ESTUDIOS SOBRE LA ECONOMÍA ESPAÑOLA

Coral del Río Otero

Carlos Gradín Lago

Olga Cantó Sánchez

EEE 192

September 2004

http://www.fedea.es/hojas/publicado.html

ISSN 1696-6384

Las opiniones contenidas en los Documentos de la Serie EEE, reflejan exclusivamente las de los autores y no necesariamente las de FEDEA.

The opinions in the EEE Series are the responsibility of the authors an therefore, do not necessarily coincide with those of the FEDEA.

Coral del Río Otero Carlos Gradín Lago Olga Cantó Sánchez (Dpto. Economía Aplicada - Universidade de Vigo)

July, 2004

ABSTRACT

This paper presents the advantages of taking into account the distribution of the individual wage gap when analysing female wage discrimination. The limitations of previous approaches such as the classic Oaxaca-Blinder and the recent distributive proposals using quantile regressions or counterfactual functions are thoroughly discussed. The new methodology presented here relies on Jenkins' (1994) work and proposes the use of poverty and deprivation literature techniques that are directly applicable to the measurement of discrimination. As an illustrative example we measure female wage discrimination in Spain aggregating individual wage gaps estimated with OLS and quantile regressions.

Keywords: distributive analysis, gender, wage discrimination.

JEL Classification: I32, J16, J31, J71.

Address for correspondence: Coral del Río Otero, Departamento de Economía Aplicada, Facultade de CC. Económicas e Empresariais, Universidade de Vigo, Campus Lagoas Marcosende s/n, 36310-Vigo, Spain. Fax: +34 986812401, e-mail: crio@uvigo.es

¹ We thank the assistants to the VII Encuentro de Economía Aplicada for their comments. Further we are greatful for the finance from the Instituto de la Mujer (Ministerio de Trabajo y Asuntos Sociales) in the completion of the Project 35/02 “Mercado de trabajo, pobreza y género: nuevos enfoques”, within the IV Plan Nacional de I+D+I, as well as that from the “Programa de Promoción Xeral da Investigación do Plan Galego de IDIT” (Xunta de Galicia) and the Universidade de Vigo (“Cofinanciación de Proxectos de Investigación”).

1. INTRODUCTION

The lower wages paid to female workers in comparison with males in most labour markets can easily be checked empirically using individual data on wages. The fact that these wage differentials are not justified in terms of labour productivity is usually known as female wage discrimination, a phenomenon that affects the majority of labour markets where it has been analysed.² The always increasing interest on these matters in the literature is reflected in the large amount of different methodological proposals that, since the classical works of Oaxaca (1973) and Blinder (1973), have tried to quantify the level of gender discrimination in different countries.

The literature has underlined on the one hand, the ineficiencies that discrimination imposes in the functioning of the labour market and, on the other hand, the level of inequality it generates by seriously increasing the risk of social exclusion for discriminated individuals.³ Inequality arguments led Jenkins (1994) to propose the adequacy of the procedures used to measure poverty from the income distribution literature to the analysis of wage discrimination. Indeed, both phenomena have strong similarities. Most precisely, both imply some income or wage gap. Either individual income does not provide a minimum level of resources (Atkinson, 1998), or, similarly, the female wage is below what she would receive were she male but otherwise had identical attributes. From this perspective, precisely as the lack of a sufficient income in the poverty literature, the female wage gap reveals itself as genuinely individual, implying that its distribution in the population under study should play a crucial role in its measurement.

Following the proposals developed by Sen (1976) in the poverty literature, Jenkins (1994) underlines that there are three basic questions to answer if we aim to analyse discrimination rigorously: 1) define what we mean by direct wage discrimination; 2) identify, using the previous definition, which individuals suffer discrimination and in what quantity; and 3) sum up the wage gaps using an index that verifies a series of normative properties that make all value judgements explicit. The classical methodology, widely used in empirical work, limits the analysis to the calculation of the mean wage gap. Doing so they are, implicitly, imposing the same weight on each wage gap independent of its relative relevance or value within the wage distribution. Jenkins (1994), instead, suggests researchers to use the individual wage gap in the measurement of discrimination and centres the discussion in the analysis of the distribution of this gap using the theoretical advances in poverty and deprivation research. Thus, in this context, what is relevant is precisely the difference between what the individual would earn if she would not face discrimination and what she actually earns. The analysis then focuses on the indicators that sum all these differences using weights for the different discriminatory experiences incorporating a wide range of different judgements about how these gaps are aggregated in a systematic yet transparent way.

² Blau and Khan (2000 y 2003) present some recent empirical evidence on this matter.

Some recent lines of research that aim for the consideration of distributional aspects propose the use of quantile regressions in the estimation of wage equations in order to increase the number of points in the earnings distribution at which the wage gap is evaluated. Other proposals include a variety of techniques to estimate counterfactual earnings distribution functions in order to compare them with the original wage distribution and quantify the effects of wage differentials along the whole earnings range. Certainly both approaches allow us to obtain more information from the observed wage distributions than the classical approach does. Nevertheless, and besides what it may seem, in both cases, estimated wages are compared avoiding the complexity of considering the individual aspect of discrimination. Moreover, within the counterfactual wage distribution functions the comparison of distributions assumes implicitly the unlikely fact that each percentile includes the same individuals.

Our paper examines in detail the advantages of using a new distributional methodology already proposed by Jenkins (1994) in comparison with other available distributional approaches for the analysis of wage discrimination. We should be aware, however, that the estimation of wage equations using quantile regressions and the use of normative measures of discrimination á la Jenkins are complementary techniques. Therefore, as we will present in an empirical exercise at the end of this paper, it is posible to use quantile regressions to identify the individual levels of discrimination and then use normative measures that allow us to sum up the different estimated wage gaps.

³ This effect is usually referred to as “the feminization of poverty”.

⁴ The main limitation of the classical approach to the measurement of discrimination is that of measuring it in a single point. However, given that this point is the mean, this guarantees that the compared wage levels, with and without discrimination, belong to the same woman. This partially tackles the individual

The main contribution of this paper is that of offering a new normative framework for the study of wage discrimination based on the poverty and deprivation literature. In order to do this we rely on Jenkins (1994) and Shorrocks (1998) work. We detail the general limitations of the most usual distributive techniques and propose a variety of discrimination measures that allow us to aggregate individual wage gaps. These measures are explicit about the value judgements they imply and on which we would aim to reach an agreement. This allows us to rank, in a robust way, a list of women’s earnings’ distributions in terms of their discrimination level and compare the discriminatory experiences of women with different attributes. In order to provide empirical evidence on the theoretical contribution of the paper, we contrast the advantages of our approach using a sample of Spanish data. This last exercise lets us quantify the improvement of the use of quantile regressions respect to OLS classical regressions in the process of the individual wage gap identification. We should underline here also that, even if we recurrently refer to female wage discrimination, the theoretical contributions of this paper are readily aplicable to any other source of discrimination (race, religion, sexual orientation, origin, etc.).

