Minorities, Marriage and the Melting Pot Marco Manacorda and Alan Manning Centre for Economic Performance, LSE Houghton Street London WC2A 2AE March 2015 Preliminary – comments welcome – please do not cite Abstract Whether or not minorities in a society find it easy to reproduce affects the evolution of diversity in society. This paper is about one part of this process ‐ how group size affects marriage rates and the types of marriages. We review models of the marriage market emphasizing the different predictions the models make. We then consider the case of African Americans in the United States. A strong finding is that that the exogamy rate is an increasing convex function of the share of African Americans in the marriage market, approximately linear in log odds, while the effects on the overall marriage rate are smaller and less clear‐cut. 1 Introduction Diversity is a fashionable topic at the moment in many of the social sciences. In economics there is a sizeable literature about the costs of diversity (most commonly ethnic or religious) – see Alesina and La Ferrara (2005) for a survey ‐ and a smaller one about the benefits of diversity (see, for example Ottaviano and Peri, 2006). In sociology Putnam (2007) has argued that diversity causes a loss of social capital. In much of this literature, diversity is treated as an exogenous aspect of society and its existence as a fact of life. For example, Putnam (2007) wrote that “the most certain prediction that we can make about almost any modern society is that it will be more diverse a generation from now than it is today”. But, on reflection, this is not so obvious. The increase in diversity in recent years has largely been driven by an increase in immigration, something that may or may not continue. What is as important – and in the long‐run certainly more important – is how diversity evolves over time within a closed population. Yet this is a subject about which we know very little. This paper is about one part of this process – marriage. As lifetime death rates remain resolutely fixed at one, the evolution of diversity of a closed population (i.e. one without immigration) depends on the diversity of descendants of the current population. So marriage is important because overall marriage rates influence the number of descendants (as marriage is correlated with fertility, albeit more weakly than in the past) and who marries whom influences the characteristics of their children. For example if there are two groups in a society who never intermarry and have identical fertility rates then the descendant population will be just as diverse. But if they intermarry then there will be more mixing and less diversity. One might conjecture that without inter‐marriage diversity and group differences will persist for ever but with extensive inter‐marriage group differences disappear over time because it becomes impossible to keep track of people who are 1/8th this, 3/8th that etc. (see, for example, Alba, 1990, Waters, 1990, of the diminishing significance of the ethnic identity of white Europeans e.g. Irish, Italians and Poles). We should be clear from the outset that marriage patterns are not the only factors influencing the evolution of diversity within a society. Differences in fertility across groups obviously matters as does the fact that many categories (including ethnicity) have a social dimension to them so that individuals have a degree of choice over them in a way that they do not over their genes. But it is complicated to consider a universal model of diversity so this paper is about one part of the transmission mechanism from one generation to the next not all of it. The specific question this paper addresses is how the size of a group (with a particular focus on minorities i.e. groups which are a small fraction of the population) affects the overall marriage rate and they types of marriages that are made, paying particular attention to the impact of on exogamous and endogamous marriages. The plan of the paper is as follows. In the next section, we review the economic theories of marriage drawing out their predictions about how the size of a group affects its marriage market outcomes. We show that the main classes of models – competitive and search models – can have rather different predictions. We show that competitive models predict that a reduction in the proportion of a group in the society is predicted to raise its overall marriage rate if there is any intermarriage at all. The intuition is simple – it is best to be on the short side of a market. But this increase in the overall marriage rate is at the expense of endogamous marriage rates i.e. there is more intermarriage. Search models also tend to predict that intermarriage rates will be higher for small 2 minorities but also predict that – if frictions are sufficiently large ‐ overall marriage rates fall as group size falls ‐ because there is a preference for endogamy and finding a partner from one’s own group is hard when one is rare and the market has sizeable frictions. The second section then investigates these predictions using the marriage rates of African Americans. We show that there is a strong convex relationship between the exogamy rate and the fraction of African Americans in the marriage market that is approximately linear in log odds. It is less clear how the overall marriage rate varies – the best estimate is that the impact is very small. The third section presents a simple model that might explain the results and the fourth section discusses the implication of the results for the evolution of diversity. 1. Theoretical Models of the Marriage Market There is a sizeable economics literature relevant to the analysis of marriage, some directly on the topic, some readily applied to it (e.g. models of two‐sided matching – see Roth and Sotomayor, 1990). At the risk of some simplification these models differ along two important dimensions – whether the marriage market has frictions or not, and whether there is a possibility of transfers within marriage between the spouses (transferable utility) or not (non‐transferable utility)1. Existing applications of models of the marriage market have largely focused on the effect of gender imbalance (e.g. Angrist, 2002), on the demand for certain characteristics (see, for example, Chiappori et al, 2011, 2012, 2014; Dupuy and Galichon, 2014), on the impact of legislation (e.g. the effect of abortion laws considered in Choo and Siow, 2006) or on how the size of the market affects outcomes (see, for example, Gould and Paserman, 2003, Botticini and Siow, 2010 or Gautier et al, 2010). There is also a literature on how marriage market prospects affects other decisions e.g. investments in human capital (e.g. Lafortune, 2013). Our focus here is different – the impact of group size on marriage market outcomes which has been less studied (though see Voigtlander and Voth, 2013). We will discuss the four cases and spell out their predictions about how group size affects marriage market outcomes. There is also a vast literature in population genetics on mating models that is used to model the evolution of diversity of a population. Some of these models similarities to models used in the economics literature (some work was by people whose work is also relevant for economics e.g. Wright, 1921; Karlin, 1968). But one sort of model that is rather different is the Becker (1973, 1974) ‘competitive’ model of marriage in which there is an endogenous, market‐clearing transfer between spouses. a. The Theoretical Framework We will set up a theoretical framework that can be used to encompass most the approaches currently being used in models of the marriage market (Choo and Siow, 2006, Siow, 2008, Chiappori et al 2011, 2012, 2014, Galichon and Salanie, 2011, Dupuy and Galichon, 2014). Assume there are 1 In one‐period models the difference between transferable and non‐transferable utility is clear. But in multi‐ period models it is perhaps best to think of the non‐transferable utility case as being one where it is impossible to commit (e.g. through a pre‐nuptial) to a certain division of surplus in marriage. The possibility of divorce and renegotiation probably means that reality is somewhere between transferable and non‐transferable utility. 3 distinct groups in society denoted by i . Denote the number of group i in the population by i ‐ we will assume that every group is balanced by gender though if there were gender imbalances this would affect the equilibrium in the usual way (testing for an effect of gender imbalance on the marriage market has been a common empirical application of marriage models – see, for example Angrist, 2002). Suppose, in the absence of transfers between spouses, the average surplus for a man relative to being single (which is normalized to zero) of a marriage between an i‐man and a j‐woman is yijm . We will adopt the convention that the first subscript denotes the group membership of the man and the second the woman so denote the average surplus for a woman in that type of marriage as yijf . If utility is transferable between spouses denote the transfer from the man to the woman by ij so the average surplus of the man is yijm ij and the average surplus of the woman is y y f ij ij . Denote the vector of surplus an i‐man might have in all their possible marriages by m i. i. and the vector of surpluses for a j‐woman by y.mj . j .But we also assume there is idiosyncratic variation around these averages in the demand for marriages of different types so that the utility of a particular i‐man from a marriage with a j‐woman would be given by: yijm ij ijm (1) Where ijm is their idiosyncratic shock. We start our analysis of marriage market outcomes by considering a competitive marriage market model. b. Competitive Model with Transferable Utility If the marriage market is frictionless and competitive everybody will choose the marriage market option that maximizes this utility. This leads to the following per capita demand curves for the