trading-genetics.tex

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  pdftitle={Trading social status for genetics in marriage markets: evidence from Great Britain and Norway},
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\title{Trading social status for genetics in marriage markets:
evidence from Great Britain and Norway}
\author{Abdel Abdellaoui\thanks{Department of Psychiatry, Amsterdam UMC, University 
of Amsterdam, Amsterdam, The Netherlands. Email: a.abdellaoui@amsterdamumc.nl},
Oana Borcan\thanks{School of Economics, University of East Anglia, Norwich, 
UK. Email: O.Borcan@uea.ac.uk},
Pierre-André Chiappori\thanks{Department of Economics, University of Columbia, 
  New York. Email: pc2167@columbia.edu},\\
David Hugh-Jones\thanks{Corresponding author. Email: davidhughjones@gmail.com},
Fartein Ask Torvik\thanks{Norwegian Institute of Public Health, Oslo. 
  Email: f.a.torvik@psykologi.uio.no} \&
Eivind Ystrøm\thanks{Norwegian Institute of Public Health, Oslo. 
  Email: eivind.ystrom@psykologi.uio.no}}
\date{2024-06-20}

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\begin{document}
\maketitle
\begin{abstract}
Under social-genetic assortative mating (SGAM), socio-economic status (SES) and
genetically inherited traits are both assets in marriage markets, become
associated in spouse pairs, and are passed together to future generations.
This gives a new explanation for persistent intergenerational inequality and
``genes-SES gradients'' -- observed genetic differences between high- and
low-SES people. We model SGAM and test for it in two large surveys from Great
Britain and Norway. Spouses of earlier-born siblings have genetics predicting
more education. This effect is mediated by individuals' own education and
income. Under SGAM, shocks to SES are reflected in the DNA of subsequent
generations, and the distribution of genetic variants in society is endogenous
to economic institutions.

\par

\textbf{Keywords:} Assortative mating, MoBa, UK Biobank.
\end{abstract}

\normalem

\section{Introduction}\label{introduction}

How families are formed, and transmit traits and assets to their
offspring, is crucial for understanding inequality and social structure.
Assortative mating in marriage markets can increase inequality between
families (Breen and Salazar 2011; Greenwood et al. 2014) and contribute to
its persistence across generations, which is surprisingly high
(Clark and Cummins 2015; Solon 2018). Wealthy families pass on
advantages to their children through both genetic inheritance and
environmental influence (Rimfeld et al. 2018; Björklund, Lindahl, and Plug 2006; Sacerdote 2011).

This paper examines a plausible aspect of marriage markets: both social
status and genetics contribute to a person's attractiveness, and as a
result, they may become associated in subsequent generations.\footnote{\emph{Social status} refers to characteristics that an individual
  possesses in virtue of their social position. For example, my wealth
  is a fact about me that holds in virtue of my relationship to
  certain social institutions (bank deposits, title deeds et cetera).
  Other examples include caste, class, income, and educational
  qualifications. \emph{Socio-economic status} (SES) is a specific type of
  social status which exists in economically stratified societies,
  covering variables like educational attainment, occupational class,
  income and wealth (e.g. White 1982).} For
example, suppose that wealth, intelligence and health are advantages in
a potential spouse. Then wealthy people are more likely to marry
intelligent or healthy people, and their children will inherit both
wealth, and genetic variants associated with intelligence or health. We
call this mechanism social-genetic assortative mating (SGAM). SGAM may
be an important channel for the transmission of inequality. It creates a
genetic advantage for privileged families, which may help to explain the
long-run persistence of inequality. At the same time, this advantage is
not a fact of biology, but is endogenous to the social structure.
Indeed, under SGAM, environmental shocks to a person's social status may
be reflected in the genetics of his or her children.

Below, we first write down a theory where attractiveness in the marriage
market is a function of both socio-economic status (SES) and genetic
variants. We show that social-genetic assortative mating in one
generation increases the correlation between SES and genetic variants in
the offspring generation. This result provides a new explanation of
\emph{genes-SES gradients} -- systematic genetic differences between high-
and low-SES people (Belsky et al. 2018; Rimfeld et al. 2018; Björklund, Lindahl, and Plug 2006). The dominant existing explanation for these
gradients is meritocratic social mobility: if a genetic variant predicts
success in the labour market, then it will become associated with high
SES and will be inherited in high-SES families. While under meritocracy,
genes causes SES, under SGAM causality goes both ways, from genes to SES
and vice versa. Also, the size of genes-SES gradients depends on
economic institutions. Under institutions which increase
intergenerational mobility, like high inheritance tax rates,
genes-SES gradients become weaker. On the other hand, an increase in meritocracy
can make them stronger. SGAM also interacts with economic institutions to
determine the level of socioeconomic inequality.

Next, using data on spouse pairs from two large genetically-informed
surveys in Great Britain and Norway, we test the hypothesis that a
person's higher social status attracts spouses with genetic variants
predicting greater educational attainment. Our genetic measure, the
polygenic score for educational attainment (PSEA), derives from
large-scale genome-wide association studies (Lee et al. 2018; Okbay et al. 2022). PSEA reflects a bundle of polygenic effects on
underlying traits, including intelligence, personality, and physical and
mental health (Demange et al. 2021). PSEA predicts, and causes,
educational attainment itself, as well as intelligence and labour market
outcomes. It is already known that humans mate assortatively on PSEA
(Hugh-Jones et al. 2016; Robinson et al. 2017; Torvik et al. 2022), which
makes it a likely candidate for detecting SGAM.

The endogeneity of socio-economic status is the main challenge in
identifying the effect of SES on the spouse's genetic endowment. For
instance, people with high educational qualifications tend to also have
high PSEA, and as mentioned above, they may take partners based on
genetic similarity. Indeed, recent studies show strong assortative
mating on PSEA, much more than we would expect if spouses matched only
on observed measures of educational attainment (Okbay et al. 2022). To
isolate the causal link from own SES to partner's genes, we use a shock
to SES which is independent of own genetics. Specifically, we use a
person's \emph{birth order}. Earlier-born children receive higher parental
investment and have better life outcomes, including measures of SES such
as educational attainment and occupational status (Black, Devereux, and Salvanes 2011; Booth and Kee 2009; Lindahl 2008). At the same time, the facts of biology,
in particular the so-called ``lottery of meiosis'', guarantee that
siblings' birth order is independent of their genetic endowments.\footnote{Although Muslimova et al. (2020) find that PSEA and birth order
  \emph{interact} to produce human capital.}
Because birth order could affect partner choice through both SES and
non-SES mechanisms, we run a mediation analysis similar to
Heckman, Pinto, and Savelyev (2013), decomposing the treatment effect into effects
of measured and unmeasured mediating variables. Specifically, we
estimate a reduced-form model with spouse polygenic scores for
educational attainment (PSEA) as the dependent variable, and own birth
order as the main independent variable. We then add in to the model
measures of own socio-economic status, including university attendance
and income. Under certain assumptions, these variables can be
interpreted as mediating the effect of birth order on spouse genetics.

In both Great Britain and Norway, later-born children have spouses with
significantly lower PSEA in the reduced-form regressions. When we add
mediators, including university attendance and/or income, the effect of
birth order shrinks substantially, becoming insignificant in Great
Britain, while the SES mediators significantly increase the spouse's
PSEA. The results are robust to the inclusion of several controls,
including non-SES mediators, and a rich set of own genetic traits. Thus,
SES appears to mediate the effect of birth order on spouse genetics. The
effects of individual mediators differ between the two countries. While
university attendance explains more than a third of the effects of birth
order in both Britain and Norway, income explains about 10\% of the
effects in Britain but has little or no independent effect in Norway.
Although our main focus is on testing the basic mechanism of SGAM, this
is suggestive evidence that in a more egalitarian society, some forms of
SES are less important to the marriage market, with long-run
implications for genes-SES gradients.

Both economists and geneticists study assortative mating. The economics
literature has typically focused on educational similarities (e.g. Pencavel 1998; Chiappori, Salanié, and Weiss 2017) or social class or
caste (e.g. Abramitzky, Delavande, and Vasconcelos 2011; Banerjee et al. 2013), but also
sorting based on age, physical traits and ethnicity
(Hitsch, Hortaçsu, and Ariely 2010). Some papers have studied substitution between
different traits.\footnote{Oreffice and Quintana-Domeque (2010) show that height and BMI are associated
  with spouse earnings. Dupuy and Galichon (2014) find spouse matching on
  multiple independent dimensions, including education, height, BMI
  and personality. Chiappori, Ciscato, and Guerriero (2021) analyse matching on
  multiple characteristics and show that a three-dimensional matching
  model fits their data.} For instance, Chiappori, Oreffice, and Quintana-Domeque (2012) showed that
individuals trade off BMI for partners' income or education.

In genetics, Halsey (1958) showed that social mobility combined with
assortative mating might increase the association between genetics and
social class. Cloninger, Rice, and Reich (1979) model genetic and cultural
transmission, where assortative mating is based directly on phenotype
and culture is transmitted from parents. Assortative mating, modeled
simply as a correlation coefficient, leads culture and genetics to
be associated in offspring. Heath and Eaves (1985), following earlier
papers (Rao, Morton, and Yee 1976; Rao, Morton, and Cloninger 1979), introduce ``social homogamy'',
i.e.~assortative mating by social background. Otto, Christiansen, and Feldman (1995) extend
assortative mating to include both phenotypic and social homogamy.

More recently, interest in these topics has been revived by empirical
findings from genomics. ``Direct'' effects of individual genetic variants, estimated by
within-family studies, are different from ``indirect'' effects, i.e.~associations
found in the whole sample, and direct effects of polygenic scores can be smaller
than population-wide associations (Howe et al. 2022; A. Young et al. 2022).
Also, parental alleles which are \emph{not} transmitted to the child correlate with
child outcomes (Kong et al. 2018). Both these phenomena could be explained
by confounding from gene-environment correlation, or by assortative mating
(A. S. Young 2023; Nivard et al. 2024). Lastly, correlations between spouses'
polygenic scores for education are higher than can be explained by assortative
mating on measured phenotypic education alone (Okbay et al. 2022; Robinson et al. 2017; Torvik et al. 2022). To address this, several recent papers
papers have estimated structural models of assortative mating in family data
(Eaves et al. 1999; Torvik et al. 2022; Collado, Ortuño-Ortı́n, and Stuhler 2023; Rustichini et al. 2023). Because both cultural and genetic inheritance
proceed from parents to children, it can be hard to differentiate
them. For example, Collado, Ortuño-Ortı́n, and Stuhler (2023) derive extremely low estimates
for heritability of education, within a model in which all genetic similarity
between spouses is driven by matching either on the measured phenotype, or
on a shared cultural factor; whereas Torvik et al. (2022) estimate partner
correlation between ``true'' polygenic scores for education of 0.37, and
heritability above 50\%, in a model where environment is shared between siblings
but not across generations.\footnote{As Okbay et al. (2022) put it: ``Because the parameters of a general
  biometric model cannot be separately identified from a small number of
  phenotypic correlations among different types of relatives, researchers
  typically have to assume that some of the parameters equal zero in order to
  estimate other parameters.''} In this context, we think it is worth
taking a different approach. We cleanly identify separate environmental and
genetic contributions to assortative mating: environmental contributions using
birth order, genetic contributions by comparing polygenic scores within
siblings.\footnote{See Table \ref{tab:tbl-reversed-moba} in the appendix.}

SGAM has consequences for inequality and social mobility. Long-run estimates of
intergenerational persistence of wealth and status are surprisingly higher than
would be predicted from parent-child correlations (Clark and Cummins 2015; Barone and Mocetti 2021; Solon 2018), and distant relatives in the same
generation are also more similar than parent-child and spousal correlations
would predict (Collado, Ortuño-Ortı́n, and Stuhler 2023). Clark (2023) argues that this
can be explained by an underlying process where unobserved genetic variation
determines wealth. This requires a high degree of assortative mating. Our model
shows that genetics may itself be a mediator for the transmission of SES, via
``trading'' in marriage markets. We also show how different social and economic
institutions can affect that process. When SES is highly transmissible across
generations, this increases the long-run association between SES and genetics.
If so, institutional reforms that increase \emph{intergenerational mobility}, like
mass education or inheritance taxation, may affect not only economic but genetic
inequality. Conversely, an increase in \emph{economic meritocracy} increases the
long-run association between SES and genetics,\footnote{See Proposition \ref{prop-gamma}.} posing the problem raised by
M. Young (1958) and more recently Markovits (2019): meritocracy may be
self-limiting or even self-undermining.

In terms of cross-sectional inequality, the conventional wisdom is that it is
increased by assortative mating on SES (Fernández and Rogerson 2001; Breen and Salazar 2011; Greenwood et al. 2014). But that depends what else people
assort on. As we show below\footnote{See Figure \ref{fig:pic-heritability-inequality}.}, if the same genes that are relevant in marriage
markets also affect economic outcomes, then an increase in the role of genes
vis-à-vis SES in marriage markets may increase economic inequality: it makes
households more unequal in genetics, and these are passed on to their children
with high reliability.

SGAM can also explain a large body of evidence for cross-sectional
associations between genetics and social status. For example: from twin
studies, the heritability of occupational class and educational
attainment, i.e.~the proportion of variance explained by genetic
differences between individuals, is around 50\% (Tambs et al. 1989).
Genome-wide Complex Trait Analysis (GCTA) shows that the family
socio-economic status of 2-year-old children can be predicted from their
genes (Trzaskowski et al. 2014). Children born into higher-income families
have more genetic variants predicting educational attainment
(Belsky et al. 2018). Adoption studies show that both post-birth
environment and pre-birth conditions (genetics and prenatal environment)
contribute to the transmission of wealth and human capital (e.g. Björklund, Lindahl, and Plug 2006). There is also a genes-SES gradient in genetic
predictors of health. DNA-derived scores predicting several health
outcomes are associated with regional economic deprivation
(Abdellaoui et al. 2019). The correlation between education and health
may be mediated by shared genetic causes (Amin, Behrman, and Kohler 2015; Boardman, Domingue, and Daw 2015). Family SES correlates with several health-related
polygenic scores (Selzam et al. 2019), and genetic variants associated
with SES may explain the genetic correlations between many mental health
outcomes (Marees et al. 2021).

SGAM shows how marriage markets can lead high SES to be associated with
different genetic variants, i.e.~it can explain genes-SES gradients. The
standard explanation for these gradients is returns to human capital in
labour markets, also known as meritocratic mobility. Higher-ability
parents reap higher market returns, and they may then pass both higher
socio-economic status and their genes to their children, leading to an
association between the two (Belsky et al. 2018).\footnote{Belsky et al. (2018) offer three reasons for the association between
  education-linked genetic variants and SES, but do not consider SGAM.} This mechanism depends
on the level of meritocracy in social institutions
(Branigan, McCallum, and Freese 2013; Heath et al. 1985): in a society where social status
was ascribed rather than earned, it could not take effect. Indeed, after
the fall of communism in Estonia, the heritability of SES increased,
presumably because post-communist society allowed higher returns to
talent (Rimfeld et al. 2018). By contrast, SGAM does not require meritocracy.
Even when social status is entirely ascribed, it can still become
associated with certain genetic variants, so long as their associated
phenotypes are prized assets in marriage markets. Since meritocracy is
historically rare, while assortative mating is universal, this suggests
that genes-SES gradients are likely to be historically widespread.

Lastly, we contribute to a literature in economics that examines the
relationship between genetic and economic variables.
Benjamin et al. (2011) and Benjamin et al. (2024) are reviews. Several
recent papers use polygenic scores, in particular polygenic scores for
educational attainment (e.g. Barth, Papageorge, and Thom 2020; Papageorge and Thom 2020; Ronda et al. 2020). Barban et al. (2021) use PSEA as an instrument for
education in a marital matching model. These papers, like much of the
behavior genetics literature, take genetic endowments as exogenous and
examine how they affect individual outcomes, perhaps in interaction with
the environment. We take a different approach by putting genetics on the
left hand side of the estimating equation. Assortative mating and cultural
inheritance are social processes, so we think there are good prospects for
social scientists to contribute to understanding how genetic variants get
distributed in society -- what geneticists call ``stratification'' and ``dynastic
effects''.

The observations behind SGAM are not new. That status and physical
attractiveness assort in marriage markets is a commonplace and a
perennial theme of literature. In the Iliad, powerful leaders fight over
the beautiful slave-girl Bryseis. In Jane Austen's novels, wealth,
attractiveness and ``virtue'' all make a good match. Marx (1844)
wrote ``the effect of ugliness, its repelling power, is destroyed by
money.'' The literature on mate preference from evolutionary psychology
(Buss and Barnes 1986; Buss 1989; Buss and Schmitt 2019) confirms that
attractive mate characteristics include aspects of social status (``high
earning capacity,'' ``professional status'') as well as traits that are
partly under genetic influence (``intelligent,'' ``tall,'' ``kind,''
``physically attractive''). Despite this, to our knowledge, few papers have examined
the socio-economic consequences of assortative
mating between SES and genetics.\footnote{Specifically, Halsey (1958) and Rustichini et al. (2023).} In particular, we are the first to show
how SGAM interacts with institutional variables to affect economic inequality,
mobility and associations between genes and SES, and the first to cleanly
identify an environmental effect on spouse genetics.

\section{Model}\label{model}

People in the marriage market have two characteristics:
\(x=\left( x_{1},x_{2}\right)\), drawn from a normal distribution
\[
\mathcal{N}
\left( 
\begin{array}{c}
0 \\ 
0%
\end{array}%
,%
\begin{array}{cc}
s^{2} & \sigma \\ 
\sigma & S^{2}%
\end{array}%
\right).
\]
We interpret \(x_1\) as a genetic measure, for example of genetic variants predicting
height, physical attractiveness, health or intelligence. \(x_2\) is a measure of
socio-economic status, such as income or wealth, or social status more generally
(we sometimes use ``wealth'' as a shorthand). The correlation between
\(x_1\) and \(x_2\) is
\[
Corr = \frac{\sigma }{sS} < 1.
\]
People's attractiveness is given by
\[
i\left( x\right) =ax_{1}+\left( 1-a\right) x_{2}
\]
where \(a \in [0, 1]\) is a parameter reflecting the relative importance of
genetics to wealth in the marriage market.\footnote{Note that since the variance of the shocks to \(x_1\) and \(x_2\)
  (see below) has been normalized to 1, \(a\) also reflects this variance. That
  is, a large variance of SES shocks (compared to genetic shocks) translates into
  \(a\) being large.} If \(a = 0\),
marriage markets are highly inegalitarian, such that only SES matters. If \(a =
1\), marriage markets are economically egalitarian and only genetics matter. We
expect realistic societies to fall between these extremes, with \(0 < a < 1\).
Then, both genes and SES matter to attractiveness, and as a result,
social-genetic assortative mating (SGAM) takes place.\footnote{This model does not assume that people match \emph{directly} on
  genetics, which most observers agree would be unlikely. Instead we assume that
  genetics may contribute to an attractive phenotype which is matched on.}

Attractiveness \(i\) is
distributed \(N(0,\sigma_{I}^{2})\), where
\[
\sigma _{I}^{2}=a^{2}s^{2}+\left( 1-a\right) ^{2}S^{2}+2a\left( 1-a\right)\sigma.
\]

People form matches with transferable utility, where the surplus for a match
between \(x\) and \(y\) is \(S(i(x), i(y))\) such that \(\partial^{2}S/\partial
i\partial j > 0\), i.e.~\(S\) is supermodular. As a result there is positive
assortative mating on attractiveness: \(x\) matches with \(y\) only if they are at
the same quantile of attractiveness, i.e.~if \(i(x_{1},x_{2}) = i(y_{1},y_{2})\).
Within attractiveness quantiles, matching is random. This is the
SGAM mechanism.

