The Informational Content of Surnames, the Evolution of Intergenerational Mobility and Assortative Matin

by Maia Güell José V. Rodríguez Mora^** *** Christopher I. Telmer

Documento de Trabajo 2014-19

December 2014

University of Edinburgh, FEDEA, CEPR & IZA. ** University of Edinburgh & CEPR. *** Tepper School of Business Carnegie Mellon University

Los Documentos de Trabajo se distribuyen gratuitamente a las Universidades e Instituciones de Investigación que lo solicitan. No obstante están disponibles en texto completo a través de Internet: http://www.fedea.es. These Working Paper are distributed free of charge to University Department and other Research Centres. They are also available through Internet: http://www.fedea.es. ISSN:1696-750

Maia G¨uell

José V. Rodríguez Mora^†

Christopher I. Telmer

University of Edinburgh,

University of Edinburgh & CEPR

Tepper School of Business

FEDEA, CEPR & IZA

Carnegie Mellon University

First version: May 2007. This version: September 2014

Abstract

We propose a new methodology for measuring intergenerational mobility in economic wellbeing. Our method is based on the joint distribution of surnames and economic outcomes. It circumvents the need for intergenerational panel data, a long-standing stumbling block for understanding mobility. It does so by using cross-sectional data alongside a calibrated structural model in order to recover the traditional intergenerational elasticity measures. Our main idea is simple. If ‘inheritance’ is important for economic outcomes, then rare surnames should predict economic outcomes in the cross-section. This is because rare surnames are indicative of familial linkages. If the number of rare surnames is small this approach will not work. However, rare surnames are abundant in the highly-skewed nature of surname distributions from most Western societies. We develop a model that articulates this idea and shows that the more important is inheritance, the more informative will be surnames. This result is robust to a variety of diferent assumptions about fertility and mating. We apply our method using the 2001 census from Catalonia, a large region of Spain. We use educational attainment as a proxy for overall economic well-being. A calibration exercise results in an estimate of the intergenerational correlation of educational attainment of 0.60. We also find evidence suggesting that mobility has decreased among the diferent generations of the 20th century. A complementary analysis based on sibling correlations confirms our results and provides a robustness check on our method. Our model and our data allow us to examine one possible explanation for the observed decrease in mobility. We find that the degree of assortative mating has increased over time. Overall, we argue that our method has promise because it can tap the vast mines of census data that are available in a heretofore unexploited manner.

Key words: Surnames, intergenerational mobility, cross-sectional data, population genetics, assortative mating, siblings. TDI C31 E24 J1

JEL codes: C31, E24, J1

^∗This paper was previously circulated as G¨uell, Rodríguez Mora, and Telmer (2007): “Intergenerational Mobility and the Informative Content of Surnames,” CEPR Discussion Paper 6316. We thank Laurence Ales, Namkee Ann, Manuel F. Bag¨ués, Melvin Coles, Vicente Cuñat, John Hassler, John Knowles, Ramon Marimon, Laura Mayoral, Bash Mazumder, John Moore, Diego Puga, Gary Solon, Murat Tasci, three anonymous referees, the Editor, Imran Rasul, and numerous seminar participants for very useful suggestions. We also thank Anisha Gosh, Rasa Karapanza, Ana Mosterín and Robert Zymek for superb research assistance. We are grateful to Joan Gieseke for editorial assistance. Financial support from the Spanish Ministry of Education and Science under grants ECO2011-28965 (MG) and ECO2011-25272 (JVRM) is gratefully acknowledged. Our online appendix is available at http://www.sevirodriguezmora.com/grt/GRT online appendix.pdf

^†Corresponding author: School of Economics, University of Edinburgh, 31 Buccleuch Place, Edinburgh EH8 9JT, United Kingdom. Email: sevimora@gmail.com

Maia Güell, University of Edinburgh, FEDEA, CEPR & IZA

José V. Rodríguez Mora, University of Edinburgh & CEPR

Chris Telmer, Tepper School of Business Carnegie Mellon University

Intergenerational mobility is the other side of coin of inheritance. If parents determine very much the economic outcomes of their adult children, then mobility is low. Instead, mobility is high when the chances of moving up and down the social ladder with respect to one’s parents are high. Knowing about mobility is important to understand the workings of a society and to design social policies.

Intergenerational mobility is notoriously hard to measure, mainly due to data limitations. The traditional method for measuring it requires panel data linking economic outcomes of adult children to their parents for multiple generations. These data exists only a small number of countries. Even when these data are available, it is well known (Solon (1992, 2002)) that it is of limited use for understanding how mobility varies across either time or countries.

We propose a new methodology for measuring intergenerational mobility that circumvents the need for intergenerational panel data. Our method is based on the joint distribution of surnames and economic outcomes. The first intuition of our method is simple. Surnames are passed along from parents to children for the huge majority of people. Therefore surnames travel together across generations with the other variables that can potentially matter for individual outcomes. If mobility is low, then these inherited variables will matter very much and surnames will be informative of individual outcomes. This means that we can look at how much surnames inform about individual outcomes.

There is a second necessary ingredient for our method to work. In Spain, many people are called Garcia and are not necessarily family related. However, the process by which surnames die (when the last male holder of a particular name bears no male children) and are born (when new name enters the population via immigration) generates a skewed surname distribution, with a small number of names each being held by a large number of people and a large number of names each being held by a small number of people. This means that for the large majority of people their surname is relatively small, shared mainly with their relatives and thus can potentially carry information.

Our main claim is that by observing a cross-section of surnames and income of a society it is possible to obtain information about intergenerational mobility even if we have no explicit link between the income of parents and children.

We apply our method using the 2001 census from Catalonia, a large region of Spain. We use educational attainment as a proxy for overall economic well-being. We find evidence suggesting that mobility has decreased among the different generations of the 20th century. Thanks to the Spanish naming convention, we can also approximate siblings (i.e. people that share 2 infrequent surnames). A complementary analysis based on sibling correlations confirms our results and provides a robustness check on our method.

Also, thanks again to Spanish naming convention we can look at the assortative mating of the parents of each individual of the data, that is, how similar in terms of characteristics are the two surnames of each person. We find that the degree of assortative mating has increased over time.

Overall, we argue that our method has promise because it can tap the vast mines of census data that are available in a heretofore unexploited manner.

1 Introduction

The empirical challenges to understanding intergenerational mobility are large. The main reasons are data limitations. To address the issue directly, panel data are needed that link the economic status of adults to that of their parents for multiple generations. The existence of such data is rare. That which does exist (i) has been strongly criticized in terms of various biases, and (ii) cannot address the question of how mobility has changed over time.¹ This last point deserves emphasis. While the sheer complexity of intergenerational mobility makes its measurement at a point in time very dificult to do and to interpret, the prospect of measuring how it has changed across time may be more promising. Data limitations, however, have left this promise unfulfilled.

This paper attempts to make headway by introducing a new source of data: surnames and how they vary with measures of economic well-being. We show that the implicit intergenerational links that are inherent in surnames provide a useful stand-in for the explicit intergenerational links that would exist in multi-generational panel data. We do so using both theory and data. Our theory shows how surname data can allow one to estimate both the level and the change in intergenerational mobility even in the absence of explicit links between children and their parents. Our empirical work implements this idea. We use novel, census-based data from Catalonia — a large region of Spain — to obtain measures of mobility both at one point in time and across generations. The former yield, in a calibration exercise, estimates of the intergenerational correlation of edu cational attainment of 0.60 across several diferent specifications of our model. The latter, more strikingly, provide an estimate of how mobility has changed over time. We find that, among the diferent generations of the 20th century, it has decreased. That is, the economic status of parents seems to have become more closely related to that of their children. Why? Our methodology ofers one potential explanation. Assortative mating — the tendency for people with similar economic status to mate with one another — has increased over time. We use our surname data to establish this and our model to demonstrate how it is one potential source of a decrease in intergenerational mobility.

Our main idea is straightforward. Intergenerational mobility focuses on how children inherit aspects of economic well-being from their parents. In the data, we can observe outcomes on economic well-being, but, unless the data contain explicit details on how parents and children are related, we are unable to determine the degree to which the economic well-being has been inherited. However, suppose that the data also allow us to observe surnames. Surnames are almost always inherited from one’s father. Thus, they can serve as markers. They are intrinsically irrelevant for the determination of economic well-being, but they get passed from one generation to the next, alongside other characteristics that do matter. The more important are these characteristics in determining outcomes, the more inheritable are the outcomes, and, therefore, the more information the surnames will contain on the values of outcomes. In this way, surnames can be used to measure the importance of inheritance and thus identify the degree of mobility. The following example articulates this mechanism in more detail.

¹See Solon (1992), Haider and Solon (2006), Hertz (2007) and Lee and Solon (2009). Extensive literature surveys are available in Solon (1999) and Black and Devereux (2011).

Consider a society comprising two distinct groups of people: rich and poor. In each group there are males and females, but, because surnames are (typically) passed along the male lineage, we will ignore the females for now. Suppose that the males within each group all share the same surname: Richmanson for the rich and Poormanson for the poor. This means that if we partition society either by surname or by economic status, the results are the same. We would say that, among this initial generation, surnames are ‘perfectly informative.’ If you know a man’s surname, you know his economic status. Now, how informative will surnames be among the subsequent generation? The answer depends on the degree of inheritance. Consider two extremes. First, if inheritance is ‘perfect’ — meaning that there is no mobility whatsoever — then the economic status of all sons would be identical to that of their fathers. Surnames, being passed from father to son in exactly the same manner as economic status, would remain perfectly informative. Second, if inheritance is irrelevant, so that the sons of both the Richmansons and Poormansons are equally likely to be rich or poor, then surnames would become perfectly uninformative among the next generation. Thus, the informativeness of surnames depends on the degree of intergenerational mobility. This is the essence of the mechanism with which we are able to infer the degree of mobility.

Reality lies between these two extremes, and surnames carry some information. But, there is a tendency for surnames to become non-informative as time goes on. To understand this, suppose that economic status follows a stationary process with some degree of persistence, and that this process is the same for both the Richmansons and the Poormansons. In this case, surnames will remain informative among the sons of the initial generation, but the informativeness will not be perfect. It will become less perfect with each subsequent generation. After enough generations the cross-sectional distribution of status among both elements of the surname partition will be the same and the initial informativeness will have vanished. This is what makes our methodology somewhat less than obvious. For surnames to contain information about intergenerational mobility there must be something else going on that inhibits this convergence to a common stationary distribution.

The additional ingredient in our study is a birth-death process for surnames. Surnames die when the last male holder of a particular name bears no male children. They are born when someone mutates their name, or when a new name enters the population via immigration. This generates a skewed surname distribution, with a small number of names each being held by a large number of people and a large number of names each being held by a small number of people. The surname distribution in most Western societies takes exactly this form.

Herein lies the key to our method: skewness in the surname distribution. We cannot learn anything from the name Smith. The cross-sectional distribution among the Smiths is similar to that among society as a whole. We can, however, learn something from the multitude of rare surnames. This is because they form a partition of the population that is correlated with familial linkages. Suppose, for example, that a rich person chooses a new, unique surname and that this person’s male descendants maintain this name. Then, if mobility is low, this surname will be informative for some number of subsequent generations. It will be shared by a group of rich people. If mobility is high, it will not be as informative. The same logic holds for a poor person who changes their name, or for an immigrant with a distinct name. Over time these people’s surnames will either die or will multiply. If they multiply, then, as discussed above, they are likely to become less informative. But, overall, a stationary birth/death process will generate a stationary commonality-ordered surname frequency distribution. There will always be some rare names. These are the names from which we can extract information on intergenerational mobility.

