Who Are The Worst-Off When Preferences Matter

Transcription

1 Who Are The Worst-Off When Preferences Matter C.Sapata Preliminary Draft November 15, 2010 Abstract The criteria called conditional equality and egalitarian equivalence proposed by Fleurbaey and Maniquet[15, 16] serve to assess the degree of equity among individuals. They have not been applied as often as the criteria of equality of opportunity[28, 30], mainly because the first criteria should be used in an ordinal framework which differs from the cardinal approach usually adopted in the literature on equality of opportunity. But conditional equality and egalitarian equivalence have the advantage to explicitly distinguish between the objective of neutrality with respect to the responsibility factors and the principle of compensation for factors that are beyond the individual s responsibility. This paper proposes an empirical framework that is consistent with these ordinal criteria and enables to compare them with the cardinal criteria. We estimate a utility function that incorporates heterogeneity in preferences among individuals. We obtain ordinal measures of well-being and apply conditional equality and egalitarian equivalence. Then we use the Roemer s Identification Axiom and cardinalize the measure of utility to compute Roemer s and Van de gaer s criteria. Finally we compare the criteria through the characteristics of the worst-off displayed by each criterion for the US. Keywords: Random utility, Preference Heterogeneity, Welfare Measures, Inequality of Opportunity 1 Introduction Usually, inequality of opportunity is measured by selecting particular outcomes such as education, health, income, and by decomposing it into two kinds of de- I would like to thank Marc Fleurbaey, François Maniquet, Erik Schokkaert and Dirk Van de gaer for their very helpful comments and the participants at the Seminar of Social Choice and Welfare (CORE, Louvain-La-Neuve) in May I also acknowledge the support received from the European Science Foundation (ESF) for the activity entitled Public Goods, Public Projects, Externalities Universitat Autonoma de Barcelona, Spain. sapata@fusl.ac.be 1

2 terminants: circumstances (factors beyond the individuals responsibility) and effort (factors for which the individual can be held responsible). Equality of opportunity requires erasing the inequalities due to circumstances, but maintaining the inequalities due to effort. As a result, outcomes should be a function of effort only and there should not be aversion for inequality due to differences in effort. In this way, measuring equality of opportunity consists in disentangling the effect of effort and circumstances on outcomes. Assuming that outcomes are explained only by circumstances and effort, people can be grouped according to the circumstance they share, these are called types. Within a type, the effort is the determinant of the inequality in outcomes. But, effort is rarely observable and it is often affected by circumstances which makes difficult the measure of inequality of opportunity. Roemer s identification axiom consists in identifying effort through the percentile to which an individual belongs in the distribution of outcomes by type. Enhancing equality of opportunity according to Roemer consists in reducing the inequality between individuals at each percentile across types. This definition has emerged as the mainstream reference for measuring the unfair component of inequalities [21, 22, 23], and it has been also applied for policy evaluations [5, 27]. Within the literature on equality of opportunity, this principle belongs to an ex-post approach of inequality of opportunity as the fairness of a distribution depends on observed individual s outcome, on his observed circumstances and on effort variables. Another approach consists in examining ex-ante inequality of opportunity. To focus on this aspect, Van de gaer [30] proposes comparing the expected outcome of the individual conditional to his type. This measure may not coincide with Roemer s proposal, as Roemer s criterion focuses on the inequality of individuals exerting the same effort while Van de gaer focuses on average inequality between people having different circumstances 1 [25]. Both criteria use cardinal measures of individuals outcomes as they require aggregating the individuals outcomes. This makes the criteria easily comparable and applicable in a single framework [26], and have both given rise to substantial contributions in the measurement of inequality of opportunity 2 [29, 26, 13]. However, none of these two criteria make explicit the choice of the cardinal outcomes that should be used. As the question of equality of opportunity raises the question of the fairness in the distribution of welfare, it could be desirable to justify the cardinal measure on which relies the application of normative principles. Moreover, equality of opportunity is a concept addressed to individuals and these criteria tend to identify groups and not individuals. Indeed, the worst-off according to Van de gaer s criterion are a type. With Roemer s 1 If we define the worst-off as the individuals for which an improvement of their outcomes means a decrease in the inequality of opportunity, Van de gaer s and Roemer s criteria identify the same worst-off if the distribution of outcomes by type does not intersect 2 Other criteria have been proposed by Calsamiglia, Hild, Moreno for instance[7, 20, 24] 2

