In recent decades, fertility in European countries has been marked by two simultaneous trends: an increase in age at first birth and a reduction in the number of children per woman. While it is difficult to establish a causal link between the two, we can nonetheless examine how they are related. Given that it becomes more difficult for women to conceive as their age increases, is first birth postponement a factor behind the fertility decline? To what extent does growing recourse to assisted reproductive technology enable couples to delay their childbearing until later ages? Henri Leridon explores these questions in this article using microsimulation methods based on numerous biological and behavioural parameters observed in France for three generations of women born between the 1930s and the 1970s. He measures both the share of the fertility decline potentially attributable to birth postponement, and the extent to which it has been offset for recent generations by recourse to assisted reproductive technology.
1Given that fecundity decreases with age, postponing the decision to start a family carries the risk – all other things being equal – of lower completed fertility. While the postponement of childbearing is not the only factor involved, changes in fertility timing observed in European countries over recent decades may have contributed to the reduction in completed fertility of successive cohorts. But to what extent?
2There are two dimensions to the decline in fecundity: the onset of permanent sterility (conception is not possible above age X, except possibly via recourse to certain types of assisted reproductive technology (ART)), and a decline in fecundability, i.e. in the risk of conception for a woman who is not sterile and not using ART. At first sight, the former dimension has greater consequences for fertility: if seeking to become pregnant is postponed from age X1, when the woman was still fecund, to age X2, when she is sterile, the chance of conceiving a child will have been lost definitively. In contrast, the reduction of fecundability alone implies that conception takes longer to achieve but is still possible. The postponement of family formation can thus have purely biological effects on completed fertility; this is the first question we will seek to answer in this article.
3Another problem is that couples’ wishes may change over time: fertility depends, of course, on biological aspects, but above all on the desires and behaviours of couples. If, for example, the desired number of children is revised upward or downward, or if a woman wants to make up for the time lost at the start of her reproductive life, couples’ contraceptive behaviour will be modified as a consequence. Demographers have examined the evolution of age-specific fertility rates in successive cohorts to show these “catch-up” effects. In an article on this subject, Laurent Toulemon concluded:
Fertility trends in the cohorts born after 1960 suggest that this postponement will be associated with a decline in many countries, but this evolution will not be general and is not inevitable. On the one hand, the delay in first births does not necessarily imply a reduction in the proportion of women having children, and on the other, the number of children per mother is stable in many countries.(Toulemon, 2006, p. 41)
5The purely biological effect that we seek to quantify here is potential, since it can be cancelled out or, on the contrary, augmented by behavioural modifications. It is nonetheless a factor to be considered when discussing the advantages and disadvantages of changes in the timing of family formation.
6The potential effects of regular advances in assisted reproductive technology must also be examined, since ART provides a means to overcome certain reproductive difficulties. Two aspects will be covered here: the first consists in taking account of the observed increase in recourse to these methods in recent decades, since actual fertility integrates this dimension. The second approach is prospective, namely to assess the extent to which greater recourse to ART might compensate for the current or future biological effects of birth postponement.
7We see that numerous factors intervene and interact. A very good way to take them into account simultaneously is to use Monte-Carlo simulation. This involves describing a complete reproductive history, from the beginning of family formation until the end of the reproductive period. Month after month, the individual (a woman) is exposed to successive risks, contingent on her previous history. Age is a principal variable, as several of the risks in question are directly age-dependent. But other variables are also pertinent, such as the time since entry into first union, since the last birth, or since the decision to start a pregnancy.
8This model has already given rise to several publications (Habbema et al., 2009, 2015; Leridon, 2004; Leridon and Slama, 2008; te Velde et al., 2012). In the 2008 article, we estimated the effect of first birth postponement at the individual level, for illustrative purposes. The 2012 article focused on the consequences in terms of overall childlessness (the proportion of women who remain childless) and its potential reduction by means of assisted reproductive technology in several countries. Here we will focus principally on France, and secondarily on Austria and Italy, in each case taking account of how the timing of births has changed over the past 50 years.
9Several changes have been made with respect to our articles of 2004 and 2008. On the biological side, we have improved the assumptions about fecundity in a “natural”  regime to allow for the risk of miscarriage. Assumptions concerning the onset of permanent sterility have likewise been modified. The previous assumptions were based on data pertaining to last births in France many years ago, before birth control had been widely adopted. Critics have pointed out that health conditions are much better today, which may have reduced the risk of sterility linked, for example, to puerperal fever. We have derived a new age distribution for onset of sterility based on more recent data. These data were compiled and “optimized” by Eijkemans et al. (2014) based on the age at last birth in six populations presumed to be non-contracepting, some of which had their children in the second half of the nineteenth and the first half of the twentieth centuries.  We have also increased the efficacy of ART to take account of recent technological changes, and envisaged the possibility of several successive attempts by a couple to achieve a pregnancy. Moreover, the proportion of multiple births resulting from ART is also taken into account, which notably increases the overall efficacy of these techniques.
10Our approach is also new. Our previous calculations were “theoretical”, in the sense that we modelled hypothetical changes in behaviour to measure their potential impact on completed fertility. Here we do the reverse: based on actual changes in the completed fertility of successive cohorts, we seek to deduce the behavioural changes that could have produced them. We then proceed from data on observed recourse to ART over cohorts to estimate its impact on cohort fertility.
I. Model implementation
A trade-off between reality and simplicity
11Although simple in principle, the microsimulation model rather quickly becomes complicated in the details of its implementation (Appendix A.1). It would be an impossible task to fully represent biological and behavioural reality, so the complexity of the model must be adjusted to the nature of the question being asked. However, the model must be realistic enough to produce derived variables that have the expected orders of magnitude (like the mean and the distribution of the final number of children per woman, or the mean age at birth) to serve as the basis for more detailed analysis of the effect of a specific variable. Let us consider some examples.