marriage of an i‐man to a j‐woman: ijm yim. i. Pr yijm ij ijm max yikm ik ikm , 0 , k j (2) There will be a similar set of per‐capita demands from a j‐woman to an i‐man that will be given by: ijf y.mj . j Pr yijf ij ijf max ykjf kj kjf , 0 , k i (3) This set‐up nests the structure used in most empirical models of the marriage market that typically use a logit structure of preferences (e.g. Choo and Siow, 2006). The discrete nature of the marriage market (one can consume only one type of spouse) puts some restrictions on the nature of these demand curves. First, it must be the case that if the average return to one type of marriage goes up the demand for that type of marriage cannot go down and the demand for all other types of marriage cannot go up. However the overall demand for marriage cannot fall i.e. the proportion choosing to be single cannot rise. If there is transferable utility then, in a competitive marriage market, the transfers must be chosen such that the demand from i‐men for j‐women is equal to that from j‐women to i‐men. This market‐ clearing condition can be written as: 4 i ijm yim. i. j ijf y. fj . j (4) From (4) one can readily see that the population shares will affect the equilibrium outcome (as would gender imbalances if they are present). Note that we assume that population shares do not directly affect the returns from different types of marriages. We are now in a position to do some comparative statics but the general case is potentially complicated so we will make some simplifying assumptions to help illustrate the effects that are at play. First we will assume gender symmetry in the nature of demands for each type – this amounts to assuming that the distribution of the errors is the same for men and women of the same type. It is more convenient to express this assumption in terms of the per capita demands (2) and (3) when it can be written as: (Gender Symmetry) ijm z jif z ij z , z (5) As we shall see the model still has sufficient flexibility to allow the equilibrium outcome to be asymmetric in gender but it does assume that the elasticity of the response of demands of different genders to pay‐offs is the same and this could conceivably be restrictive. To make things a little simpler we will just consider the case with two groups. With gender symmetry, the market‐clearing conditions in this case can be written as: 111 y11m 11 , y12m 12 111 y11f 11 , y21f 21 2 22 y22m 22 , y21m 21 2 22 y22f 22 , y12f 12 (6) (7) 112 y11m 11 , y12m 12 2 21 y22f 22 , y12f 12 2 21 y22m 22 , y21m 21 112 y11f 11 , y21f 21 (8) (9) The first two conditions are the equilibrium conditions in endogamous marriages and the group sizes cancel out from both sides of the equations (the result of the assumption of gender balance). The second two equations are the equilibrium conditions in exogamous marriages and here the group sizes do matter. First consider the case where the average surplus from both types of exogamous marriages is the same i.e. the assumption: (Exogamy Symmetry) m y12m y12f y21 y21f (10) A useful simplifying result is the following: Result 1: If the demand for marriage is not completely inelastic, with gender and exogamy symmetry, the equilibrium must always have: yiim ii yiif ii 5 (11) And: y12m 12 y21f 21 (12) Independent of group sizes. Proof: See Appendix This result is unsurprising – with identical demands for marriage for men and women and (11) says that the surplus to an endogamous marriage must be split equally between men and women of the same type – the ‘price’ in these markets must be independent of the relative group sizes. But, while the returns to marriage are equalized across genders within types, they can vary across types. (12) must hold independent of group sizes but does not say that the equilibrium prices in the exogamous marriages is independent of group size. Rather it says that with exogamy symmetry, the returns to an exogamous marriage must be the same for men and women of the same type i.e. the returns do not vary with gender but do vary with type (note the difference from the result for exogamous marriages). It is now simple to do the comparative statics as we change group sizes. We will use as the key variable log 2 / 1 which is the log‐odds ratio for the share of group 2 in the population (the log‐odds specification will turn up in the empirical analysis). As the returns in endogamous markets are fixed independent of we can simply differentiate (8) and (9). From (12) we only need to focus on one of these markets as the other will be identical. From this it is trivial to produce the following result: Result 2: With gender and exogamy symmetry an increase in : a. decreases 12 with: 12 1 0 log 12 log 21 12 12 (13) b. increases 21 by the same amount c. increases the marriage rate of 1‐people and reduces the marriage rate of 2‐people d. Increases the fraction of 1‐people in exogamous marriages with the change in the log odds being given by: log 11 log 12 12 log 12 log 11 12 0 log log log 21 log 12 12 12 Proof: 6 (14) a. Differentiating (8) and re‐arranging gives (13) b. The returns to endogamous marriages are constant but the returns to exogamous marriages for 1‐people goes up and 2‐people goes down so the overall marriage rates must move in this direction. c. This comes from (13) and (6) and (8) Part (a) says that the transfers from 1‐men to 2‐women in exogamous marriages fall as the ratio of 2‐people to 1‐people rises. From (35) we have that the transfers from 2‐men to 1‐women must rise so the result is best summarized as being that things shift in favour of 1‐people as they become relatively less common. The intuition for part (a) of this result is very simple ‐ it is good to be on a short side of the market. This prediction of the competitive marriage market model is something we seek to test later in this paper. There are a number of aspects of this result worth discussing. First, although the direction of the effect is unambiguous and does not depend on how attractive or unattractive are exogamous marriages, the size of the effect of group size does depend on the demand for mixed marriages as it works through those markets. If there is no demand for mixed marriages there will be no effect of group size on marriage markets. At the other extreme if the two groups are perfect substitutes in marriage then the surpluses cannot change and again there will be no effect on overall marriage rates. So the effect will be largest when there is some intermarriage but there is some preference for endogamous marriage. The elasticity on the log odds of an exogamous relative to endogamous marriage will be greater than or less than 1 as log 11 log 21 i.e. it depends on how sensitive is group 1’s demand for ( ) 12 12 endogamous marriages to changes in the returns to an exogamous marriage relative to the sensitivity of group 2’s demand for exogamous marriages. One case of particular interest is where one group – say group 1 – is indifferent about who they marry, but the other group does have marked preferences. In this case a shift in the group sizes will not alter the returns to marriage at all for group 1 and the elasticity of the exogamy rate for group 1 with respect to the relative population sizes will be equal to 1. In contrast, if it is group 2 that is indifferent and group 1 has the preferences then the elasticity will be zero. So the size of the elasticity will be informative about the relative preferences of different groups towards marrying other groups. These are strong predictions but have been derived under the assumption of gender and exogamy symmetry. That is analytically convenient but produces the prediction that the exogamy rate will differ by type but not differ by gender. That is at variance with what we often observe e.g. we see the black man ‐ white woman more commonly than the other way round (see Fryer, 2007, for more details). It is natural to ask what happens if we relax one or both of the symmetry assumptions. It is easiest to relax the exogamy symmetry assumption i.e. to assume that one type of mixed marriage generates more surplus than the other type. Without loss of generality we will assume m m y12f y21 y21f . This naturally leads to an equilibrium in which 1‐men (resp. 2‐ that y12 women) are in more demand than 1‐women (resp. 2‐men). So the equilibrium shifts. The rate of 12‐ 7 marriages will be higher than the rate of 21‐marriages so this is the simplest way to explain the gender asymmetry in exogamous marriages that is often observed. Now what happens when increases. One can show that it is possible – if there is gender asymmetry – for the marriage rate of either 1‐men or 1‐women to go down as their group size decreases but not both. To see this, consider the extreme case in which there is no demand for m marriages between 1‐women and 2‐men which we can model as y21 y21f . In effect, this shuts down the market for marriages between 1‐women and 2‐men; 1‐women only have the option of an endogamous marriage. Now consider what happens when the proportion of 1‐people falls. The only market this affects initially is the 12‐market and here the surplus has to shift towards the 1‐ men. This causes 1‐men to exit the endogamous market for the exogamous one causing the terms of trade within the endogamous market to shift in the men’s favour. This makes marriage less attractive to 1‐women whose marriage rate then falls. But this can only happen if the marriage rate for 1‐men improves i.e. the marriage market for all 1‐people cannot fall though the overall marriage rate could. This scenario could potentially be relevant e.g. Crowder and Tolnay (2000) argue that part of the marriage squeeze for black women has been caused by rising rates of inter‐racial marriage that affects black men more than black women. c. Frictionless models with Non‐Transferable Utility The above model has been one of transferable utility – this section considers how the predictions might be altered if there is non‐transferable utility. These are models in which the