1In a number of previous studies, the effect of migration on population dynamics has been taken into account. In these studies, migration has either been assumed to occur at a constant rate (Le Bras [4], Keyfitz [3]), or the number of migrants has been assumed constant (Espenshade, Bouvier and Arthur [2], Mitra [7], Cerone [1]). Some of these studies were primarily concerned with the consequences of migration for the classical properties of ergodicity in stable population models, others were more interested in the potential role that migration could play in the regulation of total numbers, either lowering these numbers in populations with rapid natural increase, or raising them in populations with fertility rates below replacement.
2The present paper re-examines the last idea to use immigration as an instrument of demographic control, but from the point of view of regulating age structures rather than population size. Indeed, the concern with low fertility in developed countries is not only about numbers, but also about the ratio of the population of working age to older non-workers and the potential consequences for the viability of pension systems. We can ask, in this context, whether an adequate migration policy could help to avoid the increase in the pensioners/workers ratio which is implied, in the long run, by a regime of low fertility.
3As soon as this question is formulated, the answer can be seen to be negative. A large number of immigrants of working age at a given point in time may help to solve temporarily a disequilibrium between the labour force and the population of pensioners but, when these migrants will themselves reach retirement age, the problem will re-appear, and could do so in an aggravated fashion. The aim of this paper is to show that this is actually so, and that this type of migration policy would lead, under rather general conditions, to migration cycles of a large amplitude, of the same nature as the cycles that Le Bras has shown to exist for populations which are subject to constraints [6]. Furthermore, it can be shown that, for realistic values of migration parameters (age structure of migrants), these cycles will be explosive, so that the usual results of ergodicity will no longer apply. It is, therefore, dangerous to try any short-term regulation of age structure through migration. This does not mean, of course, that a regular flow of immigrants could not be of some help in equilibrating the age structure in the long run : what is under discussion here is not the general idea of compensating low fertility through migration, but the de-stabilizing consequences of a stop and go migration policy motivated only by short-run considerations.
4These results are going to be established in three steps. We shall first explore the problem by means of a very simple model with only four age groups. We then examine the more complex situation where age is treated as a continuous variable. Finally, our conclusion will illustrate these results with some projections applied to the French case, which will clearly contrast the consequences of migration policies directed by short-run and long-run demographic considerations.
1 – Model with four age groups
5Our notation will be as follows. We index the four age groups from 1 to 4 in increasing order of age. Pi will be the population of age group i (the subscript t corresponding to time will not be used explicitly). For simplicity, we shall ignore mortality at these ages. We shall write B(t) for the number of births at time t, M(t) for total in-migration at t, with an age structure of migrants represented by μi (the sum of these μis being equal to 1). The working age groups will be groups 2 and 3. Reproduction will be assumed to occur only in group 2, with a fertility rate equal to ϕ. This rate will apply only to the population already resident in the country at its entrance into age group 2, the fertility of migrants being taken into account through the amount of family migration, as measured by the ratio μ1/μ2. This assumption does not really imply that migrants have their children before migrating. In fact, due to the simplified time structure of the process, migration and birth of children occur at the same time. Our convention makes it possible, however, to assume, if necessary, that the fertility of migrants is lower than that of residents, i.e. μ1/μ2 < ϕ. We will assume, finally, that migration policy aims to maintain a constant workers/pensioners ratio, i.e. (P2 + P3)/P4 = k.
6These assumptions specify the complete age-structure of the population at t. We have :
8From these expressions we can derive a system of two equations that determine the changes in B(t) and M(t). The first is obtained by writing that the number of births B(t) at time t is equal to the fertility rate multiplied by P2 diminished by migrants of age 2, i.e. :
10The second equation is obtained by putting the ratio (P2 + P3)/P4 = k. We have :
12Let us look for solutions of this system of the general form :
14We have, from (1) and (2) :
16The values of z which are of interest are those which allow B and M to differ from zero, i.e. those for which the determinant of this homogenous linear system in B and M is equal to zero. This condition on z can be written, after rearranging terms in descending powers of z :
18Solutions to this third-degree equation are difficult to discuss in the general case where all μis are different from zero. But we can examine two special cases.
19The first, which is quite realistic, is where migration is made up only of young adults and their children, i.e. μ3 and μ4 equal to zero. In this case, equation (3) can be written :
21This equation has three roots, all real, which are, if ϕ′= μ1/μ2 is the fertility of migrants :
23What are the relative positions of these roots ? First, it is easy to see that the absolute value of the negative root z3 is always larger than the positive root z2. Next, it is natural to consider the case where ϕ2 + ϕ < k since, otherwise, demographic growth without in-migration would ensure that the workers/pensioners ratio is larger than k. This implies that :
25Therefore, if z1 happens to be positive, because ϕ′ < ϕ, z1 will be lower than both z2 and z3 in absolute value. In total, therefore : (a) either z1 is positive, and is dominated by z2 and z3, or (b) z1 is negative, and may or may not be dominated by z3 depending on whether the fertility of migrants is higher than that of residents by a small or a large amount. In any case, the dominant root is negative, implying some explosive cycles of period 2 around the positive trend represented by z2.
