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Inequality of Opportunity, Income Mobility, and the Interpretation of Intergenerational Elasticities, Correlations, and Rank-Rank Slopes

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Title: Inequality of Opportunity, Income Mobility, and the Interpretation of Intergenerational Elasticities, Correlations, and Rank-Rank Slopes
Language: English
Authors: Pablo A. Mitnik (ORCID 0000-0002-2988-7456)
Source: Sociological Methods & Research. 2025 54(4):1289-1338.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
Peer Reviewed: Y
Page Count: 50
Publication Date: 2025
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Social Mobility, Income, Correlation, Measurement Techniques, Statistical Analysis, Parent Child Relationship, Computation
Assessment and Survey Identifiers: National Longitudinal Survey of Youth, Panel Study of Income Dynamics
DOI: 10.1177/00491241251352102
ISSN: 0049-1241
1552-8294
Abstract: Although there is an extensive methodological literature on the measurement of intergenerational income mobility, there has been limited research on the conceptual interpretation of mobility measures and the methodological implications of those interpretations. In this article, I focus on the three measures of mobility most frequently used in the literature--the intergenerational elasticity (IGE), the intergenerational correlation (IGC), and the rank-rank slope (RRS)--as well as a recently introduced measure, the intergenerational elasticity of expected income (IGEE). I make two main contributions, both related to the conceptual interpretation of mobility measures. First, I specify the formal relationships between those four mobility measures and the measures of inequality of opportunity developed in the luck egalitarian empirical literature on the topic, and determine the methodological implications of the analyses. I show that (a) the IGC is a measure of relative inequality of opportunity for monetary income, (b) the RRS is both a measure of relative inequality of opportunity for income rank and a rescaled measure of absolute inequality of opportunity for income rank, and (c) the products of parental income inequality by the IGEE and IGE are both measures of absolute inequality of opportunity for monetary income that differ in how they measure the value of opportunity sets. Second, relying on a conceptual distinction that has been influential in the field of public finance, the IGE and IGEE have been characterized as "person-weighted" and "dollar-weighted" elasticities, respectively, thus raising doubts about the desirability of a recent proposal to replace the IGE by the IGEE as the workhorse elasticity of the mobility field. I show that this contrasting characterization of the two intergenerational elasticities is the joint result of a category mistake--equating quantile-specific elasticities to person-specific elasticities--and of misconstruing the nature of the IGE and the epistemic goal it has been meant to serve. Based on this analysis, I conclude that the case for replacing the IGE with the IGEE remains well-founded.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1485822
Database: ERIC
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  Value: <anid>AN0188424702;som01nov.25;2025Oct06.06:36;v2.2.500</anid> <title id="AN0188424702-1">Inequality of Opportunity, Income Mobility, and the Interpretation of Intergenerational Elasticities, Correlations, and Rank-Rank Slopes </title> <p>Although there is an extensive methodological literature on the measurement of intergenerational income mobility, there has been limited research on the conceptual interpretation of mobility measures and the methodological implications of those interpretations. In this article, I focus on the three measures of mobility most frequently used in the literature—the intergenerational elasticity (IGE), the intergenerational correlation (IGC), and the rank-rank slope (RRS)—as well as a recently introduced measure, the intergenerational elasticity of expected income (IGEE). I make two main contributions, both related to the conceptual interpretation of mobility measures. First, I specify the formal relationships between those four mobility measures and the measures of inequality of opportunity developed in the luck egalitarian empirical literature on the topic, and determine the methodological implications of the analyses. I show that (a) the IGC is a measure of relative inequality of opportunity for monetary income, (b) the RRS is both a measure of relative inequality of opportunity for income rank and a rescaled measure of absolute inequality of opportunity for income rank, and (c) the products of parental income inequality by the IGEE and IGE are both measures of absolute inequality of opportunity for monetary income that differ in how they measure the value of opportunity sets. Second, relying on a conceptual distinction that has been influential in the field of public finance, the IGE and IGEE have been characterized as "person-weighted" and "dollar-weighted" elasticities, respectively, thus raising doubts about the desirability of a recent proposal to replace the IGE by the IGEE as the workhorse elasticity of the mobility field. I show that this contrasting characterization of the two intergenerational elasticities is the joint result of a category mistake—equating quantile-specific elasticities to person-specific elasticities—and of misconstruing the nature of the IGE and the epistemic goal it has been meant to serve. Based on this analysis, I conclude that the case for replacing the IGE with the IGEE remains well-founded.</p> <p>Keywords: intergenerational income mobility; inequality of opportunity; luck egalitarianism; intergenerational elasticity; intergenerational correlation; rank-rank slope; rank correlation; inequality and mobility; person-weighted elasticity; income-weighted elasticity</p> <hd id="AN0188424702-2">Introduction</hd> <p>The methodological research on the measurement of intergenerational income mobility has primarily focused on estimators, the biases arising from imperfect data, and strategies to mitigate these biases (e.g., [<reflink idref="bib8" id="ref1">8</reflink>]; [<reflink idref="bib36" id="ref2">36</reflink>]; [<reflink idref="bib37" id="ref3">37</reflink>]; [<reflink idref="bib41" id="ref4">41</reflink>]; [<reflink idref="bib53" id="ref5">53</reflink>], [<reflink idref="bib54" id="ref6">54</reflink>], [<reflink idref="bib55" id="ref7">55</reflink>]; [<reflink idref="bib56" id="ref8">56</reflink>], [<reflink idref="bib57" id="ref9">57</reflink>]; [<reflink idref="bib68" id="ref10">68</reflink>], [<reflink idref="bib69" id="ref11">69</reflink>]; [<reflink idref="bib81" id="ref12">81</reflink>], [<reflink idref="bib82" id="ref13">82</reflink>]). By contrast, there has been little attention paid to the conceptual interpretation of estimands, despite some recent exceptions (e.g., [<reflink idref="bib26" id="ref14">26</reflink>]; [<reflink idref="bib34" id="ref15">34</reflink>]; [<reflink idref="bib45" id="ref16">45</reflink>]; [<reflink idref="bib62" id="ref17">62</reflink>], [<reflink idref="bib63" id="ref18">63</reflink>]). In particular, almost no effort has been made to specify the exact relationship between standard mobility measures and well-founded theoretical notions of inequality of opportunity.</p> <p>In this article, I focus on the three measures of income mobility that have most often been used in the mobility literature—the intergenerational elasticity (IGE), the intergenerational correlation (IGC), and the rank-rank slope (RRS)—as well as a new measure, the intergenerational elasticity of expected income (IGEE), recently introduced by [<reflink idref="bib62" id="ref19">62</reflink>]. Unlike most previous methodological work on these measures, my primary concern is with them as estimands, not with their estimators. My main goal is to clarify how these measures should be interpreted, independent of the data and methods used to estimate them.</p> <p>I address two related methodological issues. The first issue pertains to the aforementioned relationship between economic mobility and inequality of opportunity. Although mobility scholars have long assumed that a society's level of mobility is somehow informative of the extent of inequality of opportunity within that society, the nature of the relationship between these two notions has yet to be precisely specified within the mobility field. This is particularly striking, since research on mobility has, by and large, been motivated by concerns about inequality of opportunity.</p> <p>This conceptual problem has not been properly tackled outside the mobility field either. Over the past 20 years, a large, sophisticated, and influential theoretical and empirical literature on inequality of opportunity has developed alongside (although largely independent of) the mobility literature (for reviews, see [<reflink idref="bib30" id="ref20">30</reflink>], [<reflink idref="bib74" id="ref21">74</reflink>], [<reflink idref="bib76" id="ref22">76</reflink>], and [<reflink idref="bib15" id="ref23">15</reflink>]). Given the greater conceptual, methodological, and practical difficulties involved in theorizing and measuring inequality of opportunity compared to mobility, and the quite impressive progress that has nevertheless been made, it would have been natural to expect that, as a byproduct, a good understanding of the relationship between mobility and inequality of opportunity would have also emerged. Surprisingly, this has not occurred. Instead, the prevailing view among inequality of opportunity scholars about that relationship has been very similar to that of their mobility counterparts—while I show here that the one attempt at a formal analysis, based on the theoretical approach developed in the inequality of opportunity literature ([<reflink idref="bib47" id="ref24">47</reflink>], [<reflink idref="bib48" id="ref25">48</reflink>]), did not succeed.</p> <p>I considerably reduce the significant gap in both literatures regarding the relationship between mobility and inequality of opportunity. Taking a concrete rather than abstract approach, I demonstrate how the four mobility measures—the IGE, IGEE, IGC, and RRS—<emph>formally</emph> relate to the measures of inequality of opportunity developed in the theoretical and empirical literature on this topic. Additionally, I explore the methodological implications of my analyses for cross-country, cross-period, and cross-cohort comparisons.</p> <p>The second methodological issue I address concerns the appropriate characterization of the two intergenerational elasticities I consider in this article, that is, the IGE and the IGEE. [<reflink idref="bib62" id="ref26">62</reflink>] showed that the IGE has been widely misinterpreted and is affected by serious methodological problems. For these reasons, they called for replacing the IGE with the IGEE as the workhorse elasticity of the mobility field. However, a contrasting characterization of these two measures by [<reflink idref="bib20" id="ref27">20</reflink>] portrays them as "person-weighted" and "dollar-weighted" elasticities, respectively, raising doubts about the desirability of the proposed replacement.</p> <p>I show that by relying on a conceptual distinction that has played an important role in the public-finance field—the distinction between person-weighted and income-weighted elasticities—Chetty et al. unwittingly made two key missteps. First, they equated quantile-specific elasticities with person-specific elasticities, thereby committing a category mistake. Second, they treated the IGE as a behavioral elasticity, implicitly misconstruing both its conceptual nature and the epistemic role it has been intended to serve in the mobility field. Based on this analysis, I conclude that [<reflink idref="bib62" id="ref28">62</reflink>] case for replacing the IGE with the IGEE remains well-founded.</p> <p>In addition to its two main contributions, this article offers two supplementary contributions. First, as a prolegomenon to my primary analyses, I critically introduce the four mobility measures on which I focus, clarify their interrelationships in detail, and correct some widespread misunderstandings. Second, while the primary focus of the article is on conceptual interpretation, I also address an empirical issue: the possibility that the measures yield very similar estimates. A skeptic might argue that, if this were the case, the different interpretations would be interchangeable in practice. I demonstrate that this is not the case by estimating all four measures using three U.S. datasets. The results reveal that the estimates can, in fact, diverge substantially across measures. I also discuss these estimates in the context of my conceptual arguments.</p> <p>The structure of the rest of the article is as follows. I first introduce the four mobility measures central to my analysis. Next, I discuss the two key methodological issues outlined earlier. Following that, I present the empirical analysis. The final section distills the article's main conclusions.</p> <hd id="AN0188424702-3">Four Measures of Income Mobility and Their Relationships</hd> <p>The four measures I focus on here are <emph>global measures of relative mobility</emph>. They measure relative rather than absolute mobility because they aim to "characterize the distribution of opportunity by comparing outcomes across children" from different origins rather than describing "the central tendency for the full population or a particular subpopulation" ([<reflink idref="bib59" id="ref29">59</reflink>]: 1222–1223; see also [<reflink idref="bib26" id="ref30">26</reflink>]: 995–996). In addition, they are all global rather than local measures of mobility because they aim to provide a summary comprehensive assessment of mobility in a geographic region rather than "exploring how mobility varies across the joint density of income" ([<reflink idref="bib26" id="ref31">26</reflink>]:1001).</p> <p>The intergenerational elasticity (IGE) has long been a very central measure of economic mobility (e.g., [<reflink idref="bib7" id="ref32">7</reflink>]; [<reflink idref="bib40" id="ref33">40</reflink>]; [<reflink idref="bib83" id="ref34">83</reflink>]). The underlying population regression function, under the assumption that the elasticity is constant across levels of parental income, is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi><mo>,</mo><mspace width="0.25em" /></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>></mo><mn>0</mn></math> </ephtml> is the child's long-run income or earnings, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>></mo><mn>0</mn></math> </ephtml> is long-run parental income or father's earnings, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub></math> </ephtml> is the IGE, and I use expressions like Z|w as a shorthand for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow /><mi>Z</mi><mo fence="false" stretchy="false">|</mo><mi>W</mi><mo>=</mo><mi>w</mi><mo>.</mo><mrow /></math> </ephtml></p> <p>The intergenerational correlation (IGC) is the Pearson correlation coefficient between the log incomes of children and parents. The IGC and the IGE are closely related, as the following identities indicate: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">IGC</mi></mrow><mo>≡</mo><mrow><mi mathvariant="normal">Corr</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo>,</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>X</mi><mo stretchy="false">)</mo><mo>≡</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">Cov</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo>,</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">SD</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo stretchy="false">)</mo><mrow><mi mathvariant="normal">SD</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>≡</mo><msub><mi>β</mi><mn>1</mn></msub><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">SD</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">SD</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>,</mo><mspace width="0.25em" /></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">SD</mi></mrow><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is the standard deviation operator.</p> <p>Importantly, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub></math> </ephtml> is not the elasticity of the conditional expectation of the child's income, as mobility scholars have assumed ([<reflink idref="bib62" id="ref35">62</reflink>]).[<reflink idref="bib6" id="ref36">6</reflink>] This would hold as a general result only if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mrow><mi>ln</mi><mspace width="0.2em" /><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mspace width="0.2em" /><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> . But, due to Jensen's inequality, the latter is not the case. Instead, as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≡</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mrow><mi mathvariant="normal">exp</mi></mrow><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi></mrow><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">GM</mi></mrow><mo stretchy="false">(</mo><mrow><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi></mrow><mo stretchy="false">)</mo><mo>≡</mo><mrow><mi mathvariant="normal">exp</mi></mrow><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> , Equation (<reflink idref="bib1" id="ref37">1</reflink>) is equivalent to <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mrow><mi mathvariant="normal">GM</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">GM</mi></mrow><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is the geometric mean operator. Therefore, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub><mo>=</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mrow><mi mathvariant="normal">GM</mi></mrow><mo stretchy="false">(</mo><mrow><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi></mrow><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi></math> </ephtml> and the IGE is the elasticity of the conditional geometric mean, that is, the percentage differential in the geometric mean of children's long-run income with respect to a marginal percentage differential in parental long-run income. This means, in particular, that the archetypal interpretation of the IGE as a measure of regression to the arithmetic mean (e.g., [<reflink idref="bib4" id="ref38">4</reflink>]; [<reflink idref="bib67" id="ref39">67</reflink>]:24–25) is invalid.