An important issue for us is ethnicity. Early-generation immigrants, for example, tend to have distinctive surnames and relatively low economic status. Surnames, therefore, contain information on both familial linkages and ethnicity. Ethnicity, therefore, is likely to make surnames economically informative, in-and-of-itself. One might even think that the only reason that surnames are informative is because they reveal ethnicity. We show that this is not the case. Our data permit us to control for ethnicity and we find that, once we do, surnames remain informative. We provide supplemental evidence indicating that this is because our methodology is able to reveal familial linkages when applied to ethnically-homogeneous populations. Our method seems to be measuring what our theory indicates it should be measuring. We show this in Section 4 and go on to relate it to existing papers on intergenerational mobility, a literature that typically amalgamates the inheritance of ethnicity with the inheritance of unobservable economic traits. Our approach allows us to separately identify them. We therefore report two sets of parameter estimates, those that control for ethnicity and those that do not. The former serve to highlight a distinctive feature of our methodology whereas the latter draws links to the existing literature.

Our method is not without its limitations. Most obviously, we do not estimate the intergenerational elasticity directly, as is typically done with panel data. The limitations, however, are counterbalanced by some important strengths. One is that our method is quite data-friendly. We are able to measure mobility from a single, cross-sectional census. We do not require any explicit links between parents and children. As a result, the censuses that are periodically compiled by many governments contain most of the information that we require. Moreover, a great deal of confidentiality and anonymity can be maintained while still allowing access to the necessary information. Surnames can be encoded without negating what we do.

Our method can be applied to any country that follows the Western naming convention. Spain, the source of our data, uses a variant of the Western convention that is identical to its Anglo-Saxon counterpart, but with the additional ingredient of the maternal surname, which survives for one generation. We show that knowledge of the maternal name can be exploited in three ways: (i) to control for ethnicity, (ii) to determine the degree of assortative mating among the parents of an individual, and (iii) to partition our data into sets of siblings. The latter afords us a powerful robustness check on our methodology.

Existing literature that is closely related to our study is as follows. First, a number of papers have studied the distribution of first names to understand phenomena ranging from racial discrimination and economic status (e.g., Bertrand and Mullainathan (2004), Fryer and Levitt (2004), Levitt and Dubner (2005)) to mobility (Olivetti and Paserman (2011)). Endogeneity plays a key role here. Parents choose first names and these choices can be related to parental characteristics. Our study, in sharp contrast, makes use of the fact that surnames are surely much more exogenous in nature, playing the role of innocuous markers. Second, papers that have used last names to study economic phenomena include Angelucci, De Giorgi, Rangel, and Rasul (2010), Bag¨ués (2005), Collado, Ortuño-Ortín, and Romeo (2012a) and Long and Ferrie (2011). These works use surnames as family links explicitly and intensively (i.e., to determine familial links in a small sample) while we use them implicitly and extensively for the whole population. Third, a large literature on mobility has devised many creative ways to overcome limited panel data availability. Examples include Aaronson and Mazumder (2008), Dahan and Gaviria (2001), Duncan, Featherman, and Duncan (1972), Levine and Mazumder (2007), Page and Solon (2003), and Solon, Corcoran, Gordon, and Laren (1991). Again, most of these papers rely on explicit familial linkages, the lack of which is a distinguishing feature of our approach.

Finally, Collado, Ortuño-Ortín, and Romeo (2012b), Clark (2013) and Diaz Vidal (2014) share our emphasis on the use of rare surnames for the measurement of intergenerational mobility. Nevertheless, our data and methodology are quite distinct. These papers use a “grouped estimator” of the standard intergenerational mobility coeficient. They compute within-surname averages for multiple points in time, and then, based on a panel of these averages, they obtain an estimate of the mobility coeficient. It is well known (e.g., Card et al. (2000), Aaronson and Mazumder (2008) and Solon (1992)) that this estimator will be biased upward if the grouping variable is associated with economic outcomes. This is likely to be the case for grouping by surname. Moreover, the surname-grouped intergenerational correlation is likely to be inelastic to diferences in ‘true’ mobility over time and across societies.² Our approach, in contrast, exploits variation both across and within surname-groups. It deals with the reason for group averaging — the absence of explicit familial linkages — without sufering from the (above) drawbacks of group averaging. It is elastic to diferences in ‘true’ mobility. What enables our approach to do this is our use of an explicit model. The model, of course, comes with its own set of drawbacks. But averaging-away intergenerational variation is not one of them.³

²Both drawbacks of the grouped estimator can be seen with the following example. Imagine that the following assumptions hold. First, in addition to family wealth, surnames contain information on something else that is inheritable and that afects economic status. We will call it “beauty,” but it could be religion, ethnicity, or anything other inheritable trait. Second, assume that mobility is high, so that, once beauty is taken into account, parental status has little predictive power over children’s status. Within-lineage variation in status will therefore be large. Nevertheless, an estimator of the mobility coeficient based on a panel of cross-surname averages will be close to unity. This is because the within-lineage variation — that which is the essence of intergenerational mobility will have been averaged away, leaving only variation in the purely inherited trait, beauty. Note that this will be true irrespective of the degree to which beauty matters for status. Note also that any structural variation in ‘true mobility across either time or societies will be masked by the within-lineage averaging. This is the sense in which the grouped estimator is inelastic to diferences in mobility.

³Our approach is also distinguished by the use of a complete population census at a single point in time. In general, the above studies use subpopulations at multiple points in time. The latter approach uses time-series data and is thus able to obtain a direct estimate of the mobility coeficient. The drawback, however, is that the process of selecting the subpopulation may subject this estimate to selection and survivorship bias. Clark (2013), for example,

The remainder of our paper is organized as follows. In Section 2 we define an empirical measure of the extent to which surnames are informative for the economic status of their owners. We also develop and analyze a model of the joint distribution of surnames and economic outcomes. This model maps the informational content of surnames into an intergenerational correlation that is standard in the literature. Section 3 describes the census data that we use to implement our model. Section 4 makes precise how we control for ethnicity and how the parameters that we try to estimate relate to those that typify the literature. Section 5 reports results on mobility in Catalonia in the year 2001, and Section 6 uses these results to calibrate our model. Section 7 reports results on how mobility has changed over time. Section 8 presents additional theory and evidence suggesting that changes in assortative mating may have played an important role. Section 9 afirms the robustness of our methodology by constructing and analyzing an accurate proxy of sibling correlations. Section 10 concludes.

2 A Model of the Informational Content of Surnames

The population consists of N individuals. Each individual is associated with one surname, s, which is an element of the finite set of all possible surnames, Ω. A census is a list with one entry per individual in the population. The entry records individual i’s surname, a measure of their economic well-being, , and a vector of additional characteristics, , such as age, gender, ethnicity, place of birth, etc. We define the informational content of surnames (ICS) as the diference in the between two regressions. The first, with denoted , estimates economic well-being for the average individual with surname s

\[y _ {i s} = \gamma^ {\prime} X _ {i s} + b ^ {\prime} D + \mathrm{residual},\tag{1}\]

where D is an S-vector of surname-dummy variables with if individual i has surname s and otherwise. Our methodology is based on the idea that as surname s becomes more infrequent it becomes more likely that this average gets taken across individuals with familial linkages, thereby providing information about economic inheritance. Note that surnames group together diferent members of the same family (brothers, cousins, uncles, etc.) and not only fathers and sons. Our model in Section 2.1 provides a mapping between our measure and the traditional one based on father-sons.

uses data from English probates. The wealthy appear in such probates at a higher frequency than the less wealthy and, more importantly, at a higher frequency than the less wealthy whose families were once wealthy. This is survivorship bias and, if present, it will bias the mobility coeficient upward.

The second regression mixes up the surnames so that they cannot be informative. It is

\[y _ {i s} = \gamma^ {\prime} X _ {i s} + b ^ {\prime} F + \mathrm{residual},\tag{2}\]

where F is an vector of ‘fake’ dummy variables that randomly assign surnames to individuals in a manner that maintains the marginal distribution of surnames. The from this regression is denoted . The ICS is defined as

\[\mathrm{ICS} \equiv R _ {L} ^ {2} - R _ {F} ^ {2}.\tag{3}\]

The ICS is a moment of the joint distribution of surnames and economic well-being that measures the incremental informational content of surnames. In our model it will turn out to be monotonically increasing in the degree of economic inheritance. This is what will allow us to use surnames to identify mobility.

The basic idea behind the ICS measure is this. Surnames define a partition of the population. If the surname partition is informative about familial linkages — if some individuals with the same surname come from the same family — then it can be used to measure the importance of economic inheritance. The fake-surname partition is constructed to have zero information about familia linkages. By comparing the relative informativeness of the two partitions, therefore, we measure the extent to which surnames contain incremental information.

The ICS measure has a number of important advantages. Suppose, for example, that every individual had a unique surname. Then, by definition, surnames would contain zero information. In this case but ICS=0 indicating, as it should, that surnames contain no information. In addition, the ICS insures us against spuriously attributing informativeness to surnames that might be, instead, attributable to having many dummy variables in our regressions. This is important because the number of surnames in a typical census is large and grows with the population size so that the D matrix from Equation (1) has many columns.

In Section 5 we estimate the ICS using Spanish census data. Before doing so, we develop a model that allows us to map ICS units into ‘inheritance units:’ an intergenerational correlation coeficient.

2.1 Model Set-up

In Western societies the intergenerational transmission of surnames occurs primarily between fathers and sons. We therefore begin by ignoring females. At date t − 1 the population consists of males. Each individual reproduces with probability q. Conditional on reproducing, an individual gives birth to m sons. Generations do not overlap. Fathers die after reproducing (or failing to reproduce). The expected growth rate of the population is , which we assume to be zero.

Each of the individuals is associated with one surname from the fixed, discrete set Ω. The typical element of denoted , can take on one of S diferent values (so that . The date marginal distribution of surnames is . The number of active surnames at date , denoted , is therefore equal to the number of strictly positive values of . If each individual has a unique surname then . Otherwise . In reality — and in our model is far greater than . The initial distribution is denoted

The surname distribution evolves from to according to a birth/death process. The death of a surname occurs if all fathers possessing that name bear zero ofspring.⁴ Birth occurs via mutation: a son acquiring a diferent (typically new) surname than his father. Mutations are a necessary ingredient of our methodology. In Section 2.2 we show that, absent mutations, the surname distribution will neither be informative nor will it resemble the highly skewed nature of observed surname distributions. Note that there is a certain irony here. Mutations seemingly frustrate the surname researcher: they ‘destroy’ intergenerational linkages. Yet without them surnames will eventually be of no use.

Formally, mutation occurs as follows. At date t each existing name, , will have vanished, so that , if all fathers possessing that name at bear zero ofspring. This occurs with probability where , the number of fathers with name s. A surviving surname matches that of the father with probability and mutates with probability . A mutated surname is simply a new name, , chosen randomly.

Economic well-being is passed from fathers to sons according to a simple autoregressive process that is typical in the literature,

⁴The death of surname s can also occur if all sons of all fathers with surname s mutate their names. Quantitatively, however, the likelihood of this happening is dwarfed by the likelihood that these fathers simply give birth to zero sons.

\[y _ {i s t} = \rho y _ {i p, t - 1} + \varepsilon_ {i s t},\tag{4}\]

where is individual i’s father’s economic well-being, one generation removed. By definition the surname associated with ₋₁ is the same as that of (unless mutation occurs). Note that the mean of is zero, meaning that we are dealing with demeaned data relative to what is implicit in our data and in from Equations (1) and (2). Hereafter, for simplicity we refer to as income. We refer to the parameter as the inheritance parameter.

Siblings are individuals with and such that their surname s and parent are the same (again, ignoring mutation). Identical surnames can also be associated with cousins, second cousins and so on. On the other hand, identical surnames can arise purely by chance, in the absence of any familial linkages. The smaller is , the less likely this is (e.g., if it is impossible).