12If we are interested in the waiting time between the first attempt to become pregnant and successful conception, it is indispensable to work in a monthly time frame (the average waiting time is less than one year for couples with normal fecundity), to use a monthly distribution of couples’ conception risk (fecundability), to make individual fecundability age-dependent (at least that of the woman), and to assume that its value does not simply fall abruptly to zero at the end of the reproductive period. All of these conditions are dictated by empirical studies and research on methods for estimating fecundability. But it is clear that a model interested solely in the distribution of the final number of children, for example, could function with a less detailed representation of the biological factors and still produce the expected distribution.
13Here is another example. In the unfolding of each life history, a union disruption may occur at any time. We ignore here situations of widowhood, since male mortality at ages 20-45 is negligible, in the order of 3%. In any case, it is possible to include the mortality effect with other forms of union disruption. The other types of disruption can be taken into account by our model, along with entry into a new union. Nevertheless, as union histories have become rather complex, we do not seek to include them as we would have to envisage several successive disruptions, estimate the rates of repartnering (by duration and by age), and consider the number of children already born in previous unions to define the objectives in the new union. A whole series of hypotheses would be needed, most of which would be very difficult to formalize. Here, then, we limit the model to definitive separations only, to reflect the end of exposure to the risk of a birth, independently of involuntary sterility. This example shows that it would be imprudent to seek to capture behaviours – even more so their evolution over time – that cannot be adequately modelled in the simulation. However, this limitation does not disqualify the model if the “neglected” variable is neither essential nor strongly linked to the object of our study.
14Evaluating the impact of ART on overall fertility provides a very good example of the risk of excessive model simplification. Hoorens et al. (2007) claimed that the effect on completed fertility of systematic recourse to ART after failure to conceive within 12 months could be 1 or 2 children per woman, while the estimates of other authors are of the order of 0.2 to 0.3. Habbema et al. (2009) have shown that this difference is the consequence of a faulty hypothesis: Hoorens et al. assume that in the absence of ART, the fertility of the couple concerned would have remained permanently at zero after one year without conception. This ignores the fact that the time to conception can exceed 12 months, even for couples of normal or close to normal fecundity, and that many couples who use ART are not completely sterile, but simply subfecund: their chances of eventually conceiving naturally are thus not zero. This fact has been confirmed by empirical studies (Troude et al., 2012). Finally, implementing the Hoorens hypotheses in the Habbema model confirmed that the difference between their two sets of results was attributable to the simplifying hypotheses of Hoorens et al.
15The fact that the model satisfactorily reconstitutes the available fertility data (mean and distribution of the number of children, age-specific fertility rates, birth intervals, mean age at childbearing, etc.) suggests that the model functions well, despite its complexity, but this does not guarantee that all of the behaviours are reproduced well, nor that other combinations of parameters would not have yielded similar results. But the experience acquired with these simulations shows that analysis of the consequences of the variability of a specific parameter is only marginally affected by the remaining inaccuracies of the other parameters. 
Constructing a reproductive life history
16A constant age-related risk is that of dying. Although it could be easily included in the model, we will neglect it here, since the probability of dying at ages 20-45 for French women is at present only 1-2%. Besides, we are interested here in the fertility of women who are still alive at age 50. In practice, a woman’s reproductive history begins with the formation of her first union. Initially, we took only marriage into account, but now that more than half of births occur outside marriage, this milestone has become irrelevant. However, not many births occur outside a union. We have data on the age of entry into a first union, but they do not always allow us to follow changes in behaviour over time, especially in a comparable manner across different countries. What we are interested in here is the change in age at first birth, knowing that first births typically occur sometime after union formation (couples do not necessarily want a child right away). It makes little difference if we shift the age of entry into union while maintaining the desired time lag between union entry and first birth, or extend the waiting time to conception while maintaining the age at first union. In practice, we operate “in reverse” and in the following manner: based on available data, we define a distribution of age at first union, associated with a waiting period before attempting a first pregnancy so as to reconstitute the age at first birth; next, based on the change in age at first birth, we shift the distribution of age at first union. We cannot claim to reflect the exact union formation behaviours in each country and each period, but we fit a plausible distribution of age at first union to the actually observed distribution of age at first birth. Another advantage of this way of proceeding is that de facto, births outside a union will be taken into account, since we are fitting to data on age at first birth, regardless of union status.
17Our construction of the reproductive history continues as follows. Since the first birth is not necessarily desired immediately, we will assume that the couple uses a reliable contraceptive method. Since contraception is never 100% effective (except for sterilization, which is very rare in France before age 35), there is a risk of an unwanted pregnancy. This may be terminated by an induced abortion, and for the sake of simplicity, we will include this possibility in the efficacy rate of the contraceptive method used. Available data on contraceptive practice (Bajos et al., 2004; Rossier and Leridon, 2004) are converted into contraceptive efficacy rates, that is, coefficients of fecundability reduction. But we know that despite widespread contraceptive use and recourse to abortion, a non-negligible proportion of births are unwanted or occur earlier than desired (Leridon and Toulemon, 1990). The model allows us to reconstitute these proportions. After a given waiting time (the desired spacing), the couple is assumed to stop contracepting in order to conceive. This is the “biological” – and perhaps the most original – part of the model. Our simulation is very detailed in this regard, enabling us to take very diverse behaviours into account, and in particular, recourse to ART.
