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Immigration and the Stable Population Model IMMIGRATION AND THE STABLE POPULATION MODEL Thomas J. Espenshade The Urban Institute, 2100 M Street, N. W., Washington, D.C. 20037, USA Leon F. Bouvier Population Reference Bureau, 1337 Connecticut Avenue, N. W., Washington, D.C. 20036, USA W. Brian Arthur International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria RR-82-29 August 1982 Reprinted from Demography, volume 19(1) (1982) INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS Laxenburg, Austria Research Reports, which record research conducted at IIASA, are independently reviewed before publication. However, the views and opinions they express are not necessarily those of the Institute or the National Member Organizations that support it. Reprinted with permission from Demography 19(1): 125-133, 1982. Copyright© 1982 Population Association of America. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the copyright holder. iii FOREWORD For some years, IIASA has had a keen interest in problems of population dynamics and migration policy. In this paper, reprinted from Demography, Thomas Espenshade, Leon Bouvier, and Brian Arthur extend the traditional methods of stable population theory to populations with below-replacement fertility and a constant annual quota of in-migrants. They show that such a situation results in a stationary population and examine how its size and ethnic structure depend on both the fertility level and the migration quota. DEMOGRAPHY© Volume 19, Number 1 Februory 1982 IMMIGRATION AND THE STABLE POPULATION MODEL Thomos J. Espenshade The Urban Institute, 2100 M Street, N.W., Washington, D.C. 20037 Leon F. Bouvier Population Reference Bureau, 1337 Connecticut Avenue, N.W., Washington, D.C. 20036 W. Brian Arthur International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria Abstract-This paper reports ()n work aimed at extending stable population theory to include immigration. Its central finding is that, as long as fertility is below replacement, a constant number and age distribution of immi­ grants (with fixed fertility and mortality schedules) lead to a stationary population. Neither the level of the net reproduction rate nor the size of the annual immigration affects this conclusion; a stationary population eventually emerges. How this stationary population is created is studied, as is the generational distribution of the constant annual stream of births and of the total population. It is also shown that immigrants and their early descendants may have fertility well above replacement (as long as later generations adopt and maintain fertility below replacement), and the outcome will still be a long-run stationary population. Since the beginning of the twentieth country each year. The effect of these century, the population of the United regulations was to increase substantially States has roughly tripled-from approx­ the volume of immigration, and for the imately 75 million in 1900 to about 225 next decade the annual number of legal million in 1980. Both natural increase immigrants was close to 400,000. Recent (births minus deaths) and net immigra­ statistics indicate a further increase to tion (immigrants minus emigrants) have perhaps 600,000 per year, including refu­ contributed to this growth. During the gees. With this growth in numbers, the decade 1901-1910 the average annual relative contribution of net immigration number of immigrants to the United to overall U.S. population growth has States was nearly 880,000, and net immi­ once again risen; for the period 1971- gration accounted for 40 percent of inter­ 1978, it was estimated at 22 percent. censal population growth. 1 But following Falling birth rates have accentuated 1910 the importance of net immigration the rising comparative importance of net relative to natural increase declined, immigration. The U.S. total fertility rate reaching a minimum during the Depres­ crossed below the replacement level in sion decade, 1931-1940, when emigrants 1972, for the first time since the Depres­ outnumbered immigrants. The 1965 sion, and it has fluctuated around 1.8 or amendments to the 1952 Immigration 1.9 ever since. Annual births still exceed and Naturalization Law replaced the annual deaths, but that is due to a tempo­ previous annual ceiling of 154,000 immi­ rary phenomenon of large proportions of grants with a preference system permit­ females in the childbearing ages. ting 290,000 immigrants plus about We may ask what the U.S. population 100,000 relatives of citizens to enter the would look like if current conditions 125 126 DEMOGRAPHY, volume 19, number 1, February 1982 were to persist into the indefinite future. tality schedules are fixed. Here we add Specifically, suppose fertility and mor­ the assumption of a fixed annual number tality schedules were held constant so and age composition of immigrants. 