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Generalized Intergenerational Mobility Regressions

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Title: Generalized Intergenerational Mobility Regressions
Language: English
Authors: Esfandiar Maasoumi, Le Wang (ORCID 0000-0001-5918-2644), Daiqiang Zhang (ORCID 0009-0009-6617-9270)
Source: Sociological Methods & Research. 2025 54(4):1594-1623.
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: 30
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Descriptors: Regression (Statistics), Social Mobility, Statistical Analysis, Income, Parent Child Relationship
Assessment and Survey Identifiers: Panel Study of Income Dynamics
DOI: 10.1177/00491241251357586
ISSN: 0049-1241
1552-8294
Abstract: Current research on intergenerational mobility (IGM) is informed by "statistical" approaches based on log-level regressions, whose "economic" interpretations remain largely unknown. We reveal the subjective value-judgments in them: they are represented by weighted-sums (or aggregators) over heterogeneous groups, with controversial "economic" properties. Log-level regressions tend to overrepresent the experiences of middle-class children while underrepresenting those from disadvantaged families. We propose a general construction of IGM measures that can incorporate any transparent "economic" preferences. They are interpreted as the marginal effect of parental normalized social welfare on children's normalized welfare. Conventional regressions are special cases with implicit economic preferences that fail inequality-aversion and the Pigou-Dalton principle of transfers. Empirically, a variety of economic preferences, with varying inequality aversion, demonstrate a nuanced view of mobility, and perspectives on geographic-differences and dynamics of it.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1485847
Database: ERIC
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  Value: <anid>AN0188424704;som01nov.25;2025Oct06.06:36;v2.2.500</anid> <title id="AN0188424704-1">Generalized Intergenerational Mobility Regressions </title> <p>Current research on intergenerational mobility (IGM) is informed by statistical approaches based on log-level regressions, whose economic interpretations remain largely unknown. We reveal the subjective value-judgments in them: they are represented by weighted-sums (or aggregators) over heterogeneous groups, with controversial economic properties. Log-level regressions tend to overrepresent the experiences of middle-class children while underrepresenting those from disadvantaged families. We propose a general construction of IGM measures that can incorporate any transparent economic preferences. They are interpreted as the marginal effect of parental normalized social welfare on children's normalized welfare. Conventional regressions are special cases with implicit economic preferences that fail inequality-aversion and the Pigou–Dalton principle of transfers. Empirically, a variety of economic preferences, with varying inequality aversion, demonstrate a nuanced view of mobility, and perspectives on geographic-differences and dynamics of it.</p> <p>Keywords: intergenerational mobility; inequality aversion; decision theory; welfare; elasticity; heterogeneity; J62; D63; I31; C43; J13</p> <hd id="AN0188424704-2">Introduction</hd> <p>Economic mobility—particularly the extent to which children's income depends on that of their parents—remains central to evaluating fairness and opportunity in a society. The most widely used measure of intergenerational mobility (IGM) is the intergenerational elasticity (IGE) from log-level regressions. Although typically interpreted as a descriptive statistic, the IGE plays a central role in contemporary debates about social mobility and the effects of public policy.</p> <p>For example, differences in IGE across countries or groups are often attributed to differing balances between family influence, market institutions, and public policy ([<reflink idref="bib8" id="ref1">8</reflink>]; [<reflink idref="bib32" id="ref2">32</reflink>]). Similarly, the Great Gatsby Curve—a widely cited empirical relationship between inequality and mobility—is often used to motivate discussions about how inequality dampens opportunity, suggesting that "underlying trends" in mobility could change only if "public policy promotes the human capital of children in a way that offers relatively greater benefits to the relatively disadvantaged" ([<reflink idref="bib14" id="ref3">14</reflink>]). Even if not constructed as normative measures, IGEs frequently guide the interpretation of policy relevance. For example, [<reflink idref="bib21" id="ref4">21</reflink>] emphasize the prominence of the Great Gatsby Curve in policy debates. Similarly, [<reflink idref="bib33" id="ref5">33</reflink>] highlight the growing concern among social scientists and policymakers about the connection between economic inequality and social mobility, illustrating how even descriptively framed mobility measures like the IGE are regularly invoked in normative discussions.</p> <p>However, understanding, comparing, and addressing mobility is complicated by a fundamental challenge: the nature of its measurement. The conclusions we draw about whether mobility is rising or falling, equitable or unequal, often hinge on the specific methodology used. These methodological choices can profoundly shape both our perception of mobility and the policy debates it informs.</p> <p>Our article parallels the literature on the measurement of inequality in many ways, which has generated many valuable insights for us to consider more deeply the issue of measurement of IGMs.[<reflink idref="bib6" id="ref6">6</reflink>] A key lesson from the inequality literature is that summary measures are never neutral. Inequality indices, including the popular Gini coefficient and Theil indices, implicitly embed value judgments about which parts of the distribution matter most—depending on how they weight differences across the income spectrum. Atkinson (1970) argues that these value judgments should be made explicit through the specification of a social welfare function underpinning each measure. This perspective is echoed powerfully by [<reflink idref="bib18" id="ref7">18</reflink>], who write, "By collapsing the whole rainbow of the income distribution into a single statistical point of white light, it necessarily conceals much of great interest," and emphasize that "the best measures are those that match our purpose, or pick up on the places where important changes are happening."</p> <p>Inspired by these insights, this paper challenges the implicit neutrality of commonly used descriptive mobility measures and their interpretation in policy debates. The challenge stems from underlying heterogeneity and nonlinearity in intergenerational income transmission. Economic theories predict, and recent empirical work confirms, that mobility (the correlation between child and parental income or the marginal effect of parental income on child income) varies (locally) across families and the income distribution (e.g. An et al. 2022; [<reflink idref="bib7" id="ref8">7</reflink>]; [<reflink idref="bib8" id="ref9">8</reflink>]; [<reflink idref="bib11" id="ref10">11</reflink>]; Chang et al. 2025; [<reflink idref="bib16" id="ref11">16</reflink>]; [<reflink idref="bib25" id="ref12">25</reflink>]; [<reflink idref="bib26" id="ref13">26</reflink>], to name a few). We show that the IGE aggregates these local measures of mobility. Crucially, this aggregation scheme embeds value judgments: we show that the IGE tends to assign most weight to middle-income families, while underweighting those from the poorest and richest backgrounds. Thus, IGEs overrepresent the experiences of middle-class children while underrepresenting those from disadvantaged families. This imbalance skews the interpretation of mobility and results in measures that are not aligned with economic principles, such as inequality aversion. What appears as a "neutral" statistical summary is actually the result of hidden normative choices.</p> <p>Our goal and contribution is to make these value judgments transparent. We propose a novel regression framework that allows mobility measures to be constructed using social welfare functions that flexibly reflect explicit societal preferences. Given a particular societal preference and associated welfare function, our mobility measure has a clear economic interpretation: it captures the marginal effect of parents' normalized social welfare (as opposed to income) on that of their children, a societal mobility measure. This perspective represents a significant departure from traditional analyses, reframing mobility as a societal phenomenon by treating parental and child cohorts as collective entities.</p> <p>Importantly, our framework also nests the traditional IGE as a special case. We further demonstrate that the social preference implicitly underlying the IGE and its weighting scheme corresponds to a set of convex economic preferences that are not equality-minded (also fail to meet the Pigou–Dalton principle of transfers, see footnote 2).[<reflink idref="bib7" id="ref14">7</reflink>] In contrast, as an illustrative example, we examine the extended Gini family of social welfare functions, which increasingly prioritize lower-income individuals as inequality aversion rises.</p> <p>We present a substantive empirical analysis using PSID data to compare traditional IGEs with estimates from our welfare-based framework. By using the same data, we isolate the role of methodological differences in shaping our understanding of mobility. Our results demonstrate that weighting schemes and the corresponding social preferences matter. Gini-based measures, the ones that emphasize inequality aversion by assigning greater weight to disadvantaged children, lead to conclusions that diverge significantly from those based on traditional IGEs, challenging key findings in the existing literature. First, Gini-based measures yield smaller regression coefficients. This suggests a more mobile society than what traditional IGEs imply, consistent with the literature (e.g., Kourtellos et al. (2020)) finding that children from both disadvantaged and richer families may have a higher level of mobility than those in the "middle class".</p> <p>Second, when using the Gini mobility measures, we actually find that the West becomes less mobile, relative to the rest of the country, including the South. Finally, we examine how intergenerational mobility evolves across cohorts. Traditional IGEs suggest a decline in mobility for those born before 1954 and after 1968. In contrast, when applying greater weight to disadvantaged families through our welfare-based measures, we find that mobility has actually increased over time.</p> <p>The rest of the paper is organized as follows. Section "A Motivating Example" provides a numerical example to motivate our analysis. Section "Problem: Weighted Average Representations sec:regression of Traditional IGEs" exposes the conceptual issues for the traditional regression approaches by analyzing their weighting schemes. Section "A Solution: A General Theory of Mobility Measures Based on Social Preferences" presents our measures. We illustrate our proposals using the PSID data in Section "Empirical Illustration". Section "Conclusions" concludes. Proofs are collected in the online supplement.</p> <hd id="AN0188424704-3">A Motivating Example</hd> <p>We first present the simplest possible example that introduces heterogeneity both within and across groups to highlight the need for aggregation in measuring mobility. This stylized comparison between two groups—say, Group A and Group B—intentionally abstracts from other complexities to focus on a central insight: even minimal differences in mobility across the distribution can yield very different conclusions depending on the weighting scheme used. This example makes clear that aggregation is not a technical afterthought but a defining feature of a summary mobility measure.</p> <p>Let there be only two income levels, low-income, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> </ephtml> and high-income, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> </ephtml> . Let the income functions be <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>A</mi></msub><mo>=</mo><msub><mi>g</mi><mi>A</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ϵ</mi><mi>A</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>B</mi></msub><mo>=</mo><msub><mi>g</mi><mi>B</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ϵ</mi><mi>B</mi></msub></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>g</mi></msub></math> </ephtml> is child income for group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>∈</mo><mo fence="false" stretchy="false">{</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo fence="false" stretchy="false">}</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> is parental income, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>∈</mo><mo fence="false" stretchy="false">{</mo><mi>l</mi><mo>,</mo><mi>h</mi><mo fence="false" stretchy="false">}</mo></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>l</mi><mo><</mo><mi>h</mi><mo><</mo><mi mathvariant="normal">∞</mi></math> </ephtml> . Let mobility at <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>x</mi></math> </ephtml> be <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>g</mi><mi>A</mi><mo>′</mo></msubsup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.6</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>g</mi><mi>A</mi><mo>′</mo></msubsup><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.3</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>g</mi><mi>B</mi><mo>′</mo></msubsup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.9</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>g</mi><mi>B</mi><mo>′</mo></msubsup><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.2</mn></math> </ephtml> .