</p> <p>The population regression function underlying the intergenerational elasticity of expected income (IGEE), also under the assumption that the elasticity is constant across levels of parental income, is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mspace width="0.2em" /><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>α</mi><mn>0</mn></msub><mo>+</mo><msub><mi>α</mi><mn>1</mn></msub><mi>ln</mi><mspace width="0.2em" /><mi>x</mi><mo>,</mo><mspace width="0.25em" /></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>≥</mo><mn>0</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>></mo><mn>0</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mn>1</mn></msub><mo>=</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi></math> </ephtml> is the IGEE, that is, the percentage differential in the expectation of children's long-run income with respect to a marginal percentage differential in parental long-run income.</p> <p>[<reflink idref="bib62" id="ref40">62</reflink>] have called for replacing the IGE by the IGEE as the workhorse elasticity of the mobility field. A key fact behind their argument is that the geometric mean is undefined whenever an income distribution includes zero in its support, which means that the IGE is undefined as well when this is the case. They show that this has serious methodological consequences: it often leads to IGE estimates affected by substantial selection biases and badly hinders the study of gender and marriage dynamics in intergenerational processes. Crucial to Mitnik and Grusky's argument, (a) all interpretations incorrectly applied to the IGE are correct for the IGEE ([<reflink idref="bib62" id="ref41">62</reflink>]: Section 5.1), and (b) the IGEE is fully immune to the methodological problems affecting the IGE and, in particular, is very well suited for studying the role of marriage in the intergenerational transmission of advantage ([<reflink idref="bib62" id="ref42">62</reflink>]: Section 5.2).</p> <p>The rank-rank slope (RRS) was first introduced by [<reflink idref="bib25" id="ref43">25</reflink>] and became highly popular in the mobility field following the publication of [<reflink idref="bib20" id="ref44">20</reflink>]. The population regression function underlying the RRS, under the assumption that the slope is constant across levels of parental rank, is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>γ</mi><mn>0</mn></msub><mo>+</mo><msub><mi>γ</mi><mn>1</mn></msub><mrow><mspace width="0.25em" /></mrow><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mo>.</mo><mo stretchy="false">)</mo></math> </ephtml> is the percentile rank operator and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>γ</mi><mn>1</mn></msub><mo>=</mo><mi>d</mi><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><mspace width=".1em" /><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> is the RRS, that is, the differential in the expectation of children's long-run income rank with respect to a marginal differential in parental long-run income rank.[<reflink idref="bib7" id="ref45">7</reflink>]</p> <p>In addition to the relationship between the IGE and IGC specified in Equation (<reflink idref="bib2" id="ref46">2</reflink>), there are formal relationships among other pairs of mobility estimands that are worth mentioning. First, let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo>=</mo><mi>ln</mi><mspace width="0.2em" /><mi>Y</mi><mo>−</mo><msub><mi>β</mi><mn>0</mn></msub><mo>−</mo><msub><mi>β</mi><mn>1</mn></msub><mi>ln</mi><mspace width="0.2em" /><mi>X</mi><mspace width="0.25em" /></math> </ephtml> denote the population error term associated with equations (<reflink idref="bib1" id="ref47">1</reflink>) and (<reflink idref="bib3" id="ref48">3</reflink>), and assume that there are no children with zero income (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>></mo><mn>0</mn></math> </ephtml> for all children). Then, the IGE and IGEE are equal if and only if <emph>U</emph> is both mean and geometric mean independent of <emph>X</emph> (see [<reflink idref="bib70" id="ref49">70</reflink>]).[<reflink idref="bib8" id="ref50">8</reflink>] This necessary and sufficient condition for the equality of the IGE and IGEE is very demanding and is not very likely to be satisfied.[<reflink idref="bib9" id="ref51">9</reflink>]</p> <p>Second, assume that there are no children with zero income and that the conditional distribution of children's income given parental income (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi></math> </ephtml> ) follows a log-normal distribution for any value of parental income (equivalently, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mspace width="0.2em" /><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi></math> </ephtml> follows a normal distribution). In this case, the IGEE will be equal to, smaller than, or larger than the IGE if the variance of children's log income conditional on parental income is constant, monotonically decreasing, or monotonically increasing with parental income, respectively (see Online Appendix, Section A, in the online supplement). A constant conditional variance, that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Var</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi mathvariant="normal">Var</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> is equivalent to homoscedastic errors in Equations (<reflink idref="bib1" id="ref52">1</reflink>) and (<reflink idref="bib3" id="ref53">3</reflink>).[<reflink idref="bib10" id="ref54">10</reflink>] Although children's conditional income distributions are only imperfectly approximated by log-normal distributions, the analysis under this distributional assumption is heuristically illuminating and may be helpful in a variety of contexts.[<reflink idref="bib11" id="ref55">11</reflink>]</p> <p>Finally, if the incomes of parents and children are positive and follow a bivariate log-normal distribution, and the IGC is small, then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">RRS</mi></mrow><mo>≈</mo><mn>0.95</mn><mspace width="0.25em" /><mrow><mi mathvariant="normal">IGC</mi></mrow></math> </ephtml> ([<reflink idref="bib20" id="ref56">20</reflink>]:1561, fn. 7). This is of interest because, in some cases, it allows us to heuristically extend arguments that clearly apply to one of the two measures to the other, even if they do not directly apply to it (or apply less clearly). Relatedly, whereas it is easy to incorporate the IGC into theoretical models of intergenerational processes, doing the same with the RRS is very challenging. In this context, approximating the RRS with the IGC provides an appealing way to indirectly model the RRS, after making the needed distributional assumption. This assumption, however, is quite strong, and the bivariate log-normal distribution does not fit empirical joint distributions of income for parents and children very well. As a result, empirically, the approximation is likely to be rough.[<reflink idref="bib12" id="ref57">12</reflink>]</p> <p>Even if the four measures may provide different pictures of mobility at a point in time, shouldn't they vary in tandem over time, leading to similar qualitative conclusions about mobility trends? Not necessarily. Equation (<reflink idref="bib2" id="ref58">2</reflink>) is consistent with the IGE and the IGC changing in the same or different directions over time, and there is empirical evidence that these measures have moved in different directions in some countries (see, for instance, [<reflink idref="bib42" id="ref59">42</reflink>]). Moreover, the link between the RRS and the IGC discussed in the previous paragraph further suggests that the RRS and the IGE may also change in different directions.</p> <p>But what about the two elasticities? Since they differ only in the measure of central tendency used to summarize the information in children's conditional income distributions, don't they necessarily lead to similar qualitative conclusions about mobility trends? The answer is no. Figure 1 provides a "constructive proof." The figure is based on simulated data for two periods. The panels show the scatterplot of the data and the regression lines corresponding to Equations (<reflink idref="bib3" id="ref60">3</reflink>) and (<reflink idref="bib4" id="ref61">4</reflink>) in each period. I generated the data under the following assumptions: (a) the parental income distribution is log-normal and the same in both periods, (b) the children's conditional income distributions are log-normal, (c) the IGE is 0.50 in the first period and 0.35 in the second (while the intercept in Equation (<reflink idref="bib3" id="ref62">3</reflink>) is the same in both periods), and (d) the conditional variance of children's log income is constant across levels of parental income in the first period whereas it increases steadily with parental income in the second. This setup results in the IGEE increasing 30 percent across periods, from 0.50 to 0.65, despite the 30 percent fall in the value of the IGE.[<reflink idref="bib13" id="ref63">13</reflink>]</p> <p>Graph: Figure 1. Simulated data showing that the intergenerational elasticity (IGE) and the intergenerational elasticity of expected income (IGEE) may change over time in opposite directions when the relationship between parental income and the dispersion of children's income changes. In the first period (left panel), the variance of children's log income is constant across values of parental income. In the second period (right panel), the variance of children's log income increases by 0.6 for each 1 percent increase in parental income.</p> <p>I conclude my discussion of the four mobility measures by clarifying the relationship between the two elasticities and the other two measures, as misunderstandings of this relationship have underpinned the oft-made but unfounded normative claim that the latter are superior to the former. Based on Equation (<reflink idref="bib2" id="ref64">2</reflink>), it has frequently been argued that the IGC and RRS are preferable to the IGE as measures of mobility because the IGE is affected by changes in cross-sectional inequality across generations whereas the IGC and the RRS are not. (By implication, the IGEE should also be less preferred).[<reflink idref="bib14" id="ref65">14</reflink>] For example, [<reflink idref="bib6" id="ref66">6</reflink>] asserted the following:</p> <p>Although informative and interpretable, ... [the size of the IGE] depends on income dispersion in the two generations. If income inequality rises from one generation to the next, a larger coefficient is needed to account for the larger income differentials in the second generation. Arguably, the [IGC], which equals the elasticity multiplied by the ... [inequality ratio], may be preferable. ... The correlation is independent of the marginal distributions in the two generations and is therefore arguably more suitable for comparing mobility across countries, especially if the marginal distributions have changed in different ways across the countries ([<reflink idref="bib6" id="ref67">6</reflink>]:497).</p> <p>This argument, along with many similar arguments found in the literature (e.g., [<reflink idref="bib24" id="ref68">24</reflink>]:191 and 195; [<reflink idref="bib46" id="ref69">46</reflink>]: 186 and 188), which all rely on Equation (<reflink idref="bib2" id="ref70">2</reflink>), is flawed.</p> <p>The reason is easy to state: Equation (<reflink idref="bib2" id="ref71">2</reflink>) is a mathematical identity without any empirical content, that is, a tautology; therefore, there is no justification whatsoever for interpreting it as a directed causal relationship. In other words, we can write both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">IGE</mi></mrow><mo>≡</mo><mrow><mi mathvariant="normal">IGC</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">SD</mi></mrow><mo stretchy="false">(</mo><mi>ln</mi><mspace width=".1em" /><mi>Y</mi><mo stretchy="false">)</mo><mo>/</mo><mrow><mi mathvariant="normal">SD</mi></mrow><mo stretchy="false">(</mo><mi>ln</mi><mspace width=".1em" /><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">IGC</mi></mrow><mo>≡</mo><mrow><mi mathvariant="normal">IGE</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">SD</mi></mrow><mo stretchy="false">(</mo><mi>ln</mi><mspace width=".1em" /><mi>X</mi><mo stretchy="false">)</mo><mo>/</mo><mrow><mi mathvariant="normal">SD</mi></mrow><mo stretchy="false">(</mo><mi>ln</mi><mspace width=".1em" /><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> </ephtml> , but neither expression has any empirical content. There is no more reason to assume that the IGC and inequality in both generations are exogenous and the IGE is endogenous than to assume that the IGE and inequality are exogenous and the IGC is endogenous. In fact, all these quantities are almost certainly jointly determined by other structural parameters (see [<reflink idref="bib55" id="ref72">55</reflink>]: 90–91 for a closely related argument).</p> <hd id="AN0188424702-4">Mobility and Inequality of Opportunity</hd> <p>The study of economic mobility—in terms of both income and earnings—has been motivated, explicitly or implicitly, by the notion that (im) mobility rates provide information on how (un) equal opportunities are. In early studies, these two pairs of concepts were hardly distinguished from each other. For instance, in the introduction of one of their seminal articles on intergenerational mobility, Becker and Tomes wrote the following:</p> <p>The degree of regression toward or away from the mean in the achievements of children compared to those of their parents is a measure of the degree of equality of opportunity in a society. The purpose of this paper is to analyze the determinants of unequal opportunities, sometimes called "intergenerational mobility," or, as in the title of our paper, "the rise and fall of families." We use all these terms interchangeably ([<reflink idref="bib4" id="ref73">4</reflink>]:S3).</p> <p>Although not necessarily expressed so clearly, the conflation we see in this paragraph of a society's level of equality of opportunity with its level of intergenerational mobility—or, similarly, the conflation of its level of inequality of opportunity with its level of intergenerational persistence (or "degree of regression away from the mean")—was common in the early mobility literature.</p> <p>Over time, mobility scholars became increasingly aware of the need to avoid these conflations, and today they seldom appear in their writings. However, they have not been replaced by a carefully articulated understanding of the relationship between equality/inequality of opportunity and intergenerational mobility/persistence. Instead, these conflations have been replaced by the view that these pairs of concepts are (a) irreducibly different but, in some sense left mostly unspecified, conceptually related, and (b) mutually informative when empirically measured (see, e.g., [<reflink idref="bib23" id="ref74">23</reflink>]).</p> <p>A similar view can be found in the burgeoning empirical literature on inequality of opportunity (see, e.g., [<reflink idref="bib11" id="ref75">11</reflink>]). Although the inequality of opportunity field has made impressive theoretical and empirical progress, especially over the last 20 years, specifying with precision the conceptual relationship between mobility and inequality of opportunity has been almost entirely absent from its agenda (more on the one exception of which I am aware later).</p> <p>In this section, I contribute to both the mobility and inequality of opportunity literatures by specifying the relationships between the four income mobility measures I focus on in this article and the concepts and measures developed in the literature on inequality of opportunity for income. I also show that a previous attempt to do the same for the intergenerational elasticity ([<reflink idref="bib47" id="ref76">47</reflink>], [<reflink idref="bib48" id="ref77">48</reflink>]) did not succeed because it misinterpreted the underlying population regression function as a structural model. First, I briefly discuss the theoretical understanding and the measures of inequality of opportunity developed in the literature on this topic. Next, I present [<reflink idref="bib47" id="ref78">47</reflink>], [<reflink idref="bib48" id="ref79">48</reflink>]) account for the intergenerational elasticity and show why it is invalid. I then formally determine the relationships between the mobility and inequality of opportunity measures, and discuss the methodological implications of my characterizations. I conclude the section with a brief recapitulation.</p> <hd id="AN0188424702-5">Luck Egalitarianism and Inequality of Opportunity for Income</hd> <p>The empirical literature on inequality of opportunity for income has mostly adopted the "luck egalitarian" understanding of inequality of opportunity (e.g., [<reflink idref="bib2" id="ref80">2</reflink>]; [<reflink idref="bib21" id="ref81">21</reflink>]; [<reflink idref="bib27" id="ref82">27</reflink>], [<reflink idref="bib28" id="ref83">28</reflink>]; [<reflink idref="bib75" id="ref84">75</reflink>]) as its philosophical foundation. The various luck egalitarian theories of justice available (a) put individual responsibility at the center of the normative assessment of inequality, and (b) stress the ethical imperative of counteracting the distributive effects of luck on people's income, health status, educational attainment and other outcomes that have a large impact on their chances of achieving their life plans. As luck is often deemed the opposite of what individuals are responsible for (e.g., [<reflink idref="bib22" id="ref85">22</reflink>]:442), luck egalitarianism has also been referred to as "responsibility-sensitive egalitarianism."[<reflink idref="bib15" id="ref86">15</reflink>]</p> <p>Luck egalitarians argue that income and other crucial outcomes—generically referred to as "advantages"—are determined by factors that are beyond individuals' responsibility, commonly called "circumstances" (e.g., gender, race, parental income and education), and by factors for which individuals should be held responsible, typically referred to as "effort" (e.g., number of hours worked, educational attainment, occupational choice). Inequalities arising from differences in circumstances are considered ethically unacceptable or unfair. By contrast, inequalities resulting from differences in effort are seen as just—provided that the differences in effort cannot be traced back to differences in circumstances.[<reflink idref="bib16" id="ref87">16</reflink>] Thus, for any relevant outcome, the luck-egalitarian normative ideal is an outcome distribution that satisfies two principles: the <emph>reward principle</emph>, that is, the principle that efforts should be properly rewarded; and the <emph>compensation principle</emph>, that is, the principle that the effects of circumstances should be fully compensated for. In this ideal context, inequality is entirely due to differences in effort not accounted for by circumstances.