the mathematics of sex and marriage, revisited

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The mathematics of sex andmarriage, revisitedMaia Martcheva a & Fabio A. Milner ba Department of Mathematics , PolytechnicUniversity, Six MetroTech Center , Brooklyn, NewYork, 11201b Department of Mathematics , PurdueUniversity , WestLafayette, Indiana, 47907–1395E-mail:Published online: 21 Sep 2009.

To cite this article: Maia Martcheva & Fabio A. Milner (2001) The mathematicsof sex and marriage, revisited, Mathematical Population Studies: AnInternational Journal of Mathematical Demography, 9:2, 123-141, DOI:10.1080/08898480109525499

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THE MATHEMATICS OF SEX ANDMARRIAGE, REVISITED

MAIA MARTCHEVAa and FABIO A. MILNERb,*

aDepartment of Mathematics, Polytechnic University, Six MetroTechCenter, Brooklyn, New York, 11201

bDepartment of Mathematics, Purdue University, WestLafayetteIndiana 47907-1395

We analyze the problem of modeling marriages in a two-sex model of populationdynamics. We first deal with the problem of incomplete and inconsistent census data andthen use a simulator to compare the performance of a variety of marriage functions inmodeling births and couples during the ten-year period between consecutive U.S. cen-suses. Unlike most empirical methods for comparing marriage functions based ongoodness of fit, the differences in the projections of the various functions in our methodare of the same magnitude (or even smaller) than the errors between the projected andreal data. We observe that for the population of the United States, the harmonic meanfunction frequently found and used in the literature is a quite poor performer whencompared with many other functions in the family we use.

KEYWORDS: Two-sex population; Marriage; Modeling; Simulation

1991 MATHEMATICS SUBJECT CLASSIFICATION: 65M10, 65M20, 92A15,65C20

1. INTRODUCTION

A vast bibliography on a variety of population models and theirproperties already exists. See, for example Coale (1972), Hoppensteadt(1975), Keyfitz and Flieger (1971), McFarland (1975b), Parlett (1972),and the references cited therein. Most of the models are concerned

* Corresponding author. E-mail: [email protected].

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124 M. MARTCHEVA AND F. A. MILNER

with one population that may be unstructured, or may be structuredaccording to one or more important parameters such as age, gender,race, size, economic status, etc.

For human populations it is quite useful to have models that arestructured at least by gender and age, because many health care,education, and social security issues, for example, depend on thegender and age structure of the population. Yet few studies of math-ematical properties of such models have been done and, in particular, amajor gap still exists in the study and modeling of marriage functions(Hadeler, 1989; Hadeler et al, 1988; Keyfitz, 1972; McFarland, 1972;Parlett, 1972). We shall concentrate in deterministic models for mono-gamous populations, in which the past history of marriages anddivorces plays no role in future behavior with respect to theseprocesses.

Among all the processes intervening in the dynamics of humanpopulations, perhaps migration is the one least understood and mostdifficult to model. In this paper we shall ignore it entirely and assumethat the population is closed in the sense that individuals arrive into itonly by birth and leave only by death. Processes related to births anddeaths are among those we can understand and model with greatestease and, when reliable data are available, lead to fairly accuratepredictions of the evolution of the population even when modeledlinearly. In terms of a gender structured model, this means thatthe equations used for the dynamics of females and males can bedeterministic and the error, stemming from the real life randomness,can be neglected.

The equations used to model couples become necessary in order tohave some closed form for the birth functions of females and males.Ignoring couples and applying linear extrapolation of known birthdata leads to significant errors by the tenth year when a new censusis made and new data are thus acquired to compare prediction andreality. This creates a need for modeling couples separately, inwhich case linear birth processes can be described in terms of theage distribution of couples and their fertility and more accurate longterm projections are obtained.

If we choose to model couples using differential equations, thesemust include terms corresponding to divorces, separations, and mar-riages. The dynamics of divorces and separations is fairly wellunderstood and it is actually modeled very similarly to the deathprocess. In fact, both of them do mean the death of the couple. Con-sequently, they can be modeled linearly without the introduction of

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THE MATHEMATICS OF SEX AND MARRIAGE 125

great errors. Marriages are much more complex (McFarland, 1972;Parlett, 1972). Assuming that they are constant leads to very poorestimation of births from married couples and therefore should beavoided. In fact, it is expected that the mating process is nonlinear interms of available "singles" (McFarland, 1972).

