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.

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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. 6 2.0 3517.1 3387.7 ::0 60-64 4983.0 4380.0 0.4 0.8 3352. 6 3078.6 :it 65-69 4708.0 3732.0 0.2 0.3 3121. 4 2660.9 70-74 3543.0 2594.0 0.1 0.1 2798.5 2142.2 &. 75+ 5634.0 3219. 0 0.1 0.1 5994. 5 3236.4 !..

Total 111,061. 2 105,675.0 192.8 207.2 55,693.2 52,209.9

Both sexes 216,736.2 400.0 107,903.1

Swmnary Demographic Measures

1977-1982 Stationary Population

Total fertility rate 1. 826 1.826 Gross reproduction rate 0.891 0.891 Net reproduction rate (NRR) 0. 869 0. 869 Male births per 100 female births 105.0 105.0 Female life expectancy at birth (in years) 77 .09 77.09 Male life expectancy at birth (in years) 69.32 69.32

Population size 221, 241. 8 107 ,903.1 Yearly births 3,449.6 1,209.8 Yearly deaths 2,028.7 1,609.8 Yearly net immigrants 400.0 400.0 Annual rates per 1,000 population Birth rate 15.6 11. 2 Death rate 9.2 14.9 Natural increase 6.4 -3. 7 Net migration 1. 8 3.7 ... Population increase 8. 2 o.o !::I 128 DEMOGRAPHY, volume 19, number 1, February 1982 23 equals the number of females who on the right-hand side of equation (3) migrated at age 0 times the probability of does not involve the variable time t, the surviving from age 0 to age 23, plus the number of births to foreign-born women number of females who migrated at age l is some constant value that is repeated times the probability of surviving from year after year. We can represent it by age l to age 23, and so on. Expressing Bi so that this algebraically, the number of foreign­ born women who have attained age a at time t equals B(t) = J: B(t - a)p(a)m(a)da +Bi.

/(O) . p(a) + /(l) . p(a) (4) p(O) p(I) We may now ask what the long-run p(a) behavior of B(t) will be. Taking Laplace + · · · + /(a - 1) · + /(a). transforms across (4) in the usual way, p(a - 1) we have The continuous-form analog of this num­ ber is - - - Bi B(s) = B(s) · F(s) + - (4a) s a I(x) p(a) dx. where F(s) is given by fo p(x) Therefore, F(s) = {'" e-sa p(a)m(a)da. (4b) N(a, t) = B(t - a) · p(a) From (4a) we obtain + I(x) p(a) dx. (2) fa - Bi o p(x) B(s) = _ (5) s(l - F(s)) In words, equation (2) says that the number of women in the population who We now invoke the tauberian theorem are age a at time t is the number of that, providing sB(s) has no singular native-born women who have attained points for s > 0, then liml--+"" B(t) = age a plus the number of foreign-born lims--+o sB(s). This means in our case that women who have attained age a. as long as I - F(s) does not equal zero for any positive s, which from (4b) is Since the second term on the right­ 00 hand side of (2) depends only on a and guaranteed only if J0 p(a)m(a)da < 1, not on t, it is simpler to write it as Hl._a). then the birth trajectory must reach an Now we can substitute for N(a, t) in (1) asymptotic limit given by to obtain B Jim B(t) = Jim '- 1--+oo s->O (1 - F(s)) B(t) = J: B(t - a)p(a)m(a)da

(6a) 00 + J: Hl._a)m(a)da. (3) l - f0 p(a)m(a)da This equation tells us that the total num­ 00 ber of births at time t is the sum of births We recognize f 0 p(a)m(a)da as the net to native-born women and births to for­ rate of reproduction NRR. The theorem eign-born women. Since the second term thus tells us that providing the NRR < 1, Immigration and the Stable Population Model 129 births must ultimately level off to a con­ represent the total size of the foreign­ stant B given by born population, f 0 wHJ(a)da. Thus,

B1 (9) B=---- (6b) 1 - NRR or The reader may check that a stationary level B does indeed satisfy (4) if N = B1 ( eo ) + H1. (10) 1 - NRR B = L"" Bp(a)m(a)da + B1 Equation (9) shows that the total even­ tual stationary population is actually that is, if composed of two smaller constant popu­ lations. One of these arises from a con­ stant annual number of births and has an as in (6b). exact parallel in the ordinary life table To summarize, we have shown that stationary population. There, the crude the annual number of births eventually birth rate (lofT ) equals the reciprocal of becomes stationary, at a level equal to 0 life expectancy at birth (To/1 0), so that the the annual number of births to immigrant total population that would ultimately be women divided by 1 - NRR. generated by a constant yearly number Total Population of births (B) is B · eo. The second population contains HJ. To calculate total population size, we the stock of foreign-born women. We return to equation (2) and recognize that can compute H 1 simply, by summing the total number of females is obtained HJ(a)-the number of immigrants in the by adding up the number at each age, or population who are age a-across all that ages. This yields:

N(t) = Lw N(a, t)da where N(t) is the total number offt~µiales at time t, and w is the oldest age attained wfa p(a) = I(x)-dxda. (11) by anyone in the population. Substitut­ fo o p(x) ing from (2) into (7) we have

Substituting for B 1 and H1 in (10) we N(t) = f w {B · p(a) + Ht(a)}da. (8) may write the total population size, in 0 full, as Since the right-hand side of equation (8) does not involve the variable t, total population size does not change with N~ c-·~RR) time. We can therefore drop t from the left-hand side, knowing that we have a formula for the size of the eventual sta­ f3 fa p(a) · I(x) - m(a)dxda tionary population (N). a o p(x) It is possible to write equation (8) f more simply by realizing that f o"'p(a)da w fa p(a) is another way of expressing life expec­ + I(x)-- dxda. (12) tancy at birth (e0) and by letting H1 Io o p(x) 130 DEMOGRAPHY, volume 19, number 1, February 1982 NUMERICAL RESULTS fects this conclusion; a stationary popu­ To confirm our analytic results, we lation eventually emerges. have applied them to U.S. fertility and We can both generalize the above mortality schedules for 1977 and to the result and see how this stationary popu­ data in Table 1 on immigrants. lation is constructed, using a simple heu­ The annual number of female births ristic argument. Imagine a country divid­ (B) in the stationary population is given ed into halves in such a way that the by equation (6b) as population alive at time t = 0 and any of its descendants reside in the western Bi portion, and immigrant arrivals after t = B=---- 0 together with their descendants reside 1 - NRR in the eastern portion. Concentrating where B 1, the annual female births to first on the population in the west, we immigrants, can be evaluated using the can see that this population eventually second term on the right-hand side of dies away. Even though it may continue equation (3). Doing so yields B 1 = 77.29 growing for a while after t = 0 due to the thousand, and combining this with NRR momentum that a youthful age composi­ = 0.869, we have B = 77 .29 -:- .131 = 590 tion imparts to population growth, its thousand. In Table 1 annual male and below-replacement fertility is sufficient female births combined total 1209.8 to guarantee a negative stable growth thousand, but since these projections rate and, therefore, long-run extinction. assume a sex ratio at birth equal to 105 The eastern portion of the country males per 100 females, approximately develops demographically in a more 0.4878 of all births are female. There­ complex way. Any population that exists fore, the computer-based projections im­ there must either be direct immigrants or ply that B = 1209.8 x .4878 or 590. l the descendants of immigrants. Hence thousand. this population (that is, the female part of Total female population size (N) is it) will consist at any time of surviving computed from equation (9) as N = Be0 immigrant women, native-born women + H 1, where Hh the size of the foreign­ whose mothers were immigrants, native­ born female population, is equal to born women whose grandmothers were fowHI(a)da. Setting B = 590.1, e0 = immigrants, and so on. It will be useful 77.09, and H 1 = 10,201.25, we have N = to C8:~J women whose mother immigrated 55,692. l thousand. This, except for "first generation," whose grandmother rounding, is the same as the number in immigrated "second generation," whose Table 1. For the female population the great-grandmother immigrated "third crude birth rate is 10.60, the crude death generation," and so on, tagging each rate is 14.06, the immigration rate is 3.46, woman in the population by her immi­ and the rate of natural increase equals gration ancestry. We can assume, in -3.46. general, that fertility behavior differs for women of different immigration ''genera­ DISCUSSION AND FURTHER RESULTS tions," so that women of "generation" i If stable theory is expanded to include have fertility schedule m,{a), with associ­ immigration, we have shown that as long ated net reproduction rate NRR;. as fertility is below replacement, a sta­ The eastern population then builds up tionary population results by combining as follows. In a relatively short time after fixed fertility and mortality schedules time zero, say two or three generations, with a constant number and age distribu­ the stock of surviving direct immigrants tion for immigrants. Neither the level of becomes constant and stays constant, the net reproduction rate nor the size of building up in exactly the same way as a the annual immigration qualitatively af- standard life-table population, except Immigration and the Stable Population Model 131 that in this case people can enter the other words, the birth flow in the east­ population at all ages. In time, then, ern population eventually becomes sta­ there is a constant number of surviving tionary, providing only that immigrant­ immigrant