2 Fo­ that fertility was permanently below re­ cusing on females, we may extend the placement, and suppose that a constant theory to include immigration in the fol­ number of persons (with a fixed age lowing way. distribution) migrate to the United States each year. Would the population contin­ STABLE THEORY WITH BELOW­ REPLACEMENT FERTILITY AND ue to grow because of the influx of 3 immigrants and the children they would CONSTANT IMMIGRA TION bear? Would the population eventually Annual Births level off and then experience a long-term If we represent the annual number of decline owing to subreplacement f ertil­ females immigrating at age a by I(a), the ity? Or, would net immigration counter­ annual rate of bearing daughters for balance the low fertility rates, causing a women at age a by m(a), and the proba­ stationary population to evolve? This bility of surviving from birth to exact age problem takes on added significance a in the female life table by p(a), then the since immigration has been and is likely annual number of births at time t, B(t), to continue to be an important source of can be expressed as the sum across all U.S. population growth, and because ages of childbearing of the number of immigration will be a major policy con­ women at age a at time t multiplied by sideration throughout the 1980s. More­ the annual rate of childbearing at age a, over, the circumstance of below-replace­ or as ment fertility plus net immigration is one shared by numerous other industrial na­ tions. B(t) = J: N(a, t) · m(a)da (I) There are two ways to answer the question. One is with a straightforward where a and f3 denote the lower and projection of the U.S population. To upper limits of the childbearing ages, illustrate this approach, we use the esti­ respectively. Since we are interested in mated U.S. population on July 1, 1977 the long-run character of the population, and project it forward on the assumption we will restrict our attention to values of that 1977 age-specific fertility and mor­ t > {3, where t = 0 represents the time tality rates remain constant and that net after which /(a), m(a), and p(a) are held immigration totals 400,000 each year. constant. For t > {3, women in the popu­ Given these postulates, we arrive ulti­ lation at time t = 0 are no longer bearing mately at a stationary population. As children, and the youngest females in the seen in Table 1, the eventual stationary first wave of immigrants after t = 0 have population contains 107,903,100 per­ reached the end of their childbearing sons, with 1,209,800 annual births and years. 1,609,800 annual deaths to offset the The number of women at age a at time 400,000 immigrants. t depends first on the number of women A second approach is to analyze the who were born in the population a years problem in terms of stable population earlier and have survived to age a, and theory. Typically, by assuming a female second on the number of women who population closed to the influence of immigrated at all ages less than a and are migration, the stable model has investi­ now age a. The first component can be gated the shape of the long-run age distri­ written as B(t - a) · p(a). To understand bution and eventual levels for rates of the second component, consider a par­ birth, death, and natural increase when ticular age, say age 23. Then the number underlying age-specific fertility and mor- of foreign-born women who are now age Table 1.-U. S. Population, July I, 1977, and Eventual Stationary Population Achieved with Constant 1977 Fertility and Mortality and 400,000 Annual Immigrants (all numbers in thousands) i 3 ci5" Age U.S. Population, July 1, 1977 Immigration Assumptions Eventual Stationary Population c;·a Females Males Females Males Females Males ::0 Q ::0 0-4 7414.0 7760.0 16.4 17.6 2952. 2 3089. 8 0.. 5-9 8400.3 8759.0 16.4 17.6 3027.3 3168.0 :;. 10-14 9413. 0 9791.0 10.4 11.2 3090.6 3234 .1 • 15-19 10428.0 10753.0 6.4 5.6 3126.3 3261.1 if 20-24 9978.0 10111.0 20.4 8.4 3183.5 3266. 6 CT 25-29 8909.0 8837.0 48.4 47.2 3344.4 3372. 3 ii" 30-34 7776.0 7640.0 34.4 44.0 3537.7 3567.5 35-39 6309.0 6030.0 19.2 26.4 3651. 9 3703.2 .,,~ 40-44 5735.0 5465.0 10.0 14.0 3692. 8 3744.6 c 45-49 5898.0 5613.0 5.6 8.0 3680.2 3706.2 50-54 6167.0 5714.0 2.8 4.0 3622.3 3590.6 c;·a 55-59 5766.0 5277.0 1.
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