The Demographic Effect of Mixed MarriagesAuthor(s): Fjalar FinnäsSource: European Journal of Population / Revue Européenne de Démographie, Vol. 4, No. 2(Jun., 1988), pp. 145-156Published by: SpringerStable URL: http://www.jstor.org/stable/20164474 .
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European Journal of Population 4 (1988) 145-156
North-Holland
145
THE DEMOGRAPHIC EFFECT OF MIXED MARRIAGES
Fjalar FINN?S *
Abo Akademi, Vasa, Finland
Received September 1988, final version received November 1988
Abstract. This paper gives a formal expression for the demographic effect of mixed
marriages; i.e., the effect on the number of children, and thereafter illustrates the
long-term effects of these marriages with a simple simulation model
R?sum?. L effet d?mographique des manages mixtes
Cet article formalise l'effet d?mographique des mariages mixtes, c'est-?-dire leur effet
sur le nombre d'enfants, et illustre ensuite l'effet ? long terme de ces mariages en
utilisant un mod?le simple de simulation.
1. Introduction
Social scientists have shown great interest in the study of different
kinds of mixed marriages. The frequency of mixed marriages has often been considered to be the most conclusive and objective indicator of the degree of assimilation of a minority (see, e.g., Mittelbach and
Moore (1968)). Mixed marriages are of great interest from a demo
graphic point of view, too, but so far very little research has been done
in this respect. Unless one of the two spouses joins the group to which his or her
partner belongs (this is possible for example in religious groups), a mixed marriage does not by itself have any direct effect on population
size. The demographic effect of the mixed marriage appears in the
generations which follow, mainly via the classification of the children
of such marriages into subpopulations. In the case of language groups this classification is not necessarily predetermined. In this paper, I will
first derive a formal expression for the demographic effect of mixed
* Author's address: Social Science Research Unit, Vasaesplanaden 15B, SF-65100 Vasa, Finland.
0168-6577/88/53.50 ? 1988, Elsevier Science Publishers B.V. (North-Holland)
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146 F. Finn?s / Demographic effect of mixed marriages
marriages i.e., the effect on the number of children, and thereafter
illustrate the long-term effects of these marriages with a simple simula
tion model.
2. The first-generation effect
In what follows I will ignore classification problems. Further, I will consider only populations in which all individuals can unequivocally be classified according to a given variable - such as religion, citizenship, race or language. In principle the number of groups involved could be
more than two, but for the sake of simplicity and clarity I shall include
only two subpopulations. Transitions between the two groups are
allowed but, at any given time, the classification is presumed to be (and is kept) always unequivocal. To make the presentation easier, I shall
deal with the following two language groups: Finnish and Swedish. Assume that we study a closed cohort; that the fertility of this cohort
is independent of its linguistic composition, and that the proportion of its members getting married is the same regardless of the existence or
non-existence of mixed marriages. The number of males and females is
assumed to be equal, and there are no differences between the sexes in
any respect. The children of mixed marriages could belong to either
language group, but in the case of unilingual marriages they are
assumed to have the same language as their parents. If there are
language shifts between the different groups, they are assumed to take
place at the moment of marriage. After a language shift of one of the
spouses, the new homogeneous marriage is considered to be no differ
ent from the originally unilingual ones.
To study the effect of mixed marriages under the assumptions noted
above, we have to keep track of the marriages that remain mixed, the
language of the children in these marriages, and the frequency of the
language shifts involved together with their direction. Given the fairly
well-known result that the frequency of mixed marriages in a subpopu lation is to a great extent determined by the relative size of the
subpopulation (see e.g. Blau (1977)) we think it necessary to study the
effect of relative subpopulation size in differing situations. From the
study of the Swedish population in Finland it is also evident that the
other factors may depend on the linguistic composition of the popula tion. I shall introduce the following notations, where x (0 < x < 1) is
the proportion of Swedes in the population:
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F. Pinnas / Demographic effect of mixed marriages 147
nx = the proportion (among all Swedes who marry) of Swedish per
sons marrying Finns,
ux = the proportion of the children who are Swedish in mixed mar
riages that remain mixed,
kx = the proportion of mixed marriages in which one of the spouses
shifts to the language group of the other,
wx = the proportion of language shifts directed from the Finnish to
the Swedish language group. To study the effect of mixed marriages we have to compare the
number of children in a cohort where mixed marriages occur, with one
in which only unilingual marriages take place. In the latter case the number of Swedish children, Sb0(x), can be expressed as
Sb0(x) =
clc2x,
where cx and c2 are constants representing the total number of
marriages taking place and the number of children per marriage,
respectively. Since the total number of marriages is cv the number of Swedish
males as well as females getting married is cxx. Under the assumptions made, c1xnx of both sexes will marry a Finnish partner resulting in a
total of 1clxnx mixed marriages and cxx(l -
nx) unilingual Swedish ones. Out of the mixed marriages, 2clxnxkx become unilingual as a
result of language shifts, 2cxxnxkxwx become unilingual Swedish and
2cxxnxkx (1 -
wx) unilingual Finnish. The number of unilingual Swedish and mixed marriages are therefore ctx(l
- nx) + 2clxnxkxwx
and 2cxxnx (1 -
kx\ respectively. If mixed marriages occur, the total number of Swedish children is
therefore
Sb^x) =
clClx{{\ -
nx) + 2nxkxwx + 2nx(l -
kx)vx).