The sense in which relates to families and inheritance is manifest in the distinction between the conditional and the unconditional variance of . For example, the cross-sectional variance between siblings is equal to the conditional variance, . For cousins it is . For the entire population it coincides with the unconditional variance, . A larger inheritance parameter, therefore implies lower cross-sectional variance between family members relative to overall cross-sectional variance. The larger is the larger will be the tendency for a surname to link two people with similar incomes, relative to two people randomly chosen from the population with, typically, diferent surnames.

2.2 Analysis

The most important feature of our model is that skewness in the surname distribution gives rise to the informational content of surnames, and that this informational content is increasing in the inheritance parameter, . We demonstrate this in the following sequence of properties.

Property 1 : Random walk behavior

Suppose that there is no surname mutation, , and the expected growth rate of the population is zero, . Then the number of individuals with surname follows a driftless random walk with an absorbing barrier at zero.

The proof is in Appendix A. It is a simple consequence of the fact that the number of individuals with a given surname is a binomial random variable. Its importance is that it tells us that mutation is necessary for surnames to be informative. This is because a driftless random walk will, given enough time, visit all parts of its sample space. Eventually, therefore, all but one (non-informative) surname will disappear prior to the disappearance of the population (which, with , must also eventually happen).

Next we demonstrate the way in which mutation generates skewness and thus informativeness. To do so we work with the ordered frequency distribution, denoted . This distribution simply provides the relative frequency of the most common surname, the second most common surname, and so on. The long-run distribution associated with is denoted , for

Property 2 : Skewed surname distribution

Given zero expected population growth, mq = 1, and a mutation rate, , then for any initial distribution, (and the associated , there exists a such that, for all the distributions display three key properties: (i) they are highly skewed, (ii) the number of individuals per surname is a constant, (iii) the Gini coeficient is a constant.

We demonstrate this property using simulation-based evidence and a battery of sensitivity and robustness checks. We use the following baseline parameter values. First, we abstract from growth so that the expected population growth rate, is zero. To achieve this, we set the reproduction probability to and the number of ofspring to . Second, we choose the conditional variance, , and the mutation rate, . Third, the initial number of individuals, , is set to 1 million and the initial surname and income distributions are uniform. Finally, we vary the inheritance parameter, from 0.05 to 0.95. Figures 1 and 2 use these values to establish Property 2. Robustness checks are provided in our online appendix.

In Figure 1 we plot the time-series, for , of the number of individuals per surname and the Gini coeficient of the distributions . In each graph there are four time series, each one corresponding to a diferent initial condition for the number of surnames (described in the caption). These moments of the distribution have clearly converged, thus validating Property 2. It is well known that the Pareto distribution provides a good approximation of surname distributions in many societies (Fox and Lasker (1983)). Since the Pareto distribution is completely characterized by these two moments, it seems likely that , also plotted in the figure for (along with the associated Lorenz curve), is a Pareto distribution. For our purposes, however, the exact form of is not critical. What is critical is the behavior of the ICS, discussed below.

⁵Formally, order the elements of Ω (arbitrarily) so that we can write . Define the ranking function as that which ranks each surname according to its commonality so that, for each for all (ties are randomly allocated). , for is then . The long-run distribution is then defined as such that, for . Note that, since is necessarily random for all t, we define as the expected fraction of the population associated with the most popular surname.

The skewness in Figure 1 is what drives our methodology. To understand why, consider first the names that occur with a high frequency. Since the income process in Equation (4) is stationary, the cross-sectional distribution among these names is very similar to that of the overall population. These names therefore cannot be informative. In contrast, consider the very infrequent names. Many of them derive from recent mutations. They are newly created names, or the names of sons of fathers with newly created names, or grandsons, and so on. These names are markers that are likely to identify people with familial linkages.⁶ If inheritance is important, so that these familial linkages connect people with relatively similar incomes, then the markers must make the same connections and, thus, be informative for income.

To make this clearer still, consider the evolution of the surname distribution versus the income distribution. They are (to this point) independent of one another. The frequency of a surname cannot be informative for income, in and of itself. It is not the case that ‘rich people have uncommon surnames.’ Rather, it is the case that low-frequency markers are indicative of familial linkages while high-frequency markers are not. This is just as true for the rich as it is for the poor.⁷

We now turn to our main result, the behavior of the ICS. The ICS is a moment of the joint distribution of surnames and income. Even though the two are independent of one another, the ICS connects them and reveals information about the latter based on the markers inherent in the former.

Property 3 : ICS and the importance of inheritance

Under the conditions of Property 2 the ICS from Equation (3) is approximately constant for all . Moreover, for any , the ICS is monotonically increasing in the value of the inheritance parameter ρ.

⁶For the same reason, surnames play an important role in the field of population genetics. The connection actually goes even further. In our model, surnames are innocuous markers. They have no direct efect on income. Mitochondrial DNA, or the male Y-chromosome, is analogous. It does not code for any known protein and has no efect on the diferential survival or reproductive chances of the individual receiving it. Nevertheless, it is a useful marker that allows researchers to uncover familial linkages.

⁷In Section 2 of our online appendix we relax the independence assumption by allowing fertility rates and the mutation rate to depend on income. Our results do not change.

The importance of Property 3 for our paper lies in the monotonicity result. This is proved, analytically, in Appendix A. To establish convergence for we rely on simulation. Figure 2 plots the ICS against for our baseline parameter values. Aside from establishing convergence (and confirming monotonicity), what is quite striking is the level and the convexity. Relatively small values for the ICS are associated with moderately large values of and only for very high values of do we see ICS values above, say, 10%. Again, echoing comments made above, this is necessarily the case given that only the rare surnames can be informative. Our online appendix demonstrates that Property 3 is robust to substantial deviations from our baseline parameter values.

To summarize, the results of this section are as follows. First, without mutations the number of surnames will tend to become small, with each name conveying very little information about familial linkages and, therefore, inheritance. Second, mutations provide a countervailing force, allowing many rare surnames to have informational content. Finally, the informational content of these rare surnames — the ICS from Equation (3) — is, ceteris paribus, monotonically increasing in the magnitude of the inheritance parameter, This is what allows us to identify the magnitude of from data on the joint distribution of surnames and economic outcomes. In Sections 1 and 2 of our online appendix we demonstrate that these results (i) are robust to diferent parameter values, and (ii) are maintained in an extended model that features income-dependent fertility.

3 Data

We use data from two sources, the 2001 Spanish census and the 2004 Spanish telephone directory. From the census we have individual-level data from the Catalonian region of Spain on surnames, education and several other variables. From the telephone directory, obtained from Infobel, we have surname data. We describe the data and its uses below. First, however, it is important to understand how Spaniards name themselves and how this relates to our methodology.

3.1 Spanish Surnames

Spanish people have two surnames, ‘first’ and ‘second.’ The first is the first surname of their father and the second is the first surname of their mother. First surnames, therefore, are passed between generations in exactly the same manner as with the (traditional) Anglo-Saxon convention. Our methodology is based, primarily, on first surnames. It can be used in exactly the same way for males in Spanish societies as in many other Western societies. The second surname ofers additional information. Unlike the Anglo convention, each male is connected with his mother. Also, because females do not change their surnames upon getting married, each female is connected to her father. Note that this has no bearing on the evolution of the paternal lineage. The maternal surname vanishes after two generations.

We make use of this additional information in a number of ways. First, we use the second surname as a control for ethnicity, leaving the first as an indicator of the importance of familial linkages. Second, we use the combination of the two names to identify the strength of ‘assortative mating’ and its importance for economic inheritance (Section 8). Third, we use the combination of the two surnames to identify siblings and, thus, be more precise about familial linkages (Section 9).

3.2 Data Description

The census data for Catalonia covers the entire population of 6,343,110 individuals. For each individual we have their two surnames, some demographic characteristics (age, education, gender, marital status, place of birth, place of residence), as well as employment status, level of proficiency in the Catalan language and several housing characteristics (tenancy, size, inheritance, availability of a second house).⁸ The census does not record information on wealth or income. We therefore use years of education as our measure of economic well-being, y from Equation (4).

We eliminate individuals living in ‘collective households’ because the census has no educational information on them. We also eliminate individuals for whom the first or second surname is missing. This leaves us with 6,123,909 individuals. We eliminate females in order to be consistent with the literature on intergenerational mobility. We eliminate males who are less than 25 years of age so as to focus on individuals who have finished full-time education. We include only Spanishborn, Spanish citizens so as to mitigate the extent to which surnames are informative because they distinguish an immigrant who is likely to have relatively low education. Finally, we exclude individuals with a unique first surname because such names cannot, by definition, provide familial linkages with other individuals. This leaves us with our baseline population of 2,057,134 males.

We use data from the Spanish telephone directory (obtained from a commercial source) for the purpose of controlling for ethnicity (Section 4). There are roughly 14 million households in Spain and the directory contains surname information on roughly 11.4 million private, fixed telephone lines. Mobile phones, which are not included, obviously account for the majority of the diference. We assume that the geographic distribution of surnames in fixed lines replicates the geographic distribution of surnames in Spain.

⁸We define place of birth diferently for those born in Catalonia versus those born elsewhere in Spain. If Catalanborn, we use county dummies, otherwise we use Spanish province dummies. Catalan counties are administrative units somewhat smaller than a typical U.S. county. Spanish provinces are somewhat larger than a typical French departament.

3.3 The Surname Distribution

Figure 3(a) plots the commonality-ordered frequency distribution of the first surname (the empirical counterpart to from Property 2 in Section 2.2). The distribution is very skewed. There exist a large number of low-frequency surnames and the few most frequent surnames represent a large percentage of the population. The 10 most popular names cover roughly 11 percent of the population.

Figure 3(b) plots the Lorenz curve for the Spanish and Catalan surname distributions, using both the telephone directory and census information for the latter. Table 1 reports the relevant statistics for the three distributions. Notice that the number of people per surname is larger in the whole of Spain than in Catalonia. This is probably because Catalonia is a net receiver of immigration and because the Catalan language has, historically, had less orthographic rigidity than the Spanish language. There are also more surnames in the census than telephone directory, probably because each household typically has just one telephone line.

4 Ethnicity

To this point we have emphasized the ability of surnames to reveal familial linkages. However they may also reveal ethnicity and this complicates matters. That is, sons inherit from their fathers (i) unobservable economic traits, (ii) observable ethnicity, and (iii) a surname. Ethnicity and surnames are often connected.⁹ So are ethnicity and economic outcomes. Surnames might therefore be informative for economic outcomes simply because they are informative for ethnicity.

⁹Surnames must eventually reveal ethnicity even if there are no ethnic diferences in the initial distribution of surnames. This is because the surname birth/death process is independent across ethnic groups. Assume tha there are only two ethnic groups, red and blue (r and b) with the red group being richer on average. A surname mutation among the r group will generate a new name which, until the name dies of, will only be associated with the r ethnicity. A surname death among the b group will leave relatively more r ethnic-group individuals with this name, thus increasing informativeness about ethnicity. The independent birth/death process will lead the r and b surname distributions to drift apart over time. Eventually an individual’s surname will necessarily be informativ on whether his ancestors were of the r or b ethnicity. If ethnicity is related to other characteristics like income, then surnames will also be informative on these characteristics, even though they may not have to begin with. An example of a paper that uses surnames to elicit ethnic information is Rubinstein and Brenner (2011).

Catalonia is a good example. It is well known that native Catalan speakers (roughly half the population) (i) have fairly distinctive surnames, and (ii) have high education and economic status relative to non-native-Catalan speakers.¹⁰ This then begs the question, if surnames are informative, is this primarily due to ethnicity? Or do direct familial linkages also play a role?

