Mathematical Social Sciences 14 (1987) 51-57 North-Holland

51

ALLOCATIVE COST OF WEIGHTED DISCRIMINATION

Paul van MOESEKE Department of Economics, Massey University, Palmerston North, New Zealand

Communicated by K.A. Fox Received 2 May 1986

Discrimination, whether of the reverse or the usual kind, distorts the allocation of resources in favour of, respectively against, the group or individual concerned. Affirmative action and institutional distinction extraneous to work performance are examples where more weight is attached to the agent’s subsequent output than is warranted by its intrinsic merit. We introduce this extra weight 85 a discrimination factor and propose a method of measuring the allocative effect on resources. Inasmuch as the discrimination factor measures a bias in the agent’s behavior setting the article is relevant to eco-behavioral research.

Key words: Separable programming, parametric programming, discrimination, behavior settings.

1. Introduction

While discrimination may generally be expected to carry a social cost, the present paper suggests a simple way of measuring that cost for a particular type of discrimination characterized by attaching a different weight to a person’s output than its intrinsic merit warrants. The weight is smaller for the usual kind of discrimination, greater for the reverse kind. We shall consider two workers, or two groups of workers, one of which is favored by having the objective value of their output upweighted by a discrimination factor /3. Actually, the model applies to regular (/I< 1) as well as reverse (/?> 1) discrimination but to fix the ideas the discus- sion below is couched in terms of the latter.

Apart from discrimination - either way - of the usual type (minority groups) the approach applies as well to discrimination by institutional distinction unreiated to output such as membership of a political party or hierarchy, an order of nobility, a club, a kinship group and so forth.

In eco-behavioral terms we introduce and quantify discrimination as a relative change in the behavior settings of two individuals or groups (or indeed, as pointed out below, of any number of individuals or groups): since the favored individual

. (group) gains (measurably) less than the other individual or group loses, the change is Pareto suboptimal. (Compare with the improvements considered by Fox, 1985, ch. 2.3 and ch. 3.) Discrimination can readily be incorporated into so-called

0165-48%/87/$3.50 0 1987, Elsevier Science Publishers B.V. (North-Holland)

52 P. van Moeseke / Allocative cost o f weighted discrimination

Fox-van Moeseke models (cf. Fox and Moeseke, 1973 and Moeseke, 1985) by alter-

ing individual objective functions prior to aggregation. The present model instances

the manipulation of behavior settings such as 'stores, meetings, classes, and all other

behavior settings', which 'have plans for their human components and armories of

alternative ways of enforcing their plans,' cf. Barker and Associates (1978, pp.

285-287).

The analysis need not apply, however, to discrimination by distinction conferred

in recognition of merit. Indeed, Lazear and Rosen (1981) showed that, under

reasonable assumptions, wages based on ordinal rank, rather than individual out-

put, cause no inefficiency, at least in the case of risk-neutral workers, because of

the incentive provided by the chance of reaching executive status and the cost of

monitoring individual workers' efforts. On efficiency comparisons of various incen-

tives and the relative merits of reward structures based on rank order, contracts,

patents and prizes see Stiglitz (1975), Green and Stokey (1983), and Wright (1983).

Again, distinction has obvious discriminatory effects in dimensions other, and less measurable, than resource allocation. An example is the weight conferred by celebri-

ty, which is a type of distinction awarded by the media to pronouncements of, say,

movie or rugby stars on public issues unrelated to their dramatic or athletic exper-

tise. While these effects appear to be distortions of a kind somewhat similar to the

one studied here, they are not so readily amenable to quantitative treatment.

2. The model

For expository convenience and without loss of generality we shall refer in con-

crete terms to research scientists, rather than 'workers' in the abstract and to two

individuals rather than groups. The employer is initially assumed to relate marginal return, hence the value of

salary-cum-perks, to the net present discounted value of patents or new research

contracts secured on the basis of research carried out. Alternatively, return may be

tied to number of publications or some such objective measure. Research activities are limited by the availability of resources, some of which are

of the capacity type: they are proper to the scientist and can, for practical purposes,

be considered as given (the scientist's time, office space and library access, for in-

stance). More important, there are resources for which a number of scientists com-

pete, such as the use of certain laboratories, electronic microscopes, chemicals,

reactors, telescopes, and computers, as well as clerical and research assistance.

These resources can be reallocated in the short run. Economic efficiency, as well as

internal accounting practice, require that the use of the variable resources be im- puted to scientists' budgets at certain accounting prices (of which more anon). Hence every individual scientist is faced, implicitly or explicitly, with maximizing

his return net of imputed resource costs, while the firm or institution as a whole

maximizes aggregate return.

P. van Moeseke / A/locative cost of weighted discrimination 53

This is a problem in separable programming: the basic theorem states that, under

standard economic assumptions (essentially nonincreasing returns) there are (non-

negative) resource prices that will reconcile the aggregate optimality problem with

the maximization problem of each individual scientist or unit. The problem, in other

words, is one of decentralized optimization.