The problem of existence of a marriage function and its form startedbeing widely discussed sometime in the beginning of the 1970s. It wasinteresting both from the point of view of the two-sex model, becauseit was expected to help make sense of male and female marriage rates(Keyfitz, 1972), and also because people were interested in forecastingmarriages for different (mainly business-related) purposes. As demo-graphers define it (McFarland, 1972), a marriage function is a functionfor predicting the number of marriages which will occur during a unitof time between males in particular age categories and femalesin particular age categories, from knowledge of the numbers availablein the various categories.

Marriage is a complex socio-economic process influenced by manyfactors. Just to mention some of them: dominating perceptions andrituals, religion, health, economy, educational status, racial and ethnicinteractions, age and sex composition. As can be seen from the definitionof marriage function, it is assumed that the age and sex compositionare the only essential properties that should be explicitly taken intoaccount in modeling marriages.

A general perception in demography is that the marriage systemrepresents a market and is ruled by laws of competition. At a personallevel marriage is an act expected to bring more comfort in life (South,1992) - health (Lillard and Panis, 1996) and/or premarital childbearingare some of the factors which can decrease one's chance for gettingmarried (see also Bennett et al., 1992).

Several authors have proposed a variety of marriage functionsbased on what seem to be reasonable mathematical properties suchfunctions should have. See, for example Hadeler (1989), Hadeleret al. (1988), McFarland (1972) and Parlett (1972) for a discussion ofsome of these properties. But comparative studies relative to actualpopulation data involve few of the known candidates and we fre-quently find a weighted harmonic mean as the function used inmathematical models (Hoppensteadt, 1975; Keyfitz, 1972), althoughdemographers know that it is not a good choice (McFarland, 1972;Parlett, 1972).

In this paper we are interested in comparing different marriagefunctions in their predictive power, not so much with respect to the age

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distribution of couples, but rather to their total number and especiallywith respect to the progeny they generate.

The plan of this paper is as follows: In §2 we describe a model forthe dynamics of a two-sex population such as that of humans. In §3we discuss some problems associated with incomplete and/or incon-sistent census and vital statistics data. In §4 we present a wide varietyof marriage functions and compare their performances using resultsfrom numerical simulations. The simulations are run with demo-graphic data for the U.S.A. from 1970 and 1980. Finally, in §5 wemake some remarks and draw some conclusions.

2. A TWO-SEX MODEL

A fairly standard demographic model for human populations,structured by gender and age (Health Resources Administration,1974) is derived from a coupled system of partial differential equa-tions of McKendrick type (McKendrick, 1926). Let it/(x, t) andum(y, t) denote, respectively, the age densities of females and malesin the population, where x and y refer to the ages of females andmales, respectively. These age densities satisfy the following integro-differential systems:

duf . . , „-g^ = -¥.*' 0«/» X>0, t>0,

/*0O rOO

uj{0, t) = Bj{t) = / / PA*,y, t)c(x,y, t)dxdy, t > 0, t2-1)Jo Jo

ujix,0) = u}(x), y>0,

—-+-^-——Sm{y-,t)um, y > o, / > o,

um(0,t)=Bm(t)= / / 0m(x,y,t)c(x,y,t)dxdy, t > 0,7o Jo

um{y,0) ^ w^Cv)) y ^ 0,

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where c(x,y, t) is the density of couples with female of age x andmale of age y; 6/(x, t) and 6m(y, i) are, respectively, the age specificdeath rates of females and males; and /?/ and fim give, respectively,the productivity of such couples for female and male progeny (thatis, (3f(x,y,t) is the average number of daughters and (3m(x,y, t) theaverage number of sons born at time t to a couple with female of agex and male of age y). The function c = c(x,y, i) is the solution of thefollowing nonlinear initial boundary value problem:

(x,j>,f;a/,5m), x > 0 , y> 0, f > 0,

c{x,0,t) = c(0,y,t)=0, x>0,y>0,t>0, (2'3)

.^,7,0) = ^ ^ , ^ ) , x>0, ^>0,

where the first term on the right hand side of the differential equationdescribes the change in c due to the separation of couples (by death,annulment, or divorce) while the second term describes the source ofcouples - that is marriages, which depends on the distribution of"single" females and males,

roo

Sf{x, t) = Uf(x, t) - I c(x, y, t) dy,Jo

roo

sm(y,t) = um (y, t)- c{x, y, t) dx.Jo

(2.4)

The function /x is called the marriage function and is empiricallychosen in several different ways in the literature (Hoppensteadt, 1975;Keyfitz, 1972; McFarland, 1972). We discuss this further in Section 4.