18The progress of the reproductive process depends on other behaviours in addition to the decision to enter or leave a union. We have already taken account of contraceptive efficacy, including abortion. Now we must introduce a key variable, the desired number of children – assuming that couples behave rationally – based on a number of children and the desired spacing. Of course, these plans may change over the life course, in response to the vagaries of marital life, economic constraints, experience with earlier births, etc. At the same time, we note that an indicator as simple as the mean “ideal number of children” reported by women and men of reproductive age (ages 25-34) and contextualized “for their socioeconomic environment”,  is very close to the mean completed fertility of these same cohorts (Toulemon and Leridon, 1999). We thus used the distributions based on the ideal number of children so defined, considering that the fertility calculated in the model could have resulted from the perfect application of couples assumed fertility intentions. Here again, we seek to take account of plausible behaviours, in the knowledge that reality, in all its details, is much more complex.
19The ideal number of children and its distribution across couples are known from surveys like the Eurobarometer for European countries. These data allow us to distribute the couples of a given cohort by the desired number of children, from 0 to 4 (we combine values greater than four children in the last category), and to take account of potential changes in these desires. One difficulty is that, at the individual level, the desired number of children at a given moment does not necessarily determine the exact number of children born: even if contraception is 100% effective, involuntary sterility may prevent couples from reaching their desired family size. Moreover, some women never enter a union and remain childless – voluntarily or otherwise. Among women in union, very few say that they want no children (often 2-3%), but those who say at a given time that they want at least one child may subsequently change their minds (Beaujouan, 2016; Régnier-Loilier and Sébille, 2017; Toulemon and Testa, 2005). For these reasons, our distributions of desired number of children deviate slightly from the ideal numbers reported by women. But as we aim to reconstitute the total fertility of each cohort, we prefer to keep couples who do not want children in our calculations.
20When a couple is exposed to the risk of conceiving, it only achieves this goal after a certain time, in the order of a few months, depending on the monthly likelihood of conception or fecundability. Of course, this likelihood depends on the fecundity of both partners (de La Rochebrochard and Thonneau, 2003), but we do not try to estimate the respective roles of the woman and the man, for the simple reason that fecundability can only truly be measured at the couple level. The available data thus combine the contributions of both the man and the woman. Moreover, fecundability is not the same for all couples: it is well known that some couples are less fecund, or even sterile. Previous work has shown that the variability of fecundability among (non-sterile) couples can be modelled using a Beta law, which also provides a quite simple means to describe changes in the distribution of fecundabilities over successive months without a conception (Bongaarts, 1975; Leridon, 1977; Wood, 1989). We thus attribute to each woman in the simulation a reference fecundity drawn at random from the chosen Beta distribution. In case of exposure to the risk of conceiving, the initial fecundity is that resulting from the draw, modulated by age at onset of exposure (see below). During each of the following months, fecundability is defined by the Beta law, taking account of the number of months that have passed without a conception; it thus decreases rapidly.
21Moreover, fecundity declines as age increases (Bongaarts, 1975; Wood, 1989). It is measured in practice by the woman’s age, but age at onset of definitive sterility must also be taken into account. This parameter varies from one couple to the next, and should not be confused with age at menopause, which is a different variable (Stanford et al., 1987). Estimating the distribution of ages at onset of sterility is tricky. No tests exist to determine whether a woman is already sterile, except in cases of major pathologies such as obstructed fallopian tubes. In addition, data on age at last birth are of no help when couples control their fertility: from the moment when a couple decides to end all exposure to the risk of conception, there is no way of knowing when it finally becomes biologically sterile. The situation is different in a context of natural, uncontrolled fertility, in which case a relationship can be established between age at last birth and onset of sterility – the former resulting from the latter – without confusing the two distributions (a detailed description of the method is given in Leridon, 2004). In the model, we thus include a probability of becoming permanently sterile at age X.
22Let us now return to fecundability. It would be unrealistic to assume that the curve of fecundability with age is simply interrupted at the onset of sterility, with no relationship to this event. A study of the distribution of birth intervals by completed family size (Leridon, 1977), has shown that the complete network of these intervals can only be satisfactorily reconstituted by assuming that fecundity decreases as the age of sterility onset approaches. Moreover, this result is consistent with biological knowledge: there is no abrupt failure of reproductive capacities but rather a gradual decrease as menopause is approached, due, for example, to less frequent ovulation (O’Connor et al., 1998 and 2001; te Velde and Pearson, 2002). We will thus assume that once the age of sterility onset is determined (by a random draw), fecundability diminishes over several years as that age is approached. 
23Following a conception, the pregnancy may not end in a live birth: the risk of spontaneous fetal mortality is non-negligible, and increases significantly with age beyond age 30. It is, in fact, the primary factor in the reduction of fecundity as age increases (Leridon, 2008). Fortunately, we have data on this risk from contemporary populations (Leridon, 1977). A conception resulting in a live birth is followed by a non-susceptible period: the nine months of pregnancy, plus about two months of post-partum sterility if the mother does not breastfeed, and a longer period if she does. In the event of a miscarriage, the non-susceptible period is shorter, in the order of four to five months (Leridon, 2005).
24All of this allows us to define for each woman (since her initial fecundability is drawn at random) and for each age, the risk of a conception resulting in a live birth within a given period and by age at onset of exposure.