3 criterion, the worst-off are individuals within every type but their identification depends on the way types and effort variables are built. These issues have found an echo in Fleurbaey and Maniquet [14] [15] [16] [17] whose work in the field of welfare economics is strongly related to the concept of equality of opportunity [19]. They propose another methodology for assessing equity which is briefly summarized as follows. They adopt the distinction between circumstances and preferences (understood as a representation of individuals goals in life) for monitoring the unfair distribution of welfare. It will take into account the variability of the objectives among individuals and it still permits compensating people for the factors that are beyond the realm of their responsibility. As this approach allows individuals to have distinct goals, equity should be assessed on the basis of the distribution of individuals well-being instead of particular outcomes. Indeed, focusing on one particular outcome could yield to wrong evaluations of fairness. More precisely, to compare individuals situations, their approach is not cardinal. Their methodology does not consist in using direct utility measures or multidimensional measure of well being because both require making interpersonal comparisons of well-being using a common utility function that erases the differences in preferences. Instead, the criteria of equity rely on the way people rank allocations only. And any cardinalization of well-being employed to make interpersonal comparisons is explicitly justified on normative principles. Because this approach is firstly ordinal, Roemer s and Van de gaer s criteria cannot be applied directly in the same set-up. Nevertheless, the definition of equity given by Fleurbaey and Maniquet relies on two principles of equity, compensation and natural reward, which are closely related to the approach for equality of opportunity. That is why it could be of interest to compare the ordinal approach with the cardinal approach. Compensation means we should compensate for factors for which the individual is not responsible, called circumstances or non-responsibility factors. If people have the same preferences, the principle of compensation states that these persons should achieve the same welfare as circumstances should not affect the realization of objectives. Natural reward refers to the neutral treatment of individuals with respect to their preferences. This neutrality principle means that individuals should be treated equally with respect to their preferences 3. Thus, redistribution mechanisms should be designed in such a way that individuals with equal circumstances will pay/receive the same taxes/transfers. The compensation and natural reward principle, probably equally appealing, cannot be both satisfied by any redistribution rule for more than four individuals having heterogeneous preferences [17, 6]. That is why, in the ordinal framework with heterogeneous preferences, inequity is inevitably measured through two criteria, each one giving the priority to one principle and fulfilling only partially the second one. The criterion of conditional equality fulfills the natural reward 3 This idea could be questioned if we consider that some goals or preferences should be avoided according to some prevalent values but this discussion falls outside the scope of this paper 3

4 principle and compensates partially inequalities due to non-responsibility factors. The criterion of egalitarian equivalence gives priority to the compensation principle but does not satisfy neutrality with respect to all preferences. The point of this paper is not to discuss from a theoretical perspective this approach with the traditional approach but to propose an empirical framework that is consistent with each of the approaches. In this way, we offer a methodological exercise that consists in applying each criterion in accordance with its original framework and we compare who the worst-off are in each case. Comparing the ordinal criteria with the traditional ones has been mostly a theoretical issue, mainly discussed by Fleurbaey who analyses the cardinal criteria as a fair solution between utilitarism and egalitarianism requirements whereas the ordinal criteria would be a compromise between libertarism and egalitarianism [14]. But applications of conditional equality and egalitarian equivalence criteria have been scarce and no empirical study has proposed a comparison of these criteria. Decoster and Haan [10] have proposed the first application that follows the key points of the approach, especially the identification of heterogeneous preferences and the use of an ordinal framework [10] but preferences are not individualistic, they are common to groups of people depending on their socio-demographic variables. In this paper, we use Decoster and Haan s model but we introduce slight changes to identify the individualistic component of preferences. Also Decoster and Haan obtain ordinal measures of utility in a discrete framework and then apply the criteria using indifference curves. Instead, we work with indifference sets to be consistent with the original discrete model on which relies our estimates. Concerning other empirical applications of Fleurbaey and Maniquet s criteria, they have been mainly adapted to become cardinal measures. Almas [3] and Devooght[9] have used the definition of conditional equality and egalitarian equivalence to measure unfair inequalities for income and do not explicitly account for preferences. Here, the application is developed in an ordinal framework and the outcome the individuals care about is well-being. As a result, the paper aims to fill the gap between the progress made in the theoretical field in comparing the different concepts of equality of opportunity and the empirical studies that focus mainly on the cardinal approach for equality of opportunity. We believe that comparing the ordinal criteria with the cardinal approach would contribute to a useful comprehension of their degree of divergence and similarities. Moreover, we focus on the characteristics of the worst-off for policy purpose. The worst-off are defined as the individuals who matter according to each criterion in the sense that equity would improve if the well-being of these persons increased. Because identifying the worst-off by using a variety of criteria should help in understanding who suffer the most inequalities of opportunity, this paper may help to define to whom redistribution policies should be addressed. The rest of this paper is organized as follows: Section 2 presents the model. Section 3 discusses the way of implementing the four criteria. In Section 4 are presented the data and variables. Section 5 displays the main results. Finally, Section 6 provides a conclusion. 4