[<reflink idref="bib8" id="ref15">8</reflink>] Income in group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> is less persistent and more mobile in the lower tail, while income in group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> is less persistent and more mobile in the upper tail. A weighted average measure of mobility for any group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> </ephtml> is thus <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>m</mi><mo accent="false">¯</mo></mover><mi>k</mi></msub><mo>=</mo><msub><mi>w</mi><mi>l</mi></msub><msubsup><mi>g</mi><mi>k</mi><mo>′</mo></msubsup><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>w</mi><mi>h</mi></msub><msubsup><mi>g</mi><mi>k</mi><mo>′</mo></msubsup><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mi>l</mi></msub><mo>+</mo><msub><mi>w</mi><mi>h</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><msub><mi>w</mi><mi>l</mi></msub><mo>,</mo><msub><mi>w</mi><mi>h</mi></msub><mo><</mo><mn>1</mn></math> </ephtml> . Consider two different weighting schemes for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>l</mi><mi>j</mi></msubsup><mo>,</mo><msubsup><mi>w</mi><mi>h</mi><mi>j</mi></msubsup><mo>,</mo><mi>j</mi><mo>=</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></math> </ephtml> .</p> <p> <bold>Case 1:</bold> Let the first weighting scheme (placing higher weights on the <bold>poorest</bold> family) be <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>l</mi><mn>1</mn></msubsup><mo>=</mo><mn>.95</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>h</mi><mn>1</mn></msubsup><mo>=</mo><mn>.05</mn></math> </ephtml> . Then, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>m</mi><mo accent="false">¯</mo></mover><mi>A</mi></msub><mo>=</mo><mn>.585</mn></math> </ephtml> . However, if we reverse the weighting scheme instead by placing more weights on the <bold>richest</bold> families with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>l</mi><mn>2</mn></msubsup><mo>=</mo><mn>.05</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>h</mi><mn>2</mn></msubsup><mo>=</mo><mn>.95</mn></math> </ephtml> , then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>m</mi><mo accent="false">¯</mo></mover><mi>A</mi></msub><mo>=</mo><mn>.315</mn></math> </ephtml> . These two numbers resemble the current debate on the magnitudes of the IGM for the U.S. with different approaches. The discrepancy between two approaches is large since the mobility implied by the first weighting scheme is drastically different from the second weighting scheme, even for the same group or society with the same income transmission process,.</p> <p> <bold>Case 2:</bold> Now consider how varying weighting schemes may impact the conclusions regarding the between-group comparison of mobility. Under the first weighting scheme when we place more weights on the children from the disadvantaged families with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>l</mi><mn>1</mn></msubsup><mo>=</mo><mn>.95</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>h</mi><mn>1</mn></msubsup><mo>=</mo><mn>.05</mn></math> </ephtml> , it follows <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>m</mi><mo accent="false">¯</mo></mover><mi>A</mi></msub><mo>=</mo><mn>.585</mn><mo><</mo><mn>.865</mn><mo>=</mo><msub><mover><mi>m</mi><mo accent="false">¯</mo></mover><mi>B</mi></msub></math> </ephtml> . The implied mobility of group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> is actually <emph>greater</emph> than that of group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> . By contrast, under the second weighting scheme, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>l</mi><mn>2</mn></msubsup><mo>=</mo><mn>.05</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>w</mi><mn>2</mn></msup><mo>=</mo><mn>.95</mn></math> </ephtml> , we actually observe the opposite result <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>m</mi><mo accent="false">¯</mo></mover><mi>A</mi></msub><mo>=</mo><mn>.315</mn><mo>></mo><mn>.235</mn><mo>=</mo><msub><mover><mi>m</mi><mo accent="false">¯</mo></mover><mi>B</mi></msub></math> </ephtml> . In other words, group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> is less mobile than group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> .</p> <p>Consider a policy evaluation context, where Groups <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> represent populations before and after a policy intervention, one may reach significantly different conclusions about the policy's effectiveness. This highlights how aggregation choices can shape not only academic assessments but also potentially real-world policy decisions.</p> <p>In practice, the weighting schemes should depend on the underlying income of each group as well as the social weight attached to that group. All weighting schemes are subjective. Transparency is needed and provided below.</p> <hd id="AN0188424704-4">Problem: Weighted Average Representations of Traditional IGEs</hd> <p>Having demonstrated the importance of weighting schemes, we now formally express the IGE as a weighted average of heterogeneous, local mobility measures across the parental income distribution, and rigorously characterize its underlying weighting mechanism. While many statistics can be written as weighted averages, the interpretive value of such a representation depends critically on the context and the goals of the analysis. In the case of the IGE, this formulation offers meaningful insights into its normative content and structural limitations.</p> <p>Following [<reflink idref="bib38" id="ref16">38</reflink>], let the true income-transmission process be given by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>ϵ</mi><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> is child income and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> parental income, with error term <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>ϵ</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math> </ephtml> . We choose the (log) income equation as our benchmark since the theory is usually concerned with the level, not the rank (although one can consider an alternative bechmark). The derivative, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> , is the slope which measures persistence, interpreted as the inverse mobility at a particular level of parental income <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>x</mi></math> </ephtml> . In the presence of nonlinearity, the derivative, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> , varies with parental income. The popular IGE can be considered as a summary or aggregate of the values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> , represented by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></math> </ephtml></p> <p>Graph</p> <p>The weights, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> , depend on the parental income, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>x</mi></math> </ephtml> . These weights are different for different methods. Under standard assumptions in Assumption 1, Lemma (<reflink idref="bib1" id="ref17">1</reflink>) (see Proposition 2 in [<reflink idref="bib38" id="ref18">38</reflink>]) clarifies the weighted average representation of the IGEs (the Least Squares (LS) of the projection parameter).[<reflink idref="bib9" id="ref19">9</reflink>]</p> <p>Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></math> </ephtml> be the support of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> . The marginal cumulative distribution function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> , with marginal density function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> and inverse distribution function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>F</mi><mi>X</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> , is strictly increasing. The mean and variance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>X</mi></msub><mo>=</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></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"><msubsup><mi>σ</mi><mi>X</mi><mn>2</mn></msubsup><mo>=</mo><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> , respectively, are finite.</p> <p>[<bold>Weighted Average Representation of Linear Regression</bold>] Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo>*</mo><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>+</mo><mi>β</mi><mi>X</mi></math> </ephtml> denote the best linear predictor (projection) of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> . Under Assumption 1, we obtain <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mo stretchy="false">[</mo><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><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"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>μ</mi><mi>X</mi></msub><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo fence="false" stretchy="false">{</mo><mi>X</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo fence="false" stretchy="false">}</mo><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>X</mi><mrow><mo>−</mo><mn>2</mn></mrow></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><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><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></math> </ephtml> .</p> <p>[<reflink idref="bib38" id="ref20">38</reflink>] shows that under normal and uniform distributions, the weight function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> in Lemma 1 is maximized at the median of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> .[<reflink idref="bib10" id="ref21">10</reflink>] This implies that a regression-based IGM assigns <emph>smaller</emph> weights to the marginal effects in the lower and higher tails of the parent income distribution than at the middle, but places most weights on middle-class children. This property is not considered desirable under concave utility functions with aversion to inequality (and with concern over lower incomes). Note that this property is about the weight on the derivative of the function, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> , different from the familiar concern with higher linear LS weights on the outliers of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> .</p> <p>Figure 1 illustrates the core intuition of Lemma 1 and clarifies how perceptions of intergenerational mobility can be misrepresented when using the IGE. Consider a nonlinear income transmission function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> denotes parental income, and the marginal effect <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>x</mi></math> </ephtml> varies with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> . Here, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> is distributed according to a standard normal distribution. The top-left panel of the figure shows that the IGE—obtained via linear regression—is zero, indicated by the red dashed line. The adjacent panel plots the marginal effects <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> over the domain <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo stretchy="false">]</mo></math> </ephtml> . Using the weight function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> defined in Lemma 1 (the bottom-left panel), we observe that weights are symmetric around zero, the median of a standard normal distribution. These weights peak at the median, where the mobility for this family <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math> </ephtml> . Since the integral (or summation) of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> (the bottom-right panel) over the distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> equals zero, the resulting IGE reflects only the experience of middle-income families. In other words, although children from both low- and high-income families may face strong intergenerational persistence, the IGE collapses this rich heterogeneity into a summary statistic that exclusively reflects conditions around the median. This highlights the limitations of IGE as an economically sound measure of mobility.</p> <p>Graph: Figure 1. Example of Lemma 1: g(x)=x2.</p> <hd id="AN0188424704-5">A Solution: A General Theory of Mobility Measures Based on Social Preferences</hd> <p>Subjectivity in the implicit weights behind the IGEs leads us to consider a general aggregation approach with transparent weighting schemes. We adopt a decision theoretic approach that is well motivated by well-founded principles in the literature on inequality, poverty, and mobility. Figure (<reflink idref="bib2" id="ref22">2</reflink>) illustrates the construction of a summary mobility measure: a model or method acts as an aggregation device, combining (<reflink idref="bib1" id="ref23">1</reflink>) heterogeneous local mobility measures across families, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math> </ephtml> , with (<reflink idref="bib2" id="ref24">2</reflink>) a specified social preference <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> and its corresponding weighting scheme <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> , to produce an overall mobility measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> .