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Sources of inequality in Canada Rongve, Ian 1994

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SOURCES OF INEQUALITY IN CANADAByIan RongveB. Comm. (Economics) University of SaskatchewanM. A. (Economics) University of British ColumbiaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESECONOMICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIA1994© Ian Rongve, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. it is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of cwiThe University of British ColumbiaVancouver, CanadaDate L? 199YDE-6 (2188)AbstractThis thesis first presents a general procednre for decomposing income inequality measures by income source. The first method draws on the literature of ethical social indexnumbers to construct a decomposition based on a weighted sum of the inequality indicesfor the respective component distributions. The second method is based on the Shapley value of transferable utility cooperative games. The ethical and technical propertiesof the decompositions are examined, showing that the interactive technique has somepreviously known decompositions as special cases.In the third chapter I examine the contribution of differences in educational attainment to earnings inequality using the interactive decomposition by factor sources, introduced in chapter two, of the Atkinson-Kolm-Sen inequality index. I first use an estimatedsample-selection model to decompose predicted labour earnings of a random sample ofCanadians into a base level and a part due to returns to education. I do this decomposition once ignoring the effect education has on the probability of being employed and onceaccounting for this fact. I then calculate the contribution of these two sources of earningsto inequality measured by a S-Gini index of relative inequality for the full sample as wellas two separate age cohorts. The results indicate that approximately one half to twothirds of measured inequality can be directly attributed to returns to education whilethe interaction between the two sources post-secondary.The fourth chapter uses the earnings model from the third chapter to conduct policysimulations for broadly based policies, low targeted policies, and high targeted policies.I demonstrate that the policies targeting low education individuals produce a largerincrease in social welfare than do the other two types of policy.11Table of ContentsAbstractList of Tables viAcknowledgement viii1 Introduction 12 Decomposing Inequality Indices by Income Source 82.1 Introduction 92.2 Ethical Inequality Indices 102.3 Previous Decomposition Attempts 132.4 Decomposing Inequality Indices 172.4.1 Interactive Decompositions 182.4.2 Shapley Decomposition 262.5 A Comparison of the Decompositions 352.5.1 Interactive verses Shapley Decompositions 352.5.2 Old vs New Decomposition Methods 392.6 Potential Applications 442.7 Conclusion 463 Human Capital Models of Income Distribution 483.1 Row Education Affects Earnings 493.2 General Equilibrium Issues 521114 The Contribution of Education to Earnings Inequality4.1 Introduction.Research StrategyPrevious Empirical AnalysesInitial ComparisonsReturns to EducationThe Effect of Education on Inequality.4.6.1 Initial Estimation4.6.2 Complete Earnings Model .Differing Education EffectsShapley ContributionsConclusions5 Policy Simulations5.1 Introduction5.2 Frictions in the Education Market . .5.3 Costs of Education5.4 Constant Population Policies5.5 Constant Cost Policies5.6 Constant Probabilities5.7 Conclusion1111121131151181231251276 ConclusionsBibliographyA Data Appendix1291331384. Regression Results 144C Figures 151VList of Tables2.1 Amount of Measured Inequality Attributable to Different Sources . . .. 414.2 Estimates of the Earnings Equation 834.3 Decomposition of 1(y) for Earnings Decomposition 2 914.4 Amount of Measured Inequality Attributable to Different Sources . . 934.5 Percent of Measured Inequality Attributable to Different Sources . . 944.6 Marginal Decomposition for the OLS Model 964.7 Decomposition of I(y) for Earnings Decomposition 974.8 Amount of Measured Inequality Attributable to Different Sources . . 984.9 Percent of Measured Inequality Attributable to Different Sources . . 994.10 Marginal Decomposition for the Tobit Model 1014.11 Decomposition of I(y) for Earnings Decomposition 1024.12 Amount of IVleasured Inequality Attributable to Different Sources . . 1034.13 Percent of Measured Inequality Attributable to Different Sources . . 1044.14 Marginal Effects of Income Sources on Inequality 1054.15 Comparison of Decomposition Measures 1074.16 Percent of Inequality by Shapley Decomposition 1095.17 Government Expenditures on Education 1165.18 Costs of Education (1985-1986) 1175.19 Cost of a Change in Education Level 1175.20 Effect of Policies on Lifetime Inequality and Welfare 1205.21 Calculation of Cost of low Targeted Policy 121vi5.22 Effect of Policies on Lifetime Inequality and Welfare, Discounted at 6% 1225.23 Effect of Policies on Lifetime Inequality and Welfare, Discounted at 3% 1245.24 Effect of Policies on Lifetime Inequality and Welfare, Discounted at 6% 1255.25 Effect of Policies on Lifetime Inequality and Welfare 1265.26 Effect of Policies on Lifetime Inequality and Welfare, Discounted at 6% . 127A.27 Data Collected 140A.28 Cuts to the Sample 140A.29 Sample Means of the Data 141B.30 Estimates of the Taubman Equation 146B.31 Estimates of the Layard and Zahalza Equation 146B.32 Estimates of the Generalized LZ Equation 147B.33 Estimates of the Earnings Equation 149B.34 Estimatesof the Restricted Earnings Equation 150viiAcknowledgementI would like to thank the various members of my thesis committee, David Donaldson,David Green, Bill Schworm, and Guofu Tan for their help and encouragement. I’d alsolike to thank Herb Emery, Ann Holmes, Cherie Metcalf, Henry Thille, Wayne Thomas,Shaun Vahey, Liz Wakerly, and the rest of the UBC grad students for discussions and general silliness that made grad school relatively enjoyable. Thanks also to UBC, SSHRCCand Mom and Dad for financial support.viiiChapter 1Introduction1Chapter 1. Introduction 2It has been demonstrated time and again by empirical economists that there is a generallypositive correlation between the level of edncation of an individual and his or her labourearnings. There are several competing theoretical explanations for this phenomenon including the human capital approach of Becker (1964) and the market signalling approachof Spence (1973). This means that differences in educational attainment translate directly into differences in earnings and thns suggests that one possible method of reducingearnings inequality is to reduce the differences in educational attainment. The degree towhich this is a reasonable policy depend to great extent on the actual quantitative effectof education on earnings inequality. This thesis is concerned with determining the contribution of education to total earnings inequality in Canada. The primary assumptionbehind the empirical work is the human capital hypothesis. The general format that Ifollow is to first determine a reasonable method of allocating contributions to incomeinequality to different sources of income and then to use these results in an empiricalexamination of how inequality is affected by returns to edncation.My interest in this topic derives from my overall interest in welfare economics. Oneof the driving forces in welfare economics is how to increase the welfare of a society. Theusual method of doing this is to increase the aggregate income of society but there isanother way in which an increase in welfare can be achieved and that is to ensure thatthe existing income is better allocated, in whatever way is considered better.In this thesis I provide an answer to the question of how does differences in educationalattainment amongst individuals affect the measured inequality. As a preliminary to thatquestion two other questions must be asked. The first is how do we measure inequalityand the second being how best to allocate the effects on overall inequality to individualsources of income.There are typically two approaches to the measurement of inequality. The firstmethod is the purely descriptive, or statistical approach to measuring inequality. InChapter 1. Introduction 3this approach an attempt is made to describe the underlying distribution function thatgenerates the observed income inequality, with no attempt to decide whether or not theobserved income distribution is good or bad. The second method is the ethical approach,typified by the analysis in Blackorby and Donaldson (1978) and Chakravarty (1990). Inthe ethical approach to the measurement of inequality an attempt is made to quantifythe social desirability of the existing distribution in relation to other possible distributions which have the same total income’. This approach to inequality measurementdemands more of the researcher as it requires a formalization of the societal preferencesover income distributions and in fact there is a one-to-one correspondence between agiven ethical inequality index and a given formalization of society’s preference relationover income distributions. In this thesis I take an ethical approach to the measurementof inequality throughout. This does not mean that the results themselves depend on thisethical interpretation, only the interpretation of these results depends on this type ofinequality measurement.Once the method of measuring inequality has been decided, the second question is howthis measured inequality is to be allocated amongst different sources of income. Severalmethods of doing this exist in the literature, such as the Pseudo-Gini decompositionof Fei, Ranis and Kuo (1978), and the technique introduced in Shorrocks (1982), butthese suffer to varying degrees from several problems such as discontinuity in specificareas of the income space, or intuitively unappealing results2. Therefore the initial itemon the agenda is a general discussion of the theoretical decomposition of ethical incomeinequality indices by income source. This analysis is presented in Chapter two of thethesis.I begin chapter two by critically reviewing the existing methods for decomposing‘Sen (1992) takes this one step further by claiming that the ethical approach does not measureinequality at all, what it measures is the social badness of a given income distribution.2Chapter two considers these problems in greater detail.Chapter 1. Introduction 4income inequality indices by income source and outlining what I consider to be theundesirable features of these decompositions. I then introduce three general proceduresfor decomposing ethical inequality indices. The first general method produces what, iscalled a direct interactive decomposition because the decomposition method produces aterm which measures the extent to which inequality within the various sources of incomeis counteracted by inequality between sources of income. For example consider the caseof two sources of income, where source one and source two are negatively correlated. Inthis case an individual with a high income from source one will have a low income fromsource two. In this case the interaction term in the interactive decomposition will benegative, indicating that inequality in one source of income counteracts ineqnality in theother source of income. This decomposition answers questions of the form ‘How much ofobserved inequality is a result of income sourceThe second method of decomposing inequality indices produces a marginal interactivedecomposition. In this decomposition I consider how measured inequality would changeif I removed a given income source from aggregate income. The resulting value is themarginal contribution of an income source to overall inequality. This decomposition alsohas a term which may be interpreted as an interaction effect. In this case it provides abase level of inequality from which the marginal effects are measured. This decompositionis useful in answering questions like ‘How much would inequality change if we removedincome sourceThe last general procedure for decomposing inequality indices is derived from theShapley value of transferable utility games. With this decomposition I first decomposethe welfare measure using the Shapley value and then use the decomposed welfare todecompose the inequality index. Unlike the previous decompositions, the Shapley methodallows an exact allocation of inequality to individual income sources, with no need foran interaction term. Thus this decomposition is able to make statements like, ‘The totalChapter 1. Introduction 5effect of income source j on inequality is x.’After introducing the three methods of decomposing inequality indices I compare thenew methods amongst themselves, then compare these methods to the previously knownmethods. I demonstrate, through the use of examples, the ethical and technical propertiesof the new decompositions. I then conclude the second chapter with a discussion of someof the possible applications of the decomposition techniques developed.The third chapter is a theoretical discussion of the possibility of a human capitalapproach to earnings inequality. I first discuss the main competing explanation for thecorrelation between education and earnings, the signalling hypothesis, and state why myapproach requires the human capital approach. I then construct a simple theoreticalmodel which allows me to determine the effect of general equilibrum considerations oninequality. I maintain that even should the simple model be inadequate, and it likely isinadequate, the exercise in the rest of the thesis is still a valuable one to do.The fourth chapter builds directly on the material presented in the second chapter.It is an application to a specific problem of the decompositions in the second chapter.The major question that I ask is ‘What is the contribution of returns to education toearnings inequality in Canada?’To appropriately answer this question I first must decompose labour earnings into apart due to returns to education and a part due to other personal characteristics. Laboureconomics has identified two primary ways in which education can affect the returns toeducation; the first is the human capital approach, typified by Becker (1964), whereeducation is seen as actually increasing the productivity of the worker, and the second isthe market signalling approach of Spence (1973), where education is acquired to signal topotential employers a high productivity person. The technique that I use to decomposeearnings themselves is based on the human capital model and is not appropriate for amarket signalling model.Chapter 1. Introduction 6The first step in determining the impact of education on inequality is to estimatean empirical earnings equation. Therefore in chapter three I begin by drawing on theempirical labour economics literature to estimate an empirical earnings equation, usingeducation as one of the explanatory variables, for a cross section of individuals in Canadafor the year 1986. All subsequent analysis uses this sample.With the estimated earnings equation I proceed to decompose earnings, into a partdue to returns to education and a part due to other factors, by conducting a series ofcounterfactual experiments. I consider what would happen to the observed earnings distribution if, instead of their actual education level, the individuals had a counterfactualeducation level and earnings were generated by the same earnings process. The counterfactual level of education varies but is usually the lowest education level possible,less than nine years of schooling. I then consider the returns to education to be thedifference between the actual earnings level and the constructed counterfactual earningslevel. I then have two vectors, one a return to education and the other a return to allother personal factors. An appendix considers the possible impact of general equilibriumconsiderations on the inequality analysis.With the decomposition of earnings just described I then use the direct interactivedecomposition of the S-Gini inequality index to evaluate the effect of the various sourcesof income on overall measured inequality. I compare the results for several cohorts, thoseless than sixty-five years old, those between thirty and forty years old, and those betweenfifty and sixty years old. I also examine two ways of estimating the earnings equation,straightforward Ordinary Least Squares, and a tobit specification that takes into accountthe response of the probability of being employed to different education levels. Theresults indicate that approximately one third to one half of inequality is due to educationFinally for the most general earnings decomposition, I consider how the results changewhen the marginal interactive and the Shapley decomposition of the 5-Gini index areChapter 1. Introduction 7used instead of the direct interactive decomposition.The fifth chapter presents some experiments to determine the effect of various education policies on the contribution of education to earnings inequality. I concentrate onthree types of policies; these are broadly based policies, policies targeted at low educationindividuals, and policies targeted at high education individuals. In the fifth chapter Iconsider only the effect of the policies and not the specific form that they take.The policies are examined in terms of their effect on the distribution of lifetimeearnings and its effect on both inequality and social welfare. To do the lifetime analysisI use the earnings generating model developed in chapter four.Two cases are considered for each policy; the case where the number of individualsaffected by each policy remains constant and the cost of implementing the policy isallowed to vary, and the case where the cost of implementation is constant and thenumber of individuals affected varies. I show that, in both cases, the policy targetinglow education individuals keeps earnings inequality approximately the same but doesincrease social welfare. The final chapter concludes.Chapter 2Decomposing Inequality Indices by Income Source8Chapter 2. Decomposing Inequality Indices by In come Source 92.1 Introduction.Most people obtain income from a variety of sources, such as returns to education, property income or government transfer payments. This fact is often not considered in empirical studies of income inequality that consider only the distribution of total income. Atthe same time these studies are frequently cited as justification for economic and socialpolicies which attempt to decrease income inequality by targeting one particular sourceof income. In examining the effectiveness of such policies, a researcher needs a measureof income inequality that can be decomposed into the amount of inequality attributableto different sources of income.Many papers have attempted to construct inequality indices that can be decomposedinto a weighted sum of sub-indices, defined over the different factor components of income.Attempts of this sort include Fei, Ranis and Kuo (1978), Shorrocks (1982) and Westand Theil (1991). In this chapter I first examine the previous theoretical attempts toperform decompositions and then introduce two different methods of performing thedecomposition for arbitrary numbers of income sources. The new methods are bothintuitively appealing and empirically tractable. Two of the methods that I introduceexplicitly recognize that, for example, in an economy with two types of income, there arethree potential sources of income inequality, income source one, income source two andthe interaction between the income sources. The third method successfully allocates thisinteraction effect to the individual income sources.I adopt an explicitly ethical approach to decomposing income inequality indices andbegin by decomposing an equally distributed equivalent income measure for a given income distribution. Then the decomposed equally distributed equivalent income is used toconstruct a decomposition of the Atkinson-Kolm-Sen (AKS) family of inequality indicesand the Kolm-Blackorby-Donaldson (KBD) family of inequality indices. I show how theChapter 2. Decomposing Inequality Indices by Income Source 10ethical approach to decomposition allows for an easy and intuitive interpretation of theterms in the decomposition.I compare the new decompositions presented here with some previously known decompositions and present in detail the differences between my decompositions and previouslyknown decompositions. Then it is demonstrated that two of the previously known decompositions, that of the Theil inequality index and the standard decomposition of thevariance of income, can he considered special cases of one of my decompositions for theAKS inequality index.The rest of the chapter is as follows. In section two I introduce the basic notationand the ethical inequality indices that will be used. Section three discusses the previousliterature on the decomposition of inequality indices by income source. The interactiveand Shapley decompositions of the inequality indices are introduced in section four andin section five they are compared with each other and the previously known methods.Section six contains a discussion of some potential applications of the techniques andsection seven concludes.2.2 Ethical Inequality Indices.An index number is said to be ethical if the value judgements inherent in the index aremade explicit in the formulation of the index. In this section of the paper, I brieflyexplain the main results in the theory of ethical income inequality indices. This materialforms the background of the later analysis.I assume that society has preferences over alternative income distributions that canbe represented by a social evaluation function (SEF), W Q —+ R where 12 will bea different subset of RN depending on which type of index is used. W is continuous,increasing along the ray of equality and has all social indifference curves crossing the rayChapter 2. Decomposing Inequality Indices by Income Source 11of equality at some point1. Social evaluation functions of this type are called regular. Ican use the SEF to construct the equally distributed equivalent income function (EDE)2defined implicitly by equation 2.1:= E(y) <—* W(1N) = W(y) (2.1)where 1N denotes the N-vector of ones. The EDE gives the per capita level of incomewhich, if distributed equally throughout the population, would provide society with thesame level of welfare as does the current income distribution. The EDE function is acontinuous ordinal transform of the SEF. Thus, for two income distributions and ,E() E() if and only if W(Q) W().The EDE function is the foundation of the two types of ethical inequality indices thatI consider in this paper. The Atkinson-Kolm-Sen (AKS) inequality index for an arbitrarySEF isI(y)=1- (2.2)where p(y) is the mean of the income vector y and Q = {RN I Zy > O}. The AKSinequality index 1(y) has many well known specific inequality indices as special cases.Setting E(y) = (fL12i —1))/N2,where is a ranked permutation of y such that jgives the Cmi inequality index I°(y), setting E(y) = jz(y) —1(y/N) log(y1/N)produces the Theil inequality index and setting E(y) = p(y) — [L1(y —yields the coefficient of variation inequality index. Corresponding to the AKS inequalityindex is the AKS equality index, defined by E(y) 1 — 1(y) =The AKS index is the percentage of aggregate income that can be given up by thesociety and still be able to obtain the same level of well-being as with the original distribution by distributing the remainder equally among the individuals in the society.1The ray of equality is the ray defined by the equation y = alN, where is an N-vector of ones.2See Sen (1973) or Atkinson (1970) for more on the equally distributed equivalent income.Chapter 2. Decomposing Inequality Indices by Income Source 12Thus it can be interpreted as the percentage of total income that is wasted because ofinequality and does not contribute to social welfare.The AKS index is the most commonly used type of inequality index in appliedwork but there is another general inequality index. This index is the Kolm-BlackorbyDonaldson (KBD) inequality index defined byA(y) = i(y) — E(y) (2.3)where in this case={y I Z y,j > O} The KBD index gives the amount of incomeper capita that could be given up from the current aggregate income, provided that theremainder is distributed equally, and still preserve the same level of societal well being.It is the dollar value of the loss of potential welfare arising from income inequality3.An index with the feature that 1(y) = I(Ay) for every positive constant A is termed arelative index. A relative index is used if the society is concerned only about percentagedifferences in income, or income shares, and not the actual differences in income. Forexample a change in the units that income is measured in has no effect on inequalitymeasured by a relative index. It has been shown by Blackorby, Donaldson and Auersperg(1981) that the AKS index in equation 2.2 is a relative index if and only if the SEF Wis homothetic.Similarly an index for which A(y) = A(y--61”), where 6 is a scalar, and y+6l” C Q, iscalled an absolute index. An absolute index is appropriate for a society which is concernedonly with the differences in income measured in specific units. It can also be shown thatthe KBD index in equation 2.3 is an absolute index if and only if W satisfies a conditioncalled translatability4.A function W is translatable if and only if the SEF satisfies thefollowing condition, W(y) = q5(W(y)) where W(y) satisfies T/(x + 61”) = W(x) + 6 for3Because the KBD index measures inequality in dollars, intertemporal comparisons require y to be avector of real incomes.4See Chakravarty (1990) and Blackorby and Donaldson (1980).Chapter 2. Decomposing Inequality Indices by Income Source 13all (y,6).2.3 Previous Decomposition Attempts.In this section I discuss previous attempts at decomposing income inequality measuresand provide examples to demonstrate some undesirable features of these decompositionmethods. In subsequent sections I develop an alternative decomposition technique thatdoes not share these drawbacks.Income received as payment for a particular service or as a return to a particular assetis defined to be the factor component of that service or asset in income. The service orasset is the source of that factor component. Assume that an economy contains J sourcesof income. The j’th income component of the income distribution is a vector y C RN,where N is the number of individuals in the economy5. The i’th element of y3, denoted4 gives the income of person i from income source j. Let the total income distributionin the economy be denoted by y C RN where p = > y.The Cmi index of inequality is the most familiar inequality index to most economists.The Cmi index can be written asI(y) =1- (2.4)where is a permutation of the income vector p such that th i+i, i = 1, ..., N andjt(y) is the mean income. Fei, Ranis and Kuo (1978) decompose the Cmi index in thefollowing way,J II°(y) = 1’YJ3PGi) (2.5)j=1 IkYiWhere S(y’,y) = [1— Z1f1(2i — l)W/,4y)N2]. The term in the square brackets isknown as the pseudo-Gini for income source j. The pseudo-Gini is similar to the Cmi but5j must also assume that the means of the components of income and the means of the total incomedo not equal zero.Chapter 2. Decomposing Inequality Indices by Income Source 14the rank assigned to person i’s income from source j is person i’s rank in total income,not his or her rank in the distribution of factor component j. The value of the pseudoGini for factor source j, weighted by the percentage of total income coming from sourcej, is defined to be the amount of inequality due to source jTo see the problem with this decomposition of the Gini index consider the followingexample with two people and two sources of income. Let the vector of incomes fromsource one be y1 = [1, 2] and from source two be y2 = [2, 1]. Thus total income isa vector y = [3, 3]. The decomposition of the Gini index using the technique of Fei,Ranis and Kuo gives p(y’) = = .5, 5P(y1,y) = —1/6 and 5P(y2,y) = 1/6 orvice-versa. Notice that the value of the pseudo-Gini is negative for the first source andpositive for the second source even though one is just a permutation of the other. Thereason for this is the arbitrary way of breaking a tie in the ranking of total income. Inthis case, the signs of the pseudo-Gini’s for the individual factor component distributionscan be reversed simply by reversing the method used to break the tie.The problem above is related to the continuity of the pseudo-Gini. The pseudo-Giniis not continuous in the factor component at the amount of income that will generate atie in the amount of total income between individuals. Consider the above distributionof income with the exception that y2 becomes y2 = [2, 1 + e]. Then as e approaches 0from below, the pseudo-Gini for income source 2 approaches 1/6 and for income source1 the pseudo-Gini approaches —1/6. As 6 approaches 0 from above the pseudo-Gini forincome source 2 approaches —1/6 and for income source 2 the pseudo-Gini approaches1/6.Layard and Zabalza (1979) use the variance as a measure of income inequality. Thefollowing decomposition is obviousVar(y) = Var(y’) + Var(y2)+ 2Cov(y’, y2) (2.6)Chapter 2. Decomposing Inequality Indices by Income Source 15This decomposition has a drawback in that it is only valid when either the variance orcoefficient of variation is the measure of inequality. A second problem is that, when nsingthis decomposition, the nnmber of terms in the decomposition increases with the nnmberof income sources at a greater than proportional rate because the covariance betweeneach possible pair of factor components mnst be included.The next decomposition rnle I examine in this section is the general decompositionrule proposed by Shorrocks (1982). In this paper Shorrocks states that the decompositionrules described above are not nniqne, then sets out six axioms that do lead to a uniqnedecomposition rule. Of these six axioms there are two that, when combined, lead to aresult that is undesirable. As well, a third axiom that Shorrocks uses casts doubt on thereasonableness of this decomposition for a broad class of inequality indices.The six axioms used by Shorrocks areAssumption 1: 1(y) is a continuous and symmetric inequality index.Assumption 2: a) S3(y1,..., yJ; J) is continuous in y,b) S(y, ..., yJ; J) = Sj(y, ..., yrJ; J) if wl / 2TJ is any permutation of 1, ...,j.Assumption 3: Si(y’, ..., y”; J) = Si(y1,y—y; 2) = S(y’, y) andyJ; J) = S(y, y).Assumption 4:>’.. S(y, y)=I(y)Assumption 5: a) If P is any N x N permutation matrix, then 5(yiP, yP) = S(y, y)and b) S(z-’1”,y) = 0 for all v.Assumption 6: S(y’, y’ +y1P) = S(y1P,y’ +y1P) Given these assumption Shorrocks’main result may be statedProposition 1 Defining the contribution of source j to inequality to be S(y°, y), thenassumptions 1—6 imply— S(y,y) — Cov(y,y)27— I(y) — Var(y)Chapter 2. Decomposing Inequality In dices by Income Source 16for all y 0 yiN.Shorrocks’ result is that the percentage contribution to inequality of a given incomesource, J, is equal to the value of the slope coefficient in the ordinary least squares regression between total income and the income source j. Thus the percentage contributionof a given source is the same for all inequality indices. Assumption 4 is called ConsistentDecomposition. It states that the sum of the terms in any decomposition must equal thevalue of the inequality index for total income. Assumption 6 is called Two Factor Symmetry. Now consider again the previous example. The overall inequality index equalszero and y2 = y1P; therefore consistent decomposition and two factor symmetry lead tothe conclusion that S(y1,y) = S(y2,y) = 0. This is so even though it is obvious thatthere is inequality present in the distribution of income from each source and without agiven source there is inequality. Shorrock’s axiom means that his decomposition ignoresinformation such as this. It would be nice to have a decomposition rule which providedan indication of the amount of inequality present in the component distributions.Consistent decomposition and two factor symmetry combine to produce an undesirable result but not an extremely bad result. The third axiom of Shorrocks’ that I considerin detail is difficult to reconcile with relative indices of inequality although it does seemto be reasonable for an absolute index of inequality. This axiom is assumption 5, partb and is called Normalization for Equal Factor Distributions This axiom states that thecontribution to inequality of an equally distributed factor component is zero. At firstglance this may sound reasonable but consider what happens in the following exampleusing the mean of order 1/2 index of relative inequality. Let y’ = [1, 2], y2 = [1, 1].Then SS(1,)=I(y) = .01. Now change y2 to 2 = [2, 2]. Again, by Normalizationfor Equal Factor Components, S8(y1,y)=1(y) but now note that SS(yl,y) = .005.Chapter 2. Decomposing Inequality Indices by Income Source 17Out of two sources of income, only one has changed but this does not affect the contribution to inequality of the changed income source but does change the contribution toinequality of the factor component distribution that does not change. This is, perhaps,counterintuitive. The same objection does not apply to the Gini absolute index of inequality. Consider the previous example. In the first case A(y) = A(y’) = 1/4 and in thesecond case, since the only change is that everyone’s income has increased by the sameamount, A(y) = A(y’) = 1/4. Thus for the KBD absolute index this axiom seems moreappropriate.The last decomposition rule that I examine is for the Theil inequality index. TheTheil Inequality index is given by the following equationIT(y) = !