26The existence of these explosive cycles does not depend on the fertility of migrants, since they appear even if ϕ′= 0. But it evidently depends on the distribution of migrants between the two working age groups. Without considering the general case where μ1, μ2 and μ3 all differ from zero, we can check this point in the case of a pure migration of workers, where only μ2 and μ3 differ from zero. In this case, equation (3) becomes, taking into account the fact that μ2 + μ3 = 1 :
28One of its roots is :
30and the two others are :
32We can check that z3 will always be negative and z2 always positive. If μ2 > kμ3 then, as before, the negative root dominates the positive. If this negative root is higher in absolute value than z1 = ϕ, then we are back at the previous case (the limit situation where μ3 = 0 and μ2 = 1 being equivalent to the limit situation μ1 = 0 and μ2 = 1 of this previous case). On the other hand, as soon as there are enough migrants in age group 3, i.e. for kμ3 > μ2, the negative root z3 will be dominated by the positive root z2. The long-term evolution is therefore non-cyclical, but this assumes a very unrealistic age-pattern of migration, with a large migration of older workers and no family migration. Furthermore, the resulting population growth rate will be much larger : migrants reach retirement age much earlier, and a given migration wave must be compensated by an echo-wave much sooner to compensate for their entrance into age group 4. For instance, for k = 3, μ2 = μ3 = 0.5, we get 
, i.e. nearly a doubling of migration in each generation, while, if μ1 and μ2 are non-zero, for the same value of k, we get 
.
2 – A continuous-time model
33Results in discrete-time models, whether convergent or not, cannot systematically be applied to continuous-time models, because in the former, mixing between age-groups which affects the ergodicity of the latter is ignored (Le Bras, [5]). It is, therefore, necessary to reproduce the computations of the first section in this continuous-time framework. Following the previous notation, we write P(a, t) for population aged a at time t, ϕ(a) for the fertility rate at age a, B(t) and M(t) for total births and total entries of migrants at time t, μ(a) for the age structure of these migrants. We next assume that migrants adopt the fertility of natives on arriving in the receiving country. On the other hand, their fertility before migration, which partly determines the profile of μ(a) at younger ages, can take any value. Finally, we introduce a mortality schedule defined either by the series of probabilities of dying q(a), or by the survivorship function l(a).
34Let us determine first the age structure of the population at time t, for given past records of M(t) and B(t). We can write, in a given cohort :
36of which the solution is :
38whence, changing variables :
40From this point, as before, we can write equations that determine B(t) and M(t). The first is simply :
42The second term of this sum can be written :
44since, in the first integral, the term M(t – u) appears, for a > u, with the series of coefficients :
46Therefore we obtain, by inverting a and u :
48As regards the equation for M(t) we must have, at any period :
50denoting by α and β the ages of entering and leaving the labor force, and k the desired ratio between workers and retired persons. Whence :
52I(a) being defined by :
54and :
56This implies, taking into account the value of P(a, t) :
58whose second term can be transformed by changing variables as we did in the second member of the expression for B(t). Regrouping the two equations for B(t) and M(t) we obtain :
60This is a system of Lotka equations whose general structure is :
62equivalent to those obtained in models of two populations with inter-migration (Le Bras, [4]). We know that this system can result in some explosive periodic solutions if the functions ψij(a) are not positive, which is the case here. We can check this by a numerical search. As in the discrete case, we look for particular solutions of (5) of the form :
64which leads us to the system :
66This system has some solutions (Y1, Y2) which are non-zero only if z is such that its determinant is equal to zero. This determinant is :
68Let us apply this to the system in B(t) and M(t). The determinant D(z) becomes :
70Looking for an analytical solution of D(z) = 0 is, of course, impossible, but we can find the main roots by systematic search. Given the results for discrete time, we have done so by making two assumptions about the function μ, one allowing for migration at the beginning of working life and accompanying children, the other one allowing for migration of older workers only, without children. The detailed profiles have been obtained from Rogers’s model (Rogers, Raquillet and Castro [8]), with the two parametrizations :
72for the first assumption, and :
74for the second. The two standardized profiles are shown in Figure 1. For the mortality and fertility schedules, we have chosen those of French women in 1982. The level of fertility has been fixed at 1.7 children per woman, and we took a value of three for the desired ratio between workers and retired persons, which is approximately its present value in France.