We address this question by controlling for ethnicity and thereby examining mobility among ethnically-homogeneous partitions of the population. This allows us to answer the above question and ask how the answer has changed over time. It also represents a departure from much of the literature, so we must take care to interpret our measurements of the inheritance parameter, ρ from Equation (4), appropriately. The literature has typically estimated a parameter that amalgamates the efects of direct familial linkages with those of ethnicity. We prefer to separate the two so that (ideally) ρ measures only the direct familial efects. This will tend to underestimate the parameter that is typically estimated in the literature and therefore overstate the conventional measure of mobility. As such, we also report more comparable results in which we do not control for ethnicity.

Our methodology works as follows. First, by considering only Spanish-born, Spanish citizens we immediately eliminate non-Spanish ethnicities. This is because Spain (including Catalonia) saw very little foreign immigration prior to the 1990’s. This leaves us with a Spanish population for which the primary ethnic trait is regional origin: the region of Spain from which one’s family originates. We measure this with the following index of the “Catalonianess” of each particular surname s:

\[C a t a l a n D e g r e e (s) = \frac {\text {Number of telephones under surname} s \text {in Catalonia}}{\text {Number of telephones under surname} s \text {in Spain}},\]

which we interpret as an estimate of the fraction of people with surname s that reside in Catalonia. For example, since the 2001 Catalan population is about 16% of the total Spanish population, then, if surnames were uniformly distributed throughout Spain, this ratio would be roughly 0.16. The extent to which it is higher (lower) indicates a concentration of people with surname s residing inside (outside) of Catalonia.

We go one step further and interpret the CatalanDegree(s) variable as a proxy for the extent to which a person with surname s has Catalan regional origin. In Appendix B we demonstrate that this variable is indeed a good proxy of ethnic regional origin. It has strong predictive power of both knowledge of the Catalan language and place of birth within Spain. Like any proxy for the complex phenomenon of ethnicity, CatalanDegree is blunt. However it is the best proxy available to us and it is better than self-reported ethnic identity (which is not available).

¹⁰ Possible reasons are the initial conditions of migrants, the linguistic advantage that native Catalan speakers have in the educational system (see Aspachs-Bracons, Clots-Figueras, Costa-Font, and Masella (2008)), and a variety of forms of discrimination that non-Catalan speakers may be subject to.

We use the CatalanDegree(s) variable to control for ethnicity two diferent ways. First, in our ICS regressions we include on the RHS the CatalanDegree(s) value associated with an individual’s second surname (their maternal surname). A dummy variable for their first surname will be included for measurement of the ICS (Section 2, Equation (3)). Using the first and second surnames in this manner is meant to mitigate multicollinearity. Second, in Section 5, we consider sub-populations of regionally-homogeneous groups. We calculate the geometric mean of the CatalanDegree for both the first and second surname and then order the population according to the associated values. We then identify the upper 50% quantile as a homogeneous group of individuals with Catalan regional origin.

5 Cross-Sectional Results

In this section we present results that are ‘static’ in nature. This means that we pool together individuals from diferent birth cohorts. We obtain cross-cohort average measures of the ICS. In Section 6 we use calibration to map them into measures of intergenerational mobility that are comparable to those in the literature. In Section in contrast, we condition on birth cohort and obtain measures of how these things have changed over time.

Table 2 reports a benchmark set of estimates of Equations (1–2) and the associated ICS from Equation (3). Column 1 begins by including only individual controls — dummy variables for age and place of birth. The adjusted is 0.2652. Column 2 adds our CatalanDegree variable. The coeficient is positive and highly significant. It is also economically significant. The standard deviation of CatalanDegree is about 0.3. Therefore, the estimate of 1.706 translates into an additional 0.5 years of education for a one-standard-deviation increase in a surname’s ‘Catalonianess.’ The mean and standard deviation of education are 8.4 and 4.6, respectively. So Catalan regional origin is associated with higher educational attainment equal to about 10% of the overall dispersion.

Column 3 of Table 2 adds paternal surname dummies to the regression (recall that maternal surnames are used to define CatalanDegree). We note that (i) the surname dummies are jointly significant (given the large number of RHS variables involved this is not obvious in spite of the large population size), (ii) the coeficient of CatalanDegree is smaller but remains economically meaningful, with a one-standard-deviation increase translating to 4 extra months of education, and (iii) the increases to 0.2980. Surnames are thus informative. Knowledge of the particular surname of an individual is informative for predicting their educational attainment.

Column 4 replaces the actual surname dummies with ‘fake’ dummies as in Equation (2) of Section 2. The fake dummies are not jointly significant and their presence increases the very little. The estimate of the CatalanDegree coeficient is largely unafected. Our estimate of the ICS (controlling for ethnicity) from Equation (3) is, therefore, 2.45%. Columns 5 and 6 are analogous to columns 3 and 4 except that the CatalanDegree variable is omitted. Since surnames now capture both ethnicity and familial linkages, the ICS increases to 3.02%.

Tables 3(a) and 3(b) repeat the exercise of table 2, but restricting the population to those born in Catalonia (in table , immigrants are not included, even if their children are) and to those born in Catalonia before 1950 (table 3(b)). The results are very similar to our benchmark in Table 2, the only exceptions being that the are smaller and the ICS is slightly higher. Both are to be expected given that the population is more ethnically homogeneous.

One concern is that, in spite of our use of CatalanDegree, our results are dominated by ethnic and not familial linkages. To examine this, Table 3(c) restricts the population to be the 50% of those who have surnames with the highest CatalanDegree. The qualitative nature of our results is unafected, with the ICS being just slightly higher. Note that, encouragingly, the ICS is basically unafected by the inclusion of the CatalanDegree variable.

To summarize, our results show that surnames are informative for educational attainment. In part, this is because surnames are informative for Spanish regional origin. But an important part remains, even after controlling for regional origin in a variety of diferent ways. Our interpretation — that elucidated by our model — is that the surname partition is informative about familial linkages. We now present two additional pieces of evidence that support this interpretation.

First, our model predicts that, if inheritance is important, surnames will be informative because the partition of rare surnames should group together people with familial linkages. This then implies that the ICS should increase as we exclude common names. Checking this in the data provides a valuable check on our interpretation of our results. Table 4 repeats the exercise of Section 5 but includes only the 50% of the population with the least-frequent surnames. As our model predicts, the ICS increases. It does by a factor of roughly 1.5 (with and without ethnicity controls) relative to Table 2. Figure 4 provides additional evidence. It is based on a series of regressions, analogous to those in Table 2, that sequentially include people with more and more common surnames, as we move from left to right on the horizontal axis. The left vertical axis reports the ICS — the downward-sloping line — and the right axis reports the average number of individuals per surname in each sub-population. Figure 4 provides strong evidence in favor of our model’s interpretation of the ICS. Moving from least frequent names to the most frequent, the ICS monotonically falls by a factor of five. This suggests that our controls for ethnicity are working and that our findings are driven by the informativeness of surnames for familial linkages.¹¹

Second, Appendix C provides evidence that our results are not sensitive to any random (but suficiently large) partitioning of the surnames set. This suggests that our findings are structural and that they depend on deeply rooted social and economic mechanisms.

6 Calibration

We now calibrate the model of Section 2.1 using the results of Section 5. The main task is to obtain a value of the inheritance parameter, that (i) results in our model’s ICS matching that of the data (Table 2), while at the same time (ii) results in our model’s surname distribution matching key aspects of the observed surname distribution from Catalonia.

It is important to be clear about the precise interpretation of the parameter that we estimate, especially in relation to the existing literature. What distinguishes the ICS from the traditional approach is that it identifies familial linkages and then asks how much of the cross-sectional variation in the data can be attributed to these linkages. This stands in contrast to the traditional approach, which estimates the father-son correlation directly. The ICS is a diferent empirical moment than the direct father-son correlation. Nevertheless, the economic parameter that we calibrate is the same. In our model, is the father-son correlation. All other familial correlations are functions of The grandson-grandfather correlation, for example, is . For cousins, it is . The ICS is a complex amalgamation of these correlations that depends on all of the model’s parameters (mutation, fertility, etc.). In spite of this complexity, however, what we attempt to measure is simple. It is the same father-son correlation that the literature has empirically estimated using direct observations on fathers and sons.

¹¹ In Section 3 of our online appendix we address the question ‘is someone with a rare surname likely to be highly educated?’ We find that (i) surname frequencies are mildly informative, and that (ii) the important properties of the ICS are nevertheless unafected. In other words, surname frequency is informative in and of itself, which stands in contrast with Figure 4, where we see that the ICS decreases with average surname frequency.

Our baseline model has five parameters. They are the inheritance parameter, the surname mutation rate, , the number of sons conditional on giving birth (‘family size’), the probability of having sons (‘fertility rate’), and , the conditional standard deviation of educational attainment. We set the latter so that the unconditional standard deviation matches that of our data.¹² In our baseline model the fertility rate is pinned down by the zero-growth condition at . Our enhanced model, discussed below, features a more prominent role for

The remaining parameters are and . We calibrate these parameters to match the ICS and the surname distribution. Regarding the latter, it is well known that the Pareto distribution provides a good approximation of surname distributions in many societies ((Fox and Lasker (1983))). The Pareto distribution (for surnames) is fully characterized by two moments, the Gini coeficient and the number of persons per surname (PPS). Our calibration exercise, therefore, consists of choosing three parameters, , to match three moments, (ICS, Gini, PPS).

The structure of our model allows us to proceed recursively. The following properties are established, numerically, in Section 4 of our online appendix. First, the Gini coeficient is humpshaped in the mutation rate, , with the value of that maximizes the Gini coeficient being (essentially) independent of both and . This maximum falls short of the observed Gini for all parameter values, so we begin by setting to attain the maximal Gini coeficient. Second, given a value for the PPS does not depend on so we choose m to match the observed PPS. Finally, we choose to match the ICS. The results are shown in Column 2 of Tables 5 and 6. The latter is based on the ethnicity-controlled ICS. For the former, no such controls are employed.

The most important findings in Tables 5 and 6 are our calibrated values for . We find that the observed ICS maps into an inheritance parameter of for ethnicity-adjusted ICS and for ICS without ethnicity controls. These values are near the top of the range of estimates from the literature, which tend to be in the neighborhood of 0.4 to 0.6.¹³

Our calibrated value for , , is obviously unrealistic. However this is primarily a reflection of the simplicity and restrictiveness of our model’s demographics structure.¹⁴ If, for instance, m were to be minimal (subject to the zero-growth condition) at , this would still imply that the average family with children would have two sons, implying a far larger family size than that prevailing in Catalonia in 2001. The simplifying assumptions that we have made imply that our model cannot be thought of as a good model of the joint distribution of family composition and surnames/education. Ours is a model of the joint distribution of surnames and education. What is reassuring is that, as we show below, our model’s implications for are largely independent of this issue and the particular value for m that we use.

¹²For given a value of we set , where the standard deviation of education in our data is 4.7. ¹³For Spain, Kalkbrenner and Villanueva (2007) estimate educational mobility from the “Encuesta de Conciencia Clase de Biografia” for 1990-91 and obtain estimates between 0.42 and 0.52 among fathers and sons. Also for Spain, Pascual (2009) examines income mobility using the European Community Household Panel and reports an elasticity of 0.3 among fathers and sons. More broadly speaking, a value between 0.4 and 0.6 is typical for U.S. data for the latter third of the 20th century. Solon (1992), in the context of the various biases that plague panel-data estimation, argues for a value of 0.4 or higher.

Our calibrated value of lends credibility to our exercise in the following sense. We can obtain an empirical estimate from our data. In 2001, in all of Spain, there were 1,570 applications to change one’s surname. Of these, 1,426 were granted. Assuming that 2001 is a representative year, and a population of forty million with a live expectancy of 70 years, this translates to a mutation rate of around 0.0025. Our estimate of 0.0067 is 2.68 times larger. The empirical estimate, however, is surely lower than the mutation rate that would be a good analog for our model. Immigration, for example, is entirely absent from the empirical estimate but is implicit in our model and is explicitly an important factor for our ICS estimates. Whether or not the incorporation of such efects would reflect well on our estimate of 0.0067 is dificult to say. We are not aware of data with which we could examine this issue. We simply emphasize that our estimate arises in an unconstrained fashion in order to most closely match a key moment in our entire exercise, skewness in the surname distribution. The value that we arrive at is certainly in the set of plausible values.