In this model the total population p(t) is given by

/•oo roo

Pit)= I u/{x,t)dx+ um(y,t)dy. (2.5)Jo Jo

If the system (2.1)-(2.4) is to have a classical (differentiable) solu-tion, the initial and the boundary conditions must be compatible;

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1


moreover, from obvious biological considerations we must have allfunctions involved non-negative:

u}(0)=Bf(0) and u°m(0) = Bm(0),

cO,u},u°m>O,

c°(x,0) = c°(0,y) = 0, x,y>0,(2.6)

c\x,y)dy<u%x) (i.e., s°f > 0),

/•OO

/ co(x,y)dx<u°m(y) (i.e., s°m > 0).Jo

Also, the initial age distributions of individuals and couples, w9, iPm, c°,must be compactly supported (for biological reasons). These condi-tions clearly imply that c > 0, which in turn implies that Uf, wm > 0 andthat uf, um have compact support. If we set n(s/, sm) = 0 for s/ orsm < 0, then in fact s/, sm > 0. To see this for s/, for example, note that

^ + £ = ~ 6 f S f + ( 1 "/

OO i»O

a^j.Oc^j,/)^- /Jowhere a(x,>-, r) = a(x,y, f) - 6f(x, i) - 6m(y, t) + 6f(x, t)6m(y, i) is the

annulment and divorce rate; since J9 > 0 and the characteristic slope ispositive when s/ is negative, s/ must stay nonnegative. Finally, then, chas compact support for any t (Arbogast and Milner, 1989).

3. PROBLEMS WITH AVAILABLE DATA

We use the model (2.1)-(2.4) to simulate the dynamics of thepopulation of the U.S. To estimate the parameters of the model weinterpolate data from the U.S. census and Vital Statistics (HealthResources Administration, 1974-75; National Center for HealthStatistics, 1984-85; U.S. Bureau of the Census, 1972; 1973; 1981;1984). The most recent years for which we have census data are 1970,1980 and 1990 but since some of the vital statistics data for 1990 arenot available (especially those related to marriage and divorce) welimit our computations to data from 1970 and 1980.

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The available data, however, are inconsistent and can not be usedin the form given in the tables. This is a problem demographersfrequently face, and they work to solve it (Luther and Retherford,1988). In what follows we discuss some of the problems we faced andhow we corrected them.

1. Births to married women of age 10-14. The number of births towomen under age of 15 can be obtained from tables in the Vital Stat-istics concerning natality (U.S. Bureau of the Census, 1981). There is adifference between the total number of births in this age group and thenumber of births to unmarried women of ages 10-14. That suggeststhat some of the mothers in this age group are married. However,Volume II of the Vital statistics (U.S. Bureau of the Census, 1984),which reports statistics about marriage and divorce assumes thatmarriages in this age bracket are so scarce that they can safely be takenas zero. As a result we have a nonzero number of births to marriedwomen of age under 15 but we don't have any married women of thisage. To resolve this logical and computational inconsistence we makethe following hypothesis

(HI) the number of married women of ages 10-14 is the same as thenumber of the births in this age group.

This assumption seems reasonable since most of the marriages atthis young age take place because of a pregnancy.

2. The Vital Rates in the Statistics. Besides the number of births,the statistics also give the vital rates, including birth rates. However, itis not always clear which data were used for computing these rates.For example, for 1980 and for the cohort of women of age 25-29 years,the number of never married women is reported to be 2,111,744, thenumber of widowed women is equal to 51,751, and the number ofdivorced women is 890,566. This produces a total of 3,054,061 singlewomen (U.S. Bureau of the Census, 1984) Table 1. In (U.S. Bureau ofthe Census, 1981), Table 1-34 reports 99,583 births to unmarriedwomen. From the ratio of these two values one can compute the birthrate to unmarried women to be 32.6 per thousand. However thereported birth rate, Table 1-32 in (U.S. Bureau of the Census, 1981) is34.0. We could not establish what data were used for these birth ratesand to avoid inconsistency we computed new birth rates using thegiven data. It should be noted that birth rates are especially prone tothis variation since different sources give different numbers for themarried/unmarried women. This modification in the coefficients made

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TABLE 1Distribution of males and females

Age bracket (yrs.)