5 2 The Model 2.1 The Ordinal Framework: Conditional Equality and Egalitarian Equivalence Well-being is assumed to depend on disposable income, c and leisure l and can be represented through an utility function u(c, l). The determinants of wellbeing are also decomposable into responsibility factors and non-responsibility factors. Concerning circumstances, skills, that are equal to the individual s wage rate, are supposed to be beyond individuals responsibility. The wage-rate is likely to represent skills that are mostly the product of genetics, family background, luck but also effort. Nevertheless, as Fleurbaey and Maniquet use the gross wage rate as the variable of non-responsibility, we take the same hypothesis to be consistent with their model. Regarding individuals preferences, they are restricted to the preferences for leisure and for consumption as preferences affect utility through consumption and leisure. In a population of N agents, we have a set of skills s N and a set of utility u N, that depends on consumption and leisure. An economy is e = (s N, u N ) where every agent maximizes his utility u i over consumption and leisure (C i, L i ) X. Given every individual maximizes his utility subject to a budget constraint B, the utility function derived from any subset B X is: u i (B) = {max u i (x) x B} (1) Precisely, the budget constraint set is determined by a skill s and a lump-sum transfer t as follows: B(s, t) = {(C, L) X c s l + t} (2) where l = 1 L is the amount of working hours. The individual s budget constraint is actually not linear. However, the neutrality principle states that the treatment of individuals should be neutral with respect to preferences. As a result, a neutral budget set would be a lump-sum transfer. Finally, for simplification, Fleurbaey and Maniquet propose transforming the actual budget set into a linear budget set where we replace the actual transfer t that depends on the amount of leisure by ˆt, the minimum lump-sum transfer that permits the individual to reach u i (B). To be coherent with their approach, we work with a simplified budget set with a lump sum transfer defined by ˆt i where: u i (x i ) = u i (B(s i, ˆt i )) (3) Regarding the equity criteria, conditional equality permits to fulfill completely the neutrality principle and achieve partially the compensation for circumstances. By consequence, a situation is fair if whatever are the changes in individual s preferences, his budget set remains the same. Moreover, to fulfill 5

6 partially the compensation principle, all the individuals with the same preferences should have, in the largest extent, the same budget set. Formally, defined in Fleurbaey [14] conditional equality can be computed as follows: Define a reference value for responsibility characteristics and give priority (according to the leximin criterion) to individuals, who, with their current resources and circumstances and this reference value of responsibility characteristics, would be the worst-off. Here, it requires defining a reference value for preferences so that the utility function ũ is the same for all individuals. Then, individuals are ranked according to: where ˆt i is defined by ũ(b(s i, ˆt i )) (4) u i (x i ) = u i (B(s i, ˆt i )) (5) This allows us to rank all the individuals from the worst-off to the better-off according to the criterion of conditional equality. But, we focus on the first decile of this resulting distribution to characterize the worst-off according to the different criteria. The second equity criterion, called egalitarian equivalence, stresses the compensation principle. According to it, for any wage rate, the individuals budget sets should be the same as circumstances should not cause unequal access to well-being. As defined by Fleurbaey [14], it requires the following: Define a reference type of circumstances and give priority (leximin) to individuals whose current level of well-being would be obtained with the least resources if their circumstances were of the reference type Fleurbaey proposes to vary the measures for egalitarian equivalence by using different reference value for the wage rate as the reference values affect the ranking of people 4. Among possible measures, we focus on three values for the wage rate. If the reference wage rate is equal to zero, we obtain the zero egalitarian equivalence. It means to rank ˆt i in ascending order as defined by : u i (x i ) = u i (B(0, ˆt i )) (6) If the reference wage rate is equal to the minimum wage rate, we will obtain the min egalitarian equivalence: It consists in ranking in ascending order the ˆt i as defined by u i (x i ) = u i (B(min j N s j, ˆt i )) (7) Lastly, the wage egalitarian equivalence or equivalent wage is defined as [14]: 4 Indeed, the higher is the wage rate, the more priority will be given to the people having lower preferences for leisure, that is to say the hard-working people. 6

7 For each individual, compute the counterfactual wage rate (with no transfer) that would make the individual as happy as in his current situation, and give priority (leximin) who are the worst-off in these terms. Formally, it is equivalent to rank the individuals in ascending order according to ŝ i that satisfies: u i (x i ) = u i (B(ŝ i, 0)) (8) 2.2 The Cardinal Framework: Roemer s and Van de gaer s criteria For the application of Roemer s and Van de gaer s criteria, we suppose that utility is the outcome that individuals want to achieve which is consistent with the hypothesis made above about individuals behavior. Also, these criteria require dividing between effort and circumstances. To make the comparison more reliable between ordinal and cardinal criteria, the circumstance is unique and corresponds to the wage rate 5. Regarding effort, from the previous framework, effort is multidimensional, it is composed by the preferences for leisure and consumption but also the amount of leisure chosen and the form of the utility function so it is impossible to find one single continuous value for effort. A way out is to use Roemer s Identification Axiom to determine one index of responsibility. Firstly, the population is partitioned into types according to the wage rate. Then, using a cardinalization of utility, we rank the individuals according to their level of utility. We obtain as many distribution of utility as number of types. Finally, the individuals belonging to the same percentile of the distribution of utility, whatever their types, are assumed to exert the same effort. This alternative is coherent with Roemer s definition of effort and allows us to compute Roemer s and Van de gaer s criteria consistently with their original framework. Here the worst-off are the individuals who have the lowest well-being at each percentile whatever their types. Indeed, as people exerting the same effort should obtain the same outcome, any policy that would improve the well-being of these individuals would reduce inequality of opportunity. In order to obtain the same number of worst-off, we define ten types that correspond to the ten deciles of the distribution of the wage rate. Then, we define the ten levels of effort as the ten deciles of the distribution of utility by type. As we have ten types and ten levels of effort, once grouped the worst-off for each level of effort, we obtain the same number of worst-off as the one obtained after the implementation of conditional equality and egalitarian equivalence criteria. 5 More complex specifications could be of main interest but are left to other studies since here the purpose is to offer a consistent comparison among several criteria and not a detailed implementation of one specific criterion. 7