The effect of mixed marriages, i.e., the relative change in the number of Swedish children, is then
dix)=Sbl%(SX)?{X) ="?kx(2?x-l) + (l-kx)(2vx-l)).
In any given population we have to estimate the values of the functions nx, kx, wx and vx to calculate the expression for d(x). If we
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148 F. Finn?s / Demographic effect of mixed marriages
Table 1 The relative effect of mixed marriages on the number of Swedish children under different
linguistic conditions with vx =
(1 + x)/3, wx =
(1 4- 3x)/5 and nx =
(1 -
x)/2.
0.1 0.2 0.3 0.4
??5 -0.154 -0.165 -0.177 -0.188
0.10 -0.130 -0.139 -0.149 -0.158
0.25 -0.068 -0.073 -0.078 -0.083
0.50 0.000 0.000 0.000 0.000
0.75 0.023 0.024 0.026 0.028
0.90 0.014 0.015 0.017 0.018
0.95 0.008 0.009 0.009 0.010
study not just one, but several populations under widely varying
conditions, it may be necessary to estimate their entire functional
forms. This was the case in the study of the Swedish population in
Finland, since the Swedish proportion of the total population varies
very much in the different municipalities (Finn?s (1986)). To illustrate the magnitude of the expression d(x), I will now present some calcula tions for different linguistic situations and varying proportions of
language shifts. To make the assumptions about the functions realistic,
I have based them on the results from the Finnish study. I assume that the proportion intermarrying is nx
= (l? x)/2\ that a proportion
wx =
(1 + 3x)/5 of the language shifts is directed from Finnish to
Swedish, and that the Swedish proportion of the children in the
remaining mixed marriages is ?x-(1 + x)/3. The frequency of lan
guage shifts is assumed to be independent of the linguistic conditions,
i.e., kx = k. Note that these assumptions imply that the language
groups in question are equal in the sense that the outcome is indepen
dent of which group makes up the majority. Now let us look at the situation where the Swedes are one tenth of
the whole population, and there are language shifts in 30 per cent of
the originally mixed marriages, i.e., fc = 0.3. Under the assumptions
made, we then have n01 =
0.45, i.e., 45 per cent of the Swedes marry
Finnish partners, and since v01 =
0.367, 36.7 per cent of the children in
the remaining mixed marriages becomes Swedish. Further, w0l =
0.26,
which means that 26 per cent of the language shifts take place from the
Finnish to the Swedish language group. Taken together, the effect of
the mixed marriages is a reduction of the number of Swedish children
by 14.9 per cent. With language shifts in 40 per cent of the mixed
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F. Finn?s / Demographic effect of mixed marriages 149
marriages, the number of Swedish children is reduced by a total of 15.8
per cent. Out of this total reduction, 7.2 per cent is the result of the
marriages that remain bilingual, while the rest, or 8.6 per cent, arise
from the marriages in which language shifts among the parents occur.
These figures show that if we are interested in the total effect of mixed
marriages, and if transitions between the groups are possible, then it is
of decisive importance to start from the contracted marriages instead of
studying the remaining mixed marriages only.
Although the effect for both groups is of equal size in absolute
figures, the relative effect is much smaller for the majority. The reduction of 15.8 per cent mentioned above corresponds to an increase
of only 1.8 per cent for the majority. Another result of the shift in favour of the majority is that the Swedish proportion of all children is reduced to 8.4 per cent.