</p> <p>Graph: Figure 2. The process for constructing a summary mobility measure.</p> <p>Below, we will propose and examine a generalized one-step framework to flexibly embed an arbitrary social preference in constructing our mobility measure, without having to calculate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math> </ephtml> and their corresponding weights, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math> </ephtml> . Within this framework, we will also explicate the (unreasonable) social preference function underlying the IGE and its weighting scheme in Lemma (<reflink idref="bib1" id="ref25">1</reflink>). As an illustrative example, we examine a special family of our measure that is based on the parametric Gini evaluation function. This showcases the flexibility of our measures in accommodating a wide range of economic criteria.</p> <hd id="AN0188424704-6">A General Social Welfare-based Summary Measure of Income Mobility</hd> <p>A general approach to construct summary measures of mobility is based on the axiomatic characterization of desirable social welfare properties. To begin, [<reflink idref="bib36" id="ref26">36</reflink>], [<reflink idref="bib37" id="ref27">37</reflink>] show that a preference relation, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> , defined on income distribution, which satisfies a set of standard axioms, can be represented by the following social welfare <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>X</mi></msub><mo>=</mo><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mi>x</mi><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msubsup><mi>F</mi><mi>X</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>F</mi><mi>X</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> is the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> </ephtml> -th quantile of the income distribution.[<reflink idref="bib11" id="ref28">11</reflink>] Social welfare <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>X</mi></msub></math> </ephtml> is taken as a weighted average of individual incomes in which the weight <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is a function of income ranks <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mi>P</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math> </ephtml> . This accommodates interdependencies. The functional form of the preference/evaluation function, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> , reveals a policy-maker's inequality aversion and determines the weights, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> in measuring social welfare.[<reflink idref="bib12" id="ref29">12</reflink>] Due to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> </ephtml> 's dependence on the rank <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>X</mi></msub></math> </ephtml> is also called a rank-dependent social welfare (e.g., [<reflink idref="bib3" id="ref30">3</reflink>]), and widely adopted in the literature. For example, based on the rank-dependent social welfare, [<reflink idref="bib2" id="ref31">2</reflink>] analyzes ordering relations on Lorenz curves, and [<reflink idref="bib3" id="ref32">3</reflink>] propose a general approach to ranking intersecting distribution functions.</p> <p>To explicitly incorporate the policy maker's preference function, or her attitude towards inequality in rank-dependent social welfare, the following summary measures of IGM are considered: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mrow><mfrac><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mrow><mfrac><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>.</mo></math> </ephtml></p> <p>Graph</p> <p>where there is no restriction on the functional form of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> </ephtml> except for satisfying some standard smoothness conditions,[<reflink idref="bib13" id="ref33">13</reflink>] and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> </ephtml> is the weight in social welfare (<reflink idref="bib2" id="ref34">2</reflink>).[<reflink idref="bib14" id="ref35">14</reflink>] The mobility measures in (<reflink idref="bib3" id="ref36">3</reflink>) are a significant economic construct—it can be nicely interpreted as the marginal effect of parents' (normalized) social welfare on children's (normalized) social welfare , which correspond to the numerator and the denominator in (<reflink idref="bib3" id="ref37">3</reflink>), respectively. To see this, note that the numerator can be rewritten as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo></mtd><mtd><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo fence="false" stretchy="false">{</mo><mo stretchy="false">[</mo><mi>Y</mi><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd><msub><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mi>X</mi></msub><mo fence="false" stretchy="false">{</mo><mo stretchy="false">[</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo stretchy="false">)</mo><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd><msub><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mi>X</mi></msub><mo fence="false" stretchy="false">{</mo><mo stretchy="false">[</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo stretchy="false">)</mo><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><msub><mrow><mover><mi>y</mi><mo>~</mo></mover></mrow><mi>x</mi></msub><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mover><mi>W</mi><mo>~</mo></mover></mrow><mi>Y</mi></msub><mo>,</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo>~</mo></mover></mrow><mi>x</mi></msub><mo>=</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math> </ephtml> is the normalized conditional expected children income given parents income <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mi>x</mi></math> </ephtml> , and the second equality follows the law of iterated expectations. In terms of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>X</mi></msub></math> </ephtml> , we can interpret <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>W</mi><mo>~</mo></mover></mrow><mi>Y</mi></msub></math> </ephtml> not only as a weighted average of children's normalized conditional incomes given parents' income but also as a normalized children's social welfare. We can similarly show that <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mrow><mover><mi>x</mi><mo>~</mo></mover></mrow><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mover><mi>W</mi><mo>~</mo></mover></mrow><mi>X</mi></msub><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>x</mi><mo>~</mo></mover></mrow><mo>=</mo><mi>x</mi><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> is the normalized parents income, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>W</mi><mo>~</mo></mover></mrow><mi>X</mi></msub></math> </ephtml> can be considered a normalized parents social welfare and a weighted average of normalized parents incomes. This perspective represents a significant departure from traditional analyses, reframing mobility as a societal phenomenon by treating parental and child cohorts as collective entities. We summarize this important result in the following proposition.</p> <p>For the mobility measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> , we obtain the following properties:</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><msub><mrow><mover><mi>W</mi><mo>~</mo></mover></mrow><mi>Y</mi></msub><mo stretchy="false">(</mo><msub><mrow><mover><mi>W</mi><mo>~</mo></mover></mrow><mi>X</mi></msub><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> </ephtml> can be interpreted as the marginal effect of parents' normalized social welfare <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>W</mi><mo>~</mo></mover></mrow><mi>Y</mi></msub><mo>=</mo><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><msub><mrow><mover><mi>y</mi><mo>~</mo></mover></mrow><mi>x</mi></msub><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> </ephtml> on children's normalized social welfare <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>W</mi><mo>~</mo></mover></mrow><mi>X</mi></msub><mo>=</mo><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mrow><mover><mi>x</mi><mo>~</mo></mover></mrow><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo>~</mo></mover></mrow><mi>x</mi></msub><mo>=</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></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"><mrow><mover><mi>x</mi><mo>~</mo></mover></mrow><mo>=</mo><mi>x</mi><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> are children's normalized conditional income and parents' normalized income, respectively.</p> <p>Here, we also provide a general result of the weighted average representation of mobility measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> for a general form of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> </ephtml> . Using arguments similar to [<reflink idref="bib38" id="ref38">38</reflink>], <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> can be expressed as a weighted average of the individual mobility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> . This will facilitate our examination of the properties of the Gini family of IGM measures in Section ("Mobility Measures based on Gini Evaluation Functions")</p> <p>The summary measure of mobility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> can be rewritten as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where the underlying weight scheme satisfies <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></math> </ephtml> , and <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mo fence="false" stretchy="false">{</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><mo>.</mo></math> </ephtml></p> <p>Graph</p> <hd id="AN0188424704-7">Economic Meaning of IGEs</hd> <p>Using our framework, we can now actually interpret the economic meaning of the conventional IGE and uncover its underlying <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> (in addition to its weighting scheme). We note that the popular IGEs are special cases of our proposed measure, defined with a constant marginal effect <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mrow><mi mathvariant="normal">Cov</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>/</mo><mrow><mi mathvariant="normal">Var</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi mathvariant="normal">Cov</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>/</mo><mrow><mi mathvariant="normal">Cov</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> . Using (<reflink idref="bib3" id="ref39">3</reflink>), we can see that the corresponding evaluation function has the derivative of the form, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math> </ephtml> , or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>F</mi><mi>X</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> . This in turn implies the social preference underlying the IGEs is given by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><msubsup><mi>F</mi><mi>X</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi></math> </ephtml></p> <p>Graph</p> <p>This is familiar to a poverty "count measure," the number in population below t.</p> <p>The evaluation function underlying the IGE is convex due to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mo>″</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><msubsup><mi>F</mi><mi>X</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub></math> </ephtml> is the density function of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> . As a result, the weight <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>P</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> in (<reflink idref="bib2" id="ref40">2</reflink>) increases with the rank <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> </ephtml> . In other words, the policymaker who employs this IGE cares more about the rich than the poor. This is in contrast to the Pigou–Dalton principle of transfers, which states that a transfer of income from the rich to the poor is preferred (Atkinson 1970), so long as it does not affect anybody's position in the income ranking. Indeed, a preference function in (<reflink idref="bib8" id="ref41">8</reflink>) underlying the conventional IGE does not satisfy the Pigou–Dalton principle. This is because the Pigou–Dalton principle cannot hold for a convex <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> </ephtml> in (<reflink idref="bib8" id="ref42">8</reflink>) ([<reflink idref="bib37" id="ref43">37</reflink>]). Consequently, the policy maker is not <emph>equality minded</emph>.[<reflink idref="bib15" id="ref44">15</reflink>] Since the Pigou–Dalton principle is overwhelmingly used in the literature to introduce a concern for inequality into judgments about income distribution ([<reflink idref="bib6" id="ref45">6</reflink>]), the current linear IGEs do not represent a desirable metrics, especially if the policy maker's preference is consistent with the Pigou–Dalton principle.