8 Van de gaer s criterion consists in comparing the mean well-being by type.we use the same cardinalization method as the one used for the implementation of Roemer s criterion. As every type is a decile of the distribution of the wage rate, the worst-off are the individuals who compose the type with the lowest average well-being. 2.3 Empirical approximation of the responsibility and non responsibility factors We borrow from Aaberge et al. [1, 2] the specification of the utility function. We suppose that individuals utility function depends on disposable income C, leisure L, socio-demographic variables X i and a random error term ɛ ij which varies independently among individuals and alternatives. It is not observed but affects individuals choices. Given his wage rate and the taxes he pays, the individual i maximizes his utility by choosing his amount of worked hours from a set of the alternatives on worked hours j J. This problem can be written as the following utility function: V ij (C, L) = U(C ij, L ij, X i ) + ɛ ij (9) It has two sub-indexes because utility is different across individuals since individuals have distinct preferences for leisure and consumption and because utility is affected by the amount of worked hours through the simplified budget set. The maximization problem is subject to two constraints. The first one is the budget constraint: C = s l t(s, l) (10) Where t(s, l) is the tax function that transforms gross earnings into net earnings. The second constraint indicates that people can choose their amount of worked hours among the following set: l J = (0, 5, 10, 15,.., 55) (11) We adopt this discrete choice framework because it allows us to work with non linear budget sets so as to represent better the tax and transfer system the individuals face. Omittimg indexes, the deterministic part of the indirect utility function has the following form: U(C, L) = β C C α C 1 α C + β L L αl 1 α L (12) The parameters β C, α C, β L and α L indicate preferences for consumption and leisure. 8

9 β C, α C and α L are common to all individuals but heterogeneity among individuals is introduced through taste shifters β L = β L0 + β L1 X i (13) Where X i are gender, age, education and ethnic group. These variables have been widely used in the literature to explain the determinants for labour supply. They are not really under the individual s control but they are expected to explain differences in preferences for leisure. This corresponds to Dworkin s, Rawls, Fleurbaey s and Maniquet s arguments according to which as long as people identify with their preferences, no matter what explains these preferences, we should be neutral with respect to them. As we aim to implement the criteria following the philosophy of their corresponding authors, β L will represent individuals preferences. After estimating the model, we can have two individuals, with the same wage rate, the same amount of worked hours, the same value for β L, and with different working time. Therefore, the error term is the key variable that explains the differences in individuals choices. As β L does not include information enough on individuals, the error term, ɛ ij, captures the individualistic component of preferences not explained by β L. For this reason, epsilon is a component of individuals preferences. As a consequence, we have two preference parameters: β L and ɛ ij. 2.4 Empirical approximation of the individuals utility function To obtain the parameters of the deterministic part of the utility function, the estimation relies on a rationality assumption that states that if the ith individual makes the choice j in particular, it means that V ij is the maximum among the j alternatives. In other words, the probability that the the ith individual makes the choice j is: P rob(y i = j) = P rob(v ij > V ik ) k j (14) We replace V ij and V ij by its expression and obtain: P rob(y i = j) = P rob(ɛ ik ɛ ij < (U ik U ij )) k j (15) The resulting multinomial model depends on the assumptions made on the joint distribution of the epsilon. If we assume that ɛ ij is i.i.d random value with type I extreme value distribution, differences in epsilon follow a standard logistic distribution [8]. In this case, we estimate the parameters of the utility function by maximum likelihood as a conditional logit model where: P rob(y i = j) = exp U(Ci j, Li j ) n k=0 exp U(Ci k j (16) k, Li k ) We obtain as many utility functions as numbers of groups having the same socio-demographic characteristics. Formally, we have: 9