0-45-9

10-1415-1920-2425-2930-3435-3940-4445-4950-5455-5960-6465-6970-7475-7980-8485 and over

Total

Calculated(lOOO's)

851089648661

1012210467953875966417542551945419521444933667274817661030758

105989

Males

Actual(lOOO's)

836285399316

1075510663970586776862570853885621548246703903285418481019682

110053

Error

(%)

1.85.07.05.91.81.7

12.56.55.03.63.64.93.86.03.74.41.1

11.1

3.7

Calculated(lOOO's)

8088853783469787999791558181674857565564591558865254457637732268

425426

108682

Females

Actual(lOOO's)

798681618926

10413106559816888471045961570260896133541848803945294619161559

116493

Error

(%)

1.34.66.56.06.26.77.95.03.42.42.94.03.06.24.4

23.077.872.7

6.7

our initial data consistent which, as we well know, could be veryimportant especially for nonlinear models.

3. Undistributed data in the tables. Some of the tables have acolumn "not stated" for the number of those who have not reportedtheir status. This is particularly frequent for those densities and rateswhich are cross-tabulated by two, three, or more characteristics. Thenumbers in those columns are often in the range of 10-15% ofthe total. If this column is simply neglected then the initial data will beunderestimated by at least 10%.

Consequently, for the birth rates of boys and girls by age of father,we adjusted the values given in Table 1-55 of (U.S. Bureau of theCensus, 1981), Natality, by distributing the numbers from the extracolumn of those fathers who did not report their age not only pro-portionally to the sizes of the age groups, but also by using the avail-able birth rates by age of father from which we computed the modifiedtotal number of births in every age group of the father.

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THE MATHEMATICS OF SEX AND MARRIAGE

TABLE 2Distribution of couples (numbers in thousands)

131

Age ofhusband(yrs.)

15-24

25-34

35-44

45-54

55-64

All ages

15-24

1681.02638.8

36.3%

2680.82226.7

20.4%

284.9119.9137.6%

71.918.5

288.6%

60.67.2

741.7%

4797.05014.5

4.3%

25-34

993.2332.5198.7%

7007.79010.4

22.2%

2602.33094.9

15.9%

181.6263.8

31.2%

61.144.537.3%

10858.212756.1

14.9%

Age of wife (yrs.)

35-44

103.914.4

621.5%

416.5479.9

13.2%

5685.16346.1

10.4%

2397.42725.4

12.0%

181.3263.7

31.2%

8811.69868.7

10.7%

45-54

34.65.0

586.2%

36.828.529.1%

243.4374.335.0%

5169.95423.4

4.7%

2442.42725.8

10.4%

8133.28827.4

7.9%

55-64

41.33.1

1232.3%

32.85.9

455.9%

25.126.7

6.0%

407.4495.8

17.8%

4533.34900.6

7.5%

7045.17720.8

8.8%

All ages

2865.72995.6

4.3%

10182.411754.4

13.4%

8845.69967.1

11.3%

8258.38965.5

7.9%

7648.28392.3

8.9%

43937.449513.9

11.3%

CalculatedActualRelative

errorCalculatedActualRelative





error

4. A SIMULATION BASED APPROACH TO COMPARINGMARRIAGE FUNCTIONS

We present and discuss some of the major marriage models knowntoday and accumulated during the last 25 years or so of intensivediscussions of the problem. The purpose is not so much to acquaint thereader with those models but to create a feeling about the diversity ofmethods used in their derivation, as well as the differences in theresults obtained and the lack of tools for their comparison.

Two major trends can be identified in the literature as bases for thecomparison of marriage functions - one of them is the theoreticaland conceptual criteria and the other one is the goodness of fit to data.The main problem in the empirical comparison of the marriage functions

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is that the differences in the predictions of the alternative methods forprojecting marriages are much smaller than the differences between thepredicted and actual values. This effect is especially strong over longtime intervals. In contrast with most of the other empirical methodsour approach here spans a 10 year period and gives differencesbetween the various projections of comparable magnitude to (ofteneven smaller than) the errors between the simulated and real data (seeTables 3-6).