Discrimination is now introduced in the following way: the objective return of a

member of the favored group is henceforth upvalued by a discrimination factor /j> 1. If, under certain types of affirmative action, a worker is paid more ex post than his ex ante market value, p can be taken as the ratio between the two wage

rates.

3. Allocative distortion

The analysis below assumes familiarity with the standard interpretation of dual

variables as accounting values assigned to resources according to their marginal con-

tribution to the objective being maximized.

Denote the activities of two scientists by, respectively, xi and x2, and the cor-

responding returns by f, (x,), fi(xZ). Individual capacity limitations, adverted to

above, are denoted X,, X, so that necessarily

x1 EX,, x+X2. (1)

The activities use up amounts ui(x,), u2(x3) of the common resources, which are

available in quantities r so that necessarily

ui(xt)+Ux&r.

Optimally, the employer or institution so allocates r as to

(2)

maximize ft (xi)) +f2(x2)

subject to (l), (2). Observe that xl may denote the level of a single activity or an

n,-tuple of such activities, i.e. it may be a real number or a vector. Similarly x2

may be a number or an n,-tuple. Again, ut, u2 may be numerical functions or m-

tuples of such, and r is, accordingly, a scalar or an m-tuple. The analysis is the same

whether the symbols x1, x2, ul, u2, r are interpreted as scalars or as vectors. In the

latter case a second index is used for the ith coordinate: uli etc.

Problem (P) is a standard problem in separable programming: it can be decen- tralized over both scientists by using the duals (saddlepoint values) of(P) and charg-

ing, or at least imputing, the m-tuple of nonnegative prices p for the use of the

common resources so that the individual scientists face the problem of maximizing

return net of resource cost:

maximize fr (x,) -pu, (x1), x1 E X, , (PI)

maximize f2 (x2) - pu2(x2), x+X2. (P2)

54 P. van Moeseke / Allocative cost o f weighted discrimination

Adapted to the present problem and notation, the relevant theorems are as follows (proofs in Moeseke and de Ghellinck, 1969):

Theorem 3.1. I f xl, x2 solve (P1), respectively (P2), and satisfy (2) as well as the condition

Uli(Xl ) q- U2i (X2) < ri --~ Pi = O,

then x 1, x 2 solve (P).

Theorem 3.2. Assume (P) & convex. Assume further that (P) is either

homogeneous, or satisfies the Slater condition, or both. I f &, x 2 solve (P) then there exists a semipositive p such that Xl, x2 respectively solve (P1), (P2).

It was shown by the author (1974) that, under homogeneity of (P) (i.e. of fl,J2, ul, u2) the Slater condition is not required for (P) to have a saddlepoint. Regarding generality note further that neither proposition requires differentiability, that the individual constraint sets X1, )(2 need not be functionally specified and that Theorem 3.1 does not require convexity of (P): here Xl, X2 might be, say, sets of discrete points.

Again, under convexity of (P) (nonincreasing returns) both theorems hold so that activity levels xl, x2 are optimal for the overall problem (P) iff they are respectively optimal for the individual programs (P1), (P2). That is to say, the activity levels x~, x2 are optimal for the overall problem (P) iff they are respectively optimal for the individual problems (P1), (P2).

Let us now introduce discrimination by attaching a weight f l> 1 t o l l (Xl) so that the new overall program becomes

maximize/3fl (xl) +f2 (x2) (P')

subject to (1), (2), while the individual problems read

maximize 13fl(xl)-p ' ul(xl), Xl eX1, (P ' I )

maximize f2 (x2) - p ' u2(x2), x2 ~ X2, (P'2)

bearing in mind that the new prices p ' will normally differ from p. The weight f l> 1 attached to the actual contribution f~(x~) of the 'favored' scientist is the discrimination factor: it will normally distort efficient allocation by diverting resources to him (and away from his colleague). The inequafities derived below

amplify and specify this statement. The solutions to (P) and to (P') will, of course, generally be different. Let us

denote the corresponding optimal values of the functions involved bY fl, f2, ui, u~ for (P) and fl', f2', u(, u£ for (P'), By A fl we denote f j ' - f l and define A f2, Au,, Au 2, Ap similarly.

By definition of optimality in (P), respectively (P') ,

P. van Moeseke / Ailocative cost of weighted discrimination 55

fi +f2 ++f;, (3)

pf;+f;UG +LP (4)

By (3) discrimination normally decreases efficiency, i.e. total return. However, the actual return to scientist 1, even when valued without the discrimination factor, in- creases: indeed, adding (3), (4) and cancelling yields

(P- l)&%(P- 1)A

so that

Afi 10. (5)

But since (3) can be written

Af,+Aj&O (6)

formula (5) implies

Af2s0 (7)

so that the second scientist loses out, and indeed loses more than the first one gains. Further, by combining (4), (6), (7):

01 -Af2@sAf,s -Af2, (8)

which sets lower and upper bounds to scientist l’s gains, which are smaller than 2’s loss but greater than 2’s discounted loss’. In fact, weighting ft by the discrimina- tion factor /3 is equivalent to discounting f2 by the factor l//3< 1.