10 V ij (C, L) = β C C α C 1 α C + (β L0 + (β L1 X i )) LαL 1 α L + ɛ ij (17) The estimation does not allow to observe epsilon. However, it is crucial to account for epsilon in the computation of the ordinal criteria of equality of opportunity as explained above. To this end, we use the assumptions of the model to get a proxy for differences in epsilon. In particular, we use the rationality assumption, that is to say, the fact that if the individual has picked the alternative j it means that the utility he obtains with this option is superior to any other alternative. Formally, it means that: ɛ ik ɛ ij < (U ik U ij ) k j (18) Omitting indexes, we can say that because individuals are rational, we have to satisfy: ɛ < U (19) We also use the hypothesis of the estimation according to which differences in epsilon follow a standard logistic distribution that means that the density function is: f( ɛ) = exp ɛ 1 + exp ɛ 2 (20) Satisfying the rationality hypothesis permits us to restrict the values for ɛ, in particular, to rescale the density function of ɛ so that: f( ɛ ɛ < U) = f( ɛ) P rob( ɛ < U) = f( ɛ) P ɛ < U (21) where P is: f( ɛ ɛ < U) = 0 ɛ U (22) P = 1 exp U 1 + exp U (23) To this end, we generate one million drawings of a random variable that follows a standard logistic distribution so as to obtain possible values for ɛ. We compute for every individual and every alternative the value for U ij. Then, we restrict the possible values of the random variable following the assumptions explained above and take the expected value. We obtain as many expected value for ɛ as values for U ij when U ij < 0. When the utility for the observed choice is superior to the utility for another alternative, the expected value of ɛ is zero, as stated by the standard logistic distribution. This strategy makes use of the assumptions of the model only and allow us to compute measures of inequality of opportunity that account for individuals preferences and not only groups preferences. As a result, we obtain individuals utility functions that depend both on a group and an individualistic component for utility. 10

11 3 Application of the four criteria 3.1 Conditional Equality Assessing equity in the ordinal framework requires comparing the individual s well-being for each possible amount of working hours since they are free to change this amount to maximize his utility. The starting point has to be the definition of the indifference set that will give all the points of an hypothetical budget set that give to the individual the actual well being for the twelve alternatives of working hours. After estimating the individual s utility function, we obtain the actual level of utility for the chosen working hours. k j, we define IC k the level of consumption that gives to the individual the same utility as the one observed for the observed choice on working hours j. It means that V ik = V ij k j. By replacing V ik and V ij by their expressions and omitting the index i, we have: IC k = [C αc j + α c ( ɛ + β l (L α l j L α l k β c α ))] 1 αc (24) l With this expression and using the actual choice made by the individual as well as the expected value for ɛ (given ɛ is inferior to U), we obtain the twelve points of the indifference set. Then, we deduce the simplified budget set. For every individual and k, we compute t so that: t = IC k w i k (25) The minimum value of t among the twelve k is t. It permits to compute the simplified budget set BC k defined as: BC k = w i k + t (26) Now we fix a reference value for β L and ɛ and obtain what would be the amount of worked hours that would maximize utility given the simplified budget constraint of every individual.then, we rank this maximal level of utility across individuals so as to obtain the ranking of individuals according to the conditional equality criterion. The worst-off are the persons belonging to the first decile of the distribution of the utility. We can describe them in terms of gender, education and ethnicity. As the results may change depending on the reference value for β L and ɛ, in the next section we will use different reference values. 3.2 Egalitarian Equivalence Departing from the indifference set IC k, egalitarian equivalence requires fixing a reference value for the wage rate and computing the minimum lump-sum transfer that gives to the individual his actual level well-being with this reference value for the wage rate. To do this, we compute k: 11

12 EE k = IC k w k (27) where w is the reference wage rate. It is equal to 0 in the case of the zero egalitarian equivalence criterion and w is equal to the minimum wage rate (fixed to 5 dollars per hour) when computing the min egalitarian equivalence criterion. We select the minimum lump-sum transfer called EE among the twelve EE k for every individual and rank it in ascending order. The worst-off are the persons belonging to the first decile of the distribution of EE. For the computation of the wage egalitarian equivalence, we find which wage rate the individual should receive to achieve the lowest point of his actual indifference set if there were no tax. For each point of the indifference set IC k, except for k = 0, we generate: W EE k = IC k (28) k We take the minimum W EE k among the eleven values for k, it is called W EE, we order them and we describe the characteristics of the worst-off who are the individuals at the first decile of the distribution of W EE. 3.3 Roemer s criterion or the Mean of Mins We first have to propose a cardinalization of the utility function. To be able to make reliable comparisons with the ordinal criteria, we use two types of cardinalizations 6. On the one hand, we use the point of the indifference set when k = 0. It is the consumption level that would give to the individual his actual level of well-being if he did not work, called CI 0. This measure corresponds to the observed level of well-being of the individual. Therefore, it is directly comparable with the egalitarian equivalence criterion as it corresponds to this criterion the reference wage rate is equal to 0 and it is also a way of cardinalizing utility. It is also related to a special case of conditional equality when individuals have such a large preference for leisure so as to choose not working. On the other hand, we cardinalize utility by using the reference value of β L and ɛ we have used for the conditional equality criterion. These two cardinalizations will make the cardinal criteria more comparable alternatively with the egalitarian equivalence or with the conditional equality criterion. As a result, the comparisons of the worst-off should be more reliable. Finally, we build ten types corresponding to each decile of the distribution function of the wage rate. The worst-off are the individuals with the ten percent lowest utility at each decile of the distribution of the utility across types. They are the worst-off in the sense that inequality of opportunity would be reduced if their well-being were improved. 6 We plan to use other types of cardinalizations in order to make a sensitivity analysis of the results depending on the cardinalization that has been used 12

