Statistical discrimination and competitive signalling

Economics Letters North-Holland

36 (1991) 93-97

93

Statistical discrimination and competitive signalling

*

Rein Haagsma Institute Received Accepted

of Economics/CIA

V, Utrecht Unwersity,

3512 JE Utrecht, Netherlands

27 December 1990 71 February 1991

This paper examines the nature of statistical discrimination It concludes that banning discrimination always widens signalling costs.

arising from different signalling costs in fully confirmed equilibria. the gap in net incomes at the expense of workers with higher

1. Introduction

Spence’s (1973, pp. 368-374) discussion of ‘indices’ in labour markets with a signalling externality is often cited as an alternative explanation of statistical discrimination. According to Spence there are multiple signalling equilibria, and workers with different indices (such as race or sex) may well end up in different equilibrium positions - notably even if their underlying signalling cost functions are the same. Yet such equilibrium differences do not withstand plausible wage experiments of firms [Riley (1975)] or trial period proposals of workers [Cain (1986)J. Statistical discrimination therefore only persists if signalling cost functions diverge. For these so to call fully confirmed discriminatory equilibria, this paper examines the nature of statistical discrimination.

2. Analytical

framework

We employ the competitive-signalling model of Spence (1974), adapted by Riley (1975). Consider labour markets where workers have heterogeneous abilities that cannot be observed by firms. Individual labour productivity s is an increasing, twice-differentiable function of individual ability n and education y, s=.S(n,y) (subscripts

denote

with partial

S,,>O.

derivatives).

S,.>O. Let us distinguish

* This research has been supported by the Foundation which is part of the Dutch Organisation of Scientific 0165-1765/91/$03.50

0 1991

(1) workers

for the Promotion Research (NWO).

Elsevier Science Publishers

by sex and assume

of Research

B.V. (North-Holland)

in Economic

that education

Sciences

(ECOZOEK),

94

R. Haagsma

/ Statistical discrimnation

costs c, of a sex i individual (i =f, ability and education, such that c, = C’( n, y) So besides that additional education to be a signal), for men. 2.1. Fully confirmed

with

and competh.w

signolhg

m) are given by a twice-differentiable

Ci,. < 0,

C,: > 0 (i =f,

m); and C;>

Cr.

function

of individual

(2)

education is more costly for less-able individuals (the critical condition for it is assumed that marginal costs of education are higher for women than

equilibrium

with statistical

discrimination

Since individual abilities are unknown to firms, wage offers depend on education and, as we will see, on sex as (costlessly) observed but imperfect indicators of labour productivity: w, = w’(y). A sex i worker of ability n chooses education y in order to maximise income net of education costs, that is, he or she maximises w’(y) - C’( n, y) with respect to y. Competitive firms pay wages equal to actual, ex post determined, labour productivity, or w’(y) = S( n, y). Provided that the second-order condition of income maximisation is satisfied, in a signalling equilibrium the wage schedule offered to sex i workers is thus found by solving a pair of equations for w’(y) (i = f, m):

w;,= c; ,

(3)

w’(y) = s(n, y).

(4)

It is obvious that sex differences in marginal education costs generally lead to different wage schedules. Inverting (4) with respect to n and substituting for n in (3) we get a first-order differential equation for w’(y), of which the solution can be expressed as w, = MI’(y; k,)

with

w: > 0, WI, > 0,

(5)

where k, is a constant of integration. Each value of k, involves a potential equilibrium wage schedule of sex i workers. Riley demonstrates that given a certain lower limit of ability there exists always a maximum value of k,, here denoted by k,P. Since all lower values of k, yield lower wages for given education levels, w, = w’( y ; ky) represents the Pareto-dominating wage schedule of sex i workers. Among all pairs of equilibrium wage schedules, only the Pareto-dominating members of both sexes seem to be persistent. Pareto-inferior wage schedules will not survive because, as spelled out by Riley (1975, pp. 181, 182) firms are likely to experiment with new wage offers in an attempt to fully confirm their prior expectations about the relationship between labour productivity and education. Moreover, as argued by Cain (1986, pp. 727, 728), if workers are sufficiently aware of their individual ability, they can offer a trial period of employment in order to reveal their true labour productivity. Therefore, the subsequent analysis will be concerned with, what Riley calls, fully confirmed signalling equilibria. The determination of the parameter k,? can only be summarised here [see Riley (1975, pp. 178-180). Let both male and female abilities be defined over [no, n’]; no > 0. In a signalling equilibrium we have S(n, y) = w’(y ; k,), which can be converted and written as n=n’(y;k,) [using dn’/dy

