Immigration and the rate of population mixing: explorations with a stylized model

We start with a hypothetical country (region) with overall population N. The population is distributed among two sexes and five age groups: “children” (0–19), “young adults” (20–39), “middle aged” (40–59), “old” (60–79), and “very old” (80–99), with no survivors at 100. The age-sex distribution is captured by a 10-element vector n = [n _x ], with x = 1 to 5 representing the female age groups, x = 6 to 10 the male age groups. All children have young adult parents—mothers in group n ₂, fathers in n ₇. The fertility rate for young adult females is thus equivalent to the conventional total fertility rate. In addition, restricting fathers to the same age range allows the calculation of a male fertility rate. (The model is a one-sex model in the classical sense of female lineage, and the male fertility rate is an easily calculated concomitant of the female rate.)

In the absence of migration, in or out, the population is augmented by births, depleted by deaths, has a 10 × 10 projection (Leslie) matrix Q, and a stable age distribution. The female fertility rate is set at the natural replacement level, survival rates are unchanging, and n _t + 1 = Qn _t  = n _t for all t where the time interval is 20 years, the same as the age intervals. For convenience, we shall refer to this interval as a generation. (To keep notation as simple as possible, we add a time subscript when necessary but avoid doing so otherwise.)

Now introduce immigration (still with no emigration, which we shall take to be zero) and assume that immigrants have the same stable age-sex distribution, proportionately, as the original non-immigrants and the same projection matrix Q. In our simulations below, immigration may be one-time or repeated, but to develop the framework, assume for the present that it is a one-time event with immigrants arriving at t = 0. The question of interest is how rapidly will the populations of immigrants and non-immigrants mix where mixing in our context means cross-parenting—initially, the bearing of children with one immigrant parent and one non-immigrant parent, although the descendants of such a union will also be regarded as mixed. (It is perhaps well to recognize, before proceeding, that realistically in virtually any place in the world, all of the population will be descended from immigrants if one goes back far enough in time. In what follows, we will simply identify t = 0 as a point at which new entrants will be classified as immigrants and the existing population as non-immigrants.)

We identify then three separate populations within the overall population N, each evolving in its own way: (1) the original non-immigrant population and its non-mixed descendants H; (2) the population of immigrants and their non-mixed descendants M; and (3) a mixed population U, including all children of mixed lineage—children of mixed parents, grandchildren of mixed parents, and, in general, all persons with lineage traceable back to a mixed union. (Mnemonically, H is for “home,” M for “migrant,” U for “union.” Where the meaning is clear, we shall sometimes use the word “immigrants” to refer to members of the M population, thus including both those who immigrated originally and their descendants.) These three populations have age-sex vectors h, m, and u corresponding in structure to n (and aggregating to n). They also have the same projection matrix Q. While initially immigrants and non-immigrants have the same proportionate age-sex distribution, they may differ in other characteristics. The non-immigrant population will be augmented in each generation by births and depleted by deaths, but those births to non-immigrant mothers mated with immigrant or mixed fathers will be transferred (reclassified) to the mixed population. Similarly, the immigrant population will be augmented by births and depleted by deaths, but all births to immigrant mothers mated to non-immigrant or mixed fathers will be transferred to the mixed population. The mixed population will be augmented by births, depleted by deaths, and augmented also by the cross-parenting transfers from the other populations. If cross-parenting continues freely and indefinitely—if individuals choose to mate randomly and bear children without preference as to population membership—the non-immigrant and immigrant populations will vanish in the limit; all residents of the country will eventually be of mixed lineage. The proportion of mixed population in the total population of the country serves as an indicator at any given time of the degree of mixing that has occurred. (Note that the overall population N continues to have the same stable age-sex distribution; that is not affected by transfers among its component populations.)

The foregoing assumes one-time immigration. If immigration is repeated—at a constant rate proportional to the total population, let us say—the framework is the same as before except that the immigrant population will now be augmented by new immigrants each generation. The non-immigrant population will still vanish, in the limit, under random parenting, but the immigrant population will be continuously replenished, and the mixed proportion in the overall population will always be less than one.