The paper is organised as follows. Section 2 presents the classic approach to the measurement of discrimination and gives a sound justification of the importance of considering distributive aspects in discrimination measurement. In section 3 we discuss the limitations of a variety of distributional techniques recently used in the study of wage discrimination. Section 4 presents our proposal for the measurement of discrimination and details its main contribution. In section 5 we provide empirical evidence on the advantages of our techniques on a sample of Spanish wage microdata. Finally, section 6 concludes by presenting our main findings.

2. THE RELEVANCE OF THE DISTRIBUTIVE APPROACH IN ANALYSING WAGE DISCRIMINATION

dimension of discrimination. The techniques used in the decomposition of the wage gap using quantile regressions maintain this property but at different points of the wage distribution.

2.1 The problem of finding a satisfactory definition of wage discrimination

Human capital theory assures that wages are directly linked to workers’ productivity levels. Therefore, in any competitive labour market, a workers’ earnings should equal her marginal productivity and the presence of gender wage differentials, easily observable in empirical work, is not sufficient to presume the existence of discriminatory practices. The reason is that the different salary paid for female work could be justified, at least in part, by the existence of differences in productivity. Therefore, nowadays there is a wide consensus on identifying gender wage discrimination as the difference in earnings between male and female workers who are otherwise identical in their attributes and thus in their expected productivity.

2.2 Wage discrimination: The identification problem.

Once the discriminatory experience is defined, detecting its presence and quantifying it presents some further difficulties: in the first place productivity is not directly observable. Thus, who are those female workers actually earning below their productivity level? Traditionally, in order to overcome this problem, researchers have used the information on workers’ observable characteristics to approximate individual expected productivity. This strategy has generated a large international empirical evidence on the factors that generally determine workers’ wages and has helped to contrast the predictions of human capital theory. The attributes considered to be most significant in determining workers’ productivity are years of schooling, age, labour market experience and tenure. Nevertheless, wage equations usually include other explanatory variables that are directly related to labour market demand and supply such as occupation, type of labour contract, firm sector, firm size and ownership (public or private), type of collective agreement, level of unionization and firm geographical situation.

⁵ This procedure is not applicable when using the counterfactual functions approach given that this method does not provide us with individualised wage gaps.

⁶ For two representative examples that illustrate this definition see Cain (1986) or McConnell, Brue and Macpherson (2003).

⁷ See Willis (1986) for a detailed explanation of the main determinants of workers’ wages.

Once we select the a priori potential determinants of the individual wage level, using the estimated regression coefficients we can first identify which of them are most relevant in the individual wage determination and, secondly, estimate how, in our particular labour market, a certain attribute is remunerated. Thus, once we have obtained the estimated values of male and female wages we will be able to quantify which part of the estimated wage differential is due to the different attributes of each group and which is due to a differentiated remuneration of otherwise identical characteristics related to different market premiums for male or female workers. These premiums, that directly depend on gender, and are not explained by endowments or productivity differences of individuals, are what we can assign to discriminatory practices based on gender.

Two separate mincerian log wage equations for males and females are estimated:

\[\begin{array}{r l} & {\ln (y _ {h i}) = Z _ {h i} ^ {'} \beta_ {h} + u _ {h i}} \\ & {\ln (y _ {m i}) = Z _ {m i} ^ {'} \beta_ {m} + u _ {m i}} \end{array}\]

where h refers to males, m to females, and stands for the worker hourly wage, is the vector of individual characteristics that we consider relevant in explaining wages, are the characteristics’ rates of return, and is the corresponding error term. Once the model is estimated we are able to predict both the estimated wage of a female worker, , and her potential wage if her attributes were remunerated as if she were male, :

\[\hat {y} _ {m i} = \exp (Z _ {m i} ^ {\prime} \hat {\beta} _ {m})\]

\[\hat {r} _ {m i} = \exp (Z _ {m i} ^ {\prime} \hat {\beta} _ {h})\]

The individual wage gap reflects the estimated wage discrimination experienced by a female worker i, being the distribution of the estimated discrimination in the female workers group.⁸

⁸ In his classic survey Cain (1986) offers a detailed reference to the most important theories that try to explain discrimination and discusses mincerian models. Since then, a large list of papers have tried to

2.3 Wage discrimination: The aggregation problem.

At this point it is now time to face the most difficult issue in the measurement of discrimination: decide on how to aggregate the individual discriminatory experience. We need to choose some statistic that is able to make the best use of the information contained in the previously estimated wage equations in order to measure the extent of overall discrimination in the population.

Traditionally, and based on OLS estimations of mincerian equations, discrimination has been evaluated in the mean distribution of the characteristics, and has thus quantified the wage discrimination suffered by the mean female worker when compared to the mean male worker. This is precisely the approach proposed by Oaxaca (1973) and Blinder (1973) in their seminal articles, which has been recurrently utilised in the literature on wage discrimination. In the original Oaxaca-Blinder decomposition the mean observed wage gap is divided in two components relying on the well-known property of OLS estimated regressions.⁹ A first component would quantify the labour market premium on the mean differences in characteristics between genders while the second component would show how differently the labour market rewards gender evaluated at the mean female characteristics:

\[\overline {{\ln (y _ {h})}} - \overline {{\ln (y _ {m})}} = (\overline {{Z _ {h} ^ {'}}} - \overline {{Z _ {m} ^ {'}}}) \hat {\beta} _ {h} + \overline {{Z _ {m} ^ {'}}} (\hat {\beta} _ {h} - \hat {\beta} _ {m}) = A + B.\]

Graph 1 shows, in the unidimensional case, that the first component (A) identifies the wage gap we would observe in the absence of discrimination, that is, if the characteristics of males and females were rewarded at the same return rates. Here the same rate implies that male rates of return are identified as the non-discriminatory experience.¹⁰ The second component (B), denotes the wage penalty the mean female worker faces given that she has a different remuneration of attributes compared to males. Even if seldomly noted, it is easy to check that B is the mean of the differences of predicted male and female wages estimated for each woman in the population (in our example: . The choice of the male wage structure as the non-discriminatory reference is equivalent to considering discrimination as the disadvantage of any group respect to the most advantaged group. This would not be true in the case of choosing some other reference.

improve the robustness of mincerian wage equations estimations by reducing their selection biases (in relation with female labour market participation), the potential endogeneity biases (basically related to educational attributes) or the incorrect specification. Recently, in Kunze (2000) we find a revision of the most relevant empirical literature in trying to achieve a consistent estimation of the parameters in wage equations.

⁹ Property that guarantees that the OLS estimated wage evaluated at the mean values of all attributes is equal to the observed mean wage.

Graph 1. Wage discrimination using OLS

Building on this second component, Oaxaca (1973) aggregated measure of discrimination is:

\[D _ {O} = 1 0 0 * \left[ \exp (\overline {{Z _ {m} ^ {\prime}}} (\hat {\beta} _ {h} - \hat {\beta} _ {m})) - 1 \right].\]

¹⁰ Assuming that individual attributes are exogenous and that they would not experience any change in the absence of discrimination.