—log(i) (2.8)West and Theil (1991) decompose equation 2.8 in the following wayIT(y)= _LEt_iog(’) (2.9)Notice that the first term is a weighted summation of Theil indices defined for the component incomes. The second term is an interaction term that gives a measure of how thevarious factor component distributions cancel out or reinforce the inequality present inthe other distributions. Thus the philosophy of this decomposition is similar to that forthe coefficient of variation inequality index.2.4 Decomposing Inequality IndicesIn this section I present two general methods for decomposing both the AKS and KBDinequality indices for arbitrary SEF’s by income source. The method used is to firstdecompose the EDE income and then to use this to decompose the respective indices.In decomposing the EDE functions two basic methods are suggested, the first methodChapter 2. Decomposing Inequality Indices by Income Source 18produces what I call an interactive decomposition and the second method produces whatI call a Shapley decomposition.2.4.1 Interactive DecompositionsIn this subsection I discuss and derive a decomposition which includes a direct effect oninequality for each income source and an interaction effect similar to a covariance term inthe standard decomposition of the variance. The approach taken is to state some properties that would be nice for a decomposition to have and then to propose some candidatedecompositions. I make no claim that this characterizes all possible decompositions withthese features but instead provides one intuitively appealing decomposition.The initial feature I would like my decomposition to have is that it has numericalsignificance. That is I want a decomposition where it makes sense to speak of a particular element causing a particular percentage of inequality. The immediate effect of thisrequirement is that the decomposition must therefore be additive. Any other structure(such as a multiplicative one) means that the effect of one element of the decompositioncannot be separated from the effect of any of the other elements and therefore a givenelement cannot be said to contribute a certain percentage of inequality6.The second feature that I would like to have the decomposition satisfy is that thecontribution of a given income source should depend, at least partially, on the inequalitypresent in the income source itself, that is I would like the contribution of a source toinequality to be an increasing function of the amount of inequality present in the source.This seems to be a reasonable requirement for a decomposition.Finally, consider what would happen if society starts initially with J sources of incomeand eliminates inequality in each source of income in turn with the following thought6j have not claimed that additivity is sufficient for numerical significance of the decomposition, onlythat it is necessary.Chapter 2. Decomposing Inequality Indices by Income Source 19experiment; regard each source of income as the only source of income and construct theEDE income for this source in the usual way. This gives a measure of the welfare thatsociety would receive from income source j if that source were the only possible source.This could then be considered a measure of the direct effect of income source j on totalwelfare. It is extremely unlikely that the contributions of individual income sources willsum to the actual level of welfare because, in the aggregate distribution, some of thereduction in welfare because, for example, person i has a very low amount of incomesource j may be counteracted because person i has a very high amount of income fromsource j’. This effect is not accounted for in summing the direct effect on welfare over allincome sources so, besides the direct effect on welfare, there will be an interaction effectas well which takes this into account. It seems reasonable that I ask this decompositionof inequality indices to include a term similar to this interaction effect as well.I now provide a candidate decomposition of the AKS and KBD inequality indicesreflecting the three requirements above. Later in this subsection I discuss whether thisis a reasonable decomposition in light of the suggested requirements.Recall that the EDE income for a given income distribution is given by the function= E(y). The EDE function can be trivially rewritten asJ JE(y)=E(y) + E(y)— > E(y), (2.10)j=Iwhich, defining ?=,4y2), p=p(y) and .s’=,u2/p, where s is the share of source j intotal income, can in turn be rewritten asJ JE(y) = E(y’) + [s2E(y)—E(y’)}. (2.11)j=1 j=1This decomposition of E reflects requirement three above. To construct the decomposition of the AKS inequality index, multiply equation 2.11 by —1/p and add 1 to bothChapter 2. Decomposing Inequality Indices by In come Source 20sides to obtainJ‘=( i’i =1 ‘ •=( i’\i‘— s—‘j, (2J2)j=1 j=1 Itwhich can be written asI(y)lZs3E (2.13)Remembering that zL1 s = 1,J J1(y) = s3I(y3) + s3[I(y)— 1(y3)]. (2.14)j=1 j=1Now, defining C’(y’, ..., y, y)=[1(y) — I(y)], equation 2.14 may be rewritten morecompactly as1(n)= E siJ(y) + C’(y’, ...y, y). (2.15)The first term in equation 2.15 is a weighted sum of the relevant inequality indices forthe different factor component distributions, with weights equal to the fraction of totalincome generated by source j. Note that these are the actual indices, not pseudo-indicesas in Fei, Ranis and Kuo (1978) and thus do not depend on total income y . The termyJ, y) is an interaction term which measures how much inequality in the J factorcomponent distributions is counteracted by inequality in the other J — 1 distributions offactor component incomes. If this value is negative then it means that the interaction isincome equalizing and if it is positive then the interaction is income disequalizing.The KBD inequality indices can he decomposed in a similar way to the AKS. Recallfrom equation 2.11J JE(y) = E(y’) + [sE(y)— E(yj], (2.16)i=1 j=1the KBD index is defined as A(y) = It — E(y) so substituting in from 2.16 I getJ JA(y) = — 3(y)——p +s3E(y) — E(y2)], (2.17)j=1 j=1Chapter 2. Decomposing Inequality Indices by Income Source 21which can be rewritten asJ J JA(y)=— E(y3)] + [3(— E(y))}—— E(y3)j, (2.18)j=1 2=1 j=1and now substituting in from the definition of the KBD index leaves the decompositionJA(y) = A(y2)+ Z[sA(u) — A(y2)j. (2.19)j=1 j=1The decomposition in equation 2.19 is again a weighted sum of the relevant inequalityindices for the different distributions of factor component income plus an interactionterm. In contrast to the AKS index, however, the weights on the inequality indices inthe KBD decomposition are equal to one. This is intuitively because the units that theindices A(y) and A(y) are measured in are the same, whereas in the case of the AKSindex, the units of 1(y) and i(yi) are different.I(y) is measured in terms of the fractionof total income but I(yi) is measured in terms of the fraction of income from source j.Both A(y) and A(y) are measured in terms of income.Defining QA(yl,...,yJ,y) = L1[siA(y)— A(y)], equation 2.19 can be expressedmore compactly asA(y)= E A(y3)+ 0A(Yl ..., y, y). (2.20)The term CA(yl, ..., y, y) is again an interaction term composed of a scaled down KBDindex for total income and a KBD index for income from source j. The scaling downof A(y) is done by multiplying by the share of total income arising from source j. Thisapportions the EDE income in a way consistent with every factor component distributionhaving the same pattern of inequality. If GA(yl, ..., y”, y) is positive, this has the interpretation that the interaction between income sources reinforce the inequality present inthe distribution of income.The previous decompositions of the AKS and the KBD inequality indices are basedon considering the individual factor component distributions. Suppose the interestingChapter 2. Decomposing Inequality In dices by Income Source 22question is not how much a given source of income contributes to inequality but insteadis how does inequality change as a result of income source j. The current decompositionsdo not necessarily answer this question. An alternative, and perhaps equally plausible,decomposition can be obtained by using the distribution of income from all sources exceptsource j. Denote by y the distribution of income from all sources but source j, that is=y — y3.I follow the same general method for decomposing the EDE income. Thus the counterpart to equation 2.11 isJ JE(y) = E(y_) — E(y_) + E(y). (2.21)j=1 j=1Given 2.21 the AKS inequality index isJo = 1 — — E(y_) + (2.22)j1 j=1 ItMultiplying terms by ti/i yields= 1—ui(y_i)_Ey) + (2.23)orJ =‘—j’ J =(—j’ =1(y) = 1—-I- — (2.24)IL IL ftNow by definition s = 1 — .s’ so by substitutionJ (-i) J (-i) ()I(y) = 1 — (i—+ (1—— (2.25)or by adding and subtracting one and rearrangingJ J1(y) = ,s3J(y) + s[J(y) — I(y)]. (2.26)j=1 j1Which is more compactly expressed asJJ(y) = C’(y’, ..., y) + i[1(y) — J(y_)]. (2.27)j=1Chapter 2. Decomposing Inequality Indices by Income Source 23Equation 2.27 is the final decomposition of the AKS index 1(y). The terms in thisdecomposition have the following interpretation. The first term is a weighted sum of therelevant inequality index defined over the distributions y5. This provides a base level ofinequality and is similar to the interaction term from the previous direct interactive decomposition. The second term is the difference between the inequality in the distributiony and the distribution y5 and is the contribution to total inequality of source j given thedistribution y5. It thus represents the marginal contribution of source j to inequality. Ifthe inclusion of the jth factor component results in a more unequal distribution then thissecond term is positive and 1(y) is increased by y5. The value of this marginal term ispositive if income source j is unequalizing and is negative if income source j is equalizing.The KBD index can be decomposed in a similar way. Again starting with the decomposition of the EDE incomeJ JE(y) = E(yj— > E(y) + E(y) (2.28)5=1 j=1Adding and subtracting c5 gives the expressionJ JE(y)=— E(y5)]—— E(y5)] + E(y) (2.29)5=1 5=1Then recalling the definition of the KBD index givesJ JA(y) = A(y3)— A(y3)+ A(y) (2.30)5=1 5=1Rewriting equation 2.30 leavesJ JA(y) = A(y3)+ [s3A(y) — A(y3)] (2.31)5=1 5=1orJA(y) = CA(y_l, ..., y, y) + [.s2A(y) — A(y2)] (2.32)5=1Chapter 2. Decomposing Inequality In dices by Income Source 24Equation 2.32 has an interpretation similar to the interpretation of equation 2.27.The first term is a base level of inequality generated by the distributions. This baselevel is then modified by the second term which again has the interpretation as themarginal effect of the jth source of income on the inequality distribution. If this value ispositive, then the marginal effect of introducing the jth source of income is to increasethe inequality in the total distribution.To help in justifying the decomposition in this section consider the generalization ofthe decomposition 2.15 for the relative AKS inequality index given byJ J JI(y)= > a’I(y’) + MI(y)— Ea31(y),. (2.33)i=1 j=1 j=1where >[..1 a = = 1. Now take the example with two individuals and J incomesources. The vector of incomes from source 1 is y1 = [gii, y], from source j, j 0 1 is= [cyj, c?y], and the vector of total income isy = [(1+2&)y}, (1 +Z2ajy]. (2.34)Thus 1(y)=1(y2)=1(y’) and the interaction term in the decomposition is identically zero. What then would be the percentage contribution to total inequality assignedto income source j? It would be equal to aI(y)/I(y) = a3. Now since the measuredinequality in each source y is the same, it seems reasonable that the percentage contribution to inequality be equal to the percentage of total income supplied by income source j.For this to be true for all vectors of income it must be true that a’ = ctk/(1 +j2 a) = .sifor k $ 1 and a1 = 1/(1 + ZJ=2 cS) = s’. The value of the interaction term C(y1,..., y3)is unchanged with different values of bi, and so there is no harm in setting b = .si aswell.The AKS inequality index is a cardinal measure of inequality, that is it makes senseto say that one distribution is twice as unequal as another distribution. Does this implyChapter 2. Decomposing Inequality In dices by Income Source 25anything about the decomposition? Cardinality means that the two functions 1(y) andAI(y), A > 0 are informationally equivalent. Thus the decompositionsJ J J1(y) = > a31(y2)+ > frI(y)— Ea31(y)), (2.35)j=1 j=1 j=1andJ J JAI(y) = Aa’I(y’) + AMI(y)— E Aa’I(y’)) (2.36)j=1 j=1 j=1are informationally equivalent and the individual terms of the decomposition have numerical meaning. The implication is that it makes sense to speak of source j contributinga given percentage to inequality.I now present an example of the decompositions which should help highlight theintuition behind the interactive decompositions. I will focus on the AKS index for theSEF, mean of order one half7. Consider a society of two individuals who each have twopossible sources of income. The first source is the return to personal characteristics whichare not education, such as experience. Suppose individual 1 has a return to experience of$2 and the second person has a return to experience of $4. Denote this income source 1and it is given by the vector y1, y’ = [2, 4]. The second source of income is the returnsto education. Suppose individual 1 has a return to education of $1 and individual twohas no income from education so the vector of income from source 2 is y2 = [1, 01.Overall income is the sum of these two vectors so is y = [3, 4].The above vectors give values for E(y) = 3.4821, E(y’) = 2.9142, and E(y2) = .25.The direct interactive decomposition yields a contribution of .0245 for source 1, .0714for source 2, and a interaction term of .0908. The intuition behind these results are7The mean of order 1/2 index is derived from the following EDE function=[Yi’I2 + Y21”2]Chapter 2. Decomposing Inequality In dices by In come Source 26that source two contributes more to total inequality than source 1 does. This is so, eventhough source 2 is a much less important source of income than source 1 (in terms ofmagnitude), because the measured inequality in source 2 is much higher than that insource 1. The inequality in source 1 counteracts the inequality of income from source 2,and vice versa. Overall the interaction of the two sources reduces measured inequalityby .0908.The marginal interactive decomposition gives a value of - .4242 for the marginal contribution of source 1, -.0034 for the marginal contribution of the returns to education,and a base term of .4327. Thus here we see that source 1 is also a much more importantsource of reductions in inequality than is source 2. This results from a combination of theaddition of source 1 counteracting some of the inequality in source 2, and the dramaticincrease in mean income when we add source 1. The increase in mean income meansthat for the same difference in levels of income, the measured inequality is lower. Thecross term of .4327 provides a base from which the marginal effects are measured.The previous discussion highlights the reasons why the interactive decompositiontakes the form that it does instead of some other similar form. Although I concentratedon the interactive decomposition given in equation 2.15, the same procedure will producesimilar justifications for the other interactive decompositions.2.4.2 Shapley DecompositionThe interactive decompositions presented in the previous subsection are intuitively appealing and have some well known decompositions as special cases. In one way howeverthey may not provide us with the answer that we want. They provide only a partial answer to the question of how much of inequality is a result of a given income source. Thisproblem arises because of the non-separability of the interaction effect. I now present acomplete characterization, based on the Shapley value of transferable utility games, ofChapter 2. Decomposing Inequality Thdices by Income Source 27an ethical decomposition that does allow for unambiguous allocation of the effects8.I again assume J income sources in random order j = 1, ..., J. Note vectors of incomedo not have to be indepeudent of each other, only that they be in random order2 Thevector of incomes from source j is the N-vector y. The aggregate income distribution isdefined by y= [_ fr’.Let M denote the set of vectors of income sources, that is M={y’, y2, ...,y’}. Asnbset of income sources is denoted C C {l, ..., J}. The grand subset containing allelements of {l, ..., J} will be denoted C. Denote by c = C I the number of elementsin the subset C. For a given subset the aggregate income from subset C is denotedyC= Zc y°.In an ethical decomposition of an inequality index the first step is to decompose theEDE income function E(y). Let uo(M, E) be the contribution of subset C to the EDEincome determined by the function E and the set M. Denote by c the contribution ofsource j to the value of the EDE. Let the sum all j of the contributions equal the EDEincome E(y). ThusJZa(M,E) = E(y) (2.37)j=1Since the SEF and thus the EDE income function are functions of aggregate incomeonly, the decomposition of the EDE function should be symmetric with respect to incomesources. This means that the name of the income source or how it was earned does notmatter10. Formally, symmetry with respect to income sources requiresAxiom A) For any two sets of income sources, M and MW, where MW is a permutation81 am indebted to an anonymous referee who provided the idea for this decomposition.9For example suppose I have a vector of returns to education and another vector of returns to otherpersonal characteristics. Requiring them to be in random order only requires that it not matter if vector1 is the returns to education and vector two the returns to other personal characteristics or vice versa.10This property is an implication of welfarism in the SEF.Chapter 2. Decomposing Inequality In dices by In come Source 28of M, and income source Ac in M is Kk in rUk(M,E) = Uwk(Mw,E) (2.38)The second axiom is borrowed from Shorrocks (1982) who calls it independence of thelevel of disaggregation. Suppose there are two income sources from the same population.This axiom states that the sum of the individual contributions must equal the totalcontribution of a new income source constructed by adding the original two sourcestogether. FormallyAxiom B) For any two subsets of income sources C’ and C”uc(M, E) = uc’(M, E) +a0n(M, E) (2.39)where C = C’ U C”, and C’ fl C” = 0.Define the welfare from a subset y° to be equal to (yC) Now define an incomesource’s marginal contribution to subset C as E(y°) = E(f)_E(yG_y). The marginalcontribution is a measure of the change in the welfare from a particular subset, that isproduced when the jth income source is included compared to when it is excluded. Whenthe marginal contribution is positive then the income source would increase the welfarefrom the subset and when it is negative then the source reduces welfare from the givensubset. In general the counteraction of inequality in one source by inequality in othersources will mean that the marginal contribution of a source is not equal to the welfarefrom the source.The last axiom that I use in deriving the Shapley decomposition is that the contribution of source j to inequality be a function of the marginal contributions of source j tothe various subsets of income sources. What is ultimately of concern is the total effectthat an income source has on welfare. The simplest measure of this effect is comparingwelfare with, and without, the given income source11. The most fundamental measure11Such as in the Marginal interactive decomposition.Chapter 2. Decomposing Inequality Indices by In come Source 29of this effect is the marginal contribution, to total welfare from all sources of income,of a given income source. But that is not the only way that a source can contributeto measured welfare. Each income source interacts with all other income sources bothindividually and in combination to produce the total measure of welfare. For examplereturns to education can counteract inequality in transfer payment income by itself, canreinforce inequality in non-labour income by itself, and could have a neutral effect on theinequality in transfer payment plus non-labour income. All three effects are legitimatecomponents of how returns to education contribute to overall welfare. Thus a decomposition to find the total effect of a given income source on welfare should consider themarginal effect of a given income source on all possible subsets of income sources. AxiomC puts a little more structure on the way in which the marginal effect contributes to thedecomposition.Axiom C For two sets of income sources M and M, with y° the vector of incomes fromcoalition C in M, and g’ the vector of incomes from coalition C in M , if Ei(yC) E(y°)for all C then u(M,E) a(M,E); orb) for a given set of income sources and two EDE functions E and E2, if E(yC) E(y°),for all C then a(M,E1) o’(M,E2).The first part of this axiom says that if there are two different economies with differentsets of income sonrces, and if an income source consistently contributes a larger marginaleffect to all subsets from one economy than to subsets from the other economy, then thecontribution of that income source to welfare in the first economy must be greater thanit’s contribution in the second economy. The second part of this axiom may also needsome explanation. As a society, it is by no means obvious what our social preferences are.This part of the axiom says that if we compare the same economy with two different socialpreferences and one set of preferences consistently attributes a higher marginal effect toChapter 2. Decomposing Inequality In dices by Income Source 30one income source, then the decomposition of welfare using the social preferences whichattributes a higher marginal effect will also yield a higher contribution to total welfarefor the given income source.The next step is to construct an equivalent function to E(y°), denoted E, which isdefined on the power set of C and the set of income sources M, for which E(C, M) =E(y°) for all C. Consider the function(C, M) = E(Iy1 + Jy2 + ... + JyJ) (2.40)whereri ifjeC,I0= (2.41)0 otherwiseIt is easy to see that E(C, lvi)) = (C) for all C. Thus any decomposition of willimply an equivalent decomposition of E. I will therefore concentrate on a decompositiona(C, E, M) of Now consistency requires the sum of the ã must equalJ= E(C,M) (2.42)In turn the axioms A, B, and C have the following implications for a(E)Axiom A’) For any two vectors of income sources M and M where income source k inM is Kk in M, and CWJ is the appropriate permutation of the grand coalition,Uk(C, E, M) = ark(Cr, E, M) (2.43)Axiom B’) dc(C,E,M) = Uc’(C,E,M) +d011(C,,M) where C = C’ U C” andC’ n C” = 0Axiom C’)a) For any two sets of income sources M and M, E(C, M) E(C, M), for all C whereM) = (C, M)— (C \ j, M) implies a(C, , M) o(C, , M) and,Chapter 2. Decomposing Inequality Indices by Income Source 31b) For any given M and two EDE functions Ei and E2, C, M) E?2(C, M) for all Cimplies o(C,’,,M) a(C,E2,M).Actually axiom B’ turns out to be redundant. Axioms A’ and C’ are enough touniquely determine the decomposition rule as the Shapley rule. Since the Shapley ruledoes satisfy B’ there is no contradiction.Proposition 2 There is only one decomposition rule satisfying A’ and C’. It is theShapley decomposition given by(c—1)!(J—c)!> (C) (2.44)1<c<J ICI=c,—.1CCProof: The proof of the proposition is from Young (1985)12. QEDSince E(C, M) = E(y°) for all C, it must be the case that a(C, E, M) is a decomposition of E(y) and it will satisfy the two axioms A, and C. It is easy to demonstratethat a(C,E,M) = a(M,E) whereu(M, E)= E (c - ‘)RJ - c)! [E(yC) - E(y° - y)] (2.45)1<c<J ICI=c,—— CCCJ,jECAn alternative characterization of the Shapley value ej given in Moulin (1988) demonstrates that ã satisfies axiom B’ and thus a satisfies b.A potential problem with the Shapley decomposition of welfare may be that not allsubsets of income sources are feasible. For example consider the case where there are threesources of income which are returns to high school education, returns to post—secondaryeducation, and returns to other potential factors’3. In this example it is unreasonable to‘2Actually Young (1985) assumes a stronger version of axiom C’ than I use but his proof only requiresthe weaker version.13This case is examined in the next chapter.Chapter 2. Decomposing Inequality In dices by Income Source 32talk about positive returns to post-secondary education if the individual does not firsthave a return to high school education. This is admittedly a drawback in the analysis. Inthe case where not all subsets of income are reasonable it may be the case that the Shapleydecomposition is not the only decomposition that satisfies the three axioms presented butit will still be the case that the Shapley decomposition does satisfy the axioms.The decomposition of the EDE function allows a corresponding decomposition of theAKS and the KBD inequality indices. Recall that the AKS index is given by 1(y) =1 — E(y)/p. Substitute the Shapley decomposition a into the AKS index to getJ(y) =1— (2.46)Multiplying and dividing by 1u gives‘1(y)=l_>__1 (2.47)j=1 P /1which in turn equalsJ j 0•I(y)LL 1—- (2.48)3=111 1tThe formula in equation 2.48 can be broken down into two parts. The first part isthe share of income source j in total income. This is multiplied by the second part whichresembles an AKS inequality index but instead of using the EDE function it uses theShapley contribution of source j to the EDE income.Now I use the Shapley decomposition of the EDE function to decompose the KBDinequality index. Recall that the KBD index is given by A(y) = p — E(y) which can bewritten asJA(y) = p — >Zcrj (2.49)j=1or in turnJA(y) = [p’ — (2.50)3=1Chapter 2. Decomposing Inequality In dices by Income Source 33Equation 2.50 has the following interpretation. It is the sum of KBD inequality indicesfor individual income vectors with the modification that only the Shapley contributionof income source j to the overall EDE income is used instead of the actual EDE income.The intuition behind the Shapley decomposition can best be thought of in referenceto the direct interactive decomposition. In the interactive decomposition the interactionterm is present and makes it impossible to allocate all of the inequality to one sourceor another. The Shapley decomposition gets around this problem and one way to thinkof what it is doing is separating the interaction effect and allocating it to the individualcomponents. To separate the effect of a given income source on the interaction effect,it must consider all possible sources of interaction. This is the reason for examining allpossible coalitions of income sources.The intuition for the complete decomposition provides some additional intuition aboutaxiom C. What this axiom says is that if two economies are compared, and in one worlda given source has a greater marginal effect on all sources of interaction than in the otherworld, then the total effect of that source in the first world must be greater than that inthe second world.The usefulness of the Shapley decomposition arises from two things. First many people may be uncomfortable using either of the interactive decompositions because of thepresence of the interaction term which cannot be allocated between the different sources.The solution to such philosophical problems is to use a method which has no troublesome interactions. Although alternative methods to the Shapley decomposition exist thepseudo-gini approach has serious technical problems, and the Shorrocks decompositionhas some properties that may make it undesirable for the AKS inequality indices. TheShapley decomposition is an attempt at providing a decomposition which avoids theproblems of having an unpleasant interaction term but at the same time, since it is basedon an explicit decomposition of the EDE function, should be equally applicable to AKSChapter 2. Decomposing Inequality Indices by In come Source 34and KBD indices.As a help in understanding the Shapley decomposition I present the example with twoindividuals and two income sources, returns to education and returns to other sources,that I constructed previously. Recall that the vector of incomes from other personalfactors, which I denote y’ is y’ = [2, 4], and the vector of income from education isdenoted y2, y2 = [1, 0]. The vector of total income is again y1 = [3, 4]. I againuse the mean of order 1/2 SEF. With this SEF the EDE values are E(y) = 3.4821,E(y’) = 2.9142, and E(y2) = .2500. The Shapley contributions of the two sources towelfare are respectively a1 = 3.0732 and a2 = .4090. These numbers correspond to acontribution of source 1 to the AKS index of - .0290 and a contribution of returns toeducation to inequality of .0260, for a total AKS index of .0051. The interpretation ofthese contributions is that the returns to characteristics other than education reduceinequality. The intuition behind why this is true can be seen by examining the vectors ofincome sources themselves. Notice that while the absolute difference in income is greaterfor the first vector, the difference as a proportion of mean income from the source isgreater for the second vector. The inequality index is a function both of the contributionto welfare from differences in income and the contribution to welfare from the mean ofincome. For income source 1 the increase in welfare because of the increased mean incomeoutweighs the decrease in welfare from the greater differences