transfers ij are fixed, possibly at zero. The most common way to find equilibrium in this case is the Gale‐Shapley (or deferred acceptance) algorithm (Gale and Shapley, 1962 but also see, for example, the applications of Hitsch et el, 2010, and Banerjee et al, 2013 to marriage and dating markets). But it is also fruitful to think of this case as a market model in which not all markets clear so some agents are rationed in the type of marriage they can have. This then has consequences for demands (e.g. see Malinvaud, 1977; Neary and Roberts, 1980). The outcome of the Gale‐Shapley algorithm is in the core so corresponds to a case of efficient rationing though one could have other models e.g. random rationing. In our two‐group case it is possible that men are rationed or not on each of the four marriage markets (2 exogamous and 2 endogamous) leading to 16 possible types of equilibria. If we knew which one these equilibria was likely to be relevant then we could restrict our analysis to that but, unfortunately, we do not. Perhaps the most plausible argument is that the terms in marriage markets reflect customs that change only slowly as demands change and are not responsive to local conditions such as the fraction of a group in the society. For example, one might think that the traditional assignment of household work to women is changing but not as quickly as would be justified on economic grounds ‐ in this case one might think it is the men who tend to be rationed. But this is little more than speculation and as it is extremely tedious to work through all the possible cases and do the comparative statics on how the groups sizes affect the equilibrium, we will confine ourselves to some general remarks. To keep things simple we will restrict attention to the case of gender and exogamy symmetry. As one’s group becomes smaller this reduces the supply of one’s group into the exogamous marriage market so that any excess supply of one’s group to that market falls and any excess demand rises. 8 From this one can deduce that if one’s group is rare enough one will be in excess demand on the exogamous markets so that the fraction that are in exogamous marriages will be determined by the demand for that type of marriage from one’s own group. This per capita demand is not affected by changes in the group sizes so that the fraction in exogamous marriages will not be affected by the group size. If there is excess supply to the exogamous market then the number of exogamous marriages is determined by the demand from the other group. In this case a fall in the group proportion will result in the number of exogamous marriages not changing but the proportion of the own‐group in exogamous marriages will rise. What will be the comparative statics in this case? It is hard to overturn the predictions of the competitive marriage market model because being in relatively short supply can only be to one’s advantage. If one is rationed on an exogamous market then becoming relatively rarer can only lessen the rationing. And if one is not rationed it will have no effect. d. Marriage Models with Frictions There are quite a large number of models of the marriage market with frictions, both with transferable and non‐transferable utility (see, for example, Adachi, 2003, Burdett and Coles, 1997, 1999, Burdett and Wright, 1998, Shimer and Smith, 2000, Smith, 2006, and Smith 2011 for a review)2. We know (see Adachi, 2003, for a formal proof of this result in the current context with non‐ transferable utility) that the equilibrium of models with frictions often approaches the competitive outcome as frictions become small. In this case models with frictions will have the same predictions as the models discussed earlier. Here, we want to emphasize how models with frictions can have different predictions so will simply consider one simple but extreme model to make that point. To that end, consider the simplest possible model with frictions in which people have only one shot at love i.e. they are matched with a possible partner and need to decide whether to marry but a decision to remain separate means that individual is single forever. We assume everyone has this shot at love but the comparative statics would be identical if some individuals have no choice but to remain single. For the matching process – which is important ‐ assume that the probability of an i‐person being matched with a j‐person is ij (for simplicity, assume this is the same for both genders). For this to add up we must have i ij j ji . A common model used in mating (first introduced into biology by Wright (1921) but also used by Cavali‐Sforza and Feldman (1981), Boyd and Richerson (1985) and in the economics literature by Bisin and Verdier (2000)) is that individuals match with their own type with probability and with probability 1 the unmatched individuals are matched at random – this device means one is more likely than random to meet someone of one’s own type if 0 . This has. This means we will have: ii 1 i , ij 1 j , j i (15) 2 Some of these models are explicitly about the marriage market, others about two‐sided search in general. 9 First consider the case where there is non‐transferable utility. In this case a marriage will only result from a match if both parties desire it i.e. if their returns are bigger than the zero return from remaining single. So the probability a match between a 1‐man and a 1‐woman results in a marriage m 11 11m 0 and y11f 11 11f 0 . Using the notation for the is the probability that y11 demands introduced earlier this probability, 11 , can be written as: 11 11 y11m 11 , 11 y11f 11 , (16) This way of writing it is valid because there is no option of an exogamous marriage so that this is formally equivalent to the case where the pay‐off from an exogamous marriage would be so low no‐ one would choose it. Similarly we can write the probability of a match between a 1‐man and a 2‐ woman as resulting in a marriage as: 12 12 , y12m 12 21 , y12f 12 (17) Which means the overall marriage rate for 1‐men will be given by: m1m 1 1 11 1 1 1 12 (18) How does the marriage rate vary with 1 ? We have that: m1m 1 11 12 1 (19) It is plausible that endogamous matches are more likely to result in a marriage than exogamous marriages so that 11 12 in which case (19) says that the marriage rate will decline as groups become a smaller proportion of the population i.e. as 1 falls. This contrasts to the prediction of the competitive model in which the marriage rate rose. The intuition is simple – one’s chances of marriage are higher if one is matched with an individual from the one’s own group but this is less likely if one’s group is rarer in the population. What about the exogamy rate? From (16) and (17) we have that the log‐odds from the exogamy rate will be given by: 1 1 1 ln 12 , y12 12 21 , y12 12 ln 1 1 11 y11m 11 , 11 y11f 11 , m f (20) This says, unsurprisingly that the exogamy rate will be decreasing in the group size. But, if 0 the log odds of the exogamy rate will tend to a fixed amount as the group size goes to zero – this occurs because the specification of the matching process means there is always a prospect of finding a partner of one’s own group, however rare one is. Bisin and Verdier (2000) adapt this model to make endogenous and argue that as a group becomes rare it takes efforts to increase which only strengthens the effect just described. In contrast, if 0 so matching is random then the log odds of the exogamy rate is linear in the log odds of the group sizes with a slope coefficient of 1. 10 These results have been derived under the assumption of non‐transferable utility but the predictions if there is transferable utility are identical. In this case a match will become a marriage if there is any joint surplus i.e. if that yijm ijm yijf ijf 0 which is a condition independent of group sizes so all the previous results will follow – the only thing that changes is the probability of matches becoming marriages. But these results have been derived under the assumption there is only one shot of love. If there is a chance of more than one marriage opportunity then the decision whether to marry this partner will be affected not just be the returns from being single but by future marriage prospects that will be affected by group sizes. So the reservation utility levels for marriage will no longer be zero and the model will be much more complicated. But, for high levels of frictions the above decision rules will be close to what happens but as frictions get very small the model will go to the competitive model. We can summarize the predictions of the different marriage models for the effects of group size on marriage and exogamy rates in Table 1. All models predict that exogamy becomes more likely as the group size falls but it is the nature of that relationship in which the models differ. We now turn to examination of these predictions in the form of an empirical application of how marriage market outcomes vary with group size. 2. The Marriage Market for African Americans Our empirical application is to the marriage patterns of African Americans in the US, exploiting the fact that there is considerable geographical variation in the proportion of African Americans in the population. In addition this has the advantage of being a group that is culturally relatively homogeneous as compared, say, to immigrant communities from different source countries (see Kalmijn and Tubergen, 2010, Chiswick and Houseworth, 2008, Furtado, 2012). On the other hand, there are some aspects of the black‐white interactions in the US that do not naturally carry across to interactions between other groups. The distinction between black and white has remained salient in American society whereas some distinctions between white ethnic groups have become of little practical importance (see, for example, Alba, 1990; Waters, 1990) and interactions between Asians and whites or Hispanics and whites may all be very different. The conclusion briefly discusses issues of external validity