The expression for the total effect presented above is based on a
comparison of the number of children in a cohort in which mixed
marriages occur with the number in a situation where all the contracted
marriages are unilingual. For that reason this effect may be called a
first-generation effect. If we regard mixed marriages as part of an
assimilation process, it is also interesting to study the effect in a
long-term perspective. We should then be aware of the fact that the
effect presented above may change from one generation to the next.
Note first that even in a closed population, the relative sizes of the
subpopulations will probably change, as an effect of the mixed mar
riages. Note also, at least for religious (Thomas (1951)) and language groups (Finn?s (1982)), that persons with a homogeneous background tend to choose their partners more endogamously than do persons with
a heterogeneous background. This means that the function nx, intro
duced before, will change in successive generations, and that the effect
of mixed marriages will accumulate. One way of studying how the
function nx, as well as the total effect d(x), changes is to specify a
model for the mating process, and to study consequent developments in
successive generations. In the next section, I give a short presentation of such a model and the main results obtained from its use.
3. The long-term effect
The model constructed for the mating process is based on the
assumption that all individuals can be classified as 'endogamous' or
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150 F. Finncis / Demographic effect of mixed marriages
'exogamous' with respect to their behaviour in the mate selection
process. 'Exogamous' persons are assumed to select their partners
randomly, ignoring the factor of language, while a marriage between two 'endogamous' persons from different groups is taken to be impossi
ble. This means that 'endogamous' persons choose their partners only
among persons from their own language group and among 'exogamous'
persons from the other one. A much more easily understood descrip tion is obtained in our case if we replace the terms 'endogamous' and
'exogamous' by 'unilingual' and 'bilingual', respectively, thus taking into account actual ability to use language. The model then implies that
mate selection is done randomly as regards language; the combination
'unilingual Swedish' and 'unilingual Finnish' being considered impossi ble. The process is assumed to be independent of age and sex, and all
the persons are assumed to be equally active in the marriage market.
Family background enters the model through the classification of individuals as 'endogamous' or 'exogamous'. I assume that all persons
with a heterogeneous background and a certain proportion of those
with a homogeneous background are 'exogamous', while the rest of the
latter group is 'endogamous'. Further, the division of persons with a
homogeneous background into 'endogamous' and 'exogamous' is as
sumed to be dependent on the relative sizes of the subpopulations. The mate selection process under discussion can also be described
mathematically as an urn model with two urns. Assume that we have
one urn with blue (men) balls and one with red (women) ones. In both urns the balls have a number, 1, 2, 3 or 4, corresponding to 'endoga mous Swedish', 'exogamous Swedish', 'exogamous Finnish' and 'endo
gamous Finnish', respectively. For every pair to be formed we first pick an urn at random, and then draw a ball (a 'suitor', who may be a
female) from it also at random. From the other urn we draw another
ball at random. If the two balls form a forbidden combination (1-4 or
4-1), the latter ball is put back into the urn, and a new ball is drawn
from it. We continue until a permitted pair is obtained. This pair of
balls is put aside before we start to form a new pair. If it is impossible to find a permitted 'partner' for a given 'suitor', the ball corresponding to the latter has to be put back and we have to draw another one
instead. The process terminates when new permitted pairs cannot be
formed, or when a predetermined proportion of the balls has been
drawn.
The urn model is simple, though no explicit mathematical expression
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F. Finn?s / Demographic effect of mixed marriages 151
for the expected outcome of the process has yet been found. In
principle we can calculate the expected outcome recursively, but this
requires an enormous amount of computer resources in practice. Another and much simpler method is to simulate the process, and that
is what I have done.
Although I have simulated the mate selection process in stochastic
fashion, all the other calculations measuring the effect of mixed mar
riages are deterministic. I started from the outcome of the process, i.e., the number of marriages (pairs) of different combinations, and as
sumed that the number of children was independent of the linguistic composition of the marriage. Language shifts are permitted, and the
language of the children in the remaining mixed marriages is de
termined by a function vx. Assuming that children choose partners of
their own generation, and that this happens according to the same
pattern as their parents, it is possible to study how the effect of mixed
marriages changes in successive generations as the result of changes in
the composition of the population with respect to family background. In my own calculations I have started from a cohort in which all
persons share the language of their parents. There are no differences
between the sexes - neither in behaviour nor in quantity. Looking at
the standard deviations I concluded, after some experiments, that a
cohort size of 2000 persons and 100 repetitions of the process would render results reliable enough to my purposes. The calculations were
carried out for cohorts with different linguistic compositions. All
generations were supposed to have the following characteristics:
- The proportion that marry is 90 per cent. - The proportion of 'exogamous' persons, of those with a homoge
neous background, is directly proportional to the relative size of the other language group. Denoting this proportion ux, we can write
ux ?
h(\ -x) for the Swedish population. Here h is a parameter, and I have used values for h in the interval 0.1-0.6.