</p> <p>The empirical literature has focused on the compensation principle, which has received two main interpretations. In the <emph>ex-post perspective</emph>, the principle requires equalizing outcomes among individuals exerting the same level of effort but subject to different circumstances. In the <emph>ex-ante perspective</emph>, it requires equalizing people's opportunity sets. Inequality of opportunity has typically been measured by determining how far a society is from fully satisfying the compensation principle. This has involved (a) measuring and suitably aggregating into one index of inequality of opportunity the inequalities that exist across individuals with the same effort levels (under the ex-post perspective), or (b) measuring the inequality in the value of the opportunity sets of people with different circumstances (under the ex-ante perspective).</p> <p>Most studies in the empirical literature on inequality of opportunity for income have adopted a notion of inequality of opportunity based on a specific variant of the ex-ante perspective. This variant posits the following: (a) <emph>circumstances</emph> are all the factors that account for people's incomes and are beyond their control (and for which, therefore, they cannot be held responsible); (b) <emph>types</emph> are groups of individuals who share the same circumstances; (c) the individuals belonging to a type share a common <emph>opportunity set</emph>, i.e., a set of income prospects; (d) the <emph>value</emph> of each opportunity set is measured by the mean of the realized incomes of those belonging to the type; (e) <emph>absolute inequality of opportunity</emph> is the inequality in opportunity-set values across individuals; and (f) <emph>relative inequality of opportunity</emph> is the share of a society's overall inequality that is deemed unfair under the luck-egalitarian perspective, that is, absolute inequality of opportunity as a share of overall inequality. Using this approach, the literature has produced estimates for many countries and—similar to the economic mobility literature—has made cross-country comparisons a central goal (e.g., [<reflink idref="bib6" id="ref88">6</reflink>]; [<reflink idref="bib11" id="ref89">11</reflink>]; [<reflink idref="bib16" id="ref90">16</reflink>]; [<reflink idref="bib29" id="ref91">29</reflink>]; [<reflink idref="bib39" id="ref92">39</reflink>]; [<reflink idref="bib52" id="ref93">52</reflink>]; [<reflink idref="bib71" id="ref94">71</reflink>]; [<reflink idref="bib86" id="ref95">86</reflink>]).</p> <p>Four additional observations are in order, as they will be relevant to my analysis of the relationships between the measures of mobility and inequality of opportunity. First, the approach of indexing the value of an individual's opportunity set by the mean income of all individuals with the same circumstances was first proposed in the theoretical literature by Van de gaer (1993) and has been widely adopted in the empirical literature. Nevertheless, it does not seem to have any strong theoretical justification; arguably, it would be equally legitimate to use other measures of central tendency (e.g., the median) with the same purpose.</p> <p>Second, although any inequality measure may be used to measure absolute inequality of opportunity and overall inequality (and therefore relative inequality of opportunity), until recently the preferred inequality measure among scholars in this field was the mean logarithmic deviation due to its attractive theoretical properties (see, e.g., [<reflink idref="bib29" id="ref96">29</reflink>]).[<reflink idref="bib17" id="ref97">17</reflink>] This inequality measure, however, is unattractive for both methodological and pragmatic reasons ([<reflink idref="bib9" id="ref98">9</reflink>]; [<reflink idref="bib64" id="ref99">64</reflink>]). In recent times, these considerations have led inequality of opportunity scholars to move away from the mean logarithmic deviation, typically in favor of the Gini coefficient (e.g., [<reflink idref="bib10" id="ref100">10</reflink>]).</p> <p>Third, scholars of inequality of opportunity have frequently conducted cross-country comparisons of both absolute and relative inequality of opportunity—even though only the former is relevant for comparative normative assessments ([<reflink idref="bib64" id="ref101">64</reflink>]). An example will help clarify this point. Suppose we want to compare how countries A and B are performing in terms of inequality of opportunity for income, and we know that the Gini-based measures of absolute and relative inequality of opportune for these countries are those presented in Table 1. The distribution of income opportunities (i.e., the distribution of opportunity-set values) in country A is much more egalitarian than in country B (the Gini for absolute inequality of opportunity in the latter country is twice as large). According to the theory of distribute justice motivating the analyses conducted in the inequality of opportunity literature, this inequality, the ethically unacceptable or unfair inequality, is the one that needs to be minimized. Clearly, country A is doing substantially better (i.e., twice as well) than country B in this regard. The fact that relative inequality of opportunity—that is, the share of inequality that is unfair—is substantially larger in country A than in country B (50 versus 33.3 percent) is literally irrelevant to this assessment.[<reflink idref="bib18" id="ref102">18</reflink>]</p> <p>Table 1. Income Inequality and Inequality of Opportunity for Income in Two Hypothetical Countries (Gini Coefficient).</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="char" char="." /></colgroup><thead><tr><th align="left" /><th align="left">Country A</th><th align="left">Country B</th></tr></thead><tbody><tr><td>Income inequality</td><td>0.2</td><td>0.6</td></tr><tr><td>Absolute inequality of opportunity for income</td><td>0.1</td><td>0.2</td></tr><tr><td>Relative inequality of opportunity for income (%)</td><td>50</td><td>33.3</td></tr></tbody></table> </ephtml> </p> <p>Lastly, the all-encompassing nature of the theoretical notion of circumstances—which includes <emph>all</emph> factors beyond people's control—presents a fundamental methodological challenge. Specifically, this notion implies that, in any empirical study, absolute inequality of opportunity is always measured with respect to an incomplete set of circumstances (a predicament sometimes referred to as "the partial observability of circumstances"). As a result, any estimate of absolute inequality of opportunity will be a lower-bound estimate, provided that (a) the type-specific means are estimated nonparametrically, (b) the inequality measure is Lorenz-consistent, and (c) the sample is sufficiently large (see [<reflink idref="bib51" id="ref103">51</reflink>]: Prop. 1). Although this is not well appreciated in the literature, the same does not necessarily hold if the type-specific means are estimated parametrically.[<reflink idref="bib19" id="ref104">19</reflink>] Nonetheless, it seems reasonable to assume that the results will still be lower-bound estimates if the parametric models include only a few circumstances and are not severely misspecified. Although comparing lower-bound estimates of absolute inequality of opportunity across countries is very challenging ([<reflink idref="bib64" id="ref105">64</reflink>]), this issue has largely been overlooked in the literature.</p> <hd id="AN0188424702-6">The IGE and Inequality of Opportunity for Income: An Invalid Interpretation</hd> <p>[<reflink idref="bib47" id="ref106">47</reflink>], [<reflink idref="bib48" id="ref107">48</reflink>]) claimed that they were able to establish very simple relationships between the IGE and measures of absolute and relative inequality of opportunity. I show next that their analysis is invalid. Their main result for what is relevant here can be derived by starting with my Equation (<reflink idref="bib1" id="ref108">1</reflink>) for the IGE and doing some straightforward algebraic work (see Online Appendix, Section B, in the online supplement). This result is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msubsup><mi>β</mi><mn>0</mn><mo>′</mo></msubsup><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mspace width=".1em" /><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>β</mi><mn>0</mn><mrow><mi mathvariant="normal">′</mi></mrow></msubsup><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mi>ln</mi><mspace width="0.2em" /><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>−</mo><mi>ln</mi><mspace width="0.2em" /><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mo>.</mo><mo stretchy="false">)</mo></math> </ephtml> is the mean logarithmic deviation operator. Lefranc et al.'s interpretation of Equation (<reflink idref="bib6" id="ref109">6</reflink>) is the following (I have slightly altered their notation):</p> <p>Income inequality among descendants measured by the [mean logarithmic deviation] can be written as a linear affine function of inequalities of circumstances measured by the [mean logarithmic deviation] among the fathers. The constant <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><msubsup><mi>β</mi><mn>0</mn><mrow><mi mathvariant="normal">′</mi></mrow></msubsup></math> </ephtml> can be interpreted as residual inequality if there were no inequality of circumstances, namely, any parents came from the same group.</p> <p>The linear regression model, joint with the [mean logarithmic deviation] to measure inequality, leads to a quite simple expression of inequality of opportunity, that is: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow><mrow><mi mathvariant="normal">oppt</mi></mrow></mrow></msub><mo>=</mo><msub><mi>β</mi><mn>1</mn></msub><mspace width=".1em" /><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>.</mo></math> </ephtml></p> <p>It is the part of inequality that would remain if the only disparity factor among descendants were their father's earnings. It is the product of the intergenerational earnings elasticity and inequality of circumstances. The share of inequality of opportunity in inequality of outcomes is given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub><mspace width=".1em" /><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>/</mo><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math> </ephtml> ([<reflink idref="bib48" id="ref110">48</reflink>]:149).</p> <p>This construal of Equation (<reflink idref="bib6" id="ref111">6</reflink>) is unwarranted. The term <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><msubsup><mi>β</mi><mn>0</mn><mrow><mi mathvariant="normal">′</mi></mrow></msubsup></math> </ephtml> cannot receive a counterfactual interpretation. It is the difference between the actual values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub><mspace width=".1em" /><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> and it is not constant across values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub><mspace width=".1em" /><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> .[<reflink idref="bib20" id="ref112">20</reflink>] It follows that it cannot be interpreted as "residual inequality if there were no inequality of circumstances." Moreover, if all parents had the same income ("any parents came from the same group"), it would be the case that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Var</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>0</mn></msub><mspace width="0.25em" /></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>β</mi><mn>0</mn><mrow><mi mathvariant="normal">′</mi></mrow></msubsup></math> </ephtml> would all be undefined. Therefore, the proposed justification for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub><mspace width=".1em" /><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> as a measure of absolute inequality of opportunity has no basis. Presumably, this term measures absolute inequality of opportunity because it is "the part of inequality that would remain if the only disparity factor among descendants were their father earnings," that is, if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>β</mi><mn>0</mn><mrow><mi mathvariant="normal">′</mi></mrow></msubsup></math> </ephtml> were zero. This counterfactual-based justification assumes that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub></math> </ephtml> is a structural parameter and that Equation (<reflink idref="bib6" id="ref113">6</reflink>) is something else than an accounting identity, which they clearly are not.</p> <hd id="AN0188424702-7">Relationships Between Mobility and Inequality of Opportunity Measures</hd> <p>In this subsection, I formally specify the relationships between the four mobility measures I focus on in this article and the measures of absolute and relative inequality of opportunity developed in the luck egalitarian empirical literature on opportunity. I use <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>q</mi><mi>a</mi></msubsup><mo stretchy="false">(</mo><mo>.</mo><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>q</mi><mi>r</mi></msubsup><mo stretchy="false">(</mo><mo>.</mo><mo stretchy="false">)</mo></math> </ephtml> to denote measures of absolute and relative inequality of opportunity, respectively, computed with the inequality index <emph>q</emph> (e.g., Gini coefficient, mean logarithmic deviation). The argument I left unspecified in each measure is the income variable of interest. The use of this notation implies that opportunity sets are indexed by within-type mean incomes, a stipulation whose relevance will become clear below. Figure 2summarizes the results of my analyses.</p> <p> <emph>Intergenerational elasticity of expected income (IGEE)</emph>. I start with Equation (<reflink idref="bib4" id="ref114">4</reflink>), re-express the equation in terms of random variables instead of values, and take standard deviation with respect to the distribution of parental income on both sides. This yields:(<reflink idref="bib7" id="ref115">7</reflink>)</p> <p>Graph</p> <p>The standard deviation of logarithms is often used to measure income inequality. The left-hand side of Equation (<reflink idref="bib7" id="ref116">7</reflink>) can then be interpreted as the inequality in within-type mean incomes across individuals, with the types defined by the values of parental income. In other words, the left-hand side of Equation (<reflink idref="bib7" id="ref117">7</reflink>) measures absolute inequality of opportunity for income when the only circumstance is parental income, the inequality measure is the standard deviation of logarithms, and the value of opportunity sets is measured by the mean income of people with the same parental income (as is standard in the literature). Therefore, <emph>absolute inequality of opportunity for income is simply the product of the IGEE and inequality in parental income:</emph><ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mrow><mrow><mi mathvariant="normal">SDL</mi></mrow></mrow><mi>a</mi></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi mathvariant="normal">IGEE</mi></mrow><mo>*</mo><mrow><mi mathvariant="normal">SDL</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="0.25em" /></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">SDL</mi></mrow><mo stretchy="false">(</mo><mo>.</mo><mo stretchy="false">)</mo></math> </ephtml> is the standard deviation of logarithms operator. Dividing the above expression by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">SDL</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math> </ephtml> in both sides, we have: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mrow><mrow><mi mathvariant="normal">SDL</mi></mrow></mrow><mi>r</mi></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi mathvariant="normal">IGEE</mi></mrow><mo>*</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">SDL</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">SDL</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>that is,<emph> relative inequality of opportunity for income is the product of the IGEE and the ratio between the income inequality among parents and children</emph>.</p> <p> <emph>Intergenerational elasticity (IGE) and intergenerational correlation (IGC)</emph>. I now begin with Equation (<reflink idref="bib3" id="ref118">3</reflink>) and proceed in a similar manner as above. This yields:(<reflink idref="bib8" id="ref119">8</reflink>)</p> <p>Graph</p> <p>The left-hand side of Equation (<reflink idref="bib8" id="ref120">8</reflink>) can be interpreted as the inequality in within-type geometric mean incomes across individuals.[<reflink idref="bib21" id="ref121">21</reflink>] Here the types are again defined by the values of parental income but the value of each opportunity set is the geometric mean rather than the mean income of people with the same parental income. It follows that <emph>absolute inequality of opportunity for income is the product of the IGE and inequality in parental income</emph>, that is, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>φ</mi><mrow><mrow><mi mathvariant="normal">SDL</mi></mrow></mrow><mi>a</mi></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi mathvariant="normal">IGE</mi></mrow><mo>*</mo><mrow><mi mathvariant="normal">SDL</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where I have substituted <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi></math> </ephtml> for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> </ephtml> on the left-hand side to reflect the fact that opportunity sets are indexed by the geometric mean of income.</p> <p>Quite interestingly, after dividing both sides of the last expression by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">SDL</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math> </ephtml> , we have that here <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>φ</mi><mrow><mrow><mi mathvariant="normal">SDL</mi></mrow></mrow><mi>r</mi></msubsup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi mathvariant="normal">IGE</mi></mrow><mo>*</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">SDL</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">SDL</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mrow><mi mathvariant="normal">IGC</mi></mrow><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>that is, in this context, <emph>relative inequality of opportunity for income is measured by the IGC.</emph> Although the IGE and IGC have both been used to measure mobility under the assumption that they provide broadly similar information about inequality of opportunity, my analysis shows that they capture distinctly different aspects of this inequality. The IGE offers an estimate of the share of parental inequality transmitted from parents to children and, when multiplied by parental inequality, an estimate of how much inequality of opportunity exists (as does the IGEE). By contrast, the IGC offers an estimate of how much of the inequality that exists is unfair.