Here is a listing of some formal requirements on the marriagefunction which are widely accepted by demographers and mathemat-icians. These conditions imposed on the marriage function arereferred to as properties of the marriage function. Following manydiscussions and on the basis of the predominant opinions, McFarland

TABLE 3(a = -2)

Kx,y;sf,sm) = ^x,y)sfKx,y)sn

yjplg(x,y)sf? + (1 - P)[h(x,y)smf

Relative errors in newborn and couples

PopulationGroup

Girls (%)Boys (%)Couples (%)

0.2

2.52.04.1

0.3

2.52.14.2

0.4

2.52.04.2

0

0.5

2.41.94.2

0.6

2.11.74.1

0.7

1.91.54.0

0.8

1.61.13.8

TABLE 4(a = -1)

g(x,y)sfh(x,y)sm

PopulationGroup


0.0

1.81.43.4

Relative

0.2

2.11.73.8

errors

0.3

2.11.73.9

in newborn and

0.4

2.01.63.9

0

0.5

1.91.53.8

couples

0.6

1.81.33.8

0.7

1.51.13.7

0.8

1.20.83.5

1.0

0.30.22.7

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THE MATHEMATICS OF SEX AND MARRIAGE

TABLE 5

133

li(x,y;sf,sm)=-g(x,y)s/h(x,y)sm

V - P)s/h{x,y)smf


PopulationGroup


0.2

2.01.53.7

0.3

1.91.53.7

0.4

1.81.43.7

0

0.5

1.71.33.7

0.6

1.51.03.6

0.7

1.30.93.5

0.8

1.00.63.3

TABLE 6(Q -»0")


PopulationGroup


0.2

1.81.43.5

0.3

1-71.33.5

0.4

1.61.23.5

0.5

1.51.03.4

0.6

1.30.93.3

0.7

1.10.73.2

0.8

0.90.43.1

(1972) summarizes the properties of the marriage functions as follows,which are mathematically formulated also by Inaba (year 1996):

(PI) n(x,y, t;sf,sm) > 0;(P2) /i(x, y, t; 0, sm) = /x(x, y, t; sf, 0) = 0.

This condition is natural and reflects the assumption that marriage(for reproductive purposes) occurs only between individuals of differ-ent sex;

(P3) n(x, y, t; Xsf, Xsm) = X/i(x, y, t; sf, sm) for A > 0.

This condition reflects the idea that if the total population of singlesincreases by a factor A and the sex ratio is preserved, then the birthsshould also increase by a factor A. Since the births and marriages aredirectly related in this model, the same is true for the marriage func-tion. This property is called the homogeneity property.

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(P4) / n(x,y,t;sf(-),sm(-))dy<Sf{x),Jo

/•oo

/ fi(x,y, t;sf(-),sm(-)) dx < sm[y).Jo

In other words, the number of marriages which involve individuals ofcertain age (and sex) should be smaller than the number of unmarriedindividuals of that same age (and sex).

i»oo 1*00 roc roc

(P5) / / n(x,y,f,Sf,sm)dxdy< / / ti{x,y,t;s'fJm)dxdy,JO JO JO JO

for {sf,sm)< tys/J.

This property says that total number of marriages increases when thethe number of the single individuals increases. We note here that froma mathematical point of view this implies that the total number ofmarriages is a monotone functional of the density of single males andsingle females.

(P6) The number of marriages in an age class for a given sex cannotincrease if the number of singles in other age-classes of the samesex increases while the number of singles in the other sex is heldfixed. This condition reflects the competitive nature of themarriage market and essentially means that the marriageoperator is not a monotone operator. Property (P6) is typicalfor age-structured models.

We now list five functions which were first proposed (togetherwith their age-independent counterparts) and investigated as possiblecandidates for a marriage function.

(El) fj,(sf,sm) = p(x,y)s/(x); (female dominance)(E2) n(sf,sm) = p{x,y)sm(y); (male dominance)

(E3) n{sf,sm) = 2P{x,y) ^ " ^ • (harmonic mean)Sf [X) •+- sm(y)

(E4) n(sf,sm) = p(x,y)y/sjs^; (geometric mean)(E5) fi(sf, sm) = p(x, y) min(s/, sm).

These models have stood a lot of criticism but they and their exten-sions are still at the center of discussions for marriage models. Parlett

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(1972) criticizes the lack of additivity properties, that is the number ofmarriages in a five-year age group should be the sum of the corres-ponding number of marriages in one-year age groups. To furtherdiscriminate among the candidates, McFarland tested them againstthe formal properties. He established that neither of the five functionssatisfies all six conditions. However, the harmonic mean satisfies allof the conditions desired of the marriage function except for the lastone (although Robert Schoen (1988) argues that the lack of sens-itivity of the harmonic mean to the number of single males (orfemales) in the other age-groups is only apparent). To remedy thatsituation the following generalization of (E3) was proposed

(E3')

Because it fitted the desired properties best, the harmonic mean andits generalization were the favorite among the five.