in income. This yields anoverall negative effect in the contribution of source 1 to lost welfare. Likewise the lowermean income outweighs the decreased income differences in the first source, leaving apositive contribution of this source to welfare loss.The approach that I use in this paper is to first decompose the EDE income functionand then use this decomposition to construct a decomposition of the inequality index.An alternative approach is to use the same axioms and to apply them directly to theinequality indices themselves. This will provide a contribution to the AKS index for theChapter 2. Decomposing Inequality Indices by In come Source 35first of two income sources ofcrj = I(y’) + [I(y)-1(y2)] (2.51)and of the KBD index4= A(y’) + [A(y) - A(y]. (2.52)Comparing this to the ethical Shapley decompositions‘—(2.53)and— (2.54)it is obvious that, in general, the results of the direct decompositions are not the sameas the results of the Shapley decompositions presented in this paper. This is in contrastto the results of the interactive decomposition where a straightforward decompositionof the actual inequality indices can produce the same decomposition as the one that Iobtain from the ethical approach.2.5 A Comparison of the Decompositions.In this section I consider the main features of the decompositions introduced in the previous sections. First the two general approaches, interactive and Shapley, are compared,then these two new approaches are compared with the previously known decompositionmethods. I outline the main issues involved in choosing which of the decompositions ismost appropriate in a given situation.2.5.1 Interactive verses Shapley DecompositionsThe comparison of the two new decompositions suggested in this paper is easiest ifI consider only two income sources. Because the Shapley decomposition of the EDEChapter 2. Decomposing Inequality Indices by In come Source 36income requires the calculation, for each income source, of the marginal contribution ofthat source to each possible coalition, it can require a lot of computational effort. Theinteractive decomposition always has 2J terms so the calculation of the contributions is ingeneral much easier for the interactive decomposition than for the Shapley decomposition.An index that is homogenous of degree zero in incomes is said to be a relative index14.A desirable feature for any decomposition of a relative index is that each component ofthe decomposition is homogenous of degree 0 as well. Recalling the two decompositionsof the AKS indexJ J1(y)= > sI(yj—.s[I(y)— 1(y°)] (2.55)j=1andJ J1(y) = s’I(y’) + s2[I(y)—I(y’)] (2.56)j=1 j=1If 1(y) is a relative index then so are i(y) and I(y). If each y is multiplied by aconstant A then the overall distribution is also multiplied by A. Thus the numerator andthe denominator of the shares s are multiplied by A and the shares are homogenousof degree zero. Therefore the elements of the decompositions of the AKS index arehomogenous of degree zero if and only if the overall index is homogenous of degree zero.The Shapley decomposition is also a relative decomposition if and only if the AKSindex is a relative index. To see this note that Blackorby, Donaldson, and Auersperg(1981) have shown that the AKS index is a relative index if and only if the EDE functionis homogenous of degree one in income. If the EDE income itself is homogenous then theShapley decomposition of the EDE income is also homogenous of degree one, because itis additive in homogenous functions. The mean of an income source is homogenous ofdegree one so the function 1 — a/,? is homogenous of degree zero. I showed above thatthe share of income source j in total income is homogenous of degree zero. This shows14See Blackorby and Donaldson (1978) or Chakravarty (1990).Chapter 2. Decomposing Inequality Indices by Income Source 37that the Shapley decomposition of an AKS index is a relative decomposition if and onlyif the index itself is a relative index.I turn now to the decomposition of the KBD index that I introduced in the previoussection. Since the KBD index is, in general, used in different circumstances than theAKS index, it is natural that the decompositions should have different properties as well.Recall that the decomposition of the KBD index can be written asJ JA(y) = A(y3) + L[sA(y) — A(y1)] (2.57)j=1 j=1andJ JA(y) = A(yJ) + [sA(y)— A(y)J (2.58)j=1 j=1Both of these are obviously continuous functions of both y and y. As well it can quicklybe verified that both decompositions are symmetric with respect to the different incomesources.Consider what happens if, given a distribution y with component distributions y’and y2, a new total distribution is constructed so that ! = y + UrN. Denote by Qthe amount of extra income attributed to source j. If w is translatable then A(y) is anabsolute index and A() = A(y) and A(Q) = A(y). The first decomposition of AQflcan be written asJ JA() = A(°) + E[2A() — A(ñ°)], (2.59)jtrl j=1which equalsJ JA() = A() + [A() — A(’)]. (2.60)j1 3=1Similarly for the second decompositionJ JA() = ZA(r3)+ [A() — ZA(rS)]. (2.61)3=1 3=1Thus the decomposition of the KBD index is invariant with respect to the addition ofthe vector [0’ ... 0’ j if and only if the index itself is an absolute index.Chapter 2. Decomposing Inequality Indices by In come Source 38In the case of the Shapley decomposition of the KBD index the result is the same.Assume that the EDE function is translatable so that E(y + eiN) = E(y) + 0 and thateach income source j has 0j1N added to it, then the contribution to measured inequalityof each source is unchanged. To see this begin with the Shapley decomposition of A(y),JA(y)=— uj] (2.62)i1This formulation makes it easy to see that if uj(y’ +01N y+ OilT) = a(y1, ..., y) +0?, the Shapley decomposition is an absolute decomposition.Uj(y1 +01N, ••, y + =— 1”J — 1. [E(y° + 00) — E(y° + 00 — — Os)] (2.63)1<c<J——jECwhich equals=(c — 1)!(J — c)! [0) + 00 — E(y° — yi) _Oc + 0]. (2.64)1<c<J— CCCJ,jE CThus— l)’(J —= .“ [(y°) — E(y°— y3) + Oil, (2.65)1<c<JCcCJjECwhich can be rewritten as=u(y’,...,y)+(1)j)!O. (2.66)1<c<J ICIc—jECNow recognising that there are J — 1 choose c — 1 coalitions of size c which contain sourcej as= u(y’,...,y)+ (11_((Oi (2.67)1<c<JChapter 2. Decomposing Inequality Indices by In come Source 39= +a=O+u. (2.68)1<c<JThis demonstrates that the Shapley decomposition of the absolute KBD index will bean absolute decomposition, no matter what the values of O are.2.5.2 Old vs New Decomposition Methods.As I explained previously, a major objection to the use of the Fei, Ranis and Kuo decomposition is that the decomposition is not continuous at any point where there are ties inthe ranking of overall income. For the decompositions that I have suggested, this is nota problem. The problem with the Fei, Ranis and Kuo decomposition is that the rankingof overall income determines the ranking of the income source. The decompositions presented in this paper are functions of the actual inequality index defined on the incomesources and thus as long as the actual index is continuous in income, the decompositionis continuous in the component distributions.An index that is homogenous of degree zero in incomes is said to be a relative index’5.A desirable feature for any decomposition of a relative index is that each component ofthe decomposition is homogenous of degree 0 as well. Recalling the two decompositionsof the AKS indexJ J1(u) = Z sI(yj— E .s[J(y) — i(y)j (2.69)j=1 jraandJ JI(y) = shI(y3) + s2[I(y)—I(y2)J (2.70)j=1 j=1If I(y) is a relative index then so are i(yf) and I(y). If each y is multiplied by aconstant A then the overall distribution is also multiplied by A. Thus the numerator andthe denominator of the shares s are multiplied by A and the shares are homogenous15See Blackorby and Donaldson (1978) or Chakravarty (1990).Chapter 2. Decomposing Iriequality Indices by Income Source 40of degree zero. Therefore the elements of the decompositions of the AKS index arehomogenous of degree zero if and only if the overall index is homogenous of degree zero.For the moment now consider only the first decomposition=sI(y) + E[’(y) - 1(y3)] (2.71)I will now show that this decomposition has, as two special cases, the decomposition of theTheil index that is presented in West and Theil (1991) and the standard decompositionof the coefficient of variation inequality index used by Layard and Zabalza (1979).The Theil inequality index can be written asJT(y) = iLlog(i) (2.72)and the decomposition of this index is given to be=-log (yir1)—L p-log(‘) (2.73)Applying the first interactive decomposition technique for the AKS index gives an expression for the decomposition ofJT(y)=s log() + C’(y1,..., y, y) (2.74)Comparing 2.73 to 2.74 and setting=_-log(,P.) (2.75)shows that the two decompositions 2.73 and 2.74 are identical. Thus the decompositionpresented in West and Theil (1991) is a special case of the decomposition presented inthis paper.Now consider the variance of income as an inequality index. The variance isI”(y)=(y )2 (2.76)Chapter 2. Decomposing Thequality Indices by In come Source 411 1 2 3Row Index Decomposition y’ y2 Interaction1 Gini Interactive 1/12 1/12 -1/62 Gini Shapley 0.0 0.0-3 r=1/2 Interactive I .018 0.0 -.0084 r=1/2 Shapley I .016 -.0114-5 r=1/2 Interactive A .05 0.0 -0.026 r=1/2 Shapley A .04 -.01-7 r=1/2 Interactive I .013 0.0 -.0088 r=1/2 Shapley I .01 -.003-9 r=1/2 Interactive A .05 0.0 -0.00310 r=1/2 Shapley A .035 -.01-Table 2.1: Amount of Measnred Inequality Attributable to Different SourcesThe standard decomposition of this inequality index isff(y)= f EEkZ (2.77)j=1 it:1 i=1Applying the interactive decomposition technique for the KBD index yieldsJ j N (2 j\2= N +CA(yl y, y) (2.78)j=1 / i=1Inspection of 2.77 and 2.78 reveals that the standard decomposition of the varianceinequality index is a special case of one decomposition method explored in this paper.It will be helpful in comparing my decompositions, both interactive and Shapley, withthe Pseudo-Gini and Shorrocks’ decomposition, to calculate my decompositions for theexamples used in section three as part of the justification for looking for different decomposition methods. Thus Table 2.1 gives the values of these decompositions for the threeexamples used in section three. The first two rows of table 2.1 give the decomposition ofthe Gini inequality index for y’ = [1, 2], y2 = [2, 1], andy = [3, 3].As can be seen from rows one and two, the interactive decomposition of the Gini indexin this case attributes the same direct effect on inequality to both income sources. Sincethe overall measured inequality is zero, the interaction effect shows that the interactionChapter 2. Decomposing Inequality Indices by Income Source 42between sources of income serves to reduce overall inequality. In contrast, the Shapleydecomposition attributes zero to both income sources in this case. This demonstratesthat both the interactive decomposition and the Shapley decomposition satisfy continuityin the vicinity of ties in the rank of overall income. This is because an individuals weightdepends only on his or her rank in the income source j for the interactive decompositionor his or her rank in coalition C for the Shapley decomposition.Rows three through six of table 2.5.2 give the values of the interactive and Shapleydecompositions of the mean of order 1/2 AKS index and KBD index. For the incomedistribution given by y’ = [1, 2], y2 = [1, 1], and y = [2, 3]. Since y2 is equallydistributed, both of the interactive decompositions show a direct effect of y2 on measuredinequality of zero. Both of these decompositions also show that there is a negativeinteraction effect. The Shapley decompositions show something different. The overalleffect of source 2 is to reduce measured inequality while that of source 1 is to increaseinequality. Thus source 2 has a non-zero contribution to inequality even though it itselfis equally distributed. This obviously violates Shorrocks’ axiom Normalization for EqualFactor Components.Rows seven through ten of table 2.5.2 show the results of the mean of order 1/2 decomposition of the distribution given by y1 = [1, 2], y2 = [2, 2], and y = [3, 4].The interesting thing to notice in these results is the relationship between these resultsand the corresponding results in rows three to six. In the case of the interactive decomposition the contribution to inequality of source 1, and the interaction term havechanged while the contribution of source 2 has not changed. Thus it would seem thatthe interactive decomposition suffers to some extent from the criticism that I levelled atShorrocks’ decomposition that in comparing the two distributions the only contributionnot to change is the contribution of the income source that did change, at least for therelative AKS index. The Shapley decomposition does not suffer to the same extent sinceChapter 2. Decomposing Inequality In dices by Income Source 43the contribution of both sources changes’6.One of the axioms that Shorrocks uses in his construction is the axiom of symmetry. This axiom states that the name of the income source does not affect the valuation of the contribution to income inequality. To see how this holds for the interactive decompositions of the AKS index, consider an economy with two sources of income. Consider now two income distributions y and which differ because ‘=and p2= y’. These two distribution have the same aggregate distribution so theonly to have changed is the name of the income sources. Symmetry therefore requiress’I(y’) + s’[I(y)—1(y)] =2I(p)+2[I() — 1(p2)] which is demonstrated by substituting in y’ = 2. Similarly the second decomposition can he shown to be symmetric as well.The symmetry with respect to income sources is imposed on the Shapley decompositionas one of the axioms used in its characterization.I have presented two plausible interactive decompositions of both the AKS and theKBD inequality indices, the first based on the distributions y and the second based onthe distributions y. The two decomposition approaches provide slightly different information and thus are suitable in different applications. The first decomposition providesinformation about the degree of inequality within a given factor distribution and howthe different factor component distributions interact to cancel out some inequality. Thesecond decomposition shows the way in which the individual sources of income increaseor decrease inequality in the aggregate distribution hut does not say anything about theinequality within a factor component distribution. I also provided a method, based onthe Shapley value, of exactly decomposing inequality indices which does not have aninteraction term providing a much cleaner answer to how much inequality is a result ofa given income source. In this section I have compared the different approaches to each16This seems reasonable since the actual value of measured inequality changes from one case to anotherfor all of the indices.Chapter 2. Decomposing Inequality Indices kv Income Source 44other and to the previously known approaches to decomposing inequality indices.2.6 Potential Applications.In this section I outline some possible applications of the decompositions of the inequality indices that I have presented. I also discuss the relation between the use of thedecomposed inequality indices and the existing literature on these applications.The first application is to applied taxation theory. Most people agree that progressivity is a desirable feature for a given tax system to have. A progressive tax system isone where the tax payments for an individual as a percentage of his or her income are increasing in that individual’s income. A substantial literature has developed in measuringthe degree of progressivity present in a given tax system’7.One observable feature of a progressive taxation scheme is that after—tax incomeis distributed more equally than before—tax income. This characteristic means that,given the same amount of revenue collected, the after—tax welfare’8 of a society witha progressive taxation system is higher than the welfare of a society with a neutral orregressive taxation system’9.Define y to be after—tax income, y’ to be before—tax income and y2 to be tax payments.Then by definition y=y’—y2. Blackorby and Donaldson (1984) derive a relative indexof global tax progression to beTR(t, y’)= ‘2 (2.79)where t is the rate of neutral taxation that will produce the same level of revenue asthe actual tax system. This index is zero if the taxation scheme is proportional, and is‘7See Blackorby and Donaldson (1984) or Pfingsten (1987) for example.15Given some degree of inequality aversion‘°A regressive tax system is one where relative payments decline with income and in a neutral systemthe relative payments are constant.Chapter 2. Decomposing Inequality Indices by In come Source 45positive if the tax system is progressive.Now consider the first decomposition of the AKS. In this case it is given byJ J1(y) = suI(y3) + s3[I(y)—1(y3)] (2.80)j=1 j=1The first interaction term contains the same information as the Blackorby and Donaldsonindex of tax progressivity. It is.s’[I(y) — 1(y)] =_S1TR(t,yl)E(yl) (2.81)Therefore the Blackorby—Donaldson index of tax progressivity has a strong connectionwith the kind of decomposition that I use in this paper. The decomposition, however, provides more quantitative information about the degree of equalization of income achievedby the tax system because it includes not just the progressivity of the tax system, butalso information about the rest the components of income.One possible problem with using this decomposition arises in the case where the taxsystem is a purely redistributive tax—transfer scheme. In this case the mean tax level iszero and the decomposition is undefined. In this situation it is necessary to separate thetaxes and transfers and calculate the effects of the individual sources alone.The second decomposition of the AKS index can also be used to produce the Blackorby—Donaldson relative index of tax progressivity. In this case it is the expression for themarginal contribution to inequality that is of interest.Blackorby and Donaldson also provide an absolute index of tax progression that willprovide some of the same information as the decompositions provided here.Another area where the decomposition of income inequality indices is helpful is indevelopment economics, in fact the Fei, Ranis and Kuo decomposition of the Gini wasintroduced as a tool in the economic development literature. Many theories of economicdevelopment such as Lewis (1954) predict that a particular pattern of inequality willChapter 2. Decomposing Inequality hi dices by Income Source 46occur over the development process. For example in the Lewis model income inequalityis predicted to rise in the early stages of development as the income from capital increasesand then fall as the supply curve for labour becomes more inelastic and the wage rateincreases. Traditional tests of the Lewis model compare how inequality varies with meanincome. An alternative to the traditional test of the Lewis model would require anincome inequality index that can be decomposed by income source. If a time series ofdecompositions can be constructed then a rise in income inequality due to capital incomefollowed by a decline in inequality from capital income in later periods would be supportfor the Lewis model. This test of the Lewis model makes use of a prediction that thetraditional tests are unable to use.The final application that I suggest is an investigation into the effect that humancapital can have on income inequality. To the extent that earnings are increased byhuman capital, differences in education between individuals in a society will result indifferences in income. This means that one possible method of reducing inequality isto change the distribution of education levels amongst the population. In determiningthe effectiveness of such policies, decomposable inequality indices are required to showexactly how the pattern of inequality changes when the distribution of education levelschanges. A thorough examination of this question is left for the next chapter.2.7 ConclusionIn this paper I have presented several ways of decomposing both the AKS inequality indexand the KBD inequality index by income source. All of the decompositions share the samegrounding in ethical inequality theory which allows for easy and intuitive interpretationsof the individual terms of the decompositions. In contrast to most previous attemptsto decompose by inequality source, I provide a general method that is appropriate for aChapter 2. Decomposing Inequality Indices by Income Source 47wide class of inequality indices. I showed that several previously known decompositions ofspecific inequality indices can be considered special cases of the decompositions presentedhere. I concluded with a discussion of the possible uses of the decompositions in appliedeconomics.Chapter 3Human Capital Models of Income Distribution48Chapter 3. Human Capital Models of Income Distribution 49In this chapter I discuss the theoretical conditions which will ensure that the incomedecompositions of chapter four are correct. By correct I mean that not only does itmake sense to speak of returns to education being part of earnings but also that thequantitative number that I determine for the returns to education is correct. For theearnings decomposition to be strictly valid two things have to be true; the first is thatthe human capital model of earnings must be an adequate explanation of earnings, thesecond is that the construction of the counterfactual incomes must capture all the general equilibrium effects of the change in the distribution of education. In this chapterI first discuss the main alternative to the human capital explanation of earnings, themarket signalling hypothesis, and then illustrate, using a simple theoretical model, thearguements pertaining to whether a decomposition of the form I use is able to adequatelyreflect general equilibrium matters.3.1 How Education Affects EarningsThe idea behind the human capital explanation of earnings is that people undertakeeducation as an investment in future earning power. With more education an individualwill have higher productivity at his or her job and thus will be paid a higher marketwage. Any person who achieves a given level of education will have the same increase inproductivity, and thus income, as any other person with the same personal characteristics.If the human capital model is an accurate description of the world then it makes sensethat individual earnings can be separated into a part that is a return to education, anda part that is actually a return to other factors such as experience or gender.The main competing hypothesis to the human capital hypothesis is the market signalling hypothesis of Spence (1973). This explanation maintains that education does notChapter 3. Human Capital Models of Income Distribution 50necessarily contribute to increased productivity, although it may1, but that it does act asa signal to employers that the person will have high productivity at a given job. In themodel of Spence (1973), individuals with high ability also have high marginal productsand thus high incomes. The low ability individuals have lower productivity and a highercost of education than the high ability individuals. Firms can use a high educational attainment as an indication that the individual is of high ability. Since the cost of achievinga given education level is lower for the high ability individuals, it is possible to set thepay for high ability people so that it is worthwhile for them to purchase education, butthat it is not worthwhile for low ability types to purchase the education. The resultis that only people with high ability will want to purchase education. Individuals willrealize this and thus, when they are a high ability person, will purchase the education toindicate this fact to the employers.The signalling model, like the human capital model, produces a high correlationbetween educational attainment and earnings but the regression of earnings on educationdoes not yield the return to education. What this regression does yield is a measure ofthe return to ability. The reason that it cannot be considered a return to education is thepossibility that, in the absence of education being used as a signal of ability, it is likelythat another signal would be found. In this case no matter what the distribution of abilityis, the same distribution of earnings would be observed. Constructing a counterfactualdistribution of income as I do in this thesis will, in the case of the signalling hypothesis,result in two vectors of income. One is a return to other personal factors, as in the humancapital model, but the other vector will not be a return to education, it will instead bethe return to ability.1See Spence (1974), PP. 21, 22 for a model in which education does actually increase productivitybut also acts as a signal of inherent ability.Chapter 3. Human Capital Models of Income Distribution 51As noted above, it is possible that education could provide both a signal of inherent ability and an actual increase in productivity. In this case the correlation betweenearnings and education is partly a result of signalling behaviour and partly a legitimateincrease in productivity. This is not enough to save the construction of the counterfactualdistributions uuless the increase in productivity is much greater than the signalling effectin which case the signalling effect would be relatively unimportant. As the increase inproductivity from education increases it becomes more likely to be so large relative to thecost of eduction that no signalling equilibrium exists. It is thus best to restrict attentionto the pure human capital case.For the analysis in the body of the thesis to be correct it is thus necessary that thehuman capital hypothesis be maintained throughout the analysis. Empirical tests ofthe human capital model which contain separate measures of ability, such as IQ scores,have been implemented in the literature2.These studies generally show that education,even after controlling for the effect of ability, is a significant determinant of earnings. Inaddition Albrecht (1981) in a direct test of the signalling model is unable to find supportfor the signalling explanation of the correlation between education and productivity3These results at least partially support the human capital hypothesis. It is also true thatmany high paying jobs do require a certain level of education, regardless of the individualsability4. This type of evidence also tends to support the human capital model of earnings.I therefore adopt the human capital model as a reasonable theoretical model on whichto pin the rest of the analysis in the thesis.2See for example Taubman (1975), Taubman and X’Vales (1974)30f course not supporting the signalling model is not the same as supporting the human capitalmodel but Albrecht’s results do also show a positive correlation between education and productivity.4For example medical doctor or engineer.Chapter 3. Human Capital Models of Income Distribution 523.2 General Equilibrium IssuesThe second major condition that must hold for the decomposition analysis to be valid isthat the constrnction of the counterfactual income levels must contain all of the generalequilibrinm effects on earnings of a change in the distribution of education. In general,we can expect that changing the distribution of the educational attainment throughoutsociety will change the distribution of returns to education. As the supply of specific skillschange so will the returns to these skills that are generated in the market. Thus changingthe distribution of skills will change not only the level of returns to education, it will alsochange the relative returns to different types of education. I explain in this section thetheoretical conditions under which these effects on the distribution are unimportant inthe analysis. I then outline a reasonable model for which evidence on the direction ofbias exists. This section draws heavily from Lucas (1977).Suppose that earnings for person i vary according to the function= /3 + ra + ej (3.82)Where yj is person i’s income, r is the return to person i’s education, a is the amount ofperson i’s education, ej is a random error term, and is an individual specific intercept,intended to capture the effect of personal characteristics, other than education on theindividual’s earnings. A human capital earnings equation, run on cross sectional datawill result in the following predictive equation= +ia1u (3.83)Equation 3.83 has the following interpretation. b is the average level of returns to otherfactors than education, 1 is the average return to education within the sample and u isthe residual.Chapter 3. Human Capital Models of Income Distribution 53Equation 3.83 is a linear equation so the expression for the variance of income as afunction of the variance of the other terms is easy to determine. Suppose equation 3.83is used to calculate the variance of income5 then the resulting expression for the varianceof predicted income is= 2j(a) (3.84)where in general a() is the variance of the predicted income and u(a) is the variance ofthe education variable. Corresponding to this is the expression for the variance of actualincome, obtained from the linear equation 3.82,u(y) = a(a(a) +a2c(r) + (c(a)u(r)) + u(f) (3.85)where o-(y) is the variance of y, C) is the variance of j, a(r) is the variance of returnsto education r, and a(a) is the variance of educational attainment. Thus I now have anexpression for the variance of predicted income and another expression for the varianceof actual income, relating both to the variance of actual educational attainment.The next step is to see how these two expressions vary with the distribution of theeducation variable. Accordingly assume that the variance of a changes but the mean ofthe distribution of a does not change6. The change in the variance of predicted incomeis therefore calculated as the derivitive of equation 3.84 with respect to u(a), which isôu(y)= (3.86)Du(a)and the change in the variance of actual income isDu(y) =+ 2a(a)( ‘ ) + [a2 + a(a)J (8u) + u(r) (3.87)dcr(a) ôcr(a) da(a)5The variance of income is used in this subsection because it makes the exposition much easier. Thelessons learned using the variance apply equally well to other measures of inequality such as the AKSS-Gini index.6This differs from the construction of the counterfactual distribution in the body of the thesis becausethere I change both the mean education level and the variance of education. This would not affect thecalculation of the variance since the variance is not mean—dependant.Chapter 3. Human Capital Models of Income Distribution 54Constructing the counterfactual distributions the way I do and then analyzing how theinequality changes when the counterfactual distributions are used is analogous to usingequation 3.86 instead of equation 3.87 to determine how a change in the distribution ofeducation will affect the distribution of earnings. With equations 3.86, and 3.87 the biasinherent in using equation 3.86 to estimate the effect of a change in education on thechange in the distribution of income isB=___-___=- [2u(a)(3°)) + [a2 + u(a)](t)+ u(r)] (3.88)From equation 3.88 it is apparent that the degree of bias in the estimation of the effectof the change in the distribution of education is dependarit on the sign and magnitudeof the expressionsda(a) (3.89)andba(r) (3.90)da(a)A special case deserves some mention. It is when the return to education is the samefor all individuals. When this is the case, cr(r) is zero, and both expression 3.89 and 3.90are zero. In this case the bias is zero.Only a very restricted form of an economy production function will result in a biasof zero in equation 3.88. For the bias to be zero the schooling of the workers must enterinto the economy production function separately from the supply of labour. Lucas (1977)demonstrates that an economy production function of the following sort is the only typewhich will generate a bias of zero. Suppose the economy production function isQ=F(K,L,A) (3.91)where Q is the quantity of an aggregate output, K is the aggregate capital input, L is thequantity of raw labour, and A is the sum of schooling over all individuals in the economy.Chapter 3. Human Capital Models of Income Distribution 55In this case the amount of schooling enters separately into the production function. Aneconomy such as this will have everyone with the same return to education. In this casethe variance of the return to education is zero and everyone in the economy has the meaneducation level. In this case the bias is zero. In any other case the bias introduced byusing the estimated human capital earnings equation is most likely non—zero.Since the technology that generates a zero bias term is so restrictive, it is unlikelyto hold in practice and I must accept the liklihood that there is a non-zero bias in theconstruction of the counterfactual distributions.Given a non—zero bias it is worthwhile to try and analyse the terms of the biasto attempt to see what signs for the bias may he reasonable. As an illustration of areasonable case where the sign of the bias can be determined consider the case of anaggregate production function for an economy of two people7 such asQ = F(aili,a21) (3.92)where, without loss of generality, a1 > a2. a1 is the amount of education of person 1, ljis the labour supply of person 1, with similar notation for person 2. Assume that theaggregate production function is symmetric with respect to labour efficiency units andis strictly increasing and concave. Given these assumptions= =I1F(aili,a2)> 0 (3.93)andDQr2 = =12F(aili,a)> 0 (3.94)c’a2where r1 and r2 are the returns to education for individuals 1 and 2 respectively. Thesetwo expressions can be combined to yield the average return to education in the economy,—1F(aili,a21) +12F(aili,a21)3 95___________________2 (.7Alternatively two types of people, skilled and unskilled.Chapter 3. Human Capital Models of In come Distribution 56Now an increase in the variability of education a, given these assumptions is equivalentto an increase in a1 and a decrease in a2. To keep the mean education level the same,the amount added to a1 must be the same as the amount subtracted from a2. The totaldifferential of f for the changes to a1 and a2 is therefore= l/2([liFii(ai i,a2!) +12F21(aili,a21)Jdai+ [12F2(aili,a2!)