Assisted reproductive technology
25Taking account of possible failures of contraception, and even though we increase the efficacy of contraception once the desired number of children is reached (“stopping” efficacy as opposed to “spacing” efficacy), couples may end up with more children than desired, and this can be taken into account in the model. But they may also have fewer children than desired in the case of early onset of sterility or insufficient fecundability. If the couple encounters such difficulties, that is, if the waiting time becomes too long, they may decide to use assisted reproductive technology (ART). The main method used today is in vitro fertilization (IVF) or its variant, intracytoplasmic sperm injection (ICSI). IVF is used mainly to treat female problems of ovulation and fertilization by fertilizing the woman’s ovum outside the uterus (with the partner’s sperm or, possibly, that of a donor) and then reinjecting the fertilized egg into her uterus. ICSI addresses the problem of male infertility due to oligospermia (decreased sperm concentration) because only one sperm cell is needed for direct in situ injection into the ovum. ICSI has become widespread during the past decade and is replacing artificial insemination by donor (AID) because there is no need for a sperm donor. Since the practical efficacy of IVF and ICSI are similar, we consider them together. This efficacy is well known, as is the frequency of repeat attempts in case of failure. We can thus envisage in the model that after a certain period without conception, a given proportion of couples will resort to ART with a defined level of efficacy.
26Although other methods of assisted procreation exist, some of which are more sophisticated (such as frozen embryo transfer (FET) involving the implantation of frozen fertilized eggs) or less invasive but not widely used (like intra-uterine insemination), we do not distinguish them. Our ART indicators are assumed to encompass all available techniques. We can also take account of improvements over time in the efficacy of these techniques, and of the proportion of couples willing to use ART.
27How is recourse to these methods taken into account in the model? Since we are following the reproduction of a couple month by month, we could decide that after a certain duration of exposure to the risk of conception, DMAX (exposure that begins after a potential voluntary birth spacing time), the couple chooses to seek medical assistance. Effective implementation of an ART method may take some time, involving visits by the couple to specialized physicians, time for reflection, medical preparation for the intervention (for example, cycles of hormonal stimulation), even waiting times for treatment in a specialized centre. Consequently, several years may go by between the couple’s initial decision and a first attempt at IVF or ICSI. French studies have observed a waiting time of up to 4-5 years on average (FIVNAT, 2001); a more recent study reported a median waiting time of 4 years, with a mean of about 3.5 years (Troude et al., 2012). We will use waiting periods of this magnitude, and assume that they shorten rapidly as age increases.
28Next we need to decide how the couple’s fertility is affected by the treatment. We have chosen to consider that thanks to effective technical procedures, ART results in a return to non-zero fecundability for totally sterile men and women and in improved fecundability of subfecund couples. Under the first hypothesis, we consider that the couple is exposed to the same risk of conception as would have been the case before the onset of early permanent sterility. Under the second, we apply a multiplier coefficient k (for example, 5) to the “natural” period fecundability such that fecundability of 0.1 can be raised, for example, to 0.5, and for an older couple, fecundability of 0.01 increases to 0.05. In fact, it is known that the success rates of all of ART techniques decrease with age because they are applied in a physiological context that is largely determined by the woman’s age.  The two parameters that we use, KSTER for the proportion of sterile couples who manage to conceive and EART for the efficacy of ART (multiplier coefficient of fecundability), were chosen so as to replicate the observed success rates with the main ART methods (Appendix A.1.).
29Our objective is to determine the extent to which the increasing age at family formation, absent of any other factors, and for purely biological reasons, has influenced completed cohort fertility. We focus our analysis on France and, for illustrative purposes (the available data being less detailed), we briefly examine two countries – Austria and Italy – with rather different recent fertility histories.
30France is an example of a country where fertility has remained relatively high in comparison to the rest of Europe. Completed cohort fertility fell from 2.6 children per woman for the 1930 cohort – which had its children principally between 1955 and 1970, during the height of the baby boom – to 2.0 for the 1975 cohort (estimation based on fertility reached in 2015), whose children will have been born between 2000 and 2020. Between these two, the 1945 cohort (2.2 children) is representative of the behaviours of the post-baby-boom period, after most of the decline in completed fertility and, more especially, just before the sharp increase in age at first birth, which rose from 24 to 28 years between the 1945 and 1975 cohorts. The difference between these two cohorts allows us to estimate the impact of the change in the timing of births.
31In Austria, the fertility decline was early and strong: completed cohort fertility fell below 2 between the cohorts of 1940 and 1945 and continued to decline steadily (1.64 for the 1975 cohort). This low fertility is the result of some very specific preferences: according to Eurobarometer 2001 (Goldstein et al., 2003), 37% of women aged 20-34 reported an ideal family size of less than 2 children, compared with 16% in Italy and 8% in France.  The mean age at birth was lowest in the 1945 cohort (25.2 years), with a slow increase up to the 1955 cohort (25.8 years) followed by a rapid rise (28.8 years for the 1975 cohort). For reasons of data availability, we will use the 1955 and 1975 cohorts.
32Italian history is comparable to that of Austria in terms of fertility trends, but ages at entry into first union and at first birth are high. The latter increased by nearly 6 years between 1975 and 2013 (30.6 years in 2013 in cross-sectional data), and by more than 4 years between the 1955 and 1975 cohorts that we also use here.
33To simulate the cohort life histories, we began by fitting the model to the 1945 French cohort, which will serve as our reference: it is the cohort for which we have the greatest amount of information useful for the model. Women of that cohort entered a union earlier (22.5 years on average) and had their first child more quickly (24.0 years) than any of the cohorts that came after 1930 (Prioux, 2005; Toulemon and Mazuy, 2001). Of these women, 7% never entered a union (Prioux, 2005), and 21% were not living in a union at age 50 (Toulemon and de Guibert-Lantoine, 1998). In the end, they had 2.22 children on average, fewer than the 1930 cohort (2.63) but more than the 1975 cohort (2.01). This is also a little below what they reported as the ideal number, even if we limit the analysis to women aged 25-34 in a union (or ever in a union), who are the most immediately concerned by childbearing choices (Toulemon and Leridon, 1999). As indicated above, we will use a slightly different distribution of the desired number of children (which serves as the childbearing objective in the model, and beyond which contraception is assumed to be much more effective), although with the same mean. The proportions of pregnancies that were unplanned (i.e. that occurred during the desired spacing interval) and unwanted (that exceeded the desired total number of children), known from numerous French surveys (Leridon and Toulemon, 1990; Régnier-Loilier and Leridon, 2007), allow us to verify the plausibility of our hypotheses on birth spacing and contraception efficacy.
