimination coefficient would produce the total output, since it could undersell all others [1971b, p. 44]. Unfortunately, this prediction of the disappearance of discrimination with the passage of time does not seem to have been borne out by the data. This has led Arrow [1973, p. 10] to state, "Only the least discriminatory firms survive. Indeed, if there were any firms which did not discriminate at all, these would be the only ones to survive the competitive struggle. Since in fact racial discrimination has survived for a long time, we must assume that the model ... must have some limitation." Similar criticisms have been leveled against Becker's * The author acknowledges the assistance of George Borjas, Ira Goldberg, and especially Sherwin Rosen, whose lectures in Labor Economics provided the basic framework for this analysis. C?1982bythe PresidentandFellowsof HarvardCollege.PublishedbyJohnWiley&Sons,Inc. The Quarterly Journal of Economics, May1982 CCC0033-5533/82/020307-13$02.30 This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions 308 QUARTERLY JOURNAL OF ECONOMICS theory in recent survey articles by Cain [1976, pp. 1219, 1232], Freeman [1974, pp. 517-20], and Marshall [1974, pp. 852, 854]. The point of this paper is to argue that the objections raised against Becker's model are essentially misdirected. It will be demonstrated that by shifting the origin and dealing with "nepotism" toward whites in contrast to "discrimination" against blacks, one arrives at a theory that is consistent with not only the existence but also the persistence of wage differentials. That is, while it is a wellknown implication that discriminatory firms tend not to survive, it will be shown that firms exhibiting nepotism will not only survive but in fact thrive in the long run. II. THEORY OF DISCRIMINATION It will prove convenient to present in this section a slightly more formal statement of Becker's discrimination model than is currently available in the literature. Once this has been accomplished, the modifications necessary to yield a theory of nepotism will follow in a straightforward fashion. According to Becker, a firm facing a market wage Wbfor black workersacts as if the wage were Wb(l + db),where db > 0 is the firm's "discrimination coefficient" against black workers. In the simplest case, db is taken to be a constant for all employment levels within a given firm, although db may vary across firms. For this to be the case, the firm's utility function must be of the form,1 (1) U = X-dbWbLb, where U = utility level, r = profit level, and Lb = black employment in the firm. We further define profit as (2) wr= Q(LW + Lb) - WwLw- WbLb, where We = market wage for white workers, Lw = white employment in the firm, and Q( ) = production function. Note that output is taken to be the numeraire, and also that blacks and whites are perfect substitutes in production, so that only total employment enters into Q( ). The production function is assumed to be strictly concave. We may combine (1) and (2) to obtain 1. The notion that firms maximize utility rather than profit is central to the managerial discretion literature pioneered by Williamson [1967]. Williamson asserts that the utility-maximization hypothesis is most relevant for large firms, firms in highly concentrated industries, and firms with diffuse ownership. Williamson's assertion has received empirical support in studies by Kamerschen [1968];Monsen, Chiu, and Cooley [1968]; Hindley [19701;Palmer [1973]; and Edwards [1977]. This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions DISCRIMINATION,NEPOTISM, AND WAGES 309 W Wbl + db') WbOl+ db) Wb Q'(L) 0 R(WW) R[Wb(1 + db)I L FIGURE I (3) U = Q(Lw+ Lb)- WwLw- (1 + db)WbLbIt follows from (3) that the firm's behavior is identical to that of a profit-maximizer with a black wage of Wb(1 + db) rather than Wb.2 We assume initially that the market wages for black and white labor satisfy the condition Wb< Ww;it will be seen presently that this condition is in fact consistent with full labor market equilibrium. Firms take these wages as parametrically given, and since both types of labor are perfect substitutes, firms hire all-white or all-black work forces as (4) Ww : Wb(1 + db) or db > (Ww- Wb)lWbIf Ww< Wb(1 + db), whites are hired until the point at which Ww= Q'(Lw). By the strict concavity of Q( ), this may be inverted to yield Lw = R(Ww), where R'( ) < 0. By similar reasoning, firms for whom Ww> Wb(l + db) hire Lb = R[Wb(I + db)] black workers.The hiring decision is illustrated in Figure 1. 