+12F2(aili,a21)]da (3.96)which since da1 = —da2, is equal to= [112F(a,a2!)— 122F2(a,a2!)jde/2 (3.97)where de is the common absolute value of da1 and da2. Since the production function issymmetric in efficiency units, the same number of efficiency units will be hired from eachtype, that is a1! = a2!. This implies thatF11(a,a2!) =F22(ai!1,a2!) andsigncW = —sign(112— 122) > 0 (3.98)Thus a good case can be made that the average return to education is increasing in thevariance of the distribution of education. This occurs because concavity of the productionfunction implies that the decrease in the return to education for person 1 is of a smallerabsolute value than the increase in the return to education of person 2. It remains tobe shown how the second moment of the distribution of returns to education varies withthe variance of the distribution of education.Recall that the return to education for the first type of individual is given by thefollowing expression=I1F(aili,a2!) (3.99)and the derivitive of r1 with respect to a1 is= li2Fii(ai!i,a2!) <0 (3.100)Chapter 3. Human Capital Models of Income Distribution 57Since r1 is a monotonic function of a1, an increase in a1 will cause a correspondingdecrease in the value of r1. Similarly a decrease in the value of a2 will result in anincrease in the value of r2. Thus an increase in the value of the variance of a will resultin a corresponding increase in the variance of r, so that> 0 (3.101)The results so far indicate that, in the special case above, B, the bias inherent inignoring the general equilibrium effects is negative. This means that the inequality ofthe resulting predicted distribution of income is greater than the corresponding result forthe actual income. The way that this affects the decomposition of earnings in the body ofthe paper is that the variability of the base returns in my constructed decomposition ofincome will be greater than the the true variance of base returns. Since the distributionof actual earnings does not change, an increase in the variability of the base returnsresults in a decrease in the variability of the estimated returns to education, relative tothe true distribution of returns to education.The above has made a case for being able to sign the bias resulting from ignoring thegeneral equilibrium effects when constructing the counterfactual distributions. Even theexample that I presented, where production depends on efficiency units has some heroicassumptions. The first and most important being whether the efficiency units formulation of the aggregate production function is an acceptable one. To begin with thereis the question of whether or not an aggregate production function exists at all. If theaggregate production function does exist then the next problem is whethter the efficiencyunits formulation is appropriate. As some authors have pointed out8 the efficiency unitshypothesis implies that raw labour and education are perfect substitutes. We can achievethe same output with a lot of labour and little education or with a lot of education and8See Lucas (1977)Chapter 3. Human Capital Models of In come Distribution 58little labour. For many occupations, such as labourer, this is probably an inadequatedescription of reality.An important assumption in the theoretical model of this chapter that is certainlynot true in my empirical model is that the return to education is independant of otherpersonal characteristics. In the empirical model in the other chapters I explicitly assumethat the return to education interacts with experience and gender. This means thateven if the other assumptions are correct in the analysis signing the bias, there is stillsome uncertainty about it’s applicability to signing the bias in my analysis of returns toeducationEven given the probability that the analysis above, that signs the bias, is not directlyapplicable to my situation, the exercise in this thesis is still worth doing. There is verylittle econometric work done that is truly general equilibrium, everything isolates somesubset of the variables that do affect a particular dependent variable because a truegeneral equilibrium model is impossible to estimate and a partial equilibrium model isbetter than nothing at all. Thus even if the signing of the bias above is not accurate Ijustify the analysis presented in this thesis by saying that the question is an importantone to analyze from a social perspective and when a true general equilibrium model isunavailable, then a partial equilibrium model is the next best alternative.Chapter 4The Contribution of Education to Earnings Inequality59Chapter 4. The Contribution of Education to Earnings Inequality 604.1 Introduction.Empirical evidence and economic theory both indicate that education is an importantinfluence on an individuals expected earnings. An implication of this is that differencesin educational attainment have an important influence on differences in labour earnings.This means that social policies which affect the education system, such as funding cutbacks, can have important effects on the distribution of income in later time periods.Much of the current public discussion about equality of access to education is predicatedon the assumption that restricting access to education for individuals with low earningswill result in a perpetuation of earnings inequality from generation to generation. Modelssuch as Loury (1981), Becker (1964), and Spence (1973) provide a theoretical justificationfor these beliefs. Empirical work has examined in great detail the connection betweeneducation and average earnings but very little analysis has been done of the effect ofthe distribution of education on the distribution of earnings. Since society cares, notonly about the level of national income, but also about the distribution of income, animportant policy question has been largely ignored. In this chapter I present an analysisof the contribution of differences in education levels to earnings inequality in Canada.The second chapter of this thesis examined the decomposition of income inequalitymeasures by income source; thus the next step in the empirical analysis is to decomposeearnings itself into factor components. I draw on the empirical labour economics literatureto estimate earnings equations using a sample drawn from the 1986 Survey of FamilyExpenditures compiled by Statistics Canada. These estimated equations are then usedto determine how the earnings distribution would change if everyone had the same baselevel of education. The resulting distribution is used to decompose earnings into the baselevel and an amount attributable to education. First I consider what would happen ifan individual’s actual education level was replaced with a counterfactual education level.Chapter 4. The Contribution of Education to Earnings Inequality 61The estimated earnings equations are used to predict what the resulting earnings wouldbe if the individual had this counterfactual level of education. The difference betweenhis or her current earnings and the counterfactual earnings level is the contribution ofeducation to the person’s earnings.The estimated vectors of personal earnings are used in a decomposition of the SGini index of relative inequality to determine the contribution of education to measuredearnings inequality. The S-Gini index allows different degrees of inequality aversionto be used in the measurement of inequality and this flexibility is one of the majordifferences between this work and the previous literature. The earnings decompositionis first performed ignoring any effects that education may have on the probability ofbeing employed and then again considering the change in the probability of working aspart of the return to education. The inclusion of this second effect is another of thedistinguishing features of this analysis compared to previous work on the same subject.I further examine the differing effects by level of education to see if there is a differencebetween the effect of secondary education and post—secondary education. The last partof the chapter is a comparison of the results using the different methods, introduced inthe second chapter, of decomposing the inequality index.The question of the contribution of differences in education to earnings inequality hasbeen addressed empirically in previous work, notably Taubman (1975) and Layard andZabalza (1979). A major difference between my work and theirs is that I use the explicitlyethical decomposition of the S-Gini inequality index introduced in the previous paper.This ethical approach allows a numerically meaningful measure of the effect of educationon earnings inequality and yields both different results, and a different interpretation ofthese results than the analysis done by Taubman as well as Layard and Zabalza.The rest of the chapter is as follows. In section two I discuss three important issuesthat must be dealt with in any attempt to determine the contribution of education toChapter 4. The Contribution of Education to Earnings Inequality 62earnings inequality. In section three I discuss the previous empirical studies of Taubman(1975) and Layard and Zabalza (1979). The techniques used by the previous studiesare applied to my data in section four to provide a reference point for the later ethicalanalysis. In section five I estimate the earnings equations and construct the counter-factual earnings distributions used in the inequality analysis. Section six presents theestimated contribution of the two sources of earnings to measured earnings inequality fortwo methods of determining the contribution of education to earnings. In section sevenI examine how the contribution of education to earnings inequality varies with education level. Section eight presents a comparison of the results from the direct interactivedecomposition with the Shapley decomposition. Section nine concludes.4.2 Research StrategyIn this section of the paper I outline the three major problems which should be dealtwith in an examination of the contribution of returns to education to earnings inequality.The first problem is a question of what is the appropriate index to use in measuringinequality, the second is how to measure the effect of one source of income on the overallinequality, the last is how to properly decompose earnings into a part due to educationand a part due to other factors.The first consideration that must be made in an attempt to quantify the effect ofeducation on earnings inequality is how to measure earnings inequality. There are basically two general methods of doing this. The first are the statistical methods whichinvolve estimating the variance of earnings or some other measure of the dispersion ofthe earnings distribution. This type of analysis is useful in a purely descriptive exercisewhere the objective is to describe an income distribution, or compare two distributions,but not to determine their desirability. The other general approach to the measurementChapter 4. The Contribution of Education to Earnings Inequality 63of inequality is the ethical approach. In the ethical approach the first step is to decideon a social evaluation function which incorporates the ethical values of society in regardsto income inequality. Then one can construct a suitable inequality index from the datausing a transformation of the social evaluation function.It is my contention that the appropriate way to measure inequality in this paper isby the ethical approach. I make this claim for two closely related reasons. The firstbeing that, as a society, it is social welfare that we are ultimately concerned with andearnings inequality is part of the social welfare and our social ethics must be included inour measure of inequality. Thus if one wants to make any policy recommendations, onemust use an ethical index. The statistical indices are just descriptions of the earningsdistribution with no way to determine whether one value is better or worse than another.Any attempt to rank statistical indices must rely on ethical considerations and then theethical judgements should be made explicitly instead of hiding them behind a veneer ofobjectivity. The ethical index I use and its implications for the social preferences areoutlined in section six.How to determine the effect of education on measured inequality is simply a questionof how best to decompose an earnings inequality index by earnings source. To do thisproperly for an ethical index of inequality requires that the decomposition itself be ethicaland thus includes the explicit ethical judgements of the society as well as having theinterpretation of the loss of welfare in the current distribution as a result of incomeinequality. Without using the ethical approach to decomposition, there is no natural wayof selecting a decomposition rule from the infinity of possible decompositions. Insistingon an ethical approach does provide a natural way to select an appropriate decompositionrule. Appropriate methods for doing this were suggested in the previous chapter of thethesis and are briefly outlined in section six of this chapter.The question of how best to measure the returns to education in the context ofChapter 4. The Contribution of Education to Earnings Inequality 64determining earnings inequality arises because at any given time the population withwhich I am concerned consists of two groups, the employed and the unemployed. Theunemployed will have earnings of zero while the employed will have some positive valueof earnings. This has a very strong implication for the shape of the earnings distribution.Even with a continuous distribution of potential earnings, an individual will have astrictly positive probability of a realization of zero earnings, due to the possibility of beingunemployed. The density function of observed earnings then has a mass point of positiveprobability at zero earnings and some continuous distribution over the positive part ofthe real line. In examining how the distribution of education affects the distribution ofearnings the effect of education on the mass point at zero must explicitly be considered. If,for example, increased education results in an increased probability of being employed,then part of the return to education must be this increase in the probability that anindividual will be employed. If this effect is not included in the return to education,then certainly the share of earnings that are due to education will be underestimatedand to the extent also that the amount of underestimation of the return to education iscorrelated with education level, this will affect the distribution of the return to education.Both of these possibilities have substantial effects on the amount of earnings inequalitythat can be attributed to education.A typical distribution that might arise in this coutext is demonstrated in figure 1. Thisdistribution may arise from the following process. Desired labour supply is determinedby the equation= g(aj, x) + vi, (4.102)where a is the education level of person i, x is a vector of non-education personalcharacteristics, and v is a mean zero, homoskedastic error term. Positive hours areworked if and only if hI > 0 and then h = h7, actual hours worked equal desired hours.Chapter 4. The Contribution of Education to Earnings Inequality 65If h 0 then zero hours are worked, h = 0. If and only if hours are positive, anindividual receives earnings determined by the equation= =f(a, x) + fj, (4.103)where again e is a mean zero, homoskedastic error. If hours are zero then y = 0. Themeasurement of the contribution of a given set of personal characteristics to earningsinvolves two steps. The first step is to estimate the parameters of the earnings generatingprocess, the second is to use these estimated parameters to decompose earnings. Thecensoring in the observed earnings distribution creates a problem in both the estimationof the earnings process, because of the need to estimate both the mass point and thecontinuous part of the underlying density, and in the decomposition of observed earningsinto a part due education and another part due to other factors.The first step, the estimation of the earnings process is a step well understood bylabour economists. A problem arises because of the likely correlation between the errorterms v and e. This correlation, if it is non-zero results in a non-zero expected error termin the regression determining the parameters of f. The result of the non-zero expectederror term is a misspecification bias of the estimated parameters for the function f.Three methods can be considered for handling the presence of the mass point andthe subsequent estimation of the function f. The first possible method is to completelyignore the observations with zero earnings and to estimate f with only the non-zeroobservations. The omission of the zero earnings obviously determines nothing at allabout the mass point, in addition the shape of the estimated earnings distribution is notcorrect. If figure 1 shows the true shape of the observed earnings distribution, and if c andv are positively correlated (a reasonable situation), then the estimated distribution of ywill be like figure 2. In this the estimated distribution is skewed to the right, comparedto the actual distribution. This will have the result that the mean of the distribution isChapter 4. The Contribution of Education to Earnings Inequality 66larger, the left tail of the continuous portion of the distribution is smaller and the righttail is larger. This is in addition to the errors introdnced becanse the mass point is notnsed at all. Both the errors in the estimation of the continuous portion and the omissionof the mass point will lead to underestimation of earnings ineqnality with an index withany amount of ineqnality aversion at all.The next possibility is that the zero earnings observations are used but with norecognition that they are part of a mass point or that the process generating a zero isin any way different from the process generating a positive y. In this case the resultantestimated distribution will resemble figure 3. The mean of the distribntion will be close tothe mean of the censored distribution but the estimated shape is very different. The masspoint will not be present and the distribution is substantially flatter than the censoreddistribution. It is not clear exactly how this will affect measured inequality but it willaffect the results.The third, and in this situation correct, way of estimating the earnings process is tojointly estimate the probability of a zero earnings observation, conditional on a and x,and the parameters of the function f.I have so far established that the estimation of earnings must take into account boththe spike of individuals earning zero and the rest of the density of possible earnings.There are several ways of incorporating the possibility of some individuals working zerohours into a model of earnings’. I will concentrate on four; the simple tobit specification,the fixed cost of working specification, the minimum hours of work specification, and atwo stage tobit model. All four approaches concentrate on the relationship between thedecision to participate in the labour market and the decision on the number of hoursthat are worked.The simple tobit specification postulates that desired hours of work are determined1See for example Zabel (1993) or Heckman and Macurdy (1986)Chapter 4. The Contribution of Education to Earnings Inequality 67by an equation which depends on personal characteristics. If desired hours of work arepositive then the individual enters the workforce and works the desired hours. If thedesired hours are zero or negative the individual does not enter the workforce. Sincethe same reduced form equation determines both the hours of work decision and theparticipation decision, the effect of a given characteristic on the participation decisionmust be a proportion (constant across characteristics) of the effect on hours of work.There is evidence2that this model tends to overstate the effect of wages on labour hours.It remains a popular choice of model for applied work because it is relatively simple toimplement.The second model I consider is the fixed cost model of Cogan (1981). In this modelthe worker is assumed to undergo some fixed costs of entering the labour force. The effectof this is that the worker will not enter into the labour force unless a minimum numberof hours will be worked, any less and the benefit from working does not cover the fixedcost of entering the labour force. Empirically this model loosens the restriction on theeffect of an individuals characteristics on the labour supply and participation decisionthat is present in the simple tobit specification.The third model to be considered is the minimum hours model of Moffit (1982). Thismodel recognises that the number of hours worked is a result of two factors, demandfor labour, and supply of labour. The demand for labour is often institutionalized,especially in regards to hours of work. For example it is very difficult to find a job wherethe employer will allow an employee to work only half an hour a week. The minimumhours model has employers insisting that employees work a minimum number of hours ifat all. A potential worker whose desired hours are above the minimum is not constrainedin his or her choice but a worker who’s desired hours fall between zero and the minimumfaces the choice to either work the minimum or not work at all. This model provides2See Mroz (1987).Chapter 4. The Contribution of Education to Earnings Inequality 68another way of loosening the restiction imposed on the effect of personal characteristicson the labour supply decision and the participation. It does however greatly increase thecomplexity of the model.The last model to be considered is a two stage tobit specification. In the first stage adecision is made about whether to participate in the labour force or not. In the secondstage the desired hours are determined. The actual number of hours worked depends onthe results of both of the equations. Since different equations determine participationand labour supply this model also breaks the connection between the labour supply andparticipation decisions that is imposed in the simple tobit. Again the drawback is theincreased complexity of the estimation.The choice between the models then is one where increasing flexibility is traded offwith increasing difficulty. In my case this increased difficulty is doubly important sincethe major purpose of estimating the earnings is the construction of counterfactual earnings distributions when education levels are changed. In the case of the simple tobitspecification this is relatively straightforward, anyone with predicted negative earningsdoes not work, while anyone with positive predicted earnings does work3. With the nexttwo models the potential for fixed effects or minimum hours would greatly complicate theconstruction of counterfactual earnings. The two—stage tobit specification adds a furthercomplexity in the interaction between the hours of work equation and the labour forceparticipation.Thus simplicity is the overriding factor in the choice of earnings model. I get thissimplicity with some cost however. As discussed above the simple tobit model tends tooverstate the effect of wages on hours of labour supplied. This may have an impact onmy counterfactual distributions. Decreased education tends to produce lower wages. Theoverstatement of the elasticity of labour supply could mean that, for those not driven out3This is covered more completely in later sections.Chapter 4. The Contribution of Education to Earnings Inequality 69of the labour force, the decrease in earnings may be overstated. On the other hand thepartcipation decision should not be adversely affected4.The simple tobit specification ofearnings is dealt with more formally in section five.Although the above discussion has been about labour supply models, in what followsbelow I do not actually estimate labour supply, I instead estimate an earnings equation.When I estimate an earnings equation the dependant variable is the product of laboursupply and wages. Thus a change in the observed earnings could be a result of a changein hours, a change in the wage rate, or a combination of the two. The parametersof the estimated equation thus serve two purposes, the first is to capture the effect ofpersonal characteristics on labour supply, the second is to capture the effect of the samecharacteristic on wages. The fact that one parameter is asked to do two things meansthat the model is a compromise. Again this compromise is imposed for tractability, butmy opinion is that it is likely to be relatively unimportant to the analysis.It is appropriate to justify the use of a particular specification of earnings by appealingto labour supply models because most of the differences between the different laboursupply models arise in the treatment of the participation decision, or the transition fromwork to non—work. With an earnings equation this decision shows up in the observationof zero or positive earnings. Because earnings are the product of hours and wages zeroearnings can arise either from zero hours, in which case the above models of labour supplyare appropriate, or zero wage. Zero wage is very unlikely5so treating observations of zeroearnings as non—particiaption decisions and using labour supply models to analyze themseems justified.The second step in attributing a part of earnings to education is to use the estimated4Zabel (1993) does find that the simple tobit performs marginally worse at this task than the othermodels. The difference however is about half of a percentage point in the probability of correct predictionswhich is around 80% for all models.5especially with minimum wage laws.Chapter 4. The Contribution of Education to Earnings Inequality 70earnings process to separate the effect of education from the other variables. Becausethe probability of each individual being in the mass point at zero earnings is determinedin part by the desired hours function g which is a function of education, the mass pointmust be allowed to change when the counterfactual education levels are constructed.Thus the size of the mass point in the observed distribution is determined in part by thedistribution of education.So far in this section I have been referring to the non—workers as a homogenous group.In fact this is not the case. The non—workers actually consist of two distinct types ofindividuals. The first group is those individuals who are not in the labour force, ie notworking and not looking for work, while the second group are those people who are inthe labour force but cannot find work, ie the unemployed. The data that I use makeno distinction between the unemployed and those not in the labour force and so I amforced to treat these two groups as one. Since I consider both heads of households andspouses in this thesis it is likely that both the unemployed and not in labour force are wellrepresented in the sample. If the two groups react differently to labour market stimulithen the results of the study may be affected. Recent work by Gonul (1992) suggests thatfor men out of the labour force and unemployed are not distinct states but for women,individuals in the two groups behave differently. Thus the homogenous treatment of thetwo groups is likely to be a limitation.4.3 Previous Empirical Analyses.In this section I present a brief review of two previous studies of the effect of educationon earnings inequality. There have been a number of theoretical studies of how educationaffects earnings, and thus earnings inequality, beginning with Becker (1964). The actualeffect of education on earnings inequality is an empirical issue, consequently I confineChapter 4. The Contribution of Education to Earnings Inequality 71myself to the applied work that has been done on this question.A major study of the effect of education on earnings inequality was undertaken byTaubman (1975), building on work done by Taubman and Wales (1974). In this bookTaubman is concerned with determining the existence and shape of the wage earningsdistribution when various sources of earnings are removed from the overall distribution.The primary objective is to determine from which probability distribution the wageearnings is being drawn. Thus Taubman does not consider a normative welfare analysisof the earnings distribution.The first step in Taubman’s analysis is the estimation of a human capital earningsequation from a sample of individuals who entered the U.S. Army Air Cadet programduring 1943. The data on educational attainment, personal characteristics and job experience was collected by survey in 1955 and 1970. This provides a rich data source onindividual characteristics, including a measure of ability, as well as data on job experiences but, because of the fairly stringent conditions that had to be