Multigenerational Social Mobility: A Demographic Approach

Abstract

Most social mobility studies take a two-generation perspective, in which intergenerational relationships are represented by the association between parents’ and offspring’s socioeconomic status. This approach, albeit widely adopted in the literature, has serious limitations when more than two generations of families are considered. In particular, it ignores the role of families’ demographic behaviors in moderating mobility outcomes and the joint role of mobility and demography in shaping long-run family and population processes. This paper provides a demographic approach to the study of multigenerational social mobility, incorporating demographic mechanisms of births, deaths, and mating into statistical models of social mobility. Compared to previous mobility models for estimating the probability of offspring’s mobility conditional on parent’s social class, the proposed joint demography-mobility model treats the number of offspring in various social classes as the outcome of interest. This new approach shows the extent to which demographic processes may amplify or dampen the effects of family socioeconomic positions due to the direction and strength of the interaction between mobility and differentials in demographic behaviors. I illustrate various demographic methods for studying multigenerational mobility with empirical examples using the IPUMS linked historical U.S. census representative samples (1850 to 1930), the Panel Study of Income Dynamics (1968 to 2015), and simulation data that show other possible scenarios resulting from demography-mobility interactions.

1. INTRODUCTION

Studies on social mobility are dominated by a two-generation perspective, in which researchers analyze the extent to which one’s socioeconomic status, in terms of education, income, occupations, and the like, is associated with that of one’s parent (Blau and Duncan 1967; Breen 2004; Erikson and Goldthorpe 1992; Featherman and Hauser 1978; Hout 1983). The most common method of analysis uses mobility tables, a contingency-table technique that summarizes the probability a child will be in a certain social position given his parent’s position (e.g., Ginsberg 1929; Glass 1954). From a statistical view, mobility tables are equivalent to a single transition matrix of a Markov chain, which describes the transition probability of moving from one social class to another in one generation step (Bartholomew 1967; Hodge 1966; Prais 1955; Svalastoga 1959; White 1963).

Mobility tables provide an elegant and effective approach to summarizing the transmission of social status across two generations, but this method’s limitations are widely discussed and debated. Duncan (1966b: 17), for example, cautioned mobility researchers more than 50 years ago that,

What is fundamental is that the process by which occupation structures are transformed—the succession of cohorts and intracohort net mobility—are not simply translatable into the processes one may observe in a so-called intergenerational occupation mobility table.

Duncan did not explicitly use the phrase “demography,” but his critique points to the importance of accounting for demographic processes that govern the transmission of social status from parents to offspring and the succession of generations in a population. More specifically, as Duncan (1966a) noted, the conventional mobility approach relies on a sample of respondents and their reports of their own parents. The parents are not representative of a previous generation or any cohort in “some definite prior moment in time” because the sample (1) necessarily excludes individuals who never had children; (2) overrepresents parents who have many offspring; and (3) includes parents born into different birth cohorts who vary by childbearing age.

From a demographic perspective, generations within families are linked not only by their socioeconomic statuses but also by their fertility, mortality, and marriage, among other demographic behaviors. These demographic outcomes, often stratified by social class, lead to variations between families in resources allocation, household formation, and changes in kinship structure, which, in turn, limit and condition the amount of family capital that can be inherited by subsequent generations (see examples in Lam 1986; Mare 1997; Mare and Maralani 2006; Maralani 2013; Preston and Campbell 1993). Compared to the traditional approach based on mobility tables, the demographic approach provides a more complete account of intergenerational processes, shifting attention from “how likely that offspring’s status resembles that of their parents” to “how intergenerational effects transpire” (Mare 2015: 101). By doing so, researchers are no longer restricted to the analysis of parents and offspring conditional on the existence of a given offspring but are now also able to consider the degree to which offspring will come into existence as an integral part of intergenerational influences (Mare and Maralani 2006). The demographic view of social mobility, albeit long established in the literature (e.g., Matras 1961, 1967), has been largely overlooked by major studies on social mobility until recently (Breen and Ermisch 2017; Lawrence and Breen 2016; Maralani 2013; Mare 1997; Mare and Maralani 2006).

The present study generalizes the demographic approach, which has hitherto focused on social mobility between two generations, to multiple generations. Multigenerational mobility research has proliferated in recent years, with new studies leveraging the increased availability of longitudinal, genealogical, and linked administrative data that provide information on family members over three or more generations (reviewed in Ruggles 2014; Ruggles et al. 2015; Song and Campbell 2017). Yet, most of these studies follow the tradition of mobility tables, examining the association of social status across three generations, especially the role of grandparents in their grandchildren’s status attainment, net of the widely-studied effects of parents (Chan and Boliver 2013; Ferrie et al. 2016; Jæger 2012; Mare 2011, 2014; Pfeffer 2014; Song 2016; Zeng and Xie 2014). Despite the considerable merit of these studies, the complexity of multigenerational influences has not been fully explored. To pass on their advantages or disadvantages, families must first have at least one offspring in each generation who can carry the family legacy. In the long run, families’ demographic behaviors may mute or exacerbate the effect of social mobility, leading to varying numbers and types of offspring in families. Eventually, some families may grow and account for a disproportionately large share of the population after several generations or hundreds of years, whereas others may decline or even become extinct (Song, Campbell, and Lee 2015). This paper illustrates multigenerational models that not only account for the circulation of elites in society due to social immobility but also provide aggregate-level inferences about long-term population renewal and change.

Building on previous theoretical constructs and contributions of social mobility and population renewal models, I introduce several joint demography-mobility models. The models incorporate a few new features into conventional discrete-state Markov chain mobility models (Bartholomew 1967; Blumen et al. 1955; Hodge 1966; Matras 1961; Singer and Spilerman 1973) by (1) adding multigenerational effects; (2) combining demographic processes with the transmission of social status; (3) addressing population heterogeneity in social mobility; (4) allowing the transition matrix to evolve over time; and (5) differentiating between one-sex and two-sex approaches. These models are extensions to the two-generation social reproduction model that focuses on female populations.

The rest of the paper proceeds as follows. In section 2, I describe traditional methods based on discrete-time Markov chains in which time is measured as “generations” and social status is measured by a finite number of discrete, qualitatively different categories.¹ Section 3 introduces a joint demography-mobility model, also known as the social reproduction model, which reflects the evolution of socioeconomic distributions over generations in a population. It also provides examples of higher-order social reproduction models that include additional parameters for ancestral influences. Section 4 introduces various definitions of multigenerational effects based on models in Section 3 and shows how to decompose the effects into demographic and mobility components. Section 5 shows long-term equilibria of multigenerational social reproduction models compared to those implied by simple Markov models. Section 6 illustrates a mixture model that allows for heterogeneous mobility and demographic regimes among subpopulations. Section 7 describes a two-sex version of the multigenerational social reproduction model that accounts for interactions between males and females, namely, the process through which two sexes mate and produce offspring with others of similar social statuses and jointly influence the social mobility outcomes of their offspring. Section 8 provides empirical examples of various types of multigenerational mobility and demographic models using data from the IPUMS linked representative samples of U.S. census data (IPUMS linked, 1850 to 1930), the Panel Study of Income Dynamics (PSID, 1968 to 2015), and simulation data that show a range of hypothetical demography-mobility interactions. Section 9 concludes the paper by identifying areas for future research on multigenerational methodology.

2. CLASSICAL SOCIAL MOBILITY MODELS BASED ON MARKOV CHAINS

From the outset of studies on social mobility, important theoretical and empirical advances have accompanied the development of new methods of data collection, measurement, and analysis (Ganzeboom et al. 1991; Treiman and Ganzeboom 2000). In one of the earliest studies on social mobility, Prais (1955) showed that the Markov chain representation of mobility processes has methodological advantages over contingency tables (e.g., Ginsberg 1929; Glass 1954). The Markov chain is a simple form of stochastic modeling in which the outcome state of the present generation depends only on that of the parent generation, not any other preceding generation. The model provides new measures of mobility—such as equilibrium distribution of social classes and the average time spent in a social class—beyond measures used in contingency tables, such as vertical and horizontal mobility rates (Sorokin 1959 [1927]), inflow and outflow percentages (Lipset and Bendix 1959), and mobility ratios (Carlsson 1958; Glass 1954; Rogoff 1953; Tyree 1973). These early endeavors, widely considered to be the first generation of mobility research, all relied on descriptive, global measures to summarize mobility patterns (Ganzeboom et al. 1991; Boudon 1973).

Below, I provide a brief overview of classic mobility models based on Markov chains. These models typically start with a mobility table in which rows refer to fathers’ positions and columns refer to sons’ positions (with I and J categories, respectively, and typically, I = J) (Bartholomew 1967). Mobility tables can be converted into a Markov chain transition matrix by standardizing mobility rates between categories as follows:

$$\sum _{j=1}^J{p}_{Y_2=j\mid Y_1=i}=\sum _{j=1}^J\frac{n_{ij}}{n_{i+}}=1$$ (1)

where $P_{Y_2=j\mid Y_1=i}$ denotes the probability that the son (G2) of a father (G1) in social position i ends up in position j; n_ij denotes the number of father-son dyads in positions i and j; and n_i+ denotes the total number of fathers in position i regardless of their sons’ positions.

Suppose we observe f_i fathers in social position i and s_j sons in position j. The transition matrix P that transforms the distribution of fathers into the distribution of sons satisfies

$$s_j=\sum _{i=1}^I{f}_i\cdot p_{Y_2=j\mid Y_1=i}(j=1,2,\dots ,J).$$ (2)

In matrix notation, fathers and sons in different positions are denoted by vectors F = [f₁, f₂, …, f_i, …, f_n] and S = [s₁, s₂, …, s_i, …, s_n], respectively. The matrix of mobility probabilities P with p_j|i in the ith row and jth column is represented as a square matrix. A transition matrix has all entries as mobility probabilities between 0 and 1. The sum of entries in each row equals 1. The matrix shows the probability of change in social position from one generation to the next.

The matrix form of the intergenerational transmission of social classes is written as

$$S_{1\times n}=F_{1\times n}{P}_{n\times n}.$$ (3)

Assuming mobility rates are fixed over time, we can derive the distribution of men after two generations as

$$S^{(2)}=\left(F^{(0)}\cdot P\right)\cdot P=F^{(0)}\cdot P^2.$$ (4)

Furthermore, the distribution of descendants, namely the expected proportion of men in various social positions after t generations, can be projected by taking the matrix P to the t^th power,

$$S^{(t)}=S^{(t-1)}\cdot P=F^{(0)}\cdot P^t.$$ (5)

where F^(t) and S^(t) refer to the status distribution of fathers and sons in the t^th generation, respectively. This equation shows the process through which the initial progenitor distribution is transformed into subsequent generations after several generations of social mobility. The process retains no memory, in the sense that a man’s social position entirely depends on that of his father. If the position of one’s father has been taken into account, then his grandfather, great-grandfather, and earlier ancestors have no impact on his probability of attaining a specific position. A grandfather who fails to transmit his position to his son is incapable of influencing the outcomes of his grandson independently of his son. The memoryless property also makes it possible to predict how the Markov process behaves in the long run; that is, the eventual distribution of descendants after a sufficient number of generations. Provided that the transition matrix is regular, as time progresses, the process will “forget” its initial distribution and converge to a unique equilibrium distribution of the descendants that is unrelated to the initial distribution (Norris 1998).² This property implies that

$$\underset{t\to \infty }{\text{lim}}{F}^{(0)}\cdot P^t=\pi$$ (6)

where π is called the equilibrium vector of the Markov chain. This property suggests that in the short run, the initial distribution of progenitors influences future generations, but the influence diminishes as time passes. In the long run, the descendant distribution is only determined by the transition matrix P.

According to Coleman (1964a: 462), “the intent of the (Markov) model is not to mirror reality in all its facets. It is, instead, to see just how much of reality can be mirrored by a highly constrained process. That is, our question will be: How well does this rather restrictive assumption allow us to account for the data on intergenerational mobility?” To evaluate the suitability of a Markov chain model for representing a multigenerational process, it is important first to identify assumptions implied in this model. Below, I list five key assumptions that are modified from Pullum (1975: 16–17).