2. This particular utility function was implicit in Becker's earlier work, although it seems to have been made explicit first in Becker [1971a, p. 71]. Arrow [1972, 1973] has suggested a somewhat more general specification in which U = U(7r,L ,Lb), with U, > 0, U,,, ?a0, Ub < 0. One can then define db as the marginal rate of substitution between profits and black labor, evaluated at the point of equilibrium. We have taken db to be a constant in order to preserve the spirit of the original Becker model. This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions 310 QUARTERLY JOURNAL OF ECONOMICS L R(Wb) R(Ww) BLACKS WHITES 0 Ww - Wb db Wb FIGURE II Employment, or firm size, is plotted as a function of db in Figure II. We observe that, even in the short-run analysis, firm size is monotonically non-increasing as a function of the desire to discriminate.3 To obtain wages Ww and Wb that are consistent with market equilibrium, we re-express the hiring criterion so that the work force is all-whiteorall-blackas Wb/Ww ; X, whereX = 1/(1 + db). If db is distributed across firms by the rule f(db), then by a change-ofvariables, X must be distributed across firms by the rule g(x) = (1/ x 2)f[(1/x)- 1]. Let the supply functions of white and black workers be S.(W.) and Sb(Wb), respectively, where S' ( ), S'( ) > 0. These supply functions depend upon relative prices, in view of our earlier 3. The employment function is continuous at the critical value, d* = (W,Wb)/Wb. To see this, recall that the wage relevant for the firm's decision is W(db) = min[Ww,Wb(l + db)]. For the "marginal" firm that is located at d*, we have W(d*) = = Wb(1 + di). The function W(db) has equal limits at dbwhether approached from the right or from the left: lim W(db) = lim W, = Ww, and db-db*+ db-db*+ lim W(db) = lim Wb(l + db) = Wb(1 + d*) = Ww. db-db*- db-db*Hence W(db)is continuous at dt. But L(db) = R[W(db)], and since R( ) is a continuous function, it follows that L(db) is continuous at d*as well. This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions DISCRIMINATION, NEPOTISM, AND WAGES 311 Wb/Ww SUPPLY | \EM~~DEAND 1~ 0 1 Lb Lb + LW FIGURE III choice of output as the numeraire. Then Wwand Wbare determined by the equations, WblWw (5) SW(WW)= R( Ww)g(x)dx; (Wb~ (6) Sb(Wb)= J/ R g(x)dx. In interpreting these equations, recall that the all-white firms are of size R(Ww), while the all-black firms are of size R[Wb(1 + db)] = R(Wb/x). Market equilibrium is illustrated in Figure III forthe special case in which S' ( ) = S',( ) = 0, so that supply is inelastic.4 It is obvious that as long as db > 0 (orequivalently, X < 1) for all firms, then 4. The convenient geometrical representation of Figure III is invalid of S' ( ) and S5( ) are different from zero, since then the relative supply cannot in general be expressed as a function solely of the ratio of Wbto Ww.Equations (5) and (6), of course, remain valid in any case. This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions 312 QUARTERLYJOURNAL OF ECONOMICS H V(Wb) V(WW) 0 WW- Wb db Wb FIGURE IV equilibrium must occur with the wage ratio Wb/WW being less than unity. Having established the existence of short-run wage differentials, we now move on to a long-run analysis. Consider first the equilibrium profit levels, and then the equilibrium utility levels, achieved by firms that are indexed by alternative values of db. If db > (WW- Wb)/Wb, the firm has been shown to hire LW= R(Ww) white workers at the wage Ww.Its profits are given by (7) 7w = Q[R(Ww)] - WwR(Ww). Define the (indirect) profits of a single-input competitive firm at the wage W by the function, (8) V(W) = max [Q(L) - WL]. L From the envelope theorem [Samuelson, 1947, pp. 34-36], we have V'(W) < 0. Thus, the all-white firm enjoys a profit level of V(Ww). The all-black firm, on the other hand, hires R[Wb(1 + db)] black workers at the wage Wb. This leads to profits of (9) 7rb = Q[R(Wb(l +db))]- WbR[Wb(l +db)]This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions DISCRIMINATION, NEPOTISM, AND WAGES 313 Figure IV plots profits as a function of db.5 It would seem that Figure IV captures the classical argument in the literature for why discrimination should disappear in the long run. This argument, as exposited by Alchian and Kessel [1962,pp. 