We also consider random matching as a benchmark to compare against SGAM. Under
random matching, the distribution of couples' characteristics is normal with
mean 0 and covariance matrix
\[
\mathbb{C}\left( 
\begin{array}{c}
x_{1} \\ 
x_{2} \\ 
y_{1} \\ 
y_{2}%
\end{array}%
\right) =\allowbreak \left( 
\begin{array}{cccc}
s^{2} & \sigma & 0 & 0 \\ 
\sigma & S^{2} & 0 & 0 \\ 
0 & 0 & s^{2} & \sigma \\ 
0 & 0 & \sigma & S^{2}%
\end{array}%
\right). \allowbreak 
\]

Our first proposition shows that if SGAM is taking place, i.e.~if \(0 < a < 1\), then
there is a positive correlation between one partner's wealth and the other
partner's genetics.

\begin{proposition}\label{prop-couples-SGAM}
Under SGAM, the distribution of couples' characteristics is normal, with mean 0
and covariance matrix
\begin{equation}\label{cov-couples-SGAM}
\mathbb{C}\left( 
\begin{array}{c}
x_{1} \\ 
x_{2} \\ 
y_{1} \\ 
y_{2}%
\end{array}%
\right) =\allowbreak \left( 
\begin{array}{cccc}
s^{2} & \sigma  & A^{2} & AC \\ 
\sigma  & S^{2} & AC & C^{2} \\ 
A^{2} & AC & s^{2} & \sigma  \\ 
AC & C^{2} & \sigma  & S^{2}%
\end{array}%
\right) \allowbreak 
\end{equation}
where:
\begin{align*}
A &= \frac{as^{2}+\left( 1-a\right) \sigma }{\sqrt{a^{2}s^{2}+\left(
1-a\right) ^{2}S^{2}+2a\left( 1-a\right) \sigma }} &= \frac{as^{2}+\left( 1-a\right) \sigma }{\sigma_I}; \\
C &= \frac{a\sigma +\left( 1-a\right) S^{2}}{\sqrt{a^{2}s^{2}+\left(
1-a\right) ^{2}S^{2}+2a\left( 1-a\right) \sigma }} &= \frac{a\sigma +\left( 1-a\right) S^{2}}{\sigma_I}.
\end{align*}

In particular, the covariance between $x_2$ and $y_1$, $AC$, is positive if
either $x_1$ and $x_2$ are already correlated ($\sigma > 0$) or if they are
uncorrelated ($\sigma = 0$) and the attractiveness parameter $a$ is strictly
between 0 and 1.

\begin{proof}
See Appendix.
\end{proof}
\end{proposition}

We consider the distribution of couples' wealth. Under random matching this has
mean \(0\) and variance \(2S^2\). Under SGAM, the variance is:
\[
V(x_{2}+y_{2}) = 2S^{2} + 2C^{2} \ge 2S^{2} 
\]
with strict inequality if \(a < 1\) or \(\sigma > 0\). The variance is decreasing in
\(a\) and equals \(4S^2\) if \(a = 0\). Thus, SGAM increases cross-sectional inequality,
but less so than pure matching on wealth.

\subsection{Children}\label{children}

All couples have the same number of children. A child's characteristics are
given by:
\begin{eqnarray}
x_{1}^{\prime } &=&\frac{\tau }{2}\left( x_{1}+y_{1}\right) +\varepsilon
\label{Chil} \\
x_{2}^{\prime } &=&\frac{\theta }{2}\left( x_{2}+y_{2}\right) +\eta 
\nonumber
\end{eqnarray}
where \(x\) and \(y\) are the child's parents, and \(\varepsilon\) and \(\eta\) are
independent normal random shocks with mean \(0\) and variance \(1\).

Parameter \(\tau \approx 1\) reflects genetic inheritance. Under standard biological
assumptions \(\tau = 1\) and characteristics show no regression to the mean. In our
model this leads the variance of \(x_1\) to grow without limit over generations.
In reality, we expect \(\tau < 1\) because very extreme characteristics are
selected against, a process known as stabilizing selection
(Schmalhausen 1949; Sanjak et al. 2018).

Parameter \(\theta \in [0, 1]\) reflects inheritance of SES. Unlike \(\tau\) it may vary
between societies. \(\theta\) is high when there is high intergenerational
transmission of SES. Thus, \(\theta\) captures social and economic institutions
that affect this intergenerational transmission, from taxation and
public education to hereditary nobility. If we interpret \(x_2\) narrowly as
wealth, \(1 - \theta\) can be thought of as the rate of inheritance tax.

For the time being, we assume that a person's genetic endowment has no impact on
their SES. Technically, thus, \(x_{2}^\prime\) does not directly depend on
\(x_{1}^\prime\). In a meritocratic society we would expect adult SES to partly depend
on genetics. We show that even absent meritocracy, correlations between \(x_1^\prime\) and
\(x_{2}^\prime\) can arise. In an extension below, we relax this assumption and allow
meritocracy.

We can now calculate the covariance matrix for
\(x^\prime = (x_{1}^{\prime }, x_{2}^{\prime })\) under SGAM as:

\begin{eqnarray}
\mathbb{C} &=&\left( 
\begin{array}{cc}
1 & 0 \\ 
0 & 1%
\end{array}%
\right) +\left( 
\begin{array}{cccc}
\frac{\tau }{2} & 0 & \frac{\tau }{2} & 0 \\ 
0 & \frac{\theta }{2} & 0 & \frac{\theta }{2}%
\end{array}%
\right) \allowbreak \left( 
\begin{array}{cccc}
s^{2} & \sigma  & A^{2} & AC \\ 
\sigma  & S^{2} & AC & C^{2} \\ 
A^{2} & AC & s^{2} & \sigma  \\ 
AC & C^{2} & \sigma  & S^{2}%
\end{array}%
\right) \allowbreak \left( 
\begin{array}{cc}
\frac{1}{2}\tau  & 0 \\ 
0 & \frac{1}{2}\theta  \\ 
\frac{1}{2}\tau  & 0 \\ 
0 & \frac{1}{2}\theta 
\end{array}%
\right) \nonumber \\
&=&\left( 
\begin{array}{cc}
\frac{1}{2}A^{2}\tau ^{2}+\frac{1}{2}s^{2}\tau ^{2}+1 & \frac{1}{2}\theta
\sigma \tau +\frac{1}{2}AC\theta \tau  \\ 
\frac{1}{2}\theta \sigma \tau +\frac{1}{2}AC\theta \tau  & \frac{1}{2}%
C^{2}\theta ^{2}+\frac{1}{2}S^{2}\theta ^{2}+1%
\end{array}%
\right) \allowbreak \label{cov-children-SGAM}
\end{eqnarray}
\newline

We now explore two issues. First, under SGAM, genetic characteristics are no
longer exogenous; because of assortative matching, they are (partly) socially
determined. In particular, even if genetics and SES are uncorrelated among
parents, the expected genetic endowment of the child is positively related to
parental SES. Second, as a result, in the long run a correlation appears between
traits; that is, high SES people inherit genes that are attractive in marriage
markets.

Regarding point 1, we compute the expected genetic characteristic of the
child, conditional on parental wealth:
\[
\mathbb{E}\left[ \frac{\tau }{2}\left( x_{1}+y_{1}\right) +\varepsilon \mid
x_{2}=v,y_{2}=w\right] 
\]
Given the symmetry of the model, this conditional
expectation only depends on the parents' total wealth, i.e.~\(v+w\).

\begin{claim}\label{claim-E-children-RM}
Under random matching, the expected genetic endowment of the children is
proportional to the parents' SES and to the covariance between SES and genetics
for the parents. In particular, if $\sigma = 0$ (i.e. genetics and SES are
uncorrelated for the parents), then the expected genetic endowment of the
children does not depend on parental SES.
\end{claim}

\begin{claim}\label{claim-E-children-SGAM}
Under SGAM, if $\sigma = 0$ (i.e. genetics and
SES are uncorrelated for the parents), then the expected genetic endowment of
the children is linearly increasing in parental SES. The relationship
increases with the ratio of genetic variance to SES variance, is zero for 
$a = 0$ or $a = 1$, and is highest for intermediate values of $a$.
\end{claim}

Next, we study the correlation between children's traits 1 and 2 as a function
of \(\sigma\), the covariance of parents' traits. We first consider the
general case, then concentrate on \(\sigma = 0\), i.e.~when traits are initially
uncorrelated.

\begin{claim}\label{claim-corr-children-RM}
Under random matching, the correlation between characteristics is smaller
for children than for parents. In particular, if genetics and SES are
uncorrelated for the parents, then they are uncorrelated for the children.
\end{claim}

\begin{claim}\label{claim-corr-children-SGAM}
Under SGAM, if genetics and SES are uncorrelated for the parents, then they
are positively correlated for the children so long as $0 < a < 1$. The 
correlation is increasing in $\theta$.
\end{claim}

Whether characteristics are more or less correlated for children than for
parents depends on whether the initial correlation between parents'
characteristics is larger or smaller than the asymptotic one, derived below.

Figure \ref{fig:pic-intuition} shows the intuition behind the model. Parents
match on downward-sloping attractiveness isoquants given by \(a x_1 + (1-a) x_2 = u\).
Their children are in between them on both dimensions. This compresses the
distribution along the attractiveness isoquants, which leads to a
positive correlation between genetics and SES. The correlation between \(x^\prime_1\)
and \(x^\prime_2\) is 0 when \(a = 0\) or \(a = 1\), because then spouses don't trade off
SES for genes. It is highest for intermediate values of \(a\).

\begin{figure}

{\centering \subfloat[Parents\label{fig:pic-intuition-1}]{\includegraphics{trading-genetics_files/figure-latex/pic-intuition-1} }\subfloat[Children\label{fig:pic-intuition-2}]{\includegraphics{trading-genetics_files/figure-latex/pic-intuition-2} }

}

\caption{Theory. The shaded area is the population distribution. Parents (solid circles) match along attractiveness isoquants (dotted lines). Children (hollow circles) are between them. As a result, the children's distribution is squeezed along attractiveness isoquants, and $x_1$ and $x_2$ become associated.}\label{fig:pic-intuition}
\end{figure}

These results show that SGAM can lead to a genes-SES gradient, i.e.~a positive
correlation between genes and SES. Also, the strength of the genes-SES
correlation is affected by economic institutions, as captured in \(\theta\).
When \(\theta\) is high, the genes-SES correlation is high too.

We now consider the asymptotic distribution of \(x_1\) and \(x_2\) when the matching
process is repeated over many generations. As we would expect, our main results
continue to hold.

\begin{proposition}\label{prop-asymptotics-RM}
Under random matching, the dynamics converges to a stationary distribution that 
is normal with mean zero and covariance matrix 
\[
\mathbb{C}\left(\begin{array}{cc}
x_1  \\
x_2
\end{array}
\right)
=
\left( 
\begin{array}{cc}
\frac{2}{2-\tau ^{2}} & 0 \\ 
0 & \frac{2}{2-\theta ^{2}}%
\end{array}%
\right) \allowbreak 
\]%
In particular, the traits are asymptotically uncorrelated and children's
expected genetic endowment is independent of parents' wealth.

\end{proposition}

\begin{proposition}\label{prop-asymptotics-SGAM}
Under SGAM, for $\theta <1$ and $\tau <1$, the dynamics converge to a stationary
distribution that is normal with mean zero and covariance matrix 
\[
\mathbb{C}\left(\begin{array}{cc}
x_1  \\
x_2
\end{array}
\right)
=
\left(
\begin{array}{cc}
\bar{s}^{2} & \bar{\sigma} \\ 
\bar{\sigma} & \bar{S}^{2}%
\end{array}%
\right) 
\]
Moreover, the asymptotic correlation between characteristics, 
$corr = \bar{\sigma}/\bar{s}\bar{S}$, is non-negative, positive for $0 < a < 1$, 
increasing in $\theta$ and increasing then decreasing in $a$. 
The coefficient of parents' wealth on children's genetics is also positive 
for $0 < a < 1$.

For $\theta = 1$, the dynamics diverge and $\bar{S}^{2}$ goes to $+\infty$; for 
$\tau = 1$, the dynamics diverges and $\bar{s}^{2}$ goes to $+\infty$.

\end{proposition}

\begin{figure}

{\centering \includegraphics{trading-genetics_files/figure-latex/pic-asymptotic-corr-1-rgl} 

}

\caption{Long-run correlation between genetics $x_1$ and SES $x_2$, by weight of genetics in spouse matching ($a$) and strength of inheritance of SES ($\theta$). $\tau$ = 0.95.}\label{fig:pic-asymptotic-corr}
\end{figure}

Figure \ref{fig:pic-asymptotic-corr} plots the asymptotic squared correlation
between \(x_1\) and \(x_2\). In genetic terms, this is the heritability of SES. It is
maximized for intermediate levels of \(a\). Note that both \(\bar{S}^{2}\) and
\(\bar{\sigma}\), as well as the correlation between characteristics and the
conditional expectation of genetics given wealth, are increasing in \(\theta\).
Higher transmission of SES increases inequality of SES in equilibrium
(unsurprisingly), but also the heritability of SES.

\subsection{Extensions}\label{extensions}

We consider three extensions. First, the relative attractiveness of genes and
SES might differ for men and women. Our basic result extends to this setup.

\begin{claim}\label{claim-men-women-different}
Suppose that men's and women's attractiveness is given by
\begin{align*}
i(x) &= ax_1 + (1-a)x_2, \\
j(y) &= by_1 + (1-b)y_2
\end{align*}
respectively, with $0\le a \le 1$, $0 \le b \le 1$. Then if $\sigma = 0$, children's 
characteristics $x^\prime_1$ and $x^\prime_2$ will be positively correlated 
unless $a = b = 0$ or $a = b = 1$. The correlation is increasing in $\theta$.

\end{claim}

Interestingly, the \(x_1\)-\(x_2\) correlation is highest when \(a\) and \(b\) are most
different from each other. So gender differences in what counts as attractive
make the effects of SGAM stronger. Intuitively, if one sex only assorts on
SES while the other sex only assorts on genetics, this induces a very reliable
correlation between genes and SES in couples, since (e.g.) every high-SES male
is matched for sure with a high-genetics female.

Second, in modern meritocracies, people's adult SES depends not just
on their parents' social status and on chance, but also on their own effort and
skills, which might be related to their genetics. So, let
\begin{align}
x^\prime_1 &= \tau \frac{x_{1} + y_{1}}{2} + \varepsilon   \nonumber \\
x^\prime_2 &= \gamma x^\prime_1 + \theta \frac{x_{2}+y_{2}}{2}+\eta \label{eqn-gamma}
\end{align}
where \(\gamma > 0\) represents the effect of own genetics on own SES. \(\gamma\)
can be thought of as measuring meritocratic mobility, but alternatively as the
degree of overlap between genes that matter in marriage markets and in labour
markets. The basic result continues to hold, and also, \(\gamma\)
increases the correlation between genes and SES; a highly meritocratic
society may in the long run lead to a steep genes-SES gradient.

\begin{proposition}\label{prop-gamma}
Under SGAM and equation \eqref{eqn-gamma}, if genetics and SES are 
uncorrelated for the parents, then they
are positively correlated for the children so long as $0 < a < 1$ or
$\gamma > 0$. The correlation is increasing in $\gamma$. Also, so 
long as $\gamma > 0$ and either $0 < a < 1$ or $\sigma > 0$, the coefficient 
of parents' wealth on children's wealth exceeds $\theta$.
\end{proposition}

Figure \ref{fig:pic-heritability-inequality} plots computed equilibrium values
of two interesting outcomes: economic inequality, i.e.~the standard deviation of
\(x_2\), and heritability of SES, i.e.~the squared correlation between \(x_1\) and \(x_2\).
Meritocratic mobility \(\gamma\), which opens a pathway from highly transmissible
genetics to SES, always increases both inequality and heritability
(cf. Rimfeld et al. 2018). The effect of \(a\) is more complex. At low levels of
meritocracy, increasing \(a\) decreases inequality by reducing assortative mating
by SES, and intermediate values of \(a\) increase heritability by assorting
SES with genetics. As meritocratic mobility increases, it acts as a multiplier
for the effects of \(a\) on both outcomes, and \(a\) starts to increase both
heritability and inequality, for almost all values. Again, this is because
\(a\) increases assortation by highly transmissible genetics which in turn
affect SES. In particular, consider a broad social ``modernization'', in which
the direct transmission of SES and the role of SES in matching decrease, while
meritocratic mobility increases since careers become open to talents. If the
decrease in \(\theta\) does not outweigh the increase in \(a\) and \(\gamma\), both
equilibrium heritability and inequality of SES may increase.

\begin{figure}

{\centering \includegraphics[width=1\linewidth]{trading-genetics_files/figure-latex/pic-heritability-inequality-1} 

}

\caption{Equilibrium inequality (standard deviation of $x_2$) and heritability (squared correlation of $x_1$ and $x_2$) for different values of assortative mating $a$, meritocratic mobility $\gamma$ and transmission of SES $\theta$. $\tau = 0.95$. The x axis is scaled differently between the left and right plots.}\label{fig:pic-heritability-inequality}
\end{figure}

Third, we consider non-normal distributions of \(x_1\) and \(x_2\), non-normal
shocks \(\varepsilon\) and \(\eta\), and non-linear attractiveness functions. Suppose
\begin{equation}\label{non-linear-f} 
i(x) = f(ax_1, (1-a)x_2) 
\end{equation}
with \(f\) strictly increasing in both its arguments. Our sole condition on the
distribution of \(x\) is that not everybody with attractiveness \(i\) is both
genetically and socially identical. In particular, this allows for discrete
distributions, like some kinds of social status for \(x_2\), and monogenic or
oligogenic attractiveness phenotypes for \(x_1\).

\begin{proposition}\label{prop-non-normal}
Let attractiveness be given by \eqref{non-linear-f}. Let $(x_1, x_2)$ have
any distribution such that a positive measure of the population has
$i(x) = i$ where the conditional distribution of $(x_1, x_2)$ given $i(x) = i$ is 
non-degenerate. Let $\eta$ and $\varepsilon$ be mean 0 and independent of
$x$ and each other. If genetics and SES are uncorrelated for the parents, then
the correlation among children is non-negative, and strictly positive if 
$0 < a < 1$.
\end{proposition}

Other extensions are possible. We assumed that all couples have the same number
of children. If fertility increased with \(x_1\) or \(x_2\), we would expect this to
reduce the variance of traits in the children's generation and possibly also
their covariance. Here, matching preferences, summarized by the \(a\) parameter,
are exogenous; it would be natural to model \(a\) as an equilibrium outcome.
For example, if parents care about their children's wealth, \(a\) might decrease
in \(\theta\) and increase in \(\gamma\). Indeed, below we see suggestive evidence that
income is valued differently in Norwegian and British marriage markets.