Whatever the non-discriminatory remuneration structure of reference, the use of the wage distribution mean is a large waste of information. In the first place the mean does not allow for differences in the discriminatory experience at different points of the wage distribution. Further, and most importantly, it implies assuming that to give the same weight to each different individual discrimination experience is a desirable way of aggregating wage gaps, independently of the actual degree of discrimination suffered by each individual. This all implies implicitly, and in an obscure way, the imposition of value judgements that are rather implausible from a normative point of view. Moreover, there has been little, if any, discussion in the literature on the adequacy of these assumptions. This is all most probably due to the attractive mathematical properties of the mean and also to the general lack of discussion of normative implications in discrimination measurement. In this context, we consider that the study of discrimination should aim to rely on flexible and complete measures that allow us to identify the differences in results when we incorporate, explicitly, the different judgements in the aggregation of individual information.

A number of papers have utilised a wide range of econometric techniques in order to incorporate distributive aspects in the comparative analysis of wage distributions. Since the Juhn, Murphy and Pierce (1991, 1993)¹¹ seminal papers, a large list of works have suggested that the market remuneration to individual endowments is not constant along the wage range.¹² Buchinsky (1994) presented empirical evidence using quantile regressions in the study of the evolution of wages in the US. Di Nardo, Fortin and Lemieux (1996) quantified the effects generated by the change in the distribution of workers’ characteristics on wage density using non-parametric regression techniques to estimate counterfactual wage distributions (which permited them to combine one period’s population attributes with the returns structure of another). More recently, in their analysis of Portuguese wage inequality, Machado and Mata (2001) used quantile regressions to model the conditional wage distribution on workers’ characteristics allowing for the measurement of different returns for each attibute at different points of the wage range.

¹¹ These authors use OLS regressions in providing alternative disaggregation of the estimated and counterfactual wage differences for different time periods.

Within the studies that aim to measure gender wage discrimination, Blau y Khan (1996, 1997) explained the international differences in female wage gaps and their evolution in time using the methodology proposed by Juhn, Murphy and Pierce (1991).¹³ Fortin and Lemieux (1998) analysed the wage gap along various years using rank regressions in order to estimate the probability that an individual receives a salary within a certain wage interval. More recently, Bonjour and Gerfin (2001) applied the methodology proposed by Donald, Green and Paarsch (2000), which uses wage distribution’s flexible estimators based on duration models, to decompose the wage gap in Switzerland. Finally, the most recent literature has very often used quantile regressions in order to decompose the gender wage gap at different points of the wage distribution. Examples of this are Reilly (1999) and Newell and Reilly (2001) in the analysis of excommunist countries in transition, Albrecht, Björklund and Vroman (2003) in their study of the “glass-ceiling” in Sweden,¹⁴ and García, Hernández and López-Nicolás (2001), Gardeazábal and Ugidos (2004), and Dolado and Llorens (2004) for gender discrimination in the Spanish labour market.¹⁵¹⁶

We sustain that all these recent approaches to the analysis of discriminatory practices are a clear improvement to other previous approaches to the measurement of discrimination but present, nevertheless, some important limitations. In some cases, problems arise from the conceptual confusion of the distributive aspects of measurement with the distributive effects of discrimination. Also, the fact that all the procedures proposed try to avoid incorporating value judgements in the aggregation of the different discriminatory experiences, under a normative shelter, make much more difficult any comparison on the matter. Given the interest on both issues: distributive aspects and value judgements in the aggregation of the gaps, we should pay special attention to the arguments that sustain them.

¹² We identify this remuneration within a given firm type and sector. All these effects are included in the model’s estimated parameters.

¹³ This methology allowed them to take into account the role played by the wage structure in the explanation of the gender wage gap.

¹⁴ These authors use techniques developed by Machado and Mata (2004) where quantile regressions are used in order to estimate counterfactual density functions.

In García, Hernández and López-Nicolás (2001) female wage discrimination in the Spanish labour market increases along the wage range both in absolute and in relative terms (in relation to the total wage gap). These authors use instrumental variables in order to endogenise education and other econometric techniques that allow us to avoid a selection bias. In contrast, Gardeazábal and Ugidos (2004) obtain that relative female wage discrimination in Spain decreases as wages increase. Here authors estimate the discrimination at each quantile using the “corresponding” quantile characteristics and not mean population characteristics, as García, Hernández and López-Nicolás (2001) did. Dolado and Llorens (2004), instead, used (even if partially) Albrecht, Björklund y Vroman (2003) proposal and identify the highest wage discrimination levels for Spain in the last deciles of the female wage distribution for those women with a high level of education.

¹⁶ Other recent works that have tackled distributive issues from significantly more simple methodologies are, inter alia, Li, Gerry and Kim (2004), Méndez and Hernández (2001) and Vartiainen (2002).

3. THE LIMITATIONS OF RECENT DISTRIBUTIVE APPROACHES

3.1 The comparison of conditional wage distributions: distributive aspects and conceptual errors in measuring discrimination

In order to provide an illustration of the problems that arise when using counterfactual distribution functions in the estimation of wage discrimination, let us undertake a simple comparative exercise. We estimate a Generalised Lorenz Curve (GLC) of and the corresponding Generalised Concentration Curve of and compare them. Both curves use the same ordering of female workers: ascending order of , and accumulate or , respectively.¹⁷ Thus, the analysis of their differences offers us the discriminatory pattern as we incorporate more and more female workers, that is, as we aggregate individual discriminatory experiences.

Empirically speaking, it is not difficult to think that moving from one distribution to another there will be a number of reordered females. However, GLC and GCC curves, are not affected by these reorderings given that, once we preserve the initial order we guarantee that, if the curves are identical, there is no posible direct wage discrimination.¹⁸ Unfortunately, not all discrimination measures are inmune to these changes in order. Equality between wage distribution or density functions, y , does not guarantee the absence of discrimination in all females’ salaries given that there is no guarantee that will preserve the ordering in . Theoretically, it could be the case, that the discrimination suffered by most female workers may be compensated by the “advantage” or “priviledges” of just a few females, as presented in Graph 2, where women A and B, suffer discrimination, while woman C obtains a higher salary than male workers with otherwise identical attributes.

_m yˆ

4

¹⁷ The GLC curve is calculated in each accumulated proportion of the sample of female workers as the sum of their estimated wages divided by sample size once all women have been ordered in ascending order in terms of their estimated wage level The GLC curve was proposed by Shorrocks (1983) as a , mi yˆ . Social Welfare criteria in the comparison of income distributions. A GLC is nothing more than the corresponding Lorenz curve multiplied by the mean of the variable under study. The GCC is calculated using the values of rˆ , in a similar way to the GLC, but maintaining the ordering unde In its non- LC, r _m yˆ . generalised version the concentration curve is often used in the study of tax progressivity of using indices that reflect the differences between the pre-tax Lorenz curve and the Concentration Curve of disposable income after tax payment.