Assumption 1 [No Demography].

Families’ social mobility, $P_{Y_2\mid Y_1}$, is assumed to be independent of demographic behaviors, such as mortality, fertility, adoption, mating, and migration, as well as the timing of these events, in any generation, R. In particular, families’ social status, Y, does not affect their number of children or long-term reproductive success.

Assumption 2 [Markov Property].

The father mediates all multigenerational influences on his son, $P_{Y_n\mid {\overline{Y}}_{n-1}}=P_{Y_n\mid Y_{n-1},Y_{n-2}\cdots Y_1}=P_{Y_n\mid Y_{n-1}}$. The grandfather, great-grandfather, remote ancestors, and wider kin network do not affect the son when accounting for the father’s influence. Thus, the total influences of one’s ancestors are equal to the total influence of the father.

Assumption 3 [Homogeneous Mobility Regime].

A single mobility regime, P, in the society is assumed, so that all individuals in a population are subject to the same set of mobility probabilities given their fathers’ social class, $p_{Y_2\mid Y_1}$. This assumption also implies that the population is homogeneous concerning characteristics other than the measure of the social class under consideration.

Assumption 4 [Transition Stationarity].

The intergenerational transition matrix does not change as the history unfolds, that is, P(t) = P. All multigenerational relationships can be derived from the time-invariant two-generation mobility table.

Assumption 5 [One-Sex Mobility].

The model includes only fathers and sons; it ignores women’s social statuses and the potential influence of mothers and maternal ancestors, namely, $P_{Y_{g_n=\text{son}}\mid Y_{g_{n-1}= \text{father}}}=P_{Y_{g_n}=\left\{\text{son, daughter}\right\}}\mid Y_{g_{n-1}=\left\{\text{father, mother}\right\}}$. The role of mating rules, such as assortative mating according to social status in determining the number of marriages and families’ reproductive behaviors in a population, is not considered.

Mare (2011) provided examples of social contexts in which these assumptions may be violated and discussed the implications of these violations to clarify multigenerational mechanisms. In the following sections, I modify each of the five assumptions and show variants of stochastic models that may better characterize multigenerational processes under different circumstances.

3. A JOINT DEMOGRAPHY-SOCIAL MOBILITY MODEL

3.1. A Two-Generation Setup

The mobility table in Markov chain models provides a straightforward way of assessing the degree of social mobility between generations. Yet, as discussed earlier, mobility tables represent fathers’ and sons’ occupational distributions by giving equal weight to sons from families of unequal size, ignoring the fact that some fathers may have many sons while others have none. The transmission of social status from fathers to sons is not a simple, one-to-one mapping; instead, demographic processes, such as births, deaths, and migrations, may all influence the number of offspring observed in a mobility table (Kahl 1957; Pullum 1970).

Thus far, we do not have an agreed-upon solution for translating a Markov mobility model into a model that illustrates changes in social structure and population renewal simultaneously. Conceptually, the Markov model is flawed by the lack of a distinction between generation and birth cohort. If we define the son generation as a birth cohort whose occupational distribution is observed at a recent point in time, then fathers’ occupations—represented by the marginal distribution of the mobility table—do not comprise the occupational distribution of any birth cohort at any prior point in time (Duncan 1966a). As fathers’ levels and timing of fertility vary, a generation of fathers consists of a group of men whose birth years are not well-defined—and often not even reported in retrospective surveys of sons. Such ambiguity also occurs in the definition of the sons’ birth cohort when mobility tables are constructed from a prospective perspective (Song and Mare 2015; Yasuda 1964). Methodologically, a mobility table is not equivalent to a population projection matrix that can be used to describe population dynamics. Mobility tables often exclude age-specific information that could be used to predict the progression of birth cohorts. They also exclude individuals’ life history events—such as the school-to-work transition, job promotion and changes, retirement, or even death—that could be used to predict occupational compositions of fathers and sons in the labor market. Despite all these potential difficulties in combining a mobility model with demographic components, a few studies have proposed variants of Markov models that allow for differential demographic rates (Chu and Koo 1990; Matras 1961, 1967; Mare 1997; Mare and Maralani 2006; Maralani 2013; Preston 1974; Preston and Campbell 1993; Lam 1986, 1997). These models provide a good starting point for future work.

To relax Assumption 1, Matras (1961) first proposed a Markov model that incorporates differential population growth as follows:

$$s_j=\sum _{i=1}^I{f}_i\cdot r_i\cdot p_{Y_2=j\mid Y_1=i}(j=1,2,\dots ,J).$$ (7)

where f_i denotes the number of fathers in social class i; s_j denotes the number of sons in class j; r_i denotes the expected number of sons born to a man in class i who survive to adulthood or are old enough to acquire a social position;³ and $P_{Y_2=j\mid Y_1=i}$ denotes the probability that a son born to a father in class i will attain class j.⁴

Relying on the recursive form of the model, we can model the socioeconomic distribution of descendants given that intergenerational fertility and mobility processes are fixed over time (namely, a time-homogeneous Markov chain). Set a diagonal matrix for the differential fertility component, R = [r_ij], where r_ij = r_i for i = j and r_ij = 0 for i ≠ j. P is the same mobility matrix defined in equation (2). Let R · P = C, and we obtain the intergenerational relationship, S⁽²⁾ = S⁽¹⁾ · C = F⁽⁰⁾ · C². In general, the generation-to-generation change is represented by

$$S^{(t)}=F^{(0)}\cdot C^t.$$ (8)

Subsequent work has extended this basic model in several ways. Matras (1967) introduced a model that incorporates the age structure of each generation, which was later analyzed empirically by Lam (1986) and Mare (1997). Preston (1974) developed a model that separates white and non-white families. Mare and his collaborators further decomposed the differential reproduction rates into marriage, fertility, and mortality components (Kye and Mare 2012; Maralani 2013; Mare and Maralani 2006; Maralani and Mare 2005; Mare and Song 2014).⁵

Overall, these models show the effect of a person’s social class in one generation on the expected number of offspring in various social classes in the next generation; that is, the joint effects of a man’s social class on his demographic behaviors and his offspring’s socioeconomic attainment. Therefore, these models illustrate the transformation from F to S as a sociodemographic process rather than strictly a social mobility process. In subsequent sections, I refer to these models as social reproduction models or sociodemographic mobility models.

As discussed earlier, the model specified in equation (7) may simplify demographic processes in social mobility, especially by relying on the concept of generation rather than using a real-time scale (Duncan 1966a). Projections from the model often do not mirror observed empirical processes that continuously evolve. Yet, taken qualitatively, conclusions from these models may still reflect general trends in family dynamics in the long run.

3.2. Multigenerational Models

One central assumption of the Markov chain model is that each generation directly influences only the immediately following generation, exerting no direct effect beyond its offspring (Assumption 2). No matter how much influence parents have on their children’s outcomes, they do not influence their grandchildren’s outcomes independently of their own children. Therefore, the social system has no memory: if a family loses its existing advantages, it has to start from scratch. This assumption may be invalid under some social circumstances, such as when multigenerational social processes are non-Markovian. In particular, individuals’ social mobility may depend on various forms of multigenerational influence, such as direct influences from grandparents and great-grandparents, cumulative advantages (or disadvantages) of prior generations, legacy influences of remote ancestors who experienced extreme hardship or success, or supplementary influences of nonresident kin in extended families (Mare 2011). Some of these processes can be represented by second- or higher-order Markov chains (see, e.g., Goodman (1962) on attitude change and Hodge (1966) on three-generation mobility). I will refer to them as non-Markovian generational processes.

The simplest way to relax the Markovian assumption is to incorporate the effect of grandparents into Matras’s model shown in equation (7):

$$s_j=\sum _i\sum _k{f}_{ik}\cdot r_{ik}\cdot p_{Y_3=j\mid Y_2=i,Y_1=k}$$ (9)

where s_j denotes the number of men in the offspring generation who are in class j (j = 1, …, J); f_ik denotes the number of men in the paternal generation who were in class i and whose fathers were in class k; r_ik denotes the expected number of sons born to each man in f_ik; and $p_{Y_3=j\mid Y_1=i,Y_2=k}$ denotes the probability that a son with a father in class i and a grandfather in class k will attain class j. More generally, if the model parameters depend on the socioeconomic status of all prior generations, ${\overline{Y}}_{n-1}=\left\{Y_1,Y_2,\cdots ,Y_{n-1}\right\}$, the model can be written as

$$s_{Y_n}=\sum _{Y_1}\cdots \sum _{Y_{n-1}}{f}_{{\overline{Y}}_{n-1}}\cdot r_{{\overline{Y}}_{n-1}}\cdot p_{Y_n\mid {\overline{Y}}_{n-1}}$$ (10)

Despite the importance of validating the Markovian assumption in mobility models, only a few studies have tested the assumption empirically (Hodge 1966; Warren and Hauser 1997; see a review of these studies in Appendix Table S1). The increasing availability of longitudinal data in recent years has facilitated a growing body of scholarship that investigates the Markovian assumption more thoroughly from a wide range of countries.⁶ The findings from these studies are far from conclusive, suggesting that the validity of the Markovian assumption may vary across time and social context. Even within a society, patterns of social mobility may vary based on the form of social status classification, be it stocks of social advantages, such as business, land, or estate ownership, or flows of advantages, such as income, occupation, and education. Moreover, any test of the Markovian assumption may be subject to the “lumpability” problem (Kemeny and Snell 1960: pp.123–139): a non-Markovian chain may become Markovian if we combine or divide some of the transition states. Therefore, any conclusion regarding the mobility pattern is valid only under the condition that the states are defined the way they are (McFarland 1970).

3.3. Demographic Change and Structural Mobility

Social mobility analyses, typically in the form of mobility tables, have a dual character: the number of parents and offspring in different occupations reflects both the relative chances of social movement and the changing availability of job opportunities. Most analytical techniques have focused on separating circulation mobility, also known as exchange or relative mobility, from mobility caused by changes in social structures. These structural changes may emerge from exogenous shocks, such as industrialization, technological revolution, economic growth, growing new jobs, and the increasing division of labor, which reflect the changing occupational needs of the economy. For example, new technology reduced the number of agricultural jobs, and thus some offspring of farm workers would be forced out of occupations in the primary industry. However, structural changes may also result from the process of “social metabolism” produced by birth, death, migration, and retirement from economic activities (Duncan 1966b). These fundamental demographic forces change the supply and demand for workers in different age groups and of different qualifications, and interact with macroeconomic changes in ways that contribute to structural change (White 1963: p.27).⁷ As Watkin, Menken, and Bongaarts (1987) argue, demography is the foundation of social change. A demographic perspective will provide insights into the amount of mobility generated by structural change, which is often eliminated in previous mobility analyses (reviewed in, e.g., Hauser 1978).

4. SOCIAL MOBILITY EFFECT VS. DEMOGRAPHIC EFFECT

Using the models described above, we can estimate the effect of one generation on the next or several generations later in terms of the pure mobility effect based on the classic Markov models in equation (2) and the joint mobility and demography effect based on the social reproduction model in equation (7). The effects can be defined either in ratio measures or difference measures. The ratio measure refers to arithmetic quotients of mobility or demographic outcomes between two types of families; the difference measure refers to the difference score between the two. Both measures are widely used in social sciences (see a recent review and critique by Stolzenberg 2018). Ratio measures are more popular in the social mobility literature, especially in mobility models based on log-linear analysis and odds ratios (Agresti 2013; Powers and Xie 2000), whereas difference measures are widely used in the demographic literature for decomposition analyses. This section illustrates how to quantify various components of mobility and demography effects using the following definitions and decomposition techniques.