met to enter into theAir Cadet program, the data is not a random sample of the U.S. population.The earnings equation estimated by Taubman is of the formy=X+u (4.104)where y is labour earnings, X is a vector of personal characteristics, j3 is a vector ofparameters to be estimated and u is an error term. Included in the vector X is educationlevel, a measure of ability, some indications of family background and occupation. Thisequation is estimated twice, once for each year in which the survey was undertaken.With the estimated earnings equations, Taubman proceeds to separate the effectof various subsets of the explanatory variables on the earnings distribution. The mainexperiment with which I am concerned is the one where the effect of education is removed.Define Y2 = y — X2j3, where X2 is the set of education variables, to be the earnings withChapter 4. The Contribution of Education to Earnings Inequality 72the effect of education removed. Taubman then compares the inequality in Y2 with theinequality in total earnings y.Taubman uses the standard deviation as his measure of inequality. This has the badfeature that it is not invariant to scale; a change in the units of account will change theamount of measured inequality. The coefficient of variation, which is the ratio of thestandard deviation of earnings to the mean of earnings is scale invariant and is anotherof the measures of inequality used by Taubman. In 1955 the mean earnings in the samplewas $7300 and the standard deviation was $3800 for a coefficient of variation of 0.52. In1969 the mean earnings was $14500 with standard deviation $9400 for a coefficient ofvariation of 0.65. All earnings are calculated using 1958 prices.For 1955 the standard deviation of the non education—related earnings, Y2, was $3740compared to $3810 for total earnings. In 1969 the standard deviation of non—educationearnings was 9110 compared to 9420 for total earnings. Thus if all effects of educationwere removed, the standard deviation of earnings would have fallen in 1955 by $70 and in1969 it would have fallen by $310. This indicates that differences in educational attainment increase the variance of the earnings distribution. The interpretation of these resultsmust be influenced by the recollection that the sample is restricted and the extension ofthese results to the general population must be done with caution.A possible objection to these results is that the specification of the earnings equationis very restrictive and allows no interaction to occur between the different independentvariables. A more general earnings equation may produce differing results. Anotherproblem arises because only the variance of earnings is reported for Y2. The variance ofearnings is an absolute measure of inequality and as such is not changed by a change inthe earnings distribution of the form § = y + alN, for all a. It will change if the incomesare all multiplied by a constant, such as happens when Taubman converts all earnings in1955 and 1969 to real earnings in 1958 dollars. His results are sensitive to the year thatChapter 4. The Contribution of Education to Earnings Inequality 73he chooses6.The second major study of the effect of education on the distribution of earnings thatI consider is Layard and Zabaiza (1979). In this paper Layard and Zabaiza attempt to usefamily earnings in their analysis of earnings inequality. They use two related measures ofinequality, the coefficient of variation and the squared coefficient of variation. The dataused in this study is from the general social survey for 1975 and consists of 4027 Britishhouseholds.The overall inequality index for family earnings is 0.26 where the index is the squaredcoefficient of variation. To decompose family earnings by factor source, Layard andZabalza first estimate the equationf = a1 + a2si + a3s2 + a4p1 + asp2 +a6X + a7X + v (4.105)where .s1 is the age at which formal schooling ended for the male, p1 is the profession ofthe male and X is an experience variable for the male. Variables with a subscripted 2pertain to the female. Using the estimated equation, the contribution of the educationvariables to the squared coefficient of variation of family earnings is 0.052 compared to anoverall value of 0.26. Thus about 20% of total family earnings inequality can be explainedby differences in education.Layard and Zabalza continue with some analysis of possible policy changes using theirmodel. The first policy they consider is an increase in the quality of education represented by an increase in per pupil expenditure. Assuming rates of return on schoolingexpenditure of 3% and 6% , this policy results in a less unequal earnings distribution.Raising the age of compulsory education serves to increase earnings inequality slightly.Grants to those individuals who stay in school serve to increase inequality in earnings.61t is possible, given the price index used, to determine how Taubman’s results are affected by thisscalingbut a different price index will change the results of his analysis.Chapter 4. The Contribution of Education to Earnings Inequality 74Neither study accounts for any interaction between education and other personalcharacteristics, such as job experience. This is likely to have significant impacts on theresults, as it forces the experience profile of earnings to be identical across educationlevels.4.4 Initial ComparisonsIn this section I replicate the techniques used in Taubman (1975) and Layard aud Zabalza(1979). The results can be used to give context to the later results and to provide someinformation about how much of the difference between my results and these previousstudies are attributable solely to the different data.The data that I use are drawn from the 1986 Survey of Family Expenditures compiledby Statistics Canada. The personal characteristics of the head of the household and thespouse were selected and individuals not reporting an education level, and those over theage of sixty-five were cut from the sample. It may be more traditional to consider onlyfull time workers when doing the estimation. I do not do this since I are concerned, notonly with the workers in society, but also with the non—workers. Using a sample of onlyworkers would result in selection bias being a problem. Further details are provided inthe data appendix. Detailed results for each of the regressions run in this section areprovided in appendix two.In the study done by Taubman, the first step is to estimate an earnings equation ofthe formy = X3 + (4.106)where X is a vector of personal characteristics, y is the level of earnings, and e is a randomerror term. I estimate a version of this equation for my own sample where the vector ofpersonal characteristics contains education level, the experience measure, marital status,Chapter 4. The Contribution of Education to Earnings Thequality 75geographic location, sex and first language. I do not include any interactions in thisequation. This is as close to Taubman’s equation as I am able to get with my data. Theequation I estimate is thereforey = cvo +a1ed +a2ex +a3ex2+ ct4mt1+a5que1+ cr6sk1 + ct7al1+/38bc1 +a9sex + ci10otl +a11en +a12otrn + !13s1 (4.107)where ed1 is years of education7,ex is years of work experience8,quj is a dummy variablefor individuals living in Quebec, mt1 is for the maritimes, sk1 denotes Manitoba andSaskatchewan, ab1 is Alberta, bcj is BC, sex1 is 1 if person i is male, fr1 and oH1 aredummies for french and non—french and non—english as first languages, otrn1 and s denotesingle and non—married and non—single respectively.The next step in replicating Taubman’s technique is to construct a new vector Y2 bythe followingY2 = y — X2/3 (4.108)where the matrix X2 contains all of the explanatory variables related to education, including the interaction with age. The vector y2 is then interpreted as the amount ofearnings that is not a return to education. Then I compare the standard deviation ofY2 to the standard deviation of earnings y. The difference between these two standarddeviations is assumed to be the contribution of education to earnings inequality.The results show that the standard deviation of y is 16780 while the standard deviation of Y2 is 15849. Thus by this measure the returns to education increase inequalityby 6%. Compare this with the increase of 4% reported by Taubman.The technique used by Layard and Zabalza is similar to that of Taubman. Layard7See data appendix8See data appendixChapter 4. The Contribution of Education to Earnings Inequality 76and Zabaiza’s technique involves estimating the following modelf = Xcv + ii (4.109)where f = y/p and p = y/N is the arithmetic mean of the earnings distributionand X is a vector of personal characteristics. Initially it contains only education andexperience, with no interaction. That is= ao + cv1ed +a2gej + v (4.110)where ed1 is number of years of education for person i and agej is the age of person i.The next step is to calculate the portion of f generated by the education variables.This is denoted by X2cv. The final step in the Layard and Zabalza analysis is to comparethe variance of X2cv with the variance of f. The variance of X2cv is considered thecontribution of education to measured inequality. The results of my analysis show thatthe squared coefficient of variation of X2ct is 0.0341 while the index for f is 1.0022.Thus education is said to contribute 3.4% of the earnings inequality. Compare this tothe analysis of Layard and Zabalza which indicates that education contributes about20%.I repeat the analysis with a much more general specification of the earnings equation.The specification is given by= + 132ed +/33ed1age + fi4age1 + /35age + 36mt + /37que+ 58sk1 + 139ab + /31obcj +311.sex + /3j2fr +313ot1 + I3i4otrrz1+ j315s + Xi (4.111)In equation 4.111 the variables are ed is estimated number of years of schooling9,agej isthe age of person i, quj is a dummy variable for individuals living in Quebec, mt1 is forthe maritimes, sk1 denotes Manitoba and Saskatchewan, ab1 is Alberta, bc1 is BC, sex9See data appendixChapter 4. The Contribution of Education to Earnings Inequality 77is 1 if person i is male, fr1 and oH1 are dummies for french and non—french and non—english as first languages, otrn and s denote single and non—married and non—singlerespectively. The squared coefficient of variation of f is again 1.0022, with the education variables, including the interaction, contributing .1746. The results therefore havechanged substantially when a more general earnings specification is used with educationnow contributing 17%. The likely reason for the drastic increase is the inclusion in thismodel of an interaction between education and age. Since age varies by 45 years over thesample, this will obviously increase the variability of the returns to education calculatedin Layard and Zabalza’s way.The analysis done by Taubman and the analysis done by Layard and Zabalza are essentially the same, except in the measure that they use for the contribution to inequality.Taubman defines the contribution of education asC = Var(y) — Var(y2)= Var(y— y2) + 2Cov(y,y— y2) (4.112)while Layard and Zabalza define it asC = Var(X2) (4.113)Taubman attributes all of the covariance between the return to education and the returnto other factors, to education while Layard and Zabalza attribute none of this covarianceto education.The preceding discussion highlights some problems with these empirical strategies.The theory behind the decomposition of the inequality measure used in the analysishas not been adequately explained. Without this explanation it is not obvious whetherthe decomposition methods used by these authors is appropriate for the questions theyaddress. As well the covariance between earnings sources or more generally any measureof the interaction should, if considered, be treated as a source of inequality distinctChapter 4. The Contribution of Education to Earnings Inequality 78from the individual earnings sources and not simply arbitrarily assigned to one source ofearnings or another. Taubman obviously does not do this, Layard and Zabaiza have abetter approach that is very similar to the approach I use in the later sections.The results of the two papers cited, along with the replication of their analysis presented here indicate several things. Firstly, education does affect inequality but there isdisagreement about how much it affects it. From 4% from Taubman’s analysis to 20%for Layard and Zabalza’s measure. This disagreement is partially a result of the differentways of measuring the effect of different income sources on inequality indices. Taubman’smeasure of the contribution of education includes all of the covariance between returns toeducation and other returns while Layard and Zabalza do not attribute this covariance toeducation. The quantitative differences between my analysis and Layard and Zabalza’s islikely because they use family income instead of individual earnings and also consider theeducation level of both spouses in their measure of education. Both of these differenceswould increase the amount of inequality explained by education.The first paper in this thesis is a step in the direction of standardizing the techniquefor estimating the contribution of an income source to income inequality. Also, fromthe quantitative differences obtained when moving to a more general specification of theearnings equation with the Layard and Zabaiza technique, it seems that the specificationof the earnings equation has an important effect. Subsequent sections therefore addressthese issues.4.5 Returns to EducationIn this section I describe the method I use to decompose labour earnings into two factorcomponents, a part due to differences in education and a part due to other factors. Theprocedure I adopt is to first estimate an earnings equation and then to use the estimatedChapter 4. The Contribution of Education to Earnings Inequality 79earnings equation to remove the effect of differing education levels.The estimation of human capital earnings equations has a long history in laboureconomics, including work by Mincer (1974) and Welch (1970). These models typicallyspecify a potential earnings equation for person i of the general form’°= f(a1,x1) + Ej (4.114)where y is the potential earnings of person i, a is the education level of person i, x is avector of personal characteristics for person i and q is a random disturbance term. Theparameters of the function f are to be estimated.A problem arises in estimating equation 4.114. The true values of y are observedonly when y is positive, otherwise the observed value of y is zero. This results in thespike in the distribution that I discussed in section two. If I ignore this feature of the dataand just run the regression on the observed j5, the resulting estimates of the parametersa will be biased.Various methods of dealing with this problem exist in the literature. The method Ichoose to deal with the problem is the Tobit method proposed by Tobin (1958)11. Thistechnique makes the method of dealing with the spike in the decomposition of earningsstraightforward and intuitive’2The Tobit method involves constructing a likliehood function for observed y whichincludes both the probability of being in the spike and having zero earnings, and theprobability density function for the positive earnings. More specifically note that observedyj will be zero from equation 4.114 only if ej —f(a, xi). The probability of thishappening, given normality of the q is (—f(a1,x)/a) where a is the standard deviationof the error and I’ is the standard normal distribution function. If y is not zero then10See Willis (1986) for a discussion of estimating earnings equations.11See also Greene (1990)‘2See section six.Chapter 4. The Contribution of Education to Earnings Inequality 80the error follows a normal distribntion with density ç&((yj — f(a1,x1))/a). The densityfnnction for a given observation is then given byI(viO)aj,x, ) ‘2: ) (4.115)where I(y 0) is an index function that is 1 if y < 0 and 0 otherwise. The abovedensity is used to construct a likliehood function and the parameters of the function fare estimated by maximum likliehood.The Survey of Family Expenditures does not contain direct observations on the levelof education, instead the data has only categories of education level attained. Thus Iestimate a version of the earnings equation where the effect of education is captured bythe use of dummy variables. Conspicuous in their absence for the earnings regression arevariables such as industry, occupation, and union status. The reason that I do not includethese variables, even though they have often been shown to be significant determinantsof earnings, is that I want my measure of returns to education to include, as a potentialreturn to education, the ability to work in some jobs that may be higher paying.Consider occupation, as part of my construction of the connterfactnal incomes, Ipredict the earnings distribution that would result if everyone had the same educationlevel. The presence of occupation variables would confuse the issue of what is a returnto education. Suppose we did live in a world where everyone had the same level offormal education. Then formal education would not qualify anyone to work as a medicaldoctor. Nevertheless it is likely that some sort of informal training in medicine wouldarise. Thus the return to working in the medical profession would include a return tothis informal training. It would be impossible to separate the training effect from theoccupation effect in the construction of the counterfactual earnings levels. Even allowingfor different education levels it is still likely that the measured return to occupation hasa training effect.Chapter 4. The Contribution of Education to Earnings Inequality 81Due to the difficulty in separating the legitimate occupational effect on earnings fromthe education effect, I leave information on occupation out of the estimation completely.The estimated coefficients on education are unconditional on being in a particular occupation or industry. Thus my measure of returns to education includes such things as theability to work in some industries that may pay higher or which are more likely to beunionized, but that also may require a greater level of training’3. The bias introducedby not being able to separate the training effect from the occupation effect is a potentiallimitation of the analysis.The tobit regression of the earnings equation f is based on the following earningsequation= a, + cr2d2 + ct3d3 +a4d4 +a5d51 +a6d2age +ct7d3age1+ cv8d4age1+a9d5age-1- cr,omd2 + ajmd3 + cr,2rnd4 + a,3md5 +a,4md2age +a,5md3age+a,6rnd4age +ct,7md5age + a,gagej +a,9ge + cx20mt + cr2iquej+a22sk + cr23ab + cr24bc +a25sex + a26fr +a27otl +a25otrrt + a29s + Xi. (4.116)Where dl is a dummy variable that indicates that person i is in education category 1, ageis the persons age, quj is a dummy variable for individuals living in Quebec, rnt is forthe maritimes, sk denotes Manitoba and Saskatchewan, ab is Alberta, bc is BC, sex is1 if person i is male, fr and ot1 are dummies for french and non—french and non—englishas first languages, otrn and s denote single and non married and non single respectively.dlage is the product of dl and agei, while variables beginning with in are ones which arethe product of the gender variable and the indicated variable. There is thus 5 separatecategorical variables with separate dummies for each. To avoid the dummy variabletrap each categorical variable should have a reasonable number of people omitted. The‘3Lemieux (1993) provides some evidence for Canada that there are definite patterns to unionizationwith respect to skill level.Chapter 4. The Contribution of Education to Earnings Inequality 82proportion of people omitted goes from approximately 14% for the edncation statns toapproximately 82% for the marital status dummy. Overall there are over 200 individualswho are in the base category of female, married, english speakers, living in Ontario, withless than nine years of schooling.The tobit estimates of equation 4.116, for the data set already described, are given incolumn three of Table 4.2 with the asymptotic standard errors in column four. Note thatthese are the estimates of the parameters divided by the estimated standard deviationof x. Thus for example the reported coefficient for y is the reciprocal of the standarderror of the estimate. The OLS estimates of equation 4.116 are in columns one and two.Inspection of the tobit estimates shows that education increases the expected earningsand that this effect increases with age. Of the rest, most of the explanatory variablesdecrease expected earnings, with the exception of being male and single, from the basecase of a female, living in the Ontario, with English as a first language, and with amarriage status that is married. Both language variables reduce earnings as well asbeing neither married nor single. Several of the variables, notably d2age, d3age, d5age,at, fr otrn, and .s are insignificantly different from 0 at a size of .05. Examining theinteraction between the education variables and the gender dummy it can be seen thatmen have a returns to education that starts lower but increases at a faster rate with age.The results of this regression are consistent with previous studies, the return to educationis positive and earnings are increasing in experience. The results are essentially the samefor the OLS estimates except for the interaction of education categories three and fourwhich are insignificantly different from zero. The estimated values for both estimationtechniques appear to be in accord with previous Canadian studies of wage and earningsdetermination, for example Green (1991), Simpson (1985) and Kumar and Coates (1982).Since the focus of this study is on the contribution of education to income inequalityspecial attention should be taken with these parameters. The above specification ofChapter 4. The Contribution of Education to Earnings Inequality 83OLS Tobitparameter value standard errorconstant -35089 2208 -3.0980 .18315d2 9275.5 1592 .7440 .1363d3 12166 2054 .9510 .1710d4 18152 1987 1.3590 .1646d5 16333 2498 1.1395 .2025d2age -127.03 32.68 -.0051 .0029d3age -113.21 47.31 -.0025 .0040d4age -207.57 45.39 -.0079 .0038d5age -30.220 59.15 .0065 .0048md2 -108.17 1304 -.5687 .1068md3 -6639.7 2317 -1.0461 .1843md4 -10731 2294 -1.3432 .1809md5 -8053 2854 -1.1333 .2239md2age 101.60 27.48 .0130 .0022md3age 260.85 57.73 .0222 .0046md4age 403.77 56.61 .0311 .0045rnd5age 387.88 69.33 .0282 .0054age 2092.6 83.33 .1645 .0069age2 -24.050 .8510 -.0021 .00007mt -4939.9 355.4 -.3885 .0285que -3035.3 485.2 -.2820 .0390sk -2430.9 399.8 -.1722 .0318al -638.08 418.5 -.0541 .0330be -2910.8 419.4 -.2366 .0335sex 11258 613.2 1.2520 .0537oti -2194.8 373.7 -.1256 .0301fr -358.86 448.6 -.0106 .0361otm -4.7781 400.5 -.0118 .033135.889 422 .0425 .0332y = 1/cr -- .00006 .0000005R2 .3957-SEE 13070 15951LLF -140873 -110770.8observations 12929Table 4.2: Estimates of the Earnings Equation.Chapter 4. The Contribution of Education to Earnings Inequality 84the earnings equation allows the return to education to be different for the two mostimportant demographic groups. This difference is allowed to affect both the level of thereturns to education and the way that the return to education changes with the experienceof the individual. The results from the tobit regression indicate that the difference doesseem to be important. Men seem to have a lower initial return to education but afaster rate of increase over time. This is a surprising result, seeming to run counter toconventional wisdom. Part of the explanation may lie with the gender dummy. Beingmale tends to increase earnings, irrespective of education, by at least the amount thatmen have a decrease in the return to education.Given the importance of the return to education for the subsequent analysis a hypothesis test is performed to test the hypothesis that the returns to education for menand women are the same. That is, I am testing the null hypothesis that a10 through a17in equation 4.116 are equal to zero. In appendix two I provide the results for a regressionwhere the restriction is imposed.—2(LLFu — LLFR) is approximately distributed x2with 8 degrees of freedom. The value of this statistic is 146.4 and the critical value ofthe x2 distribution at size .05 is 15.507 so the null is rejected, in fact it is rejected atall reasonable sizes, and I conclude that the structural difference in returns to educationbetween men and women is likely to be important.Given the presence of some coefficients that are insignificantly different from zeroit may be questioned whether these variables should be used in the generation of thecounterfactual earnings. The question is really whether the variables with insignificantcoefficients should be dropped and the model reestimated. Simply dropping the insignificant coefficients and using the remaining estimates is clearly wrong since the model beingused to predict would not be the one that was used to estimate the parameters.Is it appropriate to simply drop variables when they are insignificant? The answer,both economically and econometrically is no. If the economic theory suggests that aChapter 4. The Contribution of Education to Earnings Inequality 85given variable be included in a regression, then to leave it out is to subject the modelto possible misspecification bias. Sometimes this is necessary, such as when the requireddata is not available, but as general rule it is something that should be avoided if possible.The fact that a coefficient is insignificantly different from zero does not mean that thevalue of the parameter in the true model is zero. The insignificance merely means thatthe variance of the estimated parameter is large relative to the estimated value. In theabsence of other reasons for the variable to be excluded it would be incorrect to excludeon the basis of low t-statistics.While the economics provides the most compelling arguement for not omitting variables with low f-statistics there is another, statistical, arguement. If all variables witht-ratios less than a given number in absolute valne are dropped, then the remaining estimates no longer have the standard t-distribntion. They have a censored distribution thatis related to the t. If this is the case then the meaning of significant t-ratios is unclear.Given these two arguements for using all of the variables, regardless of significancelevel I proceed with the construction of the decomposition of earnings using the regressionresults from the earnings equations. Denote person i’s kth counterfactual earnings levelto be 0, ic is the earnings distribution when everyone who’s actual level of educationis at or below level Ic is given a counterfactual level of education equal to their actuallevel. Everyone else has a counterfactual education level of Ic. gc is constructed by firstgenerating the kth counterfactual education dummy variable, cUt, in the following way.If dl = 1 for some 1 Ic then Jkt = 1 and Ji = 0 for 1 # Ic. If dl = 0 for all 1 lv thendlt = dl for all 1 = 1, ..., 5. The predicted counterfactual earnings 0 is generated fromthe estimated earnings equation, using the education variables Jl$ instead of the actualeducation variables, and including the estimated error term in the following way; denoteby the vector of base earnings with the lvth education level as the base education level.For the initial analysis I ignore the effect that education has on the probability of beingChapter 4. The Contribution of Education to Earnings Inequality 86employed. Thus the counterfactual distribution for the initial empirical analysis assumesthat an individual who is employed with the actual education level will also be employedwith the counterfactual edncation level. The vector of final counterfactual earnings isgenerated by equation 4.117=— &2d2 — &3d3— &4d4— &5d5 — &6d2age—&7d3age—&sd4age — &9d5age — &10ind2 — &11rnd3— &12md4 — &13md5—&14md2age—&15nid3age — &i6rnd4age —&17rnd5age + &2cf2 + &3cf3 + &4cf4 + &scf5+à6cf2age +&7cf3age+&sdcf4age +&9cf5age + &10md2 + &iimef3+&i2mcf4 +&i3mcf5 +&i4rncf2age +ãi5rricf3age + &iemcf4age +&i7mcf5age. (4.117)In equation 4.117 the ‘hat’ denotes the estimates of the parameters for equation 4.116.In subsection 3.6.1 1 use the OLS estimates from table 4.2 to construct the earnings, insubsequent sections, 3.6.2 and later, I use the tobit estimates to construct the counter-factual earning so the values from table 4.2 must be multipiled by the standard error ofthe estimate (SEE from the table) to get the appropriate value of the parameters foruse in constructing the counterfactual earnings.While equation 4.117 may look intimidating all it really does is remove the effectof actual education from earnings, to leave the earnings that cannot he attributed toeducation, and then add back the predicted effect of the counterfactual education levels.Note that in the case where the counterfactual education level is less than nine years ofschooling, this reduces to the earnings decomposition method of Taubman (1975). I nowdefine the vector of earnings due to differences in education to be y= y — gk• Theinequality analysis in the next section is undertaken using y and‘.In other words thebase level of education is category 1, less than nine years formal training.Chapter 4. The Contribution of Education to Earnings Inequality 87The preceding method of constructing the counterfactual distribution does not alloweducation to affect the probability of employment. I now outline a method of constructionof the counterfactual earnings which does allow for this effect. Consider the sampledistribution of earnings. This distribution has two important parts. The first is theregular distribution of earnings for those who are employed. The second is the spike ofindividuals who have earnings of zero because they are not working. In the sample of12,929 people, 3,167 are not working and 9,762 are employed. I want a decompositionof earnings that takes into account the change in the probability of being in the spike atzero earnings when the education level changes. The use of the tobit model allows me toinclude the fact that one of the more important elements of the return to education isthat increased education can increase the chances of an individual being employed andearning a positive income instead of being unemployed and having an income of zero.To take this effect into account the construction of the counterfactual income levelis changed slightly from that outlined previously. When the vector of counterfactualearnings is calculated from equation 4.117, it is regarded as potential earnings. If thesepotential counterfactual earnings are negative the individual is assumed unemployed atthe counterfactual education level and receives 0 of zero. If, however, the potentialcounterfactual earnings are positive then the individual is assumed to be employed atthe counterfactual education level and receives income.The return to education forindividual i is then determined by y =—One initially appealing method of determining the contribution of education to inequality is to use the R2 of the regression of y on the personal variables, includingeducation, with the R2 of the same regression excluding education variables. The reasonthat this is not appropriate is that, if any of the personal characteristics, such as maritalstatus, are correlated with the education variables, then some of the variation due toeducation will be explained by the personal characteristics in the restricted model andChapter 4. The Contribution of Education to Earnings Inequality 88the resultant R2 will not be true measure of the amount of variation due to education.Thus the