of our findings for other divisions by race/ethnicity or religion. There is a considerable existing literature on the marriage patterns of African Americans and the level of exogamy (see, for example, Monahan, 1976; Kalmijn, 1993, 1998; Landale and Tolnay, 1991; Lichter et al, 1992; South, 1993; Brien, 1997; Crowder and Tolnay, 2000; Fryer, 2007). a. The Data The data we use on comes primarily from IPUMS (Ruggles et al, 2010). We use the Censuses for 1910‐2000 inclusive and the ACS for 2009‐2011 that we use as a single observation for 2010. We group individuals into six ten‐year age groups 0‐9, 10‐19 etc up to 50‐59. This is somewhat arbitrary but does allow us to follow our age groups from one census to the next i.e. those aged 10‐19 at one census will be aged 20‐29 at the next. We label cohorts so that the ‘1950’ cohort is those aged 0‐9 in 1950 i.e. born between 1941 and 1950. We supplement the IPUMS information (which is never more than a 5% sample) with count information on population by age, race, and area from NHGIS 11 (Minnesota Population Center, 2011) although the way in which we use this information is limited by the tabulations provided. These are counts from the census so we would expect them to be more precise than estimates of population shares coming from the IPUMS data. In our estimation we restrict attention to those born in the 1910‐1990 cohorts inclusive – the exogamy rate was essentially zero for earlier cohorts and the later cohorts are too young to have marriage rates much above zero. Our primary data is on race which one should not assume is an unproblematic classification3. Starting from the 2000 Census, individuals can describe themselves as being of more than one race. We include as black anyone who identifies black as one of their races – in practice this makes little difference. We have data on those who are currently married and those who have ever been married. We cannot identify in a consistent way cohabiting couples, a group that has obviously become larger over time (see, for example, Stevenson and Wolfers, 2007; Isen and Stevenson, 2008; Lefgren and McIntyre, 2006). We can only identify race of spouse and hence identify exogamous marriages for those individuals who are currently married and living with their spouse. We start with some descriptive statistics, first on overall trends. Figure 1 presents some figures on the fraction of blacks who have ever married by birth cohort for two age categories (20‐29, and 40‐ 49). Figure 2 does the same for those currently married. As is well‐known the fraction of African Americans who are married has fallen over time (see, for example, Ellwood and Crane, 1990, South, 1993, Seitz, 2009). The most popular hypothesis for this trend here has its origins in Wilson (1987) that this can be attributed to a decline in the supply of ‘marriageable’ black men, because of declining economic opportunities or increased incarceration. There is quite a lot of research examining this hypothesis (Lichter et al, 1992, Brien, 1993, Charles, and Luoh, 2010, Mechoulan, 2011, Staub, 2012). Explaining these trends in marriage is not the purpose of this paper (and we cannot do so as our regressions contain time dummies or the equivalent) but it should be noted that the decline in marriage rates for black women is slightly greater than that for black men, not perhaps what one would expect if the problems were on the male side.4 Figure 3 documents the fraction of marriages among blacks where the spouse is not black (primarily white) i.e. the marriage is exogamous. One can see the rise in the fraction of exogamous marriages that has been documented elsewhere (Fryer, 2007). For those aged 20‐29 in 2010 over 25% of marriages for black men are exogamous though the overall marriage rate is low (see Figure 2)5. Also, as is well‐known the exogamy rate is higher for black men than white women so the black man‐ white woman combination is found more frequently than the black woman –white man combination6. Again we are not interested or able in this paper in explaining these trends over time. 3 See, for example, Loveman and Muniz (2007) , Schwartzman (2007) or Duncan and Trejo (2007). One should note that all the samples we use here include the institutionalized population and that marriage rates for the incarcerated are, while lower than for those not imprisoned, not zero. 5 This may seem extremely high but in the UK where blacks form about 1% of the population the exogamy rate is over 30%. 6 It is perhaps worth recalling that the black man – white woman combination was what in the past led to a lynching while white man – black woman unions were not uncommon in the era of slavery (though not 4 12 What we are interested in is how the marriage and exogamy rates vary with the fraction black in the population. Figure 4 presents the scatter plot of the exogamy rates against the share of the population that is not black (all done for those aged 20‐49 inclusive) where the variation is across state of current residence and the data come from the 2000 Census. We also include a regression line where the observations are weighted by population size. For the exogamy rate there is a very striking negative, convex relationship. This has been noted before e.g. Kalmijn (1993) presents essentially the same graph but with different data. However, there is little discussion of the relationship perhaps because it is thought obvious that the relationship would have the observed slope because it is harder to find a black partner when there are fewer blacks around. But what is perhaps of more interest is the striking convexity in the relationship – as we have seen not all popular marriage models predict this and some predict the opposite. Although the relationship is highly convex in levels, Figure 5 presents it in log‐odds and the linearity is then quite striking. Figure 6 present similar graphs for the fraction of the black population that are currently married and ever married, the relationship is less clear‐cut and the slope coefficient not significantly different from zero. While these graphs are suggestive, they do not control for other possible influences on marital outcomes and do not deal with any potential endogeneity issues. For this reason we turn to an empirical analysis. We are interested in the relationship between some marriage market outcome and the fraction white, wi , in the population. Most of the outcomes we look at are binary – whether married, whether a marriage is exogamous or endogamous – and given the graphical evidence in Figure 5 our basic model is a logit model where the variable of interest is measure in log odds. That is we estimate the model: 1 wi Pr yi 1 F 0 1 ln 2 xi wi (21) Where the outcome for individual i is yi , F . is the logistic function and other covariates are xi . The estimate of 1 can be interpreted as the elasticity of the log odds of the outcome with respect to the log odds of the white share. In all our specifications we report standard errors clustered on birth state. b. Results: Marriage We start by estimating models for being currently married. In our baseline specification we measure the white share as the current white share in the current state of residence among your birth cohort. Ideally the theory says we would like to measure the black share in the state of marriage at the time of marriage but we do not observe this. To alleviate this problem we consider only individuals aged generally consensual) resulting in a non‐trivial proportion of African Americans having Northern European ancestry on the male side. 13 20‐49 inclusive but we do consider robustness to a number of other measures and samples to try to alleviate concerns on this. Table 2 presents some estimates of the impact on the marriage rate. The first row includes no other covariates at all and appears to indicate that the marriage rate for all groups (men and women, black and white) are significantly higher in states with a lower proportion of blacks. The estimated effect is larger for whites than for blacks. This is implausible as a channel of influence from the white share through the marriage market and is perhaps suggestive of other omitted factors correlated with both marriage and the black share. For the marriage outcomes we will use as a plausibility test the requirement that the effect on blacks be bigger than the effect on whites i.e. that the impact of the minority share on a minority might be expected to be bigger than the impact on a majority. However, this result is not robust to the inclusion of dummies for birth cohort and age category as is shown in the second row. Cohort effects are likely to be important because attitudes to marriage have been changing over time. And, the probability of being married obviously varies with age. Now all the estimated coefficients are negative, and only the estimated effects for blacks are significantly different from zero. The third row reports the results when controls for individual education are introduced – this makes the effects for blacks larger. The fourth row then includes a variety of other regressors that might be expected to affect the marriage market. First we include some measures of the laws about inter‐racial marriage in the US. Some states never had laws against inter‐racial marriage, some repealed such laws in the nineteenth century, some in the twentieth and some only after the Supreme Court ruling in 1967 in Loving vs. Virginia, 1967 that made such laws illegal (see Fryer, 2007, for details) – though it should be noted that Southern states do not stand out as outliers in Figure 5. We include a variable indicating whether such a law was in place when you were aged 10‐19 (this varies by state and age cohort) and a variable indicating the year of repeal (to have some idea of liberality of the state in general). We also include some attitudinal variables from the General Social Survey (GSS) which is only available to us at the census division level. Following Charles and Guryan (2008) we include a cohort specific measure of attitudes to inter‐racial marriage7. To capture changing attitudes to marriage in general we also include a measure of attitudes to sex before marriage. We include some variables related to economic conditions – the employment‐population rate in the cell and the fraction of the population in prison. We also include measures of residential segregation between blacks and white (see, for example, Peach 1980, Cutler, Glaeser and Vigdor, 1999, Card, Mas and Rothstein, 2008, Glaeser and Vigdor, 2012). The inclusion of these variables means that the marriage rate for all groups seems significantly higher for whites in areas with a higher white share, and the effects on blacks are very small. However this changes in the next row when we include state fixed effects for state of birth. The coefficient of interest is now identified from changes in the white share in different states and how this is related to marriage rates. The estimated elasticities for all groups are now small and not significantly different from zero. Rows 6 and 7 augments this specification by including region‐ specific trends by cohort (row 6) and state‐specific trends (row 7) – these make very little difference to the results. 