4.1. Net and Total Mobility Effects

In traditional mobility models, researchers typically measure mobility by estimating differences in the probability of children who grew up in rich versus poor families to achieve high social status.⁸ The total mobility effect (TME) of having a parent in high status (social class k) relative to low status (social class j) on children’s attainment of high status can be defined using the following ratio measure,

$${\text{TME}}^P=p_{Y_2=k\mid Y_1=k}/p_{Y_2=k\mid Y_1=j}.$$ (11)

Likewise, the total mobility probability effect of having grandparents in high status relative to low status is estimated by combining the mobility from grandparents to parents and from parents to offspring:

$${\text{TME}}^{GP}=\sum _i{p}_{Y_3=k\mid Y_2=i,Y_1=k}\cdot p_{Y_2=i\mid Y_1=k}/\sum _i{p}_{Y_3=k\mid Y_2=i,Y_1=j}\cdot p_{Y_2=i\mid Y_1=j}$$ (12)

If we remove the effect of grandparents on parents by fixing the status of the parent generation to i, the mobility ratio represents the net mobility effect (NME) of grandparents:

$${\text{NME}}^{GP}=p_{Y_3=k\mid Y_2=i,Y_1=k}/p_{Y_3=k\mid Y_2=i,Y_1=j}.$$ (13)

The total and net effects are often unequal for the grandparent generation, but they are the same for the parent generation by definition in equation (11).

4.2. Net and Total Social Reproduction Effects

Next, we consider the joint effects of demography and mobility on the number of offspring and descendants in various social classes. The net Social Reproduction Effects (SRE) of parents are defined as the effects of a targeted social class relative to a baseline class on the number of offspring in the targeted social class. For example, the expected number of high-status individuals from high-status rather than low-status parents is specified as:⁹

$${\text{SRE}}_{k\mid j}=\frac{s_{Y_2=k\mid Y_{1=k}}}{s_{Y_2=k\mid Y_1=j}}=\frac{f_k\cdot r_k\cdot p_{Y_2=k\mid Y_1=k}}{f_j\cdot r_j\cdot p_{Y_2=k\mid Y_1=j}}.$$ (14)

We do not differentiate between net and total SRE because they are the same by definition. If we consider multiple generations, we define the net social reproduction effect (NSRE) of grandparents as the comparative advantage of a low-status parent and a high-status grandparent over a low-status parent and a low-status grandparent on producing high-status grandchildren:

$${\text{NSRE}}_{k\mid j}^{GP}=\frac{s_{Y_3=k\mid Y_2=j,Y_1=k}}{s_{Y_3=k\mid Y_2=j,Y_1=j}}=\frac{f_{jk}\cdot r_{jk}\cdot p_{Y_3=k\mid Y_2=j,Y_1=k}}{f_{jj}\cdot r_{jj}\cdot p_{Y_3=k\mid Y_2=j,Y_1=j}}.$$ (15)

The choice of the parent’s status is, to an important extent, arbitrary. Alternatively, we can fix the parents’ social class at k. The two NSRE definitions will lead to the same estimates if there are no interaction effects between grandparents’ class and parents’ class in determining levels of r and p.¹⁰

By contrast, the total social reproduction effect is the comparative advantage of a high-status grandparent over a low-status grandparent on producing high-status grandchildren, regardless of the parent’s social status. Specifically,

$${\text{TSRE}}_{k\mid j}^{GP}=\frac{s_{Y_3=k\mid Y_1=k}^{(2)}}{s_{Y_3=k\mid Y_1=j}^{(2)}}=\frac{\sum _i{f}_k\cdot r_k\cdot p_{Y_2=i\mid Y_1=k}\cdot r_{ik}\cdot p_{Y_3=k\mid Y_2=i,Y_1=k}}{\sum _i{f}_j\cdot r_j\cdot p_{Y_2=i\mid Y_1=j}\cdot r_{ij}\cdot p_{Y_3=k\mid Y_2=i,Y_1=j}}$$ (16)

where $s_{Y_3=k\mid Y_1=j}^{(2)}$ (and $s_{Y_3=k\mid Y_1=k}^{(2)}$) refers to the number of high-status grandchildren whose grandparents are in low status j (and high-status k).

Note that we define the net and total effects of a prior generation based on a ratio measure, but this tells us nothing about the absolute difference between the number of various types of descendants from two ancestral groups of different social statuses. Revising equation (9), we can define SRE in the parent generation as a difference measure:

$${\text{SRE}}_{k\mid j}=s_{Y_2=k\mid Y_1=k}-s_{Y_2=k\mid Y_1=j}=f_k\cdot r_k\cdot p_{Y_2=k\mid Y_1=k}-f_j\cdot r_j\cdot p_{Y_2=k\mid Y_1=j}.$$ (17)

More generally, the net SRE and total SRE of an ancestor who lives n generations back are expressed as

$${\text{NSRE}}_{k\mid j}^{(n)}=f_{{\overline{Y}}_{n-1,k}}\cdot r_{{\overline{Y}}_{n-1,k}}\cdot p_{Y_n=k\mid {\overline{Y}}_{n-1,k}}-f_{{\overline{Y}}_{n-1,j}}\cdot r_{{\overline{Y}}_{n-1,j}}\cdot p_{Y_n=k\mid {\overline{Y}}_{n-1,j}}$$ (18)

$${\text{TSRE}}_{k\mid j}^{(n)}=\sum _{Y_2}\cdots \sum _{Y_{n-1}}{f}_k\cdot \prod _{i=1}^{n-1}{r}_{{\overline{Y}}_{i,k}}\cdot p_{Y_{i+1}\mid {\overline{Y}}_{i,k}}-f_j\cdot \prod _{i=1}^{n-1}{r}_{{\overline{Y}}_{i,j}}\cdot p_{Y_{i+1}\mid {\overline{\mathbf{Y}}}_{i,j}}.$$ (19)

where ${\overline{Y}}_{i,k}$ refers to {Y_i, Y_i−1 ⋯ Y₂, Y₁ = k} in which the social status of the first generation is fixed at Y₁ = k. The net effect suggests the comparative advantage of a family with an n^th ancestor in high status k rather than in low status j in producing high-status sons. Social statuses of intermediate generations are fixed to be the same for these two families. By contrast, the total effect suggests the cumulative advantage of a high-status ancestor compared to a low-status ancestor in producing high-status descendants after n generations, regardless of the social positions of intermediate generations. In equations (18) and (19), the position of the n^th generation is fixed at k, namely, $p_{Y_n=k\mid {\overline{Y}}_{n-1,k}}$ (and $p_{Y_n=k\mid {\overline{Y}}_{n-1,j}}$).

4.3. Effect Standardization and Decomposition

Based on the definition of the social reproduction effect, one may ask, “how much of the difference in social reproduction between two families is attributable to differences in their levels of fertility versus differences in social mobility probabilities?” This type of question is addressed through decomposition techniques in demography (Kitagawa 1955). Below, I decompose social reproduction effects into fertility and mobility components. The two components completely account for the original difference without introducing a residual term. The decomposition method shows the relative importance of demographic and social pathways in explaining differences between descendants from two ancestral groups of different social statuses.

To begin with, I use Kitagawa’s method to decompose the social reproduction effect of a high-status parent relative to a low-status parent in equation (17).

$${\mathit{\text{SRE}}}_{k\mid j}=s_{Y_2=k\mid Y_1=k}-s_{Y_2=k\mid Y_1=j}=\underset{ \text{demography effect} }{\underbrace{\left(r_k-r_j\right)\cdot \frac{\left(p_{Y_2=k\mid Y_1=k}+p_{Y_2=k\mid Y_1=j}\right)}{2}}}+\underset{ \text{mobility effect} }{\underbrace{\frac{\left(r_k+r_j\right)}{2}\cdot \left(p_{Y_2=k\mid Y_1=k}-p_{Y_2=k\mid Y_1=j}\right)}}$$ (20)

The demography effect reflects differences in SRE attributed to differences in reproduction rates of high-status and low-status fathers, where the mobility probability of their offspring is fixed at the mean level, $\overline{p}=\frac{p_{Y_2=k\mid Y_1=k}+p_{Y_2=k\mid Y_1=j}}{2}$; and the mobility effect refers to differences in SRE due to differences in mobility probabilities of offspring from high-status and low-status fathers, where fathers’ reproduction rates are fixed at the mean level, $\overline{r}=\frac{r_k+r_j}{2}$.¹¹

Similarly, I decompose the SRE of a high-status grandparent versus a low-status grandparent in producing high-status grandchildren into the total effects of the grandparent via differences in demographic rates r and differences in mobility probabilities p.

$${\text{TSRE}}_{k\mid j}^{GP}=\sum _i{r}_k\cdot p_{Y_2=i\mid Y_1=k}\cdot r_{ik}\cdot p_{Y_3=k\mid Y_2=i,Y_1=k}-\sum _i{r}_j\cdot p_{Y_2=i\mid Y_1=j}\cdot r_{ij}\cdot p_{Y_3=k\mid Y_2=i,Y_1=j}$$ (21)

$$\underset{ \text{demography effect} }{\underbrace{=\sum _i\left(r_k\cdot r_{ik}-r_j\cdot r_{ij}\right)\cdot \frac{\left(p_{Y_2=i\mid Y_1=k}\cdot p_{Y_3=k\mid Y_2=i,Y_1=k}+p_{Y_2=i\mid Y_1=j}\cdot p_{Y_3=k\mid Y_2=i,Y_1=j}\right)}{2}}}+\underset{ \text{mobility effect} }{\underbrace{\sum _i\frac{\left(r_k\cdot r_{ik}+r_j\cdot r_{ij}\right)}{2}\cdot \left(p_{Y_2=i\mid Y_1=k}\cdot p_{Y_3=k\mid Y_2=i,Y_1=k}-p_{Y_2=i\mid Y_1=j}\cdot p_{Y_3=k\mid Y_2=i,Y_1=j}\right)}}$$ (22)

Applying Das Gupta’s decomposition method (1993) for rates of four factors, one can further decompose compound demography and mobility effects into effects from different generations. To simplify the notations below, I use r₁ = r_k, $r_1^{\prime }=r_j$, r₁ = r_ik, $r_2^{\prime }=r_{ij}$, $p_1=p_{Y_2=i\mid Y_1=k}$, $p_1^{\prime }=p_{Y_2=i\mid Y_1=j}$, $p_2=p_{Y_3=k\mid Y_2=i,Y_1=k}$, $p_2^{\prime }=p_{Y_3=k\mid Y_2=i,Y_1=j}$.

$$\text{demography effect \hspace{0.17em}}(1)=\sum _{Y_2}\left[\frac{p_1{r}_2{p}_2+p_1^{\prime }{r}_2^{\prime }{p}_2^{\prime }}{4}+\frac{p_1{r}_2{p}_2^{\prime }+p_1{r}_2^{\prime }{p}_2+p_1^{\prime }{r}_2{p}_2+p_1^{\prime }{r}_2^{\prime }{p}_2+p_1^{\prime }{r}_2{p}_2^{\prime }+p_1{r}_2^{\prime }{p}_2^{\prime }}{12}\right]\cdot \left(r_1-r_1^{\prime }\right)$$ (23)

$$\text{mobility effect } (1)=\sum _{Y_2}\left[\frac{r_1{r}_2{p}_2+r_1^{\prime }{r}_2^{\prime }{p}_2^{\prime }}{4}+\frac{r_1{r}_2{p}_2^{\prime }+r_1{r}_2^{\prime }{p}_2+r_1^{\prime }{r}_2{p}_2+r_1^{\prime }{r}_2^{\prime }{p}_2+r_1^{\prime }{r}_2{p}_2^{\prime }+r_1{r}_2^{\prime }{p}_2^{\prime }}{12}\right]\cdot \left(p_1-p_1^{\prime }\right)$$ (24)

$$\text{demography effect } (2)=\sum _{Y_2}\left[\frac{p_1{r}_1{p}_2+p_1^{\prime }{r}_1^{\prime }{p}_2^{\prime }}{4}+\frac{p_1{r}_1{p}_2^{\prime }+p_1{r}_1^{\prime }{p}_2+p_1^{\prime }{r}_1{p}_2+p_1^{\prime }{r}_1^{\prime }{p}_2+p_1^{\prime }{r}_1{p}_2^{\prime }+p_1{r}_1^{\prime }{p}_2^{\prime }}{12}\right]\cdot \left(r_2-r_2^{\prime }\right)$$ (25)

$$\text{mobility effect } (2)=\sum _{Y_2}\left[\frac{p_1{r}_1{r}_2+p_1^{\prime }{r}_1^{\prime }{r}_2^{\prime }}{4}+\frac{p_1{r}_1{r}_2^{\prime }+p_1{r}_1^{\prime }{r}_2+p_1^{\prime }{r}_1{r}_2+p_1^{\prime }{r}_1^{\prime }{r}_2+p_1^{\prime }{r}_1{r}_2^{\prime }+p_1{r}_1^{\prime }{r}_2^{\prime }}{12}\right]\cdot \left(p_2-p_2^{\prime }\right) .$$ (26)

Mobility effect (1) refers to the effect of grandparents on grandchildren via the influence of grandparents on parents’ social mobility. Mobility effect (2) refers to the effect of grandparents on grandchildren via the influence of grandparents on grandchildren’s social mobility. The demography effects (1) and (2) can be interpreted in the same vein.