- The Swedish proportion of children in mixed marriages is:
o,?(1 + jc)/3. - The frequency of language shifts is independent of the linguistic
composition of the population, i.e., kx = k. The values for k are in
the interval 0.0-0.5.
I have not assumed a certain function wx for the direction of the
language shifts, but my starting point has been the composition of the
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152 F. Finn?s / Demographic effect of mixed marriages
marriages. In a mixed marriage between an 'endogamous' and an
'exogamous' person, it is always the latter person that has made the
shift. If both persons are 'exogamous' the proportion of language shifts
from the Finnish to the Swedish group is equal to x, i.e., the relative size of the Swedish population.
Before studying the total effect of mixed marriages, let us take a look at the development of their frequency in successive generations. For
this purpose, instead of nx, I have used a function px9 defined as
A ?
"*/(!-*)
The advantage of this function can be seen as follows. If the proportion
that marry is the same in both language groups, then we can write
"** =
>h-*(l-*)
This expression merely indicates that the number of mixed marriages is
the same in both language groups. Now, if we have nx =px (1 -
x) as
above, we also get
which means that the function p is a measure that is standardized with
respect to the linguistic composition of the population. In fact, the
function p is 1 ? k where k is the conditional kappa used by Rust and
Seed (1985) for example. The function p can consequently be consid
ered as an indicator of the level of endogamy in the population. The
higher the value of the function, the smaller the importance of language
in the mate selection process. In a situation where the selection is made
at random with respect to language, the expected value of p is one.
One effect of the mating process used here is that the proportion
that marries is somewhat lower within the minority than within the
majority. The differences are, however, so small that the effect on the
function p is almost negligible, and it is not necessary to calculate
separate values for p for the two language groups.
On the assumptions made, the function p in fact depends on four
variables, all of which have to be considered. These variables are the
Swedish proportion in the first generation (x), the generation (g), and
the parameters h and k. As expected the value of the function p
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F. Finn?s / Demographic effect of mixed marriages 153
Interpretation. The numbered points in fig. 1(a) are the values of the function p in the first five
generations. To be able to illustrate several development alternatives in the same figure, a line is
drawn through the points. To study the development in detail in figs. l(b)-l(d), a step-function
corresponding to that in fig. 1(a) should be plotted.
increases from one generation to the succeeding one (fig. 1(a)). How
ever, after 4-5 generations the increase has decreased and the value of
p is almost a constant. The proportions of mixed marriages within a
minority still increases somewhat, because its relative size decreases
successively, and /?(x, g, k, A) is a decreasing function with respect to
x (fig. 1(b)). The parameter h has a direct effect on the number of 'exogamous'
persons, and the greater this proportion, the more frequent are mixed
marriages. Thus the function p(x, g, k, h) is an increasing function
with respect to h (cf. fig. 1(d)). On the other hand, it is decreasing with
respect to fc, since an increasing number of language shifts will reduce
the number of 'exogamous' persons (fig. 1(c)). In what follows I shall present a specific case in some detail so as to
illustrate the total long-term effect of mixed marriages. In my example,
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154 F. Finn?s / Demographic effect of mixed marriages
Table 2 The expected effect of the mixed marriages on the number of children in five successive
generations for a minority that originally is one fourth of the total population, h = 0.4 and
k = 0.3. Per cent. ab
Gene- p The effect of mixed marriages (4) (5) ration
(?) (2) (3) Total
? 0.404(0.0024) H?5 ^l? -?? -7.4(0.17) 18.9 (0.12) m 2 0.587(0.0030) -1.0 -4.6 -5.6 -11.2(0.17) 28.9(0.16) 20.6
3 0.678(0.0032) -1.3 -6.5 -7.3 -15.1(0.19) 35.2(0.19) 17.4
4 0.718(0.0037) -1.4 -8.3 -8.8 -18.5(0.23) 39.3(0.24) 14.2
5 0.741(0.0045) -1.8 -9.9 -10.4 -22.1(0.25) 42.8(0.33) 11.1
a (1): the effect of a lowered proportion that marry; (2): the effect from originally mixed
marriages that become unilingual owing to language shifts among spouses; (3): the effect of the
classification of the children in the remaining mixed marriages; (4): the proportion of the
minority that has a heterogeneous background; (5): the proportion of the minority of the total
cohort. b
The numbers in parentheses are standard errors after 100 simulations
the Swedes constitute a minority of one fourth of the original popula tion, and the parameters h and k are assumed to have the values
h = 0.4 and k = 0.3. The effect in five successive generations is pre
sented in table 2.