Next we conduct a sensitivity analysis. Each cell in Table 7 shows the result of holding a given pair fixed and choosing ρ to match the ICS. We see a lot of variation across rows (values of µ), but little variation across columns (value of m).¹⁵ Our interpretation is that our model’s implications for are robust. First, in the neighborhood of the calibrated pair, (0.0067, 5), the implications of variation in m are small. This is reassuring. As is discussed above, the simplicity of our model’s fertility structure implies fairly unrealistic implications for the crosssectional distribution of children in each household. But our model’s implications for the main parameter of interest are not sensitive to this parameter. Second, the variation in across rows in Table 7 does not reveal that our model is not robust. Rather, it shows that our model is well identified in the joint directions . We are confident that the skewness in the surname distribution (as is manifest in the Gini coeficient) maps uniquely to a particular value of and this value maps the observed ICS uniquely into a particular value of

¹⁴In our model, families have either zero sons or m sons. That is, there is no variation in family size. In order to keep the population constant and produce the necessary skewness to replicate the surname distribution, the value of m is high. It is likely that in a model with diferent family sizes, enough skewness would be achieved with a more reasonable average value of m. However, the simple structure that we use has no bearing on the results as the calibrated value of is not sensitive to variation in

¹⁵The small discontinuity that is apparent in moving between column and is an artifact of the grid of value at which we evaluate the pairs. The function is more convex in the neighborhood of than for higher values of . Nevertheless, the implications for are not large. In addition the small diferences in between Tables 5 and 7 are also artifacts of this grid and the linear interpolation scheme that we use to match exactly the empirical moments.

6.1 A Model with Diferential Fertility

Our baseline model does not allow for any sort of dependency between the surname distribution and the education distribution. Such dependency would exist if any of the mutation rate, the fertility rate , or the family size, m, were to be dependent on economic status. In Section 2 of our online appendix we examine each type of dependency in turn and show that the basic properties that make the ICS a desirable measure of mobility are unafected. Here, we examine the extent to which our calibrated value for is afected. We focus on the fertility rate, , because we have an additional empirical moment with which it is identified. This moment is surname frequency. When we regress surname frequency on education we get a point estimate of −31.26 (details provided in Section 3 of our online appendix). The negative sign indicates that lower-frequency surnames are associated with higher educational attainment. This efect, while small — it translates into just 0.19 years of additional education for a one-standard deviation reduction in frequency — identifies diferential fertility in our model.

We modify our model as follows. Fertility rates are denoted , with , h denoting the lower fifth, the middle three fifths and the upper fifth of the education distribution. Family size, , does not vary across educational groups. Zero growth implies that We set , and , so that d is the incremental rate by which the highly educated are less likely to have sons than the average-educated. We call this our diferential fertility rate model. All other aspects are identical to our baseline model.

Our calibration procedure must now account for the fact that the surname distribution and the education distribution interact. We proceed as follows. There are now 4 parameters of interest, , d and As above, we find that governs the maximal Gini coeficient and that the maximum is essentially independent of the other parameters. Likewise, PPS depends on m in a manner that does not depend on the remaining two parameters. We therefore choose and m first. Next, we choose values for and simultaneously to match the ICS and the frequency coeficient from the dummy variable regression of −31.256. The calibrated values are reported in Tables 5 and 6. We see only minor changes in relative to the baseline. The calibrated value of d is −0.15, indicating that the highly educated are 3% less likely to have ofspring than the average educated.

6.2 Calibration: Summary

Our methodology results in an estimate of the inheritance parameter of about . This finding is robust to an enhancement of our baseline model that incorporates diferential fertility and that matches some evidence on the informativeness of the relative frequency of surnames. The main “overidentifying test” of our estimates is the extent to which they are associated with a plausible surname mutation rate. We argue that this is indeed the case.

The foundation of our exercise is the ICS — the incremental from the various dummyvariable regressions that we estimate. These incremental are small in variance decomposition units — on the order of 2 or 3 percent. In the more economically-interesting “inheritance units” of however, they are large — on the order of 0.60. Small values for the ICS are the nature of our exercise. We extract information from a highly-skewed surname distribution in which the most prevalent names (e.g., “Rodríguez”) cannot be informative whatsoever. Small values must necessarily arise. But they are not indicative of small economic magnitudes.

7 Dynamic, Cohort-Based Results

Our analysis to this point has treated the entire 2001 Catalan census as a single cross-section. The cross-section consists of individuals of diferent ages. We have dealt with education-related age efects using dummy variables. But we have not allowed the ICS to vary with age. The above estimates are age-averaged measures of the ICS. They are ‘static’ in the sense that they are incapable of revealing any information about how the ICS and intergenerational mobility may have changed over time. As mentioned before, little is known about the time evolution of mobility.¹⁶

¹⁶The short time dimensions of the panel datasets make it very dificult to assess the dynamics in mobility. Moreover, Lee and Solon (2009) and Hertz (2007) attribute a fairly divergent body of existing results to smallsample bias in addition to the aforementioned age bias and sample attrition problems , Mayer and Lopoo (2005) and Fertig (2004)). Taking this into account leaves the authors inconclusive about trends in intergenerational

We turn now to a dynamic analysis. We partition the cross-section into birth cohorts and ask if the ICS varies across subpopulations of diferent ages. If it does, our model suggests that intergenerational mobility may have changed over time. The reasoning is as follows. Suppose that mobility has decreased, so that the value of connecting the current young generation to their parents has increased. Our model’s prediction is that, if the surname distribution of the parents is highly skewed (which it is), then the surnames of the young should be more informative than those of the old. The change in mobility should generate a change in the ICS. Why? For the same reasons as above. A higher value of implies that that familial linkages will be more informative for economic status. A skewed surname distribution of parents means that the surname distribution of children will be informative for familial linkages.

There are, of course, alternative interpretations for why the ICS may have changed over time. One possibility is a change in the birth/death process for surnames. We discount this possibility for one simple reason. Such a change must necessarily afect the ICS via its efect on the surname distribution. This efect will occur very slowly. Its immediate efect on the ICS, therefore, must be very small. A change in in contrast, will have an immediate impact on the ICS because the channel through which it works is not the slow-moving surname distribution. Our assortative mating exercise (in Section 8.2) and our empirical robustness checks (in Section 9) serve to bolster this point and provide reassurance that we are capturing changes in

Figure 5 reports our results. We run the same set of regressions as those in Table 2 for a rolling sequence of overlapping 25-year age cohorts. Figure 5 reports the evolution, from oldest cohorts to youngest cohorts, of the ICS and the point estimate of the parameter on our CatalanDegree variable. What we see is striking. Figure 5(a) shows that the ICS is substantially higher for younger cohorts. Figure 5(b) shows that regional origin has become more important for determining educational outcomes. Figure 6 reports the evolution of the ICS when not controlling for ethnicity. As expected, the ICS is higher when it amalgamates familial and ethnic efects (Section 4).¹⁷ Finally, Figure conducts the same sort of robustness checks that were undertaken for the single cross-section in Section 5. The temporal behavior of the ICS is qualitatively the same and quantitatively larger when we restrict the population to those with high CatalanDegree surnames (a much more homogeneous group in the ethnic dimension) and those low surname frequencies.

mobility. One exception is Blanden, Goodman, Gregg, and Machin (2004), who argue for a decrease in mobility in the U.K. between two cohorts of people born in 1958 and 1970, respectively.

¹⁷Section 6.1 of our online appendix provides regression details and additional results for a more coarse set of cohorts, those born before/after 1950.

We interpret these results as indicating that intergenerational mobility has decreased in Catalonia, both because of an increased importance of ethnicity and an increased importance of familial linkages for determining educational outcomes.¹⁸ In Section 8 we ofer and substantiate a potential explanation: increased assortative mating. Before doing so we ofer some interpretative discussion of these findings as they relate to important historical trends in 20th-century Catalonia.

7.1 Public Education

Decreased mobility in Catalonia is particularly striking when viewed against the backdrop of the large, secular changes that occurred in publicly-provided education in Catalonia and Spain during the mid 20th century. Others have also found a similar result , Checchi, Ichino, and Rustichini (1999), Grawe (2010) and Parman (2011)). In Spain, prior to the 1950’s access to education was very limited. This was a consequence of both the general level of wealth and income as well as the lack of investment in public education. Starting in the late 1950’s, both the economy and the level of public education began to grow, particularly from 1975 onward. These changes show up clearly in our data. Figure 8 plots the mean and standard deviation of the years of education for the same overlapping sequence of 25-year age-cohorts that we used above. It demonstrates that the average years of education of the oldest individuals in 2001 is less than half that of the youngest. Variability around the average, in contrast, is more stable.

How is ‘more education for all’ associated with intergenerational mobility in educational attainment? Does an increase in publicly-provided education decrease the importance of how educated one’s parents are? While the intuitive answer might be yes, our results show that intuition does not fit the facts. Such intuition confuses aggregate growth with cross-sectional mobility. Mobility is a relative concept. Aggregate growth is not.

7.2 Increase in the Importance of Ethnic Background

Our point estimates of the coeficients on the CatalanDegree variable have increased, indicating an increase in the importance of ethnicity (Figure 5(b)). Note that this cannot be a direct consequence of the migration process. In our regressions we include not only the children of immigrants, but also the immigrants themselves. Thus, our result indicates that CatalanDegree is more important for measuring the education of the second than of the first generation of immigrants. Our point estimates have not increased because there are more immigrants. Note also that this does not mean that low CatalanDegree individuals have obtained less education, but that their diference vis-`a-vis high CatalanDegree individuals has increased. The increase in educational attainment has been large for both ethnic groups, but has afected Catalan speakers more than non-Catalan speakers.

¹⁸Note that our main finding is still valid if we disregard the older cohorts.

In Catalonia there are two main linguistic communities, Catalan and Castillian (Spanish), each representing roughly half of the population. Catalan speakers have enjoyed substantially larger incomes and larger levels of educational attainment during the entire period of our study (this is true for both those born before 1950 and those born after 1950). Nevertheless before the late 1970’s there did not exist any formal linguistic advantage toward Catalan speakers. The language of government, commerce and education was overwhelmingly Spanish. However, beginning in the late 1970’s the increasing political power of Catalan nationalism has translated into a series of drastic legal and administrative reforms that have reversed the relative importance of both languages in society while changing its overall language composition only marginally.¹⁹ For example, since the beginning of the 1980’s all education is provided exclusively in Catalan in all public and practically all private schools. Catalan is now the sole language of the regional and municipal governments, and proficiency in Catalan has been the key requirement for working in public administration since the beginning of the 1980’s. Further legal change has made Catalan an important (albeit perhaps not the main) business language.

Governmental and institutional changes in the use of Catalan are, at best, a partial explanation for what we find. A deep understanding of the increase in the value of ethnicity is beyond the scope of this paper. Note, however, that in the next section we extend our analysis and show that assortative mating seems to have increased in Catalonia along ethnic lines. This makes ethnicity more inheritable and serves to magnify the efects of any institutional changes on the educational outcomes of the ofspring of those who assortatively mate.

¹⁹See Miley (2004) for a study of the politics of nationalism and language in Catalonia. The increasing power of Catalan nationalism might be explained (i) by the larger levels of income and education of the Catalan speaking community and (ii) because Spanish electoral law has allowed Catalan nationalism to operate as a third party in Spanish politics, allowing it to obtain high leverage from its successive alliances with either left or right leaning governments. See also Aspachs-Bracons, Clots-Figueras, Costa-Font, and Masella (2008) for a study of the efects of linguistic legislation on the educational system on identity.