A generalization of the decomposition method to n generations is discussed in Appendix C. Note that the decomposition method assumes that families’ social mobility does not affect their reproduction, or vice versa. If this assumption is violated, such that the level of parents’ fertility influences the offspring’s mobility outcomes, the effects of demography and mobility cannot be completely separated into two independent components. Instead, the decomposition results can be roughly interpreted as the effect of differences in reproductive rates (not via mobility) and the effect of differences in mobility rates (not via reproductive behaviors).

5. SHORT-TERM EFFECT, LONG-TERM EFFECT, AND EQUILIBRIUM EFFECT

Based on the recursive form of the multigenerational model in equation (7), one can quantify the relative importance of mobility and demography in shaping short-term and long-term family inequality dynamics. The equilibrium properties of the model show the eventual population composition of individuals descended from different families. A critical property of the Markov model is that if the transition matrix is time-invariant, the distribution of social classes after t generations, that is, S^(t) = F⁽⁰⁾P^t, will gradually converge to a stationary distribution that is determined only by the transition matrix, not by the initial distribution of F⁽⁰⁾.¹² This property suggests that a family’s initial social class may influence the social mobility probability for several generations, but eventually, its influence will fade away. As such, after enough generations, the social class of a descendant will eventually become independent of the social class of the lineage founder. The social class distribution between descendants from high-status and low-status origin lineages will become identical. Therefore, the social advantages associated with any generation will not permanently change the prospects of social attainment in future generations. Any short-term effect of one generation does not translate into long-term inequality between families that originate from unequal social statuses. This property is illustrated by equation (27). The long-term mobility effect is defined as the ratio of descendants in high-status k from a high-status rather than low-status lineage founder:

$$\text{LSRE}=\underset{t\to \infty }{\text{lim}}\left(\frac{S_{k\mid k}^{(t)}}{S_{k\mid j}^{(t)}}\right)=\underset{t\to \infty }{\text{lim}}\left(\frac{F_k^{(0)}\cdot P^t}{F_j^{(0)}\cdot P^t}\right)=\frac{\pi }{\pi }=1.$$ (27)

where π is the equilibrium distribution for a Markov chain with the transition matrix P. If social mobility follows a second- or higher-order Markov process with a time-invariant transition matrix, the LSRE will also converge to 1 in the long run. Such a conclusion, however, may not hold when we examine the social reproduction effect rather than the mobility effect alone. Assume S^(t) = F⁽⁰⁾ · C^t, where C = R · P, a combination of the reproduction and mobility components. Suppose the number of social classes is the same for fathers and sons, and the reproduction and mobility matrices have no structural 0s (i.e., p_j|i > 0 and r_i > 0 for all i and j). According to the Perron-Frobenius theorem, if C is a square matrix with positive entries and a unique dominant eigenvalue, the long-term behavior of S^(t) would depend on the largest eigenvalue of C.

After enough generations, families starting with high-status k will eventually produce $\frac{a_{1k}}{a_{1j}}$ times as many descendants in high-status k as low-status families do. The terms a_1k and a_1j refer to coefficients associated with the first eigenvector when $F_k^{(0)}$ and $F_j^{(0)}$ are decomposed into multiple vectors using the eigenvectors of C. A proof of this equilibrium effect is shown in Appendix D. Therefore, the asymptotic distribution of descendants is path-dependent in that not all families produce the same number of descendants in the long run. By contrast, regular Markov mobility models with no demographic effects are ergodic. The transition matrix P can be viewed as a special case of matrix C as the sum of each row is constrained to 1. This constraint also guarantees that λ₁ = 1 and a_1k = a_1j. As a result, the equilibrium distribution of the Markov mobility model will not depend on the social class of the initial generation.

6. HETEROGENEOUS MOBILITY REGIMES

The regular Markov mobility model assumes that all individuals in a population have identical transition probabilities conditional on their parents’ social status (Assumption 3). This assumption overlooks many possible sources of heterogeneity associated with individual-level social attributes and macro-level social structures (Blau 1977). Below, I group sources of population heterogeneity discussed in previous mobility research into three types: (1) individual, time-invariant heterogeneity, (2) individual, time-dependent heterogeneity, and (3) heterogeneity in mobility regimes.

The idea of mobility heterogeneity with time-invariant properties can be traced back to Blumen et al.’s (1955) pioneering study on intragenerational labor mobility. A similar notion can be applied to the analysis of intergenerational mobility of family lines (e.g., White 1970). Blumen et al. (1955) identified two types of individuals in a population: movers, who change jobs over time according to a time-constant Markov transition matrix, and stayers, who remain in the same job category with probability 1. The model includes the proportions of movers and stayers and the transition probability matrix for movers. A person’s attribute—as either a mover or a stayer—does not change during the entire period of study. The model is formulated as follows:

$$S=F\cdot (\Lambda +(I-\Lambda )\cdot P)$$ (28)

where S and F are social class distributions of sons and fathers, respectively; Λ is a diagonal matrix with the proportions of stayers in each social class on the diagonals; the diagonal matrix I − Λ includes the proportions of movers as diagonal entries; and the matrix P refers to the transition mobility matrix for the movers. The mover-stayer model considers one of many possible types of time-invariant heterogeneity by assuming only two distinct subpopulations. In general, individuals may conform to miscellaneous transition probabilities, or in a more extreme scenario, each person follows a mobility process governed by a unique set of mobility probabilities (Xie 2013).¹³

To model individual heterogeneity, Spilerman (1972a) proposed a Markov model that estimates individual transition probabilities with regression models. We first convert the father-son transition matrix into a person-transition dataset, where each observation is represented by a possible transition.¹⁴ For each father in social class i and son in social class j, define an indicator variable $Z_{ij}=\left\{z_{Y_1=i,Y_2=1},z_{Y_1=i,Y_2=2},\cdots ,z_{Y_1=i,Y_2=J}\right\}$, which equals 1 if a person born into class i moves to class j, and 0 otherwise. We then fit I × J (often I = J) linear probability equations, and in each equation, Z_ij is predicted by a set of social attributes X.

For individual c from social class i with attributes (X_1c, X_2c, ⋯ , X_Vc), his probability of moving into class j is

$$\text{Pr}\left(Z_{ijc}\mid X_c\right)={\widehat{a}}_{ij}+\sum _{v=1}^V{\widehat{b}}_{ijv}{X}_{vc}$$ (29)

where ${\widehat{a}}_{ij}$ and ${\widehat{b}}_{ijv}$ are regression coefficient estimates.¹⁵ If we estimate a separate transition matrix P_c(X_c) for each person c using the predicted probabilities based on attributes X_c, intergenerational mobility from fathers to sons can be represented in the following matrix form:

$$S=F\cdot \left(\frac{1}{C}\cdot \sum _{c=1}^C{P}_c\left(X_c\right)\right)$$ (30)

where P(X_c) is known as the individual transition matrix (McFarland 1970; Spilerman 1972a). The overall transition matrix is estimated from the sum of all individual transition matrices divided by the population size C.¹⁶

The second type of heterogeneity assumes that the mobility matrix operates as a function of time. Studies on intragenerational mobility have proposed the “Retention Model” (Henry 1971) and the “Cornell Mobility Model” (McGinnis 1968) in which movers’ transition probability is assumed to change over time. Below, I modify these models for the analysis of intergenerational mobility. Stayers are defined as individuals who remain in the same occupation as their parents and movers as those who enter an occupation different from that of their parents. These models show examples when the stationarity assumption (Assumption 4) fails.

The Retention Model allows the proportion of movers and stayers in equation (28) to be time-dependent, so the mover-stayer model becomes

$$S=F\cdot \left({\Lambda }_t+\left(I-{\Lambda }_t\right)\cdot P\right)$$ (31)

where Λ_t is a diagonal matrix with the diagonal cells indicating the proportion of stayers in each social class as a function of time, and P refers to the transition matrix of the movers.

The Cornell Mobility model postulates that individuals’ tendency to leave a social class declines as a strictly monotonic function of the duration of staying in that class. In the context of intergenerational mobility, this “cumulative inertia” property implies that the longer a family has been in its current social class, the higher its probability of remaining there for yet another generation. Following the notation in McGinnis (1968), let ${}_d{P}_{Y_2=j\mid Y_1=i}(t)$ refer to the transition probability from class i to class j at generation t when a family has remained in class i for d consecutive generations prior to generation t. The distribution of social classes in the father’s generation is F = [₁f₁, ₁f₂, ⋯ , ₁f_I, ₂f₁, ⋯ , ₂f_I, _df₁, ⋯ , _df_I]. The duration-specific transition matrix can be partitioned into the transition matrices of movers and stayers, both of which are also duration specific:

$${}_d{P}_{Y_2=j\mid Y_1=i}(t){=}_d{P}_{Y_2=j\mid Y_1=i}^{\text{mover}}(t){+}_d{P}_{Y_2=j\mid Y_1=i}^{\text{stayer }}(t)$$ (32)

The stayers’ transition matrix ${}_d{P}_{Y_2=j\mid Y_1=i}^{\text{stayer}}(t)$ is the diagonalization of ${}_d{P}_{Y_2=j\mid Y_{1=i}}(t)$, which satisfies that

$${}_d{P}_{Y_2=j\mid Y_1=i}^{\text{stayer} }(t)=\{\begin{array}{ll}{}_d{P}_{Y_2=j\mid Y_1=i}(t)\hfill & if j=i\hfill \\ 0\hfill & \text{otherwise} \hfill \end{array}$$

The relationship between the movers’ and stayers’ transition matrices satisfies that

$${}_d{P}_{Y_2=j\mid Y_1=i}^{\text{mover}}(t)=\left(I{-}_d{P}_{Y_2=j\mid Y_1=i}^{\text{stayer} }(t)\right)\cdot R$$ (33)

where R = [r_ij] subject to r_ii = 0 is shown to always exist and does not vary by d. Because of the cumulative inertia property, the model also requires that ${}_{d+1}{P}_{Y_2=j\mid Y_1=i}^{\text{stayer} }{>}_d{P}_{Y_2=j\mid Y_1=i}^{\text{stayer} }$.¹⁷ Note that this model violates the stationarity assumption in the regular Markov model by introducing a time component into the transition probability. It also violates the Markovian assumption by linking individuals’ mobility outcomes with the entire history of moves in previous generations.