34To ensure a perfect fit with the model, we can use the following tools: distribution of the number of children actually desired, desired spacing, and efficacy of spacing and stopping contraception. The following tables have three columns: the observed data for the cohort in question, to which the simulation must fit as closely as possible; the data used as parameters of the model; and the results of the best simulation. Supplementary information on the model and the data used is given in the Appendix.
35We proceed likewise for both the 1930 cohort, to verify that the model can capture quite different situations, and the 1975 cohort, which will also serve to estimate the effect of birth postponement.
France, 1930, 1945 and 1975 cohorts
36Table 1 presents the results of the simulations designed to provide the best possible representation of the observed data for the 1930 and 1945 cohorts. In both cases, completed fertility (the principal objective) is correctly replicated. The distribution of families by size is slightly less accurate: the simulation yields more childless families (even though the proportion of women not wanting any children is very small), fewer families with one or two children, and a slight over-estimation of large families. Involuntary sterility thus appears to be more frequent in the simulation, although the proportion of women who are biologically sterile from age 25 is small, at 3.5%. The other parameters are well-depicted, notably mean age at first and last birth, the proportion of never-partnered women and that of women not in a union at age 50, as well as the proportions of unwanted and unplanned births.
Table 1. Life histories of the 1930 and 1945 cohorts, France
Table 1. Life histories of the 1930 and 1945 cohorts, France
37We will now look at the effect of birth postponement observed between the 1945 and 1975 cohorts, whose mean age at first birth rose from 24.0 to 28.0 years (Table 2). To begin with, we shifted the distribution of entry into union of the 1945 cohort by 4 years without changing any other parameter to measures the effect of postponement alone (Table 2, left-hand side). Completed fertility is 2.008 children per woman, compared to 2.211 in the simulation for the 1945 cohort, i.e. –0.2 children (or –9%). This result is very stable when one or other starting parameter is modified slightly.
Table 2. Life histories of the 1975 cohort. Three simulations (without ART), France
Table 2. Life histories of the 1975 cohort. Three simulations (without ART), France
38It turns out that this level of completed fertility corresponds exactly to the actual fertility of the 1975 cohort. While postponement alone produces this outcome, it is nonetheless possible – and even certain – that other factors also changed, with compensating effects. We did another reconstitution (right-hand side of Table 2) using various data available for the cohort, like more effective contraception (which goes in the direction of a decline in fertility) and a higher ideal number of children (implying higher fertility). The distribution of ages of entry into union is now shifted by only 3 years (instead of 4) to make the age of entry into union closer to that observed, but this effect is offset by longer spacing to maintain the same age at first birth and at all births. We end up very close to the expected mean number of children, i.e. 1.996 children instead of 2.010.
39We also tested the effect of shorter interval between successive births which might be associated with a later first birth (results not shown in the tables). Beginning with the G75i data used in the first columns of Table 2, we reduced the periods of contraceptive use (before stopping) from 15 to 9 months for second and third births and from 12 or 9 to 6 months for fourth or higher births (the time before attempting a first pregnancy is unchanged, since it determines the age at first birth). Completed fertility is 2.053 children (instead of 2.008), i.e. fertility declines slightly less (about 25%) than in the case of simply postponing the age at first birth by 4 years. It is clear that a shortening of birth intervals is not sufficient to offset the effects of this delay.
40The preceding results do not take account of ART. For the 1930 cohort, which had its children between 1950 and 1970, ART was not available. For the 1945 cohort, births took place between 1965 and 1985, when ART was still limited primarily to methods of hormonal stimulation. We applied the parameters cited above (rather optimistic in this context). The result (Table 3), with a third of women using ART, was a very modest increase (+0.008) in completed fertility. For the 1975 cohort, we strengthened the hypotheses (greater efficacy and shorter waiting times) to take account of progress in these treatments. The effect of considering ART is thus logically greater: with 33% of women using ART, completed fertility reaches 2.013 (+0.017), which is again very close to the observed completed fertility.
Table 3. Results when ART is taken into account in the 1945 and 1975 cohorts, France
Table 3. Results when ART is taken into account in the 1945 and 1975 cohorts, France
41In conclusion, we have, on the one hand, a loss of 0.2 children due to postponement of childbearing, and on the other, a possible gain thanks to ART that is less than one-tenth the size, estimated at between 0.01 and 0.02 children. The latter is very far from offsetting the former.
42Table 3 gives some complementary results on use of ART. Differences in completed fertility, with and without ART, are reported, comparing a scenario in which all women eligible for ART (under the conditions defined in our model) effectively use it, and another in which only one-third do so (which is certainly more realistic).  For each hypothesis, we then indicate the number of births resulting from ART in the different simulations, expressed in the (theoretical) impact on completed fertility. The values obtained with systematic recourse to ART in case of difficulties conceiving are well above the other two. This is because a large share (and even a substantial majority) of the births obtained via ART would have occurred in any case without ART in this scenario, since many of the women seeking assistance from ART are simply subfecund and not completely sterile. The births would have happened naturally, but much later, which explains why couples did not wish to take the risk of waiting. So ART births do not all translate into increased completed fertility.