</p> <p> <emph>Rank-rank slope (RRS)</emph>. A similar analysis may be carried out for the rank-rank slope (RRS). Beginning now with Equation (<reflink idref="bib5" id="ref122">5</reflink>):(<reflink idref="bib9" id="ref123">9</reflink>)</p> <p>Graph</p> <p>Because income rank variables are insensitive to changes in the units used to measure the underlying monetary income, the standard deviation is a fine measure of inequality in this context. The left-hand side of Equation (<reflink idref="bib9" id="ref124">9</reflink>) can then be interpreted as the inequality in within-type mean income ranks across individuals, with the types defined by the values of parental income rank. In other words, the left-hand side of Equation (<reflink idref="bib9" id="ref125">9</reflink>) is the absolute inequality of opportunity for income rank when the only circumstance is parental income rank, the inequality measure is the standard deviation, and the value of opportunity sets is measured by the mean income rank of people with the same parental income rank. Therefore, <emph>absolute inequality of opportunity for income rank is simply the product of the RRS and inequality in parental income rank</emph>, i.e., <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mrow><mi>S</mi><mi>D</mi></mrow><mi>a</mi></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mi>R</mi><mi>S</mi><mo>*</mo><mi>S</mi><mi>D</mi><mo stretchy="false">(</mo><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mi>R</mi><mi>S</mi><mo>*</mo><mn>0.29</mn><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where SD <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>0.29</mn></math> </ephtml> because income rank variables follow a uniform distribution. After dividing both sides of the last expression by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>D</mi><mo stretchy="false">(</mo><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> we also have that <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mrow><mi>S</mi><mi>D</mi></mrow><mi>r</mi></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mi>R</mi><mi>S</mi><mo>*</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>S</mi><mi>D</mi><mo stretchy="false">(</mo><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>S</mi><mi>D</mi><mo stretchy="false">(</mo><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mi>R</mi><mi>R</mi><mi>S</mi><mo>*</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mn>0.29</mn></mrow><mrow><mn>0.29</mn></mrow></mfrac></mrow><mo>=</mo><mi>R</mi><mi>R</mi><mi>S</mi><mo>,</mo></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>that is, in this context <emph>relative inequality of opportunity for income rank is measured by the RRS</emph>.</p> <hd id="AN0188424702-8">Some Methodological Implications</hd> <p>Beyond formally specifying the relationships between mobility and inequality of opportunity measures, my analysis has important methodological implications. Given that the only circumstance considered is parental monetary income or parental income rank, and that, however imperfect, the linear specifications upon which the mobility measures rely are acceptable first-order approximations, we should expect estimates of absolute and relative inequality of opportunity based on the mobility measures to be lower-bound estimates. Therefore, interpreting IGEE or IGE estimates as informative for the relative performance, or at least the ranking, of countries in terms of absolute inequality of opportunity would require making two assumptions.[<reflink idref="bib22" id="ref126">22</reflink>] First, it would need to be assumed that absolute inequality of opportunity estimated with parental income as the only circumstance provides a first-order approximation to true absolute inequality of opportunity. Or, alternatively, that the ratio between the two is similar across countries. Both assumptions are very strong.[<reflink idref="bib23" id="ref127">23</reflink>] Second, it would need to be assumed that there is a strong correlation between income inequality among parents and the IGEE or IGE. If this is not the case, relying on the elasticities alone to conduct cross-country comparisons would likely lead to misleading conclusions, even regarding rankings. (This issue can be avoided by multiplying the IGEE or IGE by parental income inequality and comparing countries along this measure rather than the elasticity itself.)</p> <p>In addition, interpreting the IGE as informative requires switching from mean income to the geometric mean of income to measure the value of opportunity sets. Although this does not appear to be theoretically problematic—since, as I noted earlier, there is no strong justification for indexing opportunity sets by within-type income means rather than other measures of central tendency—relying on geometric means (i.e., estimating the IGE) is often methodologically problematic given the available data (see [<reflink idref="bib62" id="ref128">62</reflink>] for details).</p> <p>Of course, the RRS is not informative about the relative performance (or even the ranking) of countries in terms of absolute inequality of opportunity for income. However, it may provide useful insights into their relative performance regarding absolute inequality of opportunity for income rank. Moreover, because parental income rank inequality is the same across countries, interpreting the RRS as informative only requires making one of the following assumptions: (a) absolute inequality of opportunity for income rank, with parental rank as the only circumstance, provides a first-order approximation to true absolute inequality of opportunity for income rank, or (b) the ratio between the two is similar across countries. These are also strong assumptions.</p> <p>The characterization of the IGC as a measure of relative rather than absolute inequality of opportunity for income is highly consequential. It indicates that the IGC is simply not useful for assessing the relative performance of countries in providing equal income opportunities to their children. This, in turn, means that there is no inequality-of-opportunity justification for using the IGC to compare countries (at least from a luck egalitarian perspective). As I argued earlier, there is no normative justification for comparing countries in terms of the share of overall inequality that is unfair—what matters is the amount of unfair inequality itself. If interest lies in the relative performance of countries in terms of absolute inequality of opportunity for income rank, the RRS should be estimated instead. There is no reason for relying on an approximation, which may be quite imprecise, when the estimand of interest can be directly estimated. Additionally, computing the IGC is very problematic when the income measures available include a non-negligible mass of zeros, which is often the case.</p> <hd id="AN0188424702-9">Recapitulation</hd> <p>This section has shown that although a previous attempt ([<reflink idref="bib47" id="ref129">47</reflink>], [<reflink idref="bib48" id="ref130">48</reflink>]) to formally characterize the relationship between the IGE and the measures advanced in the inequality of opportunity literature is invalid, not only the IGE but also the IGEE, IGC and RRS are straightforwardly and precisely related to those measures (see Figure 2). The formal relationships between the measures of mobility and inequality of opportunity become highly apparent once the relevant inequality indices (standard deviation or standard deviation of logarithms), circumstances (parental income or parental income rank), outcomes (income or income rank), and approaches for measuring the value of opportunity sets (by the within-type mean or geometric mean of the outcome of interest) are identified. Beyond their intrinsic interest, the formal characterizations I provided here have important implications for the interpretation of cross-country, cross-period, and cross-cohort comparisons, based on the four mobility measures, in terms of inequality of opportunity.</p> <p>Graph: Figure 2. Relationships between measures of intergenerational income mobility and inequality of opportunity for income. The four mobility measures (IGEE, IGE, IGC and RRS) are precisely related to measures of inequality of opportunity that vary in terms of the outcomes of interest (monetary income or income rank), the circumstances they consider (parental income or parental income rank), the inequality index on which they rely (standard deviation or standard deviation of logarithms), or the way they measure the value of opportunity sets (by the within-type mean or geometric mean of the outcome of interest).</p> <hd id="AN0188424702-10">"Person-Weighted" Versus "Dollar-Weighted": A Flawed Characterization of the IGE and IGEE</hd> <p>As I pointed out earlier, the IGE has been misinterpreted ([<reflink idref="bib56" id="ref131">56</reflink>]). While it has been assumed to refer to the expectation of the children's earnings or income conditional on their parents' income, Equation (<reflink idref="bib3" id="ref132">3</reflink>) clearly shows that it actually pertains to the conditional geometric mean of children's earnings or income. If this were the only issue with the IGE, mobility scholars might perhaps address it by simply changing the way they interpret their IGE estimates. Unfortunately, the unwitting reliance on the geometric mean also generates serious methodological problems, which do not have easy solutions ([<reflink idref="bib62" id="ref133">62</reflink>]). Both the conceptual and methodological problems, however, can be solved at once by replacing the IGE (the de facto estimated elasticity) with the IGEE (the elasticity that mobility scholars thought they were estimating) as the workhorse intergenerational elasticity. As I also noted earlier, [<reflink idref="bib62" id="ref134">62</reflink>] have called for such a replacement.</p> <p>Although the case for this replacement seems strong, [<reflink idref="bib20" id="ref135">20</reflink>] offered a contrasting characterization of the two elasticities—the IGE and the IGEE—that raises doubts about its desirability.[<reflink idref="bib24" id="ref136">24</reflink>] They argued that while the IGE is a "person-weighted" elasticity that weights all individuals equally, the IGEE is a "dollar-weighted" elasticity that weights individuals in proportion to their income. This suggests that the IGEE is conceptually inadequate given the purposes pursued by mobility scholars when using the IGE.</p> <p>The main goal of this section is to show that the contrasting characterization of the two elasticities advanced by [<reflink idref="bib20" id="ref137">20</reflink>] is the joint result of a category mistake—equating quantile-specific elasticities to person-specific elasticities—and of misconstruing the nature of the IGE and the epistemic goal it has been meant to serve. I start by presenting their characterization and explaining its connection to the public finance literature, where the distinction between person-weighted and income-weighted elasticities originated. I then elaborate on my criticism and conclude with a brief recapitulation.</p> <hd id="AN0188424702-11">The IGE and IGEE as Weighted Behavioral Elasticities</hd> <p>The distinction between person-weighted and income-weighted elasticities is not new. It has played an important role in the field of public finance, where it has been shown that the policy-relevant elasticity of taxable income with respect to tax rates is the income-weighted elasticity (see, e.g., [<reflink idref="bib77" id="ref138">77</reflink>]). The following example illustrates well the difference between the two types of elasticities and their role in public finance:</p> <p>[A]n income-weighted ETI [elasticity of taxable income] should reflect the percent change in total taxable income (associated with a 1% increase in the NTR [net-of-tax rate])—instead of the average of individual percent changes. For example, consider two taxpayers: Person 1 has income of $10,000, and person 2 has income of $1 million. In response to a 1% decrease in the NTR, suppose that person 1 reduces his income (by 0.2%) to $9,980 and that person 2's income falls (by 1%) to $990,000. In this instance, the unweighted ETI [i.e., the person-weighted measure] equals 0.60 ..., whereas the income-weighed measure is nearly two-thirds larger, at 0.99... ([<reflink idref="bib33" id="ref139">33</reflink>]:419).[<reflink idref="bib25" id="ref140">25</reflink>]</p> <p>Chetty et al.'s (2014) comparative characterization of the IGE and IGEE assumes that the distinction between person-weighted and income-weighed elasticities is also relevant in the intergenerational mobility field. Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>,</mo><mi>τ</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> denote the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mi>t</mi><mi>h</mi></mrow></msup></math> </ephtml> quantile of the conditional distribution of the child's long-run income when long-run parental income equals <emph>x</emph>. [<reflink idref="bib20" id="ref141">20</reflink>] showed that, <emph>at any value of parental income x</emph>, the IGE and IGEE can be written as follows: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">IGE</mi></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>ln</mi><mspace width="0.2em" /><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>ln</mi><mspace width="0.2em" /><mi>x</mi></mrow></mfrac></mrow><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mn>1</mn></munderover><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><msub><mi>Q</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>,</mo><mi>τ</mi></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /></mrow><mrow><mi>d</mi><mi>ln</mi><mspace width="0.2em" /><mi>x</mi></mrow></mfrac></mrow><mspace width=".1em" /><mi>d</mi><mi>τ</mi></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">IGEE</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mi>d</mi><mspace width=".1em" /><mi>ln</mi><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mspace width="0.25em" /></mrow><mrow><mi>d</mi><mi>ln</mi><mspace width="0.2em" /><mi>x</mi></mrow></mfrac></mrow><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mn>1</mn></munderover><mspace width="0.2em" /><mrow><mfrac><mrow><msub><mi>Q</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>,</mo><mi>τ</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mrow><mfrac><mrow><mi>d</mi><mi>ln</mi><mspace width="0.2em" /><msub><mi>Q</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>,</mo><mi>τ</mi></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /></mrow><mrow><mi>d</mi><mi>ln</mi><mspace width="0.2em" /><mi>x</mi></mrow></mfrac></mrow><mspace width="0.25em" /><mi>d</mi><mi>τ</mi><mo>.</mo></math> </ephtml> </p> <p>Graph</p> <p>From the representations in Equations (<reflink idref="bib10" id="ref142">10</reflink>) and (<reflink idref="bib11" id="ref143">11</reflink>), [<reflink idref="bib20" id="ref144">20</reflink>] argued that the IGE "can be interpreted as the average elasticity of child income with respect to parent income in a model with heterogeneous elasticities, while... [the IGEE] is a dollar-weighted (i.e., child-income-weighted) average of the same elasticities" [one typo corrected]. As a result, they also claimed that the IGE "weights all individuals with positive income equally" and referred to it as a "person-weighted" elasticity, while they described the IGEE as a "dollar-weighted" elasticity where the weights "are an increasing function of the child's income" ([<reflink idref="bib20" id="ref145">20</reflink>]:1574 and their Online Appendix C).[<reflink idref="bib26" id="ref146">26</reflink>]</p> <p>The foregoing entails that the intergenerational elasticity on which mobility scholars have been interested (the IGE) is an average of person-level elasticities while the IGEE is something else, which in turn suggests that (a) the IGEE is conceptually inadequate given the purposes pursued by mobility scholars when using the IGE, and (b) we may be better served by redoubling our efforts to solve the methodological problems affecting the IGE than by replacing it with the IGEE, as [<reflink idref="bib62" id="ref147">62</reflink>] have proposed.</p> <hd id="AN0188424702-12">Criticism</hd> <p>Let's begin by considering what Chetty et al.'s ([<reflink idref="bib20" id="ref148">20</reflink>]) characterization implies about the nature of the IGE and the epistemic goal it is supposed to serve. As a person's long-run parental income is a fixed attribute of that person, a person-specific intergenerational elasticity must, by necessity, be a causal parameter. This is the case because such elasticity cannot simply reflect an empirical association across people or across times "within" a particular individual.[<reflink idref="bib27" id="ref149">27</reflink>] Therefore, although Chetty et al. did not appear to recognize this, their characterization implies that the IGE is also a causal parameter—an average of marginal proportional effects resulting from marginal proportional treatments.</p> <p>This is clearly inconsistent with how the IGE has been understood in the mobility literature. Indeed, because of the correlation between parental income and a large number of causally relevant factors not included in specifications like that of Equation (<reflink idref="bib1" id="ref150">1</reflink>)—which can be thought of as omitted variables that have been very purposefully omitted—mobility scholars have not interpreted the IGE as a parameter measuring the causal effect of parental income. Rather, they have conceived of it as a descriptive measure—comparable in nature to, for instance, the Gini coefficient—that provides information on how much better the children's economic outcomes tend to be as their parental economic status increases. As such, it can be interpreted as an imperfect index of the advantages conferred by the circumstances of birth. For instance: (a) mobility scholars often refer to the IGE as a "descriptive statistic" (e.g., [<reflink idref="bib1" id="ref151">1</reflink>]:146; [<reflink idref="bib38" id="ref152">38</reflink>]:26), or to Equation (<reflink idref="bib1" id="ref153">1</reflink>) as capturing "a simple statistical relationship" (e.g., [<reflink idref="bib85" id="ref154">85</reflink>]:2), (b) [<reflink idref="bib5" id="ref155">5</reflink>]: Secs. 4 and 6) describe the IGE as providing information on the "association" between parental and children's income and point out that "the causal effect of parental income is conceptually different and much more difficult to estimate," and (c) the key premise of [<reflink idref="bib7" id="ref156">7</reflink>] literature review in the <emph>Handbook of Labor Economics</emph> is that the IGE does not measure the causal effect of parental income or earnings. All this is, of course, compatible with the IGE being a function of various structural parameters (see, e.g., [<reflink idref="bib84" id="ref157">84</reflink>]).