with

dn’/dy

= (Ci - S,)/.S,,

(3)]. To satisfy the second-order condition 2 0 over [no, n’]. Then in a (y, n)-diagram

(6)

of income maximisation, it is required that n’(.) must lie on or to the right of the boundary

R. Haagsma

/ Statistical

discrimination

and competitive signalling

95

curve C; - S,. = 0. Assuming that c; = S,, is a positively sloped function, dn’/d y = 0 on the boundary curve and dn’/d y > 0 to the right of the boundary curve. Higher values of k, shift n’(.) upward. Then, given the requirement: y 2 0, k,!’ is implicitly determined by no = n’(max[

y,‘, 0] ; kp),

(7)

where y,’ follows from

c;.( no , y(o)

=S,.(nO,YP>i

(8)

y,” denotes the smallest educational investment made by (the least-able) sex i workers. To establish the nature of statistical discrimination in fully confirmed equilibria, we examine the effects of higher marginal education costs on the Pareto-dominating wage schedule of female workers; we do so for plausible equilibria having y/” > 0. Let us simplify by assuming c, = C’( n, y ) = C( n/b,, C,.>O(i=f, and differentiate that k/” depends

) with C( n/b,) y < 0,

m),andO
the implied on b/):

(dw,/db,)

y

(2’)

female wage schedule

lL.cons,.= -Xi.

(d&Q)

w, = w’( y ; k,p, b/) totally with respect

. (d$/db,),

to b, (note

(9)

where

dy/“/db/= [n”C,(n/b,)]/[h:(C,,.s,‘y)]. Now dn//d y = 0 for y = y/” and dnf/d y > 0 for y BY/“. Further, dyfD/db, is negative under the plausible assumption that <,‘” > .Y$,,Y [compare Riley (1975, p. 179)]. Hence, except for the lowest education level, higher marginal education costs call forth more-able workers and, consequently, yield higher wages at all education levels. Note that, contrary to what is often thought, a higher return to education for women is thus quite compatible with the view that women have somehow less access to education than comparable men. Nevertheless, what counts for the individual is her wage net of education costs. With ability fixed we calculate

@by- cJ/db,) Incons,.=@y/d&) l_vconst.+ Cc,,,/,,).(n/b;).

(10)

on the other hand, due Grantmg that C,,,,,, < 0, a rise of b, increases costs of a unit of education; to the signalling content of education, it induces a higher return to education. Although under moderate signalling the cost effect tends to dominate, and net wages of women fall in response to higher marginal education costs, net wages would be lower in the absence of signalling. 2.2. Fully confirmed

equilibrium

without statistical

discrimination

For a fully confirmed nondiscriminatory equilibrium, that male and female workers with the same education

suppose that government legislation requires receive equal wages and are hired randomly.

R. Haagsma

96

/ Statistical

discrimination

and competitive signalling

Then firms offer wages equal to the average labour productivity of males and females. Assuming second-order conditions to be satisfied, the wage schedule of a nondiscriminatory equilibrium is determined by solving three equations for w*(y) (i =f, m):

w*= c,‘
(11)

y* = qY(n,, Y),

(12)