The third type of heterogeneity concerns the mixture of mobility regimes in a society or in a broader stratification system created by spatial, cultural, or institutional forces of segregation. Using occupational mobility as an example, many studies have shown variation in intergenerational mobility among industrial societies in the mid to late 20^th century (DiPrete 2002; Lipset and Bendix 1959; Featherman et al. 1975; Grusky and Hauser 1984; Erikson and Goldthorpe 1992; Xie 1992; Yamaguchi 1987). Even within a society, several mobility regimes may coexist. One instructive example provided by Mare (2011) postulates a stratification system consisting of three strata: the top, middle, and bottom classes. The persistence of social status is stronger at the top and bottom of the social hierarchy compared to the middle due to social policies and family norms that create a cumulative advantage or disadvantage for families. In cross-country comparisons, social boundaries among mobility regimes are often assumed to be impermeable, in contrast to within-society comparisons in which families that start in one mobility regime may circulate in and out of other regimes after generations of movement. These mobility processes can be represented in the following matrix form:

$$S_{1\times n}=F_{1\times n}\cdot \left(\sum _{c=1}^C\lambda (c)\cdot P_{n\times n}(c)\right)$$ (34)

where λ(c) denotes the weight of each subgroup c, λ(c) > 0, ∑_c λ(c) = 1 and P(c) denotes the mobility probability matrix for subgroup c (c = 1, 2, …, C). For example, Mare and Song (2014) analyzed the social mobility of descendants from imperial and peasant families using Chinese family genealogies and linked historical censuses (c = 1 descendants of emperors; c = 2 descendants of peasants). If subgroups are distinct and time-invariant, then p_ij(c) will be fixed over generations. If social boundaries are permeable, families’ membership c may change over time, leading to time-varying weights λ(c, t) and mobility matrix P(c, t).

Compared to regular mobility models, mixture Markov models that account for population heterogeneity often represent observed mobility processes better in terms of model goodness-of-fit, such as the χ² test (Goodman 1962). However, McFarland (1970: 475) noted about these heterogeneous mobility models that, “any real adequate model would be too cumbersome to be fitted to numerical data.” Mobility researchers will always face a trade-off between model accuracy and simplicity.

7. A TWO-SEX APPROACH

Traditional social mobility studies typically take a one-sex approach by focusing exclusively on the intergenerational association of social status between fathers and sons while ignoring the independent role of mothers or the joint role of parents (Assumption 5). The one-sex approach is also widely adopted in demographic models, which assume population dynamics are determined by the vital rates of one sex only, often women, or that the roles of both sexes are identical (Caswell 2001). In multigenerational analyses, the one-sex approach is useful when the transmission of social characteristics is sex-linked or predominantly influenced by one parent. For example, in many patriarchal societies, social positions were passed down the male lineage from paternal grandfathers to fathers and their sons. When comparing descendants who carry sex-linked characteristics from families of unequal origins, we only need to count male descendants in the population.

In most western societies, however, this theoretical assumption may be invalid in practice (Song and Mare 2017). First, mothers, grandmothers, and maternal grandparents may influence individuals’ socioeconomic outcomes, in addition to the influences of patrilineal kin. The increasing roles of mothers and grandmothers are associated with the rise in female labor force participation, gender economic equality, and the growth of single-parent or skipped-generation households (e.g., Beller 2009). Second, mobility probabilities and demographic rates also vary by gender for both sociological and biogenetic reasons (e.g., Reskin 1993; Jacobs 1989; Preston et al. 2001). If we apply the one-sex social and demographic mobility model in equation (7) to men and women separately, the results may disagree. Third, the one-sex approach does not account for interactions between men and women through the formation of marriages, a mechanism that determines the social makeup of families and creates the family background of the next generation (Mare and Schwartz 2006). The formation of marriages depends on the abundance and preferences of mates in a population. On the one hand, the number and types of marriages are constrained by the “marriage squeeze,” or the imbalance between the number of men and women considered marriageable (Akers 1967; Schoen 1983). The marriage squeeze may produce significant changes in the timing and patterns of marriage and fertility levels in a population. On the other hand, men and women tend to marry others with similar socioeconomic characteristics. The degree of assortative mating influences not only socioeconomic resemblance between couples but also the amount of social advantage or disadvantage transmitted to the offspring generation.

Previous demographic studies have proposed various two-sex models that account for the interdependence of demographic behaviors of both sexes in determining the number of marriages and births in a population. These models have been used to predict the size, composition, and growth of future populations (Caswell 2001; Caswell and Weeks 1986; Goldman et al. 1984; Goodman 1953, 1968; Jenouvrier et al. 2010; Kendall 1949; Miller and Inouye 2011; Pollak 1986; Pollard 1973). Below, I illustrate how to adapt these demographic models for multigenerational mobility research using the Birth Matrix-Mating Rule (BMMR) model developed by Pollak (1986, 1987, 1990a,b) as an example. In parallel to the one-sex model in equation (7), the two-sex model for men and women is specified as

$$s_k=\sum _i\sum _j{\mu }_{ij}\left(N^m,N^f\right)\cdot r_{ij}^m\cdot p_{Y_2=k\mid Y_1=\{i,j\}}^m$$ (35)

$$d_k=\sum _i\sum _j{\mu }_{ij}\left(N^m,N^f\right)\cdot r_{ij}^f\cdot p_{Y_2=k\mid Y_1=\{i,j\}}^f$$ (36)

where s_k (d_k) denotes the number of sons (daughters) in the offspring generation who are in social class k; μ_ij(N^m, N^f) denotes the number of marriages between fathers in class i and mothers in class j;¹⁸ and $r_{ij}^m\left(r_{ij}^f\right)$ denotes the mean number of surviving sons (or daughters) born from each union of class i fathers and class j mothers with completed reproduction history. In general, the differences between $r_{ij}^m$ and $r_{ij}^f$ are determined by male-to-female sex ratios at birth in a population and differential survival rates of boys and girls to adulthood. In most populations, the two estimates can be considered equal. Finally, $p_{Y_2=k\mid Y_1=\left\{i,j\right\}}^m$ and $p_{Y_2=k\mid Y_1=\left\{i,j\right\}}^f$ refer to the probability of obtaining class k for sons and daughters born to class i fathers and class j mothers, respectively.

To model the mating rule term, μ_ij(N^m, N^f), I adopt Schoen’s harmonic mean mating rule (Schoen 1981, 1988), which assumes that the number of marriages between two social groups depends on the relative number of single women and men in these groups and the attractiveness of these group members to each other. The harmonic mean mating rule specifies that

$${\mu }_{ij}\left(N^m,N^f\right)=\frac{{\alpha }_{ij}{N}_i^m{N}_j^f}{N_i^m+N_j^f},{\alpha }_{ij}>0,\sum _j{\alpha }_{ij}\le 1,\sum _i{\alpha }_{ij}\le 1$$ (37)

where α_ij is the “force of attraction” between males in class i and females in class j, which results from constraints imposed by the abundance of mates as well as preferences among all class groups (Schoen 1988). In empirical studies, α_ij is often estimated from the number of marriages and single individuals in different social classes (Qian 1998; Qian and Preston 1993; Raymo and Iwasawa 2005). $N_i^m$ is the total number of eligible men in class i, and $N_j^f$ is the total number of eligible women in class j.

Using the two-generation model in equations (35) and (36), we can derive the socioeconomic distribution of the grandchild generation. Specifically, the number of granddaughters (grandsons) in class k, denoted as $d_k^{(2)}\left(s_k^{(2)}\right)$, can be estimated as:

$$s_k^{(2)}=\sum _{i^{\prime }}\sum _{j^{\prime }}{\mu }_{i^{\prime }{j}^{\prime }}(S,D)\cdot r_{i^{\prime }{j}^{\prime }}^m\cdot p_{k\mid i^{\prime }{j}^{\prime }}^m$$ (38)

$$d_k^{(2)}=\sum _{i^{\prime }}\sum _{j^{\prime }}{\mu }_{i^{\prime }{j}^{\prime }}(S,D)\cdot r_{i^{\prime }{j}^{\prime }}^f\cdot p_{k\mid i^{\prime }{j}^{\prime }}^f$$ (39)

where the number of parents μ_i′j′(S, D) is generated by men in class i′ in the father generation, s_i′, and women in class j′ in the mother generation, d_j′. These men and women in the parent generation can be estimated by men and women in the grandparent generation recursively.¹⁹ The formulas above show a nonlinear, compound relationship between the distributions of grandparents and grandchildren. Given that there is no simple analytical form of the distribution of descendants after n generations, I simulate the two-sex long-term social reproduction effect (LSRE) in the next section and compare it with its one-sex counterpart discussed in Section 3.

In most societies, individuals tend to choose spouses with similar socioeconomic characteristics more frequently than would be expected under random mating (Schwartz 2013). Two other mating patterns, random mating and endogamous mating, which assume individuals either select mates irrespective of social background or marry only within their own social classes respectively, are less common in practice but have important theoretical implications. Formally, the random mating rule is specified as

$${\mu }_{ij}\left(N^m,N^f\right)=\frac{N_i^m{N}_j^f}{\left(N^m+N^f\right)/2}$$ (40)

where $N^m={\sum }_i{N}_i^m$ and $N^f={\sum }_j{N}_j^f$. The above equation assumes the number of marriages between men in class i and women in class j is constrained by the abundance of both males and females. Other possible definitions of random mating functions are described in Appendix E. For endogamous mating, I assume marriages only happen between men and women within the same social class and thus are constrained by the gender group with fewer members:

$${\mu }_{ij}\left(N^m,N^f\right)=\{\begin{array}{ll}\text{min}\left(N_i^m,N_j^f\right),\hfill & \text{if\hspace{0.17em}} i=j\hfill \\ 0,\hfill & \text{if} i\ne j\hfill \end{array}$$

To evaluate the role of assortative mating in multigenerational processes, I compare long-term multigenerational social reproduction effects estimated from various two-sex mating and mobility scenarios in the next section.

It is worth noting that although the two-sex models provide a more complete set of demographic mechanisms compared to the one-sex models, the two-sex approach is not always preferred over the one-sex models. The two-sex models rely on additional assumptions about mating patterns and require data from individuals’ paternal and maternal ancestors. Such data are often not available for some subgroups in a population (e.g., children born outside of marriage or from single-parent families) and for ancestors beyond the grandparent generation.²⁰ One limitation of most assortative mating functions is that they assume no competition among different classes (“zero spillover mating rule”) (Pollak 1990a) and thus do not provide a dynamic perspective on the formation of marriages. Miller and Inouye (2011) provided a list of candidate two-sex mating rules and evaluate their pros and cons using empirical data. In addition, the one-sex models are still useful when the transmission of socioeconomic status is sex-linked. For example, in some historical populations, high social status, religious positions, and royal titles were inherited only through male lines (Lee and Campbell 1997; Mare and Song 2014; Goldstein 2008). The two-sex models are not simply an extension of the one-sex models. The two approaches imply different institutional frameworks and social processes underlying the inheritance of social status and kinship networks formed by blood and marriage.

8. ILLUSTRATIVE EXAMPLES: MULTIGENERATIONAL SOCIAL MOBILITY AND REPRODUCTION IN THE UNITED STATES

8.1. Data Description

In this section, I illustrate two-generation and three-generation mobility models, with and without demography, using two sources of empirical data: (1) the IPUMS linked representative samples of U.S. censuses (1850 to 1930) (Ruggles et al. 2019) and (2) the Panel Study of Income Dynamics (1968 to 2015) (PSID Main Interview User Manual 2019). The IPUMS linked data are constructed from linking the 1880 complete-count database to 1% samples of the 1850 to 1930 U.S. censuses of the population. The data combine samples from seven pairs of years—1850–1880, 1860–1880, 1870–1880, 1880–1900, 1880–1910, 1880–1920, and 1880–1930—where parents’ information is available in the first census year and offspring’s in the second. Each year contains three independent but linked samples: one of men, one of women, and one of married couples.²¹ Given that the female data contain many missing cases in occupational variables, the following illustration focuses only on male mobility. Occupations in the historical census data are coded using the 1950 Census occupation classifications scheme.²²

The empirical analysis also includes three-generation data from the 1968 to 2015 Panel Study of Income Dynamics. The PSID began in 1968 with a household sample of more than 18,000 Americans from roughly 5,000 families. Original panel members were followed each year prospectively through 1997 and then biannually. The study follows targeted respondents according to a genealogical design. All household members recruited for the PSID in 1968 carried the PSID “gene,” and their detailed socioeconomic information was collected. Members of new households created by offspring of the originally targeted household heads retained the PSID “gene” themselves and became permanent PSID respondents. The PSID Family Identification Mapping System (FIMS) provides a tool to create multigenerational linked samples.²³ I supplement the analysis with simulation data to illustrate a wide range of scenarios that are theoretically important (e.g., perfect immobility or random mating) but generally not observed in empirical data.