The one-sex fertility structure of the model and the requirement that all parents must be young adults allows us to calculate an implicit male fertility rate. The female fertility rate F is the ratio of live births to the number of young adult women, the same for all populations. Similarly, a male fertility rate G can be defined as the ratio of live births to young adult males, the number of live births being the same in both cases. With overall population vector n, the male fertility rate would be G = (n ₂/n ₇)F, where n ₂ and n ₇ are the numbers of young adult females and males, respectively. Assuming G to be the same for all populations (as is F), the overall number of births will be distributed according to the population of the fathers in proportion to the numbers of young adult males. Furthermore, the Q matrix incorporates a fixed sex ratio at birth and an adjustment to allow for infant and early-childhood mortality in calculating the numbers of surviving male and female children in a population (see the Appendix). The calculations are common to all three populations, and so, the distributions of male children and female children by population of father will be the same and proportional to the numbers of young adult males, as are births. They will be proportional also to the numbers of young adult females, for the same reasons, and thus, any conflict between the two proportionality distributions must be resolved. Females from one population who would otherwise parent with males from another cannot do so if the latter population has an insufficient number of young adult males.

A preferential distribution provides a starting point, but the relative availability of young adults in the three populations will dictate the feasibility of any cross-parenting pattern. To modify a preferential pattern to accommodate supply restrictions but remain as close as possible to the original preferential distribution, we make use of a method known variously by the names biproportional adjustment, iterative proportional fitting, and others. The method was first given formal mathematical treatment by Deming and Stephan (1940), based on a restricted least-squares criterion, in the context of the adjustment of sample estimates to fixed census marginal totals in the construction of contingency tables. It was subsequently taken up in the construction of economic input-output tables (Bacharach 1965; 1970), where it has had extensive application over the years under the name RAS Method. (See Lahr and de Mesnard 2004, for a history of RAS applications.) Note that under the isolation preferential pattern there are no supply constraints—the “demand” for and supply of young adult males are automatically in balance in each population, so the procedure we are about to describe is required only for other patterns.

We wish to construct a 3 × 3 matrix B = [b _ij ], i, j = H,M,U, showing the realized distributions of children by mothers’ population (rows) and fathers’ population (columns). (B is 2 × 2 if there is no adult mixed population, as is the case temporarily when immigration is first introduced and before any cross-parenting can occur.) Having constructed B, we then want to convert it into a matrix A that is similar in structure to P but shows the actual (realized) proportionate distribution of births among fathers’ populations for each population of mothers (row totals are thus equal to 1). The overall number of children, all populations combined, is given by calculation based on the Q matrix (all populations have the same Q matrix). The row totals of the B matrix (b _i •) are calculated by distributing the overall number of children in proportion to the numbers of young adult females in the three populations and the column totals (b _• j ) by distributing the same overall number of children in proportion to the numbers of young adult males. Given some initial matrix B* = [\( {b}_{ij}^{*} \)], which in general will not satisfy the requirements that elements must sum to both row and column totals, the biproportional adjustment algorithm proceeds iteratively to find 3 × 3 (or 2 × 2) diagonal matrices D and C such that B = D(B*)C satisfies those requirements; it thus calculates B as a transformation of the initial matrix B* by forcing row and column adding-up consistency. As a final step, the A matrix is calculated by expressing each element of B as a proportion of its row total; P and A are thus comparable, P showing preferences, A showing realizations. The sex ratio of children is constant for each population, under the assumptions of the model, and thus, the proportionate distributions provided by the A matrix are the same for male and female children.