160-61], for example, is phrased in terms of the acquisition of the highly discriminatory firms by firms displaying a lesser desire to discriminate. In the presence of perfect capital markets, if one firm has a smaller value of db than its rival, then due to the spread between their money profit levels, the first firm can buy out the second firm. Indeed, the firm with the smallest value of db in the market can profitably buy out every other firm in the industry. Hence discrimination must disappear in the long run. The above argument leads to the correct conclusion, but unfortunately, the reasoning is slightly flawed. The sellout price of a firm is not equal to its money profit level, but rather to its utility level as given in equation (1).6Recall that db was interpreted as the marginal rate of substitution between profits and black labor. Hence db converts units from employment into dollars, so that equation (1) expresses utility as a monetary equivalent. For example, a firm with a positive value of db that hires an all-black work force is willing to sell out for some amount less than its profit level, since its "real" rate of return lies below its money rate of return due to the nonpecuniary disadvantages of remaining in business. With this in mind, what is required is an expression for equilibrium utility levels as a function of the parameter db.The all-white firm avoids contact with black workers; hence from equation (1) with Lb =0, (10) U. = V(Ww). For the all-black firm, we must subtract dbWbR[Wb(1 + db)] from rb as given by equation (9), yielding (11) Ub = Q[R(Wb(1 + db))] - Wb(l + db)R[Wb(1 + db)] = V[Wb(1 + db)]- 5. The profit function is discontinuous at db. For the marginal firm with W, = Wb(l + dt), total employment is equal to R(WW)= R[Wb(1 + db)]. But money profits are higher if blacks are hired rather than whites, since the former group is cheaper, Wb 0 is the "nepotism coefficient" measuring the marginal rate of substitution, assumed constant, between profits and white labor. The firm acts as if the white wage were Ww(1 - dw), and hires 7. The functionU(db)is continuousat d*.Inthe notationof footnote3, U(db) - V[W(db)].Butit wasalsoshowninfootnote3that W(db)iscontinuousatdt, hence so must be U(db)This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions DISCRIMINATION, NEPOTISM, AND WAGES 315 exclusively blacks if Wb < Ww(1 - d), or if dw < (Ww - Wb)/Wu. In this case we have Lb = R(Wb), and profits are equal to V(Wb). Since no whites are hired, potential utility gains from nepotism are absent, and hence utility and profit levels are equal to (13) Ub =7rb = V(Wb). If d, > (Ww - Wb)/WW,whites arehired in the amountLW= R[W (I - d)]. Profits are equal to (14) 7rw = Q[R(Ww(1 - dw ))]-WwR [Ww(1 - dw)]. Adding in the utility gains from nepotism, we arrive at (15) Uw = Q[R(Ww(1 -dw))] - Ww(1 -dw)R[Ww(l-dw)] = V[Ww(1 - dw)]. Figure VI depicts the crucial profit and utility functions.8 Utility is given by the horizontal segment in the interval 0 < dw < d , and by the rising curve for d% 0 throughout the unit interval. The easiest way to see this is to suppose that Q'(L*) = 0 for some L* < + . Then consider the firm for whom dw = 1, so that it acts as if the white wage were zero and hires L* white workers. Total revenue equals the area under the marginal productivity curve, or j Q'(L)dL, while total costs equal WWL*. The difference between these two numbers may be of either sign. The final issue concerns the asymptote in U(dw) as dw- 1. If Q'(L) remains positive for all L > 0, then lim V[W,(1 - dw)] = lim V(W) =+ dt, 1- W-O since the firm may expand indefinitely and still remain within the region of positive marginal productivity. If we insist upon finite employment and utility levels, then we must restrict the admissible set of values of dWso that d, < 1. On the other hand, if Q'(L*) = 0 for some L* < + ', then even firms with dw > 1will choose finite employment levels. Note that utility becomes an increasing function of Ww in this case, OU,16W, = V'(1 - dw) > 0, so that firms may actually increase their utility levels by paying white wages above the market level. However, this practice will reduce money profits, irw1/6Ww= (dw - l)R'Wwdw - Lw < 0, until the budget constraint is eventually reached. The final outcome will be finite values of Wwas well as LW. This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions 316 QUARTERLYJOURNAL OF