¹⁸ However, the comparison of GLC and GCC curves is not free of criticisms. Favaro and Magrini (2003) underlined that the existence of females whose reduces the precision of this indicator. mi mi yˆ > rˆ Moreover note that the aggregation of discriminatory experiences is undertaken under the ordering of their predicted wage level and not under their estimated discrimination. This has implications on the normative properties of discrimination indices that are a function of the area between the curves which

Graph 2. Permutations in female wage distribution The former is clearly an extreme case, but regarding more plausible empirical situation in which , it may well be that part of the earnings differentials for women evaluated at each decile disguise a number of reorderings. Thus, lets assume that we depart from a wage distribution such as the density fuction on the left hand side of Graph 3. Suppose that once we eliminate direct wage discrimination the new density function moves uniformly to the right. In this particular case, the distributive analysis using quantile differences would conclude that all female workers experience the same absolute level of discrimination, whatever their wage.

may not imply unanimity. Further, as defended by Jenkins (1994), the construction of families of indices that allow for the parametrization of the value judgements included in the measurement of discrimination is a clearly superior strategy, both in terms of transparency and in terms of the analysis of results, as we will see in the next paragraph.

Graph 3. Transfer of female wage distribution Nevertheless, this may not be necessarily true. It may be the case, as depicted in Graph 4, that all type A women, that initially earned , earn when eliminating the discriminatory component. Additionally, a similar number of those female workers that were earning could be experimenting a lower wage change once we eliminate discrimination and thus appear in The rest of type B women would reach the same wage level than females in A, the level Obviously, the level of discrimination suffered by group A is much larger than that suffered by group B but neither the study of the differences in the mean (as expected) nor the comparison of quantile counterfactual distributions would detect it.

Graph 4. Wage discrimination using counterfactual densities In other words, when comparing density functions we are not only quantifying discrimination but also considering the reorderings in the wage distribution when discrimination is taken into account. In this way, the measurement of discrimination is contaminated in the presence of mobility between quantiles. The reason for this was already noted by Jenkins (1994):

“The root of the problem is that discrimination depends on the distribution of wages differences, not on the differences between two wage distributions. (It is only when using means that these concepts coincide). We should be interested in whether each and every woman is equitably paid , and there is some discrimination in aggregate as long as at least one woman is unfairly remunerated Equality of means of and (or higher moments) is a necessary but not sufficient condition for the absence of discrimination” (pp. 86).¹⁹

The comparison of the mean, the variance or quantiles of the wage distribution functions does not allow for the consideration of the individual discriminatory experience. It only allows us to quantify the “anonymous” mean differences, moving the focus of the analysis from the individual to the distribution. This strategy makes it impossible to assure that a certain decile suffers more or less discrimination than another one, given that the women that initially were placed into each of them may not be the same women, once individual discrimination is taken into account. Thus, the existence of differences between both distributions does not allow us to assure that women with high earnings experience higher (or lower) discrimination than women with a low wage. This comes about simply because we do not know where these women are placed in the non-discriminatory distribution of wages. In any case, various techniques in the literature of gender wage discrimination are based implicitly on the assumption that these are the same women.²⁰ Their estimations will be more or less wrong depending on the number of reorderings that actually take place when discounting discrimination. Clearly, these papers should be cautious in the interpretation of some of their results. The use of these techniques should remain within the interesting study of the distributive effects of discrimination but should never be confused with a way of identifying the actual level of discrimination in a wage distribution.²¹

¹⁹ This argument was used in Jenkins (1994) to criticise the procedures developed by Dolton and Makepeace (1985) based on the study of wage distribution differences using higher statistical moments

3.2 The need for normative measures in the measurement of wage discrimination

Regarding the second issue related to value judgements mentioned in section 2, it is important to be aware that neither the methodologies based on wage distribution functions nor those using quantile regressions consider how to weight the different levels of discrimination estimated along the wage range. Implicitly, they avoid the construction of a single aggregated indicator which prevents from any comparison of discrimination levels between distributions.²² This decission may be argued as adequate in the aim of incorporating the least value judgements possible in the analysis as, implicitly sustained by Gardeazábal and Ugidos (2004):

instead of the mean. Other pioneering works that incorporate distributive aspects and include this assumption in the study of wage discrimination are those of Munroe (1988) and Stewart (1983).

²⁰ Some recent works that suffer this problem within the literature of the analysis of gender wage discrimination are Albrecht, Björklund and Vroman (2003) and Bonjour and Gerfin (2001).

²¹ Note that the decomposition of wage discrimination using quantile regressions does not suffer from this problem given that it quantifies the level of discrimination experienced by females situated at different wage quantiles and does not evaluate the wage difference between them and those that occupy the same position in the non-discriminatory distribution. Dolado and Llorens (2004) avoid the construction of the counterfactual wage distribution and that is why their work is not affected by this problem, even if they follow Albrecht, Björklund and Vroman (2003).

²² Appart from the trivial case in which a given wage distribution presents more discrimination in all estimated quantiles.

“The measures of gender wage discrimination used in the literature summarize in a scalar descriptive statistic the degree of discrimination in the distribution of wages. There is a good reason for doing so, as a scalar statistic may be used to infer the overall level of wage discrimination of the population under study. However, the use of a scalar statistic may not be appropriate for comparisons among two or more populations, as two wage distributions might exhibit the same value of the scalar statistic while discrimination could be very differently distributed in the two populations. This problem has been raised a large number of times in the studies of income inequality. Measures of income inequality such as the popular Gini coefficient give a general view of the degree of income inequality, but two income distributions may have the same value of the Gini coefficient while income might be radically differently distributed. It is well known that two income distributions with the same Gini coefficient may have crossing Lorenz curves indicating differences in the distribution of income. We propose a measure of relative gender wage discrimination at each quantile of the distribution of wages which allows us to analyze how is discrimination distributed within the population” (pp. 2-3).

In our view, these authors are right but their arguments do not imply that we should avoid the use of aggregated discrimination measures. We agree that the classical discrimination index, is constructed using, at least debatable, aggregation assumptions. It is, nevertheless, true that the advantages of quantile regression in the analysis of discrimination are scarce if compared to the improvement of Lorenz criteria in the analysis of inequality. Quantile regressions allow us to obtain a more precise estimation of the discriminatory experience but only offer us some punctual measurements of discrimination at different quantiles avoiding the discussion of any aggregation criteria. This implies solving the judgements issue in a paradoxical way: no aggregation is undertaken and therefore no value judgements are incorporated. This is clearly an option, but, we should not forget that a Lorenz dominance criteria, taken as a reference, also aggregates income levels in order to compare different income distributions in terms or inequality. In fact, it does so under a minimum amount of value judgements on which we can reach some consensus.²³ This adds robustness but incompleteness to the orderings. It is precisely in those cases in which Lorenz criteria cannot order functions, when complete inequality indices (Gini, Theil or Aktinson index) are most interesting. The latter incorporate a larger number of value judgements but allow us to undertake slightly more delicate orderings. Often, the results offered by these battery of indices do not coincide but differences are not at all random. They agree with the normative properties of each of them. A deep analysis of these permits us the best comprehension of the analysed phenomenon.

Jenkins (1994) approach advances in this direction and proposes discrimination measures that allow for the aggregation of wage gaps. Some of them are based on Lorenz criteria, and others allow for a complete ordering of discrimination levels assuming a larger number of value judgements. Further, as we will shortly see, some of these complete measures are decomposable allowing us to deepen the analysis searching for the roots of wage discrimination.²⁴ Our proposal extends his approach incorporating some improvements. We propose a normative framework in which to insert discrimination measurement following the literature on deprivation.