7 Other studies looking at attitudes to interracial marriage and dating are Fisman et al (2006, 2008) and McClintock (2010). 14 One concern that one might have is that this result is caused by mismeasurement of the white share in the state at marriage, a problem that becomes worse when state fixed effects are included. To investigate this row 8 confines the estimates to the youngest age cohort (aged 20‐29) for whom state of marriage is more likely to be the current state. There are now significant positive effects of the white share on the marriage rate among blacks. We might also be concerned about the endogeneity of the current white share in the current state of residence because it reflects not just conditions at birth but the patterns of migration, both individual and in aggregate. Once we include state effects most of the variation in the white share reflects the differential pattern of migration of blacks and non‐blacks into and out of states. In the present context the most important factor is the Great Migration of blacks from the South to the North‐East and Mid‐West (Lemann, 1993; Gregory, 2005) and associated movements of the white population (see Boustan, 2010). Figure 8 shows the evolution over time in the fraction of the total population that is black by birth and current residence by census division. The basic pattern one observes is that the fraction black first fell and then perhaps rebounded slightly in the South while the fraction rose in the North. As it is clear these are long‐run trends and a motivated migration, primarily by declining economic opportunities in the rural south and rising opportunities in Northern cities, we might be more concerned that this variation in the contemporaneous black share is affected by factors that also affect marriage market outcomes. There is also an academic debate about the importance of selection in the Great Migration and the economic gains from this (see, for example, Tolnay, 2003, Vigdor, 2002, Eichenlaub et al, 2010; Chay and Munshi, 2012; Collins and Wanamaker, 2014). If the state’s economy is booming this is likely to increase net migration into the city that, depending on the circumstances, might increase or reduce the proportion of blacks. But if economic conditions also affect marriage market outcomes directly (and there are studies claiming such a link, e.g. Landale and Tolnay, 1991; Lichter et al, 1992) and one fails to control adequately for economic conditions it is clear that the coefficient on the black population share may be biased. These problems may well be accentuated in regressions with state fixed effects as most of the change in the black population share over time is driven by migration. All of these are reasons why one might be concerned about the endogeneity of the current white share in the current rate of residence. To investigate this further, we report estimates using the white share in the cohort in the state of birth at the time of birth, both for the whole sample (row 9) and the sample that currently live in their birth state (row 10). The results are very similar but now suggest a significant negative effect of the white share on the black marriage rate. But, although one might regard the white share in the birth state at birth as exogenous to the individual it is not necessarily highly correlated with the white share in the market at marriage. A natural approach to this is to instrument the current white share in the current state of residence by the fraction of your birth cohort birth (this is also used by Botticini and Siow, 2010)8. The identifying 8 Another approach would to find an instrument for the migration flows as is done in the literature on immigration (Card, 2001). Migration is influenced in part by economic conditions in this city but also be economic conditions in the source and destination places. It is the economic conditions there that can serve as the instrument. This is what is done in Boustan (2010) who investigates the impact of black migration on ‘white flight’. 15 assumption is that, conditional on covariates, marriagibility is unrelated to the white share in the state of birth at the time of birth. Because of the linearity in log‐odds shown in Figure 5, we want to maintain the logit specification even as we allow for the exogeneity of the current white share. To do this we adopt the ‘control function’ approach described in Blundell and Powell (2004). Essentially this approach involves estimating a first stage in which the endogenous variable is regressed on the exogenous variables and the instrument – in our case this is a simple linear regression. The residuals from this equation are then included in a logit model for the outcome of interest. The resulting logit model will have a coefficient on the variable of interest but this needs to be transformed to be conformable with the estimate from the non‐instrumented equation, essentially because the standard deviation of the residuals in this logit equation will be different. There are a number of ways of reporting the results but the measure of the sensitivity we will used will be the average value of the derivative of the log odds with respect to the log odds of the white share. In a standard logit model this is simply the coefficient but in the IV model it is more complicated –Appendix B works through the algebra. As computation of the standard errors of the resulting estimate is not straightforward, we bootstrap using a block –bootstrap on the state of birth and 100 repetitions. The results of the first‐stage are reported in Table 3. The first three four show the first‐stage without state effects but with a variety of other controls – in this case the instrument is extremely strong. The fifth row shows it with state effects and the sixth row when the sample is restricted to those aged 20‐29 – the instrument is still strong. However, the seventh and eighth rows shows that this instrument is not strong enough to survived the introduction of region‐specific trends essentially because the population flows documented in Figure 8 are well‐approximated by regional trends. The first‐stage suggests the instrument is a reasonable one. The ‘control function’ estimates of the effects are shown in row 11 of Table 2. The estimates suggest a negative impact of the white share on the marriage rate for all groups though only significantly different from zero for black men. However, for this group the estimated effect is very large though standard errors are large. There is a simple intuition for the negative effect – the estimates in row 9 can be thought of as the reduced form and they have a negative effect of the black share on black marriage rates. Overall, the pattern is quite mixed without a very clear pattern of the impact of the white share on marriage rates emerging. It is reassuring that the impact on black marriage rates seems to be larger than the impact on white marriage rates. Models treating the white share as exogenous indicate a small positive impact of the white share on marriage rates but the model to control for potential endogeneity makes this significantly negative and very large, at least for black men. Table 4 presents the same specifications as Table 2 but now changing the outcome to whether one has ever been married. The pattern of results is very similar to the ‘currently married’ outcome so we will not discuss them in detail. c. Results: Intermarriage Table 5 presents the results for how the white share affects the probability of having a white partner for blacks. We only show the results for blacks because there is an accounting identity relating the 16 exogamy rate for blacks, the marriage rates for whites and blacks and the white share. Unlike the results for marriage rates, the results here are very robust across specification, sample, and estimation methodology. We always find that a higher white share is strongly associated with a higher exogamy rate for blacks. The strong graphical cross‐sectional evidence of Figure 5 holds up in a regression setting. A possible concern with the pattern shown in Figure 5 and the regression results is that it is driven by classification error. If a certain fraction of whites are erroneously classified as black then a higher fraction of those who are recorded as black in low black share states will actually be white. These individuals may appear to be in exogamous marriages with one black and one white partner when in fact both individuals are white. One way to check this possibility is to look at the racial classification of the children of the exogamous marriages. If more of the apparent exogamous marriages in low black share states are classification error we would expect a higher proportion of the children to be classified as white. Figure 