We simulated the growth of the population of the United Statesfrom 1970 to 1980, mainly to see how the distribution of couplesdepends on the marriage function and affects births. We used theinitial distributions of males, females, and couples from the 1970population census (U.S. Bureau of the Census, 1972; 1973). Vital stat-istics for births, deaths, marriages, and marriage annulments/divorceswere taken from the U.S. Department of Health and Human Services(and its precursor the Department of Health, Education, and Welfare)(Health Resources Administration, 1974-75; National Center forHealth Statistics, 1984-85). Based on these figures, we constructed thefunctions needed by the two-sex model. We used the numerical methodof Arbogast and Milner (1989), based on the finite difference methodof characteristics.

The vital rates (mortality, natality, and divorce) were not takenconstant in time. The time dependence was simply taken to be a linearinterpolation of the values from 1970 to those of 1980. The marriagefunctions we used were generalized means with age preferences pro-posed by Hadeler (1988):

n(x,y,sf,sm) = [(1-/3) (g(x,y)sf)a+0 (h(x,y)sm)a]1/a,

where 0 < /3 < 1 and a e [-co, 0]. Here g and h are the preferencefunctions giving, respectively, the age density distribution of thespouses of females and males at the time of marriage. We took

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g(x, y)Ay to be the probability that a female of age in the interval[x, x + Ax) chooses a husband with age in the interval [y,y + Ay). Wenote that we do not assume that this female has already made adecision to get married. Hence we approximated that probabilitywith the proportion of females of age [x, x + Ax) who marry males ofag e [y> y + A>") among all single females of age [x, x + Ax). We inter-preted and estimated in a similar way the male preference functionKx,y).

When preference functions are not differentiated according to gen-der, we have g(x,y) = h(x,y) = p(x,y). Then, for any a, the extremecases /} — 0 and f3 = 1 correspond, respectively, to the single genderdominance functions (El) and (E2). When 0 < 13 < 1, then for a = - 1we obtain a biased harmonic mean, which is the true harmonic mean(E3) for (3 — 1/2; for the limiting case a —»0~ we obtain a biasedgeometric mean [h(x,y)smf[g(x,y)sf]l~^, which is the true geometricmean (E4) for /? = 1/2. Similarly, for the limiting case a —> —oo, weobtain the minimum function (E5).

Because the fertility of unmarried females is significant (and known)(U.S. Bureau of the Census, 1981), we modified the model to include asource of births from single females. In the modified model, this sourceof additional newborn girls, for example, consists in the addition ofthe term

rpf{x,t)SJ{x,t)dxJo

to the births in (2.1), where 0f(x,i) is the fertility rate of unmarriedwomen for the birth of female progeny. A similar term is added for thebirths of males in (2.2). In Arbogast and Milner (1989), a simulationwas done using for marriage function the harmonic mean multipliedby the age preference functions, which is the choice most frequentlyfound in the literature. The results were quite good in the age dis-tributions of females and males, but very poor for the density ofcouples with respect to the age of the spouses and, consequently, nottoo good in predicting the number of newborn. We present in Tables 1and 2 the results from Arbogast and Milner (1989), giving the agedistributions of females, males, and couples.

The main thrust of the simulations in this paper is to determinewhether a different choice of marriage function would lead to a betterapproximation of the age density of couples which, in turn, shouldlead to a better approximation of births from married females. In

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order to compare the performances of the various marriage functionschosen, we look at the relative error in the total number of newborngirls, newborn boys, and couples. The relative error is found by com-paring the results of our simulations with the cumulative numberstaken from the actual distributions (given by the 1980 census (Parlett,1972; Smith and Keyfitz, 1977)). We ran our simulations with a time-step At = 0.0625 years (« 22.8 days) to obtain the predicted distribu-tion of males, females, and couples in 1980, for several choices of themarriage function.

First we simulated the limiting case a —> —oo for which, asindicated earlier, we have the marriage function n(x, y; s/, sm) =min{g(x,y)sf,h(x,y)sm}. In this case the relative errors in the numbersof newborn girls, newborn boys, and couples are, respectively, 5.2%,4.8%, and 6.1%. For other values of a and /3 we present the errors inthe simulations in Tables 3-6.