ECONOMICS U, H / | [I~~(d)/l V(Wb) n(dW) IX I ~ ~~~~I \ I 0 dw* dw 1 dWFIGURE VI correctionterm d,W L. As do increases, Lo will increase along with it, so that the correction term increases a fortiori. What we have demonstrated is that the increase in the correction term outweighs the reduction in profits, so that total utility is an increasing function of dw. This result may be explained both intuitively and mathematically. Intuitively, consider two firms that both hire all-white work forces, db > daI> d , and suppose that Firm A is in equilibrium. Now Firm B could employ the same amount of white labor as Firm A, in which case Firm B's profit level would equal that of Firm A, but Firm B's utility level would be greater than that of Firm A. Moreover, any adjustment toward optimality by Firm B would yield still higher utility levels. Therefore, utility is an increasing function of dw.9 Mathematically, the key is to look back at equation (12). The objective function for the utility-maximizer characterized by dw is identical to that of a profit-maximizer facing wages WW(1 - dw) and 9. I owe this explanation to an anonymous referee. This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions DISCRIMINATION,NEPOTISM, AND WAGES 317 Wbfor the two types of labor. But profit-maximizers achieve a higher maximum when the wages that they face are lower. Thus, the maximized value rises as do rises, since a rise in do implies a fall in the net wage, W (1 - dx). This may also be seen in equation (15), where the value of the utility maximum equals that of a profit maximum when the wageis given by WW(1- dx). Clearly,bU,/?1dw =- WwV' > 0. It has been suggested by Becker [1962] that firms face a "budget constraint" in that their actions must yield at least zeroprofits in order to ensure economic survival. If so, then defining do by the equation wx(dw)= 0, only firms for whom do < do will survive even in the short run.10Survival in the long run is a more interesting and more subtle question. Consider a comparison of extremes:-the neutral firm located at dw = 0 and the break-even firm located at dw = do. Could the neutral firm buy out the break-even firm? The maximum offer that the neutral firm is willing to make equals V(Wb), or the amount of profit (= utility) that the neutral firm could earn by running its rival's business. The break-even firm, however, requires compensation not only for its money profits, but also for its nonpecuniary income, which equals U(dw). Since U(dw) > V(Wb), no transaction will take place. Turning the question around, could the break-even firm buy out the neutral firm? The neutral firm, which cares only about money income, requires V(Wb)in cash before it will sell out. The break-even firm, on the other hand, would earn zero money income only if it were to take over its rival's business. Thus, the break-even firm would violate its budget constraint if it attempted to pay out V(Wb)in cash when it only had zero money income. Again, we find that no transaction is possible. In general, consider any two firms: Firm A with a nepotism coefficient da and Firm B with a nepotism coefficient db. For Firm A to buy out Firm B, it must be the case that the money income of Firm A exceeds the utility level of Firm B, 7r(d') > U(dZb).Only if this condition holds, will Firm A be able to pay out enough cash to com- 10. Williamson |1967] suggested an alternative constraint under which profit must exceed some minimally acceptable level iF,where fi is determined by stockholders and is independent of the variables over which the firm optimizes. Imposition of this type of constraint leaves our analysis unaffected except for narrowing the range of values of do that satisfy the constraint when iF> 0. It may appear that excess profits in the industry are necessary for the short-run survival of the all-white firms, since if ir(dw) d , then the all-white firms are driven out of business. However, the reduction in the number of all-white firms would reduce the demand for white labor and thereby reduce the white wage. This would increase the profitability of the all-white firms. Clearly, some all-white firms must survive, or else white labor would be unemployed in equilibrium. This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions 318 QUARTERLYJOURNAL OF ECONOMICS pensate Firm B for both the money income and the nonpecuniary income that the latter would earn if it