\subsection{Discussion}\label{discussion}

The meanings of both social status, and ``good genes'' in the
marriage market, are likely to vary across societies. Social status could
encompass variables like social class or caste; ethnic identity in
``ranked'' ethnic systems; or in modern societies, SES, including wealth, income
and occupation. Regarding genetics, standards of physical attractiveness, and
other genetically-influenced characteristics which make someone a ``good match'',
vary across societies and over time. The central prediction of the model is that
whatever those characteristics, in the long run they will become correlated with
SES.

Recent empirical work shows high persistence of SES over time, in particular at
the top. Clark (2023) argues that this could be explained by unobserved
genetic variation.
Proposition \ref{prop-gamma} shows that if genes affect own wealth directly,
under assortative mating, the regression coefficient of parents' wealth on own
wealth exceeds the ``direct'' coefficient \(\theta\), because parents' wealth
correlates with parents' genetics and via that with own wealth. Thus,
regressions of wealth on wealth may include the effect of unobserved genetic
variation. This may be a confound due to pre-existing gene-SES correlation (if
\(\sigma > 0\)). But under SGAM (\(0 < a < 1\)) it can also be a mediating
variable, since changes in someone's wealth may indeed affect the identity of
their spouse, hence the genetics of their offspring, and from that their
offspring's adult wealth.

The converse also holds: regressions of children's characteristics on their
genetics alone risk overestimating the effect of genetics, by confounding it
with the effects of correlated socio-economic status. Recent work in genetics
has shown this. Polygenic scores for educational attainment have smaller effects
in between-sibling regressions, where between-family variation in SES is
partialled out and where genetic variants are guaranteed to be randomly
allocated, than in regressions which pool the whole sample (Howe et al. 2022).
Parents' genetic variants which are \emph{not} passed on to children predict
children's characteristics, partly due to social stratification in the geneticists'
sense of non-random mating (Kong et al. 2018; A. S. Young 2023).

The model predicts variation in the effects of SGAM. In particular, in
``caste societies'' where there is complete endogamy within social status
groups, there is no scope for SGAM, because marriage partners do not trade off
genetics for social status (\(a = 0\)). Also, the association between genes and
SES is increased by the institutional variable \(\theta\), which captures
intergenerational persistence of SES. This implies that policy has long-run
effects on biosocial structure: reducing \(\theta\) not only increases
intergenerational mobility, but reduces the correlation of genes with SES, and
hence the bias of what Harden (2021) calls the ``genetic lottery''.
Conversely, reforms that increase meritocracy (\(\gamma\), Proposition
\ref{prop-gamma}) may strengthen the genes-SES gradient.

\section{Data and methods}\label{data-and-methods}

The central insight in our model is that higher SES and good genes
assort in the marriage market. We wish to test this directly, i.e.~to
test whether \(0 < a < 1\) in the attractiveness equation \[
i(x) = a x_1 + (1-a) x_2
\] where \(x_2\) is social status and \(x_1\) is genetic endowment. Consider
the effect of a change in \(x_2\) holding \(x_1\) constant. If \(a = 1\) then
this will not change \(i(x)\) and therefore will not change the expected
characteristics of the spouse. So, if we regress spouse's \(x_1\) on own
\(x_2\), and reject the null of no effect, we can reject \(a = 1\).\footnote{Conceivably, if \(a = 0\) but there is a pre-existing correlation
  between \(x_1\) and \(x_2\) in the population, then an increase in own
  \(x_2\) will increase spouse's expected \(x_2\) and therefore spouse's
  expected \(x_1\), even though the latter does not enter the
  attractiveness equation. We can separately test the null that
  \(a = 0\) by regressing spouse's \(x_2\) on own \(x_1\), holding own \(x_2\)
  constant. Existing work has already linked own genetics to spouse's
  SES, e.g.~education, so we focus on the other direction and treat
  this direction as a robustness check below.}

We use data from two sources: Great Britain and Norway. This allows us
to check our basic result in two different societies, and also to make
(tentative) comparisons between them. Our Great Britain data comes from
the UK Biobank, a study of about 500,000 individuals born between 1935
and 1970 (Bycroft et al. 2018). The Biobank contains information on
respondents' genetics, derived from DNA microarrays, along with
questionnaire data on health and social outcomes. The Biobank does not
contain explicit information on spouse pairs. We categorize respondents
as pairs if they had the same home postcode on at least one
occasion;\footnote{A typical UK postcode contains about 15 properties.} both reported the same homeownership/renting status,
length of time at the address, and number of children; attended the same
UK Biobank assessment center on the same day; both reported living with
their spouse (``husband, wife or partner''); and consisted of one male and
one female. We also eliminate all pairs where either spouse appeared
more than once in the data. This leaves a total of
35,682 pairs.\footnote{In the appendix, we test the validity of our matching process by
  counting the proportion of pairs who had a shared genetic child, in
  a subsample of the data. We also check whether any misidentified
  pairs might have biased our results, by constructing a dataset of
  ``known fake pairs''.}

Our Norway data comes from the Norway Mother, Father and Child Cohort
Study (MoBa), a population-based study of pregnant women and their
partners and children (Magnus et al. 2016; Paltiel et al. 2014).
Participants were recruited from all over Norway from 1999-2008. 41\% of
women consented to participation. In this paper, we use about 100,000
genotyped individuals and about 45,000 genotyped spouse pairs. The
Norway data has some advantages over UK Biobank, including higher
participation, larger sample size, and spouse pairs which are known
rather than inferred. On the other hand it is missing some variables,
including IQ measures and self-reported health.

Our key dependent variable is spouse's \emph{Polygenic Score for Educational
Attainment} (PSEA). A polygenic score is a DNA-derived summary measure
of genetic risk or propensity for a particular outcome, created from
summing small effects of many common genetic variants, known as Single
Nucleotide Polymorphisms (SNPs). We focus on PSEA rather than other
polygenic scores for two reasons. First, educational attainment plays a
key role in human mate search. People are attracted to educated
potential partners (Buss and Barnes 1986; Belot and Francesconi 2013); spouse
pairs often have similar levels of educational attainment, as well as
similar PSEA (Vandenberg 1972; Schwartz and Mare 2005; Greenwood et al. 2014; Hugh-Jones et al. 2016; Torvik et al. 2022). Second,
PSEA predicts a set of important socioeconomic variables, including not
only education but also social and geographic mobility, IQ, future
income and wealth (Belsky et al. 2016; Barth, Papageorge, and Thom 2020; Papageorge and Thom 2020).\footnote{See Papageorge and Thom (2020) for a detailed discussion of polygenic
  scores aimed at economists.}

PSEA in the UK was calculated using per-SNP summary statistics from
Lee et al. (2018), re-estimated excluding UK Biobank participants; in Norway,
using statistics from Okbay et al. (2022). The score was normalized to
have mean 0 and variance 1. Because polygenic scores are created from
estimates of many small effects, they contain a large amount of noise
relative to the true best estimator that could be derived from genetic
data. For instance, PSEA explains only 11--13\% of variance in educational
attainment (Lee et al. 2018), whereas the true proportion explained by
genetic variation -- the heritability -- is estimated from twin studies
to be about 40\% (Branigan, McCallum, and Freese 2013). Also, polygenic scores are no
more guaranteed to be causal than any other independent variable. For
example, social stratification by ancestry may lead genes to be
associated with educational attainment even if they play no causal role
(Selzam et al. 2019).

Despite these points, PSEA has non-trivial estimated effects on
educational attainment. PSEA correlates with measures of education,
including university attendance and years of full-time education. Effect
sizes are smaller but still non-trivial in within-siblings regressions
(Lee et al. 2018), where they can be interpreted as causal, since genetic
variation across siblings is guaranteed to be random by the biological
mechanism involved -- the ``lottery of meiosis'' (see below). We recheck
these facts within the UK Biobank sample. In a simple linear regression
(N = 408,524) of university attendance on PSEA, a
one-standard-deviation increase in PSEA was associated with a
9.2 percentage point increase in the
probability of university attendance (\(p < 2 \times 10^{-16}\)). In a
within-siblings regression among genetic full siblings (N =
36,748), the increase was
4.5 percentage points (\(p < 2 \times
10^{-16}\)). This suggests that about half of the raw correlation of PSEA
with university attendance is down to environmental confounds like
parental nurture, while the remainder is causal (cf. Lee et al. 2018).
Still, the causal effect remains substantial: for a rough comparison,
the (ITT) effect on college attendance of the Moving To Opportunity
experiment in the US was 2.5 percentage points (Chetty, Hendren, and Katz 2016).

We use two measures of socio-economic status: income, and university
attendance. Income is a direct measure of SES. University attendance is
a predictor of income over the whole life course, and a form of SES in
itself. The MoBa data includes both university attendance and income. UK
Biobank includes university attendance, but only has a direct measure of
current household income, which is inappropriate for our purposes
because it includes income from both spouses and is measured after
marriage. Instead, we estimate income in the respondent's first job, by
matching the job's Standard Occupational Classification (SOC) code with
average earnings by SOC from National Statistics (2007). Job codes are only available
for a subset of respondents. We convert income to a z score among each
group of respondents with the same gender and year of birth.

Figure \ref{fig:pic-basic-corr} illustrates the core idea of SGAM
within the UK Biobank data. The X axis shows a measure of one partner's
socio-economic status: university attendance or income. The Y axis plots
the other partner's mean PSEA. Both males and females who went to
university had spouses with higher PSEA. So did males and females with
higher income in their first job. Since DNA is inherited, these people's
children will also have higher PSEA.\footnote{Figure \ref{fig:pic-basic-corr-moba} in the appendix shows the
  same plot for the MoBa sample.}

\begin{figure}

{\centering \subfloat[\label{fig:pic-basic-corr-1}]{\includegraphics{trading-genetics_files/figure-latex/pic-basic-corr-1} }\subfloat[\label{fig:pic-basic-corr-2}]{\includegraphics{trading-genetics_files/figure-latex/pic-basic-corr-2} }\newline\subfloat[\label{fig:pic-basic-corr-3}]{\includegraphics{trading-genetics_files/figure-latex/pic-basic-corr-3} }\subfloat[\label{fig:pic-basic-corr-4}]{\includegraphics{trading-genetics_files/figure-latex/pic-basic-corr-4} }

}

\caption{Spouse polygenic score for educational attainment (PSEA) against own university attendance and own income in first job (Great Britain). Lines show 95\% confidence intervals. PSEA is normalized to have mean 0 and variance 1. Income is estimated from the respondent's first job, as the average income of the SOC job code.}\label{fig:pic-basic-corr}
\end{figure}

These plots do not prove that SGAM is taking place. Since an
individual's own PSEA correlates with both their educational attainment,
and their income, both figures could be a result of genetic assortative
mating (GAM) alone (Hugh-Jones et al. 2016). Indeed, recent studies show
much higher levels of GAM than could be explained by matching on the
observed education phenotype alone (Okbay et al. 2022). So, to
demonstrate SGAM, we need a source of social status which is exogenous
to genetics. Also, the link between social status and spouse genetics is
likely to be noisy, for three reasons: first, polygenic scores contain a
large amount of error, as discussed above; second, causal mechanisms
behind variation in social status are likely to be noisy; third, to
paraphrase Shakespeare (1595), the spouse matching process is
highly unpredictable. So, we need a large N to give us sufficient power.
This rules out time-limited shocks such as changes to the school leaving
age (Davies et al. 2018).

We use \emph{birth order}. It is known that earlier-born children receive
more parental care and have better life outcomes, including measures of
SES such as educational attainment and occupational status
(Lindahl 2008; Booth and Kee 2009; Black, Devereux, and Salvanes 2011).\footnote{Earlier work was ambiguous on the effects of birth order (e.g. Hauser and Sewell 1985; Hanushek 1992). However, this work often used
  unrepresentative samples and/or did not control for family size or
  parental age. More recent work improves on this and shows clear
  birth order effects. Kantarevic and Mechoulan (2006) show that parental age is
  an important confound for birth order. Black, Devereux, and Salvanes (2005) show
  substantial birth order effects in the whole Norwegian population,
  even in a family fixed-effects specification, and after controlling
  for mother's age. Booth and Kee (2009) examine UK families, controlling
  for family size and for parental age at birth, and show significant
  and substantial birth order effects on education. Some studies (e.g. Black, Devereux, and Salvanes 2005; De Haan 2010) use twin births or the gender mix of
  children as instruments for family size. They too typically find
  that birth order has large negative effects.} On the other
hand, all full siblings have the same \emph{ex ante} expected genetic
endowment from their parents, irrespective of their birth order. This is
guaranteed by the biological mechanism of meiosis, which ensures that
any gene is transmitted from either the mother or the father to the
child, with independent 50\% probability (Mendel 1865; Lawlor et al. 2008). For example, siblings' expected polygenic score
is equal to the mean of their parents' polygenic scores.\footnote{Although genetic variation is randomly assigned to children at
  birth, genetics and birth order could be dependent if parents'
  choice of whether to have more children is endogenous to the genetic
  endowment of their earlier children. We check for this below.
  Isungset et al. (2021) also find that birth order differences in
  education are not genetic.} We can
therefore use birth order as a ``shock'' to social status. ``Shock'' is in
quotes because we do not claim that birth order is exogenous to all
other variables. For example, it naturally correlates with parental age,
and it may also correlate with household SES at the time of birth. We
only claim that birth order is exogenous to genetic variation.

Although birth order within a given family is independent of genetics,
birth order in the whole sample might be correlated with other
environmental and genetic factors. In particular, birth order naturally
correlates with family size, i.e.~the total number of siblings born,
because e.g.~third-born children must be in a family of at least three
siblings. It also correlates with parental age, which may affect the
income and maturity of the parents. To control for family size, we use
dummies for each possible value of family size. In most regressions, we
use only respondents with between 1 and 5 siblings, i.e.~with a family
size of 2-6: beyond that, estimation would get noisy because of small
cell sizes. To control for parental age, we use father's and/or mother's
age at the respondent's birth. This is calculated from the relevant
parent's current age. In the UK Biobank, this data is only available if
the respondent's parent was still alive. For other controls we use
respondent's month of birth, year of birth, and own PSEA. Our claim is
that controlling for these variables, birth order is independent of
genetic variation.

\subsection{Decomposing the birth order effect on spouse genetics}\label{decomposing-the-birth-order-effect-on-spouse-genetics}

Birth order offers a way to test the central assumption in our
theoretical model -- that SES is traded for genes in the marriage
markets -- by providing a shock to SES which is exogenous to own
genetics. Ideally, we might prefer to use birth order as an instrument
for SES. However, our measures of social status are noisy and
incomplete. For example, we know whether subjects attended university,
but not which university. Birth order likely affects both measured and
unmeasured aspects of SES. So, an instrumental variables approach would
fall foul of the exclusion restriction.

Instead, we conduct a mediation analysis, following the strategy of
Heckman, Pinto, and Savelyev (2013). We first confirm that birth order affects our
measures of respondents' SES (income and education). Then, we regress
spouse's PSEA on birth order, with and without controlling for SES.
Under the assumption that birth order is exogenous to own genetics,
these regressions identify the effect of birth order, plus other
environmental variables that correlate with it, on own social status and
spouse's genetics. Most importantly, if the estimated effect of birth
order on spouse's PSEA changes when SES is controlled for, that is
evidence that SES mediates the effect of birth order.

We follow Heckman, Pinto, and Savelyev (2013) to decompose the aggregate treatment
effect into components due to observed and unobserved proximate channels
affected by the treatment. Our aim is to estimate the effect of SES (as
an effect of birth order) on spouse PSEA.

Assume \(B\) is a multivalued variable indicating birth order. Let \(Y_b\)
be the counterfactual outcome (spouse PSEA) for the first-born,
second-born etc. Given \(b\), spouse PSEA is assumed to be independent
across observations conditional on some predetermined controls which are
assumed not to be affected by \(B\).

Let \(m_{b}\) be a set of mediators, i.e.~proximate outcomes determined by
\(b\), which account (at least in part) for the \(b\) treatment effect on
spouse PSEA. We can think of \(m_{b}\) as all the effects on
attractiveness, such as increments to SES, health, cognitive and
non-cognitive skills, that individuals receive due to their birth rank.
We can split the mediators in \(m_b\) into a set \(J_m\) of measured
mediators, including university attendance and income in first job, and
a set \(J_u\) of mediators that we cannot measure.

Our linear model is:

\begin{equation}
\label{eq:linear-model}
Y_b = \kappa_b + \sum_{j \in J_m} \alpha_b^j m^j_b+\sum_{j \in J_u} \alpha_b^j m^j_b + \mathbf{X^\prime} \symbf{\beta_b} + \tilde{\varepsilon}_b = \tau_b + \sum_{j \in J_m} \alpha_b^j m^j_b + \mathbf{X^\prime} \symbf{\beta_b} + \varepsilon_b 
\end{equation}

where \(\tilde{\varepsilon}_b\) is a mean-zero residual assumed
independent of \(m_b\) and \(\mathbf{X}\);
\(\tau_b = \kappa_b + \sum_{j \in J_u} \alpha_b^j E(m^j_b)\); and
\(\varepsilon_b = \tilde{\varepsilon}_b + \sum_{j \in J_u} (m^j_b - E(m^j_b))\).
We simplify by assuming that \(\beta_b = \beta\) and \(\alpha_b = \alpha\)
for all \(b\), i.e.~that the effects of \(\mathbf{X}\) and \(m_B\) don't
differ by birth order.\footnote{Under the assumption that measured and unmeasured mediators are
  uncorrelated, we can test these assumptions by running an OLS
  regression of an extended model \eqref{eq:model-to-estimate} where we
  interact the measured mediators and controls with the treatment \(B\),
  and test the significance of the coefficients on the interaction
  terms (\(\symbf{\alpha_b}=0\) and \(\symbf{\beta_b}=0\)). See
  Heckman, Pinto, and Savelyev (2013) and Fagereng, Mogstad, and Rønning (2021) for details and
  different applications. When we run the model with interactions,
  only one interaction is significant after Bonferroni correction at
  \(p < 0.05/34\): the interaction of income in first job with the dummy
  for birth order 6. So overall, the uninteracted model seems a good
  enough approximation.} We assume differences in unmeasured
investments due to \(b\) are independent of \(\mathbf{X}\).