7 presents this relationship for the 2000 Census – the fraction of children in exogamous marriages being recorded as white does not seem to vary with the black population share. As one more piece of evidence that there is a robust empirical correlation between the exogamy rate and the population share, Table 6 presents some evidence from administrative data on marriages that is available through ICPSR for 1970, 1979, 1986, 1987 and 1990. This records the state in which the marriage took place, the current state of residence of both bride and groom and the state of birth for both bride and groom. It also records age at marriage, the number of previous marriages, race and education (though the last two variables are missing for a sizeable proportion of marriages. The advantage of this data is that we know when and where the marriage is taking place but the disadvantage is that we only have marriages at a few points in time. Table 6 shows estimates of a logit model for being in an exogamous marriage where the regressor of interest is the log‐odds of the non‐black population share in the relevant state. We report results using the black population share in the marriage state, state of current residence and state of birth and with and without controls (which are education, age at marriage, number of previous marriages, state and year effects). In all cases we find a significant positive relationship i.e. in states with fewer blacks there are more exogamous marriages. From this we conclude that it seems a robust empirical finding that the exogamy rate rises as the proportion black in the population falls and that this is reasonably well approximated by a relationship that is linear in log‐odds. This suggests that factors working for endogamy fail as the share of the population that is African American goes to zero. One tentative explanation of why the findings on exogamy are so strong when those on marriage are weak findings is that the relationship between the exogamy rate and the white population share is so strong that it swamps any other omitted factors. But the relationship between marriage rates and the white population share is not so strong or clear so the estimated relationship can easily be affected by omitted factors. d. Spousal Education So far we have looked at the quantity of marriage. But it might also be interesting to look at the quality of marriages as one sign of difficulty in the marriage market might be that one accepts a less 17 desirable spouse. Table 7 presents estimates where the dependent variable is the years of education of the spouse. The preferred IV specifications with state effects do suggest a negative significant effect of the non‐black share on spousal education for both men and women i.e. as blacks become a smaller minority they have to settle for a less educated spouse. The estimated effect is large though – as before – with a large standard error. This is perhaps tentative evidence that the marriage market becomes more difficult as one becomes a smaller minority. 3. A Simple Model for Explaining the Results In this section we present a very simple model capable of explaining our results, namely a very strong effect of the population share on the exogamy rate but a weak, possibly zero, effect on the marriage rate. The model is a nested logit in which the lower nest is the type of marriage 9exogamous or endogamous) and the upper nest is whether one marries at all. We should emphasize that there are other models consistent with the data so this is intended as a sketch of the type of model that could explain the results. Define the log odds for an endogamous marriage as: 11m e m y 12 e m 0 1 y11 11 m 1 1 m 12 12 (22) Where 0 represents the preference of type‐1 for endogamy and 1 the sensitivity to the surplus. We will assume that there is gender and exogamy symmetry so that 11 is invariant to population shares. From (22) we can derive straightforwardly that: ln 1m 1 12 (23) As is well‐known the inclusive value will be given by: IV1m ln e m 0 1 y11 11 e y 1 m 12 12 y m ln 1 1 0 1 11 11 m 1 m (24) From which, using (23) we can derive: IV1m 1 m 12 1 1 (25) In the upper nest we have that the overall marriage rate can be written as: e 0 1IV1 m m 1 1 e0 1IV1 m This allows for the demand for marriage to be inelastic i.e. 1 2 0 so that a zero effect on marriage can be consistent with a large effect on the exogamy rate. From (26) we can derive: 18 (26) ln 1m 1 1 1m m IV1 (27) Now consider the equilibrium in the exogamous market: 112m 2 21f (28) 11m 1m 12m (29) 1m 11m 12m 1 1m 12m (30) 11m 2 2f 1 1m 1 2f (31) By definition we have that: And: Using (30) in (28) we have that: Taking logs of (31) and differentiating with respect to ln 2 / 1 we have that: ln 1m 1m ln 1m ln 2f 2f ln 2f 1 1m 12 1 2f 12 12 12 12 1 ln (32) Which using (27) and (23) can be written as: 1 1m 1m 1 2f 2f 1 2 2 2 1 1 1 1m 1 1m 1 2f 1 2f 12 1 ln (33) Now, suppose the minority is small so that 2f . Then (33) combined with (23) implies that: ln 1m 1 1m ln 1m (34) Which implies that a very small minority will have an elasticity in log odds of the exogamy rate which is greater than one. 4. Models of evolution of diversity This section considers what our results imply about the evolution of the diversity of a population i.e. as the minority (here, African Americans) become a smaller proportion of the population are there forces that tend to make them a still smaller proportion in which case the society will tend to become less diverse. Or are there factors tending to increase the size of the minority population that will tend to increase diversity over time. 19 One might measure diversity as the biologists do and look at the evolution of the distribution of genetic characteristics in the population – this has the advantage that we know the process by which genes are passed from one generation to the next but the considerable disadvantage that genes are less relevant in most cases than social and cultural characteristics. But the transmission of culture (in the present application, think of who is classed as ‘black’) is a process we know much less about. So this section restricts its attention to possibilities and which of these possibilities apply in practice is left for further research. Our empirical results suggest that as the share of blacks in the population falls the exogamy rate rises and the impact on the marriage rate is less clear. For the sake of discussion let us assume that the overall marriage rate is unaffected though the stability of mixed marriages may also be important (Zhang and van Hook, 2009). How society evolves then depends critically on what happens to the children of the mixed marriages (see, for example, Montgomery, 2011). There are a number of possibilities. For example, if the ‘one drop’ rule is applied the children of any exogamous marriage will be classified by society as ‘black’. In this case exogamy is a way to increase the proportion of the population is black and, if exogamous marriages produce as many children as other marriages then the black population will grow in size over time. However, there is good reason to think that the ‘one drop’ rule is not strictly applied – some children (and later generations) will be able to and will choose to ‘pass’ as white in the society. Table 8 shows a simple tabulation of the race of children of exogamous marriages. This is the race as reported by parents, not necessarily as seen by the children themselves or the wider society9. We report the frequencies for the 1990 census where it was not possible to report multiple races and in the 2000 census where it was. In the 1990 census we see that 60% of these children were classified by their parents as ‘black’ but 27% as ‘white’ and 13% as ‘other’ an indication perhaps that they felt the census form was an inadequate way to describe their children. In contrast, in the 2000 census 43% are described as having multiple races. The bottom line is that the evolution of diversity is going to be greatly affected by the racial assignment of the children of exogamous marriages (and, one generation later, how their children are described)10. Some have argued that the children of exogamous marriages are accepted by neither of the main races and have ‘problems’ as a result (see, for example, Fryer et al, 2012). If, to give an extreme example, those of mixed race were social pariahs whom no‐one would marry then the high exogamy rate in populations where the black population share was small would cause the population share of African Americans to fall in the next generation. The bottom line is that what happens to the evolution of diversity depends a lot on what happens to the children of mixed marriages. That is a topic for a different paper. 5. Conclusions This paper has studied how the fraction of African Americans in a population affects their outcomes in the marriage market. This is interesting because the understanding of this process will help us to 9 And this may make a difference – Barack Obama is much more commonly referred to as ‘black’ than as having multiple races. 10 For how ethnicity can have a social component, see Duncan and Trejo (2007) on how this affects the measure of the economic progress of Mexican Americans. 