8 Assortative Mating

In this section we develop and test one possible explanation for our finding that mobility has decreased over time: an increase in assortative mating.

Assortative mating refers to the tendency of people with similar characteristics to marry each other.²⁰ At first, it may seem intuitive that assortative mating can give rise to the ICS because, ostensibly, it can generate “organization” in the distribution of surnames. If, for instance, today’s rich and poor have distinct surnames, and if the rich marry the rich and the poor marry the poor, then one might think that the rich and poor surnames will remain distinct among future generations, thus generating informativeness. One might apply a similar argument to ethnicallymotivated assortative mating. In either case, this intuition is deeply misleading because the degree of assortative mating does not have any direct efect on the marginal distribution of surnames in the population. The reason is simple. Surnames are passed along the male lineage. For surname determination, it does not matter why one’s father married one’s mother; all that matters is one’s father’s name. It is as if females had no surnames.²¹

Assortative mating does matter for the joint distribution of surnames and characteristics because, if inheritance occurs along both the maternal and the paternal lineage, more assortative mating amounts to the increased prevalence of inheritance. In the language of our model, more assortative mating increases the father-son inheritance parameter, ρ, and, via this mechanism, it increases the ICS. We now enrich our model to make this mechanism precise. The model is richer in that it articulates the mapping between assortative mating and the inheritance parameter that we have estimated above. The model does not, however, have any bearing on the estimates themselves because, in our enriched model, if we take ρ as given, the process that generates the joint distribution of surnames and income is identical to that of Section 2.1.

8.1 A Model of How Assortative Mating Afects Inheritance

We now treat Equation (4), which correlates the income of fathers and sons, as endogenous. What is exogenous is (i) the manner in which the income of sons and daughters depends on the income of both their father and their mother, and (ii) the manner in which boys and girls sort at the time of mating. There is a continuum of males and females who form households and bear ofspring. Expanding on the notation, and denote the incomes of male and female children who inhabit household i with paternal surname s at date t. This household was formed at date The children’s incomes arise as

²⁰The existing literature on mobility that incorporates assortative mating includes Lam and Schoeni (1993), Chadwick and Solon (2002), Ermisch, Francesconi, and Siedler (2006) and Holmlund (2006). There is also a rich literature in macroeconomics that focuses on assortative mating and inequality (e.g., Fernández and Rogerson (2001), Fernández, Guner, and Knowles (2005) or de la Croix and Doepke (2003)).

²¹We are thankful to Melvin Coles for this insight.

\[y _ {i s t} ^ {m} = r z _ {i p, t - 1} + e _ {i s t} ^ {m}; y _ {i s t} ^ {f} = r z _ {i p, t - 1} + e _ {i s t} ^ {f},\tag{5}\]

where z is family income, which we assume to be the average income of the father and mother’s income: . The e innovations are i.i.d. and is a household inheritance parameter. The distribution of z is endogenous. We prove its existence below. The parameters r and are exogenous and determine the process of income transmission given the income of the two parents.

Mating determines the distribution of family incomes for the subsequent generation, given the distribution of incomes of sons and daughters of the current one. The incomes of the current generation’s male children become that of the next generation’s fathers: . Each father forms a household with a female who becomes a mother. The income of the mother is described by a mating technology, a function that combines each father’s income, , with a mating shock, , to assign to each father a spousal income, such that the distribution implied by coincides with that implied by the inheritance process (5) for the population of female children at date t. The function that we use is

\[y _ {i p t} ^ {f} = \lambda y _ {i p t} ^ {m} + u _ {i p t}; u _ {i p t} \sim N (0, V _ {u}),\tag{6}\]

where — the correlation between spousal incomes — is the degree of assortative mating. Note that Equation (6) is silent on the particular assignment mechanism that ‘mates’ the distributions of and from Equation (5). For our purposes, it is suficient to simply form a set of ordered pairs, , that satisfy two properties: (i) they capture the notion of assortative mating that we are interested in, and (ii) they are consistent with the distributions implied by the inheritance processes (5). Examples of more fully-articulated assignment mechanisms are in Becker (1973), Gavilán (2012), Kremer and Maskin (1995), Marimon and Zilibotti (1999) and Shimer and Smith (2000).

In Appendix A we show that Equations (5) and (6) imply a stationary distribution for family income, z. Inspection of the inheritance processes, Equation (5), then implies that, since 2 the stationary distributions of male and female income must be the same,

\[y _ {i s t} ^ {m} \sim N (0, V _ {y}); y _ {i s t} ^ {f} \sim N (0, V _ {y}),\tag{7}\]

where is a unique function of the model’s structural parameters, and λ. This function is characterized in Appendix A. We assume that the variance of the mating shock is This guarantees that the distribution of brides has the same cross-sectional distribution as that of female income.

Note that the inheritance parameter from Equation (5) relates male children’s income to the income of their parents’ household. The model of Section 2.1 and most of our empirical work, in contrast, refer to the correlation between children and their father because surnames are passed along only the male lineage. Therefore, in order to understand how assortative mating afects the ICS we must describe how the parameters r, and λ are manifest in both the variance of the income and the parameter from the following expression:

\[y _ {i s t} ^ {m} = \rho y _ {i p, t - 1} ^ {m} + w _ {i s t} ^ {m},\tag{8}\]

where the variance of is denoted . This equation links the income of sons to their fathers, a relationship that depends on both the mating process, (6), and the household-level inheritance process (5).

Note also that, for issues of intergenerational mobility, the appropriate measure of inheritance is and not This is because associates comparable variables — the incomes of children with their father — whereas from Equation (5) does not. The latter associates the income of one individual with the consolidated income of their childhood household, something that arises from the noisy lottery of mating.²²

In Appendix A we prove the following property:

Property 4 There exists a unique stationary distribution for and that is characterized by

\[\rho = \frac {r (1 + \lambda)}{2}; \quad V _ {w} = V _ {e} \left(1 + \frac {r ^ {2} (1 - \lambda)}{4 \lambda}\right); \quad V _ {y} = \frac {V _ {e}}{\lambda (1 + \lambda)}\]

²²One could, alternatively, use an analogous parameter that associates the consolidated income of each household with the consolidated income of that household’s children’s households. Indeed, a number of existing studies on mobility and assortative mating do just this. We choose to focus on from Equation (8) because (i) it is perfectly coherent, (ii) it is the parameter that is estimated in the most of the existing literature on mobili and (iii) it is tightly linked to the process of surname difusion.

A larger degree of assortative mating — as measured by a larger value for λ, the correlation of spousal income — thus translates into a larger value of Stronger assortative mating implies less intergenerational mobility in the population of fathers and sons. This is true even if the correlation between the income of sons and the joint income of their parents, r, is held constant. The intuition is straightforward. More assortative mating implies that the father’s income is more informative for the income of the mother. Both father and mother contribute to the characteristics of their son. Thus, the more the income of the father explains the income of the mother, the more it must explain the income of his son. Stronger assortative mating translates into lower intergenerational mobility.

The same intuition applies to any other inheritable trait, like ethnicity. Females play an important role in determining the ethnicity of a household’s children. Keeping in mind that surnames capture ethnicity only insofar as it is transmitted across the male lineage, it becomes clear that assortative mating is pivotal. It is the only way with which a mother’s ethnicity can be correlated with her children’s surname. Consider, for example, Judaism in which (ignoring conversion) ethnicity is solely passed along the maternal line. Absent assortative mating — e.g., if Jewish women marry men randomly drawn from the entire population of males — surnames must eventually become uncorrelated with the Jewishness of their holders. On the other hand, if Jewish women marry only Jewish men, then the surnames of Jews will become increasingly distinct from those of gentiles, owing both to the initial distribution of Jewish male surnames and to the surname birth/death process described previously. This mechanism applies to virtually any other ethnicityrelated characteristic. Since females are in almost all cases fundamental for the inheritance of ethnic characteristics, assortative mating and the degree of ethnic information contained in surnames go hand-in-hand.

To summarize, surnames are passed exclusively along the male line. They do not provide any direct information about the mother. Any information that is indirectly associated with the mother must arise because the characteristics of the father are correlated with those of the mother. This is the mechanism through which assortative mating can afect the ICS. In the language of our model, the ICS depends only on the correlation and conditional variance of the incomes of fathers and sons: the parameters and , respectively. But assortative mating afects and, therefore, it afects the ICS. This lends valuable interpretation to our empirical findings in the next subsection.

8.2 Assortative Mating: Evidence

We have seen that surnames contain information on two characteristics: ethnicity and educational attainment. This, combined with the Spanish naming convention, allows us to obtain measurements of the level and change in ethnic/educational assortative mating in Catalonia. As we have seen, an increase in the degree of assortative mating translates into an increase in the prevalence of inheritance, and of the ICS.

Our identification strategy is best illustrated with an example. The surname Casals is associated with a high value of our CatalanDegree variable. The same applies to the surname Pujol. A person whose complete–surname is “Casals Pujol” — a person whose father is Casals and mother is Pujol — is therefore almost certainly a person with two parents of Catalan regional origin. Ethnic assortative mating, then, can be measured by the incidence of such complete surnames relative to those that are more ethnically-heterogeneous. The measurement is a simple correlation between the ethnicity index of each person’s first and second surname.

In our data, we do not know who is married with whom. We do not have explicit links between husbands and wives. Therefore, we cannot directly compute the correlation between them. But, the Spanish naming convention enables us to estimate the analogous correlation for each person’s parents. Note that this measurement applies to each individual’s parents, not to each individual’s spouse. That is, if we find evidence of an increased correlation of the characteristics of the first and second surnames from the 25-50 year-old cohort in the 2001 Catalan census with respect to the 45-70 year-old cohort, this means that the parents of the individuals in the 25-50 year-old cohort exhibited more assortative mating than the parents of those in the 45-70 year-old cohort.

The data are constructed by first associating to each surname two characteristics: the average level of education and the average value of CatalanDegree, where the average is taken across all individuals with that particular surname. We then run two sets of regressions, one for each characteristic. The LHS variable is each individual’s first surname’s characteristic and the RHS variables are the set of controls used above along with their second surname’s characteristic.

Figure 9 displays the results. We plot the value of the coeficient of the second surname’s characteristic (education or the CatalanDegree) for the moving window of cohorts described above.²³ Assortative mating in both characteristics is clearly increasing, and the timing of education resem-

²³Section 6.3 of our online appendix reports the same exercise, splitting the population into those born before/after 1950.

bles the timing of the increase in the ICS.

In order to make sure that our results are not driven by ethnicity we run the same regressions on ethnically homogeneous populations and on populations with very infrequent surnames (Figure 10). The results are qualitatively identical.

To summarize, we have found evidence that intergenerational mobility in educational attainment has decreased in Catalonia in the 20th century. One possible explanation is that assortative mating has increased. Surname data are consistent with this explanation, suggesting an increase in the likelihood that people mate with others of similar educational levels and ethnic backgrounds.²⁴

9 Robustness Check: Analysis of Siblings

An established alternative to father-son correlations is sibling correlations (Solon, Corcoran, Gordon, and Laren (1991)). The reasoning is simple. If economic inheritance is important then the outcomes of siblings should be correlated because they share parents and, thus, they share the same inherited economic traits. In this section we make use of this logic to provide an important robustness check on our results.

We make use of Spaniards’ maternal surnames to approximate sibling relationships. We argue that this approximation is very accurate. We use it to construct an alternative measure of mobility, one which is conceptually distinct from the ICS. We then compare the time-trend in the siblingsbased measure to that of the ICS (from Section 7). This comparison serves as our robustness check.

We identify siblings in the following manner. Recall that all Spaniards have two surnames, one from their father and one from their mother. Thus, all siblings (irrespective of gender and marital status) share not one but two surnames, as well as their ordering. This allows us to construct a partition of the population that groups together individuals who have a high likelihood of being siblings.