13 3.4 Van de gaer s criterion or the Min of Means Concerning the cardinalization of the utility, we use the same method as used for the implementation of Roemer s criterion. The difference concerns the way of identifying the worst-off. They correspond to the type with the lowest average well-being. Thus, once the cardinalization is obtained, we partition the population into the same ten types we used for the previous criterion. The worst-off are the type with the lowest average utility as inequality of opportunity in the sense of Van de gaer would be reduced if their well-being were improved. 4 Data and descriptive statistics 4.1 Data The empirical analysis is based on US data from the Panel Study of Income Dynamics (PSID) for 2005 that provides information on incomes in It includes detailed information on socio-demographic variables that enables us to select the appropriate sample given we do not have at our disposal a microsimulation model describing the exact model of tax and transfers the individuals face. As the OECD report on income taxes gives detailed information on taxes for singles without children, we focus on this population. As a result, our measures are restricted to this specific sample and the results cannot be generalized to the whole population 7. We use a static labour supply model estimation to show empirically the heterogeneity in preferences among individuals. For practical reasons, we do not account for possible restrictions in the labour market as Aaberge et al. [1, 2] does. People are assumed to choose freely among twelve alternatives on worked hours. We also restrict our sample to individuals aged between 25 and 65 years old who are not self-employed, retired or fully engaged in education. This gives us a sample of 662 individuals. Distribution of working hours is given in Figure 1. The graph shows that most people work between 20 and 55 hours with a pick around 40 hours. there is no other pick for an amount of working hours inferior to 40, so it does not appear that part-time jobs require working a specific amount of time. The fact density increases around 55 hours a week is due to the fact that the maximum amount of working hours is fixed at 55. To build the budget constraint, we use the 2004 OECD report for the US to derive the net income for each possible working hours. Precisely, we first calculate the gross wage rate by dividing the individual gross labour earnings by the annual working hours. For non-working people, we take the predicted value of the wage rate after estimating a wage equation 8. Then, we vary the 7 A micro-simulation model would be very useful to describe precisely the actual individual s budget constraint but it is still not available, thus, our data set is likely to be modified once we can have access to a micro-simulation model. 8 We estimate the wage rate as a function of education, labour experience, gender, region of residence and race with the Heckman procedure. Details of the estimation are available 13

14 Figure 1: Distribution of working hours number of worked hours and obtain the gross total earnings. Finally, we use the report on tax income to simulate the corresponding net earnings. For every individual, we fix a maximal amount of time endowment T=80, and leisure is T minus the amount of weekly working hours. Then, in the empirical estimation, leisure is normalized to one and net income is expressed in units of dollars per year. We do not include capital income in the estimation. The gross amount of capital income the individual receives is available but there is no desegregation by type of capital income and no data on net capital income. As taxes depend on the sources of this income, net capital income cannot be properly computed 9. Table 1 indicates the distribution of the population among the twelve working hours, the corresponding net labour income and the distribution by education, gender and ethnicity. 4.2 Estimation Results Table 2 presents the estimates of the parameters of the utility function. We find individuals have positive preferences for consumption (α c is positive) and leisure because β l is found to be positive for every individual. On the other hand, parameters α c and α l are inferior to one which indicates concavity of the utility function. Regarding the significativity of the parameters. α c, α l and β c are in this model significantly different from zero, but β l is not. In fact, we find no significant heterogeneity among individuals explained by observable characteristics. upon request. 9 Capital income would affect the level of the budget constraint and probably the decision of working or not. Therefore, not taking into account capital income should affect the slope of the utility function for low level of working hours. As most individuals of the sample work full time, the slope around the point the individuals make should not be affected by the capital income. 14