</p> <p>It follows that Chetty et al.'s ([<reflink idref="bib20" id="ref158">20</reflink>]) characterization of the IGE, which would make it a causal parameter, (a) is poorly aligned with the epistemic goals that that IGE has been meant to serve, and (b) implies that each quantile-specific elasticity can itself be legitimately interpreted as a causal parameter, despite the omission of many variables known to be causally relevant.</p> <p>The latter fact suggests that something more fundamental may be amiss in Chetty et al.'s analysis, and indeed, this is the case. Equation (<reflink idref="bib10" id="ref159">10</reflink>) does show that the IGE is equivalent to a simple average of all quantile-specific IGEs (at each value of parental income). But quantiles are not people or, more precisely, quantile-specific IGEs are not the same as person-specific IGEs. As an individual would not necessarily remain—and most likely would not remain—in the same quantile of the conditional income distribution if their parental income were different (see [<reflink idref="bib80" id="ref160">80</reflink>]:1103), the proposed interpretation of Equation (<reflink idref="bib10" id="ref161">10</reflink>) is simply incorrect. In fact, results by [<reflink idref="bib80" id="ref162">80</reflink>]; especially Lemma 1) on what is identified by the quantile partial derivative show that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><msub><mi>Q</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mspace width="0.25em" /><mi>τ</mi><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi></math> </ephtml> at each quantile <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi></math> </ephtml> is a remarkably complex function of person-level elasticities, with the person-level elasticities being far from equally weighted. Therefore, the characterization of the IGE as a <emph>person</emph>-weighted elasticity, or as an IGE that weights <emph>individuals</emph> equally, is fundamentally flawed.</p> <p>It may still be concerning that the IGEE is equal to a weighted average of all quantile-specific elasticities, with the weights increasing with the quantile. Does this mean that there is something to the notion that the IGEE is a "dollar-weighted" elasticity while the IGE is not? Not at all. Equations (<reflink idref="bib10" id="ref163">10</reflink>) and (<reflink idref="bib11" id="ref164">11</reflink>) can be rewritten as follows: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">IGE</mi></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mi>ln</mi><mspace width="0.2em" /><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>ln</mi><mspace width="0.2em" /><mi>x</mi></mrow></mfrac></mrow><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mn>1</mn></munderover><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mi>x</mi><mrow><msub><mi>Q</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>,</mo><mi>τ</mi></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /></mrow></mfrac></mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>d</mi><msub><mi>Q</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>,</mo><mi>τ</mi></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mrow><mi>d</mi><mi>τ</mi><mo>.</mo></mstyle></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">IGEE</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mspace width="0.25em" /></mrow><mrow><mi>d</mi><mi>ln</mi><mspace width="0.2em" /><mi>x</mi></mrow></mfrac></mrow><mo>=</mo><mrow><mfrac><mi>x</mi><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><munderover><mo>∫</mo><mn>0</mn><mn>1</mn></munderover><mspace width="0.2em" /><mrow><mfrac><mrow><mi>d</mi><msub><mi>Q</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>,</mo><mi>τ</mi></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mrow><mspace width="0.25em" /><mi>d</mi><mi>τ</mi><mo>.</mo></math> </ephtml> </p> <p>Graph</p> <p>In the representation provided by Equation (<reflink idref="bib13" id="ref165">13</reflink>), the IGEE is equal to a simple average of all quantile-specific derivatives, multiplied by a scaling or standardizing factor. By comparing Equations (<reflink idref="bib12" id="ref166">12</reflink>) and (<reflink idref="bib13" id="ref167">13</reflink>), one could therefore conclude—following Chetty et al.'s logic—that the IGE is a dollar-weighted average of derivatives, with the weights being a decreasing function of children's income, while the IGEE is an average of those same derivatives, where all individuals receive the same weight.[<reflink idref="bib28" id="ref168">28</reflink>] However, this alternative characterization would be as uninformative as that based on Equations (<reflink idref="bib10" id="ref169">10</reflink>) and (<reflink idref="bib11" id="ref170">11</reflink>).</p> <p>The reason both characterizations are uninformative is that the relevant elementary units for the descriptive goal at hand are neither quantile-specific derivatives nor quantile-specific elasticities—incorrectly equated by [<reflink idref="bib20" id="ref171">20</reflink>] to person-specific elasticities—but rather conditional distributions of children's income. The IGE and IGEE summarize the information in these conditional distributions using a measure of central tendency, either the geometric mean or the expectation. Both of these measures always increase with changes in density that favor larger incomes, even if they are affected differently by those changes. But this is beside the point: there is no meaningful sense in which the distinction between person-weighted and dollar-weighted elasticities can be applied to the two intergenerational elasticities.</p> <hd id="AN0188424702-13">Recapitulation</hd> <p>The analysis of this section has shown that the distinction between person-weighted and dollar-weighted elasticities, which is certainly important in a context in which the elementary units are individual-level "behavioral elasticities" (i.e., public finance), does not map into the differences between the IGE and the IGEE. More broadly, this distinction is not relevant in the intergenerational mobility context where elasticities are not meant to measure causal effects. It follows that Chetty et al.'s (2014) characterization of the two intergenerational elasticities is invalid, and thus [<reflink idref="bib62" id="ref172">62</reflink>] case for replacing the IGE with the IGEE remains justified.</p> <hd id="AN0188424702-14">An Empirical Illustration: U.S. Earnings Mobility</hd> <p>Although this article focuses on the conceptual interpretation of mobility estimands, this section addresses a potential empirical challenge. A skeptic might argue that differences in conceptual interpretation would not matter much in practice if the various measures produced very similar estimates of mobility. I address this challenge by using three key U.S. datasets to estimate all four mobility measures, with labor earnings as the outcome of interest and parental income as the index of origin, for men and women separately.[<reflink idref="bib29" id="ref173">29</reflink>] I also discuss the estimates in light of my conceptual arguments in previous sections.</p> <hd id="AN0188424702-15">Data and Variables</hd> <p>For several decades, research on intergenerational income mobility in the United States relied almost exclusively on survey data, primarily from the National Longitudinal Survey of Youth 1979 (NLSY79) and the Panel Study of Income Dynamics (PSID). Over the past two decades, however, the use of administrative data for studying mobility has expanded significantly (see [<reflink idref="bib59" id="ref174">59</reflink>]:1226–1232 for a review); studies using administrative data include [<reflink idref="bib54" id="ref175">54</reflink>], [<reflink idref="bib25" id="ref176">25</reflink>], [<reflink idref="bib20" id="ref177">20</reflink>], [<reflink idref="bib19" id="ref178">19</reflink>]), Chetty and Hendren ([<reflink idref="bib17" id="ref179">17</reflink>], [<reflink idref="bib18" id="ref180">18</reflink>]), and Mitnik et al. ([<reflink idref="bib61" id="ref181">61</reflink>], [<reflink idref="bib60" id="ref182">60</reflink>], [<reflink idref="bib59" id="ref183">59</reflink>]). To reflect this shift, in addition to samples from the PSID and NLSY79, I also use data from the Statistics of Income Mobility (SOI-M) Panel; this panel is constructed from U.S. tax returns, W-2 forms, and other administrative sources and represents all children born between 1972 and 1975 who were living in the U.S. in 1987 (for a detailed description, see [<reflink idref="bib59" id="ref184">59</reflink>]).</p> <p>Figure 3 shows the birth cohorts covered and the specific years and children's ages at which children's earnings and parental income are measured in each sample. Descriptive statistics for the samples are provided in Table 2.</p> <p>Graph: Figure 3. Cohorts included, and children's ages and years at which children's earnings and parental income are measured, in the samples from the National Longitudinal Survey of Youth 1979 (NLSY79), the Panel Study of Income Dynamics (PSID), and the Statistics of Income Mobility (SOI-M) Panel. The cohorts included in each sample are indicated on the left. The years when parental income and children's earnings are measured are indicated at the top. The children's ages at which their earnings and their parents' income are measured are the values inside the figure.</p> <p>Table 2. Descriptive Statistics (Weighted Values).</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="left" /><col align="left" /></colgroup><thead><tr><th align="left" /><th align="left">SOI-M</th><th align="left">NLSY79</th><th align="left">PSID</th></tr></thead><tbody><tr><td>Child's gender (% female)</td><td>48.9</td><td>48.9</td><td>49.7</td></tr><tr><td>Child's age</td><td /><td /><td /></tr><tr><td> Mean</td><td>36.5</td><td>40.2</td><td>39.4</td></tr><tr><td> Standard deviation</td><td>1.1</td><td>0.8</td><td>3.1</td></tr><tr><td>Child's earnings</td><td /><td /><td /></tr><tr><td> Mean</td><td>36,547</td><td>46,656</td><td>52,039</td></tr><tr><td> Standard deviation</td><td>56,438</td><td>51,938</td><td>60,000</td></tr><tr><td>Average parental age</td><td /><td /><td /></tr><tr><td> Mean</td><td>45.4</td><td>43.2</td><td>43.8</td></tr><tr><td> Standard deviation</td><td>6.2</td><td>6.5</td><td>6.8</td></tr><tr><td>Average parental income</td><td /><td /><td /></tr><tr><td> Mean</td><td>64,183</td><td>62,988</td><td>88,207</td></tr><tr><td> Standard deviation</td><td>91,709</td><td>42,039</td><td>60,468</td></tr><tr><td>Number of observations</td><td>12,872</td><td>2,596</td><td>13,564</td></tr><tr><td>Number of observations with positive earnings</td><td>10,124</td><td>2,289</td><td>11,847</td></tr><tr><td>Number of children</td><td>12,872</td><td>2,596</td><td>2,424</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>: Monetary values in 2010 dollars (adjusted by inflation using the Consumer Price Index for Urban Consumers—Research Series).</p> <p>The sample from the NLSY79 includes children who were 14–16 years old in 1979 (and thus born between 1963 and 1965) and were present in the survey in 2006. I use information on the average annual earnings of these children over 2003 and 2005 (when they were 38–42 years old), parents' average total family income in 1978–1980, and parents' age in 1979. I construct children's annual earnings from two NLSY79 variables—I sum "total income from wages and salary" and 65 percent of "total income from farm or business." To measure annual parental income in 1978, 1979, and 1980, I use "total net family income" in those years. Children's earnings and parental income for any given year are from the following year's survey (e.g., children's earnings in 2005 are from the 2006 survey).[<reflink idref="bib30" id="ref185">30</reflink>]</p> <p>The sample from the PSID pertains to household heads and spouses (or partners) born between 1954 and 1965 and observed at least once when they were between 35 and 45 years old. The observations in this sample are children-years (as in Lee and Solon 2009), with a separate record for each year in which a child appears in the PSID. I measure children's annual earnings at ages 35–45 and parents' average total family income and age when the children were 13–17 years old. I measure children's annual earnings as the PSID measured "income from labor" up to 1992.[<reflink idref="bib31" id="ref186">31</reflink>] To measure annual parental income in 1967–1982, I use the PSID notion of "total family income"; the corresponding variables are directly available or can be constructed from information in the 1968–1983 surveys. As with the NLSY79, the earnings and income information for any given year is from the following year's survey.[<reflink idref="bib32" id="ref187">32</reflink>]</p> <p>The sample from the SOI-M Panel I use pertains to children in 2010, when they were 35–38 years old. I use information on their annual earnings in that year, parents' average after-federal-tax income (including refundable tax credits) when the children were 15–23 years old, and parents' average age during the same period. Children's earnings are measured as the sum of "Medicare wages and tips" reported in Form W-2 and 65 percent of self-employment income reported in Schedule SE.[<reflink idref="bib33" id="ref188">33</reflink>] The annual income of parents is computed from Form 1040, by summing pre-tax "total income" (which includes labor earnings, capital income, unemployment insurance income, and the taxable portion of pensions, annuities, and social security income) and nontaxable interest, and subtracting out net federal taxes (which include refundable credits).[<reflink idref="bib34" id="ref189">34</reflink>]</p> <p>The selected samples and measures reflect constraints stemming from both (a) the data availability in each dataset, including the fact that the NLSY79 and PSID have been fielded biennially rather than annually after 1994 and 1997, respectively, and that the SOI-M data are only available through 2010, (b) the sample size required to produce estimates with sufficient statistical precision, and (c) the methodological imperative to average parental income over multiple years to reduce attenuation bias, and to measure both children's earnings and parental income at an age as close to 40 as possible to minimize lifecycle biases (see, e.g., [<reflink idref="bib37" id="ref190">37</reflink>]; [<reflink idref="bib56" id="ref191">56</reflink>]; [<reflink idref="bib57" id="ref192">57</reflink>]). As a result, the estimates derived from each dataset correspond to different birth cohorts, which may vary in their levels of intergenerational mobility.</p> <p>The datasets also differ in other important ways. The PSID sample only includes household heads and their spouses or partners. Those not included (i.e., those who make do by living with relatives) are more likely to be non-earners or low-earners. Moreover, the PSID does not cover institutionalized individuals (i.e., those in correctional facilities, mental institutions, or institutions for the disabled or poor), who almost certainly have zero or (very) low earnings. In addition, attrition in the PSID has been disproportionately high among low-income adult children with low-income parents (Schoeni and Wiemers 2015), which most likely further reduces the share of children with zero or low earnings in the sample. In contrast, the SOI-M sample does cover institutionalized people. However, it almost certainly includes an excess of people with zero earnings, as any person with positive earnings obtained exclusively in the informal economy would appear as having zero earnings (as the earnings would not be captured in Form W-2). The NLSY79 sample occupies a middle ground between the other two. On the one hand, like the PSID sample and unlike the SOI-M sample, it excludes institutionalized individuals but captures earnings obtained in the informal economy (rather than coding them as zero). On the other hand, unlike the PSID sample and like the SOI-M sample, it does include individuals who are not household heads or spouses/partners.</p> <p>Another difference between the SOI-M and the two survey samples relates to the quality of their income and earnings measures and their coverage of the upper tail of the earnings and income distributions. It is well established that survey income data are affected by both systematic underreporting and random measurement error (see, e.g., [<reflink idref="bib65" id="ref193">65</reflink>]). And while the SOI-M sample can be expected to capture high earners well, the PSID and NLSY79 most likely underrepresent them substantially ([<reflink idref="bib32" id="ref194">32</reflink>]).</p> <p>All the differences discussed above are reflected in the mean earnings and income of children and parents (as well as their dispersion) across the three samples, and in the proportion of children with zero earnings. The latter, in particular, is 21.3 percent in the SOI-M sample, compared to 12.7 percent in the NLSY79 sample and 11.8 percent in the PSID sample (see Table 2).</p> <hd id="AN0188424702-16">Estimation</hd> <p>The IGEE can be estimated using several approaches. These include nonlinear least squares (NLLS) and pseudo-maximum likelihood based on various distributions in the linear exponential family (especially Poisson and gamma).[<reflink idref="bib35" id="ref195">35</reflink>] I use the Poisson pseudo-maximum likelihood (PPML) estimator ([<reflink idref="bib79" id="ref196">79</reflink>]). This is the estimator employed by mobility scholars (e.g., Helsø 2020; [<reflink idref="bib26" id="ref197">26</reflink>]; [<reflink idref="bib60" id="ref198">60</reflink>]). It is also generally preferred for estimating constant-elasticity models due to its favorable theoretical properties and strong performance in simulation studies.[<reflink idref="bib36" id="ref199">36</reflink>]</p> <p>I estimate the IGE and RRS using ordinary least squares (OLS), as is the norm in the literature. To be able to compute standard errors, I estimate the IGC through an indirect approach that combines the results of two OLS regressions (see Online Appendix, Section C, in the online supplement). For the same reason, I also use an indirect approach to estimate elasticity-based lower-bound estimates of absolute inequality of opportunity. I combine the results of three OLS regressions to estimate the standard deviation of the log of the conditional geometric mean of children's earnings, and the results of a Poisson regression and two OLS regressions to estimate the standard deviation of the log of the conditional expectation of children's earnings (see Online Appendix, Section C, in the online supplement).