4. NORMATIVE DISCRIMINATION MEASURES

So far we have shown that, firstly, when analyzing discrimination we should focus on the “experience of each individual”. Given the bidimensional nature of this information, summarised by , any measure which tries to quantify it should be written as a function of , rather than as a function of and taken separately. Secondly, we need to aggregate this individual experience. This implies taking value judgements into account, and these are, necessarily, of a subjective nature. Is this a problem? Not if we accept that discrimination is a bad in the same way as poverty or the duration of unemployment are. Hence the question is: What properties should a measure of discrimination satisfy? We propose that the properties the literature on economic poverty has widely accepted as satisfactory requirements for any poverty measure, are also adequate in the case of the study of wage discrimination.

²³ Basically resumed in two axioms: simetry (or anonimity) and the Pigou-Dalton Principle of Transfers.

24 Surprisingly, few papers have followed Jenkins (1994) approach. We only know of the empirical works of Denny, Harmon and Roche (2000) analysing wage discrimination of inmigrants in the UK; Makepeace, Paci, Joshi and Dolton (1998), analysing gender wage discrimination for the UK; Hansen and Wahlberg (2001) for the Swedish case; and Ullibarri (2003), for the Spanish case. In all these works the indices are used just as proposed by Jenkins (1994). In Favaro y Magrini (2003), differently from the rest, we find some criticisms to Jenkins’ approach and authors propose the estimation of bivariant density functions as an interesting alternative. In our opinion this is not an alternative to Jenkins’ techniques but an useful previous descriptive tool to the deeper distributive analysis of discrimination we present here. It is true, however, that this could be complemented with some index which aggregates wage changes experienced by females (for example an index based on transition matrices). In doing this we incorporate ad hoc value judgements associated with the index’s aggregation properties in an obscure way.

4.1 Normative properties of discrimination indices

Consider two vectors of wage gaps, and , where and , n and s being respectively the total number of female workers in each distribution. represents the level of discrimination which corresponds to distribution for a given measure d. The minimal set of normative properties or axioms that should satisfy are the following:

1) Continuity Axiom. must be a continuous function for any vector of wage differences, , of its domain.

2) Focus Axiom. If we can obtain from by rises in wages of nondiscriminated women, , then .²⁵

3) Symmetry (or Anonymity) Axiom. If can be obtained from by a finite sequence of permutations of individual discrimination levels, then

4) Replication Invariance Axiom. If we can obtain from by replications of the population, then

5) (Weak) Monotonicity Axiom. If can be obtained from by increasing the discrimination level of a woman, then

_{i i} yˆ ≥ rˆ

²⁵ The existence of female workers with , should not be used to balance discrimination suffered by the rest. In the same way, within the literature on poverty measurement, an increase in non-poor income does not change the poverty level (keeping the same poverty line). Nevertheless we should indicate that the analysis of non-discriminated women is also interesting but a different topic. Note that our approach also allows for the analysis of male discrimination using female wage structure as a reference.

6) (Weak) Transfer Axiom. If we can obtain from by a sequence of “regresive transfers” between two discriminated female workers, so that the one with the highest discrimination suffers an increase in her wage gap equal to the decrease experienced by the other, then

The Continuity Axiom is a reasonable property for any index in order to guarantee that small changes in wage differences do not lead to big changes in discrimination levels. The Focus Axiom requires the index to be dependent on the distribution of discriminated women while disregarding completely the wage level of the rest of female workers. This does not mean that measures verifying this axiom are necessarily independent of the existence of women with wage advantages with respect to male workers,²⁶ but it does require that these salary advantages are not taken into account when measuring aggregate discrimination.²⁷

The Symmetry Axiom guarantees that the index does not favour any particular woman. The Replication Invariance Axiom is a technical property that allows for comparisons between distributions of different size. The two other final axioms lead to two basic properties. The Monotonicity Axiom refers to discrimination intensity, so that a worsening in the position of a discriminated woman yields a higher level of aggregate discrimination. And, finally, the Transfer Axiom implies that a higher inequality level between discriminated women, in terms of their discrimination sharing, turns into an increase in the discrimination index.²⁸

Accepting the axioms above, we will be able to construct discrimination profiles by accumulating individual wage gaps and develop some dominance criteria to rank wage distributions according to their discrimination level. Subsequently we could make a correspondence between these rankings and those obtained by using complete discrimination indices that also satisfy these properties. This is the case in the inequality and poverty field, where there are valuable theorems that establish a relationship between the income distributions ranking obtained by Lorenz or TIP’s dominance criteria and those obtained by complete inequality and poverty indices compatible with those criteria. Thus, by using a minimal set of judgements, summarised in the above properties, we will be able to identify particular empirical cases where the discrimination distribution ranking is independent of the index chosen, since all indices yield the same result. This makes our analysis of discrimination significantly more robust.

²⁶ In fact, the share of these women over total female workers will be taken into account in all indices that verify continuity, monotonicity and replication invariance axioms (see Zheng (1997) for the poverty case).

27 This is similar to considering that the existence of famous Gypsy musicians or Afro-American sportsmen from discriminated groups should not offset the inferior position of most individuals in these groups.

²⁸ This axiom is the result of applying Pigou-Dalton’s Transfer Axiom to the group of discriminated women.

This was the approach followed by Jenkins (1994) when he used the Inverse Generalised Lorenz Curve (IGLC),²⁹ and defined discrimination indices consistent with the dominance criterion, parameterised to take into account different discrimination aversion degrees.³⁰ Later, Shorrocks (1998) generalised these relationships in the continuous case and summarised previous results obtained by different authors in the deprivation field.³¹

4.2 Dominance relations between Discrimination Curves

Let us define be the vector of individual wage discrimination, which corresponds to the wage gap

\[g _ {i} (x _ {m}) = \max \left\{(\hat {r} _ {m _ {i}} - \hat {y} _ {m _ {i}}), 0 \right\}\]

The Discrimination Curve represents for each the sum of the first per cent of values divided by the total number of female workers, n, once these have been ranked from a higher to a lower wage discrimination level. Hence, g satisfies that , and for each value of the curve can be written as:

²⁹ This curve represents the per capita wage gap, for each cumulative proportion of women, once they have been ranked from higher to lower wage differentials.

Xm,

³⁰ Note that Jenkins (1994), when defining the IGLC on absolute values of x_m, does not impose the focus axiom to the indices he proposes. However, as it has been shown, it seems reasonable to newly define the variable and the indices he proposes taking that axiom into account.

³¹ These results are derived from works by Spencer and Fisher (1992), and their “absolute rotated Lorenz curve”, and from Jenkins and Lambert (1997, 1998) and their TIP (“Three ‘I’s of Poverty”) curves in the analysis of poverty. Jenkins (1994) refers to the “inverse generalised Lorenz curve” in the analysis of discrimination, while Shorrocks (1993) applied that approach to the unemployment duration profiles. Also, Blanke and Shorrocks (1994) use this approach to study the length of time spent in poverty.

\[D (g; p) = \sum_ {i = 1} ^ {k} \frac {g _ {i}}{n}\]

where is any integer number such that The Discrimination Curve is the IGLC defined for , rather than for absolute values of wage gaps, as in Jenkins (1994). The latter implies, counterintuitively, considering positive and negative wage gaps as equivalents.