We use a linear model for each observed mediator variable:
\begin{equation}
\label{eq:mediator-model}
m^j_b=\mu_{0,j}+\mathbf{X^\prime}\symbf{\mu_{1,j}}+\mu_{2,j}\cdot b+\eta_j, j \in J_m  
\end{equation} where \(\eta_j\) is a mean-zero residual. We also assume
the treatment-specific intercepts are linear in \(b\): \begin{equation}
\label{eq:linear-intercept}
\tau_b=\tau_{0}+\tau b.  
\end{equation}

With the simplifying assumptions above and substituting
\eqref{eq:mediator-model} and \eqref{eq:linear-intercept} into
\eqref{eq:linear-model} we obtain: \begin{equation}
\label{eq:simplified-model}
Y_b = \tau_0+\tau b + \sum_{j \in J_m} \alpha^j (\mu_{0,j}+\mathbf{X^\prime}\symbf{\mu_{1,j}}+\mu_{2,j}\cdot b+\eta_j) + \mathbf{X^\prime} \symbf{\beta} + \varepsilon_b 
\end{equation}

Using equation \eqref{eq:simplified-model}, we can decompose the average
treatment effect of a change from birth order \(b\) to \(b'\) into the
effect of measured mediators \(m^j\) and unmeasured mediators on the
outcome:

\begin{equation}
\label{eq:decomposition}
E(Y_b' - Y_b) = \tau(b' - b) + \sum_{j \in J_m} \alpha^j E(m^j_{b'} - m^j_b) \\
=\underbrace{\tau(b' - b)}_{\text{Direct effect + unmeasured mediators}} + \underbrace{\sum_{j \in J_m} \alpha^j \mu_{2,j} (b' - b)}_{\text{Effect of measured mediators}}
\end{equation}

We are primarily interested in estimating the effect of SES on spouse
PSEA, amongst the measured mediators, and furthermore we would like to
measure the relative importance of SES compared to other factors in
predicting spouse PSEA.

We therefore estimate: \begin{equation}
\label{eq:model-to-estimate}
Y = \tau_0 + \tau B + \sum_{j \in J_m} \alpha^j m^j_b + \mathbf{X'} \symbf{\beta} + \varepsilon
\end{equation}

Estimating the above by OLS will generate unbiased estimates of
\(\alpha^j\) if \(m^j\) is measured without error and is uncorrelated with
the error term \(\varepsilon\). Since \(\varepsilon\) contains both
individual disturbances and differences in unmeasured investments due to
birth order, there are two identifying assumptions that need to hold for
unbiased OLS estimates: (a) the measured investments (specifically SES)
should be independent of unmeasured investments generated by birth
order. Failing this, the estimates of \(\alpha^j\) will be conflated with
the effects of unmeasured investments. Second, (b) the measured
investments should be uncorrelated with other shocks
\(\tilde{\varepsilon}_b\). With respect to assumption (a), in our
regressions we control for a set of potential alternative mediators
available in the data: height, BMI, and (in UK Biobank only) fluid IQ
and self-reported general health. With respect to assumption (b), we use
further controls, such as parental age, year of birth, and own PSEA, to
reduce unobserved variation in the error term.

By running a least square regression of \eqref{eq:model-to-estimate}, we
can estimate \(\tau\) and \(\alpha^j\). If assumption (a) holds, the part of
the birth order treatment effect on spouse PSEA that is due to measured
mediators, including SES, can be constructed using the estimated
\(\alpha^j\) and the effects of birth order on measured mediators. We can
estimate these effects from OLS regressions based on equation
\eqref{eq:mediator-model} for each measured mediator (in particular
university attendance and income) on \(\mathbf{X}\) and \(B\). The part of
the birth order effect that is due to university attendance (or income)
on spouse PSEA will be the coefficient of university/income in the
regression of spouse PSEA in equation \eqref{eq:model-to-estimate},
multiplied by the coefficient of birth order on university/income from
equation \eqref{eq:mediator-model}. We now apply this framework to each
of our two samples.

\section{Results: Great Britain}\label{results-great-britain}

 
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\caption{\label{tab:tbl-bo-first-stage} Regressions of mediators on birth order (Great Britain)}
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\end{table}
 

We first regress our measures of socio-economic status, university
attendance and income from first job, on birth order in the UK Biobank
spouse pairs. We also do the same for four non-SES mediators that could
be affected by birth order: fluid IQ, height, body mass index (BMI) and
a measure of self-reported health. We control for respondent's own PSEA
and their parents' age at birth (see below). Table
\ref{tab:tbl-bo-first-stage} shows that birth order significantly
predicts all the mediators. Effects are quite substantial: on average,
one extra elder sibling reduces the chance of attending university by
about 7.9 percentage points, income
by about 0.077 standard deviations, fluid IQ by
about 0.27 points on a 13 point test, height
by about 0.7 centimeters, and self reported
health by 0.043 points on a 4-point scale; and
increases BMI by 0.19.

Next we run regressions of spouse PSEA on birth order. Table
\ref{tab:tbl-bo-psea-basic} reports the results. Column 1 reports
results controlling only for family size (using dummies). As expected,
higher birth order is negatively associated with spouse's PSEA, though
the estimated effect size is small and insignificant. Column 2 reports
results controlling for the respondent's own PSEA, as well as dummies
for birth year to control for cohort effects, and dummies for birth
month to control for seasonal effects. The effect size of birth order is
not much changed.

Column 3 reports results controlling for parents' age at birth. Within a
family, later children have older parents by definition. Older parents
have more life experience and may have higher income, which may help
later children.\footnote{We often only have data only for one parent. We use this, or take
  the mean if we have both. There are also potential genetic effects
  from parental age, though recent research has rejected these in
  favour of ``social'' explanations (Kristensen and Bjerkedal 2007; Black, Devereux, and Salvanes 2011). Cochran and Harpending (2013) report that mutational load
  is approximately linear in father's age, while it is constant in
  mother's age. We observe very similar results if we control only for
  father's age at respondent's birth.} Kantarevic and Mechoulan (2006) show that mother's age at
childbirth indeed mechanically offsets the negative effect of birth
order. Including parents' age means we can separate the effect of
parental age from birth order.\footnote{A possible critique is that later siblings by definition have
  older parents, so the parental age control is inappropriate. This
  may be true if we are estimating overall inequality between
  siblings. But here we are interested in using birth order as a
  shock, so it's reasonable to separate out its effects from those of
  parental age. The resulting estimate gives the counterfactual impact
  of having one more elder sibling, holding the age of one's parents
  constant.} This reduces the N by a lot, since
only respondents with a live parent reported the necessary data.
However, the effect of birth order jumps in size and becomes significant
at the 5 per cent level. Meanwhile, parents' age has a positive effect.
This suggests that estimates in columns 1-2 mixed two opposite-signed
effects: having older parents versus being later in birth order.

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-basic} Regressions of spouse PSEA on birth order (Great Britain)}
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\end{threeparttable}\par\end{centerbox}

\end{table}
 

Having tested that birth order affects spouse's PSEA, we now look for
potential mediators of this effect. Despite the lower N, we continue to
control for respondents' parents' age, since this removes a confound
which would bias our results towards zero.\footnote{Table \ref{tab:tbl-bo-psea-no-par-age} in the appendix reports
  results without controlling for parents' age.}

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea} Regressions of spouse PSEA on birth order and mediators 
                   (Great Britain)}
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(\ref{eq:model-to-estimate}). PSEA is the polygenic score for educational attainment, which is normalized with mean 0 and standard deviation 1. We include own PSEA, mean of parents’ ages at birth, potential non-SES mediators (fluid IQ, height, BMI, self-reported health) and further controls (family size, birth year, and birth month dummies) to ensure the balance of covariates across birth order. All data is from the UK Biobank for a sample of UK individuals born between 1935 and 1970.  *** p < 0.001;  ** p < 0.01;  * p < 0.05;  + p < 0.1. Standard errors clustered by spouse pair in parentheses.}\huxbpad{2pt}}} \tabularnewline[-0.5pt]


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\end{threeparttable}\par\end{centerbox}

\end{table}
 

Table \ref{tab:tbl-bo-psea} shows the results. Column 1 shows the
effect of birth order, using the same specification as column 3 of the
previous table. The remaining columns add potential mediators of birth
order effects. Column 2 controls for our first measure of socio-economic
status: university attendance. We also include potential non-SES
mediators, which are affected by birth order and might affect spouse
matching: fluid IQ, height, BMI and self-reported health. Column 3 adds
our second measure of socio-economic status, income in first job. Column
4 includes both.

When we add university attendance and other mediators (column 2), the
effect of birth order drops and becomes insignificant at 5\%, while the
coefficient for university is positive and highly significant. Fluid IQ,
height and BMI are also positive and significant, while self-reported
health has the right sign but is insignificant. Controlling for income
instead of university attendance (column 3), again the effect of birth
order shrinks and becomes insignificant, while income has a positive and
highly significant effect. Lastly, the same pattern holds when we
control for both university and income (column 4).

 
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\begin{table}[ht]
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\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-mediation-prop} Percent of birth order effects accounted for by mediators (Great Britain)}
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\end{threeparttable}\par\end{centerbox}

\end{table}
 

Under the assumptions discussed above, we can estimate the proportion of
the birth order effect that is mediated by these variables. Table
\ref{tab:tbl-mediation-prop} reports this for each model in columns
2-4. Each estimate is the coefficient of birth order on the mediator,
times the coefficient of the mediator on spouse PSEA, divided by the
coefficient of birth order on spouse PSEA estimated from column 1, i.e.
without mediators. Education explains about 38-55 percent of the effect,
much more than all the other mediators. Income, fluid IQ, height and BMI
all explain between 6 and 17 percent of the effect, depending on the
specification.

These results provide evidence that birth order affects spouse PSEA via
education and income, with education being especially important. The
effect size of birth order is small (a few percent of a standard
deviation of PSEA), but what matters is the effect size of education and
income. The effect of education in particular is quite large as measured
here, and since it also appears to mediate the purely environmental
shock of birth order, it cannot just be due to an unobserved correlation
with own genetics.

Our next regressions split up the data into subsets. Cultural
stereotypes often assume that the link between status and genes is not
symmetric across the genders, for example, that males with high SES are
particularly likely to marry attractive spouses. Claim
\ref{claim-men-women-different} showed that these differences would
strengthen the effects of SGAM. To test for this, we separately regress
female spouses' PSEA on male birth order, and male spouses' PSEA on
female birth order. We also rerun regressions among the subset of
individuals who had children. A significant result here will confirm
that the association between status and genetics is carried over into
the next generation.

Table \ref{tab:tbl-bo-subsets} shows the results. Columns 1 and 2
present results using birth order of male respondents to predict female
spouses' PSEA. Column 1 shows the regression of birth order plus
controls; in column 2, we add university attendance and non-SES
mediators (here, we exclude first job income so as to keep our N large).
Columns 3 and 4 repeat the exercise for female respondents, using their
birth order to predict male spouses' PSEA. The effect of birth order is
imprecisely estimated in these subsets due to the lower sample size.
However, the pattern of coefficient sizes is the same as in the main
regression: the coefficient of birth order is about -0.3 (and very
similar between the sexes), and adding university attendance reduces the
absolute size of the birth order effect. Columns 5 and 6 show results
from regressions on the subsample of couples with children. Here, birth
order is significant in the base specification, and again, university
attendance still seems to mediate the birth order effect.

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-subsets} Regressions of spouse PSEA on birth order: subsets (Great Britain)}
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\end{table}
 

\FloatBarrier

\section{Results: Norway}\label{results-norway}

Now we turn to the results from MoBa in Norway. Some of the variables
are different from those for UK Biobank. Spouse PSEA is calculated using
summary statistics from Okbay et al. (2022), aka ``EA4'', rather than
``EA3''.\footnote{The \(R^2\) on own university attendance is
  0.0814 for EA4 in MoBa, compared to
  0.04 for EA3 in UK Biobank.} Income is from all sources, reported at age 30, converted to
a z score among respondents with the same gender, year of birth and year
of reported income. In particular, some low-income individuals may be in
continuing education or in relationships already.\footnote{We also tried income at age 25. This gave odd results, with a
  significant negative beta on spouse PSEA. A possible reason is that
  in Norway many potential high earners are still in higher education
  at age 25. Income at age 25 correlated only at 0.1 with income at
  age 30, and was negatively correlated with educational attainment.
  Overall, we think income at 30 is more informative as a measure of
  SES.} Data on IQ and
self-reported health is unavailable. The sample is also younger than UK
Biobank, spouse pairs are given rather than constructed, and all couples
have at least one child.

Tables \ref{tab:tbl-bo-psea-moba} and \ref{tab:tbl-mediation-moba} are
the equivalent of Tables \ref{tab:tbl-bo-psea} and
\ref{tab:tbl-mediation-prop} for respondents in the MoBa dataset.
(Equivalents to Tables \ref{tab:tbl-bo-first-stage} and
\ref{tab:tbl-bo-psea-basic} are in the appendix.) The broad pattern of
results is similar to Britain. The larger N gives higher statistical
significance. Effects of all the variables are in the expected
direction, except that the coefficient on income changes sign when
university is also included. In particular, the point estimate of the
total effect of birth order is about twice as high as in Britain, and
the estimated effect of university attendance is also higher. The effect
of own PSEA is also about twice as high. Note that these differences
could be driven by the PSEA score containing less noise in the Norwegian
sample.

Adding university attendance again substantially and significantly
reduces the effect of birth order, though here, birth order remains
independently significant and substantively large even controlling for
university. In Norway, however, it is much less clear that income is an
important mediator on its own. Adding income barely changes the effect
of birth order (column 3). Table \ref{tab:tbl-mediation-moba} computes
the percentages of the effects that are mediated by our variables. The
percentage effects of education, when controlling for income, are about
the same in both countries. But the effect of income in the Norwegian
sample is much smaller and very close to zero.

Table \ref{tab:tbl-bo-subsets-moba} runs our regressions separately for
males and females.\footnote{We don't run separate regressions for families with children,
  since MoBa only includes families with children by design.} As in Britain, coefficients look very similar
across the sexes. The effects of birth order are highly significant in
either sex, and adding the mediators significantly reduces them.
Interestingly, the effect of BMI is about 50\% larger for women, as in
Britain, and here the difference is significant. On the other hand, in
Norway, there is no difference between the male and female coefficients
on university attendance.

Overall, the Norway results show two things. First, they clearly confirm
that birth order affects spouse PSEA, with education a key mediator.
Second, they suggest that the effects of SES in marriage markets vary
between the two countries. Education has a similar effect on spouse PSEA
in Norway, but income has a much smaller effect. Of course this is a
loose comparison, since even conditioning on our controls, samples and
measures in the two countries are different. Still, it is interesting
that in Norway, a more egalitarian country than the UK, income seems to
have a smaller effect on spouse PSEA, while own PSEA, a genetic
characteristic, seems to have a larger effect. This suggests that
genetic and SES contributions to attractiveness may vary between
countries, and perhaps be endogenous to economic institutions.

 
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\caption{\label{tab:tbl-bo-psea-moba} Regressions of spouse PSEA (Norway)}
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\end{table}
 

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-mediation-moba} Percent of birth order effects accounted for by mediators (Norway)}
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\end{table}
 

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-subsets-moba} Regressions of spouse PSEA: subsets (Norway)}
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\multicolumn{1}{p{0.2\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{3pt}\parbox[b]{0.2\textwidth-3pt-3pt}{\huxtpad{3pt + 1em}\raggedleft {\fontsize{10pt}{12pt}\selectfont 0.062\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}}\huxbpad{3pt}}} \tabularnewline[-0.5pt]


\hhline{>{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}-}
\arrayrulecolor{black}

\multicolumn{5}{!{\huxvb{0, 0, 0}{0}}p{1\textwidth+8\tabcolsep}!{\huxvb{0, 0, 0}{0}}}{\hspace{3pt}\parbox[b]{1\textwidth+8\tabcolsep-3pt-3pt}{\huxtpad{3pt + 1em}\raggedright {\fontsize{10pt}{12pt}\selectfont Estimates from OLS regressions corresponding to columns 1 and 2 in Table \ref{tab:tbl-bo-psea-moba}, separately for males and females. Spouse PSEA is the dependent variable, and own birth order and university attendance are the main independent variables. PSEA is the polygenic score for educational attainment, which is normalized with mean 0 and standard deviation 1. We include own PSEA, parents’ age at birth (the mean of parent’s ages) and further controls (family size, birth year, and birth month dummies) to ensure the balance of covariates across birth order. All data is from the MoBa dataset for a sample of spouse pairs with a child between 1999 and 2008.  *** p < 0.001;  ** p < 0.01;  * p < 0.05;  + p < 0.1. Standard errors clustered by spouse pair in parentheses.}\huxbpad{3pt}}} \tabularnewline[-0.5pt]


\hhline{}
\arrayrulecolor{black}
\end{tabularx}
\end{threeparttable}\par\end{centerbox}

\end{table}
 

\FloatBarrier

\section{Robustness}\label{robustness}

As noted earlier, our results could conceivably be explained by spouses
\emph{only} mating on SES (\(a = 0\) in our model), but with a pre-existing
correlation between SES and PSEA in the population. That said, existing
work strongly suggests that own PSEA affects spouse's education
(Robinson et al. 2017; Torvik et al. 2022). To confirm this we use a
sample of siblings from the MoBa data\footnote{The UK Biobank sample has too few siblings with spouses for this
  analysis to be informative.}, and regress spouse's
university attendance and income z-score on own PSEA, including sibling group fixed
effects. Again, this uses the lottery of meiosis to guarantee that
between-sibling differences in PSEA are exogenous to environmental
characteristics including SES. Table \ref{tab:tbl-reversed-moba} shows that
own PSEA significantly increases the probability of
spouse university attendance, by about 5\% per standard deviation; interestingly,
its effect on spouse income is insignificant and tightly bounded around
zero, supporting the hypothesis that income is not an important form of SES in
Norwegian marriage markets. Overall, these results allow us to rule out \(a = 0\).