</p> <p>Estimation of the IGE and IGC relies on restricted samples where children with zero earnings are excluded, as is standard in the literature. The estimates for the IGEE and RRS reported in the next subsection are based on the full samples, which is also standard. Nevertheless, supplementary estimates based on samples with positive earnings are reported in the online supplement, in Section D of the Online Appendix.</p> <p>Mobility scholars often include polynomial terms for children's and parents' ages in their regressions to account for the effects of age at measurement on the relationship between long-run and measured income or earnings (both for parents and children). In all PSID-based models (except those used to estimate the IGC), I include dummies for children aged 38–41 and 42–45. The variation in children's ages is quite small in the SOI-M and NLSY79 samples, which makes controlling for age unnecessary. Furthermore, because the age at which parents have their children is not exogenous to their income, and because parental age may affect their children's life chances, [<reflink idref="bib61" id="ref200">61</reflink>]:34) argued that controlling for parental age is inconsistent with the goal of measuring the gross association between parental and children's income. For this reason, the next subsection only includes results from models without parental age controls. Nevertheless, supplementary estimates from models including a quadratic polynomial in parental age are reported in the online supplement, in Section D of the Online Appendix.</p> <p>I use sampling weights and compute cluster-corrected robust standard errors in all analyses.</p> <hd id="AN0188424702-17">Results</hd> <p>I report the estimates for all mobility measures and datasets in Table 3.[<reflink idref="bib37" id="ref201">37</reflink>] Within each dataset-gender combination, there is substantial variation across measures, with the IGC providing the smallest (or near smallest) and the IGEE the largest estimate in all six cases. The maximum proportional difference between these two measures occurs for women in the PSID, where the IGEE is 171 percent larger than the IGC (0.38 compared to 0.14). The minimum difference is for men in the NLSY79, where the IGEE is 65 percent larger (0.56 compared to 0.34). The proportional differences between the IGEE and the RRS are also considerable. On average across dataset-gender combinations, the IGEE is 78 percent larger than the RRS, with a minimum difference of 56 percent (0.56 compared to 0.36, for NLSY79 men) and a maximum difference of 124 percent (0.38 compared to 0.17, for PSID women). Although the two elasticities tend to be comparatively closer to each other, the IGEE is always larger than the IGE, with the proportional difference consistently larger for women than for men in every dataset. The largest proportional difference is observed for women in the SOI-M Panel, at 50 percent (0.27 compared to 0.18), while the smallest is for men in the NLSY79, at 12 percent (0.56 compared to 0.50). On average across dataset-gender combinations, the IGEE is 30 percent larger than the IGE.</p> <p>Table 3. Earnings Mobility Measures.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /></colgroup><thead><tr><th align="left" /><th align="left" colspan="2">SOI-M</th><th align="left" colspan="2">NLSY79</th><th align="left" colspan="2">PSID</th></tr><tr><th /><th>Point estimate</th><th>Gender equality</th><th>Point estimate</th><th>Gender equality</th><th>Point estimate</th><th>Gender equality</th></tr><tr><th> </th><th>(SE)</th><th>p-value</th><th>(SE)</th><th>p-value</th><th>(SE)</th><th>p-value</th></tr></thead><tbody><tr><td>Men</td><td /><td /><td /><td /><td /><td /></tr><tr><td> Intergenerational elasticity</td><td><bold>0.35</bold></td><td>0.000</td><td><bold>0.50</bold></td><td>0.001</td><td><bold>0.44</bold></td><td>0.012</td></tr><tr><td /><td>(0.035)</td><td /><td>(0.052)</td><td /><td>(0.047)</td><td /></tr><tr><td rowspan="2"> Intergenerational elasticity of expected income</td><td><bold>0.49</bold></td><td>0.000</td><td><bold>0.56</bold></td><td>0.000</td><td><bold>0.50</bold></td><td>0.168</td></tr><tr><td>(0.028)</td><td /><td>(0.060)</td><td /><td>(0.060)</td><td /></tr><tr><td rowspan="2"> Intergenerational correlation</td><td><bold>0.23</bold></td><td>0.000</td><td><bold>0.34</bold></td><td>0.001</td><td><bold>0.29</bold></td><td>0.000</td></tr><tr><td>(0.020)</td><td /><td>(0.035)</td><td /><td>(0.030)</td><td /></tr><tr><td rowspan="2"> Rank-rank slope</td><td><bold>0.31</bold></td><td>0.000</td><td><bold>0.36</bold></td><td>0.000</td><td><bold>0.30</bold></td><td>0.003</td></tr><tr><td>(0.019)</td><td /><td>(0.031)</td><td /><td>(0.029)</td><td /></tr><tr><td>Women</td><td /><td /><td /><td /><td /><td /></tr><tr><td rowspan="2"> Intergenerational elasticity</td><td><bold>0.18</bold></td><td /><td><bold>0.25</bold></td><td /><td><bold>0.26</bold></td><td /></tr><tr><td>(0.035)</td><td /><td>(0.060)</td><td /><td>(0.057)</td><td /></tr><tr><td rowspan="2"> Intergenerational elasticity of expected income</td><td><bold>0.27</bold></td><td /><td><bold>0.29</bold></td><td /><td><bold>0.38</bold></td><td /></tr><tr><td>(0.027)</td><td /><td>(0.049)</td><td /><td>(0.063)</td><td /></tr><tr><td rowspan="2"> Intergenerational correlation</td><td><bold>0.12</bold></td><td /><td><bold>0.16</bold></td><td /><td><bold>0.14</bold></td><td /></tr><tr><td>(0.021)</td><td /><td>(0.041)</td><td /><td>(0.031)</td><td /></tr><tr><td> Rank-rank slope</td><td><bold>0.16</bold></td><td /><td><bold>0.15</bold></td><td /><td><bold>0.17</bold></td><td /></tr><tr><td /><td>(0.017)</td><td /><td>(0.030)</td><td /><td>(0.030)</td><td /></tr></tbody></table> </ephtml> </p> <p>2 <emph>Note</emph>: For each dataset, the first column includes the point estimates of the mobility measures in bold, and the associated standard errors (SE) in parentheses. The second column includes the p-values of the null hypothesis that mobility is the same across genders.</p> <p>These results show that mobility estimates can, in fact, differ substantially across measures, thus addressing the challenge posed by the hypothetical skeptic. More generally, these within-dataset-and-gender comparisons provide a strong reason to avoid treating estimates based on these four measures as interchangeable in scientific and policy contexts.[<reflink idref="bib38" id="ref202">38</reflink>]</p> <p>The differences between the two intergenerational elasticities merit further examination. Substantially uneven effects of sampling error across elasticities are possible but unlikely. Lifecycle and attenuation biases (see [<reflink idref="bib57" id="ref203">57</reflink>]) could also differ across estimates, but significant differences from these sources are similarly improbable. The differences between the IGEE and IGE estimates are most likely arising from two sources: (a) true differences between the values of the corresponding estimands, which pertain to long-run earnings, and (b) the selection bias that can be expected to affect the estimation of the IGE when using short-run proxy earnings measures—particularly for women, due to the much higher prevalence of non-earners among them (for a detailed analysis of selection bias in this context, see [<reflink idref="bib62" id="ref204">62</reflink>]).</p> <p>Consistent with the argument that selection bias plays a central role, the proportional differences between the IGEE and the IGE are substantially larger for the estimates based on the SOI-M sample than the NLSY79 and PSID samples. The reason is that the share of children with zero earnings, who are dropped when estimating the IGE, is much larger in the SOI-M sample (see Online Appendix, Table D1, in the online supplement). Estimation of the IGE can therefore be expected to be affected by much larger negative selection biases in this sample than in the other two (see [<reflink idref="bib62" id="ref205">62</reflink>]). Also consistent with the argument, supplementary analyses using restricted samples that exclude children with zero earnings (see Online Appendix, Table D3, in the online supplement) show that the proportional differences between the IGEE and IGE remain in the same direction but are substantially smaller than those reported in Table 3 (on average, half as large). Of course, the narrower gap between the IGEE and IGE in these restricted samples does not guarantee that IGEs estimated using long-run earnings would also be smaller than, yet closer to, the IGEEs. If selection biases are sufficiently large, the true IGEs could even exceed the true IGEEs. Nonetheless, the balance of the evidence supports the view that (a) true IGEEs are larger, but (b) the differences observed in Table 3 likely overstate the gap due to negative selection biases in the IGE estimates.[<reflink idref="bib39" id="ref206">39</reflink>]</p> <p>Within dataset–mobility-measure combinations, the estimates are consistently larger for men than for women—on average, nearly twice as large. Table 3 shows that the null hypothesis of equal mobility across genders is strongly rejected in all but one case at the conventional significance level. The higher earnings mobility among women has not received nearly as much attention as it deserves, partly because it is not yet broadly recognized in the literature (see [<reflink idref="bib59" id="ref207">59</reflink>]:1211).[<reflink idref="bib40" id="ref208">40</reflink>] This gender differential is, however, consistent with theoretical expectations. Indeed, at least three mechanisms may account for the lower intergenerational transmission of advantage through the labor market among women. First, occupational sex segregation likely reduces the extent to which women from higher-income backgrounds can capitalize on their advantages (e.g., [<reflink idref="bib14" id="ref209">14</reflink>]). Second, the disproportionate share of domestic duties that women assume may lead them to select part-time or low-paying jobs that do not fully reflect their earnings potential ([<reflink idref="bib72" id="ref210">72</reflink>]). Finally, the combined effect of assortative mating and the negative income elasticity of labor supply may result in women from more affluent backgrounds marrying higher-earning partners and choosing to work fewer hours (or not at all) when they have young children ([<reflink idref="bib73" id="ref211">73</reflink>]).</p> <p>Viewed through the lens of inequality of opportunity, both the IGEE and IGE, when multiplied by parental income inequality (i.e., by the standard deviation of log income), provide lower-bound estimates of absolute inequality of opportunity for earnings, with parental monetary income as the sole circumstance beyond people's control. The two elasticities only differ in how they index opportunity sets. As a result, the corresponding measures of inequality of opportunity are the standard deviation of the log of the conditional expectation (for the IGEE) and geometric mean (for the IGE) of children's earnings, respectively (see Equations (<reflink idref="bib7" id="ref212">7</reflink>) and (<reflink idref="bib8" id="ref213">8</reflink>)).</p> <p>Table 4 shows the elasticity-based lower-bound estimates of absolute inequality of opportunity for earnings. Unlike double-bounded measures like the Gini coefficient, which provide an intuitive way of assessing how much unfair inequality there is, measures of absolute inequality of opportunity that are not upper-bounded are difficult to interpret in substantive terms (see [<reflink idref="bib64" id="ref214">64</reflink>]). Nevertheless, examining the results from Tables 3 and 4 together illustrates the risks involved in using elasticities, and especially the IGE, to advance comparative assessments about inequality of opportunity (at least when the latter is interpreted in broadly luck-egalitarian terms).</p> <p>Table 4. Elasticity-Based Lower-Bound Absolute Inequality of Opportunity for Earnings.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="left" /><th align="left">SOI-M</th><th align="left">NLSY79</th><th align="left">PSID</th></tr></thead><tbody><tr><td>Men</td><td /><td /><td /></tr><tr><td> Based on the intergenerational elasticity</td><td><bold>0</bold><bold>.</bold><bold>29</bold></td><td><bold>0</bold>.<bold>33</bold></td><td><bold>0</bold>.<bold>26</bold></td></tr><tr><td /><td>(0.026)</td><td>(0.033)</td><td>(0.028)</td></tr><tr><td rowspan="2"> Based on the intergenerational elasticity of expected income</td><td><bold>0</bold>.<bold>43</bold></td><td><bold>0</bold>.<bold>38</bold></td><td><bold>0</bold>.<bold>29</bold></td></tr><tr><td>(0.024)</td><td>(0.041)</td><td>(0.037)</td></tr><tr><td>Women</td><td /><td /><td /></tr><tr><td> Based on the intergenerational elasticity</td><td><bold>0</bold>.<bold>15</bold></td><td><bold>0</bold>.<bold>17</bold></td><td><bold>0</bold>.<bold>16</bold></td></tr><tr><td /><td>(0.026)</td><td>(0.042)</td><td>(0.036)</td></tr><tr><td rowspan="2"> Based on the intergenerational elasticity of expected income</td><td><bold>0</bold>.<bold>25</bold></td><td><bold>0</bold>.<bold>21</bold></td><td><bold>0</bold>.<bold>24</bold></td></tr><tr><td>(0.027)</td><td>(0.036)</td><td>(0.040)</td></tr></tbody></table> </ephtml> </p> <p>3 <emph>Note</emph>: Point estimates are in bold, standard errors are in parentheses. Lower-bound absolute inequality of opportunity based on the intergenerational elasticity is defined in Equation (<reflink idref="bib8" id="ref215">8</reflink>). It is the standard deviation of the log of the conditional geometric mean of children's earnings, and is equal to the product of that elasticity and the standard deviation of log parental income. Lower-bound absolute inequality of opportunity based on the intergenerational elasticity of expected income is defined in Equation (<reflink idref="bib7" id="ref216">7</reflink>). It is the standard deviation of the log of the conditional expectation of children's earnings, and is equal to the product of that elasticity and the standard deviation of log parental income.</p> <p>Suppose we are willing to adopt an assumption analogous to the fixed-ratio assumption introduced earlier, when discussing cross-country comparisons: for a given gender, circumstances, and approach for indexing opportunity sets, the ratio of measured to true inequality of opportunity is approximately the same across estimates. Under this fixed-ratio assumption, if the product of the elasticities and parental income inequality yields rankings consistent with those based solely on the elasticities, then the elasticities can be treated as "sufficient statistics" for ranking true inequality of opportunity. However, this condition does not hold for the estimates reported in Tables 3 and 4. None of the four rankings derived from Table 3—two for men and two for women across datasets—align with the corresponding rankings based on inequality of opportunity in Table 4. In other words, even qualitative conclusions about relative levels of inequality of opportunity across cohorts based on the elasticities estimated with the SOI-M, NLSY79, and PSID samples would be misleading.</p> <p>In addition, the negative selection biases affecting the IGE estimates also impact the corresponding estimates of parental income inequality, as the latter exclude the parents of non-earner children, who are more likely to be low-income. As a result, in five out of six dataset-gender combinations, the ratio between the IGEE-based and IGE-based measures of inequality of opportunity in Table 4 is somewhat larger than the ratio between the underlying elasticities in Table 3, especially in the case of women. In other words, the IGE-based lower-bound estimates of inequality of opportunity are lower than their IGEE-based counterparts for three different reasons: (a) they index opportunity sets with the geometric mean rather than with the expectation of earnings, (b) they are based on downward-biased estimates of IGEs, and (c) they are based on downward-biased estimates of parental inequality.</p> <p>In terms of absolute inequality of opportunity, the RRS estimates reported in Table 3 represent rescaled measures of absolute inequality of opportunity for earnings rank, rather than for monetary earnings. The rescaling is implicit and involves dividing by approximately 0.29 (the standard deviation of a uniformly distributed variable), which bounds the resulting measure between 0 and 1 and significantly improves interpretability. Furthermore, it is easy to see that adopting (an appropriate version of) the fixed-ratio assumption in this context would allow us to do more than ordinal comparisons—under this assumption, the RRS estimates can be treated as ratio-level measures of true inequality of opportunity scaled by an unknown constant (the ratio of true to measured inequality of opportunity). Since the RRS estimates are quite stable within gender, adopting that assumption would lead to the conclusion that true inequality of opportunity for earnings ranks is roughly constant across cohorts and approximately twice as large for men as for women. Naturally, any confidence in this conclusion must be tempered by the strength of the fixed-ratio assumption.</p> <p>The IGC and RRS both provide estimates of relative inequality of opportunity, focusing on monetary income and income rank, respectively. In Table 3, the RRS tends to be similar in value to, if not larger than, the IGC. Based on these figures, one might be tempted to conclude that relative inequality of opportunity in monetary earnings is not greater than, and perhaps is lower than, in earnings ranks. In other words, a smaller (or at least not larger) share of inequality in monetary earnings is unfair compared to the share of inequality in earnings ranks.</p> <p>Although this might be the case, there are at least two reasons to be very cautious. First, both the IGC and RRS are lower-bound estimates of relative inequality of opportunity. Drawing the above conclusion from their comparison would seem to require an implausibly strong identification assumption: namely, that for each gender and dataset, the ratio of true to measured inequality of opportunity is approximately the same across the two measures—despite their focus on different outcomes (monetary earnings versus earnings rank), different circumstances (parental income versus parental income rank), and different ways of indexing opportunity sets (based on conditional geometric means versus conditional expectations). Second, like with the IGE, IGC estimates are affected by selection bias due to the exclusion of individuals with no (short-run) earnings from the samples.