The derivation of the B matrix is as follows. The row and column totals are calculated as above. The row totals are then distributed among the elements of each row in proportion to the corresponding elements of the P matrix, thus providing the initial matrix B*, the starting point for the procedure. The elements of B* sum to the correct row totals, by construction, but not (in general) to the column totals. They are then adjusted pro rata to force them to sum to the column totals, but now, they no longer sum to the row totals. They are forced again pro rata to sum to the row totals, and so, it proceeds, iteratively, until convergence is obtained and the adding-up restrictions are satisfied. Following Deming and Stephan (1940), this simple procedure can be shown to be equivalent to minimizing the sum of squares of the differences between the B* and B elements, \( {\displaystyle \sum }{\left({b}_{ij}^{*}-{b}_{ij}\right)}^2 \). (Deming and Stephan used a weighted average for illustration; we use an unweighted average, or a weighted average with weights equal to 1, if one prefers to say it that way.) The minimization is subject to the conditions that the rows of B must add to b _i • and the columns to b _• j . The A matrix derived from B, and representing realized distributions, is thus as close as possible to the P matrix, representing preferential distributions, based on the least-squares criterion. (The final adjustment factors that convert B* to B in the iterative sequence are the diagonal elements of the D and C matrices—row adjustment factors for D, column adjustment factors for C.) The transformation of B* to B (more fully, P to B* to B to A) using the biproportional adjustment algorithm is unique, and convergence is fast and guaranteed under simple conditions that are satisfied by the model.

The application of the biproportional adjustment algorithm here is similar to its application in the construction of an economic input-output matrix, as noted above: marginal input and output (row and column) totals are known precisely, but interindustry product flows—the elements of the matrix—are not. They are represented initially by estimates that are then adjusted iteratively to enforce the adding-up restrictions. The present application involves only small matrices; input-output applications are generally on a much larger scale but the basic ideas are the same.

We provide below examples of the P matrices for selected preferential patterns and the corresponding derived A matrices for two of our immigration simulation applications. First though we lay out the formal specifications of the model on which the applications are based.

(A small point: The immigration vector at generation t includes children who are immigrating with their parents. Alternatively, the children of immigrant parents could be born immediately after the parents enter the country—in the same generational interval, that is; that would make no difference. In either case, the children would represent an addition to M, the immigrant population. ϕ would simply be defined to include both pre-entry and immediate post-entry births to immigrant parents.)

Equations (1), (2), and (3) constitute the integrated tri-population system that we use to explore, by simulation, the rate of population mixing. Given an initial population h _− 1, immigration commencing at t = 0 at an assumed rate ϕ, and a preferential cross-parenting pattern represented by P, A can be calculated, the system can be moved forward one generation at a time, and the proportionate distribution among the three populations noted. Of particular interest is the time path of the ratio of mixed to total population under alternative assumptions.

The model requires calibration for simulation application. The P matrix is chosen by assumption, and the A matrix is then derived, as above. ϕ is also chosen by assumption. However, Q must be specified realistically, and for that purpose, we use a projection matrix based on Canadian life tables centered on the year 2001 (Statistics Canada 2006). Any realistic projection matrix would serve our purposes, but this is one used in previous studies (Denton and Spencer 2014, 2015a, 2015b), and we find it convenient to use it here. The matrix is 10 × 10, representing females and males in the five broad age groups. It incorporates survival rates for those groups, a female fertility rate set at the natural replacement level consistent with the life tables (approximately 2.0745) and adjusted for infant and early-childhood mortality, and a male/female sex ratio at birth of 1.05. The Q matrix and associated stable population age-sex distribution are provided in the Appendix, with discussion.

The immigration rate is set at ϕ = 0.20 in Table 3, and simulation results for four preferential patterns are presented: indifference, isolation, partial discrimination, and adaptation. The P matrices for indifference, isolation, and partial discrimination are as shown in Tables 1 and 2 and discussed above. The matrix for adaptation (also discussed above) assumes isolation at the beginning (t = 0) but moves linearly, from one generation to the next, until it achieves the indifference form after three generations (t = 3), the form it retains thereafter.