women H/.,_a) at age a, in any descended women adopt below-replace­ year. In tum, each year thereafter B 1 ment fertility a finite number of genera­ children are born whose mothers are tions after "arrival." immigrants, where Now each of these births, whatever its "generational" status, faces the same survival schedule, and so each birth flow B1 = J: H/.,_a)m0(a)da (13) Bi generates its own stationary popula­ tion B,e0 • Counting the annual stock of and where m0(a) is the fertility schedule surviving immigrants, H1' in with the of immigrant women. Since immigrants "generational" population stocks, the are constant in number at any age, these eastern-half population levels off at the annual "first generation" births are con­ value stant too. A generation or so after the appearance of "first-generation" births, N = eoB10 + NRR1 + NRR, · NRR2 + NRR • NRR · NRR + · · ·) + H1. "second-generation" births B2 start to 1 2 3 appear. Since these are born to the con­ stant flow of "first generation" births, (18) they number We can conclude from this argument that stationarity can still come about even when immigrants and their close B2 = BJP(a)m1(a)da = NRR 1B 1 J: desendants have above-replacement fer­ tility. All we require is that from some (14) "generation" on, immigrant descen­ and each year, they too are born in dants adopt, like the native population, constant numbers. below-replacement fertility. If so, sta­ Given sufficient time, children of all tionarity is guaranteed. 5 "generations" up to "generation" Rare Returning to the special case of the born each year, and generalizing (14), we previous sections, where all net repro­ can show that each year produces a duction rates are equal and below one, constant flow Bi of "generation i" births, we see that (17) becomes where B = B,(l + NRR + NRR2 + NRR3 + Bi= NRR-1B;-1; 2 ::s i ::s R. (15) . . . ) (19) As we move indefinitely into the future, or all "generations" are represented in the eastern population, and the annual birth B = B 1 • (20) flow can be written as the infinite sum of I - NRR' "generational" births which is the same as (6b), so that (18) is a B = B 1 + B2 + B3 + ( 16) generalization of our previous result, or, substituting from (15) (10). Equations (16}-(20) provide a basis for B = B 1 (I + NRR1 + NRR1 · NRR2 + determining the ''generational'' distribu­ NRR 1 • NRR2 · NRR3 + · · · ). (17) tion of total births and of total popula­ tion. In the example in Table 1, there are This series will converge providing that 590.1 thousand female births each year NRR; is less than one for all "genera­ in the stationary population. Since NRR tions," after some finite number n. In = 0.869, the fraction I - NRR or 13 . l 132 DEMOGRAPHY, volume 19, number 1, February 1982 percent are "first-generation" births; tion. The size and other characteristics 11.4 percent (13.1 x .869) will be "sec­ of this eventual stationary population ond-generation" births, and so on. The depend only upon our assumptions re­ total stationary population includes garding fertility, mortality, and the age­ 55,693.2 thousand females, of which sex composition of immigrants, and are 10,201.3 thousand, or 18.3 percent, are not influenced in any way by the popula­ immigrants. Since we have assumed that tion we begin with. all females are subject to the same age­ Moreover, we have shown that this specific death rates, the size of the na­ long-run stationary population is actually tive-born population, Beo = (B 1 + B2 + composed of many smaller stationary · · · + B; + · · · )e0 , is distributed by populations-one of immigrants them­ generation in the same proportions as selves, one of "first-generation" descen­ total births. Thus, 10.7 percent of all dants, and so on. The composition of the females are "first-generation," 9.3 per­ total stationary population by its so­ cent are "second generation," and so called "generational status" can be com­ forth . The distribution of total popula­ puted from a knowledge of the specific tion by "generational" status is impor­ fertility, mortality, and immigration as­ tant because the preservation of native sumptions. language, tradition, and culture is likely We have shown that these results can to be infuenced by whether one is an be obtained even when some "genera­ immigrant, the child of an immigrant, or tions" have above-replacement fertility. the grandchild. Cultural heterogeneity All that is required to establish a station­ will be more pronounced the lower is the ary population in the long run is that, at value of NRR. some point in the generational chain of This kind of analysis can also be of immigrant descendants, one generation practical significance in helping to for­ and all those that succeed it adopt fertil­ mulate immigration policy. The projec­ ity below replacement. tion in Table l shows that 400,000 annual net immigrants lead eventually to a total NOTES population of 107.9 million, or 269.76 1 These and subsequent statistics on the part played by immigration in U.S. population growth persons in the stationary population for are contained in Bouvier (1981) . every annual immigrant. Suppose the 2 Since immigration is controlled in most coun­ United States wanted to arrive at a sta­ tries, assuming that the number of immigrants is tionary population as large as the 1980 constant is preferable to assuming constant rates of population of approximately 226 million. immigration. 