D(g;p) accumulates individual discrimination levels, from higher to lower discrimination, divided by n. As shown in Graph is a positive, increasing and concave function; where , and takes a constant value when is obtained, precisely when we consider the last discriminated woman,

Graph 5. Discrimination Curve

D(F;·)

³² In the continuous case we would consider a measure of individual wage discrimination, given by variable u, distributed in the female population as the distribution function F. Afterwards, and following Shorrocks (1998), we would define the discrimination curve as:

( ; ) ( ) ( ) , [ ] 0,1¹₁ ¹_{(1 )} = ₁ = ∈ ∫ ∫ ₋^{∞ −}₋ D F p ₋ udF u F q dq p _{F p p}

y

³³ This is an adaptation of Figure 1 in Jenkins y Lambert (1997), where the properties of the TIP curves are shown to measure aggregate poverty.

The shape of the above curve provides us with useful information. First, it shows the incidence of discrimination so that to identify the number of discriminated women we only need to know the percentile where the curve becomes a horizontal line, Second, it informs us about its intensity, since the height of the curve is the accumulated wage gap averaged by the number of female workers. Third, it also shows the inequality aspect of the discrimination distribution by the degree of concavity of the curve before point h.³⁴

Definition of dominance in discrimination. Given two discrimination vectors, we would say that:

\[g ^ {l} \text { dominates } g ^ {2} \text { in a discriminatory sense if }\]

\[g ^ {1} \neq g ^ {2} \text { and } D (g ^ {1}; p) \leq D (g ^ {2}; p) \text { for any } p \in [ 0, 1 ]\]

It is straightforward then to show that this dominance criterion is closely linked to the six properties mentioned above: the continuity axiom (small changes in g yield small changes in the curve); the focus axiom (the curve becomes horizontal when the first non-discriminated woman is included in the calculation of the cumulative share, so that the wage advantage of these women is not taken into account); the symmetry axiom (since the only aspect of female workers considered is their discrimination level, which makes impossible to identify them); the replication axiom (the curve does not change when the initial population has been replicated); the monotonicity axiom (the curve turns upwards when the discrimination level of any woman increases); and the transfer axiom (the curve increases its degree of concavity when the discrimination is more unevenly distributed while the average discrimination level is kept unchanged).

From this definition we can establish a relationship between dominance in the discrimination sense and the set of aggregate indices, , that satisfy in the continuity, focus, monotonicity, symmetry, transfer and replication invariance axioms.

³⁴ Note that if all discriminated women suffered the same absolute discrimination level, the first part of the curve would be a straight line where the slope would be equal to the common individual discrimination.

Theorem:³⁵

For any pair of discrimination distributions, and , it follows that,

dominates in a discriminatory sense

\[d (x _ {m} ^ {1}) < d (x _ {m} ^ {2}) \text { for any } d (\cdot) \in d ^ {*}\]

Hence, a higher discrimination curve leads, unambiguously, to a higher discrimination level.

This result also allows us to compute differences in discrimination between two wage distributions, taking Jenkins and Lambert (1998) as a reference.³⁶ Let us consider, for example, that, given curve dominance, wage distribution B has a higher discrimination level than A. In this case, it can be helpful to increase estimated wages, , in distribution B, by multiplying them by a number higher than 1, while keeping constant. This entails decreasing individual wage gaps proportionally, so that later we can check if the initial relationship of dominance still holds. If it holds, we could repeat the exercise to determine the larger interval (written in terms of wages in distribution B) where distribution A has lower discrimination levels. In this way we can study how robust and intense is our initial result even without using complete discrimination indices.

4.3 Complete indices consistent with dominance discrimination

Since the dominance criterion is not always able to give us conclusive results in empirical applications (the estimated discrimination curves can cross), it is interesting to consider the use of some of the indices belonging to . We are interested in those that satisfy the axioms above and also some property that may be of special interest for wide empirical analysis, as for example: decomposability.

35 This result was first shown in Shorrocks (1993), where it was used to study the duration of unemployment, and in Jenkins and Lambert (1993), in the poverty field. These works established the basis for later results about TIP curves (Jenkins y Lambert (1997, 1998)). The continuous case is shown in Shorrocks (1998). Jenkins (1994) first used this approach in the wage discrimination field, where he defined wage discrimination as the difference, in absolute terms, between the wages estimated with and without discrimination

³⁶ See theorems 4 and 5.

Additive Decomposability. Consider a partition within , where are the sizes of J subpopulations . A discrimination index d is said to be additively decomposable if:

\[d (x _ {m}) = \sum_ {j = 1} ^ {J} \left(\frac {n _ {j}}{n}\right) d (x _ {m} ^ {(j)}).\]

This property suggests that it may be desirable for the overall discrimination as the weighted sum of subpopulations’ discrimination levels. However, this is not a widely accepted criterion in the poverty field for example, we consider that the poverty level in a group cannot be independent of that in other groups. Despite this serious criticism to our approach, the above property is clearly very helpful in most empirical applications, since it allows us to measure the contribution of each population group to the total level of detected discrimination. This means that we can study discrimination for different female characteristics and thus not only classify individuals by earnings (as in the quantile estimations mentioned above) but also by any other variable, such as education level, age or geographical location.

Jenkins (1994) proposed the use of different families of aggregate discrimination indices. These should be conveniently defined over instead of as initially proposed. The main difference of Jenkins’ approach with respect to our proposal is whether these indices should satisfy or not the transfer axiom. Jenkins shows a preference for the use of indices that do not satisfy this axiom. In fact, the family of decomposable indices, , that he recommends and uses in his empirical analysis is a concave function that depends on the relative individual discrimination level (with respect to the average wage):

\[J _ {\alpha} = \sum_ {i \in m} \omega_ {i} (1 - d _ {i} ^ {- \alpha}) = 1 - \sum_ {i \in m} \omega_ {i} d _ {i} ^ {- \alpha}\]

³⁷ Even though he offers theoretical results for both cases depending on the sign and value of a parameter.

where is the normalised wage gap, is the earnings rate of individual i, and , where is a parameter which represents the discrimination aversion degree of the index: the higher the parameter value, the higher the weight of larger wage gaps. Note that the concavity of this function guarantees that these indices take values between 0 and 1, which is a good property.³⁸ However, this property also means that the more evenly discrimination is distributed, the higher the value of the index. And reciprocally: given a constant aggregate wage gap, the more discrimination is focussed on fewer women the lower discrimination level. It follows then that evenness in the distribution of discrimination is being penalised.

“Munroe (1988, p. 22) has argued in favour of F being convex : ‘the penalty attached to discrimination should increase as the extent of discrimination rises’. I am not wholly convinced by his argument, since a given marginal increase in a wage gap corresponds to a smaller proportionate increase for large wage gaps than for small ones, and therefore perhaps deserves a smaller penalty” [Jenkins (1994), pp. 90].

Jenkins here is not consistent with his initial approach about the individual nature of discrimination: the relevance of its distribution and its similarities with economic poverty clearly make the transfer axiom a desirable property for any discrimination index to satisfy.