43For the 1945 cohort, the number of ART births implied by these results would be equivalent to an increase in fertility of 0.056 if 100% of eligible women opted for ART, and 0.019 if 33% of women did so. This latter figure is compatible with the available estimate of recourse to ART for the 1980s (de La Rochebrochard, 2008). For the 1975 cohort, our estimate of the number of ART births is somewhat lower than expected: +0.051 (with 33% recourse) versus an expected +0.06-0.10 based on available statistics. Admittedly, the available data do not cover specific cohorts, but we used the calendar years in which births primarily occurred, i.e. around 1980 for the 1945 cohort and around 2010 for the 1975 cohort (Agence de la biomédecine, 2015; FIVNAT, 2001; Kupka, 2014). In total, the orders of magnitude are compatible with our results, given the uncertainties regarding both the statistics (especially for methods other than IVF, ICSI, or AID) and the ways these techniques should be taken into account in our model, thus helping to confirm the validity of our model formulation.
44Would the impact of ART be more significant on involuntary childlessness? In the 1975 cohort, the percentage of women remaining childless would fall from 14.9% to 13.5% with 100% recourse to ART, a relative decline of 9%. This figure may seem low, but it is the result of a rather small percentage of eligible women (the model gives an estimate of 16.4%) and the limited efficacy of ART (here, 54%). Likewise, the proportion of women reporting not having had their desired number of children is barely affected.
45We did a last simulation (right-hand side of Table 2) to determine what would happen if first births were postponed even further, as most recent data seem to suggest. We assumed a further shift of 4 years (like that observed between the 1945 and 1975 cohorts), maintaining the other parameters at the values used for the 1975 cohort. Completed fertility without ART would be 1.805 children per woman, about 0.2 below that of the 1975 cohort. The difference would thus be identical to that observed in the preceding shift. One might have expected a more substantial effect, given that pregnancies are sought at more advanced ages; in fact, while fecundity decreases at the ages concerned, most childbearing still occurs before the sharp tail-off that begins at around ages 35-40 (the mean age at childbearing would be 33.9 years). Recourse to ART, under the same conditions as previously, would here be more effective: with 33% of users, perhaps a rather low hypothesis, the catch-up in terms of completed fertility would be 0.043 children; but the number of ART births would be nearly tripled, significantly increasing the number of unnecessary interventions from a demographic perspective.
Two other experiences: Austria and Italy
46For these two countries, we do not have the same detailed data as for France, so we will examine them more briefly. In a context that may differ for at least one of the parameters we use, we want to see if noticeably different results are obtained. When necessary, we will use the French parameters as a complement to those characterizing the situation of these other countries.
47In Austria, the mean age of childbearing rose from 25.8 years to 28.9 years between the 1955 and 1975 cohorts, a significant increase of three years, and the pattern was similar for age at first birth. Completed fertility of these two cohorts was 1.77 and 1.64 children, respectively.
48Our simulation, based on data from the 1998 Fertility and Family Survey (FFS) and the 2008 Genderations and Gender Survey (GGS) for age at first union, age at first birth, and contraceptive practice, and on Eurobarometer 2001 (Goldstein et al., 2003) for the ideal number of children, gives fertility of 1.774 children for the 1955 cohort, almost identical to the actual data, with a very good concordance in the distributions by number of children. By shifting the distribution of age at first union by 3 years, we obtain 1.670 children, a shortfall of 0.1 children (–6%), which is smaller than in the French case (but the upward shift was 4 years). Compensation by ART was probably negligible, given the available data on the practices at the time (de La Rochebrochard, 2009). We nonetheless ran a simulation applying the practices of the 1955 French cohort, thereby certainly overestimating the impact of ART in Austria. The gain is modest, at only 0.004 children if a third of subfecund women use ART. In fact, under this hypothesis, the number of ART births in the model would exceed the number estimated from available statistics, whereas for the 1975 cohort our estimate of the potential gain (0.016) is compatible with the estimate of the number of births obtained through ART.
49We proceeded in the same manner for Italy, with the same cohorts and the data from the 1995 FFS survey and the 2003 GGS survey. Completed fertility fell from 1.83 to 1.42 children per woman between the 1955 and 1975 cohorts, and the mean age of childbearing rose from 27.1 years (close to the values observed in the 1945 and 1950 cohorts) to 31.6 years, an increase of 4.5 years. To fit the simulation for the 1955 cohort, the distribution of the ideal number of children (as reported in Eurobarometer) had to be substantially modified. The shift of 4 years resulted in mean completed fertility of 1.67 children, only 0.16 less than the initial value, but well above the value of 1.42 observed for the 1975 cohort. To approach this result, we again had to modify the distribution of the desired number of children, reflecting a decisive behavioural change between the cohorts of 1955 and 1975.
50With recourse to ART, the gain would be modest for the 1955 cohort (+0.006 children with one-third of women as users) and greater for the 1975 cohort (+0.014 with a third of users). This result assumes a somewhat higher number of ART births than that observed in the data of the European Society of Human Reproduction and Embryology (ESHRE), but the order of magnitude of the difference is acceptable.
51All in all, and after an initial analysis, the Italian and Austrian examples confirm the results obtained for France.
52Fertility trends in European countries since the 1960s have been characterized by a notable decline in the completed fertility of successive cohorts (and even more so in the total fertility rate), and a rapid 3-4 year increase in the age at first birth over a 30-year period, that is, within a single generation (age difference between mothers and daughters). At the same time, recourse to assisted reproductive technology (ART) has greatly increased in response to the development of new methods that are clearly more effective than traditional treatments. One might think that this growing use of ART is indicative of greater difficulties to conceive, which could, at least in part, be a consequence of the later timing of births desired by couples. ART might also reflect a decline in male and female fecundity linked to environmental factors (Leridon and Slama, 2008). We assume here that biological characteristics have remained unchanged.