75We first compute the real root of equation (6). Figure 2 shows that there exists a real root corresponding to a growth rate of 1.6 % per year in the first case, and to 6.4 % per year in the second. Periodic roots are obtained by looking for values of z which make the real and imaginary parts of (6) simultaneously equal to zero. Figure 3 shows that this is the case, on the first assumption, if we take z approximately equal to :
77and, on the second assumption :
79(Figure 3 shows the result of systematic search as a function of real parts of z only).
Assumptions on migration schedules by age
Assumptions on migration schedules by age
80In the first case, we therefore obtain some oscillations with a period of about 35 years ; these oscillations are going to be explosive, and their amplitude will increase by about 2.6 % per year, i.e. more rapidly than the general trend of B(t) and M(t) given by the real root, which is 1.6 %. This confirms the results of the previous part, the period of 35 years corresponding approximately to the difference between the mean age of adult migrants and the age at retirement. This difference represents the lag after which the entry into retirement of a given wave of migrants implies calling upon another wave to increase the labour force proportionately.
Computation of the real root
Computation of the real root
Computation of the first periodic root
Computation of the first periodic root
81In the second case, oscillations are still explosive in absolute value, but damped when compared to the general trend of M(t) and B(t) (1.4 % of annual growth for the amplitude of cycles compared with 6.4 % for the trend). The period is also shorter, approximately equal to 20 years, and always corresponds to the difference between the mean age of migrants and the age of retirement.
Conclusion : projections and an alternative policy
82We can check that the system actually behaves as forecast by these roots by making some projections, beginning with the French situation in 1982. Keeping the same assumptions for mortality, for the age structure of migrants and the desired ratio of people between 20 and 60 to those over 60, and considering three assumptions for total fertility (1.7, 2.1 and 2.5 children per woman), we obtain Figures 4 to 6 showing, for the different situations, the annual rates of net migration (ratio of the crude number of migrants to total population) which would be necessary each year, up to 2081.
83The first assumption about the age structure of migrants (Figure 4) shows the expected explosive cycles, with a periodicity of about 35 years. We see that these cycles imply, at some periods, some out-migration. This happens when there is a relative excess of the adult population, in which case, as implied by the model, migration policy tries to reduce the ratio of workers to pensioners which is “too” favourable. We can, of course, assume that this is not the case in practice, and that this ratio is only constrained to be equal to or higher than 3, and not strictly equal to 3. In this case, migration will never be negative, but truncated cycles will continue to be explosive, as shown by Figure 5.
84The projection in Figure 6 has been made with the second assumption on the age structure of migrants (older adults without children). We observe some damped cycles, with a period of approximately 20 years : there is actually a long-run stabilization of the crude migration rate. Now, as we have mentioned, this stabilization is obtained with a very unrealistic assumption about migration by age, and implies a population growth rate which is also highly unrealistic, since the final value of the crude migration rate is about 6-7 % per year. This implies, in 2081, a total population of 2.2 billion people (!). With the first assumption, we arrive at a total of 120 million people on the same date, which is less absurd, but still high : we know that, in any case, maintaining the present ratio of workers to pensioners can be done only through rapid population increase.
Projection with migration of young adults and children
Projection with migration of young adults and children
Same projection as in Figure 4 without out-migration
Same projection as in Figure 4 without out-migration
Dependency ratio with a migration rate equilibrating the age structure in the long run
Dependency ratio with a migration rate equilibrating the age structure in the long run
85Do these results mean that migration cannot help to correct the age structure of the population ? The answer is no. What is questioned is the idea of trying to achieve an immediate correction of this age structure which, though it solves problems in the short run, aggravates them in the long run, calling upon a cumulatively increasing correction.
86An alternative solution would be to adopt a policy with a fixed rate of in-migration from the beginning, without considering the transient fluctuations that this would imply for the ratio workers/pensioners. If the level of fertility is given, we can compute this equilibrium migration rate. We just need to take the real root z of (6). For the corresponding stable state, the total population in t will be, given (4) :
88and the ratio between migrants and births :
90whence, after development, the population/migrants ratio :
92Numerically, we find long-run in-migration rates of respectively 1 %, 0.6 % and 0.2 % for total fertility of 1.7, 2.1 and 2.5 children per woman. Assuming that these rates are applied continuously from 1982 for each fertility assumption, we obtain the changes in the workers/pensioners ratio shown in Figure 7. It fluctuates between 2.7 and 3.8 and stabilizes around 2040 at a level which is very near to 3.