 
  \providecommand{\huxb}[2]{\arrayrulecolor[RGB]{#1}\global\arrayrulewidth=#2pt}
  \providecommand{\huxvb}[2]{\color[RGB]{#1}\vrule width #2pt}
  \providecommand{\huxtpad}[1]{\rule{0pt}{#1}}
  \providecommand{\huxbpad}[1]{\rule[-#1]{0pt}{#1}}

\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-reversed-moba} Within-siblings regressions of spouse university attendance and income on own PSEA (Norway)}
 \setlength{\tabcolsep}{0pt}
\begin{tabularx}{0.75\textwidth}{p{0.25\textwidth} p{0.25\textwidth} p{0.25\textwidth}}


\hhline{>{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}-}
\arrayrulecolor{black}

\multicolumn{1}{!{\huxvb{0, 0, 0}{0}}p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\centering \huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\centering University\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\centering Income\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{>{\huxb{255, 255, 255}{0.4}}->{\huxb{0, 0, 0}{0.4}}->{\huxb{0, 0, 0}{0.4}}-}
\arrayrulecolor{black}

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\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft 0.0477 ***\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft $-$0.0007\hphantom{0}\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{}
\arrayrulecolor{black}

\multicolumn{1}{!{\huxvb{0, 0, 0}{0}}p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedright \huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft (0.0098)\hphantom{0}\hphantom{0}\hphantom{0}\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft (0.0174)\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{>{\huxb{255, 255, 255}{0.4}}->{\huxb{0, 0, 0}{0.4}}->{\huxb{0, 0, 0}{0.4}}-}
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\multicolumn{1}{!{\huxvb{0, 0, 0}{0}}p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedright Sibling group dummies\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\centering Yes\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\centering Yes\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{}
\arrayrulecolor{black}

\multicolumn{1}{!{\huxvb{0, 0, 0}{0}}p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedright Birth month dummies\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\centering Yes\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\centering Yes\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{}
\arrayrulecolor{black}

\multicolumn{1}{!{\huxvb{0, 0, 0}{0}}p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedright Birth year dummies\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\centering Yes\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\centering Yes\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{>{\huxb{255, 255, 255}{0.4}}->{\huxb{0, 0, 0}{0.4}}->{\huxb{0, 0, 0}{0.4}}-}
\arrayrulecolor{black}

\multicolumn{1}{!{\huxvb{0, 0, 0}{0}}p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedright N\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft 9729\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft 9755\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{}
\arrayrulecolor{black}

\multicolumn{1}{!{\huxvb{0, 0, 0}{0}}p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedright Adj. $R^2$\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft 0.182\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\huxbpad{6pt}}} &
\multicolumn{1}{p{0.25\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.25\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft 0.083\hphantom{0}\hphantom{0}\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{>{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}-}
\arrayrulecolor{black}

\multicolumn{3}{!{\huxvb{0, 0, 0}{0}}p{0.75\textwidth+4\tabcolsep}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.75\textwidth+4\tabcolsep-6pt-6pt}{\huxtpad{6pt + 1em}\raggedright Estimates from within-sibling-group regressions. PSEA is the polygenic score for educational attainment, which is normalized with mean 0 and standard deviation 1. Sibling group dummies are included to ensure exogeneity of PSEA.  *** p < 0.001;  ** p < 0.01;  * p < 0.05;  + p < 0.1. Standard errors in parentheses.\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{}
\arrayrulecolor{black}
\end{tabularx}
\end{threeparttable}\par\end{centerbox}

\end{table}
 

Although all children of the same parents have the same polygenic scores
in expectation, it might still be possible that genetics correlates with
birth order within the sample. This could happen in three ways. First,
siblings with high birth order will typically come from larger families
than those with low birth order, and parents of different-sized families
are likely to differ systematically on many dimensions, including
genetics. We controlled for this by including a full set of family size
dummies in the regression. Second, there could be selection bias. For
example, if later siblings with high PSEA, and earlier siblings with low
PSEA, are more likely to enter the sample, then this would bias our
results. Thirdly, parents might choose family size in a way related to
genetics. For example, suppose that when the first child has a phenotype
reflecting a high PSEA, parents are more likely to have a second child.
Then within the subset of two-child families, first children would have
higher-than-average PSEA, while second children would not.

To check for the latter two problems, we run balance tests on 33
different polygenic scores in the UK Biobank sample.\footnote{Polygenic scores were residualized on the first 100 principal
  components of the genetic data. Scores were for: ADHD, age at
  menarche, age at menopause, agreeableness, age at smoking
  initiation, alcohol use, Alzheimer's, autism, bipolarity, BMI, body
  fat, caffeine consumption, cannabis (ever vs.~never), cognitive
  ability, conscientiousness, coronary artery disease, smoking
  (cigarettes per day), type II diabetes, drinks per week, educational
  attainment (EA2 and EA3), anorexia, extraversion, height, hip
  circumference, major depressive disorder, neuroticism, openness,
  smoking cessation, schizophrenia, smoking initiation, waist
  circumference, and waist-to-hip ratio. For full details of score
  construction, see Abdellaoui et al. (2019). We also ran similar tests
  on MoBa data, and found no significant associations between birth
  order and polygenic scores.} We regress
each score on own birth order, controlling for family size. No scores
were significant at \(p <
0.10/33\). Four scores were significant at \(p < 0.10\), all with effect
sizes of less than 0.02 per standard deviation. Table
\ref{tab:tbl-bo-psea-pgs} in the appendix reports regressions
controlling for these scores. Results are almost unchanged. To test
whether polygenic scores might vary across birth orders within a
particular family size, we also regress each score on a full set of
birth order dummies, interacted with a full set of family size dummies.
None of the 495 birth order coefficients were significant
at \(p < 0.001\). However, among families of size 3, there is a marginally
significant positive correlation of birth order with own PSEA (effect
size 0.0277,
\(p = 0.06\)). Table \ref{tab:tbl-bo-psea-no3}
in the appendix therefore reports regressions with families of size 3
excluded. Results are substantially unchanged. Of course, there could
still be unmeasured genetic variants which correlate with birth order in
our sample. Nevertheless, a wide set of polygenic scores shows no large
or significant correlation. This makes us more confident that birth
order is indeed exogenous to genetics.

Another concern is that our chosen SES mediators might not be exogenous.
We have already seen that birth order affects intelligence, height, BMI
and health. So there might be other unobserved variables which mediate
the effect of birth order on spousal PSEA, and which correlate with
education or income, but which do not themselves capture SES. If so,
that would threaten our claim that education and income are important
mediators. However, the effects of education on spouse PSEA in Table
\ref{tab:tbl-bo-psea}, and of birth order on education in Table
\ref{tab:tbl-bo-first-stage}, are both large and highly significant. In
other literature on spouse matching, education is a common, robust and
significant predictor. For these reasons, we think that our results are
unlikely to be driven wholly by other, unobserved mediators. In the
appendix we run Oster (2019) style robustness checks where we
formally ask how strong selection on unobservables would need to be to
reduce the effect of our key mediators to zero.

A final concern is that polygenic scores, including PSEA, contain noise
from correlated environments. (That is, effect sizes of individual SNPs
may be confounded in the underlying statistical analyses used to create
polygenic scores.) It is conceivable that birth order could only affect
the noise component of spouse PSEA, rather than the component which is
truly causal for education. We think this is unlikely for several
reasons. First, our polygenic scores were calculated using per-SNP
summary statistics estimated on non-UK populations. So they will only
include noise that correlates with social environment insofar as
non-causal correlations of SNPs with social status are the same across
different countries. Second, we have residualized PSEA on 100 principal
components of the genetic data, a standard technique in genetics to
avoid confounding causal effects with population stratification. Third,
true causal effects of individual SNPs are highly correlated (r = 0.74)
with their ``population effects'' including noise (A. Young et al. 2022).
Lastly, it is hard to imagine an assortative mating process by which
people match spouses who have genetic variants that correlate with
educational attainment, but not genetic variants that cause it.

Our main specification is linear in birth order. Tables
\ref{tab:tbl-bo-psea-dummies} and \ref{tab:tbl-bo-psea-dummies-moba}
in the appendix run specifications with separate dummies for each birth
order. The pattern that birth order coefficients shrink after
controlling for SES mediators holds robustly across all birth orders in
both Britain and Norway.

UK Biobank is not a representative sample of the population. Table
\ref{tab:tbl-bo-psea-weights} in the appendix weights cases to match
the Biobank's sampling frame. Results are similar to those in the main
text. Although this is still not representative of the population as a
whole, it provides some assurance that our results are not driven by
volunteering bias.

The appendix reports other robustness checks, including replacing
university attendance with age of leaving full-time education. Overall,
while significance sometimes varies, the pattern of results is
remarkably consistent. Birth order is always negatively associated with
spouse PSEA, and this effect is always reduced in magnitude after adding
education as a mediator. Effect sizes are also consistent, with the
exception that they are smaller if we do not control for parental age
(just as in Table \ref{tab:tbl-bo-psea-basic}).

\section{Conclusion}\label{conclusion}

Our empirical analysis shows that in Great Britain and Norway, two
contemporary developed countries, earlier-born siblings had spouses with
higher PSEA. We also provide evidence that these effects are mediated by
socio-economic status, specifically income and education. We interpret
this as evidence of social-genetic assortative mating (SGAM).

Advantage is transmitted across generations by many mechanisms. Rich
parents may invest more in their children's human capital, transfer
wealth via gifts and bequests, model valuable skills, or provide them
with advantageous social networks. They may also pass on causally
relevant genetic variants. This channel has been proposed as a reason
for the surprising persistence of inequality over generations
(Clark and Cummins 2015; Clark 2023). One problem with
this theory is that in the absence of assortative mating, genetic
variation regresses swiftly to the mean, with coefficient \(r = 0.5\) per
generation. Thus to explain long-run persistence, the genetic theory
seems to require very high levels of genetic assortative mating. SGAM
may help to solve this puzzle. Persistence will be increased if, in
addition to genetic assortative mating, high SES itself attracts ``good
genes''. At the same time, SGAM changes the interpretation of genetics.
As our model shows, genetic variation is not an exogenous input into the
social system, but an endogenous outcome -- not a confound for wealth,
but a mediator.

SGAM also provides a new explanation for the genes-SES gradient -- the
observed association of genes with SES -- which is an important cause of
educational and occupational inequality, and perhaps also of health
inequalities. The leading alternative explanation is meritocratic social
mobility. Whilst meritocracy exists in modern capitalist economies, it
has been far more limited in most societies throughout history
(Smelser and Lipset 1966). On the other hand, assortative mating is likely to
be a cultural universal (Buss 1989). Thus, SGAM predicts that
genes-SES gradients should exist in all stratified societies. In fact,
people in many societies have believed that innate traits do vary by
social status.\footnote{The appendix has a selection of relevant historical quotations.} In future, it may be possible to directly test for
genetic differences across social status in ancient DNA samples.

Under SGAM, the association between SES and genetic variation depends on
economic and social institutions. Institutions that make wealth more
persistent across generations also increase the correlation between SES
and genetics. If so, then institutional differences may have long-run
effects over generations by altering the genes-SES gradient. There could
be hysteresis, with initial social differences cumulating over time via
their effect on genetic inequality. On the other hand, while lowering
the intergenerational transmission of wealth may eventually flatten the
genes-SES gradient, increases in the level of meritocracy paradoxically
make it steeper, suggesting a deep conflict between meritocracy and
egalitarianism (M. Young 1958; Markovits 2019). Lastly, the
structure of marriage markets also affects the gradient. Our empirical
analysis suggests that in relatively egalitarian Norway, income plays a
less important role in assortative mating than it does in the UK.
However, this is a loose comparison, and we see careful tests of
comparative statics across different contexts as an important challenge
for future work.

The broadest message of this paper is that genetics are a social
outcome. Both popular and scientific discourse often parse genetics as
``nature'', in opposition to ``nurture'' or ``environment'' (e.g. Chakravarti and Little 2003; Plomin 2019). This reflects the fact
that our individual genetic endowment is fixed at birth, affects our
body and brain through proximate biological mechanisms, and cannot be
changed by our social environment. But the idea that human genetics are
natural can be highly misleading. Humans inherit their genes from their
parents, along with other forms of inheritance such as economic and
cultural capital. Human parents, in turn, form spouse pairs and bear
children within social institutions. A person's genetic inheritance is a
social and historical fact about them, not just a fact of nature. As
Marx (1844) wrote, ``History is the true natural history of man''.

The theory of evolution suggests that two motivations are likely to be
central for fitness-maximizing organisms: acquiring material resources
so as to survive and raise offspring, and pursuing reproductive partners
who themselves have high fitness value. Arguably, these two motives
structure nearly all of human society. On this view, the genetics-SES
trade in marriage markets is the most basic trade there is. Genetic
endowments can be thought of as another form of capital, alongside
human, social and cultural capital: a resource to be sought, accumulated
and competed over. The analysis of this kind of capital is an exciting
area for further research.

\section{Acknowledgements}\label{acknowledgements}

The Norwegian Mother, Father and Child Cohort Study (MoBa) is a
population-based pregnancy cohort study conducted by the Norwegian
Institute of Public Health. Participants were recruited from all over
Norway from 1999-2008. The women consented to participation in 41\% of
the pregnancies. The cohort includes approximately 114,500 children,
95,200 mothers and 75,200 fathers. The current study is based on version
12 of the quality-assured data files released for research in 2019. The
establishment of MoBa and initial data collection was based on a license
from the Norwegian Data Protection Agency and approval from The Regional
Committees for Medical and Health Research Ethics. The MoBa cohort is
currently regulated by the Norwegian Health Registry Act. The current
study was approved by The Regional Committees for Medical and Health
Research Ethics.

We thank the Norwegian Institute of Public Health (NIPH) for generating
high-quality genomic data. This research is part of the HARVEST
collaboration, supported by the Research Council of Norway (\#229624). We
also thank the NORMENT Centre for providing genotype data, funded by the
Research Council of Norway (\#223273), South East Norway Health
Authorities and Stiftelsen Kristian Gerhard Jebsen. We further thank the
Center for Diabetes Research, the University of Bergen for providing
genotype data and performing quality control and imputation of the data
funded by the ERC AdG project SELECTionPREDISPOSED, Stiftelsen Kristian
Gerhard Jebsen, Trond Mohn Foundation, the Research Council of Norway,
the Novo Nordisk Foundation, the University of Bergen, and the Western
Norway Health Authorities.

This study was conducted using UK Biobank resources under application
number 40310.

AA is supported by the Foundation Volksbond Rotterdam and by ZonMw grant
849200011 from The Netherlands Organisation for Health Research and
Development.

Code to reproduce is available at
\url{https://github.com/hughjonesd/trading-genetics}.

\FloatBarrier

\newpage

\section{Appendix: for online publication}\label{appendix-for-online-publication}

\localtableofcontents
\clearpage

\subsection{Proofs}\label{proofs}

\begin{proof}[Proof of Proposition \ref{prop-couples-SGAM}]

By a change of variable, rewrite:

\[
\left( 
\begin{array}{c}
x_{1} \\ 
x_{2}%
\end{array}%
\right) \rightarrow \left( 
\begin{array}{c}
x_{1} \\ 
u%
\end{array}%
\right) \text{ where }u=\frac{ax_{1}+\left( 1-a\right) x_{2}}{\sqrt{%
a^{2}s^{2}+\left( 1-a\right) ^{2}S^{2}+2a\left( 1-a\right) \sigma }}=
\frac{ax_{1}+\left( 1-a\right) x_{2}}{\sigma_I}
\]
is the attractiveness rescaled to $\mathcal{N}(0, 1)$. Thus,
\[
\left( 
\begin{array}{c}
x_{1} \\ 
u
\end{array}
\right) = \left( 
\begin{array}{cc}
1 & 0 \\ 
a/\sigma_I & (1-a)/\sigma_I%
\end{array}%
\right) \left( 
\begin{array}{c}
x_{1} \\ 
x_{2}%
\end{array}%
\right). 
\]


Note that the means are still zero, but the covariance of $(x_1,u)$ is:

\begin{eqnarray*}
\mathbb{C}\left( 
\begin{array}{c}
x_{1} \\ 
u
\end{array}
\right) &=&\left( 
\begin{array}{cc}
1 & 0 \\ 
a/\sigma_I & \left( 1-a\right)/\sigma_I%
\end{array}%
\right) \left( 
\begin{array}{cc}
s^{2} & \sigma \\ 
\sigma & S^{2}%
\end{array}%
\right) \left( 
\begin{array}{cc}
1 & a/\sigma_I \\ 
0 & \left( 1-a\right)/\sigma_I%
\end{array}%
\right) \\
&=&\left( 
\begin{array}{cc}
s^{2} &  A \\ 
A & 1 
\end{array}%
\right) \allowbreak 
\end{eqnarray*}
where
\[
A=\frac{
as^{2}+\left( 1-a\right) \sigma 
}{
\sqrt{
a^{2}s^{2} + (1-a)^2S^{2}
+ 2a\left(1-a\right)\sigma 
}} = 
\frac{as^{2}+\left( 1-a\right) \sigma }{\sigma_I}.
\]

Under SGAM, individual $\left( 
\begin{array}{c}
x_{1} \\ 
u
\end{array} 
\right)$ is matched with $\left( 
\begin{array}{c}
y_{1} \\ 
v%
\end{array}%
\right)$ such that $u = v = t$.

The distribution of $t$ is 
$\mathcal{N}\left( 0,1\right)$. Therefore the vector $\left( 
\begin{array}{c}
x_{1} \\ 
y_{1} \\ 
t%
\end{array}%
\right)$ is normally distributed, with mean 0, and covariance
\[
\Sigma =\left( 
\begin{array}{ccc}
s^{2} & A^{2} & A \\ 
A^{2} & s^{2} & A \\ 
A & A & 1
\end{array}%
\right) \allowbreak 
\]

Finally, we are interested in 
\[
\left( 
\begin{array}{c}
x_{1} \\ 
x_{2} \\ 
y_{1} \\ 
y_{2}%
\end{array}%
\right) =\left( 
\begin{array}{ccc}
1 & 0 & 0 \\ 
-\frac{a}{1-a} & 0 & \frac{\sigma_I}{1-a} \\ 
0 & 1 & 0 \\ 
0 & -\frac{a}{1-a} & \frac{\sigma_I}{1-a}%
\end{array}%
\right) \left( 
\begin{array}{c}
x_{1} \\ 
y_{1} \\ 
t%
\end{array}%
\right) 
\]%
therefore again the means are 0 and 
\begin{eqnarray*}
\mathbb{C}\left( 
\begin{array}{c}
x_{1} \\ 
x_{2} \\ 
y_{1} \\ 
y_{2}%
\end{array}%
\right) &=&\left( 
\begin{array}{ccc}
1 & 0 & 0 \\ 
-\frac{a}{1-a} & 0 & \frac{\sigma_I}{1-a} \\ 
0 & 1 & 0 \\ 
0 & -\frac{a}{1-a} & \frac{\sigma_I}{1-a}%
\end{array}%
\right) \Sigma \left( 
\begin{array}{ccc}
1 & 0 & 0 \\ 
-\frac{a}{1-a} & 0 & \frac{\sigma_I}{1-a} \\ 
0 & 1 & 0 \\ 
0 & -\frac{a}{1-a} & \frac{\sigma_I}{1-a}%
\end{array}%
\right) ^{T} \\
&=&\allowbreak \left( 
\begin{array}{cccc}
s^{2} & \sigma & A^{2} & AC \\ 
\sigma & S^{2} & AC & C^{2} \\ 
A^{2} & AC & s^{2} & \sigma \\ 
AC & C^{2} & \sigma & S^{2}%
\end{array}%
\right) \allowbreak
\end{eqnarray*}%
where:%
\begin{eqnarray*}
A &=&\frac{as^{2}+\left( 1-a\right) \sigma }{\sigma_I}\text{ \ and} \\
C &=&\frac{a\sigma +\left( 1-a\right) S^{2}}{\sigma_I}.
\end{eqnarray*}

\end{proof}

\begin{lemma}\label{C2}
 $C^2 \le S^2$, with strict inequality if $a > 0$ and $\sigma < 1$.