20 understand better the forces within our society to increase or decrease diversity over time. It can also help us understand which is the ‘right’ model of the marriage market. Our conclusions are that there is strong evidence that exogamy rates rise as the share of African Americans in the population falls but that the impact on overall marriage rates is less clear‐cut. This implies that in areas where African Americans are a tiny minority a greater fraction of their children will be of mixed race. How they see themselves and the wider society sees them is then of critical importance in how the diversity evolves over time. That is a subject for another paper. While we hope that the study has shed some light on black‐white interactions the question naturally arises as to whether any wider conclusions can be drawn from our findings i.e. whether one can generalize to other groups. First, it should be noted that our conclusions on exogamy are very different from the well‐known studies of Bisin and Verdier (2000) and Bisin, Topa and Verdier (2004). They use a Sewall Wright model of the matching process and add to the mix an incentive for minorities to invest in making it more likely that they meet someone from their own group. Bisin and Verdier (2000, p955) summarize this as “frequency of intragroup marriage (homogamy), as well as socialization rates of religious and ethnic groups, depend on the group’s share of the population: minority groups search more intensely for homogamous mates, and spend more resources to socialize their offspring”. The outcome is that the exogamy rate is a concave function of the minority share, not the convexity we find. Bisin, Topa and Verdier (2004) present evidence for this effect for religions. There are a number of ways to reconcile these contradictory results. First, it may be that race and religion are very different. Most if not all religions (at least the ones that reproduce through time) put some emphasis on bringing up children in the faith and typically marrying someone of the same faith is part of that. Some religions will even have explicit match‐making institutions and a strongly negative view may be taken of those who marry outside the faith. In contrast, there is no organized black group, no matchmaking, perhaps only weak group pressure. So it is perhaps natural if the pressures against exogamy are stronger within religions than races. However, while we do observe tiny minorities that nevertheless manage to avoid extinction and the mechanisms emphasized by Bisin and Verdier are undoubtedly part of the survival mechanisms used, there is a real sample selection problem in focusing attention on them – the very large number of religious cults and minorities that once existed are forgotten. Our societies have not ended up as a collection of very small fragments which suggests the presence of powerful homogenizing forces. So it is likely that race is different from religion but do our conclusions about black‐white interactions carry over to other races or ethnicities? One characteristic of black‐white relations is the high degree of segregation so one should not assume that other races will behave in the same way. Differences between different European heritages were once very significant but have eroded to the point where the adoption of a particular ancestry now seems to be little significance (see Alba, 1990; Waters, 1990). But the mixing of Asians and Hispanics (whose exogamy rates exceed those of African Americans) and mixing in other societies would be interesting to study in order to decide whether general conclusions can be drawn. 21 Appendix A: Proof of Result 1 m Suppose we have y11 11 y11f 11 which implies that the return to an endogamous marriage for 1‐women is higher than the return for 1‐men. Then from (6) the market for endogamous marriages of the type 1s can only be in balance if the return to an exogamous marriage for the 1‐men is lower than that for the 1‐women i.e. we must have that: y12m 12 y21f 21 (35) Using the exogamy symmetry assumption (10) this can be re‐written as: m y21 21 y12f 12 (36) i.e. the return to 2‐men from an endogamous marriage must be less than that for 2‐women. This then implies from the market for endogamous marriages for 2‐people, (7), that: m y22 22 y22f 22 (37) This comes down to the result that the returns to men in every type (endogamous or exogamous) of marriage must be worse than the pay‐off for the women in that type of marriage. This implies that the fraction of men both types wanting to be married is less than the fraction of women of both types wanting to be married. But this is inconsistent with market‐clearing as the number of married men and women must in total be equal. A similar argument but with the roles of men and women reversed applies if y11m 11 y11f 11 . Appendix B: Deriving the Elasticity of Interest from the ‘Control Function’ Model Denote by p x the probability of the outcome variable (being married, having an interracial marriage) for someone with characteristics x . As Blundell and Powell (2004) explain this can be written as: p x p x, v f v x dv (0.38) Where v is the residual from the first‐stage regression that we need to control for. Blundell and Powell (2004) show how the estimation of p x, v will be consistent under the assumption that the instrument is exogenous. In our application p x, v is estimated as a logit function so that (0.38) can be written as: p x e x v f v x dv 1 e x v 22 (0.39) However, Blundell and Powell (2004) also show that the sensitivity of p x, v to x is not a parameter of interest – rather one would like to be able to estimate the sensitivity of p x to x . The sensitivity of ln p x with respect to one of the regressors xi can be written as: 1 p x ln i x i p x 1 p x p x 1 x p x 1 p x xi p x, v 1 p x, v f v x dv p x 1 p x (0.40) Var p x, v x i 1 p x 1 p x Where the non‐zero variance comes from the variation in v given x . One can readily see from inspection of (0.40) that if this variance is zero i.e. there is no endogeneity then this reduces to the derivative of the log odds ratio being constant for everyone. We will use the delta method to approximate the variance of p x, v so that we have: Var p x, v x 2 p x 1 p x Var v x 2 2 (0.41) Substituting (0.41) into (0.40) leads to: i x i 1 2 p x 1 p x Var v x (0.42) This model will lead to a different estimate of the elasticity of interest for each value of x . To summarize we report an estimate of the average of this across different values of x using the sample distribution of x . This leads to: i i 1 2 p x 1 p x Var v x f x dx (0.43) Which we estimate using the sample equivalent: i ˆi 1 N j 1 ˆ 2 pˆ j 1 pˆ j vˆ 2j The standard errors of this are computed using a block bootstrap. 23 (0.44) Table 1 Summary of Different Marriage Market Models for Effect of Reduction in Group Size Frictionless Models Transferable Utility Exogamy Rate Increases, possibly linear in log odds Increases but stops rising once in excess supply i.e. if group small enough Increases but concave in log odds unless α=0 Increases but concave in log odds unless α=0 Non‐Transferable Utility Models with Frictions Transferable Utility Non‐Transferable Utility 24 Marriage Rate Increases Increases but stops rising once in excess supply i.e. if group small enough Falls if Frictions Large Enough Falls if Frictions Large Enough Table 2 The Effect of the Population Share of African Americans on Being Currently Married 1 2 3 4 5 6 7 8 9 10 11 12 Black Women 0.010 (0.04) ‐0.057 (0.02) ‐0.084 (0.02) ‐0.014 (0.018) ‐0.022 (0.015) ‐0.023 (0.015) ‐0.023 (0.015) 0.043 (0.024) ‐0.047 (0.022) ‐0.049 (0.022) ‐0.143 (0.145) Men 0.048 (0.041) ‐0.066 (0.019) ‐0.095 (0.021) ‐0.003 (0.016) ‐0.001 (0.014) ‐0.001 (0.014) ‐0.002 (0.014) 0.070 (0.021) ‐0.154 (0.018) ‐0.152 (0.027) ‐0.884 (0.224) White Women 0.062 (0.019) ‐0.017 (0.022) ‐0.003 (0.023) 0.044 (0.014) 0.002 (0.006) 0.002 (0.007) 0.002 (0.007) 0.005 (0.008) 0.011 (0.007) 0.024 (0.012) ‐0.060 (0.043) Controls Men 0.047 (0.016) ‐0.034 (0.020) ‐0.021 (0.023) 0.032 (0.012) ‐0.013 (0.007) ‐0.012 (0.007) ‐0.011 (0.007) 0.006 (0.007) ‐0.012 (0.007) 0.004 (0.013) ‐0.122 (0.060) 25 None Differences from main specification Cohort, age Cohort,age,education Cohort,age,education, covariates Cohort,age,education, state Cohort,age,education, state, region trends Cohort,age,education, state, state trends Cohort,age,education, state Cohort,age,education, state Cohort,age,education, state Cohort,age,education, state Age 20‐29 only Black share at birth in birth state Black share at birth in birth state, stayers Control Function Notes to Table 2: 1. Regressions are logit models at the individual level. Reported coefficients are logit coefficients. Standard errors are clustered at the state level. Sample is restricted to 3 age groups, 20‐29, 30‐39, and 40‐49 and to cohorts born after 1910 and before 1990. 2. All regressions have both the dependent and independent variable in log‐odds form. 3. The covariates included are residential segregation, the employment‐population rate, the incarceration rate, attitudes to interracial marriage and pre‐marital sex, whether one was affected by a law against interracial marriage and the date at which the state repealed the law against inter‐racial marriage. Table 3 First‐Stage Regressions 1 2 3 4 5 6 7 8 Coefficient (s.e.) 0.546 (0.016) 0.549 (0.015) 0.538 (0.015) 0.531 (0.031) 0.156 (0.042) 0.313 (0.064) 0.075 (0.049) 0.0152 (0.026) Controls None Differences from main specification Cohort, age Cohort,age,education Cohort,age,education, covariates Cohort,age,education, state Cohort,age,education, state Age 20‐29 only Cohort,age,education, state, region trends Cohort,age,education, state, state trends Notes: These are regressions of the log odds of the current white share in the current state of residence on the log odds of the white share in the birth state at the time of birth. Standard errors are clustered on the birth state. 