appropriate method to get a measure of the base earnings is to estimate thefull model and then remove the effect of the education variables.It is important to recognize just what the respective counterfactual earnings contain.In generating these base level distributions, only partial equilibrium effects are beingcaptured. The same is true of any external effects of education. To the extent that theseare important, the subsequent analysis is limited’4.4.6 The Effect of Education on Inequality.I now use the counterfactual earnings distributions generated in the previous section toexamine how the contribution of education to earnings affects overall earnings inequality.As a first step at understanding the contribution of education to earnings inequalityexamine figure 4. This is a graph of the well known Lorenz curve for both the predictedearnings distribution and the counterfactual distribution of the base level of educationusing the estimated earnings equation from the previous section’5. The Lorenz curveshows the percentage of population, ranked from the poorest to richest, on the horizontalaxis, and the percentage of aggregate earnings earned by that segment of the populationon the vertical axis. The Lorenz curve for the actual earnings distribution is higher atall percentiles of the population than the Lorenz curve corresponding to the vector ofcounterfactual earnings. Thus the actual earnings distribution is unambiguously moreequally distributed, than the counterfactual distribution, however from the Lorenz curvethis difference appears to be relatively small.Although the Lorenz curve is useful as a visual aid to understanding inequality it doesnot provide all of the information desired. For example no scale effects are captured by a14This question is dealt with in much more detail in Chapter 3.15The counterfactual in this case includes the effect of education on employment.Chapter 4. The Contribution of Education to Earnings Inequality 89Lorenz curve. Inequality in an earnings source that is a very small part of total earningsis treated in the same way as an earnings source that is a very important part of totalearnings. Decomposing an inequality index provides more information about how theearnings sources y’ and y6 contribute to overall inequality because it provides informationabout both the scale of the source of earnings in relation to the total earnings as wellas the interaction between various sources of earnings. At this stage I will concentrateattention on the Atkinson—Kolm--Sen (AT(S) inequality index as this is the more widelyknown of the two popular ethical indices.Assume that society has preferences over alternative earnings distributions which canbe represented by the function to = w(y) where to is S-concave, increasing and everyassociated social indifference curve crosses the ray of equality defined by the equationy = aiN, where 1’ is a N-vector of ones. The equally distributed equivalent (EDE)earnings function = E(y) is defined implicitly by the equation w(y) = w(1N). TheEDE is the amount of earnings which, if distributed equally, would give the same levelof welfare as the current earnings distribution and is an exact representation of society’spreferences over earnings distributions. The AKS inequality index is defined as 1(y) =1 — E(y)/p(y) where i(y) is the arithmetic mean of the earnings distribution. The AKSindex gives a measure of the percentage of current total earnings that could be given up bythe society and still obtain the same level of welfare, providing that the remainder of theearnings is split evenly’6. I then decompose this inequality index by factor componentsas described in the previous paper. There are two components to earnings ‘, the baselevel of earnings, and ye the earnings due to education. Thus the decomposition takesthe form1(y) = s’I(’) + sel(y9 + s’[I(y) — I(y’)} + se[I(y) — I(ye)] (4.118)‘6See Blackorby and Donaldson (1978) or Chakravarty (1990).Chapter 4. The Contribution of Education to Earnings Inequality 90where s = ,w1/,u, the ratio of the mean of income source j to the mean of total income,is the share of aggregate income going to source j, I(v’), J(ye),and 1(y) are the AKSinequality indices for the connterfactnal level of income, the returns to education, andaggregate income respectively. The first step in the construction of ethical inequalityindices is the selection of an EDE function. The S-Gini EDE function was introducedby Donaldson and Weymark (1980) and axiomatized by Bossert (1990). The specificfunctional form of the S-Gini EDE function isE(y) = Z1[i6-(i - ‘)6W (4.119)In equation 4.119, is the ‘welfare ranked’ permutation of the earnings vector y such thatth th+i for all i = 1, ..., N., and 6 is an inequality aversion parameter. If 6 = 1 then theS-Gini SEF becomes a linear SEF and E(y) = p. This means that income contributes thesame to social welfare no matter who has it. The social indifference curves are planes inthe space of incomes. In the case of two individuals, as in figure 5, the social indifferencecurves are straight lines. IfS = 2 then equation 4.119 is the standard Gini SEF as definedby the AKS inequality index equal to the ratio of the area between the Lorenz curve andthe line of perfect equality out of the origin, and the total area beneath the ray of perfectequality. A representative social indifference curve for this case is also given in figure 5.As 6 tends to , the S-Gini SEF approaches the maximin SEF and the AKS inequalityindex becomes one minus the ratio of the lowest income to the mean income.The reason for the S—Gini is mostly pragmatic. The data has many observations withzero earnings, or a zero contibution of education. Given this I need an SEF which canhandle zeros. The S—Gini can handle zeros, most other common SEF’s such as the meanof order r cannot.Chapter 4. The Contribution of Education to Earnings Inequality 911 2 3 4 5 6Age I(’) [(ye) S1 S G(l,ye) J()All 1 0.0 0.0 0.6337 0.3663 0.0 0.02 0.6099 0.5388 0.6337 0.3663 -0.0550 0.52895 0.9500 0.9107 0.6337 0.3663 -0.0361 0.899530-40 T 0.0 0.0 0.6159 0.3841 0.0 0.02 0.5539 0.4507 0.6159 0.3841 -0.0477 0.46665 .9196 0.8307 0.6159 0.3841 -0.0386 0.846850-60 T 0.0 0.0 0.7247 0.2753 J 0.0 0.02 0.6448 0.7137 0.7247 0.2753] -0.0463 0.61755 0.9668 0.9822 0.7247 0.2753 -0.0160 0.9550Table 4.3: Decomposition of 1(y) for Earnings Decomposition 24.6.1 Initial EstimationTable 4.3 presents the decomposition of the S-Gini AKS index for the earnings decomposition described in the previous section by equation 4.117. It uses the estimated parametersfor the straightforward OLS regressions, based on equation 4.116 but without the tobitcorrection. This is done for the whole sample, the sample restricted to those between30 and 40 years of age, and those between 50 and 60 years of age. This separation intosubsamples is done for two reasons. The first being to see if the effect of education oninequality varies with age. The second is that there is substantial reason to believe thatthere are life-cycle inequality effects arising from life-cycle earnings patterns. Thereforeit may not be as bad if an older person earns more than a younger person than if twosimilarly aged individuals have different earnings. The indices are calculated using values of 1, 2 and 5 for the inequality aversion parameter 5. These values provide a fairlywide range of inequality aversion. To illustrate this point, a graphical depiction of aniso-welfare line for the three functions in the case when N = 2 is shown in figure 5.Column 4 of table 4.3 shows that, for the earnings equation, 37% of the total earningsin the entire sample is a return to education above the base level. This value remainsChapter 4. The Contribution of Education to Earnings Inequality 92constant for the subsample aged 30-40 and falls to 28% for the cohort aged 50-60.Examining the inequality in aggregate earnings in column 6, it is apparent that theinequality in the sample of 30-40 year olds is substantially less than the inequality inearnings for both the other subsample and the complete sample. This is probably aresult of the lower experience of these workers in relation to the older cohort. Sinceexperience interacts with the other variables, especially education, a lower experiencelevel will likely produce a more homogenous earnings distribution. Compare this with thelife-cycle pattern reported by Jenkins (1992) for the UK, where he finds that inequalityis slightly less for the older cohorts’7. The measured inequality for the 5-Gini whenthe inequality aversion parameter is 6 = 5 is, as expected, much higher for all of thesubsamples.Turning to the decompositions, as shown by columns 1 and 2, the returns to the baselevel of education are more unequally distributed than the returns to education in allcases. It seems that returns to education are less unequally distributed in the age group30-40 than they are in the other two samples. This may reflect the reduced interactioneffect in the wage equation between education and experience. At the same time thebase level of earnings is also more equally distributed for the 30-40 subsample than forthe whole sample or the 50-60 year olds which are essentially the same.The interaction term in the decomposition, column 5 is negative in all cases, indicatingthat the inequality in the base earnings and the inequality in the returns to educationpartially cancel each other out. For example the rank of person i in the distributionof returns to education has a low correlation with his or her rank in the base returns.Anything other than perfect positive rank correlation will result in some cancellation ofinequality by the two sources. The actual sign of the interaction is not a surprise sinceit can be shown that for the 5-Gini inequality index the interaction term will always be‘7The discrepancy is likely a result of Jenkins’ use of family disposable income instead of earnings.chapter 4. The Contribution of Education to Earnings Inequality 931 2 3Age 6 s’I(i’) sdi(ye) InteractionFull T 0.0 0.0 0.02 .3865 .1974 -.05505 .6020 .3336 -.036130-40 1 0.0 0.0 0.01 .3411 .1731 -.04775 0.5664 .3190 -.038650-60 T 0.0 0.0 0.02 .4673 .1965 -.04635 .8283 .2477 -.2292Table 4.4: Amount of Measured Inequality Attributable to Different Sourcesnon-positive. The extent of this interaction seems quite significant, when 6 = 2 havingthe effect of an increase of average earnings of 5% for the whole sample, slightly lowerfor the 30-40 year olds and the 50-60 year olds.Table 4.4 gives the total contribution of the base returns, the returns to education,and the interaction term to total measured inequality. Table 4.4 can be interpreted inthe following way. The AKS inequality index gives the percentage of aggregate incomethat could be removed while maintaining the same level of welfare, providing that theremaining income is distributed equally. For the full sample, when 6 = 2 inequality inearnings wastes 53% of total income. Column two in table 4.4 indicates that of that 53,20 is attributable to inequality in returns to education. Thus inequality in the return toeducation causes a loss of welfare equivalent to a decrease in aggregate income of 20%,provided that the remainder is distributed equally.Examining the cohort effects it is apparent that when 6 = 2 the lowest contributionof education is for the 30-40 year old cohort. This cohort also has the lowest contributionof the base level of education. Their value for education is 17 compared to 20 for the fullsample. The 30-40’s contribution of the base level is 34 when S = 5 compared to 39 forthe full sample. Things are slightly different when 6 = 5. In this case the 50-60 cohortChapter 4. The Contribution of Education to Earnings Inequality 94fl 1 2 3Age L shI(1) .sel(ye) InteractionFull 1 0 0 0-- 73 37 -105 67 37 -430-40 T 0 0-- 73 37 -105 67 38 -550-60 T 0 0 02 76 28 -45 73 28 -1Table 4.5: Percent of Measured Inequality Attributable to Different Sourceshas the lowest contribution of education but it also has the highest contribution of thebase level. The contribution of education is approximately the same for the 30-40 cohortand the full sample at 32 and 33 respectively. The two samples are also similar in theeffect of the base level at 60 for the full sample and 57 for the 30-40 cohort.An advantage of using the decomposition of the AKS inequality indices that I use isthat the terms of the decomposition inherit the numerical significance of the AKS index.This means that it makes sense to speak of education contributing a specific percentageto measured earnings inequality. This feature provides a nice way of examining whetherthe differences outlined above are just a result of the different measured inequality or areactually a result of systematic differences in the effect of the individual components. Italso provides a way of comparing the analysis here with that of Taubman, and Layardand Zabaiza. Table 4.5 therefore presents a percentile representation of the results forthe decomposition of earnings.In column 2 of table 4.5 it can be seen that for the overall sample, the direct effect ofeducation is to increase measured inequality by approximately 37% when = 2 and by37% when = 5. Another way of looking at this is that 37% of total measured inequalityis contributed by the returns to education. Compare this with the results of TaubmanChapter 4. The Contribution of Education to Earnings Inequality 95who found that 4% of earnings inequality were a result of returns to education or Laardand Zabalza who found 20% was a result of returns to education or to my replication oftheir analysis which found 6% and 3.4% respectively. At the same time the contributionof the base level of earnings is 73% and 67% respectively as shown by column 1. Thusthe interaction term for the full sample actually has a fairly large effect on the inequalityin the full sample reducing inequality by about 10% or 4%.The results for the sample aged 30-40 are similar, the direct effect of education contributes 37% of measured inequality when 6 = 2, rising slightly to 38% when 6 = 5. Theinteraction effect and the contribution of the base level of earnings remain approximatelythe same as in the full sample. The sample of people aged 50-60 shows a similar patternof effects as the other two samples. Returns to education contribute 28% of the measured inequality in this case. \‘Vhen 6 = 5 the results show that the return to educationcontributes approximately the same to overall inequality. The interaction effect is less inthis sample than in the other two, only reducing inequality by 4% when 6 = 2 and 1%when 6 = 5.I now briefly consider the results from the marginal interactive decomposition for thestraightforward OLS earnings model. The marginal interactive decomposition is givenby the following equation,I(y)=E s(I(y) — I(y)) + C’(y’, ..., y, y). (4.120)The first J terms in the interactive decomposition in equation 4.120 give the marginalcontribution of source j to overall inequality while the last term is an interaction termwhich provides a reference level of inequality. The results for the OLS earnings modelare given in table 4.6In column two of table 4.6 it shows that the marginal effect of the return to educationis to decrease measured inequality, equivalent to an increase in EDE of 3% for the fullChapter 4. The Contribution of Education to Earnings Inequality 96ii 2 3Age 6‘ j y InteractionFull T 0 0 02 -0.0063 -0.0297 0.56495 -0.0071 -0.0185 0.925130-40 T 0 0 02 0.0098 -0.0335 0.49035 0.0099 -0.0280 0.864950-60 T 0 0 02 -0.0697 -0.0195 0.70675 -0.197 -0.0032 0.9779Table 4.6: Marginal Decomposition for the OLS Modelsample when 6 = 2. This remains fairly constant for the 30-40 and falls to 2% for the50-60 cohort. Thus for the last subsample the equalizing effect on earnings of education isless important than it is for the sample as a whole and for the 30-40 cohort. The marginaleffect of the base returns is positive in the 30-40 case indicating that this earnings sourcetends to increase measured earnings inequality. This marginal effect is negative for thefull sample and for the 50-60 cohort.The major results of this subsection are that education contributes about 30% ofmeasured inequality. Education does however serve as an equalizing influence on earnings.This equalization effect is strongest for the younger 30-40 cohort, perhaps indicatingthe greater importance of education on earnings for this cohort relative to the othersubsamples.4.6.2 Complete Earnings ModelThe analysis in the previous section used a straightforward OLS estimation of the earningsequation to decompose income. It is well known that OLS will provide biased estimatesof the coefficients in the case where the data is censored. As well, no account was takeniii the previous subsection of the effect of education on the probability of employment. InChapter 4. The Contribution of Education to Earnings Inequality 971 2 3 4 5 6Age J(1) J(ye) e C(l,ye) J)All 1 0.0 0.0 0.5767 0.4233 0.0 0.02 0.6492 0.5228 0.5767 0.4233 -0.0688 0.52895 0.9662 0.9003 0.5767 0.4233 -0.0388 0.899530-40 T 0.0 0.0 0.5723 0.4277 0.0 0.02 0.5972 0.4394 0.5723 0.4277 -0.0631 0.46665 0.9444 0.8204 0.5723 0.4277 -0.0445 0.846850-60 T 0.0 0.0 0.6271 0.3729 0.0 0.02 0.6829 0.6660 0.6271 0.3729 -0.0591 0.61755 0.9773 0.9739 0.6271 0.3729 -0.0210 0.9550Table 4.7: Decomposition of 1(y) for Earnings Decompositionthis section I repeat the above analysis using a properly specified tobit model, outlined inthe previous section, to estimate the parameters of the earnings function and use thesein the decomposition of earnings which includes the employment effect, also outlinedpreviously.Examining column four of table 4.7 the share of returns to education in terms of totalincome is slightly less than it was with the OLS estimation. The cohort aged 50-60 hasthe lowest proportion of their earnings in the form of returns to education and the cohortaged 30-40 and the full sample having approximately the same. The overall measuredinequality in column six is identical to that previously.Turning to the within source inequality in columns one and two it is again true thatthe inequality in returns to education is lower for the 30-40 cohort than for the other twosubsamples. Looking at the case when 6 = 2 the inequality in returns to education forthe cohort aged 50-60 is half again that of the 30-40 cohort, when 6 = 5 the difference isnot so striking but the inequality in returns to education for the 50-60 cohort is still muchhigher than that of the 30-40 cohort. The larger inequality in returns to education forthe full sample is not as dramatic but is still much higher than that of the 30-40 cohort.Chapter 4. The Contribution of Education to Earnings Inequality 981 2 3Age 6 .s’I(’) .sel(ye) InteractionFull T 0 0 02 .3744 .2213 -.06685 .5573 .3810 -.038830-40 T 0 0 02 .3418 .1879 -.0631.5405 .3509 -.044550-60 T 0 0 02 .4282 .2483 -.05915 .6129 .3632 -.0210Table 4.8: Amount of Measured Inequality Attributable to Different SourcesThe inequality in the base levels of income is also lowest for the 30-40 cohort and highestfor the 50-60 cohort although the magnitude of the difference is nowhere near that forthe inequality in returns to education.Compared to the previous OLS analysis returns to education are slightly more equallydistributed but this difference is fairly minor. The interaction term in column five isagain negative but is considerably smaller compared to the sample with only the OLSdecomposition. It gets to approximately- .06 when 6 = 2 and is less when 6 = 5 (comparewith a low of approximately -.10 with the OLS analysis). The interaction effect seems tobe more important for the 30-40 cohort and the full sample than for the 50-60 cohort.This is similar to the previous OLS analysis.Column two in table 4.8 shows that, for the full sample with 6 = 2 inequality inreturns to education causes the same loss in welfare as would a 22% decrease in aggregateincome if the remainder were distributed equally. This amount increases to 38% whenthe inequality aversion rises to 5. This is larger than the effect calculated with the OLSestimates alone. Counteracting this difference is that the inequality in the base levelsis less in this case than with the straightforward OLS estimates. The inequality in thebase level of education causes the same loss of welfare as would a decrease in aggregateChapter 4. The Contribution of Education to Earnings Inequality 991 2] 3Age 6 .s1J(’) eI(ye).1 InteractionFull 1 0 0 02 71 42 -135 62 42 -430-40 1 0 0 01 73 40 -135 64 41 -550-60 1 0 0 02 69 40 -95 64 38 -2Table 4.9: Percent of Measured Inequality Attributable to Different Sourcesincome of 38% and 56% for 6 = 2 and 6 = 5 respectively. The interaction effect betweenthe two sources serves to increase welfare the same as a 7% increase in aggregate incomewhen 6 = 2 and 4% when 6 = 5.The cohort effects show a familiar pattern, even though the share of returns to education in income is higher for the 30-40 year old cohort, the contribution to inequalityis the lowest for this group. When 6 = 2, it reduces welfare the same as a 19% declinein aggregate income would if income were distributed equally. This compares with areduction of welfare of 22% and 25% for the full sample and the 50-60 year old cohortrespectively. The contribution to inequality of the base levels is also less for this cohortthan for the other two, 34% for the 30-40 compared to 38 and 43 for the full sample andthe 50-60 cohort respectively. The effect of the interaction term in terms of increasingwelfare is comparable between the two cohorts and the full sample. The effect naturally varies with the inequality aversion parameter but the same pattern emerges whenexamining the case when the inequality aversion is higher.Table 4.9 confirms the results from the previous analysis. It appears that returns toeducation are of less importance to the inequality in earnings of the 50-60 cohort thanthey are for the other cohorts, as shown in column two. This difference is fairly small,Chapter 4. The Contribution of Education to Earnings Inequality 100being only a difference of 2% between the cohorts when 6 = 2 but increasing to 4%when 6 = 5. The 30-40 cohort has the smallest contribution of education to measuredinequality. Again compare the values in column 2 of table 4.9 with the 4% and 20% foundby Taubman, and Layard and Zabalza. Inequality in the base level of income seems, fromcolumn one, to contribute between 60% and 70% of the inequality. The sample 30-40has the highest value at 73% and 64% for 6 = 2 and 6 = 5 respectively, the full samplehas values of 7% and 62% with the 50-60 cohort having values of 69% and 64%. Theinteraction effect reduces measured inequality by between 2% and 13%. The interactioneffect again seems strongest for the 30-40 cohort and weakest for the 50-60 cohort. Slightlymore variability is seen in the returns to the base level and the interaction effect than ispresent in the percentage contribution of the returns to education but the numbers arestill very similar. Compared to the straightforward OLS estimates the percentage effectof education on earnings inequality is greater with the tobit model while the effect of thebase level is smaller for this model than for the OLS model. This difference is uniformacross cohorts and inequality aversion.I again consider the marginal interactive decomposition for the tobit model of earnings. These results are given in table 4.10. From table 4.10, both the full sample and thecohort aged 30-40 reflect a similar pattern as in the OLS analysis. The major differencein these two samples is the magnitude of the effects. The marginal effects of educationare greater with the tobit model while the effects of the base returns are less. A majorqualitative difference appears when the full cohort is examined. In this sample bothearnings sources have negative marginal contributions to inequality when S = 5, meaningthat both of the earnings sources act as earnings equalizers when S = 2 however the effect of the base is to increase inequality as indicated by the positive sign on the marginaleffect. Again it seems that the equalizing effect of education is much more important forthe 30-40 cohort than for the other two cohorts. This is in contrast to the OLS analysisChapter 4. The Contribution of Education to Earnings Inequality 1011 3Age 6 ye [ InteractionFull T 0 0 02 0.0035 -0.0509 0.57635 -0.0005 -0.0282 0.928230-40 T 0 0 02 0.0156 -0.0559 0.50695 0.0151 -0.0417 0.873450-60 1 0 0 02 -0.0304 -0.0244 0.67235 -0.0119 -0.0085 0.9752Table 4.10: Marginal Decomposition for the Tobit Modelwhere the base level of earnings was a disequalizing influence on total earnings.The major conclusions from this subsection are that it appears that, when the masspoint at zero earnings is not explicitly modelled, the share of returns to education intotal earnings is under estimated and the actual inequality of returns to education willalso be underestimated. The combination of the two effects results in an estimate of thetotal effect on inequality that is biased downwards towards a lesser effect of education oninequality, when the returns to education are estimated without explicit consideration ofthe change in the probability an individual will he employed.4.7 Differing Education EffectsIn the previous section of this paper, no attempt was made to determine how the contribution of returns to education to earnings inequality varied across the level of education.That is the subject of this section. Throughout this section I use the earnings decomposition technique from the tobit model where employment effects are explicitly considered.The first step in this analysis is the decomposition of earnings into a part due topost-secondary education, a part due to secondary education, and the base level. ToChapter 4. The Contribution of Education to Earnings Inequality 1021 2 3 4 5 6 7 8Age 5 J(j’) J(s) I(yPS) C I(y)All 1 0.0 0.0 0.0 0.5767 0.3422 0.0811 0.0 0.02 0.6492 0.4849 0.8495 0.5767 0.3422 0.0811 -0.0803 0.52895 0.9662 0.8956 .9989 0.5767 0.3422 0.0811 -0.0452 0.899530-40 1 0.0 0.0 0.0 0.5723 0.3363 0.0914 0.0 0.02 0.5972 0.3903 0.7905 0.5723 0.3363 0.0914 -0.0787 0.46665 0.9444 0.8082 0.9965 0.5723 0.3363 0.0914 -0.0565 0.846850-60 T 0.0 0.0 0.0 0.6271 0.3067 0.0609 0.0 0.02 0.6829 0.6274 0.9201 0.6271 0.3067 0.0609 -0.0641 0.61755 0.9773 0.9716 0.9999 0.6271 0.3067 0.0609 -0.0220 0.9550Table 4.11: Decomposition of 1(y) for Earnings Decompositionperform this decomposition, I essentially use the technique used in section six exceptthat I consider two counterfactual distributions of education. The first is the same aspreviously considered, I give everyone the lowest level of education possible, denote theresulting earnings vector as . The second counterfactual education distribution giveseveryone with high school or less their actual education level. Everyone else is giventhe education level of some post-secondary, denote the corresponding earnings vector asThe return to post secondary education is therefore y = y — and the return tosecondary education is yS = — ‘. I then use these three vectors of contributions inmy decomposition of the earnings inequality index. The results are given in table 4.11.Column six shows that the share of earnings that is a return to post—secondary education is actually between 9% for those aged 30-40 and 6% and 8% for those aged 50-60 andthe full sample respectively. As can be seen from the table in column three, the returnsto post-secondary education tend to be distributed much less equally than the returns tosecondary education. Since many more people actually have secondary education thanhave post-secondary the number of people with zero returns to post—secondary is higherthan the number of people with zero returns to secondary education. Thus the spike atChapter 4. The Contribution of Education to Earnings Inequality 1031 2 3Age 6 s’I(i’) .sSJ(y8) s’8I(yP3) InteractionFull 1 0 0 0 02 .3744 .1659 .0689 -.08035 .5573 .3065 .0810 -.045230-40 T 0 0 0 02 .3418 .1313 .0723 -.07875 .5405 .2718 .0911 -.056550-60 T 0 0 0 02 .4282 .1924 .0609 -.06415 .6129 .2979 .0662 -.0220Table 4.12: Amount of Measured Inequality Attributable to Different Sourceszero in the distribution of returns to post-secondary education is much larger than thespike at zero for the returns to secondary. The returns are most unequally distributedfor the 50-60 cohort and are most equally distributed for the 30-40 cohort.Column three of table 4.12 shows that inequality in the returns to post secondaryeducation have the same effect on welfare as a loss of between 6% and 9% of aggregateincome. There does not seem to be any cohort differences in this case, with all ofthe cohorts having approximately the same results. The largest impact of high schooleducation is felt by the 50-60 cohort while the least effect is felt by the 30-40 cohort. Theeffect of post-secondary education compares to an effect equivalent to a loss of 13% to31% for the effect of secondary education in column two and 34% to 61% for the returnsto the base level in column one. The direct effects of secondary and post—secondary donot add up to the overall direct effect of education, given in the previous section, becausewith the additional income source the interaction effect has changed slightly.Table 4.13 presents the percentage contribution of the various sources of earnings tomeasured inequality. Examining this table the most interesting thing is that, comparingcolumn two to column three, the returns to secondary education contribute much moreChapter 4. The Contribution of Education to Earnings Inequality 1041 1 2 3 4Age s1J(’) ,SJ(y8) 8sJ(y8) InteractionFull 1 0 0 0 071 31 13 -155 62 34 9 -530-40 T 0 0 0 02 73 28 15 -165 64 32 11 -750-60 T 0 0 0 02 69 31 10 -105 64 31 7 -2Table 4.13: Percent of Measured Inequality Attributable to Different Sourcesto earnings inequality than do returns to post—secondary education in all cases. This isso even though the actual returns to education are distributed much more unequally forreturns to post—secondary education. This result seems primarily because of the muchhigher share of earnings that returns to secondary education have relative to returns topost-secondary education. Because of this, inequality in the distribution of returns tosecondary education should intuitively be more important to society than inequality inthe returns to post-secondary education. Again it seems post—secondary education ismost important for the 30-40 cohort, contributing 15% or 11% of measured inequality.It seems least important for the 50-60 cohort. No real pattern emerges when examiningthe percentage contribution of high school education.In most cases the direct contribution of secondary education is at least twice that ofthe contribution of post-secondary returns. This suggests that for social policy purposeswe, as a society, should be concerned with reducing inequality at the lower end of theeducation spectrum. Say for example implementing policies that encourage individualsto remain in high school instead of increasing the number of individuals who enter university. This would reduce the number of people who have not completed high schoolChapter 4. The Contribution of Education to Earnings Inequality 1051 2 3 4Age 6 s’I(’) .ssJ(ys) .PSi(yPS) InteractionFull T 0 0 0 01 0.0035 -0.0314 -0.0005 0.55735 -0.0005 -0.0178 -0.0006 0.918430-40 1 0 0 0 02 0.0155 -0.0325 -0.0008 0.48445 0.0151 -0.0243 -0.0012 0.857250-60 1 0 0 0 02 -0.0304 -0.0160 0.0001 0.66385 -0.0118 -0.0054 -0.0001 0.9723Table 4.14: Marginal Effects of Income Sources on Inequalityand, for those people, increase their income from secondary schooling. This would reducethe measured inequality in the low education categories. This policy would probably alsoincrease the demand for post-secondary education, requiring more funding, but the primary target of social policy, if it is desirable at all, should be secondary education. Thenext chapter provides more concrete evidence for this statement.I conclude this section with an examination of the marginal effects on inequality ofthe different levels of education. The marginal decompositions in this case are given intable 4.14. Table 4.14 confirms what has already been seen. The marginal effect of bothtypes of education on measured inequality is to reduce inequality. The marginal effect ofsecondary education is much greater in magnitude in all cases than the marginal effectof post—secondary education. This provides further support for the statement that socialpolicy should target lower education levels.Results by cohort show that the marginal effect of all types of education is mostimportant for the 30-40 cohort. This again indicates the importance of education forlower experience workers.The major conclusions from this section are that secondary education is a much moreimportant determinant of earnings inequality than is post—secondary. The largest effectChapter 4. The Contribution of Education to Earnings Inequality 106of post-secondary education is felt by the younger 30-40 cohort. This likely reflects anincreased importance of education, relative to other characteristics such as experience,for the the younger cohort. Marginal analysis confirms that education is more importantfor the younger cohorts.4.8 Shapley ContributionsIn this section I briefly consider the other decomposition of the AKS index that I introduced in chapter 2, the Shapley decomposition. The Shapley decomposition is givenby=s [i — ¶.] (4.121)The terms in the Shapley decomposition provide a measure of the total contribution ofsource j to inequality.The values of this decomposition are presented, for the income decomposition thatdoes account for employment effects, in table 4.15. For ease of comparison I also repeat intable 4.15 the values of the decomposition for the direct interactive decomposition fromtable 4.7. Column one gives the degree of inequality aversion, columns two and threeshow the respective contributions of the base level of education and returns to educationrespectively, column four gives the contribution of the interaction term, and column fivepresents the total measured inequality.The last six rows of table 4.15 give the Shapley decomposition for the inequalityindices. Since the Shapley decomposition decomposes the inequality index exactly, thereis no interaction term for this part of the table’8. With this decomposition it appearsthat for all of the groups, and for both inequality aversion parameters, the returns to15Note that this is not the same as the technique of Taubman, and Layard and Zabaiza who eachattributed all of the interaction to a given source of earnings. This technique can be thought of asallocating part of the interaction to each earnings source.CD C) 0 CiD 0 0 CD 0 0 Ci) 0 CD Ci) CD Ci)(IDH) •eCD. CDoCJD‘CIDI-CSt’3-Ct•eI-D00-T—10CJCtD00e-00c-00CJ0000t’D.