15 Table 1: Descriptive Statistics Working Share Men More than High White Black Other Monthly Net Hours (%) School (%) (%) (%) (%) Income ($) Parameters have the expected signs, for instance, ceteris paribus women tend to have higher preferences for leisure as well as people with less education. But none of the coefficient of β l appear to be significantly different from zero. This may be due to the fact we have restricted our sample to singles without any children. Within this particular sample, most individuals tend to work fulltime, so it is likely that this sample lacks of heterogeneity in preferences. On the contrary, Decoster and Haan [10] focus on married women labour supply in Germany, they have found similar estimates in terms of sign and most of the components of β l (education, number of children) have been found to be significantly different from zero. Here, when singles with children are included in the estimation, parameters on gender, education and number of children, are still positive and they turn to be significantly different from zero. However, as the budget constraint is not correctly computed by using the OECD report that provides incomplete information on taxation of singles with children, we prefer to display these results. Again, a micro-simulation model would allow us to enlarge our sample but still, we need to investigate in further detail these results. Nevertheless, the fact we account for unobserved heterogeneity through the expected value of differences epsilon in the computation of the welfare measures should correct part of these unsatisfactory results concerning β l. 5 The Results 5.1 Who are the worst-off? In order to make comparisons between the characteristics of the worst-off displayed by each criterion and the composition of the sample without applying any criterion, the table 3 informs on the composition of the sample in terms of gender, level of education and ethnicity. 15

16 Table 2: Parameters of the Utility Function Coefficient Preference for Consumption β c α c Preference for leisure β l : Age Age squared Women Education (more than high school degree, being reference value) Less than high school High school Ethnicity (white, being reference value) Black Other Intercept α l z Table 3: Composition of the sample Category Share (%) Mean working hours Mean wage rate Men Women Education Less than high school High school More than high school Ethnicity White Black Other Total

17 Figure 2: Conditional Equality Regarding conditional equality, we use four different reference values for the utility function. In all cases, we assume epsilon is equal to zero for every individual and every alternative. In fact, when selecting a utility function form common to all individuals, epsilon should vary among alternatives only which can be done directly through β l. Firstly, we fix two different reference values for β l. The first one is 0.16 which corresponds to the mean of β l among individuals in the sample. This value is very small that means people do not value leisure. In this case, given taxes depend on the wage rate and on the numbers of working hours, the worstoff should be individuals with low wage rate and/or people working few hours. That is concordant with our results given in figure 2. Individuals with low education, the women, and the black people tend to be the worst-off according to this criterion and these are the categories with lowest wage rate and fewer working hours. As we increase the reference value for β l the results shown in figure 2 change significatively. If we assume strong preferences for leisure (β l = 100, the worstoff turn to be the individuals with high education and the white people. The change in the characteristics of the worst-off show how the importance of making explicit the choice of the cardinalization of the utility function. Another way to fix a reference value for this criterion is to fix directly a specific amount of working hours as suggested in a paper by Fleurbaey and Maniquet [16]. We fix the number of working hours equal to 20 and then 40 and we compute the utility of the individuals if everybody worked this amount of hours using their own value for β l and ɛ. The individuals with the lowest utility are the worst-off. This is equivalent to fix a value of β l such that everybody decides to work the same amount of time. When l = 20 the worst-off are very similar to the ones observed when β l is 17

18 Figure 3: Egalitarian Equivalence close to zero. As l increases, the proportion of men and high skilled among the worst-off decrease but the characteristics of the worst-off remain very different from the ones when the reference value for β l is high. Regarding the egalitarian equivalence criteria, results depend strongly on the reference value for the wage rate (see figure 3). When the reference value is equal to zero, the worst-off tend to be people with high distaste for working, so that they need less transfer to be as happy when they work or when they do not. Here, the worst-off tend to be women and people with a degree inferior to high school. On the contrary, we have fewer white individuals in comparison with the overall sample. On the other side, results given by the wage egalitarian equivalence criterion and by the egalitarian equivalence when the reference value for the wage rate is equal to 5 dollars per hour are very similar. This may be due to a under-estimation of the actual transfers received by the individuals so that the wage egalitarian equivalence cannot capture their unequal distributions among individuals. Regarding the cardinal criteria, we select two different cardinalizations. When we fix β l = 0.06 and ɛ equal to zero, we make the results more comparable with the conditional equality criteria. However, the worst-off according to Roemer s criteria are not comparable with the ones identified by the different conditional equality criteria (figure 4). In fact, the worst-off according to Roemer reflect the composition of the whole sample. It seems that none of the specific group tend to be more affected by inequality of opportunity according to Roemer s criterion for this value of β l. Results are different when we use the zero egalitarian equivalence criterion as a way to cardinalize well-being. In this case, women, people with a high school degree or less and non-white people would be the worst-off. This shows how 18

19 then distribution of effort, and by consequence the extent of unfairness, depends on the way we measure well-being. This shows how in this case, accounting for preferences cause strong changes in the identification of the worst-off. Figure 4: Van de gaer s criterion Figure 5: Roemer s criterion This is not observed for Van de gaer s criterion. The worst-off according to Van de gaer s criterion are the same regardless the type of cardinalization. In any case, they are the individuals who belong to the type with the lowest wagerate. The composition of the worst-off are depicted in the figure 5 and we observe that the composition of the worst-off coincides fully with the one depicted by the conditional equality when the reference value for preferences in the sample 19