The PSID survey asked household heads and wives to report their occupations in almost every wave of the survey. These data have been coded into detailed three-digit census categories since 1980. As a part of a retrospective project, PSID created the Retrospective Occupation-Industry file by collecting three-digit occupation codes for the 1968 to 1980 period (Survey Research Center 1999). I merged these data with cross-year individual files. The occupational variables in the 1968 to 2001 PSID file were originally coded using Census 1970 classification codes, and those in the 2003 to 2015 file were coded using Census 2000 classification codes. Following Hauser (1980), I converted these three-digit occupations into five major occupational groups (upper nonmanual, lower nonmanual, upper manual, lower manual, and farming).²⁴ Because the longitudinal data provide multiple-year observations of each respondent, I use the mode of the cross-year occupational variables (i.e., the occupation that appears most often) to define a person’s working-life major occupation.

The reproduction rates by fathers’ social class were calculated from the average number of sons in the data. Strictly speaking, this measure is not equivalent to the typical Gross Reproduction Rate (GRR) measure, because GRR is defined as the average number of sons who would be born to a man during his lifetime if he lives through his childbearing years and conforms to the age-specific reproduction rates of a given year.²⁵ The surveys omitted sons who died during young childhood before the next census recorded them or before they became a PSID respondent. Additionally, this measure may have underestimated some fathers’ fertility if they were not linked to some of their sons or if they did not live with their sons in the same household.

8.2. Empirical Results

To estimate the two-generation social reproduction model in equation (7), I first calculate transition probabilities and GRR using the IPUMS linked historical census data and the contemporary PSID sample. The mobility probabilities are estimated from multinomial logistic regressions shown in Appendix Tables S1 and S3. The regression models only include fathers’ and grandfathers’ occupations as predictors. If other individual-level covariates are included, the model can be considered as a heterogeneous mobility regime, which is discussed in Section 6.²⁶ The GRRs are estimated from Poisson regression models in which all fathers in each sample are treated as a synthetic cohort. For the historical sample, over 55% (= 43,079/78,133) of fathers belonged to the farming population. This number declined to 6% (=262/4,142) in the contemporary sample. In the late 19^th and early 20^th centuries, more than half of sons born into lower manual families inherited their father’s occupations, as compared to 30.1% of their counterparts in the contemporary sample. The results of GRR show that fertility has declined over time, from more than 2.6 sons per father on average at the beginning of the historical sample to less than 1.6 sons per father in the most recent data. This trend is even more pronounced for farmers whose GRR decreased from 3.1 sons per father to 1.6 sons per father. The social class gradient in fertility has also become less remarkable over time. The gap in GRR between farmers and upper nonmanual workers decreased from 0.5 in the historical sample to 0.1 in the contemporary sample.

Table 2 shows similar estimates from three-generation mobility transition matrices by taking into account grandfathers’ occupational class. Numbers in the mobility table refer to the percentage of sons who would achieve a certain occupational group conditional on the father’s and grandfather’s occupations. Numbers in the column of reproduction rates refer to the number of sons of a father conditional on his own and his father’s occupations. These numbers are estimated from the multinomial logistic regressions and Poission regressions shown in Tables S2 and S4 in the appendix. In both samples, the highest GRR is observed in families in which both fathers and grandfathers are farmers. Some estimates may not be reliable; for example, the category of farmers with fathers in lower nonmanual occupations included only one case in the PSID sample. For illustration purposes of the methods, I ignore such possible inaccuracies, but future research should be cautious about fertility estimates from surveys. The models do not account for the age structure of the population either, as age-classified models require mobility, fertility, and mortality rates by age group and social class, but such estimates are often unreliable in surveys due to small sample sizes. Further discussions about age-classified models are included in Appendix B.

Table 2.

Three-Generation Mobility Transition Matrices and Gross Reproduction Rates

Grandfather’s Occupation	Father’s Occupation	Gross Reproduction Rate (GRR)	Historical Social Mobility Son’s Occupation
			1	2	3	4	5	Total	N
1. Upper nonmanual	1. Upper nonmanual	2.5	37.6	34.1	7.4	16.1	4.7	100.0	3,204
	2. Lower nonmanual	2.3	28.9	46.5	8.0	13.4	3.2	100.0	934
	3. Upper manual	2.5	16.3	27.2	21.6	29.5	5.3	100.0	660
	4. Lower manual	2.5	13.8	21.6	10.9	48.0	5.7	100.0	753
	5. Farming	2.8	17.5	6.2	4.6	12.0	59.7	100.0	1,034
	N		1,869	1,934	585	1,322	875		6,585
2. Lower nonmanual	1. Upper nonmanual	2.4	33.6	37.0	7.9	17.6	3.9	100.0	403
	2. Lower nonmanual	2.2	25.3	49.5	8.4	14.3	2.6	100.0	637
	3. Upper manual	2.4	14.0	28.4	22.3	31.1	4.2	100.0	156
	4. Lower manual	2.3	11.8	22.4	11.2	50.1	4.5	100.0	228
	5. Farming	2.6	17.4	7.6	5.5	14.6	54.9	100.0	150
	N		371	571	154	347	131		1,574
3. Upper manual	1. Upper nonmanual	2.5	25.8	32.3	12.2	22.7	7.0	100.0	1,135
	2. Lower nonmanual	2.3	19.7	43.8	13.1	18.7	4.7	100.0	451
	3. Upper manual	2.5	9.1	21.1	29.2	34.1	6.5	100.0	4,648
	4. Lower manual	2.5	7.6	16.5	14.5	54.5	6.8	100.0	2,051
	5. Farming	2.8	9.2	4.5	5.8	12.9	67.7	100.0	1,573
	N		1,107	1,954	1,944	3,248	1,605		9,858
4. Lower manual	1. Upper nonmanual	2.6	23.6	32.2	8.6	29.1	6.5	100.0	1,142
	2. Lower nonmanual	2.4	18.1	44.0	9.4	24.1	4.4	100.0	591
	3. Upper manual	2.6	8.4	21.1	20.7	43.8	6.0	100.0	1,797
	4. Lower manual	2.5	6.4	15.0	9.3	63.6	5.7	100.0	6,056
	5. Farming	2.8	8.7	4.7	4.3	17.2	65.1	100.0	2,596
	N		1,139	2,038	1,202	5,557	2,246		12,182
5. Farming	1. Upper nonmanual	2.8	28.2	29.3	6.7	22.9	12.9	100.0	2,680
	2. Lower nonmanual	2.6	22.5	41.4	7.5	19.7	9.0	100.0	962
	3. Upper manual	2.8	10.9	20.9	17.4	37.7	13.0	100.0	2,348
	4. Lower manual	2.8	8.5	15.1	8.0	55.7	12.7	100.0	4,218
	5. Farming	3.1	6.5	2.6	2.0	8.4	80.4	100.0	37,726
	N		4,028	3,309	1,771	7,210	31,616		47,934
Grandfather’s Occupation	Father’s Occupation	Gross Reproduction Rate (GRR)	Contemporary Social Mobility from PSID Son’s Occupation
			1	2	3	4	5	Total	N
1. Upper nonmanual	1. Upper nonmanual	1.4	48.1	22.5	17.7	10.6	1.1	100.0	234
	2. Lower nonmanual	1.4	33.4	30.3	21.2	14.6	0.4	100.0	63
	3. Upper manual	1.4	26.4	25.3	31.2	16.4	0.7	100.0	109
	4. Lower manual	1.4	21.1	27.4	27.7	22.4	1.5	100.0	63
	5. Farming	1.5	25.6	17.7	29.4	15.2	12.1	100.0	13
	N		179	119	110	68	6		482
2. Lower nonmanual	1. Upper nonmanual	1.3	45.2	25.1	14.9	13.5	1.3	100.0	131
	2. Lower nonmanual	1.3	30.7	33.0	17.6	18.2	0.5	100.0	69
	3. Upper manual	1.3	24.6	27.9	26.1	20.7	0.8	100.0	89
	4. Lower manual	1.3	19.0	29.3	22.5	27.4	1.7	100.0	66
	5. Farming	1.4	23.5	19.2	24.2	18.9	14.1	100.0	1
	N		115	100	70	67	4		356
3. Upper manual	1. Upper nonmanual	1.5	34.2	25.0	25.0	14.0	1.8	100.0	206
	2. Lower nonmanual	1.4	22.1	31.2	27.9	18.0	0.7	100.0	125
	3. Upper manual	1.4	16.5	24.7	38.7	19.0	1.0	100.0	422
	4. Lower manual	1.4	12.9	26.0	33.5	25.4	2.2	100.0	200
	5. Farming	1.5	15.2	16.3	34.5	16.7	17.4	100.0	9
	N		195	248	320	184	15		962
4. Lower manual	1. Upper nonmanual	1.4	32.5	23.6	23.6	18.9	1.4	100.0	196
	2. Lower nonmanual	1.4	20.7	29.0	26.0	23.8	0.6	100.0	141
	3. Upper manual	1.4	15.4	22.8	35.9	25.1	0.8	100.0	466
	4. Lower manual	1.4	11.7	23.5	30.3	32.7	1.7	100.0	390
	5. Farming	1.5	14.6	15.6	33.1	22.8	13.9	100.0	44
	N		217	292	383	325	20		1,237
5. Farming	1. Upper nonmanual	1.6	30.9	15.2	32.4	19.2	2.3	100.0	143
	2. Lower nonmanual	1.6	19.8	18.9	36.0	24.4	0.9	100.0	75
	3. Upper manual	1.5	13.9	14.0	46.8	24.2	1.2	100.0	387
	4. Lower manual	1.6	10.7	14.6	40.1	32.0	2.6	100.0	305
	5. Farming	1.7	12.1	8.8	39.6	20.2	19.3	100.0	195
	N		169	152	454	276	54		1,105

Source: IPUMS Linked Representative Samples, 1850–1930 (final data release June 2010); Panel Study of Income Dynamics, 1968–2015.

Notes: The three-generation transition matrix shows percentages converted from mobility probabilities, e.g., $p_{Y_3=j\mid Y_2=i,Y_1=k}$; namely, the son of a father in social position i and a grandfather in social position k ends up in position j.

I estimate mobility effects and social reproduction effects discussed in Sections 4.1 and 4.2 using mobility matrices and GRRs shown in Tables 1 and 2. According to the definition in equation (11), the mobility effect is the ratio of the probability of sons from upper nonmanual fathers to that of sons from lower nonmanual fathers to become upper nonmanual workers. The results in Table 3 show that the mobility advantage of children born to upper nonmanual fathers relative to those born to lower nonmanual fathers is 1.317 in the historical sample and 1.596 in the contemporary sample. Note that the result shows only one of many possible choices of the target group (upper nonmanual) and the baseline (lower nonmanual) group in the definitions of mobility and social reproduction effects. The measures of social reproduction effects are defined analogously, except that the outcome measure is the number of sons in upper nonmanual occupations rather than their mobility probabilities (see equation (14)). Upper nonmanual fathers produced 1.456 times as many sons in upper nonmanual occupations than did lower nonmanual fathers in the historical data. The ratio increased to 1.622 in the contemporary sample, indicating stronger intergenerational effects in recent decades than in the past.

Table 1.