Note that, as indicated in Arbogast and Milner (1989), the modeldoes not take into account migration. Therefore, the net number ofimmigrants during the decade simulated should be added to the valuesobtained from the simulations. This number, according to figures fromthe Immigration and Naturalization Service (INS) (U.S. Departmentof Justice, 1981), is 4,336,001. If we subtract the 19,264,000 deathsfrom the 33,308,000 births recorded in the U.S. in those 10 years (U.S.National Center for Health Statistics, unpublished data), we obtainthe intrinsic increase of the population, 14,044,000 (where we haverounded the figures to the nearest thousand). Adding to this figure thetotal population recorded by the 1970 census, 203,210,000, we obtain apopulation of 217,254,000. The difference between this figure and thetotal population recorded by the 1980 census, 226,546,000, should givethe net migration into the U.S. in that decade, namely 9,292,000.However, even without considering emigration, the immigration figuregiven by the INS falls short by about 5 million. Hence, a percentageerror of about 9.3/226.5 « 4.1% is to be expected. This does not applyto the cohorts of girls and boys with ages in the interval [0,10) sincemost of the immigration occurs in ages outside that interval.

As we can see from Table 1, the model produces quite goodapproximations of the age distributions of both sexes as they evolve intime, which is the main objective of this model. The distribution ofcouples by age of the partners is used as a tool to try to better modelthe births in the population. On the other hand, as is very wellknown (McFarland, 1972; Parlett, 1972), modeling the evolution of thecouples distribution function in time is extremely difficult, mostly for

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lack of an accurate description of the marriage function (i. The resultsshown in Table 2 confirm that there is a substantial error in the dis-tribution of couples cross tabulated by ages; however, the distributionof couples by the age of one partner only is quite reasonable. This isrelated to the fact that, due to lack of sufficient data, the age pre-ference marriage rate u{x,y,i) in (Arbogast and Milner, 1989) wasconstructed as a separable function of x and y modeled by multiplyingthe data available for each sex separately.

5. COMMENTS AND CONCLUSIONS

For the chosen class of marriage functions we gave the gender-biasingfactor /3 values of 0.2 to 0.8 in increments of one-tenth, and we notedthat the extremes /3 equal to 0 or 1 result (independently of a) inthe gender dominance linear marriage functions g(x,y)s/ and h(x,y)sm,respectively. We show the results corresponding to these in Table 4,with the results for weighted harmonic means.

By looking at the errors in the total number of newborn girls,newborn boys, and couples, we readily see that the minimum func-tion is the worst of all marriage functions we tested, with errors atleast twice as large as for almost all the others we used. The reason forthis is that the model does not take migration into account and thus itunderestimates the population when simulating the dynamics of apopulation that is not spatially closed. The minimum function is theone that generates the smallest number of marriages (and hence ofcouples and newborns) when compared to all other functions in ourclass. It is apparent that the errors in the births for the worst per-forming function (the minimum function) are one order of magnitudelarger than the errors for the best performing function (the maledominance) and of the same magnitude as the average error. Hence,this test compares favorably to most of the other tests based on thequality of the fit where the error differences between the variousfunctions are much smaller than the average error. The fact thatthere are noticeable differences in the errors leads us to the conclusionthat, for simulations with the model (2.1)-(2.4), choosing the marriagefunction that provides the best fit is a matter of importance.Another conclusion is that if the source of this determined behaviorfor the error is removed - that is, if immigration is included - then theresulting test can be used to support or question theoretical delibera-tions about the form of the marriage function.

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Another observation that can be made is that the introduction ofpreferences leads to an improvement of the projective properties ofthe marriage functions with respect to the total number of couples.Compare here the 11.3% relative error in the total number of couplesfrom Table 2, with 2.7-4.2% for the corresponding errors in Tables3-6. Of course, this behavior has a logical explanation: preferencesintroduce more information from data into the marriage function andhence improve the fit. (This is also the reason why Henry's andMcFarland's models perform better than "traditional" marriagefunctions.)

Looking across Tables 3-6, we see that as a increases the errorsbecome smaller and they are monotonically decreasing for any fixedvalue of fil. Analogously, for a fixed value of a, we see in all Tables 3-6the errors decrease as /3 increases.