—1Ci00—I.—j—.i—-- CDIIIIIIeee3C_y—.—C000IIIIIIIICJ:0000J0boèj--000C5—TC00000cc00DccIChapter 4. The Contribution of Education to Earnings Inequality 108education contribute less to inequality than the returns to the base level. The returns toeducation contribute the least to the cohort aged 30-40 while the 50-60 cohort and thefull sample have similar results.It is quite obvious from the analysis in this section that the different decomposition methods in chapter 2 provide very different information. Of the three, the Shapleydecomposition and the direct interactive both provide information about how much ofobserved earnings inequality is a result of returns to education and how much is a resultof the base level of earnings. The difference is that, in the Shapley decomposition, I havesucceeded in unambiguously separating and assigning any interaction effect to a particular income source. The marginal interactive decomposition shows how the observedinequality would change if the given income source is removed. The difference betweenthe marginal and direct interactive decompositions is the base level of inequality which isused. In the direct decomposition the base inequality measurement, to which everythingis compared, is zero while in the marginal decomposition the base level of inequality is aweighted sum of the inequality present in each source19.In order to facilitate the comparison between the Shapley decomposition, the directinteractive, and the methods of Taubman and Layard and Zabalza, I have in table 4.16 thepercentage contribution of the two income sources to overall inequality. This table showsthat about forty percent of the measured inequality in earnings is attributed to education.Compare these numbers to the approximately forty percent from the direct interactivedecomposition for all the cohorts and the six and four percent from the Taubman andthe Layard and Zabalza techniques with this data respectively. The percentage analysisconfirms that returns to education are least important, in the Shapley decomposition,for the 30-40 cohort. This is a seeming contradiction to the interactive decompositions.19Assuming only two sources of earnings. In general the base level is a weighted sum of the inequalitypresent in the distributions y.Chapter 4. The Contribution of Education to Earnings Inequality 10912Age y1 yeFull 1 0 02 65 355 59 4130-40 1 0 02 65 355 59 4150-60 1 0 02 71 395 63 37Table 4.16: Percent of Inequality by Shapley DecompositionOne way of thinking of this apparent contradiction is that the Shapley decompositionallocates less of the interaction term to the returns to education than it does to the returnsto the base level of education. This is reasonable since, thinking back to the marginalinteractive decomposition, returns to education had a relatively strong equalizing effecton earnings for the 30-40 cohort. The results for the full sample are similar to the resultsfor the 30-40 cohort. The percentage contribution of education is highest for the cohortaged 50-60The similarity in results between the two methods I introduced is comforting especiallygiven that the calculated interaction effect in the direct interactive decomposition isrelatively small and thus the percentage contributions of the direct effect should be closeto that in the non-interactive Shapley decomposition.4.9 Conclusions.In this paper I have investigated the decomposition of measured earnings inequalityinto an effect due to education and an effect due to other factors. The paper is anempirical application of the inequality theory developed in chapter 2. 1 determine theChapter 4. The Contribution of Education to Earnings Inequality 110contribution of returns to education to earnings inequality in Canada using the S-Giniinequality index. The major empirical findings are that returns to education directlycontribute approximately one third to four tenths of measured earnings inequality. Withthe remainder a result of inequality in returns to personal characteristics other thaneducation and an interaction term that serves to reduce measured earnings inequality. Ialso demonstrate that a significant portion of the returns to education arises from theincrease in the probability of finding employment that is a result of higher education.The next section investigated the differential effect that returns to different levelsof education have on measured earnings inequality. I demonstrated that the returns tosecondary education have a much larger effect on earnings inequality than do returns topost-secondary education. A possible interpretation of this result is that social policyshould be aimed at lower education levels such as high school education rather thanhigher education. The comparison with the other decomposition methods shows that theShapley decomposition, since it measures the total contribution of a source to inequalityand not just the marginal contribution of that earnings source, gives results that aresimilar in spirit to the direct decomposition, which also measures the total contributionand not the marginal contribution.The paper provides an empirical illustration of the human capital based theories of theincome distribution, such as Loury (1981), by quantifying the amount of inequality thatmay be attributed to education. As well, the differential in the effect of post-secondaryeducation verses secondary education has a specific policy implication. Viewed solely interms of earnings inequality, the emphasis in public policy should be on policies, such aspromotion of high school completion, that target lower education groups. It is importantto note that this analysis is concerned only with the inequality of earnings and is silenton how best to affect the level of earnings in the society.Chapter 5Policy Simulations111Chapter 5. Policy Simulations 1125.1 IntroductionIn this chapter I use the earnings generating model from the chapter three, and the inequality decomposition from chapter two to examine the effect on measured inequality,the marginal decomposition of measured inequality, and the EDE of some social policy changes affecting the quantity of education that individuals obtain. These can becontrasted to quality changes, where the quality of education obtained by the individuals who attend school is changed but without changing the actual number of people atdifferent levels.The analysis of chapter four in this thesis showed that there is a large difference inthe contributions of education to earnings inequality depending on the level of education;returns to elementary and secondary education have a much greater impact on observedearnings inequality than returns to post-secondary education. It was suggested that thismay indicate that a social policy targeting lower education levels would be more effectiveat combating earnings inequality than a general policy of encouraging education at alllevels. This possibility is one of the policy options analyzed in this chapter.I consider three specific types of policies. The first type, a low targeted policy, isdesigned to increase the educational attainment of the individuals in the lowest twoeducation categories. A high targeted policy is one which encourages further educationfor individuals who have completed high school but do not have a university degree. Abroadly based policy is one which encourages individuals of all eduction levels to increasetheir educational attainment.The actual specifics of the policies are left open. I use as my starting point the effect ofthe policy on education levels. This causes a bit of a problem since without being specificabout the administration of the policies I am unable to consider how administrative costsmay affect the desirability of a particular option. To the extent that the administrativeChapter 5. Policy Simulations 113costs are small compared to the cost of education, this may not be an important omission.A more important omission is that I do not consider the effect of the opportunitycost of education. When an individual decides to purchase education for another year,one of the factors that would affect this decision is the possibility of quitting schooland entering the labour force full time with the current education level. Since I do notconsider opportunity cost in this analysis it must be recognized as a limitation of theapplicability of the results.The rest of the chapter is as follows. Section two briefly considers some frictions inthe market for education which may justify the need for social policies. In section threeI estimate the costs involved in increasing an individual education level by one category.Section four considers what I term constant population policies. These policies are onesfor which the total number of people affected by the policy remain constant when movingfrom policy to policy. Section five considers constant cost policies, where the total costof the policy is held constant. The last type of policy to be considered is one where theprobability of any one person being affected, given that they are in a targeted group, areheld constant. These are considered in section six.5.2 Frictions in the Education MarketIn this section I present some arguments that can be used to justify the need for socialpolicies designed to encourage individuals to increase their education purchases.In standard human capital models’ an individual regards education purchases as aninvestment in lifetime earning power. As an investor the individual purchases educationup to the point where the marginal cost of purchasing another unit of education is equalto the marginal increase in lifetime earning power. If the social discount rate is the same‘For example Heckman (1976) or Becker (1964).Chapter 5. Policy Simulations 114as the private discount rate, if the individual has access to perfect capital and factormarkets, and if the individual has all of the available information, then the optimalamount of education for the individual to acquire from societies point of view is thesame as the privately optimal purchase. In this case any social policy which changes theamount of education purchased will result in a decrease in welfare. These three conditionsare seldom met in reality.Consider first the assumption of perfect capital markets for students. Perhaps thesingle greatest reason for capital market imperfections facing students is the lack of collateral. Unlike most investments the purchase of education involves nothing physical. Inaddition, individuals who are purchasing education frequently have little or no physicalassets. This makes the provision of collateral on a student loan very difficult. Withoutcollateral to help prevent default, the incentives to lend money for the purchase of education are greatly reduced. In response to the imperfect capital markets a variety ofschemes for the provision of student loans have been proposed. It is schemes such asgovernment sponsored student loans that I have in mind to reduce this problem.The second major imperfection in the market for education is an information problemon behalf of the purchaser. Individuals who are making education purchase decisions areusually young. Frequently they do not have information about the consequences of theirpurchase decision or about how their circumstances may change in the future. This lack ofinformation may lead to incorrect or myopic decisions regarding the amount of educationto purchase. For example a job at minimum wage may look attractive to someone who issixteen and living at home but not nearly so when trying to support a family. To combatthis type of problem a policy could be undertaken to inform those making the decisionsjust what the likely consequences of their decisions will be in later life.The last way in which an intervention in the market for education may be beneficialis in the presence of externalities to education purchases. If the education of individualsChapter 5. Policy Simulations 115interacts with the edncation of his or her co—workers, for example if it increases notonly his or her productivity but also the productivity of co—workers, then it may be insocieties interest to promote education beyond what is optimal when considering onlyprivate benefits to education.Any of the reasons outlined above, plus many others, would provide adequate reasonfor a social policy intervention in the market for education. In reality all of them likely arepresent to varying degrees so this type of intervention is worth examining more closely.5.3 Costs of EducationIn order to properly discuss the effect of a given set of policies on social welfare, I will beconsidering the social net present value of a given policy option. This involves estimatingthe benefits as well as the costs of the policy discounted to the same base year. A policywhich has discounted benefits greater than discounted costs is said to have a positive netpresent value and is socially desirable. The costs that I consider are only the direct costsof education. In other words I do not consider any dead weight loss due to increasedtaxation to cover the increased education expenditures. Living costs of the student haveto be met whether or not a policy affects the education purchases and so are not includedas a cost of education. A potentially important cost that I ignore is the opportunity costof education. If an individual is pursuing education then he or she cannot be earning asmuch of an income as he or she could be if time spent in acquiring the education wasinstead spent full time in the labour force. The fact that opportunity costs of educationare ignored is a limitation to the analysis.I also adopt the simplifying assumption that the increased cost is financed by a proportional income tax. This has the effect of multiplying everyone’s pre-tax earnings by aconstant to arrive at post-tax earnings. Since the S-Gini AKS index is a relative index,Chapter 5. Policy Simulations 116Level ExpenditureElementary and Secondary 21,946,596Post-Secondary Non-University 2,784,226University 7,034,994Vocational Training 2,814,113Table 5.17: Government Expenditures on Educationthis does not change the amount of measured earnings inequality from pre to post-taxearnings. In reality the Canadian tax and transfer system is commonly thought to besomewhat progressive, which means that after-tax earnings inequality will be somewhatoverstated by the analysis presented here, compared to what would be the actual situation.The first step in constructing the costs of education is to determine the expenditure on education. Table 5.17 shows the government expenditures on education for fourcategories2.Government expenditures on elementary and secondary education are combined. I therefore use the technique of Constantos and West (1991) to separate the costsinto an amount for elementary students and an amount for secondary students. An initial expenditure per student is obtained of 4,437 dollars a year. Secondary education isassumed to be 1.3 times as expensive as elementary education. Using this and the enrolment weights3 of .3653 for elementary enrolment and .6347 for secondary enrolment,I obtain an expenditure per student for elementary education of 4161 dollars and forsecondary education of 5198 dollars.I divide the expenditure on community colleges by the total non-university post-secondary enrolment to get a figure for the non-university, post-secondary expenditureper student of 10,631 dollars. To calculate an expenditure per student for universities,I again follow Constantos and West and assume that each part time university student2Source:Financial Statistics on Education (1986) Statistics Canada.3These are calculated using the data in table 7 of Statistics Canada Education in Canada (1985-1 986)Chapter 5. Policy Simulations 117Level Public Percentage of EstimatedExpenditure Cost CostElementary 3999 96.1 4161Secondary 5198 96.1 5409Post-Secondary 8632 81.2 10631University 11536 81.2 14207Table 5.18: Costs of Education (1985-1986)Category Change Extra Time Total Cost, 3% Total Cost, 6%1-2 3 17220 182532-3 1 10950 112693-4 1 10950 112694-5 3 45230 47943Table 5.19: Cost of a Change in Education Levelis equivalent to one half of a full time student. The subsequent calculation gives anexpenditure per full time student of 14,207 dollars.Table 5.18 gives the subsequent calculation of the total per student cost. The percentage of total cost born by the government is from Constantos and West. The onlyconcern may be the use of the University figure for the other post-secondary. This isused to produce a conservative estimate of the cost of the education so that the use ofthe cost in the welfare analysis will in turn be conservative.The costs of education are going to be compared to the welfare arising from lifetimeincome. Thus I compound the per year cost for the additional time spent in school. Theadditional time spent is determined by the following pattern. Category one takes 9 years,category two takes 12 years, category three takes 13 years, category four takes 14 yearsand category five requires a total of 17 years of education. I do the compounding at anassumed social discount rate of 3% and 6%. The resulting total cost of an additionalstep in education are shown iii table 5.19These estimated costs are in line with other estimates such as those in ConstantosChapter 5. Policy Simulations 118and West (1991) and Vaillancourt and Henriques (1986). The assumed real interest ratesof 3% and 6% are similar to the social rates of discount assumed in Layard and Zabaiza(1976). Given the estimates of the cost of education I can continue with the policysimulations. Again it is worth reemphasizing that I have considered only the direct costsof education and have not considered the opportunity costs of education.5.4 Constant Population PoliciesIn this section I consider policies that hold the number of individuals affected by thepolicies constant. I analyze three types of policies; a policy which increases the categoryof education by one with a probability of ten percent for the individuals who initiallyare in categories one to four, a policy which increases the category of education with aprobability of 14 percent for those initially in categories one or two, and a policy whichincreases by one with a 33 percent probability for those in categories three and four.The probabilities are chosen so that the nnmber of individuals affected by each policyremains approximately the same. Since the numbers of people in the various categoriesis different, this obviously indicates that different probabilities are needed.Education is an investment which increases earning power over a persons entire working life so it is inappropriate to consider only one years earnings in analyzing policiespertaining to education. I therefore simulate, using the earnings model in the previouschapter, discounted lifetime earnings and do the welfare and inequality analysis usingthese discounted lifetime earnings.Simulated lifetime income is calculated for a group in the population. I use thepersonal characteristics, except for age, of the 30-40 sample in the simulations. I use thepersonal characteristics of individuals in this cohort because any policy that a governmentinitiates is likely to impact the younger members of society most heavily. This is especiallyChapter 5. Policy Simulations 119important for education policy since any impact of education policies will depend on thecurrent distribution of education within the targeted groups. The 30-40 cohort is usedsince they are still relatively young, and thus approximate in personal characteristics thecurrent group of individuals attending school, but are old enough so that major lifetimechoices such as education have already been completed. For the simulations I have thefirst time period for everyone be when they are twenty-five.In calculating the simulated incomes the first step is to use the personal characteristics of the age 30-40 cohort to calculate expected potential earnings from equation 4.116for each individual in the cohort. I do this for each of thirty-five years starting at agetwenty-five. I thus have, for each individual a vector of 35 yearly expected earnings.Then a random draw is taken from the normal distribution with mean 0 and standarddeviation equal to the estimated standard deviation of the error in equation 4.116, foreach individual, and added to the yearly potential earnings. This represents the individuals person specific characteristics, luck and other unmodelled determinants of earnings.Using the same error term for each year assumes that the whole estimated error term inequation 4.116 is fixed across time. It is not possible to use the actual estimated errorterm from equation 4.116 in the simulations since it is not observed for individuals withzero earnings.At this point I have, for each individual, thirty-five years of potential earnings, representing an entire working lifetime. Then every negative value of potential earnings isset to zero to reflect the fact that the person is considered unemployed in that year.Thus every individual has simulated yearly earnings for each of thirty-five years whichare either zero or positive. Finally the income in each year is decomposed in the sameway as in chapter three, into a base level and a part due to education. The resultingvectors of earnings are discounted back to age twenty-five at a rate of 3% or 6% per yearto give the discounted value of simulated earnings at age twenty-five.Chapter 5. Policy Simulations 120T 2 3 4 5 6 7 8 9Policy 6 y’ ye C I(,,1) I(2) I()None 1 0 0 0 0 0 0 423,436 -2 .021 -0.064 .512 .615 .433 .470 224,459 -5 .019 -.047 .871 .950 .809 .843 66,352 -Broad 1 0 0 0 0 0 0 430145 6,7092 .022 -0.067 .511 . .615 .465 229986 5,5275 .021 -.050 .868 .950 .801 .838 69,518 3,166low T 0 0 0 0 0 0 428,873 5,4372 .025 -.068 .507 .615 .420 .64 229,780 5,3215 .023 -.050 .865 .8950 .796 .837 69,729 3,377high T 0 0 0 0 0 0 432,814 9,3782 .017 -.068 .517 .615 .435 .466 231,193 6,7345 .015 -.050 .873 .950 .810 .839 69,622 3,270Table 5.20: Effect of Policies on Lifetime Inequality and WelfareI then assume that a S-Gini SEF defined over discounted lifetime income representssocieties preferences and continue with the decomposition analysis from chapter three.Table 5.20 shows the results of the lifetime analysis, using the marginal interactivedecomposition of the AKS index, of the broad policy, the low targeted policy, and thehigh targeted policy.Examining column seven of the table it is apparent that all of the policies reducemeasured inequality of total earnings but that this effect is quite small, being only abouta reduction of .01. The marginal effect of education with all policies is to reduce measuredinequality as shown in column three. This decrease varies from- .03 to -.07. The marginaleffect of the base level of income is to increase measured inequality by .02 to .03.In order to perform a welfare evaluation for these policies I compare the change inthe EDE income given in column six of table 5.20 with the average cost per person of thepolicies. The average cost per person of a policy is calculated by first finding the numberof people affected in each education category, then multiplying by the cost per personof the increased education from table 5.19. This gives the total cost of the policy. ForChapter 5. Policy Simulations 1211 2 3 4 5Move Prob. Pop. Cost per Move Total Cost1-2 .1384 287 17220 683,9922-3 .1384 1787 10950 2,708,16rTable 5.21: Calculation of Cost of low Targeted Policyexample the cost of the policies targeted towards low education individuals is calculatedin table 5.21. The total cost of the low targeted policy is calculated by summing thetwo values in column 5 and is therefore 2,712,155 dollars. The high targeted policy costs11,170,3742 dollars, while the broad policy costs 5,885,960 dollars. Average cost perperson is then the total cost divided by the total number of people, or 709 dollars, 2,923dollars, and 1,540 dollars for the low, high and broadly based policies respectively.The average cost per person can be compared to the increase in the EDE discountedincome to see if the given policy is welfare increasing. If the increase in the EDE incomeis larger than the average cost per person, then the social net present value of the policyis positive and the policy is welfare enhancing. Since all of the policies have an increasein the EDE that is greater than the cost, they all have a positive net present value. Thecalculated net present value of the policies when 5 = 2 is 4818 for the broadly basedpolicy, 1958 for the low targeted policy, and 2915 for the high targeted policy. Thus theresult is that the policies can be ranked in terms of net present value with the broadlybased policy most desirable, then the high targeted policy, and then the low targetedpolicy. When 6 = 5 the result is that the policies remain desirable but since the changein EDE is smaller, they are less desirable. The results when 6 = 1 are that all of thepolicies are more desirable than when 6 = 2, or 6 = 5. This is because none of the increasein individual earnings is wasted by inequality when there is no inequality aversion. Ingeneral, the more inequality aversion in the SEF, the less desirable any policies will be.Inequality aversion reduces the benefit of the policies but makes no difference for theChapter 5. Policy Simulations 122T 2 3 4 5 6 7 8 9Policy T y’ ye C J() [(1) J(,2)None 0 0 0 0 0 0 282,488 -2 .022 -0.064 .510 .616 .429 .469 150.094-5 .020 -.047 .869 .950 .807 .842 44,477-Broad 0 0 0 0 0 0 0 286,757 4,2692 .023 -0.067 .508 .616 .422 .464 153,656 3,5625 .021 -.050 .866 .950 .799 .838 46,543 2,066low 0 0 0 0 0 0 0 285,962 3,4742 .026 -.067 .504 .616 .417 .463 153,527 3,4335 .028 -.050 .863 .950 .794 .837 46,683 2,206high 0 0 0 0 0 0 0 288,496 6,0082 .018 -.067 .514 .616 .432 .465 154,469 4,3755 .017 -.050 .871 .950 .808 .838 46,636 2,159Table 5.22: Effect of Policies on Lifetime Inequality and Welfare, Discounted at 6%costs.I now repeat the above analysis but with a discount rate of 6%. Table 5.22 gives theresults of the constant population analysis with a discount rate of 6%. The same generalpattern is present here as with the discount rate of 3%. All three policies decreasemeasured inequality by a small amount. Again the marginal effect of education is toreduce inequality while the marginal effect of the base level is to increase inequality.To do the welfare analysis I use average cost per person of 1,612 dollars for thebroadly based policy, 919 dollars for the low targeted policy, and 3,081 dollars for thehigh targeted policy4. Comparing this to the change in the EDE income gives the netpresent value of the policy change. For all policies but the high targeted one, the socialnet present value is greater than zero. Again the policies are most desirable when there isno inequality aversion present in the SEF. The net present value is negative for the hightargeted policy , but in all other cases is positive, indicating that the benefits outweigh4These are calculated in an identical way to the previous values except the cost of education iscompounded by 6% over the period required to change education levels.Chapter 5. Policy Simulations 123the costs.The constant population policies in this section are all, with one exception, worthwhileby the NPV criterion. A difficult arises when comparing them since the costs of thepolicies are not the same. For example the NPV of the high targeted policy, when 6 = 2and the discount rate is 3% is higher than the NPV of the low targeted policy but thelow targeted policy costs substantially less. Therefore the next section compares policieswhich all cost the same amount.5.5 Constant Cost PoliciesThe policies that I considered in the previous section of this chapter all had the featurethat the number of individuals who are affected by the policy change remained constant.Given that the policies affected the same number of people, this meant that the cost ofthe policies had to be different. In this section I hold the cost of the policies constantand change the number of people affected by the policy.I consider only policies that cost five million dollars to implement and again concentrate on broadly based policies, policies targeting low education groups and policiestargeting high education groups. The way that I calculate the lifetime income is identical to that in the previous section except for the probability that a given person in agroup is affected by the policy. For the broadly based policy everyone in groups one tofour has a probability of 8.5% of having