Under isolation, there is no mixed population and no cross-parenting of immigrants and non-immigrants; the H/N and M/N proportions are unchanging over the eight generations (and beyond). Under the other preferential patterns, the mixed population proportion is small at first in each case but steadily increasing as cross-parenting shifts the distribution and the H/N and M/N proportions decline; it is clear from Table 3 (and simple reasoning) that the ultimate percentage distribution as the number of generations increases without bound would be U/N = 100, H/N = M/N = 0 for all patterns except isolation. The most rapid shift in distribution occurs under indifference, as one would expect: the mixed population is almost a third of the total after three generations (U/N is 32.0%) and just under half after four generations (U/N is 49.5%). The slowest shift occurs under partial discrimination, as we have defined that pattern. One might think of indifference as providing a benchmark with which the results of other patterns can be compared.

The rate of immigration obviously has a major role in determining the generational pace of population mixing. Table 4 shows the shifts in population distribution for the indifference pattern under the three alternative immigration rates that we have chosen to experiment with. The differences in shift patterns are in general as one would expect: increasing the immigration rate from 0.20 to 0.30 produces more rapid shifting (U/N is 40.0% after three generations) and a faster approach to the limiting distribution; reducing the rate to 0.10 has the opposite and much slower effect (U/N is 19.6% after three generations).

One general feature of the simulations in Table 5 (with ϕ set at 0.20) is that the proportion of immigrants in the population increases over the eight-generation span. The pace at which it increases tapers off though. M/N increases from the beginning, but the increases get smaller from one generation to the next, and for the adaptation preferential pattern, they are replaced by decreases, starting at t = 5, as the effects of indifference begin to offset the earlier effects of isolation in the adaptation process. The mixed proportion U/N rises in all cases except isolation, where of course it remains at zero.

Each preferential pattern results in an ultimate stable state for the proportionate population distribution as the number of generations increases without bound. The (proportionately) stable state for isolation is obviously M/N = 100%, H/N = 0. (The H population itself is unchanging; it simply becomes a smaller and smaller proportion of the total as the immigrant population grows.) We have calculated the stable states for the other preferential patterns (with ϕ = 0.20) by running the model for 20 generations (more than enough to achieve stability, for practical purposes). H/N is 0 in all cases. For indifference and adaptation, M/N = 41.6% and U/N = 58.4; for partial discrimination, M/N = 47.9% and U/N = 52.1. That the results for adaptation and indifference are the same is simply a consequence of the fact that once isolation is replaced by indifference in the adaptation process, the effects of isolation wear off and eventually the evolution of the population follows the same course as it would under pure indifference.

Taking a shorter-run view, and focusing on the mixed proportion, U/N is 29.2% after three generations under indifference and 41.9 after four. As with one-time immigration, the percentages are smaller for the other patterns—15.5 and 24.3 under partial discrimination, 19.6 and 32.9 under adaptation.

The rate of immigration plays an important role in determining the population distribution, as one would imagine, and as illustrated in Table 6 for the indifference pattern. However the actual results are somewhat different from what one might have expected. Lowering the rate reduces the mixed proportion somewhat in the earlier generations but increases it in the later ones and especially in the final stable state. Setting ϕ to 0.10 rather than 0.20 reduces the U/N percentage from 29.2 to 23.9 after three generations but increases it from 58.0 to 77.1 in the stable state. Setting ϕ to 0.30 increases U/N in the first two generations, but by the third, it lowers it slightly, to 28.2% and thereafter lowers it more sharply; in the stable state, the percentage falls to 43.7. The reason for the differences between early effects and later ones is that it takes time for repeated immigration to build up the immigrant population, starting from a base of zero, and then time for the mixing process to take advantage of the presence of more immigrants. On the one hand, as the immigrant population continues to grow, it forms an increasing share of the total population; how fast depends on the rate of new immigrant entrants. On the other hand, a larger immigrant population means more opportunities for mixed parenting, thus tending to raise the proportion of the mixed population and so reduce the proportion of immigrant population. These two offsetting effects eventually strike a balance and produce a stable state.