²⁴Some existing work on increased assortative mating attributes it to an increased level of education among females. As above, one needs to be careful not to confuse this story with the efect of an increase in average educational attainment. Suppose, for instance, that the primary driver of assortative mating is wealth. Suppose also that there has been no change in the tendency for people to assortatively mate. If the daughters of the rich experience an increase in education that is larger than the daughters of the poor — something that is very plausible then one might mistakenly conclude that assortative mating has increased, in the educational dimension, although in reality it has not. Our methodology does not sufer from this possible bias because our measures do not refer to the individual woman, but to her family. Education and ethnicity are imputed by the surname, not measured at an individual level.

More specifically, we define the “complete–surname” for an individual to be their two surnames, in order. That is, if a person’s father and mother are named Fernández and Caballé, respectively, then their complete–surname is “Fernández Caballé”. This is distinct from both “Caballé Fernández” and “Fernández Vila”. Next, we group each person together with those who share their complete–surname. This partition will be very similar to the actual sibling partition (which we cannot observe), with the similarity increasing in the rarity of the surnames. It is very unlikely that two males who share the same rare surname will marry two females who share the same rare surname, thus generating children who are not siblings with the same complete–surname. Much more likely is that two individuals with the same rare complete–surname are in fact siblings. We therefore focus on matches of rare, complete–surames.

We proceed as follows. We partition the population according to complete–surname. In order to increase the probability of matching siblings, we then extract the subpopulation of those individuals who share their complete–surname with either one or two other individuals.²⁵ As above, we run two regressions, one with legitimate complete–surname dummy variables and one with fake dummy variables. We define the Sibling Correlation Proxy (SCP) as the diference between the R²’s from these regressions.

Figures 11 and 12 report results that serve as a powerful robustness check on our method and its findings. Figure 11 shows the evolution of the SCP over time for those who share their complete– surname with only one other person, and for those who share it with at most two other people. Figure 11 cannot control for ethnicity because our CatalanDegree variable and the complete–surname dummy are based on a common surname. Figure 12 shows the same result for the subpopulation with the 50% most Catalan surnames, resulting in a more ethnically homogeneous population. In both cases we see a marked increase over time in the SCP, the same pattern that we observed in the ICS of Section 7. The two measures are conceptually and mechanically quite diferent from one another. Yet our model and the existing literature suggest that they are both legitimate measures of changing mobility. The fact that they tell the same story provides important reassurance to our interpretation of the ICS.²⁶

²⁵We do this in order to approximate the sibling relationship even better. We purge from the data names like “García Pérez,” which being the concatenation of two very common surnames, are more likely to be shared by non-siblings.

²⁶In Section 5 of our online appendix we explain that the SCP is a much higher number than the ICS because our proxy of the sibling partition is a very diferent (and much finer) partition of the data that approximates the sibling relationship very closely. The ICS, on the other hand, is a more coarse partition that is informative on the family relationship broadly understood. Since the SCP and the ICS are based on very diferent partitions of the

10 Conclusions

Our paper makes two contributions, one methodological and one applied. Methodologically, we develop a framework that shows how an untapped data source can shed light on a question that requires much data, but for which relatively little data exist. We show that a single cross-sectional census can reveal much about both the level and the change in intergenerational mobility. The key data objects are surnames, markers that provide intergenerational links where more explicit links are unavailable. Surnames define a partition of a population. Elements of this partition associated with rare surnames will be correlated with the partition that groups people according to familial linkages. A particular moment of these partitions — which we label the Informational Content of Surnames (ICS) — connects the familial linkages with familial economic status and thus provides information on intergenerational mobility. This method yields measures of the degree of mobility at a point in time as well as its evolution across time.²⁷

Our method would be of limited practical value in the presence of multi-country, intergenerational panel data. However, the existence of such data is quite limited. The practical relevance of our method, therefore, depends on how much data we do have on the joint distribution of surnames and economic outcomes. Here, there is reason to be optimistic. Most countries compile censuses containing such data. We have shown that one can learn much from one census. Multiple censuses, both within and across countries, can obviously yield much more. Comparability, over time and across countries, can be handled. The US, for example, has a diferent surname distribution than Spain. The essence of our method — the idea that rare surnames connect people with familial linkages — is nevertheless unafected.

Our practical contribution is to use our methodology to ask how and why intergenerational mobility has changed over time. We study Catalonia, a large region of Spain. Using the 2001 is due to the increased explanatory power of ethnicity. But the story is more complex. There is a component of the ICS that is unrelated to ethnicity and the impact of this component has also increased. This is true among very ethnically-homogeneous individuals, among siblings, and among people with infrequent surnames. Our model, alongside an extensive set of controls and robustness checks, associates this increase in the ICS with a decrease in intergenerational mobility. Our model and data also ofer one possible explanation. Maternal and paternal surnames allow us to document that assortative mating in both the educational and ethnic dimensions was larger among the parents of the younger cohorts of our data than it was among the parents of the older cohorts. An increase in assortative mating among parents should translate into a decrease of intergenerational mobility one generation later. Our ICS measure shows this. An increase of assortative mating appears to have been an important driver in the secular fall of intergenerational mobility that we document.

population, their values are not directly comparable. We also present the results using the whole population and show that they are qualitatively identical. Section 6.2 of the online appendix reports the same exercise, splitting the population into those born before/after 1950.

²⁷Several studies estimate mobility for a given country (Lillard and Kilburn (1995), Dearden, Machin, and Reed (1997), Wiegand (1997), Osterberg (2000), Osterbacka (2001), Hertz (2001), Dunn (2007), Ferreira and Veloso (2006), Leigh (2007), Ng (2007)). And a growing body of work has attempts to compare mobility across countries: several studies compare the USA to other countries (Bj¨orklund and J¨antti (1997), Couch and Dunn (1997), Checchi, Ichino, and Rustichini (1999), Bj¨orklund et al. (2002), Grawe (2004)); and Comi (2003) uses the European Community Household Panel to obtain estimates for 12 EU countries. Nevertheless, it is important to keep in mind the problems that plague cross-country analysis (Solon (2002)). Our method may represent promise in this context. Our model can be calibrated to specifically incorporate some of the cross-country heterogeneity that forms some of the basis of Solon’s critique. We leave this to future research.

Figure 1: Time series of number of agents per surname and Gini coeficient, and, for diferent values of the surname distribution and associated Lorenz curve. Notes: Model Simulations with baseline parameter values: ; V_ε=1.000; µ=0.0200; q=0.50; m=2; . Diferent initial conditions: number of surnames= 10, 1000, 100000 and 1000000. (a) Time series of average number of agents per surname

(b) Times series of Gini of surname distribution

(c) Ordered frequency distribution,

(d) Lorenz curve of surname distribution

(a) Average (solid line) and (dotted line) against 2: Surnames are informative, and their informational content increases with the degree of inheritance_{Figure 2:} Surnames are informative, and their informational content increases with the degree of inheritance that there is in society.

Notes: Model Simulations with baseline parameter values: as in Figure 1.

(a) Distribution of First Surname

(b) Lorenz Curve of the Surname Distribution in Catalonia and Spain Figure 3: Distribution of the first surname in Catalonia and Lorenz curves for Spain and Catalonia

Figure 4: ICS is larger for less frequent surnames Notes: ICS (solid line) and individuals per surname (dotted line). Regressions as in Table 2 (Columns 3 and 4) by percentiles, where percentile x corresponds to the x% least-frequent surnames. Source: 2001 Catalan Census (Idescat).

For 3(a): Population: Men with first & second surnames not missing living in Catalonia. Source: 2001 Catalan Census (Idescat). For 3(b): Population: All phones with first and second surnames not missing. Population percentage per surname (1% Steps). Source: 2004 Spanish telephone directory and 2001 Catalan census (Idescat).

(a) Evolution of ICS

(b) Evolution of parameter of CatalanDegree Figure 5: Evolution of ICS and parameter of CatalanDegree over moving windows of cohorts

Notes: Regressions as in Table 2 (Columns 3 and 4). The overlapping sequence of cohorts starts with those aged 75-100 years old in 2001, then continues with those aged 70-95 years old, and so on, ending with the 25-50 year-old cohort. Source: 2001 Catalan Census (Idescat).

Figure 6: Evolution of ICS over moving windows of cohorts. No ethnic controls.

Notes: Regressions as in Table 2 (Columns 5 and 6). Overlapping age-cohorts are described in caption to Figure 5. Source: 2001 Catalan Census (Idescat).

Figure 7: Evolution of ICS over moving windows of cohorts, subpopulations.

Notes: Regressions as in Table 2 (Columns 5 and 6). Overlapping age-cohorts are described in caption to Figure 5. Source: 2001 Catalan Census (Idescat).

Figure 8: Evolution of years of education over moving windows of cohorts.

Notes: Overlapping age-cohorts are described in caption to Figure 5. Source: 2001 Catalan Census (Idescat). Figure 9: Evolution of assortative mating over moving windows of cohorts.

Notes: Correlation between the individual’s first and second surname’s average characteristic (education and CatalanDegree). Regressions include age and place of birth dummies. Overlapping age-cohorts are described in caption to Figure 5. Source: 2001 Catalan Census (Idescat)

Figure 10: Evolution of Assortative Mating in Education over moving windows of cohorts, subpopulations.

Notes: Regressions as in Figure 9. Overlapping age-cohorts are described in caption to Figure 5. Source: 2001 Catalan Census (Idescat).

Figure 11: Evolution of Sibling Correlation Proxy, SCP over moving windows of cohorts.

Notes: Regressions include age and place of birth dummies. Overlapping age-cohorts are described in caption to Figure 5. Source: 2001 Catalan Census (Idescat).

Figure 12: Evolution of Sibling Correlation Proxy, SCP over moving windows of cohorts. 50% Most Catalan Surnames

Notes: Regressions include age and place of birth dummies. Overlapping age-cohorts are described in caption to Figure 5. Source: 2001 Catalan Census (Idescat).

Table 1: Surnames Distribution: Gini Index and People per Surname in Catalonia and Spain

Populations: Columns (1-3): All individuals/phones with first & second surnames not missing. Column 4: Men with first & second surnames not missing. Source: 2004 Spanish telephone directory (Infobel) and 2001 Catalan Census (Idescat).

Table 2: ICS. Baseline population.

Notes: All regressions include age and place of birth dummies. Fake-surnames have the same distribution as Surnames and are allocated randomly. (*) F-test if Surname dummies are jointly significant. Standard errors in parenthesis. Population: Male Spanish citizens living in Catalonia aged 25 and above, with frequency of first surname larger than one. Number of observations: 2,057,134. Number of surnames: 30,610. Source: 2001 Catalan Census (Idescat)

(a) Born in Catalonia.

(b) Born in Catalonia before 1950.

(c) 50% Most Catalan Surnames.

Notes: Regressions as in table 2. For 3(a): Number of observations: 1,328,003. Number of surnames: 28,523. For 3(b): Number of observations: 465,896. Number of surnames: 20,793. For 3(c): Number of observations: 1,028,567. Number of surnames: 23,892. Source: 2001 Catalan Census (Idescat).

Table 4: ICS. 50% Least Frequent Surnames.

Notes: Regressions as in table 2. Number of observations: 1,028,727. Number of surnames: 30,275. Source: 2001 Catalan Census (Idescat).

Table 5: Calibrated Parameter Values. No Ethnicity Controls

Table 6: Calibrated Parameter Values. With Ethnicity Controls

Table 7: Calibration Sensitivity Analysis

Notes: Values of that match the ICS for various pairs. The small diferences in between this table and Table 5 are an artifact of the grid that we use to evaluate our model and the linear interpolation scheme that we use to match exactly empirical moments.