</p> <p>The undesirable property of the evaluation function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> is also consistent with that of the weight function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> that we unravel earlier when examining the weighted average representation of the IGE. As shown in [<reflink idref="bib38" id="ref46">38</reflink>], the weight function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> under normal and uniform distributions assigns smaller weights to the individuals in the lower tails of the income distribution than the middle part of the income distribution. These results are summarized in Proposition 3.</p> <p>If the evaluation function is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><msubsup><mi>F</mi><mi>X</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi></math> </ephtml> , then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>β</mi></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>/</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> . However, the fact that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> is convex implies that the policy maker is not equality minded, and the evaluation fails to satisfy the Pigou–Dalton principle of transfers.</p> <p>Moreover, we show that the conventional IGEs correspond to the social welfare with the preference function being Lorenz curve, using our framework of mobility measure. Specifically, the evaluation function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> in Proposition 3 has a closed-form relationship with the Lorenz curve, that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>μ</mi><mi>X</mi></msub><mi>L</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> is the Lorenz curve at <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math> </ephtml> . Plugging <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>μ</mi><mi>X</mi></msub><mi>L</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> into (<reflink idref="bib4" id="ref47">4</reflink>) and (<reflink idref="bib5" id="ref48">5</reflink>), we obtain <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mrow><mfrac><mrow><msub><mi>μ</mi><mi>X</mi></msub><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><msub><mrow><mover><mi>y</mi><mo>~</mo></mover></mrow><mi>x</mi></msub><mi>d</mi><mi>L</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mi>μ</mi><mi>X</mi></msub><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mrow><mover><mi>x</mi><mo>~</mo></mover></mrow><mi>d</mi><mi>L</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mrow><mfrac><mrow><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><msub><mrow><mover><mi>y</mi><mo>~</mo></mover></mrow><mi>x</mi></msub><mi>d</mi><mi>L</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mrow><mover><mi>x</mi><mo>~</mo></mover></mrow><mi>d</mi><mi>L</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml></p> <p>Graph</p> <p>Building upon Proposition 3, we obtain the following corollary.</p> <p>The mobility measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> can be interpreted as the marginal effect of parents normalized social welfare on children normalized social welfare, with the preference function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> </ephtml> being the Lorenz curve <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> </ephtml> .</p> <p>Corollary 1 implies that the conventional regression-based IGE is associated with a specific preference function being Lorenz curve. Lorenz curve can be used to analyze a person's preference ranking (e.g., see [<reflink idref="bib2" id="ref49">2</reflink>].[<reflink idref="bib16" id="ref50">16</reflink>] However, as is well known, if two Lorenz curves intersect, it is not possible to determine which distribution has more inequality. This limitation of Lorenz curve further suggests that the conventional IGE may not be relevant when one analyzes the income mobility and inequality from the perspective of social welfare.[<reflink idref="bib17" id="ref51">17</reflink>]</p> <hd id="AN0188424704-8">Mobility Measures Based on Gini Evaluation Functions</hd> <p>Having demonstrated the unreasonable social preference underlying the IGE, we now turn to a special case of our measure that is based on the parametric Gini evaluation function (also considered in [<reflink idref="bib38" id="ref52">38</reflink>]), given by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><msup><mo stretchy="false">)</mo><mi>κ</mi></msup><mtext>\,for</mtext><mspace width="1em" /><mi>u</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><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>κ</mi><mo>></mo><mn>1</mn></math> </ephtml> is the inequality-aversion parameter; <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> is the income position or rank.</p> <p>In principle, one may use any evaluation functions in the summary measure. But the Gini family is an important special case of evaluation functions, which are commonly used in the literature on social welfare and inequality and characterized by one parameter representing the degree of inequality aversion (e.g., see [<reflink idref="bib3" id="ref53">3</reflink>]; [<reflink idref="bib38" id="ref54">38</reflink>]). As we will show below, these measures are capable of producing most weighting schemes of interest. In the example of Great Gatsby Curve, the inequality measure is indeed the Gini coefficient, and this class of mobility measures then facilitate the analysis of the relationship between comparable inequality and mobility measures.</p> <p>There are also several technically appealing properties of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> </ephtml> . Under the Gini preference function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> </ephtml> , the ordering of Lorenz curves can be represented by the extended Gini family of inequality measures ([<reflink idref="bib19" id="ref55">19</reflink>]) which includes the Gini coefficient as a special case when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>3</mn></math> </ephtml> . The higher <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> , the more inequality-averse a society. On the one extreme case <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>1</mn></math> </ephtml> , society is indifferent to inequality; on the other extreme case <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo stretchy="false">→</mo><mi mathvariant="normal">∞</mi></math> </ephtml> , society cares most about the welfare of the poor. Furthermore, the derivative of the preference function, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msubsup><mi>P</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><mi>u</mi><mo>=</mo><mi>κ</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><msup><mo stretchy="false">)</mo><mrow><mi>κ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>></mo><mn>0</mn></math> </ephtml> , reflects the weight placed on a particular income position <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> </ephtml> in the definition of welfare functions (e.g., [<reflink idref="bib1" id="ref56">1</reflink>], [<reflink idref="bib35" id="ref57">35</reflink>], [<reflink idref="bib36" id="ref58">36</reflink>], [<reflink idref="bib37" id="ref59">37</reflink>]). In addition, unlike the convex IGE preference, since <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>d</mi><mn>2</mn></msup><msubsup><mi>P</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>/</mo><mi>d</mi><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><mi>κ</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mrow><mi>κ</mi><mo>−</mo><mn>2</mn></mrow></msup><mo><</mo><mn>0</mn></math> </ephtml> , the policy maker's evaluation satisfies the principle of transfers ([<reflink idref="bib37" id="ref60">37</reflink>]), representing the inequality aversion of policy makers. Given its importance and appealing properties, we follow the literature and focus on the Gini evaluation function (e.g., [<reflink idref="bib2" id="ref61">2</reflink>]; [<reflink idref="bib3" id="ref62">3</reflink>]).</p> <p>Gini-Based Mobility Measure, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>s</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo></math> </ephtml> . The Gini-based mobility measure based on (<reflink idref="bib10" id="ref63">10</reflink>) is defined as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>s</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mfrac><mrow><mrow><mi mathvariant="normal">Cov</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>κ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">Cov</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>κ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where the denominator is the extended Gini variability index, and the numerator is the extended Gini covariance. Note that the constant of the derivative of the preference function ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> ) is dropped in the measure since it appears in both denominator and numerator. Below we present a systematic analysis of the weighting scheme behind our measure.</p> <p>Underlying Weights of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi mathvariant="bold">s</mi><mi mathvariant="bold">I</mi></msup><mo mathvariant="bold" stretchy="false">(</mo><mi>κ</mi><mo mathvariant="bold" stretchy="false">)</mo></mrow></math> </ephtml> : <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi mathvariant="bold">w</mi><mi>κ</mi><mi mathvariant="bold">I</mi></msubsup><mo mathvariant="bold" stretchy="false">(</mo><mi mathvariant="bold">u</mi><mo mathvariant="bold" stretchy="false">)</mo></mrow></math> </ephtml> . We substitute (<reflink idref="bib10" id="ref64">10</reflink>) into (<reflink idref="bib7" id="ref65">7</reflink>) and obtain the weighting scheme for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mi>κ</mi><mi>I</mi></msubsup></math> </ephtml><ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mfrac><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mi>κ</mi></msup></mrow><mrow><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><mo fence="false" stretchy="false">{</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mi>κ</mi></msup><mo fence="false" stretchy="false">}</mo><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><mo>.</mo></math> </ephtml></p> <p>Graph</p> <p>Hence, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>s</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo></math> </ephtml> can be expressed as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>s</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo><mrow><msub><mrow><mi mathvariant="script">S</mi></mrow><mi>X</mi></msub></mrow></msub><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>.</mo></math> </ephtml></p> <p>Graph</p> <p>Moreover, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> can be rewritten as a function of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> , that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>c</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><msup><mo stretchy="false">)</mo><mi>κ</mi></msup><mo stretchy="false">]</mo></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>c</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo fence="false" stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><msup><mo stretchy="false">)</mo><mi>κ</mi></msup><mo fence="false" stretchy="false">}</mo><mi>d</mi><msubsup><mi>F</mi><mi>X</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>></mo><mn>0</mn></math> </ephtml> is a positive constant, depending on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub><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"><mi>κ</mi></math> </ephtml> . As shown below, the expression of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> in terms of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> </ephtml> is convenient for analyzing its properties.</p> <p>Properties of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi mathvariant="bold">w</mi><mi>κ</mi><mi mathvariant="bold">I</mi></msubsup><mo mathvariant="bold" stretchy="false">(</mo><mi mathvariant="bold">u</mi><mo mathvariant="bold" stretchy="false">)</mo></mrow></math> </ephtml> . The first- and second-order derivatives of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> </ephtml> are given, respectively, by <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>u</mi></mrow></mfrac></mrow><mo>=</mo><msup><mi>c</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mrow><mo>[</mo><mi>κ</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><msup><mo stretchy="false">)</mo><mrow><mi>κ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>]</mo></mrow><mspace width="1em" /><mtext>and</mtext><mspace width="1em" /><mrow><mfrac><mrow><msup><mi>d</mi><mn>2</mn></msup><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><msup><mi>u</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>=</mo><msup><mi>c</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mi>κ</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><msup><mo stretchy="false">)</mo><mrow><mi>κ</mi><mo>−</mo><mn>2</mn></mrow></msup><mo><</mo><mn>0.