\end{lemma}
\begin{proof}
Write
\begin{align*}
C^{2}   &= \frac{
            a^{2}\sigma^{2}+(1-a)^{2}S^{4}+2a(1-a)\sigma S^{2}
         }{
           a^{2}s^{2}+(1-a)^{2}S^{2}+2a(1-a)\sigma
         } \\
    &\le \frac{
        a^{2}s^{2}S^{2}+(1-a)^{2}S^{4}+2a(1-a)\sigma S^{2}
      }{
        a^{2}s^{2}+(1-a)^{2}S^{2}+2a(1-a)\sigma
      }
      \textrm{, since }\sigma/sS = Corr(x_1,x_2) \le 1 \\
    &= S^{2}
\end{align*}
and observe that the inequality is strict if $a > 0$ and $\sigma < sS$.
\end{proof}

\begin{proof}[Proof of Claim \ref{claim-E-children-RM}]

Under random matching, the joint distribution of $\left( \frac{\tau }{2}\left(
x_{1}+y_{1}\right) +\varepsilon ,x_{2},y_{2}\right) $ is normal with mean 
$( 0,0,0)$ and covariance
\[
\mathbb{C}=\left( 
\begin{array}{lcr}
\frac{\tau ^{2}}{2}\left( s^{2}+\sigma \right) +1 & \frac{\tau }{2}\sigma  & 
\frac{\tau }{2}\sigma  \\ 
\frac{\tau }{2}\sigma  & S^{2} & 0 \\ 
\frac{\tau }{2}\sigma  & 0 & S^{2}%
\end{array}%
\right) 
\]%
Using the matrix formula for the conditional mean of normal variables,
\begin{eqnarray*}
\mathbb{E}\left[ \frac{\tau }{2}\left( x_{1}+y_{1}\right) +\varepsilon \mid
x_{2}=v,y_{2}=w\right]  &=&\left( 
\begin{array}{rr}
\frac{\tau }{2}\sigma  & \frac{\tau }{2}\sigma 
\end{array}%
\right) \left( 
\begin{array}{rr}
S^{2} & 0 \\ 
0 & S^{2}%
\end{array}%
\right) ^{-1}\left( 
\begin{array}{r}
v \\ 
w%
\end{array}%
\right)  \\
&=&\frac{\sigma \tau }{2S^{2}}\left( v+w\right) 
\end{eqnarray*}

In particular, if $\sigma =0$, this expectation is equal to 0.
\end{proof}

\begin{proof}[Proof of Claim \ref{claim-E-children-SGAM}]

From \eqref{cov-children-SGAM}, the joint distribution of $\left( \frac{\tau }{2}\left(
x_{1}+y_{1}\right) +\varepsilon ,x_{2},y_{2}\right) $ is normal with mean $%
\left( 0,0,0\right) $ and covariance$\allowbreak $%
\[
\Sigma =\left( 
\begin{array}{rrr}
\frac{1}{2}\tau ^{2}\left( A^{2}+s^{2}\right) +1 & \frac{\tau }{2}\left(
\sigma +AC\right)  & \frac{\tau }{2}\left( \sigma +AC\right)  \\ 
\frac{\tau }{2}\left( \sigma +AC\right)  & S^{2} & C^{2} \\ 
\frac{\tau }{2}\left( \sigma +AC\right)  & C^{2} & S^{2}%
\end{array}%
\right) 
\]%
Therefore
\begin{eqnarray}
\mathbb{E}\left[ \frac{\tau }{2}\left( x_{1}+y_{1}\right) +\varepsilon \mid
x_{2}=v,y_{2}=w\right]  &=&\left( 
\begin{array}{rr}
\frac{\tau }{2}\left( \sigma +AC\right)  & \frac{\tau }{2}\left( \sigma
+AC\right) 
\end{array}%
\right) \left( 
\begin{array}{rr}
S^{2} & C^{2} \\ 
C^{2} & S^{2}%
\end{array}%
\right) ^{-1}\left( 
\begin{array}{r}
v \\ 
w%
\end{array}%
\right)   \nonumber \\
&=&\frac{1}{2}\tau \frac{\sigma +AC}{C^{2}+S^{2}}\left( v+w\right) 
\label{P1}
\end{eqnarray}%
In particular, if $\sigma =0$, we have%
\begin{eqnarray*}
A &=&\frac{as^{2}}{\sqrt{a^{2}s^{2}+\left( 1-a\right) ^{2}S^{2}}}\textrm{ \ and%
} \\
C &=&\frac{\left( 1-a\right) S^{2}}{\sqrt{a^{2}s^{2}+\left( 1-a\right)
^{2}S^{2}}},
\end{eqnarray*}%
and (\ref{P1}) becomes 
\begin{eqnarray*}
\mathbb{E}\left[ \frac{\tau }{2}\left( x_{1}+y_{1}\right) +\varepsilon \mid
x_{2}=v,y_{2}=w\right]  &=&\frac{1}{2}\tau \frac{a\left( 1-a\right) s^{2}}{%
a^{2}s^{2}+2\left( 1-a\right) ^{2}S^{2}}\left( v+w\right)  \\
&=&\frac{1}{2}\tau \frac{a\left( 1-a\right) \lambda }{a^{2}\lambda +2\left(
1-a\right) ^{2}}\left( v+w\right) 
\end{eqnarray*}
where $\lambda =s^{2}/S^{2}$ is the ratio of genetic variance to wealth variance.
The coefficient $\frac{a\left( 1-a\right) \lambda }{a^{2}\lambda +2\left(
1-a\right) ^{2}}$ is increasing, then decreasing in $a$ and is 0 for $a =  0$ or $a = 1$.
\end{proof}

\begin{proof}[Proof of Claim \ref{claim-corr-children-RM}]

Under random matching, the covariance matrix for children's characteristics is:%
\begin{eqnarray*}
\mathbb{C} &=&\left( 
\begin{array}{cc}
1 & 0 \\ 
0 & 1%
\end{array}%
\right) +\left( 
\begin{array}{cccc}
\frac{\tau }{2} & 0 & \frac{\tau }{2} & 0 \\ 
0 & \frac{\theta }{2} & 0 & \frac{\theta }{2}%
\end{array}%
\right) \allowbreak \left( 
\begin{array}{cccc}
s^{2} & \sigma & 0 & 0 \\ 
\sigma & S^{2} & 0 & 0 \\ 
0 & 0 & s^{2} & \sigma \\ 
0 & 0 & \sigma & S^{2}%
\end{array}%
\right) \allowbreak \left( 
\begin{array}{cc}
\frac{1}{2}\tau & 0 \\ 
0 & \frac{1}{2}\theta \\ 
\frac{1}{2}\tau & 0 \\ 
0 & \frac{1}{2}\theta%
\end{array}%
\right) \\
&=&\left( 
\begin{array}{cc}
\frac{1}{2}s^{2}\tau ^{2}+1 & \frac{1}{2}\theta \sigma \tau \\ 
\frac{1}{2}\theta \sigma \tau & \frac{1}{2}S^{2}\theta ^{2}+1%
\end{array}%
\right)
\end{eqnarray*}%
so that the correlation between characteristics for children is: 
\[
Corr\left( x_{1}^{\prime },x_{2}^{\prime }\right) =\frac{\frac{1}{2}\theta
\sigma \tau }{\sqrt{\frac{1}{2}\tau ^{2}s^{2}+1}\sqrt{\frac{1}{2}\theta
^{2}S^{2}+1}} 
\]

Note that $\sigma =0$ gives a zero correlation for children as well. 
Also, because $\theta < 1$ and $\tau < 1$, the correlation is less than the 
parents' correlation of $\sigma/sS$.

\end{proof}

\begin{proof}[Proof of Claim \ref{claim-corr-children-SGAM}]

Again applying \eqref{cov-children-SGAM}, under SGAM, the correlation between 
children's traits is:
\[
Corr\left( x_{1}^{\prime },x_{2}^{\prime }\right) =\frac{\frac{1}{2}\theta
\tau \left( \sigma +AC\right) }{\sqrt{\frac{1}{2}\tau ^{2}\left(
A^{2}+s^{2}\right) +1}\sqrt{\frac{1}{2}\theta ^{2}\left( C^{2}+S^{2}\right)
+1}} 
\]
This is positive if $\sigma = 0$ so long as $AC > 0$ i.e. $0 < a < 1$. To show
it is increasing in $\theta$, strip out constant terms and take the derivative of
\[
\frac{\theta}{\sqrt{\frac{1}{2}\theta^2(C^2+S^2) + 1}}
\]
The derivative is signed by
\begin{align*}
& \left(\frac{1}{2}\theta^2(C^2+S^2) + 1\right)^{0.5} -
\frac{1}{2}\theta^2(C^2+S^2)\left(\frac{1}{2}\theta^2(C^2+S^2) + 1\right)^{-0.5} \\
>& \left(\frac{1}{2}\theta^2(C^2+S^2) + 1\right)^{0.5} - 
\left(\frac{1}{2}\theta^2(C^2+S^2)  + 1\right)\left(\frac{1}{2}\theta^2(C^2+S^2) + 1\right)^{-0.5} \\
=&\ 0.
\end{align*}
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop-asymptotics-RM}]

The fixed point condition on the covariance matrix is
\[
\left( 
\begin{array}{cc}
s^{2} & \sigma  \\ 
\sigma  & S^{2}%
\end{array}%
\right) =\left( 
\begin{array}{cc}
\frac{1}{2}s^{2}\tau ^{2}+1 & \frac{1}{2}\theta \sigma \tau  \\ 
\frac{1}{2}\theta \sigma \tau  & \frac{1}{2}S^{2}\theta ^{2}+1%
\end{array}%
\right) 
\]
which gives
\[
s^{2}=\frac{2}{2-\tau ^{2}},S^{2}=\frac{2}{2-\theta ^{2}},\sigma =0.
\]

The asymptotic conditional expectation of children's genetics given parental SES is:

\[
\mathbb{E}\left[ \frac{\tau }{2}\left( x_{1}+y_{1}\right) +\varepsilon \mid
x_{2}=v,y_{2}=w\right] =0
\]%

since the traits $x_1,x_2,y_1,y_2$ are uncorrelated.

\end{proof}

\begin{proof}[Proof of Proposition \ref{prop-asymptotics-SGAM}]
Start by characterizing the invariant distribution. This must satisfy:%
\[
\left( 
\begin{array}{cc}
s^{2} & \sigma \\ 
\sigma & S^{2}%
\end{array}%
\right) =\left( 
\begin{array}{cc}
\frac{1}{2}\tau ^{2}\left( A^{2}+s^{2}\right) +1 & \frac{1}{2}%
\theta \tau \left( \sigma+AC\right)  \\ 
\frac{1}{2}\theta \tau \left( \sigma+AC\right)  & \frac{1}{%
2}\theta ^{2}\left( C^{2}+S^{2}\right) +1%
\end{array}%
\right) \allowbreak 
\]%
where%
\begin{eqnarray*}
A &=&\frac{as^{2}+\left( 1-a\right) \sigma}{\sqrt{a^{2}%
s^{2}+\left( 1-a\right) ^{2}S^{2}+2a\left( 1-a\right) \sigma%
}}\textrm{ \ and} \\
C &=&\frac{a\sigma+\left( 1-a\right) S^{2}}{\sqrt{a^{2}%
s^{2}+\left( 1-a\right) ^{2}S^{2}+2a\left( 1-a\right) \sigma%
}},
\end{eqnarray*}%

Note that if the distribution converges, $s^{2}$ and $S^{2}$
must be above 1. Also, for $A$ and $C$ to have a real-valued
solution, it must be that $a^{2}s^{2}+(1-a)^{2}S^{2}+2a(1-a)\sigma>0$;
using this,

\[
AC=\frac{a^{2}s^{2}\sigma+(1-a)^{2}S^{2}\sigma+a(1-a)(\sigma^{2}+S^{2}s^{2})}{a^{2}s^{2}+(1-a)^{2}S^{2}+2a(1-a)\sigma}>\sigma\textrm{, by }\sigma^{2}<S^{2}s^{2}.
\]

Since $AC>\sigma$, $\frac{\theta\tau}{2}(\sigma+AC)>\theta\tau\sigma$.
If $\sigma<0$ then $\sigma=\frac{\theta\tau}{2}(\sigma+AC)>\theta\tau\sigma>\sigma$,
a contradiction. Thus $\sigma\ge0$. Also if $\sigma=0$
then 
\[
\sigma=\frac{\theta\tau}{2}\left(\frac{a(1-a)(S^{2}s^{2})}{a^{2}s^{2}+(1-a)^{2}S^{2}}\right)
\]
which implies $a=0$ or $a=1$. This proves that $\sigma$ is
non-negative, and positive if $a\in(0,1)$, so long as the distribution
converges.

From the invariant distribution, first:

\[
\sigma\left(1-\frac{1}{2}\theta\tau\right)=\frac{\theta\tau}{2}\frac{\left(as^{2}+\left(1-a\right)\sigma\right)\left(a\sigma+\left(1-a\right)S^{2}\right)}{a^{2}s^{2}+\left(1-a\right)^{2}S^{2}+2a\left(1-a\right)\sigma}
\]
or
\[
\mu\left(1-\frac{1}{2}\theta\tau\right)=\frac{\theta\tau}{2}\frac{\left(a\lambda+\left(1-a\right)\mu\right)\left(a\mu+\left(1-a\right)\right)}{a^{2}\lambda+\left(1-a\right)^{2}+2a\left(1-a\right)\mu}
\]
where
\[
\lambda=s^{2}/S^{2}\text{ and }\mu=\sigma/S^{2}.
\]
Solving for $\lambda$ gives
\begin{equation}
\lambda=\frac{1-a}{a}\mu\frac{\left(-2a+4a\mu-2\theta\tau+2a\theta\tau-3a\theta\tau\mu+2\right)\allowbreak}{\left(1-a\right)\theta\tau-2a\mu\left(1-\theta\tau\right)}\label{L}
\end{equation}

Then
\begin{align*}
s^{2}\left(1-\frac{1}{2}\tau^{2}\right) & =\frac{1}{2}A^{2}\tau^{2}+1\\
S^{2}\left(1-\frac{1}{2}\theta^{2}\right) & =\frac{1}{2}C^{2}\theta^{2}+1
\end{align*}
give
\begin{align*}
s^{2}\left(1-\frac{1}{2}\tau^{2}\right)-\frac{1}{2}A^{2}\tau^{2} & =S^{2}\left(1-\frac{1}{2}\theta^{2}\right)-\frac{1}{2}C^{2}\theta^{2}\text{, therefore}\\
\lambda\left(1-\frac{1}{2}\tau^{2}\right)-\frac{1}{2}\frac{A^{2}}{S^{2}}\tau^{2} & =\left(1-\frac{1}{2}\theta^{2}\right)-\frac{1}{2}\frac{C^{2}}{S^{2}}\theta^{2}
\end{align*}

Here
\begin{align*}
\frac{A^{2}}{S^{2}} & =\frac{\left(a\lambda+\left(1-a\right)\mu\right)^{2}}{D}\text{ \ and}\\
\frac{C^{2}}{S^{2}} & =\frac{\left(a\mu+\left(1-a\right)\right)^{2}}{D},\\
D & =a^{2}\lambda+\left(1-a\right)^{2}+2a\left(1-a\right)\mu
\end{align*}
which give a quadratic equation in $\mu$:
\[
\lambda\left(1-\frac{1}{2}\tau^{2}\right)D-\frac{1}{2}\left(a\lambda+\left(1-a\right)\mu\right)^{2}\tau^{2}-\left(1-\frac{1}{2}\theta^{2}\right)D+\frac{1}{2}\left(a\mu+\left(1-a\right)\right)^{2}\theta^{2}=0
\]
Plugging in $\lambda$ given by (\ref{L}), this can be rewritten
to
\[
F\left(\mu\right)=\frac{\left(1-a+a\mu\right)^{2}}{a}\frac{N\left(\mu\right)}{D\left(\mu\right)}=0
\]
where
\[
N\left(\mu\right)=X\mu^{2}+Y\mu+Z,\textrm{ with }X,Y,Z\textrm{ polynomials in }a,\theta,\tau
\]
and
\[
D\left(\mu\right)=\left(\theta\tau\left(1-a\right)-2a\mu\left(1-\theta\tau\right)\right)^{2}
\]

One can check that the discriminant is always positive. Therefore
this has two solutions (not shown), of which only one is acceptable
(it goes to the exact solution when the coefficient of $\mu^{2}$
goes to 0). Writing 
\[
\mu=\phi_{1}(a,\theta,\tau)
\]
 for this solution:
\[
\lambda=\psi\left(a,\theta,\tau\right)=\frac{1-a}{a}\phi_{1}\left(a,\theta,\tau\right)\allowbreak\frac{\left(-2a+4a\phi_{1}\left(a,\theta,\tau\right)-2\theta\tau+2a\theta\tau-3a\theta\tau\phi_{1}\left(a,\theta,\tau\right)+2\right)}{\left(1-a\right)\theta\tau-2a\phi_{1}\left(a,\theta,\tau\right)\left(1-\theta\tau\right)}.
\]

Finally
\begin{align*}
S^{2} & =\frac{1}{1-\frac{1}{2}\theta^{2}-\frac{1}{2}\frac{C^{2}}{S^{2}}\theta^{2}}\text{ where}\\
\frac{C^{2}}{S^{2}} & =\frac{\left(a\phi_{1}\left(a,\theta,\tau\right)+\left(1-a\right)\right)^{2}}{a^{2}\psi\left(a,\theta,\tau\right)+\left(1-a\right)^{2}+2a\left(1-a\right)\phi_{1}\left(a,\theta,\tau\right)}
\end{align*}
and
\begin{align*}
s^{2} & =\lambda S^{2}=\frac{\psi\left(a,\theta,\tau\right)}{1-\frac{1}{2}\theta^{2}-\frac{1}{2}\frac{C^{2}}{S^{2}}\theta^{2}};\\
\sigma & =\mu S^{2}=\frac{\phi_{1}\left(a,\theta,\tau\right)}{1-\frac{1}{2}\theta^{2}-\frac{1}{2}\frac{C^{2}}{S^{2}}\theta^{2}}.
\end{align*}

Conditional expectations of children's genetics given parents' wealth under SGAM
are calculated using the same formula as before, plugging in moments of the 
asymptotic distribution:

\[
\mathbb{E}\left[ \frac{\tau }{2}\left( x_{1}+y_{1}\right) +\varepsilon \mid
x_{2}=v,y_{2}=w\right]  = \frac{1}{2}\tau \frac{\sigma+AC%
}{C^{2}+S^{2}}\left( v+w\right)  
\]

\end{proof}

\begin{proof}[Proof of Claim \ref{claim-men-women-different}]

Since men and women have different distributions of attractiveness,
we have to match them by quantiles of their respective distributions. 
Men's and women's attractiveness are distributed
\begin{align*}
N(0,\sigma_{I}^{2})\textrm{ where }\sigma_{I} & =\sqrt{a^{2}s^{2}+(1-a)^{2}S^{2}+2a(1-a)\sigma};\\
N(0,\sigma_{J}^{2})\textrm{ where }\sigma_{J} & =\sqrt{b^{2}s^{2}+(1-b)^{2}S^{2}+2b(1-b)\sigma}.
\end{align*}
Thus, men with normalized attractiveness $i(x)/\sigma_{I}$ match
women with normalized attractiveness $j(y)/\sigma_{J}$.