The value 0.4 for the parameter h means that in the original cohort
30 per cent of the Swedes and 10 per cent of the Finns were 'exoga
mous' (0.4 0.75 = 0.30 and 0.4 0.25 = 0.10). If 90 per cent of the total
population marry, the expected outcome of the mate selection process
is that the corresponding proportions for the Swedes and the Finns are
89.2 and 90.4 per cent, respectively. Out of all the Swedes that marry,
30.3 per cent have a Finnish partner giving the value of 0.404 for the
function p. Since we assume that k = 0.3, there are language shifts in
30 per cent of all mixed marriages, and in 33.8 per cent of these cases it is the Finnish spouse who moves to the Swedish group. Since v025
=
0.417, 41.7 per cent of the children in the remaining mixed marriages
become Swedish. Taken as a whole this means that the expected
number of Swedish children is reduced by 7.4 per cent as a result of
mixed marriages. The corresponding relative gain for the Finns is 2.5
per cent. Note that of all the Swedish children, 18.9 per cent have a
heterogeneous background. In the succeeding generations the expected value of the function p
increases via 0.587, 0.678 and 0.718 to 0.741. This, in combination with
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F. Finn?s / Demographic effect of mixed marriages 155
Table 3 The cumulated relative reduction of the second child-generation of a minority that originally was
one fourth of the total population, owing to mixed marriages. Different values of the parameters h
and k.
0.1 0.2 0.3 0.4 0.5 0.6
O? 0.046 0.092 0.120 0.153 0.174 0.192 0.1 0.045 0.091 0.129 0.159 0.193 0.220
0.2 0.044 0.090 0.129 0.169 0.207 0.242
0.3 0.043 0.088 0.129 0.180 0.221 0.269
0.4 0.041 0.088 0.135 0.184 0.232 0.284
0.5 0.037 0.087 0.137 0.189 0.251 0.301
the decreasing proportion of the total population, results in an ever
increasing loss for the Swedish minority. From the fourth to the fifth generation the loss is 22.1 per cent, and cumulated over all five
generations it is 55.7 per cent. This means that the Swedish proportion of the fifth generation is only 11.1 per cent.
Besides the quantitative effects illustrated above, mixed marriages have a considerable qualitative impact on the minority and its composi tion. This is clearly illustrated by the increasing proportion of the
minority that has a heterogeneous background. In the example above, no less than 42.8 per cent of the Swedish children in the fifth genera tion are descended from mixed marriages.
To illustrate the impacts of h and fc, the total effect cumulated over two generations is presented in table 3. The figures in the table express the relative reduction of a minority that originally constituted one
fourth of the total population. Since the parameter h has a direct impact on the total effect through
the function j?, it is quite natural that it is an important factor in the long run. The influence of the parameter k is not equally predictable.
On the one hand, an increasing value of k has a decreasing effect on
the function /?, but, on the other hand, a higher value of k results in a greater direct effect owing to language shifts. Therefore, the latter
aspect is more pronounced for high values of A, when the influence of the linguistic conditions of the environment is greater.
Like most statistical models, the one used here is a rough simplifica tion of reality. In this case this is so especially for the classification into
'endogamous' and 'exogamous' groups. In practice one should use a
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156 F. Finn?s / Demographic effect of mixed marriages
much more refined scale than that used here, which consists of the
extremes only. As a consequence it is in general hardly meaningful to
try to estimate the parameter A, or the function ux. This also means
that the model should be used for illustrative purposes only. The
essential point in this particular case is that the model clearly illustrates
how, in a simplified situation, mixed marriages have a pronounced cumulative quantitative effect. Furthermore, that mixed marriages have
a considerable qualitative impact on the minority is also illustrated by the model, and this is certainly of great importance for an understand
ing of the assimilation process.
References
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