These last observations lead us to conclude that, for the decadesimulated, and for the population of the United States of America,the "best" marriage function in the family chosen is the one corres-ponding to /3= 1, namely n(x,y;sf,sm) = h(x,y)sm. This observationis essentially equivalent to the highly disturbing and provocativestatement that marriages were largely determined by the preferencesof the males, without any input from the females. Clearly, the waywe chose to decide which marriage function is "best" is not the onlypossible one. Moreover, as far as couples are concerned we justlooked at the error in their collective number, not in their age dis-tribution. The male dominance function may not be a good choicefrom a modeling point of view since it violates several properties ofthe marriage function, namely (P2), (P4), and (P6). However, wemight ask whether it is still possible that the better performance ofthat function over the rest of them is a consequence of an actualdemographic phenomenon. Indeed, if the marriage market obeysmarket laws, then the number of marriages should be more heavilyinfluenced by the scarcer sex. And it is a fact that after age 20 (whichincludes most marriage-eligible ages) females outnumber males.

We shall conduct further analysis by repeating these simulationsbetween 1980 and 1990 and by including (possibly a rough estimateof) the immigration numbers. We will also try to gather the necessarydata to repeat the simulations for the demography of a differentcountry.

The observed pattern in the behavior of the errors can be used as acriterion for choosing an appropriate marriage function to improvethe forecasting capability of a simulator. In fact, as it can be seen from

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our experiments the total number of newboms (girls and boys) and thetotal number of couples can be obtained with very good accuracy,provided that an appropriate marriage function is used. In fact, anew, more complex marriage function can be proposed, which assignsto each bride-groom age category as projection of the number ofmarriages for that bride-groom age category the number projected bythe marriage function in our class that gives the largest number ofmarriages for that class.

In conclusion we would like to note that abundant literature existsdiscussing the problem about the form of the marriage function.Various formulas and procedures have been proposed for the com-putation of the marriages from single male and female individuals. Butthe problem of choosing the "right" one in general or in each par-ticular situation still remains open.

Further analysis and simulation is needed to compare the perform-ances of marriage functions in predicting marriage distributions, andother types of functions should also be included (McFarland, 1972;McFarland, 1975b; Parlett, 1972).

REFERENCES

Arbogast, T. and Milner, F. A. (1989). A finite difference method for a two-sex model ofpopulation dynamics. SIAM J. Num. Anal., 26: 1474-1486

Bennett, N., Bloom, D. and Craig, P. (1992). American marriage patterns in transition.In: S. J. South and S. Tolnay (eds.), The Changing American Family: Sociological andDemographic Perspectives, pp. 89-108, Westview, Boulder.

Coale, A. J. (1972). The Growth and Structure of Human Populations, A MathematicalInvestigation. Princeton, New Jersey: Princeton University Press.

Hadeler, K. P. (1989). Pair formation in age-structured populations. Acta Appl. Math.,14: 91-102.

Hadeler, K. P., Waldstäter, R. and Wörz-Busekros, A. (1988). Models for pair forma-tion in bisexual populations. J. Math. Biol., 26: 635-649.

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Hoppensteadt, F. (1975). Mathematical Theories of Populations: Demographics,Genetics, and Epidemics. Society for Industrial and Applied Mathematics. Philadelphia.

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Lillard, L. and Panis, C. (1996). Marital status and mortality: The role of health.Demography, 33(3): 313-327.

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McKendrick, A. G. (1926). Applications of mathematics to medical problems. Proc.Edinburgh Math. Soc, 44: 98-130.

National Center for Health Statistics (1984/85). Vital Statistics of the United States,vols. I—III, DHHS publication nos. (PHS) 85-1100 to 85-1103, Public Health Service.Washington, D.C.: U.S. Government Printing Office.

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Schoen, R. (1988). Modeling Multigroup Populations. New York, London: Plenum Press.Smith, D. and Keyfitz, N. (1977). Mathematical demography. Biomathematics, 6.South, S. (1992). For love or money? Sociodemographic determinants of the

expected benefits from marriage. In S. J. South and S. Tolnay (eds.), The ChangingAmerican Family: Sociological and Demographic Perspectives, pp. 171-194. Westview,Boulder.

U.S. Bureau of the Census (1972). Census of Population: 1970, Marital Status, FinalReport PC(2)-4C. U.S. Summary, Washington, D.C.: U.S. Government PrintingOffice.

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the mathematics of sex and marriage, revisited

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