Change variables so that 
\begin{align*}
\left(\begin{array}{c}
x_{1}\\
x_{2}
\end{array}\right) & \rightarrow\left(\begin{array}{c}
x_{1}\\
u
\end{array}\right)\textrm{ where }u=\frac{ax_{1}+(1-a)x_{2}}{\sigma_{I}};\\
\left(\begin{array}{c}
y_{1}\\
y_{2}
\end{array}\right) & \rightarrow\left(\begin{array}{c}
y_{1}\\
v
\end{array}\right)\textrm{ where }v=\frac{by_{1}+(1-b)y_{2}}{\sigma_{J}}.
\end{align*}

Thus 
\begin{align*}
\left(\begin{array}{c}
x_{1}\\
u
\end{array}\right) & =\left(\begin{array}{cc}
1 & 0\\
a/\sigma_{I} & (1-a)/\sigma_{I}
\end{array}\right)\left(\begin{array}{c}
x_{1}\\
x_{2}
\end{array}\right);\\
\left(\begin{array}{c}
y_{1}\\
v
\end{array}\right) & =\left(\begin{array}{cc}
1 & 0\\
b/\sigma_{J} & (1-b)/\sigma_{J}
\end{array}\right)\left(\begin{array}{c}
y_{1}\\
y_{2}
\end{array}\right).
\end{align*}

and their respective covariance matrices are
\begin{align*}
\mathbb{C}\left(\begin{array}{c}
x_{1}\\
u
\end{array}\right) & =\left(\begin{array}{cc}
1 & 0\\
a/\sigma_{I} & (1-a)/\sigma_{I}
\end{array}\right)\left(\begin{array}{cc}
s^{2} & \sigma\\
\sigma & S^{2}
\end{array}\right)\left(\begin{array}{cc}
1 & a/\sigma_{I}\\
0 & (1-a)/\sigma_{I}
\end{array}\right)\\
 & =\left(\begin{array}{cc}
s^{2} & A \\
A & 1
\end{array}\right), \textrm{ where } A=\frac{as^{2}+(1-a)\sigma}{\sigma_{I}};
\end{align*}

similarly
\[
\mathbb{C}\left(\begin{array}{c}
y_{1}\\
v
\end{array}\right)=\left(\begin{array}{cc}
s^{2} & B \\
B & 1
\end{array}\right), \textrm{ where } B=\frac{bs^{2}+(1-b)\sigma}{\sigma_{J}}.
\]

Under SGAM, couples have characteristics $\left(\begin{array}{c}
x_{1}\\
t\\
y_{1}\\
t
\end{array}\right),$ where $\left(\begin{array}{c}
x_{1}\\
y_{1}\\
t
\end{array}\right)$ is trivariate normal with mean 0 and covariance matrix
\[
\Sigma=\left(\begin{array}{ccc}
s^{2} & AB & A\\
AB & s^{2} & B\\
A & B & 1
\end{array}\right)
% XXX do I need to expand Pierre's "just note" proof in Computations5short?
\]


Lastly, we calculate the covariance matrix of couples' original characteristics.
Since
\[
\left(\begin{array}{c}
x_{1}\\
x_{2}\\
y_{1}\\
y_{2}
\end{array}\right)=\left(\begin{array}{ccc}
1 & 0 & 0\\
\frac{-a}{1-a} & 0 & \frac{\sigma_{I}}{1-a}\\
0 & 1 & 0\\
0 & \frac{-b}{1-b} & \frac{\sigma_{J}}{1-b}
\end{array}\right)\left(\begin{array}{c}
x_{1}\\
y_{1}\\
t
\end{array}\right)
\]
we have that the mean is again 0 and the covariance matrix is 
\begin{align*}
\mathbb{C}\left(\begin{array}{c}
x_{1}\\
x_{2}\\
y_{1}\\
y_{2}
\end{array}\right) & =\left(\begin{array}{ccc}
1 & 0 & 0\\
\frac{-a}{1-a} & 0 & \frac{\sigma_{I}}{1-a}\\
0 & 1 & 0\\
0 & \frac{-b}{1-b} & \frac{\sigma_{J}}{1-b}
\end{array}\right)\Sigma\left(\begin{array}{cccc}
1 & \frac{-a}{1-a} & 0 & 0\\
0 & 0 & 1 & \frac{-b}{1-b}\\
0 & \frac{\sigma_{I}}{1-a} & 0 & \frac{\sigma_{J}}{1-b}
\end{array}\right)\\
 & =\left(\begin{array}{cccc}
s^{2} & \sigma & AB & AD\\
\sigma & S^{2} & BC & CD\\
AB & BC & s^{2} & \sigma\\
AD & CD & \sigma & S^{2}
\end{array}\right)
\end{align*}
where
\[
C=\frac{a\sigma+(1-a)S^{2}}{\sigma_{I}};D=\frac{b\sigma+(1-b)S^{2}}{\sigma_{J}}.
\]

From the above and \eqref{Chil} we can calculate the
covariance matrix of children's characteristics as
\begin{align*}
\mathbb{C}\left(\begin{array}{c}
x_{1}^{\prime}\\
x_{2}^{\prime}
\end{array}\right) & =\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)+\left(\begin{array}{cccc}
\frac{\tau}{2} & 0 & \frac{\tau}{2} & 0\\
0 & \frac{\theta}{2} & 0 & \frac{\theta}{2}
\end{array}\right)\left(\begin{array}{cccc}
s^{2} & \sigma & AB & AD\\
\sigma & S^{2} & BC & CD\\
AB & BC & s^{2} & \sigma\\
AD & CD & \sigma & S^{2}
\end{array}\right)\left(\begin{array}{cc}
\frac{\tau}{2} & 0\\
0 & \frac{\theta}{2}\\
\frac{\tau}{2} & 0\\
0 & \frac{\theta}{2}
\end{array}\right)\\
 & =\left(\begin{array}{cc}
\frac{\tau^{2}}{2}(s^{2}+AB)+1 & \frac{\tau\theta}{4}(2\sigma+AD+BC)\\
\frac{\tau\theta}{4}(2\sigma+AD+BC) & \frac{\theta^{2}}{2}(S^{2}+CD)+1
\end{array}\right).
\end{align*}

Thus $x_{1}^{\prime}$ and $x_{2}^{\prime}$ will be positively correlated
if $2\sigma+AD+BC>0$. This is always positive if $\sigma>0$; if $\sigma=0$ 
it reduces to
\[
\frac{(a+b-2ab)s^{2}S^{2}}{\sigma_{I}\sigma_{J}}
\]
which is positive unless $a=b=0$ or $a=b=1$. The correlation is
\[
\frac{
\frac{\tau\theta}{4}(2\sigma+AD+BC)
}{
\sqrt{\frac{\tau^{2}}{2}(s^{2}+AB)+1}
\sqrt{\frac{\theta^{2}}{2}(S^{2}+CD)+1}
}
\]
and taking the derivative shows it is increasing in $\theta$, as in the proof 
for Claim \ref{claim-corr-children-SGAM}.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop-gamma}]
Write
\begin{align*}
x_{1}^{\prime}  &=\tau\frac{x_{1}+y_{1}}{2}+\varepsilon \\
x_{2}^{\prime}  &=\gamma x_{1}^{\prime}+\theta\frac{x_{2}+y_{2}}{2}+\eta \\
                  &=\gamma\tau \frac{x_{1}+y_{1}}{2} + \theta \frac{x_{2}+y_{2}}{2} 
                    + \eta + \gamma\varepsilon
\end{align*}

Since
\[
\left(\begin{array}{c}
\tau\frac{x_{1}+y_{1}}{2}\\
\gamma\tau\frac{x_{1}+y_{1}}{2}+\theta\frac{x_{2}+y_{2}}{2}
\end{array}\right)=\left(\begin{array}{cccc}
\frac{\tau}{2} & 0 & \frac{\tau}{2} & 0\\
\frac{\gamma\tau}{2} & \frac{\theta}{2} & \frac{\gamma\tau}{2} & \frac{\theta}{2}
\end{array}\right)\left(\begin{array}{c}
x_{1}\\
x_{2}\\
y_{1}\\
y_{2}
\end{array}\right)
\]
we can use \eqref{cov-couples-SGAM} to derive the covariance matrix for children:

\begin{align*}
\mathbb{C}  &= \left(\begin{array}{cc}
1 & \gamma\\
\gamma & \ 1+\gamma^{2}
\end{array}\right)+\left(\begin{array}{cccc}
\frac{\tau}{2} & 0 & \frac{\tau}{2} & 0\\
\frac{\gamma\tau}{2} & \frac{\theta}{2} & \frac{\gamma\tau}{2} & \frac{\theta}{2}
\end{array}\right)\left(\begin{array}{cccc}
s^{2} & \sigma & A^{2} & AC\\
\sigma & S^{2} & AC & C^{2}\\
A^{2} & AC & s^{2} & \sigma\\
AC & C^{2} & \sigma & S^{2}
\end{array}\right)\left(\begin{array}{cc}
\frac{\tau}{2} & \frac{\gamma\tau}{2}\\
0 & \frac{\theta}{2}\\
\frac{\tau}{2} & \frac{\gamma\tau}{2}\\
0 & \frac{\theta}{2}
\end{array}\right) \\
    &=\left(\begin{array}{cc}
\frac{\tau^{2}}{2}(s^{2}+A^{2})+1 & \frac{\gamma\tau^{2}}{2}(s^{2}+A^{2})+\frac{\tau\theta}{2}(\sigma+AC)+\gamma\\
\frac{\gamma\tau^{2}}{2}(s^{2}+A^{2})+\frac{\tau\theta}{2}(\sigma+AC)+\gamma & \ \ \frac{\gamma^{2}\tau^{2}}{2}(s^{2}+A^{2})+\gamma\tau\theta(\sigma+AC)+\frac{\theta^{2}}{2}(S^{2}+C^{2})+1+\gamma^{2}
\end{array}\right)
\end{align*}

The first claim in the proof follows from the covariance:
\[
\frac{\gamma\tau^{2}}{2}(s^{2}+A^{2})+\frac{\tau\theta}{2}(\sigma+AC)+\gamma
\]
This is increasing in $\gamma$, and positive if any of $\sigma > 0$, 
$\gamma > 0$,  or $AC > 0$ (which holds if $0 < a < 1$ when $\sigma = 0$).

The correlation $Cov(x_{1}^{\prime},x_{2}^{\prime})/\sqrt{Var(x_{1}^{\prime})Var(x_{2}^{\prime})}$
is proportional to
\[
\frac{\gamma p+q}{\sqrt{\gamma^{2}p+\gamma2q+r}}
\]
where 
\begin{align*}
p & =\tau^{2}(s^{2}+A^{2})+2;\\
q & =\tau\theta(\sigma+AC);\\
r & =\theta^{2}(S^{2}+C^{2})+2.
\end{align*}
The derivative of this with respect to $\gamma$ is signed by $pr-q^{2}$, 
which equals
\begin{align*}
 & [\tau^{2}(s^{2}+A^{2})+2][\theta^{2}(S^{2}+C^{2})+2]-[\tau\theta(\sigma+AC)]^{2}\\
= & \tau^{2}\theta^{2}(s^{2}S^{2}+A^{2}S^{2}+s^{2}C^{2}-2\sigma AC-\sigma^{2})+2[\theta^{2}(S^{2}+C^{2})+\tau^{2}(s^{2}+A^{2})]+4
\end{align*}
The last two terms are positive. In the first term, $\sigma^{2}<s^{2}S^{2}$,
and
\begin{align*}
0 & <(AS-Cs)^{2}\\
 & =A^{2}S^{2}+C^{2}s^{2}-2ACSs\\
 & <A^{2}S^{2}+C^{2}s^{2}-2\sigma AC\textrm{ , again using }\sigma<sS.\\
\end{align*}

Hence the whole sum is positive.


Now we can calculate
\[
\mathbb{E}[x_{2}^{\prime}|x_{2}+y_{2}]
\]

using
\[
\left(\begin{array}{c}
\gamma\tau\frac{x_{1}+y_{1}}{2}+\theta\frac{x_{2}+y_{2}}{2}\\
x_{2}\\
y_{2}
\end{array}\right)=\left(\begin{array}{cccc}
\frac{\gamma\tau}{2}0 & \frac{\theta}{2}1 & \frac{\gamma\tau}{2}0 & \frac{\theta}{2}0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1
\end{array}\right)\left(\begin{array}{c}
x_{1}\\
x_{2}\\
y_{1}\\
y_{2}
\end{array}\right)
\]
to give
{\small\begin{align*}
\mathbb{C}\left(\begin{array}{c}
x_{2}^{\prime}\\
x_{2}\\
y_{2}
\end{array}\right) & =\left(\begin{array}{cccc}
\frac{\gamma\tau}{2} & \frac{\theta}{2} & \frac{\gamma\tau}{2} & \frac{\theta}{2}\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1
\end{array}\right)\left(\begin{array}{cccc}
s^{2} & \sigma & A^{2} & AC\\
\sigma & S^{2} & AC & C^{2}\\
A^{2} & AC & s^{2} & \sigma\\
AC & C^{2} & \sigma & S^{2}
\end{array}\right)\left(\begin{array}{ccc}
\frac{\gamma\tau}{2} & 0 & 0\\
\frac{\theta}{2} & 1 & 0\\
\frac{\gamma\tau}{2} & 0 & 0\\
\frac{\theta}{2} & 0 & 1
\end{array}\right)+\left(\begin{array}{ccc}
1+\gamma^{2} & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}\right)\\
 & =\left(\begin{array}{ccc}
\ldots & \frac{\gamma\tau}{2}(\sigma+AC)+\frac{\theta}{2}(S^{2}+C^{2}) & \frac{\gamma\tau}{2}(\sigma+AC)+\frac{\theta}{2}(S^{2}+C^{2})\\
\frac{\gamma\tau}{2}(\sigma+AC)+\frac{\theta}{2}(S^{2}+C^{2}) & S^{2} & C^{2}\\
\frac{\gamma\tau}{2}(\sigma+AC)+\frac{\theta}{2}(S^{2}+C^{2}) & C^{2} & S^{2}
\end{array}\right).
\end{align*}}%

Next
\begin{align*}
\mathbb{E}[x_{2}^{\prime}|x_{2},y_{2}] & =\left(\begin{array}{cc}
\frac{\gamma\tau}{2}(\sigma+AC)+\frac{\theta}{2}(S^{2}+C^{2}) & \frac{\gamma\tau}{2}(\sigma+AC)+\frac{\theta}{2}(S^{2}+C^{2})\end{array}\right)\left(\begin{array}{cc}
S^{2} & C^{2}\\
C^{2} & S^{2}
\end{array}\right)^{-1}\left(\begin{array}{c}
x_{2}\\
y_{2}
\end{array}\right)\\
 & =\left(\gamma\tau\frac{\sigma+AC}{S^{2}+C^{2}}+\theta\right)\frac{x_{2}+y_{2}}{2}
\end{align*}

So long as $0<a<1$ or $\sigma>0$, the coefficient on parents' wealth is thus 
higher than $\theta$.

\end{proof}

\begin{proof}[Proof of Proposition \ref{prop-non-normal}]
Without loss of generality let $Ex_1 = Ex_2 = 0$. The correlation is signed by
the covariance. Write $K$ for the set of couples in the parents' generation with
typical member $k = (x,y)$. Without loss of generality let $x_1 \ge y_1$. Then,
since the iso-attractiveness curves defined by $f$ are downward-sloping, 
$x_2 \le y_2$. (If $a = 1$ then $x_1 = y_1$; pick $x$ so that $x_2 \le y_2$.) Also,
for $a \in (0,1)$, if $x_1 > y_1$ then $x_2 < y_2$.

Since $Ex_1 = Ex_2 = 0$, the covariance among the parents' generation is

\[
\int_K (x_1x_2 + y_1y_2)/2 \ dk
\]

Write

\begin{align*}
x^\prime_1 =& \tau x^*_1 + \varepsilon & \textrm{where } x^*_1 =& (x_1+y_1)/2  \\
x^\prime_2 =& \theta x^*_2 + \eta & \textrm{where } x^*_2 =& (x_2+y_2)/2  
\end{align*}

and write the children's covariance as
\[
Cov(x^\prime_1, x^\prime_2) = Cov(\tau x^*_1, \theta x^*_2) + Cov(\tau x^*_1, \eta) + 
  Cov(\varepsilon, \theta x^*_2) + Cov(\varepsilon, \eta).
\]
By independence of the shocks, the last 3 terms are zero. So we need to
show that
\[
Cov(\tau x^*_1, \theta x^*_2) = \tau\theta Cov(x^*_1, x^*_2) > 0
\]

Write
\[
Cov(x^*_1, x^*_2) = \int_K x^*_1 x^*_2 \ dk
\]
using that $Ex^*_1 = Ex^*_2 = 0$. 

Take a typical parent, and write
\begin{align*}
x_1x_2 = (x^*_1 - \Delta_1)(x^*_2 - \Delta_2) \\
y_1y_2 = (x^*_1 + \Delta_1)(x^*_2 + \Delta_2)
\end{align*}
where 
\[
\Delta_1 = (x_1 - y_1)/2;\ \Delta_2 = (x_2 - y_2)/2.
\]
By assumption $\Delta_1 \ge 0$ and $\Delta_2 \le 0$. Furthermore, if $a \in (0,1)$, then
for a set of positive measure, $\Delta_1 > 0$ and $\Delta_2 < 0$, by our assumption 
that not all matching couples are identical.

Taking the average of the parents gives
\[
(x_1x_2 + y_1y_2)/2 = x^*_1x^*_2 + \Delta_1\Delta_2
\]
and if $a \in (0,1)$, this is strictly less than $x^*_1x^*_2$ for a set of 
positive measure. Plugging this into the integral gives

\[
Cov(x_1,x_2) \le Cov(x^*_1, x^*_2) = \int_K x^*_1 x^*_2 \ dk
\]
with strict inequality if $a \in (0,1)$. Since the parental covariance was 0
by assumption, this completes the proof.