By the decentralization result mentioned above we know that programs (Pl), (P2) satisfy

fl -Pwf;-Pu;,

f2 - PU2 If; - Pu; 9

in the nondiscrimination case, and

in the case

.

Pf;--P’u;ufi -P%

f;-P’u;~f2-PpIu2,

that (P’l), (P’2) satisfy

of discrimination. These four inequalities yield in turn:

OrAf, spdu,,

AfzrPAuz,

P’AudAf,,

(9)

(10)

(11)

’ The signs of the variations discussed here are instances of the Le Chatelier effect (cf. Leblanc and

Moeseke, 1976).

56 P. van Moeseke / Allocative cost of weighted discrimination

where (9) and (12) use also (5) and (7). Formula (9) shows that scientist 1 will normally consume more resources (since

pAu, > 0 and pro) and that the extra resource cost, at constant prices, exceeds his gain. Similarly, (12) indicates that the second scientist’s command over resources will normally decrease, and that by even more than the value of his output. Further. adding (9) to (10) and (la) to (B2), and putting u = u1 + u2 yields

indicating that total extra resource cost by (13) is ‘not worth it’ but appears so in (14) where et-ices are distorted under discrimination.

Finally, subtracting (IO) from (12) yields

ApAu2~0

so that discrimination will normally increase resource prices given the fact that, as we already know, tc2 will normally decrease.

The above reasoning can, of course, be extended almost verbatim to the case of n scientists, one of whom, say scientist k, is favoured. Then fk, z.+,. can be treated like J1;, ul above and all the corresponding inequalities hold: fk will increase at the expense of Cjfk Jfi and so on. Again, the analysis readily extends to two groups of scientists, one of which is favoured.

4. Conclusions

Attaching a greater weight, or discrimination factor, to the favoured worker’s output is a simple way of characterizing discrimination in reverse, which has receiv- ed rather less attention in the literature than the usual kind.2 Reverse discrimina- tion has become more important, in the USA as elsewhere, since the US Labor Department in 1968 began requiring that contractors set numerical goais for minorities. The approach also covers distinction extraneous to work performance.

While the actual return to the favoured group increases, other workers’ loss more than offsets the gain. Again, command over resources will to some extent shift towards the favoured group but the extra resource cost, at constant prices, exceeds their gain. Resources will be more expensive and the total extra cost, at constant prices, is not worth it in terms of the effect on value added.

- 2 See, however, Loury (1981). The reader may further note that we have abstracted from employer

risk aversion discussed by Aigner and Cain (1977) and from incompiete information (cf. Lundberg and

Startz, 1983).

P. van Moeseke / Allocative cost o f weighted discrimination 57

• ~ cknowledgments

Research initiated during my tenure of the Professorial Fel lowship in Economic Policy o f the Reserve Bank of Australia. For helpful comments 1 am indebted to l'rofessor K.A. Fox and Drs F. vande Ginste.

I/eference,~

r). Aigner and G. Cain, Statistical theories of discrimination in labor markets, Industrial and Labor

Relations Review 30 (1977) 175-187. b:.G. Barker and Associates, Habitats, Environments, and Human Behaviour (Jossey-Bass, San Fran-

cisco, 1978). K.A. Fox, Social System Accounls (Reidel, Dordrecht, 1985). K.A. Fox and P. van Moeseke, Derivation and implications of a scalar measure of social income, in:

H.C. Bos, H. Linnemann and P. de Wolff, eds., Economic Structure and Development: Essays in Honour of Jan Tinbergen (North-Holland, Amsterdam, and American Elsevier, New York, 1973) pp.

21-40. .I .R. Green and N.L. Stokey, A comparison of tournaments and contracts, Journal of Political Economy

91 (1983) 349-364. E.P. Lazear and S. Rosen, Rank-order tournaments as optimum labor contracts, Journal of Political

Economy 89 (1981) 841-864. G. Leblanc and P. van Moeseke, The Le Chatelier principle in nonlinear programming, Review of

Economic Studies 43 (1976) 143-147. equal opportunity enough, American Economic Review (Papers and Proceedings) 71 (1981) G. Loury, Is

122-126. 5. Lundberg and R. Startz, Private discrimination and social intervention in competitive labor markets,

American Economic Review 73 (1983) 340-347. 17. van Moeseke and G. de Ghellinck, Decentralization in separable programming, Econometrica 37

(1969) 73-78. P. van Moeseke, Saddlepoint in homogeneous programming without Slater condition, Econometrica 42

(1974) 593-596. P. van Moeseke, Socio-economic interface and social income, Mathematical Social Sciences 9 (1985)

263-273. .I.E. Stiglitz, Incentives, risk, and information: notes towards a theory of hierarchy, Bell Journal of

Economics and Management Science 6 (1975) 552-579. B.D. Wright, The economics of invention incentives: patents, prizes and research contracts, American

Economic Review 73 (1983) 691-707.