</mn></math> </ephtml></p> <p>Graph</p> <p>The first-order condition implies that the maximizer (the turning point) of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> </ephtml> is given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>u</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>κ</mi><mrow><mn>1</mn><mo>/</mo><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>−</mo><mi>κ</mi></mrow><mo stretchy="false">)</mo></mrow></msup></math> </ephtml> . The second-order condition implies that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> </ephtml> is strictly concave in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> </ephtml> . As the upper left panel of Figure 3 illustrates, the weight increases for lower values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> </ephtml> , reaches a maximum, and then declines. The key properties of the weighting scheme behind the Gini family are summarized in Proposition 4.</p> <p>Graph: Figure 3. Shapes of weight functions and turning points for different inequality-aversions. Note: The left panel is for the weights wκI(u). The right panel is for turning points uI(κ). The income X has standard normal distribution.</p> <p>We obtain the following properties:</p> <p></p> <ulist> <item> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>u</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo></math> </ephtml> is strictly decreasing in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>></mo><mn>1</mn></math> </ephtml> .</item> <p></p> <item> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow /><mrow><mi>κ</mi><mo stretchy="false">→</mo><mi mathvariant="normal">∞</mi></mrow><mtext>lim</mtext></msubsup><mspace width=".1em" /><msup><mi>u</mi><mi>I</mi></msup><mspace width="0.25em" /><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mn>0</mn></math> </ephtml> .</item> <p></p> <item> For a sufficiently large <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>w</mi><mi>κ</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>≈</mo><msup><mi>c</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>u</mi><mo stretchy="false">)</mo></math> </ephtml> for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>u</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math> </ephtml> .</item> </ulist> <p>Property (i) states that the location of maximum weight (turning point) decreases in the inequality aversion parameter. In other words, as a society or policymaker becomes more inequality averse, the individuals from the more disadvantaged families should receive the maximum weight. The pattern is illustrated in the upper right panel of Figure 3. Property (ii) states that when the inequality aversion tends to infinity, the largest weight is indeed placed on the poorest individuals. More importantly, the relative weight for the poor is larger than that for the rich when we increase the inequality aversion. This later feature is evident in the upper left panel of Figure 3 and formally stated in property (iii). For a sufficiently large inequality aversion parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> , the weighting scheme can be approximated by a downward slopping line for almost all the values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> </ephtml> . This result can be of practical importance as well, since this type of weighting schemes is usually consistent with what many empirical researchers have in mind. Moreover, it suggests that the researchers simply need to assign the inequality aversion parameter a relatively large value to obtain such weighting scheme.</p> <p> <bold>Robustness</bold> It is worthwhile to note that the weights are unknown and can be quantitatively different under different income distributions. These results remain unchanged under an alternative standard uniform distribution of income. As shown in Figures 3 and 4, the shapes of weight functions under a normal distribution are close to those under an uniform distribution.[<reflink idref="bib18" id="ref66">18</reflink>]</p> <p>Graph: Figure 4. Shapes of weight functions and turning points for different inequality-aversions.</p> <hd id="AN0188424704-9">Estimation</hd> <p>We provide the estimation of our proposed summary measures of mobility. Suppose that the data consist of an independent sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msub><mi>Y</mi><mi>i</mi></msub><mo>,</mo><msub><mi>X</mi><mi>i</mi></msub><msubsup><mo fence="false" stretchy="false">}</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup></math> </ephtml> of size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> </ephtml> . Since (<reflink idref="bib11" id="ref67">11</reflink>) takes the form of a Wald-IV estimand, we can conveniently estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>s</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo></math> </ephtml> using the two steps. Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>κ</mi><mo>−</mo><mn>1</mn></mrow></msup></math> </ephtml> . First, we obtain the estimate of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>i</mi></msub></math> </ephtml> by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>Q</mi><mo>^</mo></mover></mrow><mi>i</mi></msub><mo>=</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mrow><mover><mi>F</mi><mo>^</mo></mover></mrow><mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>κ</mi><mo>−</mo><mn>1</mn></mrow></msup></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>F</mi><mo>^</mo></mover></mrow><mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mi>n</mi><munderover><mo>∑</mo><mrow><mspace width=".1em" /><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mrow><mi mathvariant="double-struck">I</mi></mrow></mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mi>j</mi></msub><mo>≤</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">I</mi></mrow></mrow><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is an indicator function equal to one if the argument is met and zero otherwise. Next, we run an IV regression of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>i</mi></msub></math> </ephtml> , with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>Q</mi><mo>^</mo></mover></mrow><mi>i</mi></msub></math> </ephtml> being the IV for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>i</mi></msub></math> </ephtml> . The IV estimator of the coefficient on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>i</mi></msub></math> </ephtml> is the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mover><mi>s</mi><mo>^</mo></mover></mrow><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo></math> </ephtml> of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>s</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo></math> </ephtml> .</p> <hd id="AN0188424704-10">Empirical Illustration</hd> <p></p> <hd id="AN0188424704-11">Data</hd> <p>Our application is based on data from the Panel Study of Income Dynamics (PSID), which includes information at the household and individual levels for a nationally representative sample of the population of the United States. The data collection began in 1968 and has since continued to update information on the individuals of the original sample and their descendants. The long panel structure allows us to match children to their parents for intergenerational analysis, as well as to obtain their incomes at a wide range of stages over the life-cycle for both generations. See [<reflink idref="bib29" id="ref68">29</reflink>], [<reflink idref="bib30" id="ref69">30</reflink>] for excellent accounts of the unique advantages of the PSID data for analysis of intergenerational mobility, and over administrative tax data. Because of these advantages, the PSID data are widely used in the literature on estimation of intergenerational mobility. Use of alternative datasets does not impact the illustrative purpose of our analysis, or the central message of the paper.</p> <p>To facilitate comparison to the literature, especially those studies using the PSID, we follow closely the standard practices in the literature to construct our sample and relevant variables, and therefore provide only limited details here. Following the literature (e.g., [<reflink idref="bib31" id="ref70">31</reflink>]; [<reflink idref="bib20" id="ref71">20</reflink>]), we include only the Survey Research Center component of the PSID, but exclude the Survey of Economic Opportunity (SEO) component to prevent over-representing the poverty sample. Recent literature has also noted some serious irregularities in the sampling of SEO respondents that can "preclude easy generalization to any well-defined population" ([<reflink idref="bib9" id="ref72">9</reflink>]; An et al. 2022).</p> <p>In our analysis, we use (the logarithm of) permanent incomes for both children and parents. Following the literature (e.g., [<reflink idref="bib20" id="ref73">20</reflink>]), we define the permanent income as the average of annual family incomes, which include the taxable income of all earners in the family, from all sources, and transfer payments. We exclude zero and negative incomes. These income variables are converted to 2015 dollars using the Consumer Price Index.[<reflink idref="bib19" id="ref74">19</reflink>] We also follow [<reflink idref="bib30" id="ref75">30</reflink>] to take advantage of the very long panel structure of the PSID and center the average around age 40 (between 30 and 50). The choice of age 40 follows the rule of thumb in the literature that largely overcomes the life cycle bias ([<reflink idref="bib22" id="ref76">22</reflink>]; [<reflink idref="bib30" id="ref77">30</reflink>]). The life-cycle bias is due to the heterogenous life cycle earnings profiles, where individuals with high lifetime income often have relatively low income when younger, and use of the incomes when they are young can then bias the estimates downward ([<reflink idref="bib23" id="ref78">23</reflink>]). We also restrict the sample to those individuals with at least three observations of annual incomes (e.g., [<reflink idref="bib20" id="ref79">20</reflink>]).</p> <p>These standard practices also mitigate some of the known issues such as the issue of zero incomes that typically arise when using the administrative data due to non-employment. First, the family total income in the PSID includes sources of income such as transfers that are not available in the administrative tax record, and it is still reported "even when it may be too low to be filed for tax purposes" (An et al. 2022). Second, the PSID has a better coverage of life-cycles than the administrative records. Therefore, very few instances of zero incomes exist in the PSID, and the instances of multiple years of zero incomes are even rarer. Discussing these issues with the PSID for estimation of intergenerational mobility, [<reflink idref="bib29" id="ref80">29</reflink>] concludes that "the concerns about the sensitivity of results around how to handle years of zero income is effectively a non-issue when using family income." See An et al. (2022) for more details on this issue as well.</p> <hd id="AN0188424704-12">Results</hd> <p></p> <hd id="AN0188424704-13">Baseline Results</hd> <p>To facilitate the comparison to the literature, we report the Gini family of mobility measures, along with the traditional IGE in Table 1. The results based on the Gini evaluation function for inequality aversion <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>2.1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>51</mn><mo>,</mo><mn>101</mn><mo>,</mo><mn>501</mn></math> </ephtml> . The level linear regression using the full sample yields an estimate of about <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.5371</mn></math> </ephtml> (Panel A), consistent with the previous literature using PSID with an average of multiple years of annual incomes.</p> <p>Table 1. Measures of Immobility (Full Sample).</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" colspan="7"><bold>Panel A: Conventional IGE</bold></th></tr></thead><tbody><tr><td>Log of</td><td>0.5371***</td><td /><td /><td /><td /><td /></tr><tr><td>Father's income</td><td>(0.0292)</td><td /><td /><td /><td /><td /></tr><tr><td colspan="7">Panel B: Gini Family of Measures (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">></mo><mn xmlns="">1</mn></math></p>)</td></tr><tr><td /><td colspan="6">Low <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false" xmlns="">→</mo></math></p> Higher Inequality Aversion</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">k</mi><mo xmlns="">=</mo><mn xmlns="">2.1</mn></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">k</mi><mo xmlns="">=</mo><mn xmlns="">3</mn></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">k</mi><mo xmlns="">=</mo><mn xmlns="">11</mn></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">k</mi><mo xmlns="">=</mo><mn xmlns="">51</mn></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">k</mi><mo xmlns="">=</mo><mn xmlns="">101</mn></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">k</mi><mo xmlns="">=</mo><mn xmlns="">501</mn></math></p></td></tr><tr><td /><td>0.5814***</td><td>0.5905***</td><td>0.5641***</td><td>0.4870***</td><td>0.4535***</td><td>0.2828**</td></tr><tr><td /><td>(0.0311)</td><td>(0.0317)</td><td>(0.0380)</td><td>(0.0547)</td><td>(0.0668)</td><td>(0.1121)</td></tr><tr><td><bold>Observations</bold></td><td colspan="6">2042</td></tr></tbody></table> </ephtml> </p> <p>1 1. The data are from Panel Study of Income Dynamics (PSID). Level regression and rank-rank regression represent the estimates based on, respectively, the regression of log child income and log parental income, and the regression of the ranks of child income and the ranks of parental income.