were to remain in business. Inspection of Figure VI, however, clearly reveals that there does not exist a pair of firms A and B for which this condition holds. Therefore, firms located along the entire range of values of do will survive in the long run. It only remains to resolve the asymmetry between the model of nepotism and the model of discrimination presented earlier. Both discriminatory and nepotistic firms distort their input choices by hiring expensive white workers rather than cheap black workers. Therefore, both of these firms earn lower profits than a neutral firm. Moreover, the discriminatory firm suffers a nonpecuniary loss as well, which can be avoided only by going out of business. On the contrary, the nepotistic firm enjoys a nonpecuinary gain by remaining in business. Therefore, although discriminatory firms tend to disappear, nepotistic firms can coexist along side neutral firms in the long run. The neutral firms extract their utility in the form of money income, while the nepotistic firms reduce money income toward zero in order to earn nonpecuniary income. IV. CONCLUSIONS This paper has reformulated Becker's theory in terms of nepotism toward whites rather than discrimination against blacks. Both models predict short-run wage differentials, but only the nepotism model is consistent with both perfect capital markets and wage differentials that persist into the long run. The reconciliation of Becker's framework with the existence of long-run wage differentials is evidence that this framework is much more useful than one would surmise from a reading of recent criticisms in the literature. CENTER FOR NAVAL ANALYSES REFERENCES Alchian, Armen, and Reuben Kessel, "Competition, Monopoly, and the Pursuit of Pecuniary Gain," in H. G. Lewis, ed., Aspects of Labor Economics (Princeton: Princeton University Press, 1962). Arrow, Kenneth, "Some Mathematical Models of Race Discrimination in the Labor Market," in A. Pascal, ed., Racial Discrimination in Economic Life (Lexington: D.C. Heath, 1972). -, "The Theory of Discrimination," in 0. Ashenfelter and A. Rees, eds., Discrimination in Labor Markets (Princeton: Princeton University Press, 1973). Becker, Gary, "Irrational Behavior and Economic Theory," Journal of Political Economy, LXX (Feb. 1962), 1-13. -, Economic Theory (New York: Knopf, 1971a). This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions DISCRIMINATION,NEPOTISM, AND WAGES 319 The Economics of Discrimination (Chicago: University of Chicago Press, 1971b). Borjas, George, and Matthew Goldberg, "Biased Screening and Discrimination in the Labor Market," American Economic Review, LXVII (Dec. 1978), 918-22. Cain, Glen, "The Challenge of Segmented Labor Market Theories to Orthodox Theory: A Survey," Journal of Economic Literature, XIV (Dec. 1976), 1215-57. Edwards, Franklin, "Managerial Objectives in Regulated Industries: Expense Preference Behavior in Banking,"Journal of Political Economy, LXXXV (Feb. 1977), 147-62. Freeman, Richard, "LaborMarket Discrimination: Analysis, Findings, and Problems," in M. Intriligator and D. Kendrick, eds., Frontiers of Quantitative Economics, Vol. II (Amsterdam: North-Holland, 1974). Hindley, Brian, "Separation of Ownership and Control in the Modern Corporation," Journal of Law and Economics, XIII (April 1970), 1423-35. Kamerschen, David, "The Influence of Ownership and Control on Profit Rates," American Economic Review, LVIII (June 1968), 432-47. Marshall, Ray, "The Economics of Racial Discrimination: A Survey," Journal of Economic Literature, XII (Sept. 1974), 849-71. Monsen, Joseph, John Chiu, and David Cooley, "The Effect of Separation of Ownership and Control on the Performance of the Large Firm,"this Journal, LXXXII (Aug. 1968), 435-51. Palmer, John, "The Profit Performance Effects of the Separation of Ownership and Control in Large U.S. Industrial Corporations," Bell Journal of Economics and Management Science, IV (Spring 1973), 293-303. Phelps, Edmund, "The Statistical Theory of Racism and Sexism," American Economic Review, LXII (Sept. 1972), 659-61. Samuelson, Paul, Foundations of Economic Analysis- (Cambridge:Harvard University Press, 1947). Williamson, Oliver, The Economics of Discretionary Behavior: Managerial Objectives in a Theory of the Firm (Chicago: Markum, 1967). This content downloaded from 147.251.185.127 on Wed, 18 Mar 2015 10:00:35 UTC All use subject to JSTOR Terms and Conditions