A New Measure of Fertility Replacement Level in the Presence of Positive Net Immigration

Abstract

In most more developed countries, the total fertility rate (TFR) is below 2.1 and net immigration is positive. This paper proposes and calculates for 22 populations for 2011–15 a ‘Current Migration Replacement TFR’ which in combination with the mortality and absolute net migration for that period generates a stationary population equal in size to the mid-period population. The results show the Current Migration Replacement TFR ranges widely from 0.60 for Singapore to 2.05 for Slovakia. That the Current Migration Replacement TFR is below the 2011–15 TFR in 14 of the 22 countries shows that, when considered in combination with current migration and mortality, in most of the countries the current ‘below 2.1’ TFR is coherent with population increase, not population decline, over the long run. For New Zealand, Australia, Norway, Sweden and the UK continued current fertility in combination with constant mortality and constant absolute net migration is coherent with more than doubling of the current population size. The value of this measure for illustrating the interconnected population size implications of sub-replacement fertility and immigration, for sub-categorisation of ‘post-transitional’ populations by population growth prospects, and for guiding population policy is discussed.

Electronic supplementary material

The online version of this article (10.1007/s10680-020-09566-w) contains supplementary material, which is available to authorized users.

Introduction

In most of the more developed countries, the total fertility rate (TFR) is below the (approximately 2.1) replacement level, which in the absence of sustained immigration would prevent a long-run population decline (Espenshade et al. 2004; Rindfuss et al. 2016; Smallwood and Chamberlain 2005; Sobotka 2017). For a majority of such countries, net immigration has generally been positive throughout the twenty-first century, and in many countries, it has been so for considerably longer (UNPD 2019).

A population which experiences constant fertility below exact replacement level, constant mortality, and constant net immigration amount with a fixed age composition will converge to a stationary state with nonzero size, zero growth, and constant numbers by age (Cerone 1987; Espenshade et al. 1982; Pollard 1973). The size of the ‘terminal stationary population’ (TSP) may be seen as an indicator of the what (Ediev 2001) terms the ‘demographic potential’ of the combination of fertility, mortality, and net migration for that population and time, and is invariant to initial proportionate population age structure (Espenshade et al. 1982; Ediev 2001; Parr & Guest 2014). To date, the use of such a synthetic measure to identify which populations have combinations of fertility, mortality and migration which are consistent with long-run population growth and the identification of critical level for fertility which in combination with the prevailing values of other measures of components of growth are consistent with zero long-run growth appears to be absent from the literature.

This paper proposes a measure for fertility (henceforth the Current Migration Replacement TFR (TFR_R)) which in combination with mortality rates and positive net absolute amounts of migration by age and sex at the current levels for a specified population and time period generates a TSP size equal to the (mid-period) population size. Current Migration Replacement differs (arguably desirably so) from the conventional (Net Reproduction Rate = 1) replacement (henceforth ‘Zero Migration Replacement’) in that under the former the number of ‘replacers’ exactly equals the number of ‘replaced’, and that the ‘replacers’ are not restricted to be of the same sex (ratio) as the ‘replaced’, and, in common with the ‘replaced’, are comprised of a mixture of immigrants and others with ancestors who previously moved to that land (Bowles 2019).

The extension of the conventional (zero migration) Net Reproduction Rate (henceforth NRR) to incorporate age-specific rates of net migration, and the concept of a ‘replacement level’ corresponding to constant fertility, mortality and net migration rates (as distinct from absolute amounts) has been considered by Hyrenius (1951), Ryder (1997) and Preston & Wang (2007). However, possibly because the size of the corresponding age group in the destination country has little bearing on the number of events of immigration at that age, the use of age-specific rates of net migration as data inputs to population projections for countries in which it is assumed net migration will be positive has been considerably less common than the formulation of such assumptions in terms of absolute numbers, and the use to date of what Hyrenius (1951) termed the ‘social replacement rate’ (SRR), and Preston & Wang (2007) NRR* has been limited (UNPD 2019). Espenshade (1982) proposed a similarly formulated ‘(net) net reproduction rate’, considering only age-specific emigration rates. Smallwood and Chamberlain (2005) developed a measure of cohort ‘reproductive capacity replacement’ which considered the effects of changes in mortality on the size of cohorts of women reaching reproductive age. Del Ray Poveda and Cebran-Villar (2010) and del Ray Poveda and Ortega (2011) proposed a ‘birth replacement ratio’ which considers the combined effects of the history of fertility, mortality and migration on the sizes of female birth cohorts between birth and reproductive age. The approach proposed in this paper differs in its scope in that it considers the future implications of the (hypothetical) stability of fertility, mortality and absolute amounts of net migration for any specified window of time for the total population numbers across all age and sex groups, and can illustrate the implications of recent data.