</p> <p>2 2. The Gini family of measures are based on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>s</mi><mi>I</mi></msup><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mfrac><mrow><mi>Cov</mi><mspace width="0.2em" /><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>κ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mi>Cov</mi><mspace width="0.2em" /><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>κ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml> , with varying levels of inequality aversion <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> .</p> <p>On the other hand, the Gini measures of mobility vary drastically with respect to the inequality aversion parameter, and so does our impression of the mobility level in the U.S. For example, the correlation coefficient is between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.5905</mn></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>3</mn></math> </ephtml> ) and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.2828</mn></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>501</mn></math> </ephtml> ), with the difference being more than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> </ephtml> % ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mn>.5905</mn><mo>−</mo><mn>.2828</mn><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mn>.2828</mn></mrow><mo>×</mo><mn>100</mn><mo>≈</mo><mn>109</mn></math> </ephtml> ). As a larger <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> is associated with larger weights on poorer households, evidently, our results suggest a substantially more mobile society when focusing more on the individuals from the disadvantaged families. The IGE of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.5371</mn></math> </ephtml> falls between the coefficient using <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>11</mn></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.5641</mn></math> </ephtml> ) and that using <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>51</mn></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.4870</mn></math> </ephtml> ). These results provide strong evidence that the income transmission process is highly nonlinear, and as a result, the subjective weighting schemes matter. In a (preferred) special case of Gini measures with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>501</mn></math> </ephtml> (monotonically declining weights over the income distribution), we actually find a substantially more mobile measure than implied by the traditional IGE.</p> <p>The pattern of the changes with respect to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> is, however, not monotonic. We observe that at the relatively low level of inequality aversion, an increase in inequality aversion (from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>2.1</mn></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> </ephtml> ) leads to a larger coefficient and hence a higher level of immobility. On the other hand, for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>></mo><mn>3</mn></math> </ephtml> , when the inequality aversion parameter increases and we place more weights on the individuals from the disadvantaged families, we actually observe that the IGM coefficients decrease substantially in magnitudes, suggesting a more mobile society. This pattern appears to be consistent with ([<reflink idref="bib25" id="ref81">25</reflink>]), which examines mobility using a nonparametric approach with time-varying coefficients. Their results indicate an inverted U-shaped relationship between IGEs and parental income, where individual IGEs are lower at both ends of the parental income distribution. In other words, both children from disadvantaged and richer families may have a higher level of mobility than those from the "middle class".[<reflink idref="bib20" id="ref82">20</reflink>]</p> <hd id="AN0188424704-14">Geographic Disparities in Mobility</hd> <p>We also examine geographic differences in mobility. Following the literature, we compare four regions where an individual grew up: the Northeast, the North Central, the South, and the West. The conventional IGE results are displayed in Table 2. There exist significant geographic disparities in IGMs: the South is less mobile than the West, in line with [<reflink idref="bib13" id="ref83">13</reflink>].[<reflink idref="bib21" id="ref84">21</reflink>] The IGE results suggest that the South is at least <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>30.63</mn></math> </ephtml> percent less mobile than the West ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>30.63</mn><mo>≈</mo><mo stretchy="false">(</mo><mn>0.5855</mn><mo>−</mo><mn>0.4482</mn><mo stretchy="false">)</mo><mo>/</mo><mn>0.4482</mn></math> </ephtml> ). Furthermore, the South is the least mobile region, while the West is the most mobile region. Meanwhile, the North Central is less mobile than the Northeast.</p> <p>Table 2. Measures of Immobility: Conventional Regression Approaches (By Region).</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /></colgroup><thead><tr><th align="center" /><th align="center" /><th align="center" colspan="3">By Region</th></tr><tr><th align="center" /><th align="center" valign="bottom">Northeast</th><th align="center">NorthCentral</th><th align="center">South</th><th align="center">West</th></tr><tr><th align="center" /><th align="center">(1)</th><th align="center">(2)</th><th align="center">(3)</th><th align="center">(4)</th></tr></thead><tbody><tr><td>Log of</td><td>0.4650<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5193<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5855<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.4482<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td>Father's Income</td><td>(0.0634)</td><td>(0.0497)</td><td>(0.0544)</td><td>(0.0827)</td></tr><tr><td /><td /><td /><td /><td /></tr><tr><td><bold>Observations</bold></td><td>407</td><td>776</td><td>532</td><td>327</td></tr></tbody></table> </ephtml> </p> <p>3 1. The data are from Panel Study of Income Dynamics (PSID). Level regression and rank-rank regression represent the estimates based on, respectively, the regression of log child income and log parental income, and the regression of the ranks of child income and the ranks of parental income.</p> <p>The results based on the Gini evaluation functions are presented in Table 3. Varying the inequality aversion parameters impacts both the size and patterns of IGM across regions. First, the variation of the estimates with respect to the inequality aversion parameters differs across regions, suggesting significant between-group differences in the income transmission process and the extent of nonlinearity. For example, for the Northeast, when we place more weights on the children from the disadvantaged families, the size of the mobility decreases by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn></math> </ephtml> percent when comparing the largest value ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.1115</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>501</mn></math> </ephtml> ) and the smallest value ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.5481</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>3</mn></math> </ephtml> ). By contrast, the coefficient is only <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math> </ephtml> percent smaller for the North Central when comparing the smallest coefficient ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.4941</mn></math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>501</mn></math> </ephtml> ) with the largest ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.5906</mn></math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>3</mn></math> </ephtml> ).</p> <p>Table 3. Measures of Immobility: Gini Family (By Region).</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /></colgroup><thead><tr><th align="center" /><th align="center" rowspan="2">Parameter</th><th align="center">Northeast</th><th align="center">North<bold>Central</bold></th><th align="center">South</th><th align="center">West</th></tr><tr><th align="center" /><th align="center">(1)</th><th align="center">(2)</th><th align="center">(3)</th><th align="center">(4)</th></tr></thead><tbody><tr><td>Low</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">2.1</mn></math></p></td><td>0.5306<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5782<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5956<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.4611<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td /><td /><td>(0.0692)</td><td>(0.0535)</td><td>(0.0561)</td><td>(0.0870)</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">3</mn></math></p></td><td>0.5481<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5906<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5758<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.4938<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false" xmlns="">↓</mo></math></p></td><td /><td>(0.0711)</td><td>(0.0552)</td><td>(0.0570)</td><td>(0.0875)</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">11</mn></math></p></td><td>0.5442<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5750<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5056<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5649<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td>High</td><td /><td>(0.0851)</td><td>(0.0686)</td><td>(0.0701)</td><td>(0.1013)</td></tr><tr><td>Inequality</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">51</mn></math></p></td><td>0.4309<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5171<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.4393<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.4743<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td /><td /><td>(0.1127)</td><td>(0.0993)</td><td>(0.1075)</td><td>(0.1516)</td></tr><tr><td>Aversion</td><td /><td /><td /><td /><td /></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">101</mn></math></p></td><td>0.3408<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5643<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.3850<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.4871<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td /><td /><td>(0.1272)</td><td>(0.1212)</td><td>(0.1396)</td><td>(0.2005)</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">501</mn></math></p></td><td>0.1115</td><td>0.4941<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.3892<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mo>*</mo></msup></math></p></td><td>0.7825<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mo>*</mo></msup></math></p></td></tr><tr><td /><td /><td>(0.1936)</td><td>(0.1992)</td><td>(0.2358)</td><td>(0.4121)</td></tr></tbody></table> </ephtml> </p> <p>4 1. The data are from Panel Study of Income Dynamics (PSID). Level regression and rank-rank regression represent the estimates based on, respectively, the regression of log child income and log parental income, and the regression of the ranks of child income and the ranks of parental income.</p> <p>Second, the patterns of the changes with respect to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> differ from the full sample (i.e., when we increase the inequality aversion, the coefficient first increases and then decreases) for the South and the West. For the South, the coefficients <emph>decrease</emph> with respect to the inequality aversion parameter, while for the West, the coefficients <emph>increase</emph>, fluctuating around an increasing trend. When placing more and more weights on the children from the disadvantaged families, we actually find that the West becomes less and less mobile (the coefficient increases from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.4611</mn></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.7825</mn></math> </ephtml> ).</p> <p>Our impression of the <emph>relative</emph> mobility levels is indeed sensitive to the change of the inequality aversion parameter. It starts to change when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>11</mn></math> </ephtml> , and we observe that the West is actually the least mobile region. As inequality aversion increases, a more stable relative ranking emerges. In fact, when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>501</mn></math> </ephtml> and we place more weights on the individuals from the most disadvantaged families, the Northeast is the most mobile region and the West is the least mobile region.[<reflink idref="bib22" id="ref85">22</reflink>]</p> <hd id="AN0188424704-15">Dynamics of Mobility</hd> <p>To examine how the mobility evolves across cohorts, we consider four cohorts (those born before 1954, between 1955 and 1961, between 1962 and 1967, after 1968). The IGE results are displayed in Table 4. The results imply an increase in the magnitudes of the correlation coefficients and a decrease in mobility over time when comparing the (first) cohort born before 1954 and the (last) cohort born after 1968. Specifically, the IGE results suggest about 13% decrease in mobility ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mn>.5597</mn><mo>−</mo><mn>.4949</mn><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mn>.4949</mn></mrow><mo>×</mo><mo>≈</mo><mn>13</mn></math> </ephtml> ). Furthermore, the IGE results suggest an increasing trend between cohorts; we find that relative to the cohort born before 1954, the coefficient is larger for the cohort born during the period 1962 to 1967.