.I

Ii

w*(y) = CQL y)f,(n,) /Cf,h),

[

I

(13)

where f,(n) is the number of sex i workers of ability n. Eliminating n, or n, by equating (11) and (12), we derive the Pareto-dominating wage schedule in a similar way as before. It can be generally expressed as w = w *( y ; k p). The corresponding equilibrium relationships between ability and education are denoted as n, = n *‘( y ; kp). A note on the feasibility of a fully confirmed nondiscriminatory (NDC) equilibrium is in order. From (2), equating marginal education costs necessarily implies that a NDC-equilibrium has n, > n, for given y. As a consequence, certain women of low ability prefer less education than the lowest level chosen by man, while certain men of high ability prefer more education than the highest level chosen by women. A NDC-equilibrium is therefore only defined over particular segments [np, n:] of [n O, n’]. To determine these segments, let n /” denote the ability of those women choosing the same education (y’) as the least-able men; then nj? = nhf(yO; kp), where y” follows from no = n *m( y” ; k p). Likewise, let n’, denote the ability of those men choosing the same education ( y’) as the most-able women; then n’, = n * * ( y’ ; kp), where y’ follows from n’ = n */( y’ ; kp). We thus find that w = w*(y; kp) is defined over [y’, y’] and n, = n*‘(y; kp) only applies to [ny, n’] if i =f to be feasible we require n(/! 0. For clarity, suppose male and female abilities are uniformly distributed over [no, n’] and let jS denote the proportion of males in the worker population. Differentiating w *( y ; kP, b,, b/b,) totally with respect to b,, holding education fixed, we obtain (dw*/db,)

l_yCOnSt.=- [ P%,_. (dn*“/dy) . [ (dn*‘/dr)

. (dy’/db/) . (dy”/dbf)

+ (1 -P>%, - (n,/bf)]]

where dn*“/dy=(C,,-$z)/S,*_; and

dn*//dy

= (b/b,).

(dn*“/dy);

t

(14)

R. Haagsma

/ Statistical

discriminatwn

and competitwe

srgnalling

91

Although (14) cannot be definitely signed, a few considerations indicate a positive effect on wages. Note first that dn*‘/dy = 0 for y =y” and dn*‘/dy > 0 for y >y”. The sign of dy”/dbj depends or whether or not ability and education are complementary factors in labour production. on SVn,, are substitutes (S,,, 5 0), a rise of bf unambiguously Assuming C,._,.> S,:,, if ability and education raises nondiscriminatory wages. In the more likely case of complementarity, two opposing effects on labour productivity generally occur. As b, rises, each education level is combined with more-able female workers but less-able male workers (granting that the elasticity of female ability with respect to bf is less than one). The sum of both effects seems a monotonic function of ,B, approaching a strictly positive value as p J 0 and zero as /3 t 1. Therefore, also in case of complementarity higher marginal education costs for women tend to raise nondiscriminatory wages for given education. When comparing workers of the same ability level, we find that female wages fall below male wages. Besides, sex differences are widened if differences in education costs are accounted for. The change in net wages of male workers is simply given by the change in wages for fixed education (14) so banning discrimination generally favours net income of male workers. With respect to female wages net of education costs we calculate

const.=@~*/db~)l~, +C,,,,,,,&'b;), (d(w*-c,)/db,)ln I

COnb,.

(15)

equilibrium, are so given C, n ,h,J < 0 female net wages always lag behind and, as in a discriminatory likely to fall Moreover, since a ban may decrease the return to education for women, they can well be worse off.

3. Conclusion In fully confirmed signalling equilibria statistical discrimination generally implies lower net incomes for workers with higher signalling costs. Yet a ban on statistical discrimination always widens the gap in net incomes, as can be seen by combining the comparative-static results of Section 2.1 and 2.2. Statistical discrimination is therefore best combatted here by equalising pre-labour market opportunities themselves.

References Cain. G.G.. 1986, The economic analysis of labor market discrimination: A survey, in: O.C. Ashenfelter and R. Layard, eds., Handbook of labor economics (North-Holland, Amsterdam), Vol. I, 693-785. Riley, J.G., 1975, Competitive signalling, Journal of Economic Theory 10, 174-186. Spence. A.M., 1973, Job market signalling, Quarterly Journal of Economics 87, 355-379. Spence, A.M., 1974, Competitive and optimal responses to signals: An analysis of efficiency and distribution, Journal of Economic Theory 7, 296-332.