Since positive net immigration is integral to the process of population growth of most of the counties with low fertility, and in view of an apparently widespread misconception that a TFR below the ‘approximately 2.1’ (zero migration) replacement level of fertility with long-run population decline irrespective of the immigration level, there is a clear need for a new synthetic measure indicator of the fertility level which is consistent with long-run zero population growth for the entire population under constant positive net migration amount (and mortality) across all age and sex population subgroups and for analysis which quantifies its levels. The paper aims to fill this gap by formulating a (with) Current Migration Replacement TFR (TFR_R). It further describes and analyses its value across an extensive range of more developed countries with positive net immigration with the aim of illustrating the interdependence between migration pattern and the implication of fertility level.

Method

The Terminal Stationary Population (TSP) size (denoted P) which corresponds to sustained constant below-replacement fertility with a constant proportionate age distribution, in combination with constant absolute net immigration by age and sex, age-sex specific mortality rates, and sex ratio at birth at the levels observed for a specified population and time period can be expressed as the sum of components corresponding to generations of migrants (Schmertmann 1992). A person’s migrant generation index is based on the most recent foreign-born individual from the set comprising the person plus his/her all female line of ancestry to migrate into a specified population. A population can be partitioned into migrant generations. Thus P equals the sum of migrant generation sizes:

$$P=\sum _{i=1}^{\infty }{P}_i$$ 1

where P denotes the total size of the stationary population, i is the migrant generation index, and P_i the size of the ith migrant generation1. For constant net migration with constant, nonzero emigration parallel components of population size can be calculated, although literal correspondence between ‘migrant generation’ components and sets of people categorised by ancestry no longer applies. The calculation of the various generation sizes and total TSP size in this paper uses discrete approximations to formulae in Schmertmann (1992) which are readily calculated from widely available national and international statistical agency data. The ‘first generation’ element in Eq. (1) (P₁) is calculated as:

$$P_1=M\sum _{j=1}^2\sum _{x=0}^{\omega }{m}_{x,j}{e}_{x,j}$$ 2

where M denotes the constant annual total net migration, $m_{x,j}$ denotes the proportion of total net migration contributed by persons of age x (last birthday) and sex j (1 for female and 2 for male), $e_{x,j}$ is the (remaining) life expectancy for x and j2, and ω is the maximum age of people in the population.

The ‘second-generation’ element in Eq. (1) (P₂) is calculated by:

$$P_2=M\text{TFR}\sum _{j=1}^2{s}_j{e}_{0,j}\sum _{x=0}^k{m}_{x,1}\sum _{t=0}^{k-x}{f}_{x+t}{}_t{p}_{x,1}$$ 3

where TFR denotes the total fertility rate, f _{x + t} is the proportionate contribution to TFR from the age-specific fertility for age x + t, _tp_x,1 is the probability of a female surviving from x to x + t, k the upper limit of the female reproductive age range, s_j is the proportion of births of sex j, and $e_{0,j}$ is life expectancy at birth for sex j.

The annual births for the TSP (denoted B) is:

$$B=M\text{TFR}\sum _{x=0}^k{m}_{x,1}\sum _{t=0}^{k-x}{f}_{x+t,}{}_t{p}_{x,1}$$ 4

For all i ≥ 2

$$P_{i+1}=\text{NRR}{P}_i$$ 5

where NRR denotes the conventional (zero migration) net reproduction rate. The sum of the sizes of the generation-indexed components for generations with indices 2 and above is the sum of a geometric series with initial term P₂ and common ratio NRR. Hence substituting from Eq. (5) into Eq. (1):

$$P=P_1+\frac{P_2}{\left(1-\text{NRR}\right)}$$ 6

The TFR which in combination with the values of M, m_{x, j}, e_{x, j,} s_j, f_x+t and _t p_x,1 used in Eqs. (1–4) equates the TSP size (P) to the current population size POP (henceforth termed the Current Migration Replacement TFR and denoted TFR_R3) can be calculated by:

$${\text{TFR}}_R=\frac{\text{TFR}}{\text{NRR}}\times \frac{\text{POP}-P_1}{\text{POP}-P_1+\frac{P_2}{\text{NRR}}}=\frac{\text{TFR}}{\text{NRR}}\times \frac{1}{1+\frac{P_2}{\left(\text{POP}-P_1\right)\text{NRR}}}$$ 7

The derivation of Eq. (7) is as follows:

Reorganising Eq. (6):

$$\left(P-P_1\right)=P_2+\text{NRR}\left(P-P_1\right)$$ 6*

Substituting P = POP in Eq. (6*):

$$\left(\text{POP}-P_1\right)=P_{2,R}+{\text{NRR}}_R\left(\text{POP}-P_1\right)$$ P1

where P_{2, R} denotes the second-generation component of the stationary population with size POP_, corresponding to the values of M, m_{x, j}, e_{x, j,} s_j, f_x+t and _tp_x, and NRR_R is the (zero migration) net reproduction rate corresponding to s₁, f_x+t and _tp₀ and TFR_R.