</p> <p>Table 4. Measures of Immobility: Conventional IGEs (By Cohort).</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /></colgroup><thead><tr><th align="center" /><th align="center" colspan="4">By Birth Cohort</th></tr><tr><th align="center" /><th align="center"><bold>Before</bold></th><th align="center"><bold>1954-</bold></th><th align="center"><bold>1961-</bold></th><th align="center"><bold>Post</bold></th></tr><tr><th align="center" /><th align="center"><bold>1954</bold></th><th align="center"><bold>1961</bold></th><th align="center"><bold>1967</bold></th><th align="center"><bold>1967</bold></th></tr><tr><th align="center" /><th align="center">(1)</th><th align="center">(2)</th><th align="center">(3)</th><th align="center">(4)</th></tr></thead><tbody><tr><td>Log of</td><td>0.4949<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5303<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5226<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5597<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td>Father's Income</td><td>(0.0754)</td><td>(0.0602)</td><td>(0.0789)</td><td>(0.0422)</td></tr><tr><td><bold>Observations</bold></td><td>337</td><td>422</td><td>365</td><td>918</td></tr></tbody></table> </ephtml> </p> <p>5 1. The data are from Panel Study of Income Dynamics (PSID). Level regression and rank-rank regression represent the estimates based on, respectively, the regression of log child income and log parental income, and the regression of the ranks of child income and the ranks of parental income.</p> <p>The results based on the Gini evaluation functions are reported in Table 5. Varying the inequality aversion parameter again can drastically revise our view of mobility for a particular cohort, as well as that of the dynamics of the mobility across cohorts. First, the coefficients do not monotonically vary with the inequality aversion parameter ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi></math> </ephtml> ), and the patterns differ drastically across cohorts.[<reflink idref="bib23" id="ref86">23</reflink>] Second, our relative ranking of the mobility levels across cohorts also depend crucially on the part of the distribution which a particular weighting scheme emphasizes. For example, when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>2.1</mn></math> </ephtml> , we observe the same ranking as the level regression, where it is more mobile for the cohort born before 1954 than for the cohort born after 1968. Such impression is reversed starting when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo><mn>51</mn></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.5747</mn></math> </ephtml> (Before 1954) vs <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>.4751</mn></math> </ephtml> (After 1968)). In other words, our preferred specifications that place more weights on the individuals from the most disadvantaged families, actually indicate the society has become more mobile over time.</p> <p>Table 5. Measures of Immobility: Gini Family (By Cohort).</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /></colgroup><thead><tr><th align="center" /><th align="left" /><th align="center">Before</th><th align="center">1954-</th><th align="center">1961-</th><th align="center">Post</th></tr><tr><th align="center" /><th align="center" /><th align="center">1954</th><th align="center">1961</th><th align="center">1967</th><th align="center">1967</th></tr><tr><th align="center" /><th align="left">Parameter</th><th align="center">(1)</th><th align="center">(2)</th><th align="center">(3)</th><th align="center">(4)</th></tr></thead><tbody><tr><td>Low</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">2.1</mn></math></p></td><td>0.5054<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5706<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5505<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.6183<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td /><td /><td>(0.0798)</td><td>(0.0638)</td><td>(0.0829)</td><td>(0.0452)</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">3</mn></math></p></td><td>0.5230<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5875<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5150<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.6344<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false" xmlns="">↓</mo></math></p></td><td /><td>(0.0808)</td><td>(0.0652)</td><td>(0.0859)</td><td>(0.0459)</td></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">11</mn></math></p></td><td>0.5691<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5916<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.3741<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5914<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td>High</td><td /><td>(0.0963)</td><td>(0.0784)</td><td>(0.1121)</td><td>(0.0541)</td></tr><tr><td>Inequality</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">51</mn></math></p></td><td>0.5747<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.5171<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.2493</td><td>0.4751<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td /><td /><td>(0.1437)</td><td>(0.1149)</td><td>(0.1758)</td><td>(0.0757)</td></tr><tr><td>Aversion</td><td /><td /><td /><td /><td /></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">101</mn></math></p></td><td>0.5436***</td><td>0.4492<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td><td>0.3184</td><td>0.4074<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mrow><mo>*</mo><mo>*</mo><mo>*</mo></mrow></msup></math></p></td></tr><tr><td /><td /><td>(0.1822)</td><td>(0.1435)</td><td>(0.2195)</td><td>(0.0911)</td></tr><tr><td /><td /><td /><td /><td /><td /></tr><tr><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">κ</mi><mo xmlns="">=</mo><mn xmlns="">501</mn></math></p></td><td>0.2716</td><td>0.3696<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mo>*</mo></msup></math></p></td><td>0.9267<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mrow /><mo>*</mo></msup></math></p></td><td>0.2254</td></tr><tr><td /><td /><td>(0.2824)</td><td>(0.2012)</td><td>(0.4758)</td><td>(0.1488)</td></tr><tr><td /><td><bold>Observations</bold></td><td>337</td><td>422</td><td>365</td><td>918</td></tr></tbody></table> </ephtml> </p> <p>6 1. The data are from Panel Study of Income Dynamics (PSID). Level regression and rank-rank regression represent the estimates based on, respectively, the regression of log child income and log parental income, and the regression of the ranks of child income and the ranks of parental income.</p> <hd id="AN0188424704-16">Before Conclusions</hd> <p>The results here, both theoretical and empirical, may be uncomfortable for some. The ubiquitous heterogeneity and nonlinearity may imply that any conclusions regarding the mobility can be subjective and sensitive to the varying parameter. That is true. However, some of the <emph>qualitative</emph> conclusions do not have to be. It is important to see what consensus may arise from this kind of analysis. When no uniform conclusions can be reached, our paper points out the need to explicate the commitment to certain policy goals when measuring mobility. For example, many may agree that the measurement of mobility should reflect our care for the poor, and that monitoring the changes or policy effectiveness should place more weights on the children from more disadvantaged families. Only when such qualifying statements are made can our policy discussions be more meaningful and fruitful.</p> <hd id="AN0188424704-17">Conclusions</hd> <p>In this paper, we consider the decision-theoretic foundation of the intergenerational mobility measures in the presence of nonlinearity and heterogeneity. We first recast the dominant IGE approach as a weighted average of intergenerational income elasticities at different parts of the parental income distribution. A careful analysis of the weighting schemes underlying the IGE exposes some undesirable features of these approaches. A statistically valid procedure is inconsistent economically because the two serve fundamentally different purposes. Statistical models are data-driven tools optimized to fit patterns in the data according to specific objective functions such as mean squared error and mean absolute error, and they do not necessarily align with normative, welfare-theoretic or policy-relevant criteria. Similar points have been raised in recent work (presented at the same conference), such as [<reflink idref="bib10" id="ref87">10</reflink>], who demonstrate how absolute income mobility measures obscure the disadvantages faced by marginalized children. These findings reinforce our argument here that statistical validity does not equate to economic relevance.</p> <p>Our approach here provides a unifying structure for assessing mobility, transparently allowing for the integration of societal preference or policy criteria such as inequality aversion. More importantly, our generalized IGM measures is a significant economic one—for a given societal preference and welfare function, it can be <emph>economically</emph> interpreted as the marginal effect of parents' normalized social welfare on children's normalized social welfare. This perspective represents a significant departure from traditional analyses, and also aligns with the broader reconceptualization of mobility studies, as highlighted by [<reflink idref="bib34" id="ref88">34</reflink>] (presented at the same conference and considered for this special issue), who advocate for changing the level of analysis from the individual to the society.</p> <p>While the article primarily examines income-based intergenerational mobility measures, the problem we identify and the framework we propose are general. It is our hope that further analysis can be extended to consider other types of outcomes that play a role in policy objectives, such as educational attainment or social class.</p> <hd id="AN0188424704-18">Supplemental Material</hd> <p>Graph: Supplemental material, sj-pdf-1-smr-10.1177_00491241251357586 for Generalized Intergenerational Mobility Regressions by Esfandiar Maasoumi, Le Wang and Daiqiang Zhang in Sociological Methods & Research</p> <hd id="AN0188424704-19">Acknowledgments</hd> <p>We thank Professors Steven Durlauf, James Heckman, Stephen Jenkins, Larry Blume, Hashem Pesaran and participants at several seminars and conferences for their helpful comments and suggestions.</p> <ref id="AN0188424704-20"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref17" type="bt">1</bibl> <bibtext> The authors 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="ref22" type="bt">2</bibl> <bibtext> The authors received no financial support for the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref30" type="bt">3</bibl> <bibtext> Le Wang https://orcid.org/0000-0001-5918-2644 Daiqiang Zhang https://orcid.org/0009-0009-6617-9270</bibtext> </blist> <blist> <bibl id="bib4" idref="ref47" type="bt">4</bibl> <bibtext> The data and the code to replicate the results in the article are available at OpenICPSR https://<ulink href="http://www.openicpsr.org/openicpsr/project/233841/version/V1/view">www.openicpsr.org/openicpsr/project/233841/version/V1/view</ulink>.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref48" type="bt">5</bibl> <bibtext> Supplemental material for this article is available https://10.1177/00491241251357586.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref6" type="bt">6</bibl> <bibtext> Similar to [27], our paper is part of an attempt to connect and integrate the inequality literature more formally with the literature on IGM. On the other hand, our focus differs drastically from the inequality literature in that the latter focuses on univariate distributions, while we deal with the joint distribution of two incomes/outcomes.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref8" type="bt">7</bibl> <bibtext> In other words, those references fail to reflect diminishing marginal utility and inequality aversion. In this case, given a choice between two distributions of outcomes, society prefers a inequitable distribution over extreme disparities. Such preference is also inconsistent with the widely used Pigou–Dalton principle of transfers.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref1" type="bt">8</bibl> <bibtext> A little abuse of terminology here, referencing derivative at a particular income level as local mobility at that point.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref19" type="bt">9</bibl> <bibtext> Note that [38]'s results can be extended to the case of multiple regression.</bibtext> </blist> <blist> <bibtext> For the normal distribution, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> is equal to its density function. For the uniform distribution with the support <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><munder><mi>x</mi><mo>_</mo></munder><mo>,</mo><mi>x</mi><mo accent="false">¯</mo><mo stretchy="false">]</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo>6<mo stretchy="false">(</mo><mi>x</mi><mo accent="false">¯</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><munder><mi>x</mi><mo>_</mo></munder><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo accent="false">¯</mo><mo>−</mo><munder><mi>x</mi><mo>_</mo></munder><mo stretchy="false">)</mo><mrow><mo>−</mo>3</mrow></math> </ephtml> .