Two-Generation Mobility Transition Matrices and Gross Reproduction Rates

Father’s Occupation	Gross Reproduction Rate (GRR)	Historical Social Mobility Son’s Occupation
		1	2	3	4	5	Total	N
1. Upper nonmanual	2.6	31.1	32.2	8.0	20.9	7.8	100.0	8,564
2. Lower nonmanual	2.4	23.6	44.9	8.8	17.7	5.0	100.0	3,575
3. Upper manual	2.6	10.0	21.6	24.1	36.4	7.9	100.0	9,609
4. Lower manual	2.6	7.7	15.8	9.8	58.6	8.1	100.0	13,306
5. Farming	3.1	7.0	2.9	2.4	9.2	78.5	100.0	43,079
N		8,514	9,806	5,656	17,684	36,473		78,133
Father’s Occupation	Gross Reproduction Rate (GRR)	Contemporary Social Mobility from PSID Son’s Occupation
		1	2	3	4	5	Total	N
1. Upper nonmanual	1.5	38.5	22.5	22.5	14.9	1.5	100.0	910
2. Lower nonmanual	1.4	24.1	28.8	26.2	20.3	0.6	100.0	473
3. Upper manual	1.4	16.7	21.5	38.6	22.2	1.0	100.0	1,473
4. Lower manual	1.4	12.7	22.0	33.2	30.1	2.1	100.0	1,024
5. Farming	1.6	13.4	10.7	37.8	20.2	17.9	100.0	262
N		875	911	1,337	920	99		4,142

Source: IPUMS Linked Representative Samples, 1850–1930 (final data release June 2010); Panel Study of Income Dynamics, 1968–2015.

Notes: The two-generation transition matrices show percentages converted from mobility probabilities, e.g., $p_{Y_2=j\mid Y_1=i}$; namely, the son of a father in social position i ends up in position j (see equation (1)).

Table 3.

Ratio Measures of Mobility Effects and Social Reproduction Effects by Comparing Upper Nonmanual and Lower Nonmanual Families in Producing Offspring in Upper Nonmanual Occupations

	Mobility Effect		Social Reproduction Effect
	Net Effect	Total Effect	Net Effect	Total Effect
	(assuming fathers and grandfathers in the same occupation)	(unconditional on fathers’ occupations)	(assuming fathers and grandfathers in the same occupation)	(unconditional on fathers’ occupations)
Historial data
Parents	1.317	1.317	1.456	1.456
	(0.044)	(0.044)	(0.055)	(0.055)
Grandparents	1.490	1.133	1.720	1.344
	(0.085)	(0.058)	(0.109)	(0.080)
Contemporary data
Parents	1.596	1.596	1.622	1.622
	(0.146)	(0.146)	(0.164)	(0.164)
Grandparents	1.566	1.178	1.691	1.277
	(0.195)	(0.121)	(0.233)	(0.157)

Source: IPUMS Linked Representative Samples, 1850–1930 (final data release June 2010); Panel Study of Income Dynamics, 1968–2015.

Notes: Standard errors of the predicted net and total mobility effect and social reproduction effect are estimated from 1,000 bootstrap samples. The net mobility effect refers to the ratio between the probability of achieving upper nonmanual occupations by having upper nonmanual parents rather than lower nonmanual parents (or upper nonmanual grandparents and parents versus lower nonmanual grandparents and parents). The total mobility effect is calculated from the ratio between the probability of achieving upper nonmanual occupations by having upper nonmanual grandparents rather than lower nonmanual grandparents. The net social reproduction effect of parents compares parents in upper nonmanual occupations with those in lower nonmanual occupations in producing upper nonmanual offspring. The net social reproduction effect of grandparents compares grandparents in upper nonmanual occupations with those in lower nonmanual occupations in producing upper nonmanual grandchildren, assuming that parents are in the same occupations as grandparents. The total effect of grandparents compares grandparents who are in upper nonmanual occupations with those in lower nonmanual occupations in producing upper nonmanual grandchildren. The mobility effects and social reproduction effects are defined in equations (11)–(16).

The net and total effects are, by definition, the same for the father generation, but they may differ for the grandfather generation. The net mobility effect of grandfathers in the PSID shows that grandchildren with both upper nonmanual grandfathers and fathers are 1.566 times more likely to attain upper nonmanual occupations compared to grandchildren with lower nonmanual grandfathers and fathers. By contrast, the net social reproduction effect of grandfathers shows that upper nonmanual grandfathers and fathers produce 1.691 times as many upper nonmanual grandchildren than do lower nonmanual grandfathers and fathers. The net effects of grandfathers are bigger than the total effects of grandfathers because the former is estimated by assuming the father and grandfather are in the same occupation, whereas the latter is estimated by assuming the father’s occupation is uncontrolled. Thus, compared to the total effect, the net effect shows differences in descendants from advantaged and disadvantaged families under a more extreme condition. Furthermore, the total mobility and social reproduction effects of grandfathers are smaller than those of fathers because not all fathers stay in the same occupations as the grandfathers. Some family advantages are lost during the intergenerational transmission of status, also known as the “regression toward the mean” phenomenon.

Table 4 shows the effect decomposition based on difference measures described in Section 4.3. The total effect of parents (or grandparents) reflects differences in the total number of sons (or grandsons) in upper nonmanual occupations of fathers (or grandfathers) who are in upper nonmanual occupations versus lower nonmanual occupations. For example, in the historical census data, an upper nonmanual father produces 0.253 more sons who are in upper nonmanual occupations than does a lower nonmanual father. Consistent with our ratio measures in Table 3, parent or grandparent effects shown in the TSRE column are smaller in the contemporary data than in the historical data. The Kitagawa decomposition shows the parts of the social reproduction effects associated with fertility (26.7%) and mobility (73.3%). The proportion of the TSRE explained by the fertility effect is small in the contemporary data (3.5%) because of the small difference in GRR between men in upper and lower nonmanual occupations. As a result, the mobility effect accounts for over 96% of the total social reproduction effect. The Das Gupta decomposition method further partitions the total effect into the mobility and demography effect of the grandparent on the parent generation and of the grandparent on the grandchild generation net of the parent generation. For example, the total effect of grandparents contains the effect of grandparents’ own fertility, the effect of grandparents on parents’ fertility, the effect of grandparents on parents’ mobility, and the effect of grandparents on grandchildren’s mobility. In the historical data, the mobility advantage of grandchildren born to upper nonmanual grandparents relative to those born to lower nonmanual grandparents to achieve upper nonmanual occupations themselves accounted for 41.9% of the total social reproduction effect of grandparents. This proportion declines to 30.5% in the contemporary data. Most interestingly, in the historical data, most influences from grandparents to grandchildren operated through the influence of grandparents’ occupation on their own fertility (33.8%) and the influence of grandparents’ occupation on grandchildren’s social mobility (41.9%). In the contemporary data, however, most influences from grandparents to grandchildren work through the influence of grandparents’ occupation on parents’ mobility (38.7%) and grandchildren’s mobility (30.5%).

Table 4.

Effect Decomposition Based on Difference Measures of Social Reproduction Effects by Comparing Upper Nonmanual and Lower Nonmanual Families in Producing Offspring in Upper Nonmanual Occupations

		Kitagawa Decomposition		Das Gupta Decomposition
	Total Social Reproduction Effect (TSRE)	Total Demography Effect	Total Mobility Effect	Demography Effect(1)	Demography Effect(2)	Mobility Effect(1)	Mobility Effect(2)
	(%)	(%)	(%)	(%)	(%)	(%)	(%)
Historical data
Parents	0.253	0.068	0.185	0.068	-	0.185	-
	(100.0)	(26.7)	(73.3)	(26.7)		(73.3)
Grandparents	0.431	0.230	0.201	0.146	0.085	0.020	0.181
	(100.0)	(53.4)	(46.6)	(33.8)	(19.7)	(4.6)	(41.9)
Contemporary data
Parents	0.215	0.007	0.207	0.007	-	0.207	-
	(100.0)	(3.5)	(96.5)	(3.5)		(96.5)
Grandparents	0.157	0.048	0.109	0.011	0.038	0.061	0.048
	(100.0)	(30.8)	(69.2)	(6.7)	(24.0)	(38.7)	(30.5)

Source: IPUMS Linked Representative Samples, 1850–1930 (final data release June 2010); Panel Study of Income Dynamics, 1968–2015.

Note: Numbers in the parentheses are percentages of the total effect explained by each of the demographic and social mobility components. The decomposition methods are described in equations (20), (21) and (23)–(26). The Das Gupta decomposition method divides the TSRE of grandparents into the mobility and fertility effects from the grandparents on the parents and from the grandparents on the grandchildren net of the parents. Specifically, the TSRE of grandparents are decomposed into the effect of grandparents’ occupation on their own number of offspring (demographic effect (1)), parents’ number of offspring (demographic effect (2)), parents’ mobility (mobility effect (1)), and grandchildren’s mobility (mobility effect (2)).

Table 5 illustrates the long-term property of multigenerational processes described in Section 5. The long-term SRE suggests the degree to which an individual in an upper nonmanual occupation, compared to one in a lower nonmanual occupation, has descendants in upper nonmanual occupations. In the historical sample, the effect begins with a value of 1.46 and eventually converges to equilibrium at 1.16, rather than 1. Thus, differential reproduction rates that favor upper nonmanual men further amplify the effects of intergenerational transmission of status in the historical sample. By contrast, in the contemporary data, an upper nonmanual man has approximately 1.02 times as many upper nonmanual descendants as his counterpart in lower nonmanual occupations in the long run. Upper nonmanual men produce significantly more descendants in upper nonmanual occupations in the first few generations, but this advantage gradually disappears in the long run. This result can be explained by the fact that GRR for men in upper nonmanual occupations is only slightly higher than that for men in lower nonmanual occupations (shown in Table 1).

Table 5.

Long-term Social Reproduction Effects

		Occupation in the founding generation	Distribution of Descendants					Long-term SRE in producing upper nonmanual descendants
			1. Upper nonmanual	2. Lower nonmanual	3. Upper manual	4. Lower manual	5. Farming
Historical data
After n generations
	1	Upper nonmanual	0.81	0.84	0.21	0.54	0.20	1.46
		Lower nonmanual	0.55	1.06	0.21	0.42	0.12
	5	Upper nonmanual	17.24	25.45	10.36	35.51	30.55	1.12
		Lower nonmanual	15.36	22.87	9.23	31.52	25.29
	10	Upper nonmanual	2,190.05	3,036.50	1,288.62	4,507.76	5,566.03	1.15
		Lower nonmanual	1,906.90	2,652.90	1,124.17	3,930.59	4,775.19
	∞	Upper nonmanual	-	-	-	-	-	1.16
		Lower nonmanual	-	-	-	-	-
Contemporary data
After n generations
	1	Upper nonmanual	0.56	0.33	0.33	0.22	0.02	1.62
		Lower nonmanual	0.34	0.41	0.38	0.29	0.01
	5	Upper nonmanual	1.38	1.44	1.90	1.34	0.10	1.02
		Lower nonmanual	1.35	1.40	1.85	1.31	0.09
	10	Upper nonmanual	8.31	8.63	11.41	8.05	0.57	1.02
		Lower nonmanual	8.12	8.44	11.16	7.87	0.56
	∞	Upper nonmanual	-	-	-	-	-	1.02
		Lower nonmanual	-	-	-	-	-

Source: IPUMS Linked Representative Samples, 1850–1930 (final data release June 2010); Panel Study of Income Dynamics, 1968–2015.

Notes: Intergenerational mobility is assumed to follow a Markovian process. Similar results are valid if mobility follows higher-order Markovian processes. The long-term effect is defined as the ratio of upper nonmanual progeny per upper nonmanual ancestor over upper nonmanual progeny per lower nonmanual ancestor. When the ratio = 1, there is no long-term effect. The effect is defined in equation (27) and Appendix D.

The results in Tables 1–5 rely on one-sex models that account for only the male population. As discussed earlier, the one-sex model is useful when the intergenerational transmission of social statuses is sex-linked or identical for both sexes. Below, I describe the two-sex social reproduction model described in Section 7. This model relies on additional assumptions about mate selection behaviors. I first estimate assortative mating patterns using the “force of attraction,” which represents the likelihood that men and women from different occupation groups will form unions, sometimes known as an indicator of preferences between two occupation groups. Table 6 presents the attraction estimates and the number of marriages between husbands and wives in different occupations in the PSID data. These numbers are observed in the first generation but are estimated in the following generations as a function of the force of attraction using equation (37).

Table 6.