Taking into account all the above, we consider that it is not necessary to define new discrimination indices, as Jenkins suggests, but only to make good use of those with the best normative properties within the poverty literature.³⁹ Therefore, the family indices proposed by Foster, Greer y Thorbecke (1984), for values of their poverty aversion parameter higher than 1, satisfy our requirements. If we adapt their index to measure (absolute) discrimination we can write a discrimination index such as:

\[d _ {\alpha} (x _ {m}) = \left(\frac {1}{n}\right) \sum_ {i = 1} ^ {k ^ {*}} \left(x _ {m _ {i}}\right) ^ {\alpha}, \alpha > 1\]

³⁸ These numbers represent, respectively, the lowest and highest discrimination level.

³⁹ Zheng (1997 and 2000) offers a survey of the main poverty indices and also of the theorems which link those indices with poverty orderings based on deprivation profiles.

where denotes again the number of discriminated female workers and is the discrimination aversion parameter. It is well-known that , and also that it is additively decomposable.⁴⁰

4.4 Absolute versus relative discrimination

An additional issue in the measurement of discrimination is whether to use a relative rather than an absolute approach. As we have seen, is defined as a function of each wage gap with respect to the average wage gap, which means that when all estimated wages change in the same proportion, does not change. This is an interesting property. If we want this property to hold, we have to define new indices, , which would be a function of the wage gap vector normalised with respect to some average wage, for example : ⁴¹

\[d r _ {\alpha} (x _ {m} / \bar {r} _ {m}) = \left(\frac {1}{n}\right) \sum_ {i = 1} ^ {k ^ {*}} (x _ {m _ {i}} / \bar {r} _ {m}) ^ {\alpha}\]

To guarantee that this index satisfies the above theorem’s properties assigned to we need to define the discrimination curves over a vector such that: ⁴²

\[\Gamma_ {i} \left(\frac {x _ {m}}{\bar {r} _ {m}}\right) = \max \left\{\left(\frac {\hat {r} _ {m _ {i}} - \hat {y} _ {m _ {i}}}{\bar {r} _ {m}}\right), 0 \right\}\]

and adapting the dominance criterion and the theorem to the new case.⁴³ Hence, the Normalised Discrimination Curve, , which maintains the same graphic characteristics than , can be written as:

⁴⁰ It would be interesting to measure discrimination adapting our approach to the use of different poverty indices which satisfy other normative properties such as those proposed by Sen (1976) or Hagenaars (1987). The latter approach would allow us to measure discrimination as the social welfare loss it causes.

m

ym

⁴¹ Another possibility would be to use the mean predicted wage with discrimination, _myˆ , or the mean observed wage, y_m .

⁴² As Jenkins and Lambert (1997) construct the normalised poverty gaps vector.

\[D (\Gamma ; p) = \sum_ {i = 1} ^ {k} \frac {\Gamma_ {i}}{n}\]

once the vectors Γ have been ranked from higher to lower relative wage discrimination:

Definition of dominance in normalised discrimination. Given two normalised discrimination vectors, and , we say that:

\[\Gamma^ {1} \text { dominates } \Gamma^ {2} \text { in a discriminatory sense if }\]

\[\Gamma^ {1} \neq \Gamma^ {2} \text { and } D (\Gamma^ {1}; p) \leq D (\Gamma^ {2}; p) \text { for any } p \in [ 0, 1 ]\]

The dominance theorem for the relative case could be stated as follows:

Theorem (relative case):

For any pair of normalised discrimination distributions, and , it follows that,

dominates in a discriminatory sense

\[\begin{array}{c} \Leftrightarrow \\ d r (x _ {m} / \bar {r} _ {m}) ^ {1} < d r (x _ {m} / \bar {r} _ {m}) ^ {2} \text { for any } d r (\cdot) \in d r ^ {*} \end{array}\]

being the discrimination indices set which satisfies the aforementioned axioms in

When comparing distributions with the same average wage, estimated without discrimination, the orderings derived from relative indices do not differ from those of the absolute case. The differences will appear when the means differ. In this case, the relative approach implies comparisons of individual discrimination levels given as a proportion of their respective mean, which implies neglecting the differences between both distributions’ mean.⁴⁴

1 y 2

J_α

⁴³ This aspect was negletected by Jenkins (1994), which led him to misleading interpretations when connecting results with indices and R_v . Rv

Another interesting possibility consists in normalizing each female wage gap individually, by dividing it by her earnings without discrimination:

\[\nu_ {m _ {i}} = \left(\frac {\hat {r} _ {m _ {i}} - \hat {y} _ {m _ {i}}}{\hat {r} _ {m _ {i}}}\right)\]

When normalizing the wage gap in this way, the critical point is not the average wage anymore, but the highest discrimination level that each woman could suffer. Thus, is the proportion of the wage gap of female worker i with respect to the worst possibility she could face (that is, wage being 0).⁴⁵ Note that the above theorem could also be used in this case. For doing so, we would only need to adapt the definition of the indices, the discrimination profiles, and the dominance relationships as functions of instead of

5. AN EMPIRICAL ANALYSIS: THE CASE OF SPAIN

In this section, our goal is to show the advantages of our approach. We will compare aggregate discrimination levels for Spanish labour market data obtained using OLS and Quantile Regressions (QR) for males and females.⁴⁶ Coefficients are reported in Table A1 in the Appendix.

i

44 These differences are, however, crucial in the absolute case.

45 The role played by _mrˆ in this kind of normalization is similar to that of the poverty line in the deprivation literature. Hence, by dividing the individual wage gap by mirˆ , we do something similar to Pmi what is done in the poverty literature when constructing relative poverty gaps by using individual poverty lines for each household (depending on its size, composition, location,...). An alternative to this would need the use equivalence scales to make different household earnings comparable and the subsequent choice of a common poverty line to normalise poverty gaps.

46 Data come from the Encuesta de Estructura Salarial (Survey of Wage Structure) undertaken by the Instituto Nacional de Estadística (INE) in 1995. This survey covers employees in firms with ten or more workers and does not include any wage information for employees in Agriculture, Public Administration, Health Services or Education. Those individuals who did not work the entire month or who worked parttime were removed from the sample. The final number of observations for analysis are 27,085 women and 100,208 men.

The variable to be explained is the logarithm of hourly wage, and explanatory variables are those usually included in the related literature and available in the database: tenure, (potential) experience, level of education, region of residence, type of contract, occupation (one digit National Classification of Occupations 1994), firm size, type of collective agreement, firm property (public or private) and the market at which most of the firm production is destined (international, national or local).⁴⁷ Wage regressions results are shown to be roughly consistent with other previous empirical analyses, indicating to which extent market returns to employees’ characteristics vary across gender.

We construct wage distributions for working women estimated with and without discrimination. These estimates are denoted respectively by and in the OLS case, and and in the quantile case.⁴⁸ In the latter case, was calculated by attaching to each working woman those coefficients estimated in the female quantile regression which minimises her individual residual, was computed for each woman using the male wage structure at quantile . In this way, what we are actually doing for each woman is selecting her predicted wage, , as the closest to her actual wage and comparing it with a male wage, , estimated for a hypothetical man with her characteristics and situated in the same relative ranking within the conditional male wage distribution (as shown in Graph 6).⁴⁹

⁴⁷ It was not possible, however, to control for other relevant workers’ personal characteristics such as marital status or the presence of children in the household. Furthermore, this database only contains working women and does not allow controlling for selection bias.