</bibtext> </blist> <blist> <bibtext> Under a set of standard axioms of the preference relations, such as continuity, completeness, monotonicity, and independence, [36], [37] propose a new theory of choice. This theory is dual to the expected utility theory. The main advantage of it is that it can separate agent's attitude towards risk (increased uncertainty hurts) and attitude towards wealth (the loss hurts the poor relatively more), which are combined in the expected utility theory.</bibtext> </blist> <blist> <bibtext> For example, as [3] point out, an inequality neutral social planner would choose <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi></math> </ephtml> , which means <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>W</mi><mi>X</mi></msub><mo>=</mo><msub><mi>μ</mi><mi>X</mi></msub></math> </ephtml> .</bibtext> </blist> <blist> <bibtext> We assume that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>′</mo><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> </ephtml> is not a constant because otherwise, the denominator of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> is zero.</bibtext> </blist> <blist> <bibtext> The connection of our summary measure to social welfare parallels the literature on income inequality, in the spirit of Atkinson (1970) who advocates and proposes a measure of income inequality based on the social welfare in expected utility context. To distinguish, we refer to the social welfare in expected utility theory as level-dependent social welfare.</bibtext> </blist> <blist> <bibtext> A policy maker's preference relation is said to be equality minded if it satisfies the Pigou–Dalton principle of transfers ([37]).</bibtext> </blist> <blist> <bibtext> [2] uses Lorenz curve to define the second-degree stochastic dominance for ranking distribution functions. The distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi>1</msub></math> </ephtml> second-degree dominates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi>2</msub></math> </ephtml> if and only if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi>1</msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><msub><mi>P</mi>2</msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> for all <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo>0<mo>,</mo>1<mo stretchy="false">]</mo></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi>1</msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo>0<mi>t</mi></msubsup><msubsup><mi>F</mi>1<mrow><mo>−</mo>1</mrow></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi>2</msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo>0<mi>t</mi></msubsup><msubsup><mi>F</mi>2<mrow><mo>−</mo>1</mrow></msubsup><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi></math> </ephtml> are associated with distribution functions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi>1</msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi>2</msub></math> </ephtml> , respectively, and the two Lorenz curves <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi>1</msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi>2</msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> are defined similarly. Therefore, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi>1</msub></math> </ephtml> second-degree dominates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi>2</msub></math> </ephtml> if and only if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi>1</msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≥</mo><msub><mi>L</mi>2</msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> due to the fact that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>μ</mi><mi>X</mi></msub><mi>L</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> </ephtml> .</bibtext> </blist> <blist> <bibtext> This may explain why some theorists propose alternative concepts of mobility and inequality. For example, [24] proposes an index of inequality which decomposes into two components corresponding to vertical and horizontal equity. He uses the same approach to propose an index of social mobility, which is different from the conventional regression-based IGE.</bibtext> </blist> <blist> <bibtext> The shapes of turning points in Figures 3 and 4 are the same because the turning points do not rely on income distribution.</bibtext> </blist> <blist> <bibtext> Source: https://fred.stlouisfed.org/series/CPALTT01USA661S</bibtext> </blist> <blist> <bibtext> Please note that our statement is not directly contradictory to the literature (e.g., [15]; [28]; and An et al. 2022) because these studies rely on different measures or address different aspects of intergenerational mobility. For example, [28] and [15] focus primarily on upward mobility, emphasizing transitions from disadvantaged to higher socioeconomic statuses. An et al. (2022) tackle issues related to nonparametric and non-classical measurement errors in estimating mobility.</bibtext> </blist> <blist> <bibtext> Specifically, the IGE measures find that the coefficients are smaller in magnitude for the children from the West than those from the South.</bibtext> </blist> <blist> <bibtext> Viewing these results, there is some suggestive evidence that children from richer families may have a higher level of immobility or "affluence trap" in the South, while children from both disadvantaged and richer families may have a higher level of mobility than those from the "middle class" for the rest of the country. These results align with the findings in [17], who find that "the low mobility in the Southeast of the US documented by [13] is actually driven by low mobility by whites and that blacks who grew up in the Southeast actually experience higher mobility than blacks growing up in the Northeast and Midwest."</bibtext> </blist> <blist> <bibtext> For the cohorts born before 1954, between 1955-1961, and after 1968, we observe that the coefficients first increase and then decrease when we place more and more weights on the children from the low-income families. They peak at different inequality aversion parameters (for the cohort born before 1954, the largest coefficient is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">.5747</math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo>51</math> </ephtml> , while for the cohort born after 1968, the largest coefficient is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">.6344</math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo>3</math> </ephtml> ). On the other hand, for the cohort born between 1962 and 1967, we actually find that the coefficients first decrease and then increase, reaching the maximum when the maximum weights are placed on the children from the lowest-income families with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>κ</mi><mo>=</mo>501</math> </ephtml> .</bibtext> </blist> <blist> <bibtext> [4]</bibtext> </blist> </ref> <ref id="AN0188424704-21"> <title> References </title> <blist> <bibtext> Aaberge Rolf. 2000. " Characterizations of Lorenz Curves and Income Distributions." Social Choice and Welfare. 17: 639–653.</bibtext> </blist> <blist> <bibtext> Aaberge Rolf. 2001. " Axiomatic Characterization of the Gini Coefficient and Lorenz Curve Orderings." Journal of Economic Theory. 101: 115–132.</bibtext> </blist> <blist> <bibtext> Aaberge R., Havnes T., Mogstad M. 2021. " Ranking Intersecting Distribution Functions." 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" Generalized Gini Inequality Indices." Mathematical Social Sciences. 1: 409–430.</bibtext> </blist> <blist> <bibtext> Yaari Menahem E.1987. " The Dual Theory of Choice Under Risk." Econometrica: Journal of the Econometric Society. 55: 95–115.</bibtext> </blist> <blist> <bibtext> Yaari Menahem E.1988. " A Controversial Proposal Concerning Inequality Measurement." Journal of Economic Theory. 44: 381–397.</bibtext> </blist> <blist> <bibtext> Yitzhaki Shlomo. 1996. " On Using Linear Regressions in Welfare Economics." Journal of Business & Economic Statistics. 14: 478–486.</bibtext> </blist> </ref> <aug> <p>By Esfandiar Maasoumi; Le Wang and Daiqiang Zhang</p> <p>Reported by Author; Author; Author</p> <p></p> <p>Esfandiar Maasoumi is the Arts and Sciences Distinguished Professor of Economics at Emory University, former editor of Econometric Reviews, 1987–2024, a graduate of London School of Economics, with publications in econometrics and economics, including on inequality, poverty, and stochastic dominance.</p> <p>Le Wang is the David M Kohl Chair and Professor at Virginia Tech. His research interests are econometrics and applied microeconomics, focusing on inequality, and economic mobility.</p> <p>Daiqiang Zhang is an associate professor of economics at University at Albany, SUNY. His research interests are econometrics, empirical industrial organization, and labor economics.</p> </aug> <nolink nlid="nl1" bibid="bib32" firstref="ref2"></nolink> <nolink nlid="nl2" bibid="bib14" firstref="ref3"></nolink> <nolink nlid="nl3" bibid="bib21" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib33" firstref="ref5"></nolink> <nolink nlid="nl5" bibid="bib18" firstref="ref7"></nolink> <nolink nlid="nl6" bibid="bib11" firstref="ref10"></nolink> <nolink nlid="nl7" bibid="bib16" firstref="ref11"></nolink> <nolink nlid="nl8" bibid="bib25" firstref="ref12"></nolink> <nolink nlid="nl9" bibid="bib26" firstref="ref13"></nolink> <nolink nlid="nl10" bibid="bib38" firstref="ref16"></nolink> <nolink nlid="nl11" bibid="bib10" firstref="ref21"></nolink> <nolink nlid="nl12" bibid="bib36" firstref="ref26"></nolink> <nolink nlid="nl13" bibid="bib37" firstref="ref27"></nolink> <nolink nlid="nl14" bibid="bib12" firstref="ref29"></nolink> <nolink nlid="nl15" bibid="bib13" firstref="ref33"></nolink> <nolink nlid="nl16" bibid="bib15" firstref="ref44"></nolink> <nolink nlid="nl17" bibid="bib17" firstref="ref51"></nolink> <nolink nlid="nl18" bibid="bib19" firstref="ref55"></nolink> <nolink nlid="nl19" bibid="bib35" firstref="ref57"></nolink> <nolink nlid="nl20" bibid="bib29" firstref="ref68"></nolink> <nolink nlid="nl21" bibid="bib30" firstref="ref69"></nolink> <nolink nlid="nl22" bibid="bib31" firstref="ref70"></nolink> <nolink nlid="nl23" bibid="bib20" firstref="ref71"></nolink> <nolink nlid="nl24" bibid="bib22" firstref="ref76"></nolink> <nolink nlid="nl25" bibid="bib23" firstref="ref78"></nolink> <nolink nlid="nl26" bibid="bib34" firstref="ref88"></nolink>
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  Data: <searchLink fieldCode="AR" term="%22Esfandiar+Maasoumi%22">Esfandiar Maasoumi</searchLink><br /><searchLink fieldCode="AR" term="%22Le+Wang%22">Le Wang</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-5918-2644">0000-0001-5918-2644</externalLink>)<br /><searchLink fieldCode="AR" term="%22Daiqiang+Zhang%22">Daiqiang Zhang</searchLink> (ORCID <externalLink term="https://orcid.org/0009-0009-6617-9270">0009-0009-6617-9270</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22Sociological+Methods+%26+Research%22"><i>Sociological Methods & Research</i></searchLink>. 2025 54(4):1594-1623.
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  Data: 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
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  Data: 30
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  Data: 2025
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  Data: <searchLink fieldCode="DE" term="%22Regression+%28Statistics%29%22">Regression (Statistics)</searchLink><br /><searchLink fieldCode="DE" term="%22Social+Mobility%22">Social Mobility</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Analysis%22">Statistical Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Income%22">Income</searchLink><br /><searchLink fieldCode="DE" term="%22Parent+Child+Relationship%22">Parent Child Relationship</searchLink>
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  Data: 10.1177/00491241251357586
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  Data: 0049-1241<br />1552-8294
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  Label: Abstract
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  Data: Current research on intergenerational mobility (IGM) is informed by "statistical" approaches based on log-level regressions, whose "economic" interpretations remain largely unknown. We reveal the subjective value-judgments in them: they are represented by weighted-sums (or aggregators) over heterogeneous groups, with controversial "economic" properties. Log-level regressions tend to overrepresent the experiences of middle-class children while underrepresenting those from disadvantaged families. We propose a general construction of IGM measures that can incorporate any transparent "economic" preferences. They are interpreted as the marginal effect of parental normalized social welfare on children's normalized welfare. Conventional regressions are special cases with implicit economic preferences that fail inequality-aversion and the Pigou-Dalton principle of transfers. Empirically, a variety of economic preferences, with varying inequality aversion, demonstrate a nuanced view of mobility, and perspectives on geographic-differences and dynamics of it.
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      – SubjectFull: Statistical Analysis
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      – SubjectFull: Income
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      – TitleFull: Generalized Intergenerational Mobility Regressions
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