Two-Sex Assortative Mating and Force of Attraction (Age 25–60)

	Occupation, Women
Occupation, Men	1. Upper nonmanual	2. Lower nonmanual	3. Upper manual	4. Lower manual	5. Farming	N
1. Upper nonmanual	0.858	0.723	0.181	0.253	0.018
	(348)	(431)	(78)	(146)	(2)	1,005
2. Lower nonmanual	0.321	0.643	0.219	0.323	0.010
	(98)	(259)	(70)	(127)	(1)	555
3. Upper manual	0.274	0.677	0.642	0.673	0.164
	(126)	(487)	(316)	(465)	(19)	1,413
4. Lower manual	0.220	0.400	0.560	0.849	0.307
	(95)	(263)	(259)	(537)	(35)	1,189
5. Farming	0.078	0.142	0.194	0.409	0.863
	(13)	(27)	(33)	(77)	(69)	219
N	680	1,467	756	1,352	126	4,381

Source: Panel Study of Income Dynamics, 1968–2015.

Notes: Numbers in parentheses refer to the number of marriages within each assortative mating category. The parameter for the “force of attraction” (α_ij) represents the likelihood that men and women from two occupation groups will form unions. This value is a function of preferences between two occupation groups and constraints imposed by the sizes of the two groups. The force of attraction is defined in equation (37).

I estimate the number of marriages and the size of male and female populations in subsequent generations recursively using equations (35)–(37), assuming that the force of attraction is timeconstant. The total social reproduction effects are then calculated from the combined distributions of male and female descendants. To evaluate the effect of assortative mating on families’ social reproduction, I compare the results from three mating rules: random mating, endogamous mating, and assortative mating. The functions that characterize these mating rules are described in Section 7. The random mating rule assumes individuals sort into marriages irrespective of their occupational characteristics. The number of marriages is only constrained by the abundance of eligible men and women in each pair of occupations. The endogamous mating rule assumes men and women marry only within their own occupation groups. The number of marriages between men and women in different occupations is zero, and the number of marriages between men and women of the same occupation is constrained by the group with fewer members. I also consider three mobility rules: two-sex mobility assumes that both parents influence their offspring’s occupational attainment equally, same-sex mobility assumes that individuals are influenced only by their same-sex parent, and immobility assumes that sons inherit occupations from their fathers and daughters inherit occupations from their mothers. Although some of these mating and mobility rules are unrealistic in practice, they provide benchmarks for comparison when the roles of observed mating and mobility patterns are evaluated.

The results in Table 7 suggest that the strongest parent effect emerges when men and women marry within their own occupation group, and offspring inherit occupations perfectly from their parents. The ratio measure for this scenario is undefined because it is impossible for offspring born to lower nonmanual parents to become an upper nonmanual worker. The effect is smallest when people mate randomly, and offspring’s mobility is influenced only by their same-sex parent. The TSRE under assortative mating and two-sex mobility, the most common type of mating and mobility rules in human populations, is 0.125 based on the difference measure and 3.18 based on the ratio measure. The results suggest that both parents in upper nonmanual occupations produce 0.125 more offspring in upper nonmanual occupations compared to parents who are both in lower nonmanual occupations, or 3.180 times as many upper nonmanual offspring as their counterparts in lower nonmanual occupations do. The TSRE from the two-sex model under assortative mating rules is smaller than that from the one-sex model in Table 4 (0.125 vs. 0.215). The results imply that the interactions between males and females in union formation and intergenerational mobility weaken the intergenerational influences of parents on offspring and reduce inequality between families in the number of offspring in high-status occupations.

Table 7.

Ratio Measures of Social Reproduction Effects under Different Mating and Mobility Rules

		Total Social Reproduction Effects of Upper Nonmanual vs. Lower Nonmanual Parents
Mating Rule	Intergenerational Mobility Rule	Difference measure	Ratio measure
Random mating	Same-sex (father-son; mother-daughter)	0.034	2.014
	Two-sex	0.041	1.987
	Immobility (perfect inheritance)	0.160	∞
Endogamous mating	Same-sex (father-son; mother-daughter)	0.193	2.940
	Two-sex	0.234	2.900
	Immobility	0.695	∞
Assortative mating	Same-sex (father-son; mother-daughter)	0.103	3.225
	Two-sex	0.125	3.180
	Immobility (perfect inheritance)	0.356	∞

Source: Panel Study of Income Dynamics (1968–2015) and simulation data.

Notes: The total effect of parents compares parents who are in upper nonmanual occupations with those in lower nonmanual occupations in producing upper nonmanual offspring. The ratio = 1 means no effect. The effect accounts for probabilities that men and women in upper nonmanual (or lower nonmanual) occupations will form unions, produce offspring, and transmit their social status to their offspring. The random mating rule assumes mating between individuals where the choice of partner is not influenced by occupations. The endogamous mating rule assumes men and women marry only within their own occupation groups. The assortative mating rule assumes individuals with similar occupations mate with one another more frequently than would be expected under a random mating rule. The same-sex mobility rule assumes individuals are influenced by their same-sex parent only (namely, sons by fathers and daughters by mothers). The two-sex mobility rule assumes individuals’ occupations are influenced by occupations of both parents. The immobility rule assumes sons inherit occupations from their fathers and daughters inherit occupations from their mothers.

9. CONCLUSIONS

This paper provides an integrated methodological framework that allows researchers to analyze the combined effects of social mobility and demography in the process of multigenerational social inequality among families. It shifts the focus from a pure probabilistic view on individuals’ mobility probabilities to a distributional view that emphasizes the number of offspring and descendants who vary in their social class in succeeding generations. Families who have more high-status offspring may be different from families whose offspring have a higher probability of achieving high status because the joint effect of fertility and mortality may operate against families’ advantages in social mobility. The moderating effects of demographic forces often accumulate over time, as the intergenerational reproduction of families is a dual endeavor to achieve both reproductive success and status inheritance across generations. Table 8 summarizes statistical models used in classic mobility research and recent work that incorporates the role of demography in social mobility processes. Building on this line of previous works, I illustrated how to define and estimate various types of multigenerational processes and effects with and without the role of demography from both short- and long-term perspectives. More specifically, these methodological issues include differences between two-generational and multigenerational transition matrices, net and total social reproduction effects from one generation to succeeding generations, the effect decomposition, equilibrium states of long-term multigenerational effects, three types of heterogeneity in multigenerational mobility, and two-sex versus one-sex multigenerational models and their different implications for population dynamics. Careful and creative use of these models with appropriate multigenerational data will advance our knowledge of family processes that were occurring in the past and help forecast trends in the future.

Table 8.

A Summary of Intergenerational Social Mobility Research

Models	Composition	Mobility	Demography	Methods	Exemplary Prior Research
Classic Mobility Models	One-Sex	Two-Generation	No	Markov chain models	Prais (1955)
				Mobility tables and path analysis	Blau and Duncan (1967); Featherman and Hauser (1978)
				OLS models	Hout (2018)
				Loglinear models	DiPrete (1990); Erikson and Goldthorpe (1992); Grusky and Hauser (1984); Hout (1988); Jonsson et al. (2009); Torche (2011); Yamaguchi (1987); Xie (1992)
				Log-log regression	Behrman and Taubman (1985); Landersø and Heckman (2017); Mazumder (2005); Mitnik and Grusky (2017); Solon (1992)
	One-Sex	Multiple-Generation	No	Loglinear models	Chan and Boliver (2013)
				Survival analysis	Zeng and Xie (2014)
				Rank-rank regression	Pfeffer and Killewald (2017)
				Log-log regression	Long and Ferrie (2018); Solon (2014, 2018);
	Two-Sex	Two-Generation	No	Loglinear models	Beller (2009)
				Rank-rank regression	Chetty et al. (2014)
				Log-log regression	Lee and Solon (2009)
	Two-Sex	Multiple-Generation	No	Path analysis	Warren and Hauser (1997)
				Variance component model	Jæger (2012)
Joint Demography-Mobility Models	One-Sex	Two-Generation	Yes	Markov chain models	Matras (1961); Preston (1974); Mare (1997) Mare and Maralani (2006); Maralani (2013)
	One-Sex	Multiple-Generation	Yes	Markov chain models	Mare and Song (2014)
	Two-Sex	Two-Generation	Yes	Markov chain models	Preston and Campbell (1993)
	Two-Sex	Multiple-Generation	Yes	Markov chain models	Song and Mare (2017)

Despite their advantages over conventional two-generation mobility methods based on Markov chains, the multigenerational models proposed in this paper are far from complete. I outline several promising directions for future research when more refined statistical and demographic techniques are available to model the complex interactions among family members.

First, all models discussed in this paper assume a single, constant measure of socioeconomic status for each generation, ignoring changes in parents’ status across their own and their children’s life spans. From a life-course perspective, common indicators of social status—including education, employment, and earnings prospects—evolve over time. Life-cycle changes in labor supply, human capital accumulation, consumption, and nonmarket returns to education jointly shape individuals’ life trajectories and their offspring’s childhood skill formation (Heckman 1976; Heckman et al. 2013). Ignoring parents’ and offspring’ life cycles in the estimation of intergenerational association may lead to “life-cycle bias” (Mazumder 2005; Haider and Solon 2006). Furthermore, shared lifetimes, namely years during which two or more generations overlap, often vary across families and may predict the cumulative amount of influence from one generation to another (Song and Mare 2019). The complex linkage between within-generation status changes with age and between-generation transmission of statuses also warrants future consideration, given the growth of precarious work and earnings instability in the U.S. labor market and elsewhere (Gottschalk et al. 1994; Kalleberg 2009). Appendix B illustrates an age-classified model that refines the social reproduction model in equation (9) by adding age-specific fertility and mortality rates. The model can further incorporate intragenerational status mobility if age-specific mobility rates from high-quality longitudinal data are available (Lipset and Bendix 1952; Cheng and Song 2019).

Second, models in this paper do not address problems of causal inference, especially mechanisms and identification issues, in estimating the causal effects of grandparents and the other kin. Fast-growing literature in sociology and economics points to causality as a central problem in multigenerational research and offers various approaches to gauging biases in traditional association measures. Anderson et al. (2018) provided a meta-analysis of recent studies on grandparent effects on education, showing a wide range of effect estimates that vary by social contexts, analytical methods, and the operationalization of the concept of grandparent effects. Other problems, such as collider bias, unobserved confounders, and survivorship selection (e.g., Breen 2018; Sharkey and Elwert 2011; Song 2016), have also appeared in studies that aim to make causal claims about multigenerational influence using counterfactual analyses, causal graphical models, and methods based on inverse probability treatment weighting (IPTW).

Third, models in this paper ignore the complex role of migration in shaping multigenerational influences and population dynamics. Without migration, all changes in a closed population result only from births and deaths, as illustrated in the social reproduction model. Yet, migration, both internal and international, may change the composition of a population, household structure, and relationships among extended family members in origin and destination places. Zeng and Xie (2014) provided an example of grandparent influences in rural China with many skipped-generation households where children are living with their grandparents while their parents leave to work in urban areas. More work is needed to test whether and how the mobility process is interdependent with migration, even many generations back, as well as how big events, such as mass migration and refugee resettlement, influence social mobility of descendants of migrant and native-born populations.

Fourth, all socioeconomic measures embrace some degree of uncertainty that may result from random measurement errors or systematic biases. Substantive and methodological issues in the latent structure of variables are an old topic in sociology (e.g., Coleman 1964b), but they have recently been highlighted in analyses of multigenerational mobility (Solon 2018; Torche and Corvalan 2018; Clark 2014). Random noise in socioeconomic measures may lead to attenuation bias, the classical errors-in-variables problem, which shrinks estimated intergenerational correlations toward zero. Yet, no matter how accurately measured, indicators like income, occupation, and education are always a proxy for the underlying concept of “social status.” Almost all methodological issues described in this paper are still valid in models that incorporate measurement errors or a latent structure of underlying variables. However, as Singer and Spilerman (1976: 454) noted, dealing with the hidden structure of variables in Markov chains would lead to a considerable increase in complexity in both theory and methods.

All of these issues lie outside the scope of this paper, but they are central to the study of social stratification and mobility and to a better understanding of the social mechanisms that govern continuity and changes within families, dynasties, and populations. I leave these important modifications and challenges to future research.