⁴⁸ We compute quantile regressions in ten different points of the distribution (the mean quantile within each decile: i.e . 5^th, 15^th, 25^th, …., 95^th ).

⁴⁹ Obviously, this is an ad hoc use of predicted wages that might be forcing the interpretation of this type of estimates, but the purpose here is to explore to which extent OLS and quantile regressions differ, not only because they make estimates at different points in the distribution, but because they also yield different aggregate levels of discrimination.

Descriptive statistics for wages and gender gaps estimated with both models are reported in table A2 in the Appendix, while the corresponding non-parametric kernel densities are depicted in Figures 1a to 2b.

Fig. 1.a Observed and predicted wage with and without discrimination (OLS)

First of all, it is important to emphasise that observed wages are better predicted when using the quantile regression than when the estimation procedure is OLS, with the former showing also a greater dispersion. This results in a more accurate fit of the estimated distribution, especially evident in the tails. A greater dispersion is also observed when estimating wage gap densities with quantile regressions, despite the fact that this divergence is substantially reduced in the case of making the gap relative to the income predicted without discrimination , that is

The presence of a low degree of mobility originated by discrimination does not affect Lorenz curves or Generalised concentration curves given that both keep their original order in . Therefore, the area between both curves in Figures 3a and 3b quantifies total discrimination. However this result presents two problems: it includes nondiscriminated women gaps⁵⁰ and, above all, sums up individual wage gaps using an unattractive normative criteria.

⁵⁰ This point was raised by Favaro and Magrini (2003). In our particular empirical case, however, this is not an important problem given that the number of non-discriminated women is very small.

Fig. 3b Generalised Lorenz and Generalised Concentration Curves (QR)

Fig. 3a Generalised Lorenz and Generalised Concentration Curves (OLS)

In order to be able to compare discrimination levels captured by both procedures, we estimate absolute and normalised discrimination curves which are depicted in Figures 4a and 4b. It appears that OLS wage distribution dominates QR in discrimination. A consequence is that discrimination estimated by QR is always larger for all discrimination indices fulfilling the axioms proposed (both in the absolute and relative case).⁵¹ One can check this just looking at discrimination measures reported in Table 1.⁵²

Fig. 4.a Absolute Discrimination Curves

⁵¹ Notice that our notion of relative discrimination is based on the ratio of the estimated discrimination to the wage without discrimination, not to the total wage gap as it is usual in the literature. This should be taken into account when comparing our results with previous evidence.

⁵² In Table 1 we present two additional indices despite the fact that they do not verify all the proposed axioms: the head-count ratio of discriminated women, h, which provides information about the incidence of discrimination among working women (being larger than 99 percent in both cases), and d and dr indices for a discrimination aversion level equal to one, indicating the amount of money that one should transfer to discriminated women in order to remove discrimination (in absolute and relative terms respectively).

Table 1. Indices of Discrimination

In order to analyse the distributive aspects we compare discrimination curves estimated separately for each decile in Figures 5a and 5b. Absolute discrimination clearly increases with women wages in both OLS and QR. Relative discrimination, however, shows a more ambiguous pattern given that there are crosses between different decile curves (see Figures 6a and 6b). In any case, we find a somewhat larger level of discrimination at the bottom of the wage distribution than at the top.

Fig. 6a Normalised Disrimination Curves by deciles: OLS

However, interested as we are in a more explicit answer to the question about where along the wage range there is a higher degree of discrimination, we propose here the use decomposable indices of relative discrimination. We should bear in mind that, in contrast with the analysis based on discrimination curves, our results are now less robust.

Figures 7 and 8 display, for , the ratio of within-group discrimination to total discrimination estimated for each decile (based on observed wage) and education level. The population groups with a value of the ratio above one face a level of discrimination larger than average while those with a ratio below one face a lower level than average. Whatever the estimation method we find that females in the first earnings decile suffer the largest relative discrimination. The discrimination level tends to decrease as we move to higher wage deciles (with the exception of the highest one). Surprisingly, when the sample is broken into education levels, we find that relative discrimination is larger than average for women without studies and smaller for those with secondary school or holding an university degree.

Following Dolado and Llorens (2004) we wonder whether or not the interaction between education and wage levels would yield more convincing results. For that reason, Figures 9a and 9b break the sample into those holding a university degree and the rest of females and we analyse discrimination patterns by deciles separately for each of these two groups. Given that women without a university studies account for the majority of females, their pattern does not diverge significantly from that described for the whole distribution (only in that the last decile is no longer an exception to the decreasing trend). However, discrimination risk is increasing with the wage level among those women holding a university degree, this increase is sharper for the last decile when we use quantile regression estimates. Thus it appears that among top-wage female earners, more skilled women are those facing the greatest percentage of discrimination in their salaries. Moreover, this group faces a roughly similar relative discrimination than what we could consider the opposite group: low-wage unskilled women.

Fig. 9b Discrimination by deciles (university degree): dr₂ ratio (average = 1)

Figures 10a and 10b confirm this result by depicting normalised discrimination curves for women with university studies by deciles. Therefore the previous result holds not only for but is robust to index choice.

6. CONCLUSIONS

In this paper we have detailed the advantages of analysing wage discrimination from a distributive point of view, considering each individual discriminatory experience. We have exposed the limitations of using the classic approaches to the measurement of discrimination based on the analysis of the mean discriminatory experience and also of those that use some recent distributive methodologies based on quantile regressions and counterfactual wage distributions. Our theoretical contributions are: 1) to underline the imprecise measurement of discrimination using counterfactual functions: this is related to re-orderings as we move from the original wage distribution to a hypothetical nondiscriminatory one; and, most importantly, 2) to propose a new normative framework for the study of wage discrimination based on the poverty and deprivation literature. For the latter we provide a variety of improvements to Jenkins’ (1994) approach to the aggregation of individual discriminatory experiences by adding to its consistency and normative power.

The empirical exercise using Spanish data allows us to present the differences between OLS and Quantile Regressions in the estimation of female individual discriminatory experiences. This exercise shows that quantile regressions reveal a significantly higher level of aggregate discrimination compared to that detected using classical estimation techniques on the mean. Therefore, the choice between OLS and quantile regression is all but innocuous from an aggregate point of view. Nevertheless both methods raise roughly similar discrimination patterns throughout the wage range. It seems clear that absolute discrimination increases with observed wages, while in relative terms this result is not robust. Only females with very low wages and, at the other extreme of the distribution those holding a University degree with the highest salaries, register a relative discrimination significantly above that of the rest.

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APPENDIX

Table A1 O LS and Quanti le reg ress ion esti mates for hou rly wage i n logarith ms

* = coefficient is not sig n ificant at 1 0% . Coefficents for Reg ions om itted . O LS variances com puted us i ng W h ite esti m ator. Qu anti l e reg ress ions were perform ed also at perce nti l es 1 5 35 55 65 and 85 not d is played for s i m pl icity.

Table A2. Su m mary statistics : average and d ispersion