NRR_R may be rewritten as:

$${\text{NRR}}_R=\frac{{\text{TFR}}_R{\text{NRR}}_R}{{\text{TFR}}_R}$$ P2

and P_{2, R} as:

$$P_{2,R}=\frac{{\text{TFR}}_R{P}_{2,R}}{{\text{TFR}}_R}$$ P3

Since specific values of M, m_{x, j}, e_{x, j,} s_j, f_x+t and _t p_x are common to the calculations of NRR_R, NRR, TFR_R, TFR_, P_2,R and P₂

$$\frac{{\text{NRR}}_R}{{\text{TFR}}_R}=\frac{\text{NRR}}{\text{TFR}}$$ P4

and

$$\frac{P_{2,R}}{{\text{TFR}}_R}=\frac{P_2}{\text{TFR}}$$ P5

Substituting from Eqs. (P2–P5) into (P1)

$$\text{POP}-P_1=\frac{{\text{TFR}}_R{P}_2+\left(\text{POP}-P_1\right){\text{TFR}}_R\text{NRR}}{\text{TFR}}=\frac{{\text{TFR}}_R\left[P_2+\left(\text{POP}-P_1\right)\text{NRR}\right]}{\text{TFR}}$$ P6

Reorganising Eq. (P6) gives Eq. (7).

Equation (7) presents TFR_R as the (conventional with zero migration) exact replacement level for A $\left(\frac{\text{TFR}}{\text{NRR}}\right)$ multiplied by an index derived from the values of the first generation (P₁) and second generation (P₂) components of the TSP, and the NRR. TFR_R is strictly less than $\frac{\text{TFR}}{\text{NRR}}$ when $P_2$ is positive4. Moreover, as net migration at all ages approaches zero the value of TFR_R approaches the conventional (zero migration) exact replacement value (i.e. $\frac{\text{TFR}}{\text{NRR}}$). TFR_R cannot be calculated for some populations with very high net migration rates, specifically when the size of the current population (POP) is less than the size of the first generation component of the TSP (P₁).

Following Ryder (1997), Life Expectancy after Net Migration (eNM_i,) for sex j for population is the term used for:

$${\text{eNM}}_{j,}=\frac{{\sum }_{x=0}^{\infty }{m}_{x,j}{e}_{x,j}}{{\sum }_{x=0}^{\infty }{m}_{x,j}}$$ 8

The Index of Life Expectancy after Net Migration (I_{ENM, j}) for sex j is defined as5:

$$I_{{\text{eNM}}_j}=\frac{{\text{eNM}}_j}{e_{0,j}}$$ 9

The Index of the Effect of Migration and Fertility Ages on (TSP) Births6 (I_AB)) is defined as:

$$I_{\text{AB}}=\frac{B}{M\text{TFR}}=\sum _{x=0}^k{m}_{x,1}\sum _{t=0}^{k-x}{f}_{x+t,}{}_t{p}_{x,1}$$ 10

Equation (10) re-expresses (TSP) births [Eq. (4)] per unit of net migration and unit of TFR. The variation in I_AB between countries is affected by variation in the proportionate age distribution of total fertility, the proportionate age-sex distribution of net migration and by the (typically small) variation in probabilities of survival from age at migration to age at birth. A higher proportion of net migration formed by females, a younger proportionate age profile of female net migration and an older age profile for fertility all increase the value of I_AB.

This paper compares Terminal Stationary Population size (P) and the Current Migration Replacement TFR (TFR_R) between countries, using data for 2011–15. Excel spreadsheets which calculate the values of the various measures defined in this section from standard life table, age-specific fertility and net migration by age and sex input data are available free of charge via the internet (researchgate.net/profile/Nick_Parr2/research).

Data

The following 22 more developed populations with a population size (for 2013) exceeding 1 million and which over the 2011–2015 period had both a TFR below 2.1 and positive net migration are considered; Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Hong Kong, Hungary, Italy, Japan, Korea (Republic of), Netherlands, New Zealand, Norway, Singapore, Slovakia, Sweden, Switzerland, UK, and USA. The data were sourced from the websites of Eurostat, official national statistical offices, government departments and agencies, and the United Nations (“Appendix”).

For 2011–15 the mean TFR of the populations considered ranges from 1.23 for Singapore and 1.24 for Korea to 2.02 for New Zealand (Table 1). Mean age at birth is lowest for USA (28.7), relatively low for Hungary and Slovakia, and highest for Switzerland (31.8). Annual net migration ranges from 1.5 thousand for Slovakia to 959.8 thousand for USA, and the rate of net migration per 1000 population between 0.3 for Slovakia and 12.5 for Singapore. The male-to-female ratio for net migration is lowest for France, Hong Kong, and Netherlands and highest for Korea, Slovakia and Germany. For each sex, life expectancy at birth is significantly lower for Hungary and Slovakia than for the other countries considered (Table 1). For France, Hong Kong, Singapore and USA the proportionate age-sex distribution of net international migration was imputed from other sources, due to a lack of publicly available information from official sources with the necessary age detail.