53Our simulations show that in terms of completed fertility, the biological effect of a delay of 3 to 4 years in the timing of the first birth is quite limited, at between 0.1 and 0.2 children. The actual change in completed fertility has sometimes been similar, and sometimes much stronger due to the effects of other relevant behavioural factors (desired number of children, contraception efficacy, etc.).
54This result is difficult to compare with those of other studies as the purely biological effect can only be estimated if biological factors are taken into account, and only microsimulation models (like the one used here) are able to do this. With regard to previous publications using the same type of model (with somewhat different hypotheses and modalities), Leridon and Slama (2008) concluded that a 2.5-year postponement of the first birth could reduce completed fertility by 0.1 children, and that a postponement of slightly less than 6 years would reduce it by 0.24 children. Here, with an (observed) shift of 4 years, the fertility reduction is 0.17 children, which is compatible with the preceding results. 
55Recourse to ART, for its part, only compensates for a small part of the reduction stemming from postponement, 0.02 children at best, or 10% of the decline. While its efficacy may improve somewhat in the future, ART will not compensate for more than 20-25% of the decline in fertility. However, the low overall efficacy of ART should not obscure the fact that at individual level, medical interventions have made it possible to address the legitimate needs of growing numbers of “impatient” couples, enabling them to have a birth more quickly than their low fecundity would have allowed them to hope.
56With one-third of eligible women using ART, we estimate that completed fertility would increase by 0.017 children. Habbema et al. (2007) proposed a stronger estimate (in the order of 0.03 to 0.04, with a third of women using ART), but with strengthened hypotheses regarding ART treatments: shorter waiting time, more frequent repeat attempts in case of failure. The results of Hoorens et al. (2007), already mentioned and based on a different model, seem largely over-estimated, as discussed earlier. Other studies do not take “unnecessary” ART births into account and hence cannot be used to generate useful estimates of the number of supplementary births resulting from treatment. Sobotka et al. (2008) apply a correction (that we consider insufficient) and arrive at an estimate of between +0.05 and +0.08 for the 1975 Danish cohort. But the Danes have the highest rate of recourse to ART in Europe: almost three times higher than the French, according to ESHRE data for 2010, so their estimates are in fact quite compatible with ours.
57Finally, to measure the impact of a steady ongoing increase in the age at first birth, we modelled an age 4 years higher than the average age of 28 attained by the 1975 cohort in France. The result was a reduction of 0.2 children in expected fertility in France, with a drop in completed fertility from 2.0 to 1.8 children, all other things being equal. A continued increase in age at first birth could thus affect the completed fertility of the cohorts concerned, unless other behavioural modifications (such as greater recourse to ART, or a reduction in birth intervals) compensate – at least partially – for this effect.
Appendix A.1. Model description and data
58The model is used to run a set of simulations. With values defined for each of the parameters, a woman’s reproductive history is simulated. The results depend on the values of the parameters chosen, and on chance, since random draws come into play on several occasions. The operation must be repeated a great many times to obtain stabilized results corresponding to each single set of hypotheses. Thanks to high-speed modern computers, a large number of simulations can be run very quickly. A simulation with 100,000 women is usually sufficient,  but here, with differences in completed fertility being small between the models with and without ART, we increased the number of simulations to one million. With this number of cases, the second decimal of completed fertility is stable. The operation can then be repeated with a different set of values for the parameters.
The biological parameters
59The data come from rather diverse sources. For the biological parameters, the distribution of fecundabilities is most often estimated from the distribution of intervals between entry into union and first birth, which the Beta function greatly facilitates, as seen above, on the assumption that the couples do not seek to postpone this first birth. This is the case in populations with uncontrolled “natural” fertility, either ancient or contemporary. We assembled a set of such estimates (Leridon, 1977) and chose a hypothesis on the high side: mean fecundability of 0.25 (parameters: A = 3, B = 9), which yields an average time to conception of 4 months if the population is homogeneous, and 7 months if heterogeneity is taken into account. For fetal mortality, we have estimates based on contemporary populations. Here also, we constructed a series of age-specific rates based on these dozen observations (Leridon, 1977): 13% at age 25, 15% at age 30, 17% at age 35, 23% at age 40, and 32% at age 45.
60As already mentioned, it is difficult to estimate the progression of sterility with age. We chose to use the model itself to define the series of age-specific sterility rates that best reconstitutes the distribution of age at last birth in a natural fertility context. After using a distribution derived from historical birth data in France, we chose a distribution of sterility rates reflecting a less rapid progression of sterility with age. It is based on a study on five populations by Eijkemans et al. (2014) which involved choosing at each age the lowest risk of having a last birth observed in the five populations in order to construct an age distribution at last birth that could be qualified as “biological”. As before, we then derived the distribution of ages of onset of sterility, based on these lowest risks. The idea was to respond to certain criticisms regarding the non-transposability of data on historical populations to contemporary populations, assuming that the health context was much more unfavourable in the past. In reality, opposing arguments also exist: sexually transmitted diseases, for example, which are a source of premature sterility, are probably more frequent today than in the past, given the evolution of sexual behaviours. Indeed, data on disadvantaged contemporary populations, as in Africa, give proportions of married women remaining childless – for an identical age at marriage – which are comparable to those of historical populations in Europe, and often very low, less than 3-4% (Larsen, 2000). The series that we chose thus constitutes, in our view, a rather minimalist hypothesis for the progression of sterility with age, with 1% at age 25, 2% at age 30, 5% at age 35, 12% at age 40, and 40% at age 45.