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A Appendix: Proofs of Model’s Properties

Proof of Property 1

Consider some date and a surname such that . Define as the number of individuals with surname s so that . Note that, conditional on is a binomial random variable with support and distribution (suppressing the notation)

\[\mathrm{Prob} (Q _ {t} = k m) = \binom{Q _ {t - 1}}{k} q ^ {k} (1 - q) ^ {Q _ {t - 1} - k}.\tag{9}\]

The conditional mean and variance of are and , respectively. Therefore,

\[Q _ {t} = Q _ {t - 1} m q + w _ {t},\tag{10}\]

where is defined as the innovation, . If then follows a driftless random walk.²⁸

Proof of Property 3

Fix some date t. Partition the population into families: groups of individuals who share the same lineage (which is possible because of asexual reproduction). Suppose, first, that every family’s lineage dates back k periods and that no two families share the same surname. Then the cross-sectional mean and variance of income for each family are, respectively,

\[E (y _ {i s t} | s) = \rho^ {k} y _ {s, t - k}\tag{11}\]

\[V a r (y _ {i s t} \mid s) = V _ {\varepsilon} \sum_ {l = 0} ^ {k - 1} \rho^ {2 l}\tag{12}\]

where is the income of the patriarch of the family with surname s. Now recognize that the societywide cross-sectional variance can be decomposed into the average within-family variance and across-family variance in the conditional mean:

\[{V a r (y _ {i s t})} = {E \big (V a r (y _ {i s t} | s) \big) + V a r \big (E (y _ {i s t} | s) \big).}\tag{13}\]

²⁸This is related to the “branching process” literature, which was started by Francis Galton in 1873. He posed the following problem (our model with zero mutation):

Problem 4001: A large nation, of whom we will only concern ourselves with the adult males, N in number, and who each bear separate surnames, colonise a district. Their law of population is such that, in each generation, percent of the adult makes have no make children who reach adult life; have one such male child; have two; and so on up to who have five.

Find (1) what proportion of the surnames will have become extinct after r generations; and (2) how many instances there will be of the same surname being held by m persons.

The answer was finally determined using martingale methods, but not until 1950. Although complicated, the general idea is that, with strictly positive population growth, a fraction q of all surnames will vanish with probability 1 and a fraction will persist forever (U.S. data on q suggests about 0.8). The distribution for the surviving names is exponential (see Kendall (1966)).

The ICS from Equation (3) is proportional to the second term on the right which, according to expression (11) is monotonically increasing in This proves Property 3 for the case identical lineage horizons and unique within-family surnames.

Consider next the general case of lineage horizons that vary across families. Suppose that family j all derive from a patriarch who lived periods before date t. Then Equations (11) and (12) remain valid for each and Equation 13 takes the form of family-size weighted means and variances. Nevertheless, holding fixed the structure of the population, the second term on the right of Equation (13) remains a monotonically increasing function of

Finally, relax the assumption that surnames and families are uniquely associated. If family and family share the same surname, s, then is a family-size-weighted average of the incomes of all of the members of the two families. Such averaging will decrease the cross-sectional variance in the conditional means, , thereby decreasing the ICS. However, holding fixed the population structure, this conditional variance, and the ICS, are still increasing in

Proof of Property 4

Here, we demonstrate that our assortative mating model from Section 8.1 has a unique stationary distribution and derive expressions for the model’s variances and correlations in terms of its structural parameters.

Recall that male and female children in the household with surname s at date t have income described by

\[y _ {i s t} ^ {m} = r z _ {i p, t - 1} + e _ {i s t} ^ {m}; y _ {i s t} ^ {f} = r z _ {i p, t - 1} + e _ {i s t} ^ {f},\tag{14}\]

where is the average income of these children’s parents, who formed this household at date is the household inheritance parameter and the innovations e are i.i.d. . We now suppress the i and s notation (they are not needed here). Mating is described by

\[y _ {p t} ^ {f} = \lambda y _ {p t} ^ {m} + u _ {p t}; u _ {p t} \sim N (0, V _ {u}).\tag{15}\]

First, we guess that there exists a stationary distribution for z that has the form . If so, then parental income at date formed from the date t mating rule (15) — satisfies

\[z _ {p t} = \left(y _ {p t} ^ {m} + y _ {p t} ^ {f}\right) / 2 = \left((1 + \lambda) y _ {p t} ^ {m} + u _ {p t}\right) / 2\tag{16}\]

where the first equation is just the definition of average parental income and the second applies the mating rule (15). Applying the inheritance process (14), we get

\[{2 z _ {p t}} = {\left(r z _ {p, t - 1} + e _ {t} ^ {m}\right) (1 + \lambda) + u _ {p t}}.\]

The variance of the distribution of then (if it exists), results from taking the unconditional variance of both sides and imposing stationarity:

\[V _ {z} = \frac {(1 + \lambda) ^ {2} V _ {e} + V _ {u}}{4 - (1 + \lambda) ^ {2} r ^ {2}}.\tag{17}\]

This gives as a function of the structural parameters λ, and and the variance of mating noise,

which is uniquely determined below.

Next, note that a stationary distribution for z implies that the income of male and female children have the same distribution (i.e., by inspection of Equation (14)). Thus, we can write and , for some variance, , to be uniquely determined below. Given this, the mating rule, Equation (15), imposes that

\[V _ {u} = (1 - \lambda^ {2}) V _ {y}.\tag{18}\]

This guarantees that the distribution of female income implied by the mating rule coincides with that implied by the inheritance process.

Next, consider the income of males at date . Using (14) and (16):

\[y _ {t + 1} ^ {m} = \frac {r (1 + \lambda)}{2} y _ {p t} ^ {m} + r u _ {p t} / 2 + e _ {t + 1} ^ {m}.\tag{19}\]

Since and , then . Given the independence assumptions on u and e, and given that fertility is deterministic (with each male bearing one male ofspring), then Equation (19) gives the income of a male as a stationary Gaussian first-order autoregressive function of the income of his father. Its unconditional distribution is

\[{y _ {t} ^ {m}} \sim {\mathrm{N} \big (0, \frac {r ^ {2} V _ {u} / 4 + V _ {e}}{1 - r ^ {2} (1 + \lambda) ^ {2} / 4} \big).}\]

By a cross-sectional law-of-large numbers, this also gives the stationary cross-sectional distribution of male income.

All that remains is to solve for as a function of the model’s structural parameters. Using this last expression and rearranging, we have

\[V _ {y} = \frac {V _ {e}}{\lambda (1 + \lambda)} ,\]

where the second equation follows from Equation (18) and the third follows from solving for and rearranging. Substituting the result into Equation (17) yields

\[V _ {z} = \frac {\lambda (1 + \lambda) ^ {2} V _ {e} + 2 (1 - \lambda) V _ {e}}{\lambda (4 - (1 + \lambda) ^ {2} r ^ {2})}\]

This implies that there does indeed exist a stationary distribution for average parental income, z, that is consistent with the inheritance and mating rules, (14) and (15). The variance of the male inheritance shock, from Equation (8), is

\[V _ {w} = V _ {e} \big (1 + \frac {r ^ {2} (1 - \lambda)}{4 \lambda} \big).\]

B Appendix: Surnames as a Proxy of Ethnicity

How good is our CatalanDegree variable as a proxy of ethnicity? Several details can help determine this. First, a large percentage of intra-Spanish immigration into Catalonia occurred after 1955. Second, this immigration flow was large; without it Cabré (2004) estimates that the year 2000 population would have been 2.7 million instead of the actual value of roughly 6 million.

These facts tell us that, in the 2001 census, older people, and especially those born in Catalonia, are more likely to be of Catalan origin than younger people. If our CatalanDegree variable is a good proxy for regional origin, it should therefore reflect this. Table A1 shows that it does. The overall average of CatalanDegree is 0.34, whereas among people born prior to 1950, in Catalonia, the average is 0.57. Figure A1 elaborates. It plots the mean and standard deviation of CatalanDegree for the same rolling window of cohorts used in Figure 5. The surname distribution in Catalonia has clearly become ‘less Catalan’ over time, as the immigration flows tell us it should if our proxy is a good one.

As further support for the quality of our CatalanDegree proxy we run two probit regressions. In the first, the left-hand-side (LHS) variable takes value 1 if an individual has full knowledge of the Catalan language.²⁹ The right-hand-side variables (RHS) are, in column (1), individual-specific controls (place of birth and age dummies). In column (2) our CatalanDegree variable is added. Results are reported in Table A2(a). We estimate a large, significant, positive probability. Figure A2(a) shows the estimated probability for the relevant range of the CatalanDegree variable.

The second regression asks how well CatalanDegree predicts immigration history. The LHS variable takes value 1 if an individual 50 years of age or older immigrated into Catalonia from elsewhere in Spain. Results are reported in Table A2(b) and Figure A2(b). The estimates are negative, large and significant and the pseudo-R² increases dramatically with the inclusion of CatalanDegree. People with lower CatalanDegree surnames are much more likely to be immigrants than those with higher CatalanDegree surnames.

Notice that our dynamic results cannot be driven by a hypothetical change over time in the quality of CatalanDegree as a measure of ethnicity. The reason is as follows. If CatalanDegree had become a worse proxy of ethnicity over time, then the ICS (controlling for ethnicity) would have increased. However, the direct efect of ethnicity would have decreased. However, we observe the opposite (see Figure 5(b)). Moreover, the evolution of the ICS without controlling for ethnicity (see Figure 6) displays the same increasing profile as does the ICS that controls for ethnicity.

In summary, the CatalanDegree variable seems to approximate ethnicity quite well.

C Appendix: Invented Catalonias

Our results should not be sensitive to any random (but suficiently large) partitioning of the surnames set. One such partition is simply based on the alphabet. As as example, we divide the letters of the alphabet into two groups, “first half” and “second half”. Table A3 does exactly this. The first column reports, for comparison purposes, the overall ICS from Table 2. The second and third report the same statistics but for two “invented Catalonias:” those from the first half of the alphabet and those from the second half.³⁰ As our model predicts, neither the of the regressions nor the ICS changes across the populations.

²⁹The census question asks a resident if she speaks, reads and writes Catalan. Roughly 45% of the over-25 population responded in the afirmative.

³⁰We have conducted the experiment with other random groupings and obtained the same result.

(a) Average of CatalanDegree

(b) Standard Deviation of CatalanDegree Figure A1: Evolution of CatalanDegree over moving windows of cohorts Notes: Overlapping age-cohorts are described in caption to Figure 5. Source: 2001 Catalan Census (Idescat).

(a) Probability of Catalan language knowledge

(b) Probability of being an immigrant

Figure A2: Probabilities of Catalan language knowledge and of being an immigrant, as a function of CatalanDegree.

Notes: Regressions as in Table A2. For Figure , reference individual is a male, aged 50-55, born in the county of Barcelona. For Figure A2(b), reference individual is a male, aged 60-65. Source: 2001 Catalan Census.

Table A1: CatalanDegree Summary Statistics

Source: 2001 Catalan Census (Idescat).

Table A2: CatalanDegree & Probabilities of Catalan language knowledge and of being an immigrant (a) Probability of Catalan language knowledge (b) Probability of being immigrant

Notes: Probit Estimates. All regressions include age dummies. Regressions in table A2(a) also include place of birth dummies. Standard errors in parenthesis. For A2(a): Population: baseline population. The LHS variable Knowledge of Catalan takes value 1 for individuals who understand, can speak, can read and can write the Catalan language and zero otherwise. Number of observations: 2,057,134. For A2(b): Population: Individuals born before 1950 of baseline population. The LHS variable Immigrant takes value 1 for individuals who were not born in Catalonia and zero otherwise. Number of observations: 937,441. Source: 2001 Catalan Census (Idescat).

Table A3: ICS. “Invented” Catalonias.

Notes: Regressions as in table 2 (columns 3 and 4). Source: 2001 Catalan Census (Idescat)

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