Table 1.

Summary measures of input demographic data: selected countries 2011–2015

Country	Total population (Millions)	TFR (per woman)	NRR (per woman)	Mean age at birth (years)	Net migration (000 s)	Net migration rate (per 1000)	Sex ratio of net migration (males per 100 females)	Mean age at net migration (years)		Life expectancy at birth (years)
								Male	Female	Male	Female
Australia	23.1	1.89	0.91	30.6	201.8	8.7	85.9	23.2	25.8	80.3	84.4
Austria	8.5	1.45	0.70	30.3	58.2	6.9	117.6	24.2	25.0	78.6	83.8
Belgium	11.2	1.76	0.85	30.1	40.4	3.6	81.6	14.8	19.5	78.1	83.2
Canada	35.2	1.59	0.77	30.4	270.4	7.7	94.2	29.9	30.4	79.7	83.9
Denmark	5.6	1.70	0.82	30.8	25.1	4.5	116.2	25.0	24.3	78.3	82.4
Finland	5.4	1.75	0.85	30.4	16.2	3.0	113.8	26.1	27.4	78.0	84.1
France	65.8	1.99	0.97	30.2	59.7	0.9	41.6	23.5	22.0	79.0	85.6
Germany	80.7	1.44	0.69	30.4	556.7	6.9	148.1	25.6	25.3	78.1	83.0
Hong Kong	7.2	1.21	0.58	31.8	23.4	3.3	51.9	29.9	33.7	81.1	86.7
Hungary	9.9	1.36	0.65	29.5	11.1	1.1	137.3	28.9	30.7	72.2	79.1
Italy	60.2	1.39	0.67	31.5	201.3	3.3	90.0	27.4	32.6	80.3	85.2
Japan	127.4	1.41	0.68	30.8	71.6	0.6	111.6	30.3	32.1	80.2	86.6
Korea	51.1	1.24	0.60	31.5	64.2	1.3	157.5	29.7	34.4	78.1	84.6
Netherlands	16.8	1.70	0.82	31.0	28.8	1.7	47.4	-0.51	21.3	79.5	83.2
New Zealand	4.4	2.02	0.97	29.9	27.1	6.1	134.6	24.2	25.0	79.5	83.2
Norway	5.1	1.80	0.87	30.5	48.1	9.5	86.7	25.0	25.3	79.8	83.8
Singapore	5.4	1.23	0.59	31.3	67.6	12.5	98.8	24.9	26.4	80.1	84.5
Slovakia	5.4	1.38	0.67	28.8	1.5	0.3	167.6	31.6	18.7	72.9	80.1
Sweden	9.6	1.88	0.91	30.9	63.2	6.6	111.0	24.3	24.7	80.1	83.8
Switzerland	8.1	1.53	0.74	31.6	46.7	5.8	85.2	23.7	25.7	80.7	85.0
UK	64.1	1.85	0.90	30.0	249.3	3.9	85.7	24.3	24.8	79.2	82.9
USA	316.2	1.88	0.90	28.7	959.8	3.0	94.1	29.5	32.0	76.4	81.2
Mean	42.1	1.61	0.78	30.5	140.6	4.6	102.6	24.9	26.0	78.6	83.7

Results

Terminal Stationary Population Size

For 14 of the 22 countries considered, the TSP size (P) exceeds the current population size (Table 2). In absolute terms P is largest for the USA (430.9 million), followed by the UK (153.3 million), Australia (134.5 million) and France (112.4 million), and smallest for Slovakia (0.2 million), Hungary (1.4 million) and Hong Kong (2.8 million). The ratio of the P to current population size varies widely. It is highest for New Zealand (P is 9 times the current population), and is also above 2.0 for Australia, Norway, Sweden and UK, least for Slovakia (0.04), Japan, Korea and Hungary. Thus the long-run population growth implication of the below (zero migration) replacement fertility, when considered jointly with the current migration and mortality levels, varies widely between the countries considered.

Table 2.

Terminal stationary population size and with Current Migration Replacement TFR: selected countries 2011–2015

