Comment on Mitra’s generalization

In a recent article in this journal, Mitra (1983) extended work by Espenshade, et al. (1982) who examined the long-run implications for population size and composition if below-replacement fertility rates remain fixed in the face of a constant influx of immigrants. By differentiating the fundamental renewal equation for annual births, Mitra developed a unified framework for studying the existence of long-term equilibrium solutions when constant annual immigration is combined with fertility rates that are below, at, or above replacement. When fertility is below replacement, Mitra confirmed the conclusion reached by Espenshade et al. showing that a stationary population is the long-run outcome. He showed that annual births, and therefore total population, grow linearly if fertility is at replacement. For fertility above replacement, Mitra found that population growth rates tend asymptotically to the intrinsic rate they would exhibit in the absence of immigration. In the case where fertility is below replacement, it would be more accurate to say that Espenshade et al. have generalized Mitra's results. Mitra assumes that immediately upon their arrival, immigrant women adopt the fertility behavior of native born women. Espenshade et al. have shown that a stationary population will materialize in the long run even if immigrant women and successive generations of their descendants have fertility above replacement. All that is needed for a stationary population to result is that, at some point in the chain of immigrant descendants, one generation and all those following it must adopt belowreplacement fertility. Moreover, in the case where fertility is at replacement and a constant number of immigrants is assumed to enter the population each year, Ansley Coale (1972) preceded Mitra in showing that population growth will be linear in equilibrium. In the last part of his paper, Mitra studies a population with two homogeneous groups, one having fertility rates above replacement and one with fertility rates below replacement. He then examines "the mechanisms by which the growing population may adopt the reproductive norms of the declining population to result in the eventual stationarity of both groups" (113). It is not entirely clear how this section of Mitra's paper relates to his previous results on immigration. To link the two, it is useful to introduce migration explicitly into the analysis. To do so, imagine a closed population of females divided into two separate geographic regions. Suppose that women with below-replacement fertility reside in the north and those with fertility above replacement live in the south. Let mH(a) and mL(a) be the age-specific fertility schedules of the high and low fertility women, respectively. Assume that both regions are characterized by the same force of mortality schedule, yd(a). Assume further that women from the south can migrate to the north (but not vice versa), and that they migrate at a rate which is constant through time, but not necessarily across age. Once in the north, these migrant females take on the fertility schedule mL(a). The rate at which women move from the high fertility area to the low fertility one is formally equivalent to Mitra's concept of the rate at which high fertility women adopt the reproductive norms of the other group. Suppose we represent the rate of out-