[<reflink idref="bib41" id="ref217">41</reflink>]</p> <hd id="AN0188424702-18">Conclusions</hd> <p>Although there is an extensive methodological literature on the measurement of intergenerational income mobility, there has been limited research on the conceptual interpretation of mobility measures and the methodological implications of those interpretations. In this article, I have made two main contributions to this neglected area of methodological research.</p> <p>The first contribution concerns the relationship between income mobility and inequality of opportunity for income. The current understanding of this relationship is that mobility and immobility, on the one hand, and equality and inequality of opportunity, on the other, are two pairs of concepts that are (a) irreducibly different but, in some sense left mostly unspecified, conceptually related, and (b) mutually informative when empirically measured. This is a very unsatisfactory state of affairs.</p> <p>I have significantly improved our understanding by specifying the formal relationships between four mobility measures—the IGE, IGEE, IGC, and RRS—and the measures of absolute and relative inequality of opportunity advanced in the luck egalitarian empirical literature on opportunity. I have shown that the four mobility measures are precisely related to measures of inequality of opportunity that vary in terms of the outcomes of interest (monetary income or income rank), the circumstances they consider (parental income or parental income rank), the way they measure the value of opportunity sets (by the within-type mean or geometric mean of the outcome of interest), or the inequality index on which they rely (standard deviation or standard deviation of logarithms).</p> <p>I have also shown that (a) the IGC is a measure of relative inequality of opportunity for monetary income, (b) the RRS is both a measure of relative inequality of opportunity for income rank and a rescaled measure of absolute inequality of opportunity for income rank, and (c) the products of parental income inequality by the IGEE and IGE are both measures of absolute inequality of opportunity for monetary income that differ in how they measure the value of opportunity sets.</p> <p>One important implication of my analysis is that the luck-egalitarian understanding of inequality of opportunity offers no normative justification for carrying out cross-country comparisons of IGCs. From this philosophical perspective, there is no justification for comparing countries in terms of the share of overall inequality that is unfair—what matters is the amount of unfair inequality itself. A second important implication is that interpreting mobility measures in terms of the luck-egalitarian understanding of inequality of opportunity—even for the relatively modest goal of ranking countries—requires making very strong assumptions. This applies to the RRS and both intergenerational elasticities. However, it is a comparatively more serious concern for the elasticities, as the correlation between them and parental income inequality is unlikely to be high enough for the latter to be inconsequently ignored when comparing countries. This may be addressed by comparing countries in terms of the product of parental income inequality and the elasticities rather than the elasticities themselves (similarly to what I did in the empirical illustration). However, this raises the question of why one would then bother estimating the elasticities at all, instead of directly estimating inequality of opportunity measures using the approaches developed in the luck-egalitarian empirical literature. All this also applies, of course, to the comparisons of periods or cohorts.</p> <p>Is there an alternative normative motivation for estimating mobility measures like intergenerational elasticities and rank-rank slopes, and for comparing them across countries, periods, and cohorts, if one is not prepared to make the strong assumptions required to interpret them in terms of the luck-egalitarian understanding of inequality of opportunity? There might be. Without any additional assumptions, the RRS and the elasticities measure the shares of parental inequality—in terms of income ranks and monetary incomes, respectively—that are transmitted across generations. An alternative normative motivation could potentially be based on the complement to this share, which may be roughly interpreted as measuring the overall "effort" a country makes to contain the transmission of income inequality across generations. However, exploring this possibility further lies beyond the scope of my analysis here.</p> <p>The second main contribution of the article concerns an existing comparative characterization of the IGE and IGEE as person-weighted and dollar-weighted elasticities, which casts doubts on [<reflink idref="bib62" id="ref218">62</reflink>] proposal to make the IGEE the workhorse elasticity of the mobility field. My analysis has established that although the distinction between person-weighted and income-weighed elasticities is highly relevant in contexts like public finance, where the elementary units are behavioral elasticities, the distinction does not apply in the intergenerational mobility context, where elasticities are not meant to measure causal effects. Furthermore, I have shown that the argument is also invalid because of a category mistake: equating quantile-specific elasticities with person-specific elasticities. It follows that the characterization of the IGE and IGEE as person-weighted and dollar-weighted elasticities—rather than as, respectively, the elasticity of the conditional geometric mean and expectation of children's income—is off the mark, and that [<reflink idref="bib62" id="ref219">62</reflink>] case for replacing the IGE with the IGEE remains justified.</p> <hd id="AN0188424702-19">Supplemental Material</hd> <p>Graph: Supplemental material, sj-pdf-1-smr-10.1177_00491241251352102 for Inequality of Opportunity, Income Mobility, and the Interpretation of Intergenerational Elasticities, Correlations, and Rank-Rank Slopes by Pablo A. Mitnik in Sociological Methods & Research</p> <ref id="AN0188424702-20"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref37" type="bt">1</bibl> <bibtext> The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref46" type="bt">2</bibl> <bibtext> The author received no financial support for the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref48" type="bt">3</bibl> <bibtext> Pablo A. Mitnik https://orcid.org/0000-0002-2988-7456</bibtext> </blist> <blist> <bibl id="bib4" idref="ref38" type="bt">4</bibl> <bibtext> A replication package ([58]) containing data and code (openicpsr-231182) is publicly available for viewing and download at <ulink href="http://www.openicpsr.org/openicpsr/project/231182">www.openicpsr.org/openicpsr/project/231182</ulink>. The package enables reproduction of Figure 1 and all PSID-based and NLSY79-based results presented in Tables 2, 3, and 4 of this article, as well as Tables D1, D2, and D3 from the https://journals.sagepub.com/doi/suppl/10.1177/00491241251352102 (available in the online supplement). Because the SOI-M data are based on tax and other restricted-access information, the corresponding materials for this sample could not be included in the replication package. Access to data from the SOI-M Panel can be requested as part of an application to the SOI's Joint Statistical Research Program (see <ulink href="http://www.irs.gov/statistics/soi-tax-stats-joint-statistical-research-program">www.irs.gov/statistics/soi-tax-stats-joint-statistical-research-program</ulink>).</bibtext> </blist> <blist> <bibl id="bib5" idref="ref122" type="bt">5</bibl> <bibtext> Supplemental material for this article is available https://doi.org/10.1177/00491241251352102.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref36" type="bt">6</bibl> <bibtext> Mitnik and Grusky (2020a, Section 2) provide textual evidence of the mobility field's misinterpretation of the IGE by reproducing and analyzing several quotations from prominent scholars of economic mobility. Two of those quotations are the following: (a) the IGE "measures the percentage differential in the son's expected income with respect to a marginal percentage differential in the income of the father" (Björklund and Jäntti 2011:497), and (b) the IGE provides "a parametric answer to questions like, if the parents' long-run earnings are 50% above the average in their generation, what percent above the average should we predict the child's long-run earnings to be in her or his generation?" (Solon 1999:1777).</bibtext> </blist> <blist> <bibl id="bib7" idref="ref32" type="bt">7</bibl> <bibtext> Because the four measures discussed in this section measure persistence or immobility, the corresponding measures of mobility are (1 – IGE), (1 – IGC), (1 – IGEE), and (1 – RRS). I have however followed the practice, conventional in the literature, of referring to the IGE, IGC, IGEE, and RRS as "mobility measures" (even though technically they are "immobility measures").</bibtext> </blist> <blist> <bibl id="bib8" idref="ref1" type="bt">8</bibl> <bibtext> The error term <emph>U</emph> is mean independent of parental income <emph>X</emph> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mrow><mi>U</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></math> </ephtml> whereas it is geometric mean independent when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">GM</mi></mrow><mo stretchy="false">(</mo><mrow><mi>U</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><mrow><mi mathvariant="normal">GM</mi></mrow><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> in both cases for all <emph>x</emph><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mspace width="0.25em" /></math> </ephtml> Mean and geometric mean independence are implied by statistical independence but they do not imply the latter, even jointly ([70]). For related discussions, see [66], Santos [79], and Wooldridge (2002:17).</bibtext> </blist> <blist> <bibl id="bib9" idref="ref51" type="bt">9</bibl> <bibtext> Moreover, the condition that long-run income is positive for all children ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>></mo>0</math> </ephtml> ), without which the IGE is only defined for the subpopulation with positive income, is not likely to hold strictly with any income variable and may be substantially violated when the variable is children's labor earnings, particularly in the case of women ([62]). Even when parental income is both mean and geometric mean independent of the population error term, we should not expect the IGEE and IGE to be equal, or even approximately equal, if the former pertains to the whole population whereas the latter pertains to the subpopulation with positive income, and this subpopulation is significantly smaller. Similar caveats apply in the next few paragraphs.</bibtext> </blist> <blist> <bibtext> Note, however, that homoscedastic errors alone are not sufficient for the equality of the IGE and IGEE (see [70]); the assumption that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi></math> </ephtml> is distributed log-normal for all <emph>x</emph> is also necessary.</bibtext> </blist> <blist> <bibtext> For instance, given the positive association between parental income and children's educational attainment, evidence that the variance of men's log earnings increases with education (e.g., [49]) suggests that the IGEE of men's earnings should be larger than the IGE of men's earnings. This is consistent with existing evidence (see [62], Table 2). I thank Thomas Lemieux for this point.</bibtext> </blist> <blist> <bibtext> It is also worth noting that when children's and parents' income follow a bivariate log-normal distribution, the IGE and IGEE coincide (see [44]: 22, Eq. 44.27). Under such distribution (a) the conditional distributions of children's income are log-normal for all values of parental income, and (b) the variance of the conditional distribution of children's log income does not change with parental income. It is then easy to see that the assumption of bivariate log-normality is much stronger than the assumption of conditional log-normality introduced in the previous paragraph.</bibtext> </blist> <blist> <bibtext> Given the conditional log normality assumption for children's income, it follows from a property of log-normal distributions that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi><mo>=</mo><mo stretchy="false">(</mo><mi>d</mi><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo>0.5<mspace width="0.25em" /><mo stretchy="false">(</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">Var</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> (see https://journals.sagepub.com/doi/suppl/10.1177/00491241251352102). To simulate the data, I specify <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Var</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo>1<mo>+</mo>0.6<mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi></math> </ephtml> (in the second period). Then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">Var</mi></mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>X</mi><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo>0.6</math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">IGEE</mi></mrow><mo>=</mo>0.35<mo>+</mo>0.5<mspace width=".1em" />0.6<mo>=</mo>0.65.</math> </ephtml></bibtext> </blist> <blist> <bibtext> An implicit assumption in this argument is that it is undesirable for an income mobility measure to change when inequality changes. There are good reasons to reject this assumption (e.g., [59]: Section 2). However, here I focus on what (if anything) follows for the "comparative worth" of the four measures, assuming the validity of the assumption.</bibtext> </blist> <blist> <bibtext> This characterization ignores important qualifications regarding the notion of luck that is relevant here. See [50] for a detailed analysis of this notion and its role in luck egalitarianism.</bibtext> </blist> <blist> <bibtext> This position, due to Roemer (e.g., 1998), is the dominant one in empirical research. It requires that effort be "cleaned from any contamination coming from circumstances" ([43]). See [3] and [87] for alternative philosophical positions on this matter.</bibtext> </blist> <blist> <bibtext> Of course, in order to produce a meaningful measure of relative inequality of opportunity, the same inequality measure needs to be used to measure absolute inequality of opportunity and overall inequality.</bibtext> </blist> <blist> <bibtext> Relative inequality of opportunity for income, which tells us what share of overall income inequality in a country is unjust, is an interesting and important descriptive quantity, even if it is not relevant for the comparative assessment of countries from a normative point of view. In large part, its importance comes from the fact that arguments positing that a high level of income inequality is normatively unproblematic often imply that this is the case because such inequality is (mainly) the result of differences in effort. Thus, in any given country, estimates of relative inequality of opportunity provide evidence that is directly relevant for such an argument ([64]).</bibtext> </blist> <blist> <bibtext> A large class of inequality indices are Lorenz consistent, including the Gini and Theil indices.[51] ignores that estimates are based on finite samples in conducting her proof. Therefore, as indicated in the text, her result should be interpreted as applying asymptotically. In their own formal analysis, which applies to the mean logarithmic deviation only, Ferreira and Gignoux (2011:636) assumed that their conclusion that estimates are lower-bound estimates under nonparametric estimation extends to parametric estimation. Although it is not difficult to construct numerical examples showing that this is incorrect even asymptotically, until not long ago this assumption had been broadly accepted in the literature. Recent research (e.g., Brunori et al.[13], [10]) has emphasized that estimates may be upward biased when they rely on samples that are not large enough given the number of circumstances considered and how fine-grained the partition of the population into types is. I thank Steven Durlauf for pressing me to specify better these issues.</bibtext> </blist> <blist> <bibtext> Observe, in particular, that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>X</mi><mo stretchy="false">)</mo><mo>;</mo></math> </ephtml> therefore, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">MLD</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><msubsup><mi>β</mi>0<mrow><mi mathvariant="normal">′</mi></mrow></msubsup></math> </ephtml> covary due to their joint dependence on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>.</mo></math> </ephtml></bibtext> </blist> <blist> <bibtext> Equations (7) and (8) were previously reported in [60] but without any careful discussion of assumptions, the meaning of various terms, or methodological implications, which are all included in this section.</bibtext> </blist> <blist> <bibtext> Everything I say regarding cross-country comparisons also applies to the comparison of periods or cohorts within countries. Ranking countries is much less demanding than establishing their relative performance. While the latter requires cardinal measures, the former only requires ordinal measures.</bibtext> </blist> <blist> <bibtext> In fact, the assumptions are so strong that they would justify comparing countries even if the estimates of absolute and relative inequality of opportunity are upward rather than downward biased.</bibtext> </blist> <blist> <bibtext> The IGEE was first introduced in a 2015 working paper ([61]).[20] discussed an early draft of that working paper in their https://journals.sagepub.com/doi/suppl/10.1177/00491241251352102 (see also p. 1574). That draft did not characterize the IGE as pertaining to the conditional geometric mean; it just pointed out that it did not pertain to the conditional expectation in the general case.</bibtext> </blist> <blist> <bibtext> Here, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">0.60<mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo>1<mtext>%</mtext><mo>+</mo>0.2<mtext>%</mtext><mo stretchy="false">)</mo><mo>/</mo>2<mo stretchy="false">]</mo><mo>/</mo>1<mtext>%</mtext></math> </ephtml> , while <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">0.99<mo>=</mo><mo fence="false" stretchy="false">{</mo>100<mo stretchy="false">[</mo><mo stretchy="false">(</mo>20<mo>+</mo>10<mo>,</mo>000<mo stretchy="false">)</mo><mo>/</mo></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo>10<mo>,</mo>000<mo>+</mo>1<mo>,</mo>000<mo>,</mo>000<mo stretchy="false">)</mo><mo stretchy="false">]</mo><mtext>%</mtext><mo fence="false" stretchy="false">}</mo><mo>/</mo><mspace width="0.25em" />1<mtext>%</mtext><mo>.