Country	Terminal stationary population size (P)		Current Migration Replacement TFR (TFRR)
	TSP size (P) (Millions)	Ratio of P to Current population size	TFRR (per woman)	Ratio of current TFR to TFRR
Australia	134.5	5.82	1.00	1.88
Austria	10.0	1.17	1.34	1.09
Belgium	19.1	1.71	1.55	1.14
Canada	53.4	1.52	1.33	1.20
Denmark	7.7	1.37	1.56	1.09
Finland	5.0	0.93	1.77	0.99
France	112.4	1.71	1.94	1.03
Germany	85.0	1.05	1.40	1.03
Hong Kong	2.8	0.39	1.75	0.69
Hungary	1.4	0.14	1.99	0.68
Italy	29.4	0.49	1.75	0.79
Japan	10.4	0.08	2.02	0.70
Korea	6.5	0.13	1.98	0.63
Netherlands	14.3	0.85	1.75	0.97
New Zealand	40.0	9.00	1.51	1.34
Norway	20.7	4.07	0.96	1.88
Singapore	9.4	1.73	0.60	2.05
Slovakia	0.2	0.04	2.05	0.67
Sweden	35.9	3.74	1.33	1.41
Switzerland	10.5	1.29	1.37	1.12
UK	153.3	2.39	1.56	1.19
USA	430.9	1.36	1.80	1.04
Mean	54.2	1.86	1.56	1.00

That New Zealand has the highest ratio of P to current population size despite six other countries having a higher rate of net migration, is due to its higher TFR7. High rates of predominantly female migration8 contribute to this ratio for Australia and Norway having high values. The low values of this ratio for Japan, Korea, Slovakia and Hungary are linked their low fertility rates and low net migration rates. For the two Eastern European countries, relatively low life expectancies at birth are also a factor.

The Index of the Effect of Migration and Fertility Ages on Births (I_AB) correlates negatively with the mean age of female immigrants and with the sex ratio of net migration, reflecting the shorter remaining reproductive lifespan of older female immigrants and the use of females (and not males) to calculate second-generation numbers (Tables 1 and 3). If other data inputs to Eq. (10) were equal, an older age profile for fertility would raise the value of I_AB. The lack of correlation between I_AB and mean age at birth in Table 3 reflects the association of higher mean age at birth with higher female mean age at migration.

Table 3.

Selected values of metrics related to terminal stationary population size: selected countries 2011–2015

Country	Index of effect of migration and fertility ages on births (IAB)	Life expectancy after net migration (eNM)			Index of Life expectancy after net migration (IeNM)
		Male	Female	Persons	Male	Female	Persons
Australia	0.36	57.8	59.3	58.6	0.72	0.70	0.71
Austria	0.30	55.1	59.2	57.0	0.70	0.71	0.70
Belgium	0.43	63.4	64.0	63.7	0.81	0.77	0.79
Canada	0.26	51.1	54.4	52.8	0.64	0.65	0.65
Denmark	0.33	54.1	58.6	56.2	0.69	0.71	0.70
Finland	0.28	52.8	57.3	54.9	0.68	0.68	0.68
France	0.40	56.0	64.2	61.4	0.70	0.75	0.74
Germany	0.26	53.3	58.2	55.3	0.68	0.70	0.69
Hong Kong	0.28	52.1	53.6	53.1	0.64	0.62	0.63
Hungary	0.26	45.2	50.2	47.3	0.63	0.63	0.63
Italy	0.27	53.7	53.3	53.5	0.67	0.62	0.65
Japan	0.26	50.9	55.3	53.0	0.64	0.64	0.64
Korea	0.21	49.4	51.1	50.1	0.63	0.60	0.62
Netherlands	0.57	79.6	62.4	67.9	1.00a	0.75	0.83
New Zealand	0.26	52.5	54.7	53.4	0.66	0.66	0.66
Norway	0.34	54.8	58.9	57.0	0.69	0.70	0.70
Singapore	0.34	55.8	58.5	57.2	0.70	0.69	0.69
Slovakia	0.29	43.1	62.5	50.4	0.59	0.78	0.67
Sweden	0.32	56.6	59.6	58.0	0.71	0.71	0.71
Switzerland	0.36	57.5	59.8	58.7	0.71	0.70	0.71
UK	0.40	55.7	58.7	57.3	0.70	0.71	0.71
USA	0.27	49.1	50.9	50.0	0.64	0.63	0.63
Mean	0.32	54.5	57.5	55.8	0.69	0.69	0.69

^aFor explanation of why value can exceed 1 see endnote 5

The value of I_AB is least for Korea, which has predominantly male net migration and relatively old female net migration and, hence, low post-migration fertility relative to TFR (Tables 1 and 3). Both I_AB and I_eNM are highest for Netherlands and Belgium. Both countries have low mean ages for net migration and a low sex ratio for net migration. The effects on TSP size of the differences between the age-sex profiles of net migration of Netherlands and Belgium and those of other populations are substantial (Arthur & Espenshade 1988; Schmertmann 2012). If an average age-sex distribution for the other countries is substituted for their (distinctively young) proportionate age distributions for net migration, the TSP size for the Netherlands is just 62% as large (8.9 million compared to 14.3 million), and that for Belgium just 75% as large (14.4 million compared to 19.1 million).