\end{proof}

\FloatBarrier

\newpage

\subsection{More empirical results}\label{more-empirical-results}

Figure \ref{fig:pic-basic-corr-moba} redoes Figure
\ref{fig:pic-basic-corr} for the MoBa data, plotting measures of
individual SES against spouse PSEA. While university attendance gives
results similar to the UK, the relationship between income decile and
spouse PSEA is interestingly nonlinear in both sexes. Some low-income
individuals may be out of the labour market, either in continuing
education or as a stay-at-home spouse.

Table \ref{tab:tbl-bo-first-stage-moba} shows regressions of birth
order on mediators for the MoBa data. Birth order predicts all the
mediators significantly and with large effect sizes, except for BMI.

Table \ref{tab:tbl-bo-psea-basic-moba} regresses birth order on spouse
PSEA for the MoBa data, starting with a simple bivariate regression and
adding controls as in Table \ref{tab:tbl-bo-psea-basic}. Here, the
negative effect of birth order is highly significant even before we
control for parental age. As in Britain, the effect is greatly increased
when we control for parental age.

Table \ref{tab:tbl-bo-psea-no-par-age} reruns our central regressions
in the UK, dropping the control for parents' age at birth. Results show
the same pattern as in the main text: the coefficient for birth order is
negative, but changes sign when university attendance is added as a
potential mediator. However, the birth order effect is smaller overall,
and is never significant. We also ran regressions using father's age
only: results are similar to those in the main text.

Tables \ref{tab:tbl-bo-psea-dummies} and
\ref{tab:tbl-bo-psea-dummies-moba} rerun our central regressions
estimating a separate coefficient for each position in the birth order
(with firstborn as the baseline). The basic pattern of our main result
is remarkably robust, in both Great Britain and Norway: birth order
coefficients are generally negative, and adding mediators always causes
them to increase towards zero or to change sign.

We also ran a specification with separate birth order dummies within
each family size. Figure \ref{fig:pic-bo-psea-interactions} shows 95\%
confidence intervals for the birth order coefficients, from the column 2
specification including height and IQ controls but no mediators. Not
surprisingly, coefficients are imprecisely estimated. But most birth
order coefficients are negative compared to the baseline for firstborns.

Table \ref{tab:tbl-bo-psea-weights} re-estimates Table
\ref{tab:tbl-bo-psea} using weights from Alten et al. (2022).
These weight the UK Biobank sample to match its sampling frame of 40-69
year olds living close to 22 assessment centres. Although these weights
probably bring the sample closer to the UK population, the sampling
frame is still not representative of that population: for instance,
urban areas are oversampled. Results are similar to those in the main
text, although birth order coefficients are absolutely larger in all the
specifications.

Table \ref{tab:tbl-bo-psea-pgs} reruns our regressions controlling for
several polygenic scores. Results are very close to those in the main
text.

Table \ref{tab:tbl-bo-psea-age-fte} and
\ref{tab:tbl-bo-psea-age-fte-moba} rerun our regressions using age of
leaving full-time education as a measure of educational SES, instead of
the university attendance dummy. Results are similar to those in the
main text: controlling for age of leaving full-time education shrinks
the effect of birth order substantially.

Table \ref{tab:tbl-bo-psea-no3} reruns Table \ref{tab:tbl-bo-psea}
excluding families of size 3. Results are very similar to those in the
main text.

Lastly, we run an Oster (2019) style robustness analysis on
the effect of education and income on spouse PSEA. This is a
formalization of the informal idea that if a coefficient does not change
much when a known set of controls is added, it may be robust to other,
unobserved controls. The method works by comparing a ``short'' regression,
without alternative controls like BMI and height, to a ``medium''
regression with those controls. An important input is the maximum \(R^2\)
of independent variables on the dependent variable. Here, we use our
knowledge about the noise in measured PSEA. We take the ratio of the
\(R^2\) of measured PSEA on education to the maximum \(R^2\) of all genetic
variables on education, a.k.a. the heritability. We assume that this
ratio \(s\) gives the proportion of ``signal'' in measured PSEA, with the
rest being noise (from sampling error in the construction of the
polygenic score). We then make an assumption about the maximum \(R^2\) of
any own variable on spouse's true PSEA - i.e., about how random the
spouse matching process is. We take this to be 50\%. Multiplying \(s\) by
50\% gives the maximum possible \(R^2\) of own variables on spouse's
measured PSEA. The output of the analysis is a value \(\delta^*\)
representing the degree of selection on unobservables relative to
observables (with respect to the treatment variable) that would be
necessary to eliminate the effect of the independent variable. A
\(\delta^*\) of about 1 is considered a reasonable threshold for
robustness, if we assume that measured control variables ought to be at
least as important as unmeasured ones.

In UK Biobank, we calculate the maximum \(R^2\) as 0.056.\footnote{In other regressions of spouse traits on PSEA, \(R^2\) are indeed
  below this value (Hugh-Jones et al. 2016; Okbay et al. 2022).} For
university attendance, \(\delta^*\) is 0.941, and for income in
first job \(\delta^*\) is 2.03. In MoBa, we calculate the maximum
\(R^2\) as 0.131.\footnote{The MoBa number is larger because the \(R^2\) of EA4 on education
  is higher than for EA3.} For university attendance, \(\delta^*\) is
0.99, and for income at age 30, \(\delta^*\) is
0.275. These results confirm that the results on university
attendance are relatively robust, except for results on income in MoBa.
Of course, this technique depends crucially on the input assumptions,
especially about noise in measured PSEA, the true heritability, and the
maximum possible \(R^2\) in spouse matching.

\FloatBarrier

\begin{figure}

{\centering \subfloat[\label{fig:pic-basic-corr-moba-1}]{\includegraphics{trading-genetics_files/figure-latex/pic-basic-corr-moba-1} }\subfloat[\label{fig:pic-basic-corr-moba-2}]{\includegraphics{trading-genetics_files/figure-latex/pic-basic-corr-moba-2} }\newline\subfloat[\label{fig:pic-basic-corr-moba-3}]{\includegraphics{trading-genetics_files/figure-latex/pic-basic-corr-moba-3} }\subfloat[\label{fig:pic-basic-corr-moba-4}]{\includegraphics{trading-genetics_files/figure-latex/pic-basic-corr-moba-4} }

}

\caption{Spouse polygenic score for educational attainment (PSEA) against own university attendance and own income at age 30 (Norway). Lines show 95\% confidence intervals. PSEA is normalized to have mean 0 and variance 1.}\label{fig:pic-basic-corr-moba}
\end{figure}

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-first-stage-moba} Regressions of mediators on birth order (Norway)}
 \setlength{\tabcolsep}{0pt}
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\end{threeparttable}\par\end{centerbox}

\end{table}
 

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-basic-moba} Regressions of spouse PSEA on birth order (Norway)}
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\end{threeparttable}\par\end{centerbox}

\end{table}
 

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-no-par-age} Regressions of spouse PSEA, without controls for parents' age at respondent's birth (Great Britain)}
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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{>{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}-}
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\hhline{}
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\end{tabular}
\end{threeparttable}\par\end{centerbox}

\end{table}
 

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-dummies} Regressions of spouse PSEA, separate birth order dummies (Great Britain)}
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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
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\hhline{>{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}-}
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\hhline{}
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\end{tabularx}
\end{threeparttable}\par\end{centerbox}

\end{table}
 

 
  \providecommand{\huxb}[2]{\arrayrulecolor[RGB]{#1}\global\arrayrulewidth=#2pt}
  \providecommand{\huxvb}[2]{\color[RGB]{#1}\vrule width #2pt}
  \providecommand{\huxtpad}[1]{\rule{0pt}{#1}}
  \providecommand{\huxbpad}[1]{\rule[-#1]{0pt}{#1}}

\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-dummies-moba} Regressions of spouse PSEA, separate birth order dummies (Norway)}
 \setlength{\tabcolsep}{0pt}
\begin{tabularx}{1\textwidth}{p{0.2\textwidth} p{0.2\textwidth} p{0.2\textwidth} p{0.2\textwidth} p{0.2\textwidth}}


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\hhline{}
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\hhline{}
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\end{threeparttable}\par\end{centerbox}

\end{table}
 

\begin{figure}

{\centering \includegraphics{trading-genetics_files/figure-latex/pic-bo-psea-interactions-1} 

}

\caption{Regressions of spouse PSEA: birth order dummies within different family sizes (Great Britain). Labels show birth order. Lines are 95 per cent confidence intervals. The omitted category is birth order 1.}\label{fig:pic-bo-psea-interactions}
\end{figure}

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-weights} Regressions of spouse PSEA, weighted to match 
                   UK Biobank sampling frame (Great Britain)}
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\end{tabular}
\end{threeparttable}\par\end{centerbox}

\end{table}
 

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-pgs} Regressions of spouse PSEA with controls for polygenic 
                   scores (Great Britain)}
 \setlength{\tabcolsep}{0pt}
\begin{tabularx}{1\textwidth}{p{0.2\textwidth} p{0.2\textwidth} p{0.2\textwidth} p{0.2\textwidth} p{0.2\textwidth}}


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\end{tabularx}
\end{threeparttable}\par\end{centerbox}

\end{table}
 

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-age-fte} Regressions of spouse PSEA using age of leaving full-time 
                   education (Great Britain)}
 \setlength{\tabcolsep}{0pt}
\begin{tabularx}{0.9\textwidth}{p{0.225\textwidth} p{0.225\textwidth} p{0.225\textwidth} p{0.225\textwidth}}


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\hhline{}
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\end{tabularx}
\end{threeparttable}\par\end{centerbox}

\end{table}
 

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-age-fte-moba} Regressions of spouse PSEA using age of leaving full-time 
                   education (Norway)}
 \setlength{\tabcolsep}{0pt}
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\hhline{}
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\end{tabularx}
\end{threeparttable}\par\end{centerbox}

\end{table}
 

 
  \providecommand{\huxb}[2]{\arrayrulecolor[RGB]{#1}\global\arrayrulewidth=#2pt}
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-no3} Regressions of spouse PSEA, excluding family size 3 
                   (Great Britain)}
 \setlength{\tabcolsep}{0pt}
\begin{tabular}{l l l l l}


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\hhline{}
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\hhline{}
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\hhline{}
\arrayrulecolor{black}

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\multicolumn{1}{r!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\raggedleft \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont \hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}} \hspace{6pt}\huxbpad{6pt}} &
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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\multicolumn{1}{r!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\raggedleft \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont \hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}} \hspace{6pt}\huxbpad{6pt}} &
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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\multicolumn{1}{r!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\raggedleft \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont (0.0337)\hphantom{0}\hphantom{0}} \hspace{6pt}\huxbpad{6pt}} &
\multicolumn{1}{r!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\raggedleft \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont (0.0337)\hphantom{0}\hphantom{0}\hphantom{0}} \hspace{6pt}\huxbpad{6pt}} \tabularnewline[-0.5pt]


\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\hhline{}
\arrayrulecolor{black}

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\multicolumn{1}{r!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\raggedleft \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont (0.0032)\hphantom{0}\hphantom{0}\hphantom{0}} \hspace{6pt}\huxbpad{6pt}} &
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\multicolumn{1}{r!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\raggedleft \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont (0.0056)\hphantom{0}\hphantom{0}} \hspace{6pt}\huxbpad{6pt}} &
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\hhline{>{\huxb{255, 255, 255}{0.4}}->{\huxb{0, 0, 0}{0.4}}->{\huxb{0, 0, 0}{0.4}}->{\huxb{0, 0, 0}{0.4}}->{\huxb{0, 0, 0}{0.4}}-}
\arrayrulecolor{black}

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\multicolumn{1}{c!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\centering \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont Yes} \hspace{6pt}\huxbpad{6pt}} &
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\hhline{}
\arrayrulecolor{black}

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\multicolumn{1}{c!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\centering \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont Yes} \hspace{6pt}\huxbpad{6pt}} &
\multicolumn{1}{c!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\centering \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont Yes} \hspace{6pt}\huxbpad{6pt}} &
\multicolumn{1}{c!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\centering \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont Yes} \hspace{6pt}\huxbpad{6pt}} &
\multicolumn{1}{c!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\centering \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont Yes} \hspace{6pt}\huxbpad{6pt}} \tabularnewline[-0.5pt]


\hhline{}
\arrayrulecolor{black}

\multicolumn{1}{!{\huxvb{0, 0, 0}{0}}l!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\raggedright \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont Birth year dummies} \hspace{6pt}\huxbpad{6pt}} &
\multicolumn{1}{c!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\centering \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont Yes} \hspace{6pt}\huxbpad{6pt}} &
\multicolumn{1}{c!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\centering \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont Yes} \hspace{6pt}\huxbpad{6pt}} &
\multicolumn{1}{c!{\huxvb{0, 0, 0}{0}}}{\huxtpad{6pt + 1em}\centering \hspace{6pt} {\fontsize{10pt}{12pt}\selectfont Yes} \hspace{6pt}\huxbpad{6pt}} &
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\end{threeparttable}\par\end{centerbox}

\end{table}
 

\FloatBarrier
\newpage

\subsubsection{Testing the UK Biobank spouse pair matching}\label{testing-the-uk-biobank-spouse-pair-matching}

Some of our spouse pairs in UK Biobank could be false positives, i.e.
people who are not each others' spouse but simply live in the same
postcode. To validate the accuracy of our pairs, we use genetic
relationships. Some respondents in the UK Biobank sample have a child
(inferred from genetic data) who is also in the sample. Among our spouse
pairs, 463 have a genetic child of at least one partner in
the sample. For 425 of these, at least one child is the
genetic child of both partners. If this subsample is representative,
then about 92\% of the pairs who
have had a child, have had a child together. This is a lower bound
estimate, because some of the remaining couples may have had a
genetically shared child who is not in the UK Biobank sample. As a point
of comparison, 11\% of families with dependent children included a
stepchild in England and Wales in 2011 (National Statistics 2014).

It is still possible that some pairs in our data may not be actual
spouses. These pairs might show a relationship between one partner's
phenotype and the other's genotype. For example, maybe early-born
children grow up to live in richer postcodes, along with people who have
higher PSEA scores (Abdellaoui et al. 2019). This could then bias the
results. If the coefficient for ``fake pairs'' is absolutely larger
(smaller) than for real pairs, then our results will be biased away from
zero (towards zero).

To sign the bias, we create a dataset of ``known fake pairs''. These are
opposite-sexed pairs who live in the same postcode, but do not share all
the characteristics listed for the real pairs. Specifically, from the
list of characteristics used to create our real pairs (same
homeownership status, same length of time at address, same number of
children, attended same assessment center, attended on same day, husband
reported living with spouse, wife reported living with spouse) the fake
pairs ticked exactly 5 out of 7 boxes.

We again use genetic children to confirm that the fake pairs are ``real
fakes''. Out of 786 genetic children of the fake pairs, only
32 were children of both parents. Thus, the vast majority of
fake pairs do not appear to be spouses. Table
\ref{tab:tbl-bo-psea-fake} reruns the regressions of Table
\ref{tab:tbl-bo-psea-basic} using the fake pairs. Although the
coefficients on birth order are always negative, and significant when
controlling for parent's age, they are always absolutely smaller than
the corresponding coefficient in the main text. This suggests that any
fake pairs remaining in our data will have the effect of biasing our
results towards zero.

 
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\begin{table}[ht]
\begin{centerbox}
\begin{threeparttable}
\captionsetup{justification=centering,singlelinecheck=off}
\caption{\label{tab:tbl-bo-psea-fake} Regressions of PSEA on birth order: fake pairs (Great Britain)}
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\multicolumn{1}{p{0.2\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.2\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft 0.001\hphantom{0}\hphantom{0}\huxbpad{6pt}}} &
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\multicolumn{1}{p{0.2\textwidth}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.2\textwidth-6pt-6pt}{\huxtpad{6pt + 1em}\raggedleft 0.011\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\hphantom{0}\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{>{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}->{\huxb{0, 0, 0}{0.8}}-}
\arrayrulecolor{black}

\multicolumn{4}{!{\huxvb{0, 0, 0}{0}}p{0.8\textwidth+6\tabcolsep}!{\huxvb{0, 0, 0}{0}}}{\hspace{6pt}\parbox[b]{0.8\textwidth+6\tabcolsep-6pt-6pt}{\huxtpad{6pt + 1em}\raggedright  *** p $<$ 0.001;  ** p $<$ 0.01;  * p $<$ 0.05;  + p $<$ 0.1. Standard errors clustered by spouse pair in parentheses.\huxbpad{6pt}}} \tabularnewline[-0.5pt]


\hhline{}
\arrayrulecolor{black}
\end{tabularx}
\end{threeparttable}\par\end{centerbox}

\end{table}
 

\FloatBarrier

\clearpage

\subsection{Quotations on natural inequality}\label{quotations-on-natural-inequality}

\ldots your face and figure have nothing of the slave about them, and
proclaim you of noble birth.

-- \emph{Odyssey}, Odysseus to Laertes

Citizens, we shall say to them in our tale, you are brothers, yet God
has framed you differently. Some of you have the power of command, and
in the composition of these he has mingled gold, wherefore also they
have the greatest honour; others he has made of silver, to be
auxiliaries; others again who are to be husbandmen and craftsmen he has
composed of brass and iron; and the species will generally be preserved
in the children. But as all are of the same original stock, a golden
parent will sometimes have a silver son, or a silver parent a golden
son.

-- Plato \emph{Republic}

Nature would like to distinguish between the bodies of freemen and
slaves, making the one strong for servile labor, the other upright, and
although useless for such services, useful for political life in the
arts both of war and peace. But the opposite often happens -- that some
have the souls and others have the bodies of freemen.

-- Aristotle \emph{Politics}

Sons have no richer endowment than the quality

A noble and brave father gives in their begetting.

-- Euripides \emph{Heracleidae}

Abilities come from innate talents, which differ in their capacities and
take on different responsibilities in government.

-- Liu Shao \emph{Study of human abilities}

His head by nature fram'd to wear a crown,

His hands to wield a sceptre\ldots.

-- Shakespeare \emph{Henry VI Part 3}

A daughter of a green Grocer, walks the Streets in London dayly with a
baskett of Cabbage Sprouts, Dandelions and Spinage on her head. She is
observed by the Painters to have a beautiful Face, an elegant figure, a
graceful Step and a debonair. They hire her to Sitt. She complies, and
is painted by forty Artists, in a Circle around her. The Scientific Sir
William Hamilton outbids the Painters, Sends her to Schools for a
genteel Education and Marries her. This Lady not only causes the
Tryumphs of the Nile of Copenhagen and Trafalgar, but Seperates Naples
from France and finally banishes the King and Queen from Sicilly. Such
is the Aristocracy of the natural Talent of Beauty.

-- John Adams to Thomas Jefferson, on Emma Hamilton

\newpage

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\addcontentsline{toc}{section}{References}

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