3 This development parallels earlier work by Then, assuming 1977 fertility and mortal­ Ansley J. Coale (1972) . Coale approached the ity conditions and the age-sex composi­ problem by starting with a stationary population tion of immigrants in Table 1, almost closed to migration and then inquired how much of 840,000 annual net immigrants would be a reduction in fertility would be required when immigration is added to maintain a stationary popu­ needed-a number that may not be far lation with the same number of births. We begin at from the 1980 figure. (Of course, the the other end, by assuming below-replacement population would increase to almost 300 fertility and show that, with immigration constant million before falling to 226 million.) both in volume and in age composition, a station­ ary population evolves. Moreover, any below­ CONCLUSION replacement fertility schedule, if held constant, leads to a stationary population when constant In this paper we have shown that any immigration is included. For an interesting applica­ fixed fertility and mortality schedules tion of Coale' s approach assuming a lower estimate with an NRR below one, in combination of net immigration to the U.S., see Keely and with any constant annual number and Kraly (1978). 4 Where quotation marks are used, "generation" age distribution of immigrants, will lead signifies a label on each woman marking her immi­ in the long run to a stationary popula- gration ancestry. Without quotation marks, genera- Immigration and the Stable Population Model 133 tion signifies as usual either time elapsed or a their helpful comments. The capable particular population as measured reproductively technical services of Bobbie Mathis are from some initial event or population. 5 Valuable information on this important subject also appreciated. Views or opinions ex­ has been provided by Bean et al. (1980). In their pressed in this paper are the authors' and analysis of 1970 census data for Mexican-Ameri­ do not necessarily represent those of the cans, women are distinguished according to wheth­ organizations with which they are affili­ er they were born in Mexico (first-generation), whether they were born in the United States but ated. one or both of their parents were born in Mexico (second-generation), or whether they and their parents were born in the United States (third or REFERENCES higher generation). For ever-married Mexican­ American women aged 20-34, the average number Bean, Frank D. , G. Swicegood, and T. F. Linsley. of own children under age 3 (a measure of current 1980. Patterns of Fertility Variation Among fertility) was 0.64 for first-generation women, 0.57 Mexican Immigrants to the United States. Paper for second-generation, and 0.53 for third or higher prepared for the Select Commission on Immigra­ generation. By comparison, the average was 0.45 tion and Refugee Policy, Washington, D.C. Tex­ for non-Mexican-American whites. When such as Population Research Center Paper No. 2.016, other factors as age, education, and family income The University of Texas at Austin. were controlled, first- and second-generation Mex­ Bou vier, Leon F. 1981 . The Impact of Immigration ican-Americans exhibited current fertility that was on U.S. Population Size. Population Trends and approximately 16 and 12 percent greater, respec­ Public Policy, No. I. Washington, D.C.: Popula­ tively, than that of non-Mexican-American whites. tion Reference Bureau, Inc. But for third or higher generations, the differences Coale, Ansley J. 1972. Alternative Paths to a between Mexican-American fertility and that of Stationary Population. Pp. 589-603 in Charles F. other whites was not statistically significant. The Westoff and R. Parke, Jr. (eds.), Demographic authors conclude, " Hence, with respect to current and Social Aspects of Population Growth, Re­ fertility, later generational Mexican-American search Reports, Volume I, U.S. Commission on women, ceteris paribus, do not appear to behave Population Growth and the American Future. differently from other white women" (p. 37). Washington, D.C.: U.S. Government Printing Office. ACKNOWLEDGMENTS Keely, C. B. , and E. P. Kraly. 1978. Recent Net Alien Immigration to the United States: Its Im­ The authors thank Rachel Eisenberg pact on Population Growth and Native Fertility. Braun and an anonymous referee for Demography 15:267-283.