Current Migration Replacement TFR

The values of Current Migration Replacement TFR (TFR_R) (the TFR which, in combination with current levels of net migration and life expectancy is consistent with zero long-run population growth) are lowest for the three countries with the highest net migration rates; Singapore (0.60 births per woman), Norway (0.96) and Australia (1.00) (Tables 1 and 2). The implication is that, even if these countries were to experience sustained fertility at levels in the ‘lowest low’ range, under continuation of the current high levels of (young and predominantly female) net migration and high life expectancies, larger than current populations would result. For eight countries, the TFR_R falls in the ‘very low fertility’ range (i.e. below 1.5) (Billari & Kohler 2004). The value of TFR_R is highest, and near to the (zero migration) replacement level, for Slovakia (2.05), Japan (2.02) and Hungary (1.99) and Korea (1.98). Along with France, these countries have the lowest net migration rates (Table 1). Unlike France, they also have predominantly male net migration (Table 1).

For 14 of the 22 countries the current TFR exceeds the value of TFR_R (Fig. 1). The countries which are ‘above current migration replacement’ include all the English-speaking countries and all the countries which are located in north, west and central Europe, except Finland and Netherlands (and for both these exceptions the current TFR is only marginally below ‘current migration replacement’). All the southern and eastern European countries and all the Asian countries, except Singapore, have a ‘below current migration replacement’ TFR. In absolute terms, the current TFR is furthest above TFR_R for Australia (current TFR is 0.89 above TFR_R), Norway (0.84), and Singapore (0.63), and is furthest below it for Korea (− 0.74), Slovakia (-0.67), Hungary (− 0.63), and Japan (− 0.61) (Tables 1 and 2). The ratio of current TFR to TFR_R is greatest for Singapore (current TFR is 2.05 times TFR_R), Australia and Norway (both 1.88 times), and least for Korea (0.63).

Fig. 1.

Comparison of With Migration Replacement Total Fertility Rate to Total Fertility Rate to 22 Countries 2011–15

Figure 2 illustrates the strong negative correlation between TFR_R and net international migration per 1000 current population. In other words, as the net migration rate increases the level of fertility which equates TSP size with current population decreases9. Figure 3 illustrates the negative correlation between TFR_R and Index of the Effect of Migration and Fertility Ages on Births (I_AB)_. The negative correlation reflects that the TFR which equates the TSP size with a specified population size will be lower when the composition of net migration is younger and more feminine and when the proportionate contribution to the TFR from the older reproductive age groups is higher [Eqs. (7) and (10)]. The steepness of the gradient of the trendline in Fig. 3 is reduced considerably by the influence of an ‘outlier’ (the Netherlands), which has a low rate of net migration and a high value for I_AB (Table 1)10.

Fig. 2.

Current migration replacement total fertility rate plotted against net migration rate for 2011–2015

Fig. 3.

Current migration replacement total fertility rate plotted against index of the effect of ages of migration and fertility on births for 2011–2015

Discussion

The Current Migration Replacement TFR (TFR_R) is a more pertinent indicator of the fertility level with a long-run zero population growth implication than the conventional (zero migration) replacement level for countries in which continued positive net immigration is a more likely prospect than zero migration (UNPD 2019). The results in this paper show that, far from being characterised by ‘incipient decline’, for 14 of the 22 countries considered in this paper the combination of current period fertility, migration and mortality is consistent with a larger population size over the long run; current net immigration more than compensates for fertility being below (zero) migration) replacement level (Notestein 1945). The comparison of a population’s TFR to its ‘Current Migration Replacement TFR’ (TFR_R), illustrated in this paper, may be used to subcategorise populations according to long-run implication for population growth of continuation of their (below zero migration replacement) fertility, (positive) net migration and mortality as follows:

1. Long-run extinction, i.e. where net migration is negative (e.g. Bulgaria, Croatia, Greece, Poland, Romania).

2. Long-run reduction, i.e. where the current TFR is below TFR_R (e.g. Finland, Hong Kong, Hungary, Italy, Japan, Netherlands, South Korea).

3. Long-run increase, i.e. where the current TFR is above TFR_R (Austria, Australia, Belgium, Canada, Denmark, France, Germany, New Zealand, Norway, Singapore, Sweden, Switzerland, UK, USA).

The countries in the ‘long-run increase’ category all have very high average incomes, whilst those in the ‘long-run extinction’ category generally have lower average incomes. This indicates that, due primarily to their differences in levels of net immigration, the highest income countries in the future may continue to experience population growth, whilst lower-income populations whose TFRs have fallen below the replacement level and which experience low or even negative face the prospect of population decline (World Bank 2018).

For some of the countries considered in this paper, particularly New Zealand, Australia, Norway, Sweden and the UK, the TSP size (P) associated with the hypothetical stability of fertility, migration and mortality far exceeds the current population size. Such outcomes are linked to a combination of net immigration which is high relative to population size and which substantial flow-on effects on the sizes of second and higher order generation size due to near (‘usual’) replacement fertility. This paper shows that not only can immigration counterbalance the ‘births-deficit’ results from below (zero migration) replacement fertility but also, perhaps counterintuitively, under some circumstances it is consistent with large-scale future population increase.