26 Table 4 The Effect of the Population Share of African Americans on the Ever Married Rate 1 Black Women Men 0.142 0.128 (0.053) (0.047) 2 ‐0.024 ‐0.045 (0.029) (0.023) 3 ‐0.039 ‐0.064 (0.031) (0.025) 4 0.045 0.035 (0.028) (0.020) 5 0.028 0.035 (0.025) (0.021) 6 0.028 0.036 (0.025) (0.021) 7 0.026 0.036 (0.025) (0.021) 8 0.072 0.083 (0.028) (0.021) 9 ‐0.038 ‐0.13 (0.048) (0.029) 10 0.028 ‐0.083 (0.035) (0.026) 11 ‐0.153 ‐0.801 (0.280) (0.250) Notes: As for Table 2 White Women 0.050 (0.024) ‐0.035 (0.032) ‐0.016 (0.034) 0.062 (0.018) 0.005 (0.009) 0.022 (0.022) 0.009 (0.009) 0.005 (0.010) ‐0.008 (0.013) 0.022 (0.022) ‐0.012 (0.052) Controls Men 0.034 (0.018) ‐0.045 (0.028) ‐0.028 (0.030) 0.043 (0.016) ‐0.006 (0.008) 0.023 (0.020) ‐0.005 (0.008) 0.005 (0.009) ‐0.006 (0.009) 0.023 (0.020) ‐0.052 (0.032) 27 None Differences from main specification Cohort, age Cohort,age,education Cohort,age,education, covariates Cohort,age,education, state Cohort,age,education, state, region trends Cohort,age,education, state, state trends Cohort,age,education, state Cohort,age,education, state Cohort,age,education, state Cohort,age,education, state Age 20‐29 only Black share at birth in birth state Black share at birth in birth state, stayers Control Function Table 5 The Effect of the Population Share of African Americans on the Exogamy Rate 1 2 3 4 5 6 7 8 9 10 11 Black Women 0.671 (0.067) 0.944 (0.043) 0.942 (0.043) 0.763 (0.025) 0.752 (0.026) 0.753 (0.025) 0.954 (0.033) 0.685 (0.036) 0.268 (0.058) 0.321 (0.062) 2.01 (0.30) Men 0.803 (0.055) 1.082 (0.035) 1.061 (0.033) 0.961 (0.029) 0.949 (0.034) 0.751 (0.026) 0.955 (0.026) 0.845 (0.035) 0.235 (0.058) 0.493 (0.057) 3.14 (0.99) Controls None Differences from main specification Cohort, age Cohort,age,education Cohort,age,education, covariates Cohort,age,education, state Cohort,age,education, state, region trends Cohort,age,education, state, state trends Cohort,age,education, state Cohort,age,education, state Cohort,age,education, state Cohort,age,education, state Notes: As for Table 2 28 Age 20‐29 only Black share at birth in birth state Black share at birth in birth state, stayers Control Function Table 6 Evidence from Administrative Records State Variable State of Marriage State of Residence State of Birth Controls No Yes No Yes No Yes Men 0.866 0.292 0.877 0.312 0.685 0.213 [0.0646] [0.274] [0.0688] [0.0571] [0.0919] [0.0553] Women 0.64 0.103 0.652 0.322 1.169 1.015 [0.0738] [0.330] [0.0777] [0.0608] [0.0701] [0.0569] Notes: 1. All equations are individual –level logit equations with log odds of non‐black share as a regressor. Standard errors are clustered at the state level 2. Sample sizes are about 180000 for regressions without controls, 70000 with. 3. The controls are state dummies, year dummies, education, and age at marriage. Education is the variable missing most of the time. 29 Table 7 The Effect of the Population Share of African Americans on Spousal Education 1 2 3 4 5 6 7 8 9 10 11 Black Women 0.185 (0.056) 0.276 (0.026) 0.184 (0.013) 0.131 (0.015) 0.113 (0.020) 0.104 (0.016) 0.024 (0.021) 0.024 (0.023) 0.113 (0.020) 0.114 (0.020) 0.251 (0.235) Men 0.131 (0.051) 0.178 (0.023) 0.035 (0.012) 0.022 (0.013) 0.021 (0.013) 0.040 (0.011) 0.019 (0.014) 0.037 (0.022) 0.021 (0.013) 0.022 (0.013) 0.257 (0.257) White Women ‐0.078 (0.046) 0.054 (0.032) 0.021 (0.019) ‐0.023 (0.018) ‐0.019 (0.023) ‐0.005 (0.019) ‐0.008 (0.006) ‐0.014 (0.008) ‐0.019 (0.023) ‐0.018 (0.023) ‐0.041 (0.045) Controls Men 0.006 None (0.009) ‐0.004 Cohort, age (0.013) 0.015 Cohort,age,education (0.005) 0.003 Cohort,age,education, covariates (0.004) 0.006 Cohort,age,education, state (0.004) 0.001 Cohort,age,education, state, region (0.003) trends 0.003 Cohort,age,education, state, state (0.004) trends ‐0.002 Cohort,age,education, state (0.007) 0.006 Cohort,age,education, state (0.003) 0.006 Cohort,age,education, state (0.003) 0.024 Cohort,age,education, state (0.036) Notes: As For Table 2 30 Differences from main specification Age 20‐29 only Black share at birth in birth state Black share at birth in birth state, stayers Control Function Table 8 The Racial Classification of the Children of Exogamous Marriages White Black Other Multiple Races Number of Observations 1990 26.8% 60.4% 12.8% 0 9406 2000 15.4% 28.0% 8.1% 43.5% 16281 Notes. 1. The sample is all children aged under 18 who are living with their natural mother and father and who have one parent who is classified as white and one as black. 31 Figure 1 .2 .4 Fraction .6 .8 1 The Fraction of African Americans Ever Married by Birth Cohort and Age Group 1900 1920 1940 1960 Birth Cohort Men, Aged 20-29 Men, Aged 40-49 1980 2000 Women, Aged 20-29 Women, Aged 40-49 Figure 2 0 .2 Fraction .4 .6 .8 The Fraction of African Americans Currently Married by Birth Cohort and Age Group 1900 1920 1940 1960 Birth Cohort Men, Aged 20-29 Men, Aged 40-49 1980 2000 Women, Aged 20-29 Women, Aged 40-49 32 Figure 3 The Fraction of African Americans In Exogamous Marriages by Birth Cohort and Age Group 0 .05 .1 Fraction .15 .2 .25 1900 1920 1940 1960 Birth Cohort Men, Aged 20-29 Men, Aged 40-49 1980 2000 Women, Aged 20-29 Women, Aged 40-49 Figure 4 Variation in the Exogamy Rate for African Americans Across States Black Women, Aged 20-59, 2000 Black Men Aged 20-59, 2000 Vermont Montana .8 .8 Vermont Maine Utah .6 Montana .6 New Hampshire Wyoming Hawaii Idaho South Dakota Iowa New Mexico Oregon Washington North Dakota Arizona Exogamy Rate .2 .4 Exogamy Rate .4 Maine Utah Idaho Wyoming New Hampshire South Dakota Colorado Rhode Island Nevada West Virginia Minnesota Alaska Kansas Nebraska California Massachusetts Wisconsin Oklahoma Kentucky Connecticut Indiana Pennsylvania Ohio Delaware Missouri Texas New Jersey New York Virginia Michigan Florida Illinois Tennessee Maryland North Carolina Arkansas Georgia Louisiana South Carolina Alabama Mississippi District of Columbia District of Columbia -.2 0 New Mexico Washington Alaska Arizona Minnesota Rhode Island Hawaii Colorado North Dakota Iowa Massachusetts Nebraska California Nevada West Virginia Connecticut Wisconsin Kansas New York Kentucky Oklahoma Pennsylvania New Jersey Indiana Ohio Florida Missouri Delaware Michigan Illinois Texas Virginia Maryland Tennessee North Carolina Georgia Arkansas Alabama South Carolina Louisiana Mississippi 0 .2 Oregon 0 .2 .4 Black Share in State .6 0 .2 .4 Black Share in State .6 33 Figure 5 Variation in the Exogamy Rate for African Americans Across States: Log‐Odds Version Black Men, Aged 20-59, 2000 2 2 Black Women, Aged 20-59, 2000 Vermont Vermont Montana Maine Montana Utah 0 Maine Log Odds Exogamy Rate -2 0 Log Odds Exogamy Rate -2 Utah Idaho Wyoming New Hampshire South Dakota Oregon Wyoming New Hampshire Hawaii Idaho South Dakota Iowa New Mexico Oregon Washington North Dakota Arizona -4 New Washington Mexico Alaska Arizona Minnesota Rhode Island Hawaii Colorado North Dakota Iowa Massachusetts Nebraska California Nevada West Virginia Connecticut Wisconsin Kansas NewJersey York Kentucky Oklahoma Pennsylvania New Indiana Ohio Florida Missouri Delaware Michigan Illinois Texas Virginia Maryland District of Columbia Tennessee North Carolina Georgia Arkansas Alabama South Carolina Louisiana Mississippi Colorado Rhode Nevada Island West Virginia Minnesota Alaska Kansas Nebraska California Massachusetts Wisconsin Oklahoma Kentucky Connecticut Indiana Pennsylvania Ohio Delaware Missouri Texas New Jersey New York Virginia Michigan Florida Illinois Tennessee Maryland North Carolina Arkansas Georgia Louisiana District of Columbia South Carolina Alabama -6 -4 Mississippi -6 -4 -2 Log Odds Black Share in State 0 -6 -4 -2 Log Odds Black Share in State 0 34 Figure 6 Variation in Marriage Rates for African Americans Across States Black Men Aged 20-59, 2000 Fraction Ever Married .5 .55 .6 .65 .7 .75 Fraction Ever Married .5 .6 .7 .8 Black Women Aged 20-59, 2000 Alaska Idaho New Hampshire Hawaii Wyoming South Dakota Maine Texas Colorado North Dakota Arizona Nevada Oklahoma Virginia Florida Washington New Kansas MexicoArkansas Vermont North Carolina Alabama South Carolina Georgia Oregon Louisiana California Tennessee Maryland Montana Kentucky Mississippi Delaware Nebraska Indiana Missouri West Virginia Utah New Jersey Ohio Iowa Massachusetts Rhode Connecticut Island Minnesota Michigan New York Illinois Wisconsin Pennsylvania Idaho Wyoming New Mexico Alaska Hawaii Texas Colorado Virginia North Oklahoma Utah Dakota Arkansas North Carolina Nevada Delaware Washington New Hampshire Maryland South Carolina Alabama Kansas California Florida Oregon Georgia Arizona Kentucky Tennessee Louisiana West Virginia Missouri Nebraska Indiana Wisconsin Connecticut New Illinois Jersey Rhode Island Mississippi Iowa Dakota South Ohio Minnesota Massachusetts Maine Michigan New York Montana Pennsylvania District of Columbia 0 .2 .4 Black Share in State .6 Alaska New Hampshire South Dakota New Mexico Maine Montana Colorado Virginia Texas Florida North Carolina Arizona Oklahoma South Carolina Oregon Utah Kansas Georgia Washington Arkansas Maryland Delaware Alabama Nevada Louisiana New Jersey West Virginia Mississippi Rhode California Island Tennessee Massachusetts Connecticut Indiana Kentucky Iowa Minnesota New York Nebraska Ohio Michigan Missouri Illinois Wisconsin Pennsylvania Vermont 0 .6 Hawaii Idaho New Dakota Mexico North Alaska Colorado Texas Virginia New Hampshire Wyoming Oklahoma North Carolina Utah Maryland South Carolina Delaware Arkansas Washington Georgia Florida Alabama California Kansas Nevada Rhode Island Arizona Kentucky New Jersey Connecticut Louisiana Tennessee South Dakota Oregon Wisconsin Massachusetts West Virginia Iowa New York Mississippi Maine Indiana Minnesota Illinois Nebraska Missouri Pennsylvania Michigan Ohio Montana District of Columbia .2 .4 Black Share in State .2 .4 Black Share in State Black Men Aged 20-59, 2000 Fraction Currently Married .3 .35 .4 .45 .5 .55 Fraction Currently Married .2 .3 .4 .5 .6 Wyoming Hawaii District of Columbia 0 Black Women Aged 20-59, 2000 North Dakota Idaho Vermont .6 Vermont District of Columbia 0 .2 .4 Black Share in State .6 Figure 7 0 Fraction of Kids Classified As White .1 .2 .3 .4 The Racial Classification of the Children of Exogamous Marriages 0 .2 .4 .6 Black Share 35 Figure 8 The Share of African American Population by Census and Year Middle Atlantic East North Central West North Central South Atlantic East South Central West South Central Mountain Pacific 0 .2 .4 0 .2 .4 0 .2 .4 New England 1900 1950 2000 1900 1950 2000 1900 1950 2000 Fraction Black Birthplace Current Residence Graphs by region Figure 9 Exogamy Rates by Religion catholic jewish none 0 1 0 .5 Exogamy Rate .5 1 protestant 0 .2 .4 .6 .8 0 .2 .4 .6 .8 Fraction of Population Own Religion Graphs by religion in which raised 36 References Adachi, Hiroyuki. 2003. 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