Assisted Reproductive Technology
61Data for the year 2010 gathered by the European Society of Human Reproduction and Embryology (ESHRE; Kupka et al. 2014) shows that for each IVF or ICSI attempt, the average success rate (pregnancy resulting in at least one live birth) is about 22%, slightly less for France. With two successive attempts (several months apart), the success rate reaches 39%. More attempts can be offered, but women are generally reluctant to accept, as the experience is a difficult one: assuming an average of two attempts is close to the actual behaviour of French women (de La Rochebrochard et al., 2008). In carrying out simulations with different combinations of KSTER (proportion of sterile couples who manage to conceive) and EART (efficacy of ART), we kept the combination 70% and 5, with a waiting time for access to ART of 36 months up to age 35 (and diminishing subsequently) and a duration of ART treatment of 12 months. The result is a success rate of 42% around age 30, as shown in Table 3, slightly higher than the average of the ESHRE data for two successive attempts. Of course, other combinations of the two parameters could yield the same result, but that hardly matters. For recent periods, we use a strengthened combination, fixing the parameters at 80% and 7 (diminishing with age), resulting in a success rate of 54%.
62The simulation for the 1945 French cohort, which used ART in the years around 1975-80, shows that around 8% of women are eligible to use ART under the stated conditions, if one assumes that all actually use it (Table 3). This is two to three times higher than the frequency of recourse estimated for the French population around 1980 (Leridon, 2011). In reality, then, the additional births obtained with ART will be two to three times fewer than those obtained with the “100%” model.
63ART methods, including simple hormonal stimulation, greatly increase the risk of twin births. While medical protocols have evolved in recent years to avoid the risk of multiple pregnancies (three or more foetuses), twin births remain numerous. Our model only counts deliveries (with at least one live birth), so we use a multiplier coefficient of 1.26 to evaluate the total number of births after ART treatment (Habbema et al., 2009). For births without ART, the coefficient is 1.025.
64The number of births obtained via ART in the model can be compared with the reported numbers from available statistics. For France, the number of births by IVF, ICSI, and AID represented 2.4% of all births in 2006 (de La Rochebrochard, 2008). For all methods combined (adding ovarian stimulation alone and insemination with the partner’s sperm, among other techniques), the percentage might approach 4-5%. According to ESHRE data, the share of the first group of methods (IVF, ICSI, and AID), expanded to include new freezing techniques, was 3.1% in 2010 (Kupka et al., 2014), and for all methods combined, a figure of close to 5% seems reasonable. Here we touch upon an interesting and original aspect of the model. We count the births obtained thanks to ART, and we note that the additional fertility calculated in this way often clearly exceeds the observed increase in completed fertility, because a good number of the ART births simply replaced births that would have occurred naturally, a bit later, to women who are not completely sterile. The difference is all the greater if treatment is begun earlier (low DMAX) and the women are younger at the start of treatment (for a more detailed discussion on this point, see Habbema et al., 2009).
65Thus, when we obtain an apparent contribution of ART births to completed fertility of around 0.1, this represents 5% of completed fertility of 2.0 children (which is compatible with French statistics on recourse to ART in 2010), whereas the actual additional completed fertility may not exceed 0.05 children. This point is not intuitive, but it is decisive in evaluating the real efficacy of ART methods. 
French Institute for Demographic Studies, Paris, France.
Correspondence: Henri Leridon, Institut national d’études démographiques, 133 bd Davout, 75020 Paris, France, email: email@example.com
Average monthly fecundability was raised from 0.23 to 0.25, which is closer to empirical estimates when the risk of a miscarriage is explicitly taken into account.
At each age, they used the lowest probability of having a last birth in the year, and these probabilities were then combined to construct the oldest possible distribution of ages at last birth.
By way of example, using the simulation for the 1945 cohort (last column of Table 1), if fecundability is reduced by 20% (declining from 0.25 to 0.20 on average), completed fertility falls from 2.2110 to 2.2107; if intra-uterine mortality increases by 20% (from 13.8% to 16.6%) completed fertility falls from 2.2110 to 2.1920 (–0,019); a two-year shift in the age distribution of sterility (towards younger ages) would have a more noticeable effect, reducing completed fertility to 2.123 (– 0,087), but this would be contrary to our procedure for estimating sterility.
The following contextualized question was asked: “Thinking about people with the same background as you, with the same resources, what is the ideal number of children in a family?”
The 1977 study has also shown that variability of fetal mortality across couples does not need to be taken into account in addition to the heterogeneity of fecundability, even though this variability certainly exists.
The only exception concerns the use of selected, fertilized eggs from a young woman (donor) implanted in an older woman, in which case the effect of age is largely neutralized.
These preferences barely changed over the following 10 years (Testa and Basten, 2012).
In the two situations in Table 3, 8% and 16% of women, respectively, would be eligible for ART. The actual number of women using these methods would thus be equal to a fraction (for example, 33%) of these women.
The publication of te Velde et al. (2012) concerned a very different context, that of an upturn in fertility, and hence is not very comparable.
We showed, for example, that for completed fertility of 2.0 children, 10 iterations of the model with the same parameters would result in completed fertility ranging between 2.000 and 2.007, a proportion of couples remaining childless of between 9.7% and 10.0%, and a proportion of couples eligible for ART of 11.5% to 11.8% (Leridon and Slama, 2008).
This limited “demographic” efficacy of ART does not, however, imply that a good number of medical interventions are unnecessary, because they satisfy couples after a wait that is unacceptably long, and that the absence of treatment would have prolonged even further.