This paper further shows the value of TFR_R varies widely between populations with net immigration. For some countries with low net migration rates, particularly Japan, Korea, Hungary and Slovakia, TFR_R is close to the conventional (zero migration) replacement level. However for other countries, such as Singapore, Norway and Australia, even in the event of an immediate and sustained reduction in fertility to ‘lowest-low’ levels, maintaining current net migration and life expectancies would prevent a long-run decrease in population to below the current size (Kohler et al. 2002). Such heterogeneity in the values of TFR_R shows the importance of considering the implication of fertility level jointly in conjunction with a country’s prevailing (or prospective future) migration and mortality, as opposed to applying a ‘one size fits all’ consideration of the fertility level alone (Ryder 1997). The examples of Singapore, Norway and Australia call into question whether any ‘critical’ level of fertility can be regarded as synonymous with long-run population decline, at least for small or medium-sized countries with high income levels (Kohler et al. 2002; Lutz et al. 2006).

According to UNPD (2013), 62% of the countries with a TFR below 2.1 worldwide11, (including 12 countries considered in this paper) have a policy to raise fertility. Five of the countries considered in this study in which the TFR was ‘above current migration replacement’ fertility (Australia, Austria, France, Germany, and Singapore) reported to the United Nations their aim was to raise their fertility level. However, the results in this paper show that for Singapore and for Australia fertility could fall to considerably lower than current levels and positive population growth would still be in prospect.

Certain properties of P and TFR_R are noteworthy. Firstly, although their values are derived from real data, as with other synthetic measures, such as period life expectancies, the TFR and the (zero migration) Net Reproduction Rate, the interpretation of their values is based on hypothetical stability in data input values, and are not ‘real-world’ observations (Guillot & Payne 2019). The comparison of P to current population may be used to represent the interconnected implications of fertility, migration and mortality for future population growth over an infinite time interval. The value of P may complement population projections which assume constant below (usual) replacement level fertility, migration and mortality over a finite time interval by illustrating the associated limit point for population size12. In contrast to projected population growth over a finite interval, it is unaffected by the initial distribution of population by age and sex. Thus the comparison of P to current population may be used to indicate whether projected future population growth (or decrease) over finite time is more than a matter of the ‘population momentum’ stemming from the initial population age structure and the historical patterns of birth rates, death rates and migration which have determined it.

Secondly, the potential population growth related to a specified combination of net migration, fertility and mortality is more than just a matter of the values of overall ‘headline’ measures for these variables (i.e. total net migration, TFR and life expectancy at birth): it also is affected by the interaction of their age-sex distributions. This paper proposes an Index of Life Expectancy after Net Migration (I_eNM) and an Index the Effect of Migration and Fertility Ages on Births (I_AB) as new synthetic measures of the population growth implications of the combined age-sex distributions of net migration, fertility and mortality. This paper illustrates how use of I_eNM and I_AB can augment the use of total net migration, TFR and life expectancy at birth in accounting for the future population growth implication of a combination of fertility, mortality and migration, and its results provide examples of countries (specifically Netherlands and Belgium) for which the potential population growth and the value of TFR_R are influenced considerably by high values of I_eNM and I_AB (and, underlying these measures, by a young age profile of net migration).

Thirdly, a parallel formula to the one used in this paper may be applied to calculate a measure of replacement fertility when immigration is expressed in terms of absolute amounts and emigration in terms of age-sex specific rates13 (Espenshade 1982; Ryder 1997). It is to be expected the differences between P and the current population and between Current Migration Replacement level and the current TFR will be smaller when emigration is considered as a rate than when it is considered in absolute amount terms, as in this paper (Ryder 1997). For some of the populations considered in this paper to consider emigration as a rate was not possible, due to data availability constraints. Moreover, this paper’s use of absolute amounts for net migration corresponds to format of migration assumptions used in population projections by the United Nations and official statistical agencies (ABS 2018; ONS 2019; UNPD 2019).

Fourthly, (unlike for the conventional (zero migration) replacement level) changes in TFR_R over time will be affected by changes in net migration, and for some countries such changes have been considerable in recent years (UNPD 2019). As is usual in relation to the conventional (zero migration) replacement fertility, it is expected, categorisation of fertility through binary comparison (i.e. above or below) of TFR to TFR_R will more widely used than the precise value of their ratio. As for the conventional (zero migration) replacement level, for any given country the precise values of TFR_R, may change considerably over time, however such changes will not necessarily affect the above/below replacement categorisation of fertility14 (Gietel-Basten & Sherbov 2019).

Fifthly, the value of TFR_R corresponding to specified net migration will reduce in the future if, as might be expected in view of past trends, mortality continues to improve or if fertility ages continue to increase (De Beer et al. 2017; Parr et al. 2016; Rindfuss et al. 2016). Finally, whilst under a constant TFR_R (in combination with the corresponding migration and mortality) the TSP size will boomerang over time back towards the current population size, the age profiles of both the corresponding TSPs will differ from each other and from the current population age structure (Ryder 1997; Schmertmann 1992).