20 Figure 6: Conditional Equality and others mean of β l. Indeed, these are exactly the same individuals. This is due to two phenomena. Firstly, when β l equals to 0.16, it is so small in comparison with β c that all individuals are considered to give priority to consumption. As a result, the worst-off for this criterion of conditional equality are the individuals with the lowest net income. Secondly, the mean net income of the individuals is lowest for individuals with the lowest wage rate, so they are the worst-off according to Van de gaer s criterion. As conditional equality gives priority to the individuals with the lowest wage rate as β l is small enough, the two criteria coincide in our sample. In the next section, we compare the extent to which the criteria display the same worst-off. 5.2 Differences and Similarities among the criteria The final question we raise here is the following: Is there any worst-off common to all the criteria? In other words, can we raise an unique conclusion about the persons who suffer more inequity regardless the criteria of inequality of opportunity and the cardinalization we use? The answer is no. Firstly, we focus on five measures of welfare in order to compare the criteria and to assess the impact of the change in the cardinalization on the identification of the worst-off. We take the conditional equality when β l is equal to 0.16, the min egalitarian equivalence, the Van de gaer s criterion and the two Roemer s criteria (using the two cardinalisations). Then, we identify the worstoff according to each criterion, finally we compare how many individuals are 20

21 Figure 7: Egalitarian Equivalence and others common to the five criteria, to four criteria, to three criteria, to two criteria and to one criterion only. To summarize the results, figure 6 and 7 show the percentage of individuals who belongs to multiple criteria simultaneously. We obtain that no individual is worst-off according to all the criteria, and three percent of the worst-off belong at least four of the five criteria. This exercise confirm previous results on the difference between Van de gaer s and Roemers s criteria [29]. Here, the two criteria differ substantially even when we use the same cardinalization. This means that distributions of well-being by type cross so that the individuals with the lowest wage rate ar not always the worst-off when using Roemer s criterion. Regarding the difference between the two ordinal criteria, half the worstoff are common to the conditional equality and egalitarian equivalence criteria. Here, as we select a very small value for β l, the two criteria tend to give similar results. Nevertheless, giving priority to the principle of compensation or neutrality yield to different conclusions. Finally, the difference between the ordinal and the cardinal criteria are substantial. Further research are needed to make more solid conclusions. But, this first exercise shows evidence that not only the question of the cardinalization of utility matter but also the proper fact to account or not for preferences impact on the identification of the worst-off. As we choose a small value for, we almost erase differences in preferences. In this case, conditional equality and Van de gaer s criteria coincide as in this example. But when we account for them, criteria diverge as shown through the difference displayed between the criterion 21

22 of egalitarian equivalence and the Roemer s example for instance. 6 Conclusion This paper proposes an empirical application of distinct criteria of equity. The model we define aims to identify who are the worst-off by implementing every criterion consistently with the theory of fairness on which they rely. The first results of this exercise confirm previous results about the divergence between the cardinal criteria. Also, it shows first evidence about the differences between the ordinal and cardinal criteria. The sample we use does not permit to observe significant heterogeneity in preferences due to observable characteristics. Nevertheless, the fact we approximate from individuals behavior the unobserved heterogeneous preferences allows us to account for preferences in the welfare measure we implement. The worst-off differ substantially as a function of the criteria. No individuals is a worst-off according to all the criteria. And even when we use cardinalization that make comparable the cardinal criteria with the ordinal criteria, preferences alter the composition of the worst-off. These results are only partial and uncompleted. The preliminary conclusions provide a first flavor on the divergence between the criteria of fairness from an empirical perspective. References [1] Aaberge, R., Colombino, U., Strom, S.: Joint Labour Supply of Married Couples: Efficiency and Distribution Effects of Tax and Labour Market Reforms, Labor and Demography EconWPA. (2005) [2] Aaberge, R., Colombino, U., Strom, S.: Do more equal slices shrink the cake? An empirical investigation of tax-transfer reform proposals in Italy, Journal of Population Economics. 17(4) (2004) [3] Almas, I. et al.: Measuring (un)fair inequalities, NHH Discussion Paper Series. (2008) [4] Arneson, R.J.: Equality of Opportunity for Welfare Defended and Recanted, Journal of Political Philosophy. 7(4) (1999) [5] Betts, J.R., Roemer J.E.: Equalizing Opportunity for Racial and Socioeconomic Groups in the United States Through Educational Finance Reform, in Ludger Woessmann and Paul E. Peterson (Eds.) Schools and the Equal Opportunity Problem, Cambridge, MA: MIT Press.(2007) [6] Bossert, W., Fleurbaey, M.: Redistribution and compensation, Social Choice and Welfare 13(3) (1996),