</mo></math> </ephtml></bibtext> </blist> <blist> <bibtext> Chetty et al.'s (2014) analysis and my arguments here apply in a more general context than the one provided by Equations (1) and (4), as they do not assume that the relationships between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>ln</mi><mspace width="0.2em" /><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mspace width="0.2em" /><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> on the one hand, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mspace width="0.2em" /><mi>x</mi><mo>,</mo></math> </ephtml> on the other, are linear. In this broader context, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mspace width="0.25em" /><mo>/</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="0.25em" /><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>x</mi><mrow><mo stretchy="false">)</mo><mo>/</mo></mrow><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi></math> </ephtml> may differ across values of parental income and only reduce to the constant values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi>1</msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi>1</msub></math> </ephtml> when the true relationships are linear. In the nonlinear case, summary measures of mobility are produced by computing a weighted average elasticity across values of parental income, with the weights set at the density of each value (see [59]). When the true relationships are linear, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi>1</msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi>1</msub></math> </ephtml> are identical to these average elasticities. Importantly, the discussion in this section is orthogonal to the procedure for producing a summary measure just described. It is also distinct from [45], which shows that when the true relationship is nonlinear, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi>1</msub></math> </ephtml> equals a weighted average of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>E</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="0.25em" /><mo>/</mo><mi>d</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mspace width=".1em" /><mi>x</mi></math> </ephtml> across values of parental income, with greater weight placed on the middle than on the top and bottom of the parental log-income distribution.</bibtext> </blist> <blist> <bibtext> This is, of course, consistent with how the notion of person-based elasticity has been used in public finance to refer to a behavioral elasticity that measures how a person's taxable income changes when they adjust their behavior—such as their hours of work—in response to a change in the tax rate.</bibtext> </blist> <blist> <bibtext> This point is based on comments made by Joao Santos Silva in personal communication, for which I am grateful.</bibtext> </blist> <blist> <bibtext> There are important advantages to using parental income instead of father's earnings as the index of origin (e.g., Corak [2006:54] and Mazumder [2005:250]): (a) it incorporates the income of mothers and thus better indexes the full complement of economic resources available to invest in children; (b) it reflects the ability of families to draw on income sources other than earnings in response to transitory earnings shocks; and (c) it avoids any selection bias that may result from omitting children with absent fathers (as they are likely to be comparatively disadvantaged).</bibtext> </blist> <blist> <bibtext> To obtain the average parental-income measure I employ in the analyses, I do the following: (a) apply Pareto imputation (e.g., [31]) to address top-coding, (b) express the 1978–1980 incomes in 2010 dollars using the Current Price Index for Urban Consumers—Research Series (CPI-U-RS), and (c) compute the mean of these inflation-adjusted incomes across years.</bibtext> </blist> <blist> <bibtext> Starting in 1993, the PSID did not include any longer, in its measure of income from labor, the labor portion of business income and 50 percent of farm income, which were included in previous years. So, for 1989–1992, I use the PSID-provided income from labor variable, but for 1993–2010, I compute the pre-1993 notion of income from labor using several PSID variables.</bibtext> </blist> <blist> <bibtext> For 1967–1982, I rely on the notion of total family income used in the 1968–1983 surveys. But as the income components used to compute total family income are, in my sample, effectively affected by top coding in the period 1970–1978 (i.e., top codes are not only in place but are "binding" in that period for some children), and the PSID-computed total family income for those years is based on these top-coded values, I proceed as follows: (a) I address top-coding of each income component individually in 1970–1978 by using Pareto imputation, and (b) I recompute total family income for those years with these Pareto-imputed variables. Then, after expressing all annual income variables in 2010 dollars using the CPI-U-RS, I average them to obtain the five-year parental-income measure employed in the analyses.</bibtext> </blist> <blist> <bibtext> Medicare wages and tips are those reported in Box 2 of Form W-2. Unlike the wages and tips reported in Box 1, they include deferred earnings, i.e., the earnings that employees contribute to employer-sponsored pension plans.</bibtext> </blist> <blist> <bibtext> As with the NLSY79 and the PSID data, before computing average parental income, parental income in each year is expressed in 2010 dollars using the CPI-U-RS.</bibtext> </blist> <blist> <bibtext> If the mean function is correctly specified, pseudo-maximum likelihood estimators are consistent estimators regardless of the distribution of the dependent variable ([35]).</bibtext> </blist> <blist> <bibtext> Santos Silva and Tenreyro (2006, 2011, 2022) have argued that the PPML estimator should be preferred over other consistent estimators of constant-elasticity models of expected outcomes, as both theoretical arguments and simulations indicate that the NLLS estimator is inefficient (often to the point of being useless in empirical applications) and highly sensitive to outliers, whereas the gamma pseudo-maximum likelihood estimator is very sensitive to measurement error. By contrast, the PPML estimator "is reliable in a wide variety of situations" and therefore "has the essential characteristics needed to make it the new workhorse for the estimation of constant-elasticity models" (Santos [79]:649); in addition, it behaves well even when the share of zeros is very large (Santos Silva and Tenreyro 2011).</bibtext> </blist> <blist> <bibtext> The sizes of the samples underlying the estimates reported in Table 3 can be found in https://journals.sagepub.com/doi/suppl/10.1177/00491241251352102.</bibtext> </blist> <blist> <bibtext> Estimates from models including parental-age controls (see https://journals.sagepub.com/doi/suppl/10.1177/00491241251352102) are very similar to those in Table 3 and lead to the same conclusions.</bibtext> </blist> <blist> <bibtext> See note 6.</bibtext> </blist> <blist> <bibtext> See also Mitnik et al. 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" Intergenerational Income Elasticities, Instrumental Variable Estimation, and Bracketing Strategies. " Sociological Methodology. 50(1):1–46.</bibtext> </blist> <blist> <bibtext> Mitnik Pablo.2023. " Estimating the Intergenerational Elasticity of Expected Income with Short-Run Income Measures: A Generalized Error-in-Variables Model. " Empirical Economics. 65:2779–803.</bibtext> </blist> <blist> <bibtext> Mitnik Pablo. 2025. Replication Package for Inequality of Opportunity, Income Mobility, and the Interpretation of Intergenerational Elasticities, Correlations and Rank-Rank Slopes. Ann Arbor, MI: Inter-university Consortium for Political and Social Research [distributor], 2025-07-05. https://doi.org/10.3886/E231182V2</bibtext> </blist> <blist> <bibtext> Mitnik Pablo, Bryant Victoria, Grusky David. 2024. " A Very Uneven Playing Field: economic Mobility in the United States. 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Emerald.</bibtext> </blist> <blist> <bibtext> Swift Adam.2005 "Justice, Luck and the Family: Normative Aspects of Intergenerational Transmission of Economic Status." Pp. 256–276 in Unequal Chances: Family Background and Economic Success, edited by Sam Bowles, Herbert Gintis, Melissa Osborne Groves. Princeton, NJ: Princeton University Press.</bibtext> </blist> <blist> <bibtext> Van de gaer D.1993. Equality of Opportunity and Investment in Human Capital. PhD Dissertation, Leuven: Catholic University of Leuven.</bibtext> </blist> <blist> <bibtext> Wooldridge Jeffrey.2002. Econometric Analysis of Cross Section and Panel Data. Cambridge, Mass: The MIT Press.</bibtext> </blist> </ref> <aug> <p>By Pablo A. Mitnik</p> <p>Reported by Author</p> <p></p> <p>Pablo A. Mitnik is an assistant research scientist at the University of Michigan's Stone Center for Inequality Dynamics, where he directs the Wealth and Mobility Study. His research centers on economic inequality, intergenerational mobility, labor markets, and statistical methods. His recent work advances the methodological and data foundations for measuring mobility and inequality of opportunity. Current projects examine trends in earnings inequality, the intergenerational transmission of income and wealth, the policy and institutional drivers of mobility, and the conceptual underpinnings of research on inequality of opportunity.</p> </aug> <nolink nlid="nl1" bibid="bib36" firstref="ref2"></nolink> <nolink nlid="nl2" bibid="bib37" firstref="ref3"></nolink> <nolink nlid="nl3" bibid="bib41" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib53" firstref="ref5"></nolink> <nolink nlid="nl5" bibid="bib54" firstref="ref6"></nolink> <nolink nlid="nl6" bibid="bib55" firstref="ref7"></nolink> <nolink nlid="nl7" bibid="bib56" firstref="ref8"></nolink> <nolink nlid="nl8" bibid="bib57" firstref="ref9"></nolink> <nolink nlid="nl9" bibid="bib68" firstref="ref10"></nolink> <nolink nlid="nl10" bibid="bib69" firstref="ref11"></nolink> <nolink nlid="nl11" bibid="bib81" firstref="ref12"></nolink> <nolink nlid="nl12" bibid="bib82" firstref="ref13"></nolink> <nolink nlid="nl13" bibid="bib26" firstref="ref14"></nolink> <nolink nlid="nl14" bibid="bib34" firstref="ref15"></nolink> <nolink nlid="nl15" bibid="bib45" firstref="ref16"></nolink> <nolink nlid="nl16" bibid="bib62" firstref="ref17"></nolink> <nolink nlid="nl17" bibid="bib63" firstref="ref18"></nolink> <nolink nlid="nl18" bibid="bib30" firstref="ref20"></nolink> <nolink nlid="nl19" bibid="bib74" firstref="ref21"></nolink> <nolink nlid="nl20" bibid="bib76" firstref="ref22"></nolink> <nolink nlid="nl21" bibid="bib15" firstref="ref23"></nolink> <nolink nlid="nl22" bibid="bib47" firstref="ref24"></nolink> <nolink nlid="nl23" bibid="bib48" firstref="ref25"></nolink> <nolink nlid="nl24" bibid="bib20" firstref="ref27"></nolink> <nolink nlid="nl25" bibid="bib59" firstref="ref29"></nolink> <nolink nlid="nl26" bibid="bib40" firstref="ref33"></nolink> <nolink nlid="nl27" bibid="bib83" firstref="ref34"></nolink> <nolink nlid="nl28" bibid="bib67" firstref="ref39"></nolink> <nolink nlid="nl29" bibid="bib25" firstref="ref43"></nolink> <nolink nlid="nl30" bibid="bib70" firstref="ref49"></nolink> <nolink nlid="nl31" bibid="bib10" firstref="ref54"></nolink> <nolink nlid="nl32" bibid="bib11" firstref="ref55"></nolink> <nolink nlid="nl33" bibid="bib12" firstref="ref57"></nolink> <nolink nlid="nl34" bibid="bib42" firstref="ref59"></nolink> <nolink nlid="nl35" bibid="bib13" firstref="ref63"></nolink> <nolink nlid="nl36" bibid="bib14" firstref="ref65"></nolink> <nolink nlid="nl37" bibid="bib24" firstref="ref68"></nolink> <nolink nlid="nl38" bibid="bib46" firstref="ref69"></nolink> <nolink nlid="nl39" bibid="bib23" firstref="ref74"></nolink> <nolink nlid="nl40" bibid="bib21" firstref="ref81"></nolink> <nolink nlid="nl41" bibid="bib27" firstref="ref82"></nolink> <nolink nlid="nl42" bibid="bib28" firstref="ref83"></nolink> <nolink nlid="nl43" bibid="bib75" firstref="ref84"></nolink> <nolink nlid="nl44" bibid="bib22" firstref="ref85"></nolink> <nolink nlid="nl45" bibid="bib16" firstref="ref87"></nolink> <nolink nlid="nl46" bibid="bib29" firstref="ref91"></nolink> <nolink nlid="nl47" bibid="bib39" firstref="ref92"></nolink> <nolink nlid="nl48" bibid="bib52" firstref="ref93"></nolink> <nolink nlid="nl49" bibid="bib71" firstref="ref94"></nolink> <nolink nlid="nl50" bibid="bib86" firstref="ref95"></nolink> <nolink nlid="nl51" bibid="bib17" firstref="ref97"></nolink> <nolink nlid="nl52" bibid="bib64" firstref="ref99"></nolink> <nolink nlid="nl53" bibid="bib18" firstref="ref102"></nolink> <nolink nlid="nl54" bibid="bib51" firstref="ref103"></nolink> <nolink nlid="nl55" bibid="bib19" firstref="ref104"></nolink> <nolink nlid="nl56" bibid="bib77" firstref="ref138"></nolink> <nolink nlid="nl57" bibid="bib33" firstref="ref139"></nolink> <nolink nlid="nl58" bibid="bib38" firstref="ref152"></nolink> <nolink nlid="nl59" bibid="bib85" firstref="ref154"></nolink> <nolink nlid="nl60" bibid="bib84" firstref="ref157"></nolink> <nolink nlid="nl61" bibid="bib80" firstref="ref160"></nolink> <nolink nlid="nl62" bibid="bib61" firstref="ref181"></nolink> <nolink nlid="nl63" bibid="bib60" firstref="ref182"></nolink> <nolink nlid="nl64" bibid="bib31" firstref="ref186"></nolink> <nolink nlid="nl65" bibid="bib32" firstref="ref187"></nolink> <nolink nlid="nl66" bibid="bib65" firstref="ref193"></nolink> <nolink nlid="nl67" bibid="bib35" firstref="ref195"></nolink> <nolink nlid="nl68" bibid="bib79" firstref="ref196"></nolink> <nolink nlid="nl69" bibid="bib72" firstref="ref210"></nolink> <nolink nlid="nl70" bibid="bib73" firstref="ref211"></nolink>
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  Data: Inequality of Opportunity, Income Mobility, and the Interpretation of Intergenerational Elasticities, Correlations, and Rank-Rank Slopes
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  Data: <searchLink fieldCode="AR" term="%22Pablo+A%2E+Mitnik%22">Pablo A. Mitnik</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-2988-7456">0000-0002-2988-7456</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22Sociological+Methods+%26+Research%22"><i>Sociological Methods & Research</i></searchLink>. 2025 54(4):1289-1338.
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  Data: 50
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  Data: Journal Articles<br />Reports - Descriptive
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  Data: <searchLink fieldCode="DE" term="%22Social+Mobility%22">Social Mobility</searchLink><br /><searchLink fieldCode="DE" term="%22Income%22">Income</searchLink><br /><searchLink fieldCode="DE" term="%22Correlation%22">Correlation</searchLink><br /><searchLink fieldCode="DE" term="%22Measurement+Techniques%22">Measurement Techniques</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Analysis%22">Statistical Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Parent+Child+Relationship%22">Parent Child Relationship</searchLink><br /><searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink>
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  Data: <searchLink fieldCode="SU" term="%22National+Longitudinal+Survey+of+Youth%22">National Longitudinal Survey of Youth</searchLink><br /><searchLink fieldCode="SU" term="%22Panel+Study+of+Income+Dynamics%22">Panel Study of Income Dynamics</searchLink>
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  Data: 10.1177/00491241251352102
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  Data: 0049-1241<br />1552-8294
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  Data: Although there is an extensive methodological literature on the measurement of intergenerational income mobility, there has been limited research on the conceptual interpretation of mobility measures and the methodological implications of those interpretations. In this article, I focus on the three measures of mobility most frequently used in the literature--the intergenerational elasticity (IGE), the intergenerational correlation (IGC), and the rank-rank slope (RRS)--as well as a recently introduced measure, the intergenerational elasticity of expected income (IGEE). I make two main contributions, both related to the conceptual interpretation of mobility measures. First, I specify the formal relationships between those four mobility measures and the measures of inequality of opportunity developed in the luck egalitarian empirical literature on the topic, and determine the methodological implications of the analyses. I show that (a) the IGC is a measure of relative inequality of opportunity for monetary income, (b) the RRS is both a measure of relative inequality of opportunity for income rank and a rescaled measure of absolute inequality of opportunity for income rank, and (c) the products of parental income inequality by the IGEE and IGE are both measures of absolute inequality of opportunity for monetary income that differ in how they measure the value of opportunity sets. Second, relying on a conceptual distinction that has been influential in the field of public finance, the IGE and IGEE have been characterized as "person-weighted" and "dollar-weighted" elasticities, respectively, thus raising doubts about the desirability of a recent proposal to replace the IGE by the IGEE as the workhorse elasticity of the mobility field. I show that this contrasting characterization of the two intergenerational elasticities is the joint result of a category mistake--equating quantile-specific elasticities to person-specific elasticities--and of misconstruing the nature of the IGE and the epistemic goal it has been meant to serve. Based on this analysis, I conclude that the case for replacing the IGE with the IGEE remains well-founded.
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        PageCount: 50
        StartPage: 1289
    Subjects:
      – SubjectFull: Social Mobility
        Type: general
      – SubjectFull: Income
        Type: general
      – SubjectFull: Correlation
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      – SubjectFull: Measurement Techniques
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      – SubjectFull: National Longitudinal Survey of Youth
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      – SubjectFull: Panel Study of Income Dynamics
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    Titles:
      – TitleFull: Inequality of Opportunity, Income Mobility, and the Interpretation of Intergenerational Elasticities, Correlations, and Rank-Rank Slopes
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            – D: 01
              M: 11
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 0049-1241
            – Type: issn-electronic
              Value: 1552-8294
          Numbering:
            – Type: volume
              Value: 54
            – Type: issue
              Value: 4
          Titles:
            – TitleFull: Sociological Methods & Research
              Type: main
ResultId 1