None of the aforementioned should affect preference for use of the Current Migration Replacement TFR ahead of conventional ‘2.1’ (zero migration) replacement level as a yardstick for the population growth implication of fertility in countries with net immigration.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Appendix: Data Sources by Country

Country	Population	Fertility	Migration	Life tables
Australia	ABS population clock Year: 2013 https://www.abs.gov.au/	ABS births Years: 2011–15 https://www.abs.gov.au/	ABS net overseas migration Years 2011–15 https://www.abs.gov.au/	ABS life tables Australia Year 2012–14 https://www.abs.gov.au/
Austria Belgium Denmark, Finland, France, Germany, Hungary, Netherlands Norway, Slovakia, Sweden, Switzerland UK	Eurostat database Population on 1 January Years: 2013 and 2014 https://ec.europa.eu/info/statistics_en	Eurostat database Fertility rates by age Years: 2011–15 https://ec.europa.eu/info/statistics_en	Eurostat database Immigration by age group, sex and citizenship and emigration by age group, sex and citizenship Years: 2011–15 https://ec.europa.eu/info/statistics_en	Eurostat database ec.europa.eu/Eurostat Life table Year 2013
Canada	Statistics Canada population estimates Year 2013 https://www.statcan.gc.ca/	Statistics Canada Crude birth rate, age-specific fertility rates and total fertility rate (live births) Years 2011–15 https://www.statcan.gc.ca/	Statistics Canada Estimates of the components of international migration, by age and sex, annual Year 2011–15 https://www.statcan.gc.ca/	Statistics Canada Life tables, Canada, provinces and territories Year 2012–14 https://www.statcan.gc.ca/
Hong Kong	Hong Kong census and statistics department population by sex Year 2013 https://www.censtatd.gov.hk/	Hong Kong census and statistics department (2017) fertility trend in Hong Kong Year 2011–15 https://www.censtatd.gov.hk/ Sex ratio at birth from UNPD (2019)	Total from Hong Kong census and statistics department total population 1920 to 2017 https://www.censtatd.gov.hk/ Proportions by age and sex from Hong Kong Immigration department statistics for inmovers on one-way permits	Hong Kong census and statistics department. Hong Kong Life tables Year 2013 https://www.censtatd.gov.hk/
Japan	Statistics Bureau of Japan Population Estimates Year: 2013 https://www.stat.go.jp/english/	Statistics Bureau of Japan Live Births and Live Birth Rates by Age Group of Mother Years 2011–15 https://www.stat.go.jp/english/ Sex ratio at birth from UNPD (2019)	Statistics Bureau of Japan Net Migration of Japanese and Foreigners by Age and Sex Years 2010–11 to 2014–15 https://www.stat.go.jp/english/	Statistics Bureau of Japan Life Table Year: 2013 https://www.stat.go.jp/english/
Korea	Statistics Korea Resident Population Years: 2011–15 https://www.kostat.go.kr/eng/	Statistics Korea Final Results of Births Statistics Years: 2011–15 https://www.kostat.go.kr/eng/ SRB from UNPD (2019)	Statistics Korea Long-term International Migration by Sex and Age (Nationals and Foreigners) Years 2011–15 https://www.kostat.go.kr/eng/	Statistics Korea Life Tables for Korea Year 2013 https://www.kostat.go.kr/eng/
New Zealand	Statistics New Zealand National Population Year 2013 https://www.stats.govt.nz/	Statistics New Zealand Births and Deaths Years: 2011–15 https://www.stats.govt.nz/	Statistics New Zealand Permanent and Long Term by Age, Sex and Region Years: 2011–15 https://www.stats.govt.nz/	Statistics New Zealand Period: 2012–14 https://www.stats.govt.nz/
Singapore	Statistics Singapore. Singapore Residents by Age Group, Ethnic Group and Sex https://www.singstat.gov.sg/	Statistics Singapore Births and Fertility Rates: Annual Years 2011–15 https://www.singstat.gov.sg/ Sex ratio at birth from UNPD (2019)	Total net migration from annual population change and natural increase. Proportions by age and sex based on average for other countries	Statistics Singapore Complete Life Tables for Singapore Resident Population Year: 2013 https://www.singstat.gov.sg/
USA	United States Census Bureau Population Clock Year 2013 https://www.census.gov/	CDC Births. National Vital Statistics Reports Years 2011–15 https://www.cdc.gov/	Total net migration by sex from OECD.Stat database Years 2011–15 Distribution by age imputed using Census Bureau population estimates and CDC life tables	Arias et al. (2017) National Vital Statistics Reports 66(3) Year: 2013 https://www.cdc.gov/

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Not applicable. All input data are in aggregate form and were obtained from publicly available secondary sources.