their education level increased by one. The lowtargeted policy increases those with education categories one and two with a probabilityof 20.4%, and the high targeted policy increases those with category three and four witha probability of 14.5%.The results of these simulations are shown in table 5.23. As can be seen in columnfive all of the policies again decrease the inequality in lifetime income. Very little changeChapter 5. Policy Simulations 124T 2 3 4 5 6 7 8 9Policy T y’ ye C 1(y) 1(y’) 1(y2)None 1 0 0 0 0 0 0 423,436-2 .021 -0.064 .512 .615 .433 .470 224,459-5 .019 -.047 .871 .950 .809 .843 66,352-Broad 0 0 0 0 0 0 0 429,096 5,6602 .022 -0.067 .511 .615 .427 .466 229,228 4,7695 .020 -.050 .869 .950 .803 .839 69,064 2,712low 0 0 0 0 0 0 0 431,374 7,9382 .027 -.069 .505 .615 .414 .462 232,010 7,5515 .024 -.052 .862 .950 .791 .835 71,141 4,789high 0 0 0 0 0 0 0 427,816 4,3802 .019 -.067 .515 .615 .435 .468 227,543 3,0845 .018 -.048 .872 .950 .810 .842 67,800 1,448Table 5.23: Effect of Policies on Lifetime Inequality and Welfare, Discounted at 3%occurs in the marginal components of earnings inequality in columns one and two. Thepattern of education being equalizing and the base income being disequalizing is againpresent.To examine the welfare effects of the policies I again must find the cost per personof the given policies. In this case it is easy as the total cost remains constant at fivemillion dollars. With the 3,823 people I consider, this gives a cost per person of 1,308dollars. Comparing this to the change in the equally distributed equivalent income showsthat since all three policies result in an increase in EDE income which is greater thanthe average cost per person, all of the policies are desirable. The net present value ofthe policies can he ranked to show which policies give the best social return on theinvestment. The best is the low targeted policy, followed by the broadly based policy,and then the high targeted policy.Table 5.24 gives the constant cost results for the case when the discount rate is 6%The same pattern occurs as when the discount rate is 3%, the return to education reducesinequality while the return to the base level increases inequality.Chapter 5. Policy Simulations 125T 2 3 4 5 6 7 8 9Policy 6 y’ ye C 1(y) 1(y’) 1(y2)None 1 0 0 0 0 0 0 423,436-2 .021 -0.064 .512 .615 .433 .470 224,459-5 .019 -.047 .871 .950 .809 .843 66,352-Broad 0 0 0 0 0 0 0 286,091 3,6032 .023 -0.066 .508 .616 .423 .465 153,170 3,0765 .021 -.049 .867 .950 .809 .838 46,252 1,775low 0 0 0 0 0 0 0 287,543 5,0552 .028 -.068 .501 .616 .411 .461 154,964 4,8705 .026 -.051 .860 .950 .789 .834 47,616 3,139high 0 0 0 0 0 0 0 285,281 2,7932 .020 -0.065 .512 .616 .431 .467 152,092 1,9985 .018 -.048 .871 .950 .808 .841 45,426 949Table 5.24: Effect of Policies on Lifetime Inequality and Welfare, Discounted at 6%As far as the NPV calculation, the increased discount rate makes all of the policies lessattractive. Similar to the constant population case, the high targeted policy when 6 5becomes undesirable by the NPV criterion. The policies can again be ranked accordingto their NPV as the low targeted policy the best, followed by the broadly based policy,and then the high targeted policy is least desirable by the social NPV criterion.5.6 Constant ProbabilitiesIn this section I analyze a combination of the two previous types of policies. Each individual has a ten percent probability of receiving an increase in their education regardlessof the number of people in his or her education category.Table 5.25 shows the results of the lifetime analysis of the broad policy, the lowtargeted policy, and the high targeted policy given a discount rate of 3%. Examiningcolumn seven of the table it is apparent that the policies all serve to decrease measuredearnings inequality compared to the status quo.Chapter 5. Policy Simulations 126T 2 3 4 5 6 7 8 9Policy ye C JQ) ](l) JQ2)None 1 0 0 0 0 0 0 423,436 -2 .021 -0.064 .512 .615 .433 .470 224,459-5 .019 -.047 .871 .950 .809 .843 66,352 -Broad T 0 0 0 0 0 0 430,145 6,7092 .022 -0.067 .511. .615 .465 229986 5,5275 .021 -.050 .868 .950 .801 .838 69,518 3,166low 0 0 0 0 0 0 0 427,147 3,7112 .024 -.066 .509 .615 .424 .466 227,951 3,925 .022 -.049 .867 .950 .801 .840 68,531 2,170high 0 0 0 0 0 0 0 426,434 2,9982 .019 -.065 .515 .615 .434 .469 226,475 2,0165 .018 -.048 .872 .950 .809 .842 67,312 960Table 5.25: Effect of Policies on Lifetime Inequality and WelfareThe cost of the policies is calculated the same as in section three of this chapter. Witha discount rate of 3% these are 641 per person for the low policy, and 899 per personfor the high. Thus all of these policies are welfare increasing. It is interesting to notethat the ranking of these policies, in terms of their NPV is the broad based policy, thelow targeted policy, and the high targeted policy. Part of the reason for the dominanceof the broad based policy in this case is that since everyone has a 10% chance of beingaffected, this policy affects substantially more people than do the other types of policies.For a discount rate equal to 6% the results are given in table 5.26. The general effectswhen the discount rate is 6% are similar to those when the discount rate is 3%. Thecosts of the policies are calculated as 664 dollars per person for the low, and 948 dollarsper person for the high policy. Using these numbers the NPV of all but the high targetedpolicy when 6 = 5 is positive. The ranking by NPV is the same as when the discountrate is 3%.Chapter 5. Policy Simulations 1271 2 3 4 5 6 7 8 9Policy 6 y1 ye C I(y) 1(y’) 1(y2)None 1 0 0 0 0 0 0 423,436-2 .021 -0.064 .512 .615 .433 .470 224,459-5 .019 -.047 .871 .950 .809 .843 66,352 -Broad 0 0 0 0 0 0 0 286,757 4,2692 .023 -0.067 .508 .616 .422 .464 153,656 3,5625 .021 -.050 .866 .950 .799 .838 46,543 2,060low 0 0 0 0 0 0 0 284,850 2,3622 .025 -.066 .506 .616 .421 .465 152,345 2,2515 .023 -.049 .865 .950 .799 .839 45,901 1,424high 0 0 0 0 0 0 0 284,395 1,9072 .021 -.065 .511 .616 .431 .468 151,394 1,3005 .019 -.048 .950 .808 .955 .841 45,101 624Table 5.26: Effect of Policies on Lifetime Inequality and Welfare, Discounted at 6%5.7 ConclusionThis chapter uses the earnings generating model estimated in chapter three to constructsimulated lifetime earnings distributions under several policy options. I then comparethe policies in regards to their effect on earnings inequality and welfare. I show thatin most cases, the policies increase the welfare of society more than they cost and aretherefore socially worthwhile5.Without exception, increasing inequality aversion makesthe policies less attractive from society’s perspective because benefits are reduced byinequality aversion but costs of the policies are unaffected. For the constant populationpolicies the ranking by NPV, from highest to lowest, is the broad based policy, the hightargeted policy, and the low targeted policy. For the constant cost policies, the rankingis low targeted policies, broadly based policies, and high targeted policies. Finally forthe constant probability model, the ranking is broad based policy, low targeted policy,and high targeted policy. My inclination is to say that the results for the constant cost5Assuming that some sort of friction exists as in section two.Chapter 5. Policy Simulations 128policies are the ones that are of most interest, since the results show where the socialreturn on a dollar spent are highest.This analysis has ignored the possibility of quality changes in the education that ispurchased with no change in the number of individuals being educated. I ignore qualitychanges because firstly, quality of education is not something for which there is a welldefined measure, and secondly, even if quality could be measured, it is very difficult todetermine the effect of any change in quality on individual earnings.Chapter 6Conclusions129Chapter 6. Conclusions 130Time and again level of education has been shown to be empirically related to labourearnings’. This thesis assumes that the relationship between the two is causal in thateducation determines earnings, and examines the effect that differences in education haveon observed earnings inequality. The specific approach that I use is to measure earningsinequality with an S-Gini inequality index and determine how much of the observed valueof the inequality index is a result of returns to education.The first step in the analysis is a theoretical examination of how to decompose inequality indices by income source. The second chapter initially examines some previouslyknown methods of decomposition and explains why they are unsatisfactory for the problem at hand. The chapter develops three new ways of doing the decomposition by incomesources that are both theoretically consistent and intuitively appealing. The first twomethods are termed interactive decompositions because of the distinguishing feature thatthe value of the inequality index is separated into direct effects for each income sourceplus an interaction term accounting for the fact that, in the absence of perfect correlation between the sources, inequality in one source of income has a tendency to counteractinequality in any other source. The third decomposition method introduced approachesthe decomposition problem from a slightly different perspective. I ask that the values ofthe equally distributed equivalent income be divided into contributions for each incomesource, with no interaction term. I outline three intuitive properties that I want a decomposition to satisfy and use these to derive what I call the Shapley decomposition. Thismethod is based on a decomposition of the equally distributed equivalent income functionthat is mathematically identical to the Shapley value of transferable utility cooperativegames. The three methods are compared with each other as well as with the previuoslyknown decomposition methods. I conclude the second chapter with a brief discussion ofpossible uses of the decomposition methods in applied economics.‘See Willis (1986) for a survey.Chapter 6. Conclusions 131In the fourth chapter of the thesis I proceed with one of the suggested applications.I first use the data on labour earnings from Statistics Canada’s 1986 Family Expenditure Survey to estimate an earnings model. This model takes separate account of theprobability of work and the income when working. I then use this estimated earningsmodel in a series of thought experiments which determine the effect of the distributionof educational attainments on the observed earnings distribution. I am able to dividelabour earnings into two parts; a part due to returns to education, and a part that isthe result of other personal characteristics. I use these two vectors of earnings sources inthe interactive decomposition of the 5-Gini index of relative inequality to determine theoverall effect of education on earnings inequality.An interesting feature of the earnings model used in this analysis, and one of thefeatures that separates the analysis in this thesis from similar studies of the effect ofeducation on inequality, is that the explicit modelling of the probability of an individualsbeing employed allows the change in this probability associated with a change in education level to be one of the sources of returns to education. All else equal, a change inthe amount of education will have two effects. The first is the increase in the individualswage rate, and thus his or her earnings when working. The second effect is the increasein the probability that the individual will be able to find a job. Previous examinationsof the sources of earnings inequality have considered the effect of the first of these factors but have ignored the second. One of the results of this thesis is that the change inthe probability of working is a significant source of earnings, and of earnings inequality.The final results, including this effect, indicate that, depending on the sample used, andthe degree of inequality aversion desired, between one third and one half of measuredearnings inequality is a result of differing returns to education.The fifth chapter of the thesis uses the earnings generating model presented in chapterthree to conduct some policy simulations. This chapter differs from the previous ones inChapter 6. Conclusions 132that, instead of examining things year by year, I examine the effect of education on theinequality of lifetime earnings. I examine three types of social policies, policies targetinglow education groups, policies targeting high education groups, and policies that targetall groups equally. The model predicts that all of the policies I consider are in thatthe increase in costs of education more than outweighed any increase in welfare fromlifetime earnings. No policy had large effects on measured inequality or on the amountof measured inequality that can be attributed to differences in education.The questions raised in the empirical portion of the thesis are important for the insightthat they provide to the more general problem of how best to reduce inequality. Thereis a rising interest among economists in the study of income inequality in the developedcountries. This arises mostly out of a belief that inequality of incomes has been increasingand continues to increase2. This thesis has shown that a large proportion of earningsinequality on a year by year basis arises from differences in education. Thus educationpolicy may be an important weapon in the fight against inequality3. Future researchin this area may be fruitful, especially the use of panel data to develop true personalmodels of lifetime earnings and to use these lifetime earnings models in policy simulationexperiments similar to chapter four.2See for example Jenkins (1992), Karoly (1992), and Johnson and Webb (1993).3This statement may seem to be in contradiction to the simulation results in chapter four but I don’tthink that it is. The reason for this belief is that, while education policy changed inequality by only asmall amount in absolute terms, the change in relation to the overall amount of inequality was fairlylarge. As well there is reason to believe that the amount of inequality over a lifetime is less than in ayear by year comparison. Thus even though the examined education policies had little effect on lifetimeinequality, the year by year effect may be more substantialBibliography133Bibliography 134[1] Albrecht, J. (1981): ‘A Procedure for Testing the Signalling Hypothesis,’ Journal ofPublic Economics, 15.[2] Atkinson, A. B. (1970): ‘On the Measurement of Inequality’, Journal of EconomicTheory, 2.[3] Becker, G. (1964): Human Capital. National Bureau of Economic Research, NewYork.[4] Blackorby, C. and D. Donaldson (1978): ‘Measures of Relative Inequality and TheirMeaning in Terms of Social Welfare’, Journal of Economic Theory, 18.[5] Blackorby, C. and D. Donaldson (1980): ‘A Theoretical Treatment of Indices of Absolute Inequality,’ International Economic Review, 21.[6] Blackorby, C. and D. Donaldson (1984): ‘Ethical Social Index Numbers and theMeasurement of Effective Tax—Benefit Progressivity’, Canadian Journal of Economics,17.[7] Blackorhy, C., D. Donaldson and M. Auersperg (1981): ‘A New Procedure for theMeasurement of Inequality Within and Among Population Subgroups’, Canadian Journal of Economics, 14.[8] Bossert, W. (1990): ‘An Axiomatization of the Single-Series Ginis’, Journal of Economic Theory, 50.[9] Chakravarty, S. (1990): Ethical Social Index Numbers. Springer—Verlag, Berlin.[10] Cogan, J. (1981): ‘Fixed Costs and Labor Supply,’ Econometrica,, 49.[11] Constantos, C. and E. West (1991): ‘Measuring Returns from Education: SomeNeglected Factors’, Canadian Public Policy, 17.[12] Donaldson, D. and J. Weymark (1980): ‘A Single-Parameter Generalization of theGini Indices of Inequality’, Journal of Economic Theory, 22.[13] Fei, J., G. Ranis and S Kuo (1978): ‘Growth and the Family Distribution of Incomeby Factor Components’, Quarterly Journal of Economics, 92.[14] Gonul, F. (1992): ‘New Evidence on Whether Unemployment and Out of the LaborForce are Distinct States,’ Journal of Human Resources, 27[15] Green, D.(1991): ‘A Comparison of Estimation Approaches for the Union—NonunionWage Differential,’ Discussion Paper 91-13, University of British Columbia.Bibliography 135[16] Greene, W. (1990): Econometric Analysis. MacMillan Publishing Company, NewYork.[17] Heckman, J. (1976): ‘A Life Cycle Model of Earnings, Learning, and Consumption’,Journal of Political Economy, 84.[18] Heckman, J. and T. Macurdy (1986): ‘Labor Econometrics,’ in Griliches, Z. and M.Intriligator eds Handbook of Econometrics Vol. 3 North Holland, New—York.[19] Jenkins, S. (1992): ‘Accounting for Inequality Trends: Decomposition Analyses forthe UK 1971-86,’ Discussion Paper 92-10, University College of Swansea.[20] Johnson, P. and S. Webb (1993): ‘Explaining the Growth in UK Income Inequality,’The Economic Journal, 103.[21] Karoly, L. (1992): ‘Changes in the Distribution of Individual Earnings,’ Review ofEconomics and Statistics, 74.[22] Killingsworth, M. (1983): Labor Supply, Cambridge University Press, New York.[23] Kumar, P. and M.L. Coates (1982): ‘Occupational Earnings, Compensating Differentials and Human Capital: An Empirical Study,’ Canadian Journal of Economics,15.[24] Layard, R. and A. Zabalza (1979): ‘Family Income Distribution: Explanation andPolicy Evaluation’, Journal of Political Economy, 87.[25] Lemieux, T. (1993): ‘Estimating the Effects of Unions on Wage Inequality in a Two-Sector Model with Comparative Advantage and Non-Random Selection’, Cahier 9303,Universite de Montreal.[26] Lewis. W. A. (1954): ‘Economic Development with Unlimited Supplies of Labour’,Manchester School.[27] Loury, G. (1981): ‘Intergenerational Transfers and the Distribution of Earnings,’Econometrica, 49.[28] Lucas, R.E.B. (1977): ‘Is There a Human Capital Approach to Income Inequality?,’Journal of Human Resources, 12.[29] Mincer, J. (1974):Education, Schooling and Inequality, National Bureau of EconomicResearch, New York[30] Moffit, R. (1982): ‘The Tobit Model, Hours of Work and Institutional Constraints,’Review of Economics and Statistics, 64.Bibliography 136[311 Mroz, T. (1987): ‘The Sensitivity of an Empirical Model of Married Women’s Hoursof Work to Economic and Statistical Assumptions,’ Econometrica, 55.[32] Pfingsten, A. (1987): ‘Axiomatically Based Local Measures of Tax Progression’,Bulletin of Economic Research, 39.[33] Rosen, S. (1975): ‘Human Capital: A survey of Empirical Research,’ in R. Ehrenberged Research in Labor Economics, Voll, JAI Press, Greenwich.[34] Sen, A. (1973): On Economic Inequality, Oxford University Press, London.[35] Sen, A. (1992): Inequality Reexamined,??[36] Shapley, L. (1953): ‘A Value for n-Person Games’, in H. Kuhn and A. Tucker, eds.Contributions to the Theory of Games. Princeton University Press, Princeton.[37] Shorrocks, A. (1982): ‘Inequality Decomposition by Factor Components’, Econometrica, 50.[38] Simpson, W. (1985): ‘The Impact of Unions on the Structure of Canadian Wages:An Empirical Analysis with Microdata,’ Canadian Journal of Economics, 18.[39] Spence, M. (1973): ‘Job Market Signalling’, Quarterly Journal of Economics.[40] Statistics Canada: Education in Canada (1985-1986).[41] Statistics Canada: Financial Statistics on Education.[42] Taubman, P. (1975): Sources of Inequality in Earnings, North—Holland, Amsterdam.[43] Taubman, P. and T. Wales (1974): Higher Education: An Investment and a Screening Device, The National Bureau of Economic Research, New York.[44] Tobin, J. (1958): ‘Estimation of Relationships for Limited Dependent Variables’,Econometrica, 26.[45] Vaillancourt, F. and I. Henriques (1986): ‘The Returns to University Schooling inCanada’, Canadian Public Policy, 12.[46] Welch, F. (1970): ‘Education and Production’, Journal of Political Economy, 78.[47] West, C.T. and H. Theil (1991): ‘Regional Inequality by Components of Income,’in R. Carter, J. Dutta, and A. Ullah eds Contributions to Econometric Theory andApplication, Springer-Verlag, New York.Bibliography 137[48] Willis, R. (1986): ‘Wage Determinants: A Survey and Reinterpretation of HnmanCapital Earnings Functions’ in 0. Ashenfelter and R. Layard eds. Handbook of LaborEconomics, North-Holland, New-York.[49] Young, H. (1985): ‘Monotonic Solutions of Cooperative Games’, International Journal of Game Theory, 14.[50] Zabel, J. (1993): ‘The Relationship between Hours of Work and Labor Force Participation in Four Models of Labor Supply Behaviour,’ Journal of Labor Economics,11.Appendix AData Appendix138Appendix A. Data Appendix 139In this appendix I describe the data set used iu all of the analysis in the snbsequentsections of the paper. I first describe the data set in general and then the actual variablesthat are used.The raw data comes from the 1986 Survey of Family Expenditures compiled by Statistics Canada. The initial sample consists of 15334 observations on heads and spouses ofhouseholds. These are designed to represent private households in Canada. The dataset has some important exclusions that may be important. All residents of Yukon andNorthwest Territories except for those living in Whitehorse or Yellowknife are excluded,all individuals living on Indian reservations are excluded, individuals in institutions suchas old-age homes, penal institutions, and hospitals are excluded, and finally families ofofficial representatives of foreign conntries are excluded.The data collected is given in table A.27. The data was screened as according to tableA.28.Table A.29 gives the sample means for the data that I use in the main analysis of thethesis where d11 equals 1 the education level of person i is in category 1 and is 0 otherwise,agej is the age of the head of the household in 1986, intended to proxy for experience,qu, on, sk, ab and be are geographic dummies which equal 1 if the household is in thearea, sex = 1 if the head of the household is a male, fr and en are language dummieswhich equal 1 if the first language is french or english respectively and finally m and sare 1 if the individual is married or single respectively.Because I have only categorical data, I construct an education variable that is comparable to Taubman’s and Layard and Zabalza’s in the following way. Those with lessthan grade nine are assumed to have left school at 16, so they have ed = 9, those withsome or completed high school have ed = 12, those with some post-secondary are givened = 13, post secondary diploma, ed = 14 and university degree are given ed = 17 toaccount for those who took longer than average to get an undergraduate degree as wellAppendix A. Data Appendix 140Name Possible Values MeaningWages [0, oc) wages in dollars earned bythe head of the householdAge 20-65 age in years of the head of the householdSex 0 female1 maleWeeksf [0,52] weeks worked full timeWeeksp [0,52] weeks worked part timeSelf [0, co) Earnings from self employmentEd 1 less than nine years2 some or completed high school3 some post—secondary4 post—secondary diploma5 university degree6 not statedGeo 1 Maritimes2 Quebec3 Ontario4 Manitoba and Saskatchewan5 Alberta6 British ColumbiaLang 1 english2 french3 otherMarr 1 Married2 single3 otherTable A.27: Data CollectedScreen Number left lostNone 15334-age 65 13003 2331ed = 6 12929 74age [30,40) 3823 9106age [50.60) 2196 10733Table A.28: Cuts to the Sample.Appendix A. Data Appendix 141name J y age dl d2 d3 d4value 16761 40.511 .14289 .47843 .12374 .1415Jname] d5 que mt sk al bcvalue .10767 .20788 .203015 .12705 .11174 .109905name fr oti sex otm s-value .23218 .12412 .44582 .09651 .08852-Table A.29: Sample Means of the Dataas those with graduate degrees. I then construct an experience variable in the followingway ex = age — ed— 6.Below is a listing of all of the variables that I use in the regressions and what theysignify• y yearly labour income• ed, number of years of formal education. Described above.• ex number of years of work experience. Described above.• rnt a dummy variable which is 1 if the individual lives in the maritime provinces• que a dummy variable which is 1 if the individual lives in Quebec• .sk a dummy variable which is 1 if the individual lives in Manitoba or Saskatchewan• al a dummy variable which is 1 if the individual lives in Alberta• bc a dummy variable which is 1 if the individual lives in British Columbia• sex a dummy variable which is 1 if the individual is male• oti a dummy variable which is 1 if the individual’s first language is neither frenchnor englishAppendix A. Data Appendix 142• fr a dummy variable which is 1 if the individual’s first language is french• en a dummy variable that is 1 if the individual’s first language is english• otm a dummy variable for a person with marital status that is neither married norsingle• s a dummy variable for a person who is single• age the age of the individual in years• dl a dummy variable which is 1 if the individual has less than nine years of schooling• d2 a dummy variable which is 1 if the individual has some or completed high school• d3 a dummy variable which is 1 if the individual has some post—secondary education• d4 a dummy variable which is 1 if the individual has a post—secondary degree ordiploma• d5 a dummy variable which is 1 if the individual has a university degree• diage, 1 e [1, 2, 3,4, 5] is an interaction term, the product of the education dummydl and the age variable.• mdl, 1 e [1,2,3,4,5] an interaction term which is the product of the educationdummy and the gender dummy• mdlage, 1 e [1, 2, 3,4, 5] the product between the gender dummy and the interactionbetween age and education• cf1, 1€[1,2,3,4,5] a dummy which is 1 if the counterfactual education category iscategory 1Appendix A. Data Appendix 143• cflage , 1 E [1, 2,3,4, 5] an interaction term between the counterfactual educationlevel and age• rncfl, I E [1,2,3,4, 5] an interaction between gender and counterfactual educationlevel• cflage ,i E [1,2,3,4,5] an interaction term between gender and cflage•yk is earnings when the base level of education is category kAppendix BRegression Results144Appendix B. Regression Results 145In this appendix I present the results of all of the estimation in greater detail than in thetext. All of the estimation is done with the complete sample as described in the dataappendix. The variables used in the analysis are also described in the data appendix.Discussion of the results may be found in the body of the thesis where appropriate.The first equation to be estimated is the replication of the Taubman analysis. Theequation to be estimated is the followingy = /% + flied + /92ex + /33ex2 + /34rnt + /35que + fl6sk + /37a1+/35bc +/9sex + /310ot1 + /311en + /312otm + /33s (B.122)The equation is estimated using ordinary least squares. The results are in table B.30.The parameters and the associated standard errors are shown in the table followed bythe R2 of the regression, the standard error of the estimate, the log of the likliehoodfunction and the number of observations used.The second equation is the replication of the Layard and Zabalza estimation. Thus Iestimate the equationf = /% + flied + j3ex (B.l23)where f is the ratio of the individual’s earnings to the mean earnings in the sample. Theestimates for this equation are in table B.31. The results for the more general Layardand Zabalza equation which has more personal characterstics included are given in tableB.32. These results are based on the following equationf = fib + fl1ed + beta2ed x age + /33age + /34age2+ /35rnt + /36que + /37sk + /35a1+/3gbc + /3iosex +/3notl + /312en + /3iotm + /314s (B.124)C 0)0 cf CD 0 CiD IPI CD I CD 0P C;’PCCJDCIDCDCDCDi c± . Ci)t’IIIIIIIII—‘cC;’cC.J)CD00F—aC-:CCJ—-C—4C1CDCA:ICiCi)cc‘00cOC-Y’C;’C:CDDCJ-CtDDCCJ-—1,C;’))Ci-Dt’00)I-CD Ci) CD 0 CD CI0= CD Cl) CD Ci) C IAppendix B. Regression Results 147parameter value standard errorconstant -2.96 .1900ed .1114 .0128ed x age .0008 .0003age .10864 .0060age2 -.0014 0.00005mt -.2874 .0213q’ue -.1785 .0292sk -.1525 .0240al -.0397 .0252be -.1801 .0252sex .8926 .0141oti —.1106 .0223fr .0066 .0269otm -.0077 0241s -.0078 .0025R2 .3831SEE .7867LLF -141006observation 12929Table B.32: Estimates of the Generalized LZ Equation.Appendix B. Regression Results 148For the main body of the analysis the following general specification is used whichallows the return to education to be different for men and women.y = So + /31d2 + 52d3 + /33d4 + 5d5 + /36d2age+/37d3age + /38d4age +59d5age + Sio * md2 +511md3 +512md4 +513md5+/3i4md2age +515md3age + /316rnd4age + /3i7rnd5age + /31gage + /3i9age2ex-1-fi2omt + S2lque + /322sk + /32a1 + /324bc +525sex+[326ot1 + /327en +/328otm + /32s (B.125)The estimation results for this specification are in table B.33. A restricted form ofthis specification, where the returns to education for men and women are restrictedto be equal( ie coefficients /3 through are zero), is given in table B.34. The firsttwo columns of both of these tables give the results when the equation is estimated byordinary least squares. The last two columns give the results when the equation is givena tobit specification and estimated by maximum likliehood. Note that the values in thetable for the tobit specification are the estimated values of the normalized coefficients,which are the estimates of the parameters Sk divided by the estiamted standard error ofthe equation. Thus the reported coefficient on y is equal to the reciprocal of the standarderror of the estimate.Appendix B. Regression Results 149OLS Tobitparameter value st andard errorconstant -35089 2208 -3.0980 .18315d2 9275.5 1592 .7440 .1363d3 12166 2054 .9510 .1710d4 18152 1987 1.3590 .1646d5 16333 2498 1.1395 .2025d2age -127.03 32.68 -.0051 .0029d3age -113.21 47.31 -.0025 .0040d4age -207.57 45.39 -.0079 .0038d5age -30.220 59.15 .0065 .0048md2 -108.17 1304 -.5687 .1068md3 -6639.7 2317 -1.0461 1843md4 -10731 2294 -1.3432 .1809md5 -8053 2854 -1.1333 .2239rnd2age 101.60 27.48 .0130 .0022md3age 260.85 57.73 .0222 .0046nzd4age 403.77 56.61 .0311 .0045rrid5age 387.88 69.33 .0282 .0054age 2092.6 83.33 .1645 .0069age2 -24.050 .8510 -.0021 .00007mt -4939.9 355.4 -.3885 .0285que -3035.3 485.2 -.2820 .0390sk -2430.9 399.8 -.1722 .0318al-638.08 418.5 -.0541 .0330bc -2910.8 419.4 -.2366 .0335sex 11258 613.2 1.2520 .0537otl -2194.8 373.7 -.1256 .0301fr -358.86 448.6 -.0106 .0361otm -4.7781 400.5 -.0118 .0331.s 35.889 422 .0425 .0332y-- .00006 .0000005R2 .3957-SEE 13225 15951LLF -140873 -110770.8observations 12929Table B.33: Estimates of the Earnings Equation.Appendix B. Regression Results 150OLS Tobitparameter value standard errorconstant 35924 2211 2.9673 .1817d2 9032.9 1517 .48922 .1286d3 8981.5 1812 .4765 .1500d4 13798 1786 .8129 .1476d5 10604 1968 .5024 .1602d2age -78.376 32.68 -.0007 .0027d3age 7.1473 40.47 -.0075 .0034d4age -39.672 39.65 -.0047 .0033d5age 211.21 44.12 .0228 .0036age 2062.9 83.82 .1598 .0069age2 -23.795 .8559 -.0020 .00007nit -4905.1 357.7 -.3829 .0285que -2980.7 488.4 -.2753 .0390.sk -2452.9 402.2 -.1729 .0318at -660.23 421.2 -.0563 .0330bc -2928.3 422.1 -.2370 .0335.sex 14915 236.0 1.1557 .0199ott -2014.1 375.8 -.1178 .0300fr -312.52 451.4 -.0108 .0361otm -144.01 402.5 -.0349 .0331.s -180.26 424.1 .0241 .0332y-- .00006 .0000005R2 .3874-SEE 13115 16039LLF -140961 -110844observation 12929Table B.34: Estimatesof the Restricted Earnings Equation.Appendix B. Regression Results 151t(yJf(yjyFig 2: Density Estimated IgnoringZerosfly)Fig 1: Sample Censored DensityFig 3: Density Estimated with ZerosAppendix B. Regression Results 152Lorenz Curves: Base and Education Income1.2 I IEducation —Counterfactual0.8 Fa)Bo/I—I 0.6 /‘4—Io //0 . 40.20I I I I I I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9% of Population\26= 6=2— Fig 5 : Sample Indifference Curves


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